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Tumbling asteroids
Icarus 173 (2005) 108–131 www.elsevier.com/locate/icarus
Tumbling asteroids P. Pravec a,∗ , A.W. Harris b , P. Scheirich a , P. Kušnirák a , L. Šarounová a , C.W. Hergenrother c , S. Mottola d , M.D. Hicks e , G. Masi f,g , Yu.N. Krugly h , V.G. Shevchenko h , M.C. Nolan i , E.S. Howell i , M. Kaasalainen j , A. Galád k,a , P. Brown l , D.R. DeGraff m , J.V. Lambert n , W.R. Cooney Jr. o , S. Foglia p a Astronomical Institute, Academy of Sciences of the Czech Republic, Friˇcova 1, CZ-25165 Ondˇrejov, Czech Republic b Space Science Institute, Boulder, CO 80301, USA c Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA d DLR Institute of Space Sensor Technology and Planetary Exploration, Rutherfordstr. 2, D-12489 Berlin, Germany e Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA f Physics Department, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy g Campo Catino Observatory, Guarcino, I-03016, Italy h Institute of Astronomy of Kharkiv National University, Sumska Str. 35, Kharkiv 61022, Ukraine i Arecibo Observatory, National Astronomy and Ionosphere Center, PR 00612, USA j Department of Mathematics/Rolf Nevanlinna Institute, University of Helsinki, PO Box 4, FIN-00014, Finland k Modra Observatory, Astronomical Institute, FMFI Comenius University, Bratislava, Slovak Republic l Department of Physics and Astronomy, University of Western Ontario, London, ON N6A 3K7, Canada m Alfred University, 1 Saxon Drive, Alfred, NY 14802, USA n Air Force Maui Optical and Supercomputing Facility, Kihei, HI 96753, USA o Blackberry Observatory, 1927 Fairview Dr., Port Allen, LA 70767, USA p Serafino Zani Observatory, Lumezzane, I-25065, Italy
Received 23 March 2004; revised 16 July 2004 Available online 25 September 2004
Abstract We present both a review of earlier data and new results on non-principal axis rotators (tumblers) among asteroids. Among new tumblers found, the best data we have are for 2002 TD60 , 2000 WL107 , and (54789) 2001 MZ7 —each of them shows a lightcurve with two frequencies (full terms with linear combinations of the two frequencies are present in the lightcurve). For 2002 TD60 , we have constructed a physical model of the NPA rotation. Other recent objects which have been found to be likely tumblers based on their lightcurves that do not fit with a single periodicity are 2002 NY40 , (16067) 1999 RH27 , and (5645) 1990 SP. We have done a statistical analysis of the present sample of the population of NPA rotators. It appears that most asteroids larger than ∼ 0.4 km with estimated damping timescales (Harris, 1994, Icarus 107, 209) of 4.5 byr and longer are NPA rotators. The statistic of two short-period tumblers (D = 0.04 and 0.4 km) with non-zero tensile strength suggests that for them the quantity µQ/T , where µ is the mechanical rigidity, Q is the elastic dissipation factor, and T is a spin excitation age (i.e., a time elapsed since the last significant spin excitation event), is greater by two to four orders of magnitude than the larger, likely rubble-pile tumblers. Among observational conditions and selection effects affecting detections of NPA rotations, there is a bias against detection of low-amplitude (small elongation) tumblers. 2004 Elsevier Inc. All rights reserved. Keywords: Asteroids; Excited rotation; Photometry
* Corresponding author. Fax: +420-323-620263.
E-mail address: [email protected] (P. Pravec). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.07.021
Tumbling asteroids
1. Introduction The rotational motion of an asteroid is described by = Iˆω, L
(1)
is the asteroid’s angular momentum vector, Iˆ is its where L inertia tensor, and ω is its angular velocity vector. The inertia tensor is generally a symmetric tensor containing six independent components. A convenient choice of the system of coordinates in the asteroid-fixed frame gives zero non-diagonal components. The diagonal components I1 I2 I3 are then the principal moments of inertia; the axes are called the principal inertia axes. The rotational kinetic energy is generally 1 T ˆ E= ω (2) Iω 2 and particularly for the principal inertia axes choice of the system of coordinates E=
1 I1 ω12 + I2 ω22 + I3 ω32 . 2
(3)
the basic (lowFor a given angular momentum vector L, est energy) state of rotation occurs when the asteroid rotates around its principal axis of the maximum moment of inertia (I3 ), with the rotational kinetic energy of Emin = I3 ω32 /2 = L2 /(2I3 ). Unless the body has I1 = I2 = I3 (like, e.g., homogeneous sphere, or cube), it may be in an excited state of rotation with Emin < E L2 /(2I1 ). In the excited state, unless the energy is equal to L2 /(2I1 ) that corresponds to rotation around the principal axis with the lowest moment of inertia I1 , the body’s rotation is complex as described in, e.g., Kaasalainen (2001).1 We also call the non-principal axis (NPA) rotational motion “tumbling” (the word was first used by Harris, 1994, as a suitable single-word term to describe the motion). A tumbling asteroid generally does not return to a same orientation at any single period, but it shows a period of rotation around one of the two extremal principal axes, Pψ , and a quasi-period of precession of the axis around P ¯ . (The latter is rather an average period of Euler angle L, φ ¯˙ where φ¯˙ = Pψ /2 φ˙ dt/(P /2).) φ, Pφ¯ = 2π/φ, ψ 0 In a non-rigid body, a NPA rotation results in a stressstrain cycling within the body. The excess energy is dissipated in the body’s interior and the spin state evolves to lower energy states. An approximate expression for the characteristic time scale τ of damping of the excited NPA rotation has been derived by Burns and Safronov (1973) assuming a low amplitude libration τ∼
µQ , ρK32 R 2 ω3
(4)
1 A state with E = L2 /(2I ) is unstable (see, e.g., Black et al., 1999). 2 A probability to see a body with exactly the energy of L2 /(2I2 ) is infinites-
imal and it can be neglected for real cases.
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where µ is the rigidity of the material composing the asteroid, Q is the quality factor (ratio of the energy contained in the oscillation to the energy lost per cycle), ρ is the bulk density of the body, K32 is a dimensionless factor relating to the shape of the body with a value ranging from ∼ 0.01 for a nearly spherical one to ∼ 0.1 for a highly elongate or oblate one, R is the mean radius of the asteroid, and ω is the angular velocity of rotation. Harris (1994) estimated the parameters in Eq. (4) and expressed the damping timescale as τ=
P3 , C 3 D2
(5)
where P = 2π/ω is the rotation period, D is the mean diameter of the asteroid, and C is a constant of about 17 (uncertain by about a factor of 2.5) for P in hours, D in kilometers and τ in billions (109 ) of years. He found that most of the asteroids’ rotations studied at the time had damping time scales shorter than their likely ages but he found that several small slow rotators could exhibit a NPA rotation. The first identified tumbling asteroid was (4179) Toutatis, see Hudson and Ostro (1995). Since that time, a few other cases have been recognized (Harris, 1994; Mottola et al., 1995; Pravec et al., 1998; Harris et al., 1999; Benner et al., 2002). We have examined the observational evidence for, and characteristics of, the population of tumbling asteroids, revisiting earlier publications from a recent compilation by A.W. Harris and analyzing new data obtained by us in the past few years. This paper is organized as follows. Section 2 outlines methods of analysis of observational data used. In Section 3, we give a review of observational results on NPA rotators as well as on PA rotators with long damping timescales. A statistical analysis and discussion on properties of the population of NPA rotators is given in Section 4. In Section 5, we discuss mechanisms (especially the YORP effect) that could explain the population of tumbling slow rotators. We note that a reader not interested in data and technical details on individual objects may want to jump over Section 3 and go directly to Section 4 where the characteristics of the tumbling asteroids population are discussed.
2. Observational data and analysis methods Time-resolved photometric observations, also called “lightcurve observations”, contain information on rotation of the observed asteroid. An analysis of the data for a periodicity (and effects of changing geometry) can provide an estimate of the synodic rotation period of the asteroid during the particular interval. The Fourier series method developed by Harris et al. (1989) and implemented by Pravec et al. (1996, 2000a) is a powerful technique for the period derivation. It assumes that the asteroid’s lightcurve contains just a single period. In a case where the analyzed data show systematic deviations from pure periodicity that cannot be accounted for
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by effects of changing geometry, a 2-dimensional Fourier series technique (suggested by Kaasalainen, 2001) can be used to find two periods present in the lightcurve as described in Section 2.1. New photometric observations presented in Section 3 were taken with several different telescopes/stations, all equipped with CCDs. The largest contribution was with the Ondˇrejov 0.65-m telescope equipped with the SBIG ST-8 camera and the Apogee AP7p camera (since May 2001). Other telescopes are mentioned in paragraphs for individual objects. Most of the data have been taken in the Cousins R filter with absolute calibration using the standards of Landolt (1992) with absolute errors of 0.01–0.02 mag unless stated otherwise. The CCD images were reduced in standard ways as described in Pravec et al. (1996), Rabinowitz (1998), Whiteley et al. (2002a), Krugly et al. (2002). D. De Graff used an IDL program CCDPHOT written by Mark Buie to reduce his observations of 2002 NY40 . 2.1. Two-dimensional Fourier series A lightcurve of a NPA rotator is a function of Euler angles ψ and φ. In constant geometric conditions, or if small changes of the conditions can be neglected or corrected for by another method (e.g., the magnitude-phase dependence is corrected using the H –G relation), the NPA rotator lightcurve can be approximated with Fourier series in the two variables (see Appendix A). Substituting ψ = 2πt/Pψ and approximating φ with 2πt/Pφ¯ in Eq. (A.7), we get the Fourier series in the following form . F ψ(t), φ(t) = F m (t) m 2πj 2πj Cj 0 cos t + Sj 0 sin t = C0 + Pψ Pψ j =1 m m
2πj 2πk + Cj k cos t Pψ Pφ¯ k=1 j =−m 2πj 2πk + + Sj k sin (6) t , Pψ Pφ¯ +
where C0 is the mean reduced light flux, Cj k and Sj k are the Fourier coefficients of the corresponding linear combination of the two frequencies Pψ−1 , Pφ−1 ¯ , and t is the time (for arbitrary epoch). We note that in practice we cannot tell a priori which of the periods of the Fourier series fitted to the lightcurve data actually correspond to precession and rotation. We therefore use subscripts (1, 2) rather than ¯ for the derived periods; an actual assignment of the (ψ, φ) derived periods to rotation and precession depends on a subsequent physical modeling. The series is truncated at the order m, usually the highest order for which a convergent solution can be obtained. The number of Fourier coefficients of the 2-dimensional Fourier series is (2m + 1)2 — cf. the number of coefficients of 1-dimensional Fourier series of (2m + 1). Note that the number of coefficients of
the 2-dimensional Fourier series grows so large with m of only 3 or 4 that we normally cannot use high orders that would be needed to fit minor features of the lightcurve (cf. single-periodic lightcurves that sometimes contain significant signals in harmonics of orders up to 10). Thus the 2-dimensional Fourier series fitted to the complex lightcurve data may not fit smaller lightcurve features fully just because it must be truncated at the relatively low order. Nevertheless, a use of the 2-dimensional Fourier series for frequency analysis of lightcurve data of tumblers has an advantage in that it does not introduce any a priori assumptions or idealizations of shape, scattering law and albedo distribution of the body so derivation of the periods is model-independent and also much faster than any (even much simplified) physical model which would need a number of non-linear parameters.2 When describing results obtained with the method, we use both the terms “period” and “frequency” depending on context and purpose, the conversion being always fj = 1/Pj , where j is integer. A signal (amplitude) of the harmonic (jf1 + kf2 ) is expressed as 2 2 Anorm (7) j k = Cj k + Sj k /C0 . The normalization to the mean reduced light flux is because we fit the photometric data in linear units (in “luminosity”) rather than in magnitudes. For illustrative purposes, remember that in a case of a small total lightcurve amplitude, a corresponding (peak-to-mean) variation in magnitudes is . ≈ 2.5Anorm/ ln 10 = 1.09Anorm. When solving for the periods P1 and P2 , we start with an initial period estimate from the previous single-period search of the data. The period estimate may be one of the two basic periods present in the NPA rotator, or it may correspond to a strong-signal linear combination of the two basic frequencies (usually a low-integers combination). With this assumption, we can scan the periods space much more quickly, in steps, searching for one of the two basic periods in one step while keeping the other period fixed at the estimate obtained in a previous step. 2.2. Synodic–sidereal period difference and the phase angle bisector Before ascribing deviations from strict single-periodicity of a lightcurve to non-principal axis rotation, it is important to assess the possible deviations just due to changing geometry. There are well-known effects on the amplitude of variation due to changing aspect (from equatorial to polar), 2 As already noted in Kaasalainen (2001), we point out that a use of the 2-dimensional Fourier series describing a lightcurve of NPA rotator differs from a simple sum of two Fourier series as is used for lightcurves of asynchronous binaries (e.g., Pravec et al., 2000b) which do not contain the full suite of sum and difference frequency components needed to fit the tumbling case. This may help to resolve between the two hypotheses—tumbler vs. asynchronous binary—in cases where the character is not obvious.
Tumbling asteroids
and on solar phase angle, but also the rotation phase can show shifts from strict periodicity due to changing rate of motion of the viewing aspect along the direction of longitude in the asteroid’s rotation coordinates. To quantify this we use the Phase Angle Bisector (PAB), the mean direction between the heliocentric and geocentric directions to the asteroid. As discussed by Harris et al. (1984), this serves as a convenient approximation for an effective viewing direction, as if the solar phase angle were reduced to zero. We can then consider the motion of the PAB as a measure of the possible difference between the observed synodic rotation period and the presumed constant sidereal period. In the ideal case where the PAB lies in the equator of the asteroid and moves along the equator in longitude only, the synodic–siderial period difference is: d(PAB) 2 (8) P . dt When the direction of the PAB is off the equator and motion is mainly along longitude, the observed value of Psyn–sid can be greater than given above; when the motion of the PAB is mainly in latitude, it can be less, so the above expression should be considered only a dimensional estimate of the possible difference between the observed synodic period and the sidereal one. Note that if the motion of the PAB is in the same sense as the direction of asteroid rotation, then Psyn–sid is positive and the synodic period is longer than the sidereal period. If the direction of motion of the PAB is opposite the direction of rotation, then Psyn–sid is negative and the synodic period is shorter than the sidereal one. Psyn–sid =
2.3. PAR codes Analyzing lightcurve data for a large number of asteroids listed in our database, we have found a need to establish a scheme for classification of quality of the data with respect to their ability to reveal PA or NPA rotation. We have therefore established a “PAR” codes scale as follows: −4 Physical model of the NPA rotation constructed. −3 NPA rotation reliably detected with the two periods resolved. An ambiguity of the periods solution may be tolerated provided the resulting spectrum of frequencies with significant signal is the same for the different solutions. −2 NPA rotation detected based on deviations from the single periodicity but the second period not resolved. −1 NPA rotation possible, some deviations from the single periodicity are seen but not at a conclusive level. 0 Not enough data to say if it is single- or multipleperiodic. +1 PA rotation is consistent with the data but coverage poor, thus not a reliable detection. +2 PA rotation likely, or deviations from PA rotation are small, as some overlapping data fit with a single period.
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+3 PA rotation quite likely as many overlapping data covering a few cycles fit with a single period. +4 PA spin vector solution obtained. The “PAR” name of the scale was chosen as an abbreviation of “Principal Axis Rotation” quality detection codes, with positive value meaning that the data indicate a PA rotation and negative value indicating a NPA rotation. For studies of PA vs. NPA rotations in a given sample of objects, only cases with |PAR| 2 can be considered as reliable enough detections for a statistical study. Cases with PAR of +1 or −1 are not reliable detections of a PA or NPA rotations, respectively; the two codes are intended to serve just as a guide for further observations. 2.4. Physical modeling of NPA rotation Where enough data are available for a particular NPA rotator (i.e., PAR = −3), a physical model of the NPA rotation can be attempted with the available data to infer physical meaning of the two detected periods in terms of rotation and vector, and precession, to establish an orientation of the L to estimate a shape and principal moments of inertia of the asteroid. Methods of construction of physical models of tumbling asteroids from their lightcurves were developed by Kaasalainen (2001) and Kryszczy´nska et al. (1999). One of us (P. Scheirich) has developed a model of one of the best tumblers, presented in Section 3.1, based largely on the method by Kaasalainen (2001) with a few modifications. A detailed description of the model, its refinement, and an application to lightcurve data of a few other tumblers will be given in a future paper. We outline the main points of the inversion method below. The shape of an asteroid is approximated by a triaxial ellipsoid with semiaxes a, b, and c. In following, we use the notation where c is always the axis around which the ellipsoid is seen as rotating, and it is normalized to unity. The c axis precesses around L—the angular momentum vector that is constant in a case of absence of external forces. The NPA motion can be described as divided into two classes: the long-axis mode (LAM) where the asteroid’s rotation axis c is the longest one (so, a, b < 1), and the short-axis mode (SAM) where c is the shortest axis (a, b > 1). The corresponding principal moments of inertia for the axes a, b, and c are Ia , Ib , and Ic , respectively, with Ic normalized to unity. Defining the symmetric Isym = (Ia + Ib )/2 and the antisymmetric Iant = (Ia − Ib )/2 allows one to constrain their values from observed periods (see Kaasalainen, 2001) in the inverse problem. The NPA rotation motion can be described with the following eight parameters: (its magnitude L and the di• The angular momentum L rection in ecliptic coordinates λL , βL ).
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• φ0 , θ0 , ψ0 —Euler angles describing the orientation of the ellipsoid at epoch t0 . We use the angles in the so-called x-convention (see, e.g., Samarasinha and A’Hearn, 1991). The Z-axis of the inertial frame is par and the XZ-plane contains a vector directed to allel to L vernal equinox. • Ia , Ib —principal moments of inertia (Ic normalized to unity). The NPA motion consists of the precession of the c axis (the instantaneous tilt between the two vectors around L is denoted θ ) with period Pφ , and the rotation of the body around the c axis with period Pψ . Since the two periods can be estimated from the lightcurve (though ambiguously in some cases)—see Section 2.1—we transform the above set of parameters to another one: λL , βL , φ0 , ψ0 , Pψ , Pφ , Isym , Iant . Furthermore, the ellipsoid axes (and therefore Isym and Iant , too) can be guessed from lightcurve amplitudes, so the number of initially unconstrained parameters shrinks to four. The transformation between the two sets of parameters and integration of Euler equations can be found in Kaasalainen (2001). The brightness of the ellipsoid as seen by observer can be computed in two ways: (1) At low phase angles, the brightness can be approximated with the apparent sky cross-section of the ellipsoid, evaluated using a formula from (Kryszczy´nska et al., 1999). (2) At higher phase angles, the brightness is computed numerically. The ellipsoid is represented with a number of triangular facets and the sum of individual contributions of all visible facets is computed. Several photometric laws can be used but the simple Lommel–Seeliger law is usually sufficient for ellipsoidal approximation of the real shape. The inverse problem is defined as the minimization of the function (oi − mi )2 , χ 2 (λL , βL , φ0 , ψ0 , Pψ , Pφ , Isym , Iant ) = i
where oi and mi are observed and modeled magnitudes at the epoch i, respectively. Initial values for Pψ , Pφ , Isym , and Iant are obtained from the lightcurve analysis, the remaining parameters (λL , βL , φ0 , ψ0 ) are sampled on an equidistant grid. Starting from each point of the grid, the local minimum is found using Nelder and Mead Simplex Algorithm (Buchanan and Turner, 1992). For data sets containing observations at large phase angle, an amplitude-phase correction coefficient m (see Zappalà et al., 1990) can be included into the model and solved for simultaneously with the other parameters.
3. Individual asteroids—PA vs. NPA rotators In this section, we present both a review of previous results and our new analyses of data for NPA rotators with PAR −2 (Sections 3.1 and 3.2) as well as for PA rotators with PAR +2 with long damping timescales (Section 3.3). In Table 1, a summary of the data for the NPA rotators is given. In order to facilitate a comparison with the prediction, in the table as well as in the text we give values of the logarithm of the damping time scale estimated from Eq. (5) normalized to the age of the Solar System, i.e., log τnorm = log(τ/4.5 byr). 3.1. Tumblers—the best cases 3.1.1. (4179) Toutatis—the best known case This S-type Apollo asteroid made repeated close approaches to the Earth in 1992, 1996, and 2000. It was extensively observed by both radar and photometric observers in December 1992–January 1993. Hudson and Ostro (1995) analyzed the 1992 radar data. The asteroid was observed from Goldstone from 1992-12-02 to 12-18 and from Arecibo on 1992-12-19. They established a model of the asteroid’s shape, spin state, and ratios of the principal moments of inertia. The motion was recognized as LAM (see Section 2.4), with the rotation period Pψ = 5.41 day and the long-axis precession period Pφ = 7.35 day. They
Table 1 Tumbling asteroids Object
PAR
4179 Toutatis 2002 TD60 2000 WL107 54789 2001 MZ7 2002 NY40 253 Mathilde 3288 Seleucus 3691 Bede 4486 Mithra 5645 1990 SP 13651 1997 BR 16064 1999 RH27 53319 1999 JM8
−4 −3 to −4 −3 −3 −2 to −3 −2 to −3 −2 −2 −2 −2 −2 −2 −2
a Ambiguity, see text.
log τnorm 1.2 −2.2 −3.9 −0.6 0.5 0.0 0.6 1.2 >0 0.0 0.5 1.6 0.4
P1 (h)
P2 (h)
A (mag)
Strongest f ’s
176 2.851 0.1609 37.57a 19.98 418 ∼ 75 227 (days) 30.39 33.64 178.5 (136)
130 6.783 0.2188 52.79a (18.43) (250)
1.2 1.4 1.1 1.4 1.3 0.5 1.0 0.5
2f1 , f2 , (f1 − 2f2 ), 2(f1 + f2 ) 2(f1 + f2 ), 2f1 , 2f2 f1 , 2f2 2f1 , 2f2 2f1
0.7 1.2 0.7 0.7
Tumbling asteroids
found dimensions along the axes corresponding to principal moments of inertia of 1.92, 2.40, and 4.60 km, and a di in the ecliptic rection of the angular momentum vector L coordinates of (180◦, −52◦ ). Spencer et al. (1995) made photometric observations from 25 observatories on 40 nights from 1992-12-08 to 1993-01-28 at phase angles of 0.2◦ to 121◦ using photoelectric photometers as well as CCDs. They found that the data contain periods of ∼ 7.4 and ∼ 3.1 day, using an algorithm assuming additive components for the two periods. From the radar model by Hudson and Ostro, it is seen that the main photometric period of 7.4 day is the precession period and that the second period of 3.1 day corresponds to a linear combination of the two frequencies of the asteroid’s NPA rotation. They estimated a mean absolute magnitude H of 15.3 and a slope parameter G of 0.10 ± 0.10. Hudson and Ostro (1998) used the radar-derived model to simulate the 1992–1993 lightcurve data of Spencer et al. and showed a very good correspondence. Using the lightcurve data that have a much longer time base than the radar observations, they further improved their model and also obtained Hapke photometric parameters of the asteroid. The 1992–1993 lightcurve data were simulated also by Kryszczy´nska et al. (1999), using dynamical parameters derived by Hudson and Ostro (1995). They noted that the lightcurve of Toutatis is dominated by the precession and the superposition of precession and rotation. The strongest signal of their synthetic lightcurve was in 2fφ , the next strong signal was in fφ and 2(fφ + fψ ); that exactly corresponds with the periods found by Spencer et al. considering that they saw the dominance of the second harmonics in their two lightcurve components. The same data were revisited also by Mueller et al. (2002) using the WindowCLEAN algorithm (Roberts et al., 1987) for obtaining characteristic frequencies of the lightcurve. They found four significant peaks in the spectrum, in decreasing order of power, of 2fφ , fφ , 2(fφ + fψ ), and f4 = 0.029 day−1 , where the last one corresponds to a curvature of the envelope of the lightcurve maxima so it does not correspond to a real frequency of the asteroid’s NPA rotation. The first three frequencies concur well with the previous results by Spencer et al. (1995) and Kryszczy´nska et al. (1999). We revisited the data by Spencer et al. with the Fourier series method. A goal of the exercise was to learn how the method works on the photometric data for the very longperiod tumbler. Results presented in the following two paragraphs showed that there is not a unique solution, nevertheless, a resulting solution spectrum corresponds well to the results found by the earlier studies. We analyzed the data from 1992-12-20.4 to 1993-01-27.9 taken at solar phase angles < 42◦ on the outbound trajectory after the close approach on 1992 December 8; this choice was due to the method’s requirement to limit synodic effects (∼ 1 h or less) as well as to keep the lightcurve amplitude change with phase angle small. A single-period
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. search showed a period of 175 h (= 7.3 day) dominating the lightcurve. An analysis using the 2-dimensional Fourier series of the 2nd order as described in Section 2.1 confirmed the main period of 175 h and provided a few possible solu. tions for the second period: 52.2, 61.2, 74.5 h = 3.1 day, 94.4 . h, and 130 h = 5.4 day. Two of the periods, 61.2 and 94.4 h, appear to be aliases. Checking signals in individual harmonics of the two solutions, we found that there was actually relatively little signal in both the first and the second harmonics for each of the two periods themselves. The significance of the 94.4 h period was because its second harmonic corresponds to a linear combination of the two other periods: . 2/94.4 h = (1/175 h + 2/130 h). Similarly, the significance of the 61.2 h period was because its second harmonic corresponds to another linear combination of the two periods: . 2/61.2 h = (3/175 h + 2/130 h). While we could not rule out the two solutions of 61.2 and 94.4 h for the second period on that basis only, the analysis suggested that they might be spurious. We were not able to resolve between the other three solutions, 52.2, 74.5, and 130 h, for the second period. Nevertheless, the first two solutions, 52.2 h and 74.5 h, are linear combinations of the frequencies corresponding to the periods 175 and 130 h, (2f1 + f2 ) and (f1 + f2 ), so all the three solutions correspond to the same frequency spectrum. With more abundant data allowing a higher order fit of the 2-dimensional Fourier series, we might be able to resolve between the multiple solutions, as we can in a few cases with better data presented below. From the available data, we would conclude that Toutatis is a NPA rotator and would give it PAR = −2 tending to −3. The real solution (according to the radar model) with the periods P1 = 175 h and P2 = 130 h is in agreement with the precession and the rotation periods by Hudson and Ostro (1995). It has most signal in 2f1 , the next strong signals in f2 , (f1 − 2f2 ), and 2(f1 + f2 ). 3.1.2. 2002 TD60 This Amor asteroid had a favorable apparition in November and December 2002. We observed it with the 0.65-m telescope from Ondˇrejov on 6 nights from 2002-11-07.1 to 12-11.9; with the 1.2-m telescope from Elginfield Observatory on 3 nights of 11-08.3, 12-03.2 and 07.4; with the Catalina 1.54-m telescope on two nights of 11-07.5 and 08.5; with the 0.6-m telescope from Modra on two nights of 12-08.0 and 08.9; and with the 0.8-m telescope from Campo Catino Observatory (observers G. Masi, C. Belmonte, and F. Mallia) on one night of 11-11.0. The asteroid was also imaged with the Arecibo radar on 4 nights from 2002-11-20 to 23. All but two of the photometric runs were done in R filter; the two exceptions are the two runs from Catalina which were done unfiltered. The Ondˇrejov observations were calibrated on the absolute magnitude scale within 0.01–0.03 mag, the other data are relative, their magnitude zero points have been adjusted in the analysis. We have found that the lightcurve is two-periodic. Its periods are P1 = 2.8513 ± 0.0001 h and P2 = 6.783 ± 0.002 h;
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Fig. 1. Lightcurve data of 2002 TD60 of 2002-11-07.1 to 14.1 folded with the period of 2.85119 h and G = 0.55. The best-fit full two-dimensional 3rd order Fourier series with the periods of 2.85119 and 6.7841 h is plotted as well.
Fig. 2. Lightcurve data of 2002 TD60 of 2002-11-24.0, 12-01.9 and 03.2 folded with the period of 2.8513 h and G = 0.55. The best-fit full two-dimensional 3rd order Fourier series with the periods of 2.8513 and 6.7805 h is plotted as well.
these values are means of estimates of the periods from three subsets described below. Since the asteroid’s lightcurve showed an apparent change of amplitude due to changing solar phase as well as aspect, we divided the data into three subsets: 2002-11-07.1 to 14.1 (7 nights, 756 points), 11-24.0 to 12-03.2 (3 nights, 428 points) and 12-07.4 to 11.9 (4 nights, 730 points). The subsets contain data over solar phase angles 27◦ –18◦ , 5◦ –10◦ , and 15◦ –20◦, respectively. The asteroid’s sky position (L, B) in the middle of each of the three intervals was (70◦ , −14◦ ), (65◦ , +5◦ ), and (62◦ , +14◦ ), respectively. The asteroid’s heliocentric vector moved at an almost constant rate of 0.9◦ day−1 in the ecliptic longitude while it stayed within 2◦ of Bh = 0◦ throughout the campaign. A full two-dimensional Fourier series of the 3rd order were fitted to the data with a satisfactory result, see Figs. 1–3. While we found that the 4th order would further improve the fits to the large-amplitude, low noise data, we have found that the dataset is just slightly insufficient in ex-
Fig. 3. Lightcurve data of 2002 TD60 of 2002-12-07.4 to 11.9 folded with the period of 2.8516 h and G = 0.55. The best-fit full two-dimensional 3rd order Fourier series with the periods of 2.8516 and 6.784 h is plotted as well.
tent for the 4th order to be used—experiments showed that it would slightly overfit the available data. While it would not affect the period solution, we stayed with the third order. The periods estimated from the three subsets were P1 = 2.8512, 2.8513, and 2.8516 h (errors 0.0001, 0.0001, and 0.0002 h), respectively, and P2 = 6.784, 6.781, and 6.784 h (errors 0.001 h), respectively. We saw that the period estimates were consistent to within a few times their 1-σ errors, so we took their means as the adopted periods solution mentioned above. The strongest signal is in 2(f1 + f2 ) in all the three data subsets (Anorm = 0.41, 0.33, and 0.31, respectively). The next two frequencies with consistently strong signal in all the three subsets are 2f1 (Anorm = 0.27, 0.28, and 0.20, respectively) and 2f2 (Anorm = 0.14, 0.19, and 0.16, respectively). At higher solar phase angles—in the subsets 1 and 3—there was also a high signal in (f1 + f2 ) while it was weak in the low solar phase subset No. 2 (Anorm = 0.32, 0.04, and 0.16 in the three subsets, respectively). As will become obvious with a physical model below, it is a shape effect equivalent to that seen in ordinary PA rotators—odd harmonics caused by shape become smaller at low solar phases. There were significant albeit weaker signals in a number of other linear combinations of the basic frequencies. We tested also a possibility that a true period present in the asteroid is that of 2.0075 h (equivalent to the frequency (f1 + f2 ), in the 2nd harmonic of which there is seen the highest signal). We found that it provided consistently poorer fits to the data in each of the subset. For example, in the 2nd subset, the sum of square residuals increased by 40% when f2 was replaced with (f1 + f2 ). A detailed check revealed that some significant-signal linear combinations of the basic frequencies f1 and f2 of the solution mentioned in the above paragraphs were missing in the tested solution with the period of 2.0075 h, so we concluded that it was not a real period present in the asteroid, rather it really was a linear addition of the two basic frequencies of the true solution mentioned above.
Tumbling asteroids
The best estimates of the mean absolute R magnitude HR and the slope parameter G are 19.41 ± 0.08 and 0.55 ± 0.1, respectively (errors are realistic). Having the basic lightcurve periods estimated with the two-dimensional Fourier series, we went on to construct a physical model of the NPA rotation using the method developed by P. Scheirich as outlined in Section 2.4. Considering that the strongest signal seen in the lightcurve is in 2(f1 + f2 ), we started with an assumption that it is LAM and that Pψ ≡ P2 and Pφ ≡ P1 . (We also made another modeling run starting with an assumption of SAM and Pψ ≡ P2 and Pφ ≡ 1/(f1 + f2 ); getting no satisfactory fit.) Starting values of axial ratios of the fitted ellipsoid were estimated using observed lightcurve amplitudes. The full amplitude of 1.4 mag at the phase angle 7◦ , corrected to 1.2 mag at 0◦ (see Zappalà et al., 1990) suggested that the body is quite elongated with a/c (the longest-to-shortest axial ratio) about 3. The largest difference between local maxima of the lightcurve was 0.9 mag (0.8 mag after correction to zero phase), suggesting a/b ∼ 2.1. It gave starting values of Isym = 0.591 and Iant = −0.321. For starting value of the amplitude-phase coefficient m, we used 0.02. We divided the optimization process into four subsequent steps, starting from simulating the low solar phase subset No. 2 and adding two next lightcurves adjacent to the beginning and the end of current subset on each step. At a first step we approximated the brightness of the ellipsoid with its sky cross-section and scanned λL , βL , φ0 , and ψ0 over their full intervals, while other parameters (Pψ , Pφ , Isym , Iant , m) were kept fixed at the starting values. From each point of the grid, all the nine parameters were iterated to reach a minimum of χ 2 using the simplex algorithm. At other steps we used the more realistic model using a triaxial ellipsoid approximated with 2292 triangular facets and the Lommel– Seeliger scattering law and a minimum of χ 2 in all parameters was found again using the simplex algorithm from each starting point. In each step, we did not use the whole set of local minima obtained in the previous step, just the deepest ones that appeared to encompass all possible solutions. (Tests with some of the other, higher minima indicated much poorer fits as supposed.) The allowed range of parameters shrank from one step to the following one. In the end we repeated the very last step with the whole dataset once again, now with the semiaxes a, b of the ellipsoid not coupled with the moments of inertia Isym , Iant but treated as free parameters (with all the other parameters varied, too). It is an approach to account for some effects of non-ellipsoidal shape and/or inhomogeneous density distribution. While the semiaxes a and b describe the “photometric” ellipsoid, the moments of inertia define a “dynamical” ellipsoid, which can be described with dynamical semiaxes
∓2Iant + 1 adyn , bdyn = (9) . 2Isym − 1
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Table 2 Parameters of the prograde and retrograde solutions for 2002 TD60 Parameter λL (◦ ) βL (◦ ) φ0 (◦ ) ψ0 (◦ ) Pψ (h) Pφ (h) Isym Iant m a b adyn bdyn
Best solution Prograde
Retrograde
155 32 333 317 6.7872 2.8503 1.78 −0.12 0.021 0.64 0.36 0.70 0.54
334 −31 177 317 6.7871 2.8515 1.80 −0.14 0.022 0.65 0.37 0.70 0.53
The “dynamical” semiaxes adyn and bdyn were derived from Isym and Iant . The angles φ0 and ψ0 are for the epoch JD 2452585.51766 (= 2002 Nov. 07.01766 UT), LT corrected.
We found two solutions that mirror each other. The resulting parameters are summarized in Table 2. (Errors are likely in the last digits, dominated by systematic errors of the idealized model.) We note that the fitted model has a rather large r.m.s. residual of 0.18 mag (for both solutions); it is particularly apparent as imperfections of the fit to the lightcurve maxima and minima that we deem is caused by a real shape differing from the idealized ellipsoidal model. Note also that relative differences between the “dynamical” and the “photometric” semiaxes of the ellipsoid, (adyn − a)/a and (bdyn − b)/b, are 8 and 40%, respectively. The best-solution fitted curves to four sample sessions are presented in Fig. 4. We note that the notation of semiaxes used in this paragraph (see Section 2.4) differs from the conventional notation where a b c. The estimated axial ratios in the conventional notation are a/b = 1.5 and a/c = 2.7. The tilt angle θ oscillates between 53◦ and 61◦ for both solutions. It is illustrative to look at why the strongest signal was in 2(fψ + fφ ). The lightcurve showed local maxima close to the times when the angle between the asteroid’s shortest axis and the line of sight reached its extrema (minimal or maximal values). This occurred with an average period of 1.0 h (with variations of ±0.1 h), corresponding to the frequency 2(fψ + fφ ). Thus, the strongest signal in the lightcurve was due to an apparent “rotation” of the shortest axis around L. This was due to the fact that the L vector was approximately was directed to the obperpendicular to the line of sight. If L server and the asteroid was at zero solar phase, the lightcurve would be dominated by 2fψ (rotational frequency). We considered if the asteroid may be a rubble pile with zero tensile strength or a coherent body. We examined the total acceleration (gravitational plus centrifugal) of the modeled ellipsoid body. It revealed that the body must have a non-zero tensile strength if the bulk density is less than ∼ 3 g cm−3 . We fixed lengths of the ellipsoid semiaxes at values 310, 200, and 110 m (for their estimation using radar data, see below), and computed the projection of the accel-
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(a)
(b)
(c) Fig. 4. Lightcurve data of 2002 TD60 of (a) 2002-11-14.1, (b) 11-24.0, (c) 12-01.9, and (d) 12-08.9, respectively. Curve is the best-fit synthetic lightcurve of the prograde solution of the numerical model. The other, mirror solution (retrograde) gives a curve indistinguishable from the plotted one.
min of 2002 TD for bulk Fig. 5. Extremal modeled surface acceleration atime 60 density ρ of (a) 2.0, (b) 3.0, and (c) 3.5 g cm−3 . Only negative values, which correspond to the acceleration vector in the direction outward from min . the surface, are shown as equidistant contours from zero down to asurf
eration vector to a normal to the surface, a = a · (− n), at each facet over a time span T = 1 month. Then we esmin (t) = min timated a time-resolved minimum asurf r∈S a(t, r) (where t is time, r is a position vector of the facet, and S is the whole surface of the ellipsoid) across the surface min (t) for densities 2, 3, for each time point. The values of asurf and 3.5 g cm−3 oscillated in intervals −3.1 to −6.4, −2.4 to +1.4, and −0.4 to +3.6 × 10−5 m s−2 , respectively, with period Pψ /2. We also estimated a surface-resolved minimum min ( r ) = mint ∈T a(t, r) (for each facet separately) across atime min ( r ) are prethe examined time span T . The values of atime sented in Fig. 5. Asteroid 2002 TD60 was observed by CW (continuous wave) radar at Arecibo Observatory November 20–23, 2002. A monochromatic, circularly polarized radio wave was transmitted to the asteroid. The received signal has components in both opposite-sense circular (OC) and same-sense circular (SC) polarization as that transmitted. As the asteroid rotates, one limb is moving towards the observer, and one is moving away, relative to the center. This gives 2Vlimb overall velocity change across the body, which produces a Doppler
shift in the received echo. There is an additional factor of two from the change of reference frame between the asteroid and the receiver. An observer on the asteroid would see the arriving signal already Doppler-shifted, and an additional shift occurs upon reflection. This results in a total bandwidth given by: ν =
4Vlimb , λT X
(10)
where Vlimb is the limb velocity in the line of sight, and λT X is the transmitted wavelength (12.5945 cm at Arecibo).3 See Ostro (1993) for a description of the principles of radar observations. Measured values of the radar bandwidth are summarized in Table 3. The best fit of the photometric model with the radar data was obtained for the longest semiaxis value of 310 m (for the both solutions). Dimensions of the other two 3 In a case of PA rotator, we get V limb = 2π r cos δ/P where r is the asteroid’s equatorial radius, δ is the sub-Earth latitude at the time of observation, and P is the rotation period.
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Table 3 Radar observations of 2002 TD60 Date & midtime UT
Res. (Hz)
ν (Hz)
2002-11-20, 04:33 2002-11-21, 04:30 2002-11-22, 04:14 2002-11-23, 04:15
0.1 0.1 0.1 0.1
4.8 4.7 5.3 6.0
semiaxes of the ellipsoid, computed using the estimated axial ratios (Table 2), are 200 and 110 m. Uncertainties of the estimates are on the order of a few tens meters. A geometric albedo pv of 0.12 was estimated by combining the absolutely calibrated R data of 2002-11-24.0 and 12-01.9 converted to H using G = 0.55 (see above) and assuming V − R = 0.45 (about average for NEAs) with the photometric model with the absolute dimensions derived above. An error of the albedo estimate is ∼ 0.05. 3.1.3. 2000 WL107 This ∼ 0.04-km sized4 Apollo asteroid made a close approach to 0.0171 AU to the Earth on 2000 December 3. The asteroid was favorably placed for photometric observations for a few days around the closest approach. It was observed by S. Mottola on two nights of 2000-12-02.0 and 03.9 with the 1.2-m telescope from Calar Alto (total of 255 relative R points, integration times 15 s); by C. Hergenrother with the 1.54-m telescope from Steward Observatory’s Catalina Station on two nights of 2000-12-03.5 and 04.5 (147 unfiltered points, plus additional 18, 16, and 14 points in the ECAS v, x, and b filters, respectively, on the first night, all with integration times 30 s); and by M. Hicks with 0.6-m telescope from Table Mountain Observatory on the night of 2000-12-03.2 in the Johnson–Cousins BVRI system (10, 3, 2, and 2 points in R, I , V , and B, respectively). We have found that the lightcurve is two-periodic and derived its periods P1 = 0.160916 h and P2 = 0.218834 h (both with formal errors of 0.000002 h). The full twodimensional Fourier series of the 4th order has been fitted to the data with a very satisfactory result, see Fig. 6. The strongest signal is in f1 and 2f2 with Anorm = 0.24 and 0.20, respectively. The second strongest couple with significant signal is f2 and 2f1 , both with Anorm = 0.06. There are several other linear combinations of the basic frequencies with significant signal; those with Anorm = 0.03 are (f1 + f2 ), (2f1 − f2 ), (f1 − 2f2 ), and (3f1 − 2f2 ). The normalized r.m.s. residual of the fit is 0.034. In addition to the usually tested things,5 we have also checked the possibility that the true period present in the asteroid was P1 = 0.32183 h, twice the P1 value of the above solution. In that case, the strongest signal would be in 2f1 4 Size estimated from our derived absolute magnitude H = 24.44 and assuming albedo pv = 0.18. 5 At minimum, we check the 2nd harmonics and whether some of the derived frequencies might not be actually a linear addition or difference of true frequencies of the asteroid.
Fig. 6. Lightcurve data of 2000 WL107 of 2000-12-02.0 to 04.5 folded with the period of 0.160916 h and G = 0.15 (assumed). The best-fit full two-dimensional 4th order Fourier series with the periods of 0.160916 and 0.218834 h and residuals of the fitted points are plotted as well.
(equivalent to f1 above) harmonic. We consider that possibility unlikely; the normalized r.m.s. residual would be 0.041 and, even more importantly, there would be insignificant signal in all linear combinations with odd multiples of f1 . Since this asteroid is a superfast rotator, we need to check that the finite integration times do not lead to significant smoothing of the lightcurve. Following the analysis method of Pravec et al. (2000a), the integration times of 30 s should cause a smoothing of the lightcurve by a factor of 0.990 in the fastest variation with 2f2 . Thus, systematic errors in the fit due to the smoothing are a few times less than the r.m.s. residual, therefore they can be neglected. Using the two-dimensional Fourier series fit, we derived the color indices from the ECAS photometry b − v = 0.20 and v − x = −0.02. Thanks to the fact that 14 to 18 measurements have been taken in each of the bands on 2000-12-03.5, uncertainties due to imperfections of the fit (there likely remain some systematic errors on the order of a few 0.01 mag, see also below) mostly canceled out and errors of the above color indices are < 0.02 mag. We derived also color indices in the Johnson–Cousins system from the Hicks measurements on 2000-12-03.2: V − R = 0.50, B − R = 1.42, and R − I = 0.33 but their errors are estimated about 0.06 mag due to the low number of observations (see above). The ECAS color indices suggest a SQ classification. A combination of the V and v points further suggests a rather steep phase relation, lower G values than the value of 0.15 we assumed throughout the analysis.6 For G = 0.15 ± 0.2, we get the mean absolute v magnitude of 24.44 ± 0.1. 6 Since just one of the sessions used in the period analysis given above was taken in the V band, the periods solution is not sensitive to the assumed value of G.
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The observational geometry changed significantly during the campaign; the asteroid’s solar phase angle spanned a range 2◦ –21◦ and it moved in the sky by 40◦ at almost constant rate of 16◦ day−1 and in almost constant position angle of about 100◦ . We checked the data for effects that might be related to the changing geometry. We have found that while the character of the brightness variation remained the same throughout the apparition, its amplitude varied somewhat with the solar phase—an amplitude-phase effect well known from ordinary asteroids. The fact that the asteroid’s lightcurve behavior showed no apparent change with the changing sky position might indicate that our aspect to the vector did not change much during the appariasteroid’s L tion. Since the asteroid’s apparent sky motion was at the almost constant rate and direction, it also suggests that a unique, unambiguous physical model of the asteroid’s NPA rotation might not be constructed from the available data; at least two mirror solutions like in the case of 2002 TD60 are to be expected. Nevertheless, the complex two-periodic character of the asteroid is reliably detected. We conclude that 2000 WL107 is a NPA rotator, PAR = −3. 3.1.4. (54789) 2001 MZ7 We observed this ∼ 3-km sized7 Amor asteroid on 47 nights from 2003-02-12 to 06-08. The R observations were calibrated with errors of 0.01–0.02 mag for all but two nights of 06-02 and 04 that have calibration errors of 0.025– 0.03 mag. We have found that the lightcurve is two-periodic, and derived its periods P1 = 37.57 h and P2 = 52.79 h. The full two-dimensional Fourier series of the 4th order has been fitted to the data with a satisfactory result, see Fig. 7. The strongest signal is in 2f1 with Anorm = 0.37. The second strongest signal is in 2f2 with Anorm = 0.11. There are several other linear combinations of the basic frequencies with significant signal; those with Anorm = 0.05–0.07 are f1 , 3f1 , f2 , (f1 − 2f2 ), 2(f1 − 2f2 ), (3f1 − 2f2 ), (2f1 − 3f2 ), 2(f1 − f2 ), 3(f1 − f2 ), 4(f1 − f2 ). The normalized r.m.s. residual is 0.023. The solution, however, is not unique. There exist a few other solutions, combinations of two of the four periods, 37.57, 52.79, 88.74, and 130.3 h, that provide a similarly good fit as the above solution. (Note that the two last periods correspond to (f1 − 2f2 ) and (f1 − f2 ) mentioned in the previous paragraph. Generally, all the frequencies with significant signal mentioned in the previous paragraph are present also in the other solutions, just for different linear combination of their basic frequencies.) A visual check of the resulting Fourier series expansions for the other possible solutions showed a less satisfactory behavior—implausibly deep minima—at some times not covered by the observations. This suggests that the ambiguity is actually due to insufficiency of the dataset—if some additional times were covered (primarily during daytime in Europe where all the 7 Estimated from H = 15.1 (MPO 39543) assuming p = 0.18. v
Fig. 7. Lightcurve data of (54789) 2001 MZ7 of 2003-02-12 to 06-08 folded with the period of 37.57 h and G = 0.15 (assumed). The points are means of (on average) three measurements taken in quick succession. Corresponding pieces of the best-fit full two-dimensional 4th order Fourier series with the periods of 37.57 and 52.79 h are plotted as well.
data have been obtained), we would probably resolve between the solutions. It is interesting that the fit is so good (whichever of the solutions is the true one) despite changes of the observing geometry. During the campaign, the asteroid’s solar phase angle decreased from 47◦ to 33◦ , its apparent sky position moved back and forth over an arc of 29◦ , and its heliocentric vector moved by 73◦ . We checked the data for effects that might be related to the changing geometry but found nothing prominent—the amplitude and character of the brightness variation remained about the same throughout the apparition.8 This might indicate that the asteroid’s motion was such that the effective viewing aspect, that is the “latitude” of the of the Earth and Sun vectors with respect to the vector L asteroid, did not change much during the apparition. It is likely that some smaller changes of the lightcurve caused by the changing geometry occurred and that they have been absorbed into the fitted Fourier series leading to some systematic errors in the above periods. We estimate this uncertainty to be of the order of 0.1 h for each of the periods. Nevertheless, the complex two-periodic character of the asteroid is reliably detected. We conclude that (54789) 2001 MZ7 is a NPA rotator, PAR = −3, even though the ambiguity between the few periods mentioned above needs to be resolved in future. 3.2. Tumblers with PAR = −2 3.2.1. (253) Mathilde—the largest tumbler Mathilde is the largest ultraslow rotator.9 Mottola et al. (1995) observed it on 49 nights from 1995-02-03 to 06-01. 8 In particular, a variation of G around the assumed value of 0.15 by ±0.2 had an insignificant effect on the solution periods or the Anorm values given above; they varied by < 0.01 h and by ∼ 0.01, respectively. 9 Mean diameter estimate of 58 km by Tedesco et al. (1992).
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Fig. 8. Lightcurve data of (253) Mathilde of 1995-02-03 to 06-01 folded with the period of 418 h and G = 0.12. The curve is the best fit single-period Fourier series, showing that the lightcurve data contain significant deviations from the pure periodicity.
Fig. 9. Lightcurve data of (253) Mathilde of 1995-02-03 to 06-01 folded with the period of 419 h and G = 0.07. Corresponding pieces of the best-fit two-dimensional Fourier series with the periods of 419 and 250 h are plotted as well.
They found that the lightcurve data had a period of 417.7 h but they showed significant deviations from pure periodicity and inferred that Mathilde is in a non-principal axis rotation state. They also estimated the mean absolute magnitude H = 10.28 ± 0.03 and G = 0.12 ± 0.06. We have downloaded the data by Mottola et al. archived in the Asteroid Photometric Catalogue. As the first step, we reproduced the Mottola et al.’s result. We have found the best fit for a period of 418 ± 1 h and G of 0.12 (± a few 0.01). The mean absolute magnitude derived is H = 10.28 (error of a few 0.01). The composite lightcurve for the period is presented in Fig. 8. Having confirmed the Mottola et al.’s result of about 418 h and the significant deviations indicating the NPA rotation of Mathilde, we analyzed the data for a two-periodic nature. We have found that there is present another period of 250 h that significantly improves the fit—the sum of square residuals decreased 5-times. The best-fit full twodimensional Fourier series with the periods of 419 and 250 h is presented in Fig. 9. Uncertainty of both periods appears to be about 1 h. We note that the solution is not secure enough to rate a PAR code of −3, so we assign a PAR code of −2 tending to −3. Nevertheless, the solution with the two periods appears likely and plausible, with strong linear combinations of the basic frequencies (in order of decreasing signal strength) of 2f1 , f1 , (2f1 − f2 ), 2(f2 − f1 ), (f2 − f1 ), (f2 + f1 ), 2f2 , and f2 . The estimated mean absolute magnitude H and the slope parameter G are 10.24 and 0.07, respectively, both with errors of a few 0.01.
lightcurve is similar to that of (13651) 1997 BR (see below), both showing a high-amplitude (1.0–1.2 mag) main frequency component with an additional frequency causing deviations with amplitude on the order of ≈ 0.1 mag. We assign to (3288) Seleucus a PAR = −2.
3.2.2. (3288) Seleucus This 2.2-km diameter (see Harris, 1998) Amor asteroid was observed by Debehogne et al. (1983) on the night 198203-13, by M. Carlsson on 1982-03-21, both from ESO, and by Harris et al. (1999) on seven nights from 1982-04-22 to 30. Harris et al. analyzed the complete dataset and found a best fit period of ∼ 75 h but showed that the data do not fit with any simple periodicity. They concluded that (3288) Seleucus was not simply periodic, but is a tumbling asteroid. We note that the complex character of the Seleucus
3.2.3. (3691) Bede This relatively large (6-km mean diameter, assuming a dark albedo of 0.06 suggested by its Xc classification by Binzel et al., 2004) Amor asteroid was observed by Pravec et al. (1998) on 27 nights from 1995-12-28 to 1996-03-21. They have found a synodic period of 226.8 ± 0.7 h (formal error). The data show a few deviations from single periodicity that cannot be accounted for by changing geometric conditions, which were minor to moderate at the relevant times. We conclude that (3691) Bede is likely a NPA rotator, PAR = −2. 3.2.4. (4486) Mithra This ≈ 2-km sized10 Apollo asteroid was observed by Ostro et al. (2000) with the Arecibo and Goldstone radars on 8 days from 2000-07-22 to 08-09. They revealed a doublelobed object, “apparently more severly bifurcated than any other near-Earth asteroid imaged to date.” The bandwidth of the echoes were consistently very narrow, “implying some combination of very slow rotation (evident from the barely noticeable variation in the appearance of images over several hours) and a radar line of sight not far from the apparent spin vector at any time during the experiment; the radarobserved sky arc was only about 35◦ .” They further mentioned that “No simple periodicity in the day-to-day image sequence is evident, so non-principal-axis rotation is suggested. The alignment of the two lobes is almost parallel to the projected, apparent, instantaneous spin vector in some images but almost perpendicular to it in others, providing additional evidence for a very unusual spin state.” Therefore, we consider (4486) Mithra to be another slow rotator 10 Estimated from H = 15.6 (MPO 38159) assuming p = 0.2. v
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Fig. 10. Lightcurve data of (5645) 1990 SP of 2002-03-12 to 05-07 folded with the period of 30.39 h and G = 0.0. Individual points plotted are means of (on average) six measurements taken in quick succession. The fitted 2nd-order Fourier series to the April–May data is plotted to facilitate visual comparison of the data behavior at different rotational phases, see text.
in a NPA rotation state, PAR = −2. Its main period appears to be on the order of days that places it to the range of log τnorm > 0. 3.2.5. (5645) 1990 SP We observed this ∼ 1-km large11 Apollo asteroid on two nights of 2002-03-12 and 14 and on 9 nights from 2002-0406 to 05-07. The first two nights data were taken at a large phase angle of about 54◦ and at a higher ecliptic latitude of the PAB of +24◦ ; we used them for a check of consistency of the results obtained using the later nine nights data taken at the moderate phase angles of 11◦ to 24◦ and at lower latitudes of the PAB of +4◦ to +15◦ . Analyzing the later nine nights data, we found the best fit for a period of 30.39 ± 0.02 h, a mean absolute R magnitude of HR = 16.75 ± 0.2 and G = 0.0 ± 0.1. There exists another period solution of 44.5 h (i.e., ∼ 1.5× the above solution) that is formally not ruled out but is implausible as it would require a very asymmetric lightcurve with maxima differing by 0.4 mag at the moderate solar phase angles. The two March nights data support the shorter period solution as well. A composite of the lightcurve data folded with the adopted solution is presented in Fig. 10. The data do not fit together well with a single periodicity. Particularly significant is the difference by ∼ 0.3 mag between the data of 2002-04-08 and 18 around phase of 0.15 in Fig. 10 while the calibration errors were only 0.02 mag. Noticeable is also the poor fit of the data of the nights of 2002-04-06 and 10 that would require a quite steep jump by . > 0.2 mag over just 0.02 = 7◦ in rotational phase (at phase of 0.7 in Fig. 10) if the asteroid were single periodic. We therefore conclude that (5645) 1990 SP is likely a NPA rotator, PAR = −2. 3.2.6. (13651) 1997 BR This 0.8-km sized (assuming pv = 0.18 based on its Sclassification by Binzel et al., 2004) Apollo asteroid was ob11 Estimated from H = 17.2 ± 0.2 assuming p = 0.2. v
Fig. 11. Lightcurve data of (16064) 1999 RH27 of 1999-10-31 to 200002-12 folded with the period of 178.5 h and G = −0.15. Individual points plotted are means of (on average) four measurements taken in quick succession.
served by Pravec et al. (1998) on 20 nights from 1997-02-01 to 03-07. They derived the best-fit period of 33.644 ± 0.009 h (formal error) but found that the data did not fit well, the r.m.s. residual of the fit was 2- to 3-times greater than the typical errors of the points. The largest discrepancies were apparent around phases of the lightcurve maxima, making them of quite different shapes on different dates. The reader may note also the apparent similarity between the character of the complex lightcurve of 1997 BR with that of (3288) Seleucus, see above. We conclude that (13651) 1997 BR is a NPA rotator, PAR = −2. 3.2.7. (16064) 1999 RH27 We observed this Amor asteroid on 33 nights from Ondˇrejov and one night from Kharkiv from 1999-10-31 to 200002-12. The data were calibrated with errors of 0.01 mag. They fit best with a synodic period of 178.5 h, mean HR = 16.05 and G = −0.15 (realistic errors are in the last digits; Psyn–sid max = 3 h). Assuming V − R = 0.36 and pv = 0.06 based on its C-classification (Binzel et al., 2001), we estimate a mean diameter of ∼ 2.8 km. A composite lightcurve is presented in Fig. 11. The fit is far from perfect. It may be, however, caused in large part by the significant change of observing geometry—the phase angle increased from 12◦ to 30◦ and the PAB moved from 43◦ to 113◦ in ecliptic longitude at an almost constant rate of 0.7◦ day−1 during the observational interval. A few points/runs, however, deviate by more than 0.1 mag from other points at nearby rotational phases taken within relatively short time intervals (a few weeks) while the observational geometry did not change so much. The significance of the deviations with relation to a possible NPA motion of the asteroid is strengthened by the fact that the lightcurve behavior—apart from the deviations—was overally regular, consistent with an aspect not far from equator-on during the whole apparition. For a PA rotator in such conditions, we would expect a smoother fit, with lightcurve amplitude increasing due to the amplitude-phase effect. So, we conclude that the apparent deviations indicate that (16064) 1999 RH27 is likely a NPA rotator, PAR = −2.
Tumbling asteroids
3.2.8. (53319) 1999 JM8 This large Apollo asteroid was observed extensively with the Goldstone and Arecibo radars over 18 days from 199907-18 to 08-09 (Benner et al., 2002). They found it to be a slow-rotator with an asymmetric, irregular shape with typical overall dimension within 20% of 7 km, in a NPA spin state with a dominant periodicity of ∼ 7 days. They further noted that “images obtained between July 31 and August 9 show apparent regular rotation features from day to day, suggesting that the rotation state is not far from principal axis rotation.” We observed the asteroid on 20 sessions total from 1999-07-03 to 21; 15 nights from Ondˇrejov (mostly by L. Šarounová), 2 nights by Yu. Krugly and V. Shevchenko from Kharkiv, 2 nights by S. Mottola and F. Lahulla from Calar Alto, and 1 night by M. Hicks from Table Mountain Observatory. All the data were calibrated with errors 0.01–0.02 mag. On 1999-07-18 Hicks measured the color indices B − R = 1.08 ± 0.04, V − R = 0.45 ± 0.035, and R − I = 0.40 ± 0.05; the modest red slope shown by the indices is consistent with the X-classification of 1999 JM8 (Binzel et al., 2004). The photometric data best fit with a period of 136 h and G = −0.09 but significant deviations from a simple periodicity are apparent, see the composite lightcurve in Fig. 12. While the deviations are likely partly caused by changing observing geometry during the interval covered—the phase angle increased from 85◦ to 121◦ and the PAB moved by 29◦ —some of them cannot be due to changing geometry and therefore indicate that the rotation is not single-periodic. Particularly, the two points of 1999-07-14 and 15 are brighter by 0.3–0.5 mag than points taken at similar rotational phases in earlier as well as later cycles; the behavior cannot be explained by the evolution of geometry of a singly-periodic object. Another feature, the steep rise seen in the lightcurve of 1999-07-19 in the lightcurve maximum also suggests a presence of another frequency contributing to the behavior seen (cf. lightcurves of other tumblers with similar behavior mentioned in this section). We cannot derive the second
Fig. 12. Lightcurve data of (53319) 1999 JM8 of 1999-07-03 to 21 folded with the period of 136 h and G = −0.09. Individual points plotted are means of (on average) six measurements taken in quick succession, their errors are always less than the calibration errors of 0.01–0.02 mag. The fitted 2nd-order Fourier series is plotted to facilitate visual comparison of the data behavior at different rotational phases, see text.
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period of the asteroid’s NPA motion, and we also cannot be sure whether the detected period of 136 h does not correspond to some linear combination of real frequencies of the asteroid’s NPA motion. Nevertheless an excited rotation state is indicated by the lightcurve data as well as from the radar observations. We conclude that (53319) 1999 JM8 is a NPA rotator, PAR = −2. 3.2.9. 2002 NY40 This Apollo asteroid made a close approach to 0.0035 AU to the Earth on 2002 August 18. It was favorably placed for photometric observations on its inbound trajectory during a few weeks before the closest approach, an extensive photometric campaign was organized. We observed it in 21 sessions from 2002-07-31.3 to 0816.4. Most of the photometry was done in R filter, and 16 of the 21 sessions were calibrated in the standard Cousins R system. For one (08-11.3 by W. Cooney) of the five uncalibrated sessions, a zero point of the magnitude scale was derived from an overlap with another session (by D. De Graff). The other four uncalibrated sessions were retained in the analyzed dataset as they provide additional useful information on the lightcurve shape at the particular times, and we derived their magnitude scale zero points to provide a good fit. The data were obtained by D. De Graff with 0.82-m telescope from Stull Observatory (9 calibrated sessions 2002-07-31.3 to 08-12.2; calibrations done using the list of secondary standards “Loneos Reference Star Catalog” by B. Skiff12); by L. Šarounová and P. Pravec with the 0.65-m telescope from Ondˇrejov (6 calibrated sessions 0803.0 to 14.9); by M. Hicks with 0.6-m telescope from Table Mountain Observatory (one calibrated session 08-07.4; also measured V − R = 0.423 ± 0.010); by J. Lambert with the US Air Force 0.36-m telescope from AMOS Remote Maui Experimental site (2 unfiltered sessions of 08-13.4 and 16.4); by S. Foglia, C. Cremaschini, W. Marinello, M. Micheli and S. Zubani with 0.4-m telescope from Lumezzane Observatory (one uncalibrated V session 08-08.0); by W. Cooney with 0.3-m telescope from Blackberry Observatory (one unfiltered session 08-11.3, but see above); and by G. Masi with the 0.8-m telescope from Campo Catino Observatory (one uncalibrated R session 08-13.9). There are available three additional sessions of 08-17.3 (calibrated R session by M. Hicks), 08-17.4 (unfiltered session by J. Lambert), and 08-18.1 (uncalibrated R session by J. Licandro et al., personal communication). We checked them for consistency with results reported below but did not use them directly in the analysis as observational geometric conditions changed rapidly within one day around the closest approach that would require introducing additional parameters into the solution (actually requiring a pole solution) while the added information was not enough to support the increase of the number of free parameters. Therefore 12 Available on ftp://ftp.lowell.edu/pub/bas/starcats/loneos.phot.
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we left the three additional sessions for the future modeling. We found that the main period of the lightcurve is P1 = 19.98 ± 0.01 h (realistic error). A composite lightcurve presented in Fig. 13 shows significant deviations from single periodicity on the order of 0.1 mag. Changing geometric conditions cannot account for the deviations; the observational geometry changed slowly during the interval when it was on the inbound trajectory (2002-07-31.3 to 08-16.4)— the asteroid’s solar phase angle was in a range of 11◦ –16◦ , its apparent sky position changed by 8◦ (but only by 4◦ over the 20 sessions before 08-15), and its heliocentric vector moved at an almost constant rate of 0.8◦ day−1 in the ecliptic longitude while it stayed within 1◦ of Bh = +1◦ throughout the interval. The deviations from cycle to cycle must be mainly due to the asteroid itself, and indicate that 2002 NY40 is a NPA rotator. A search for the second period in the data did not yield a unique result but a period of P2 = 18.43 ± 0.01 h appears likely. No higher than the 2nd order of the full twodimensional Fourier series could be used—an attempt to use the third order showed that the data are not sufficient to describe the lower-amplitude deviations reliably to the higher order. So, the resulting fit shown in Fig. 13 necessarily has significant residuals on the order of several 0.01 mag. Most of the signal is in 2f1 , Anorm = 0.38. The next significant linear combinations of the two basic frequencies are f2 , 2f2 , (f1 + f2 ), 2(f1 − f2 ), (f1 + 2f2 ), and (f1 − 2f2 ), with Anorm = 0.03 to 0.06. The solution is still not sufficient for a PAR code of −3, so we give it PAR of −2 tending to −3. The lightcurve shows also another interesting feature; there are apparent sharp bends in the data covering the secondary minimum around phase 0.5 in Fig. 13, creating a “plateau” at the bottom of the minimum. It was well covered at times around 2002-08-11.4, 13.9, and 16.4, and with a poorer coverage also around 08-12.2. The behavior suggests the presence of large non-convex features in the asteroid, consistent with the radar images (see below). We estimated a mean absolute R magnitude of HR = 18.81 ± 0.2 assuming G = 0.15 ± 0.2. Since V − R =
Fig. 13. Lightcurve data of 2002 NY40 of 2002-07-31.3 to 08-16.4 folded with the period of 19.98 h and G = 0.15 (assumed).
0.423 ± 0.010, the mean absolute V magnitude is H = 19.23 ± 0.2. An absolute magnitude for the maximum of the lightcurve is brighter by 0.3 mag while that for the flat minimum (with the “plateau”) is fainter by 0.5 mag than the mean H . Rivkin et al. (2003) found a striking match between their 0.3–4 µm spectrum of 2002 NY40 and spectra of LL6 ordinary chondrites. They also measured thermal flux of the asteroid and estimate a geometric albedo of 0.25 ± 0.03, assuming a non-rotating model with no thermal inertia. Müller et al. (2004) estimated an albedo of 0.34 ± 0.06 with a nearEarth asteroid thermal model and a thermophysical model. Using an average of the two estimates and our mean absolute magnitude estimate gives an effective diameter of the asteroid of 0.35 ± 0.07 km (combined error of the H and the albedo estimates). Radar images show that this is an elongated object which has two fairly spherical lobes, with diameters 0.33 and 0.24 km. The maximum dimension must be at least 0.61 km, which makes the effective diameter 0.36 ± 0.07 km (Howell et al., personal communication), which agrees well with our estimate. 3.3. Principal axis rotators with long τ 3.3.1. (3102) Krok—a singly-periodic ultraslow rotator Krok is an exceptional object. Though its log τnorm is nearly 2, i.e., its estimated damping timescale is longer by two orders of magnitude than the age of the Solar System (even more compared to its expected collision lifetime), its lightcurve appears to be singly-periodic, or nearly so, showing variations consistent with the synodic–sidereal rotation period variation of a PA retrograde rotator. Harris et al. (1992) derived a synodic period of 147.8 h from their observations taken during eight nights from 198108-27 to 09-04. We re-analyzed their data and found a synodic period of 148 ± 1 h (realistic error estimate). A lower limit of the amplitude is 1.0 mag (at phase angle of 24◦ ). The only apparent deviation from single periodicity is on the two nights of 1981-08-29 and 09-04, where the magnitude levels at the same rotation phase differ by ∼ 0.1 mag, or equivalently, the rotation phase is displaced by a couple hours, when plotted at the derived period (the period is more tightly constrained by overlapping data on other nights). We observed the asteroid in two recent apparitions; on 22 nights (19 from Ondˇrejov and 3 from Kharkiv Observatory) from 2000-07-26 to 10-29 and on 15 nights from Ondˇrejov from 2003-05-31 to 07-02. Our analysis of the first 14 nights of the 2000 apparition, taken during 2000-07-26 to 09-08, revealed a synodic period of 149.4 ± 0.1 h (formal error; real error a few times greater) and amplitude of 1.6 mag at phase angles in a range 42◦ to 55◦ . All the 14 nights data are consistent with the single periodicity. Adding the data of two and six nights taken one and two lunations later, from 2000-09-30 to 10-06 and from 2000-10-21 to 29, respectively, are consistent with the same general lightcurve but are displaced in rotation phase from
Tumbling asteroids
Fig. 14. Composite lightcurve of (3102) Krok of 2000-07-26 to 10-29 for the synodic period of 149.4 h and G = 0.35. Points of eight nights of 2000-09-30 to 10-29 show a shift in rotation phase due to a change of the synodic period with the decreasing rate of motion of the asteroid, see text.
that predicted with a constant synodic period. The shift is consistent in magnitude with an expected synodic–sidereal period difference from the PAB motion, and is in a sense indicating retrograde rotation, i.e., the synodic period increased while the asteroid PAB motion decreased (see also below). A lower amplitude of 0.7 mag was seen at the lower phase angle of 12◦ of the six nights observations in the last decade of October. A composite lightcurve of the 2000 data is presented in Fig. 14. The 2003 data revealed a synodic period of 151.8 ± 0.3 h (formal error; real error a couple times greater) and a lightcurve amplitude of 1.3 mag at the phase angles from 17◦ to 37◦ . Again, all the 15 nights data are consistent with the single periodicity. The motion of the asteroid (both apparent as well as PAB) during this apparition was several times slower than during the two previous apparitions so the synodic period of the 2003 apparition is close to a sidereal rotation period of the asteroid. A composite lightcurve of the 2003 data is presented in Fig. 15. The 20◦ phase angle interval covered during the 2003 apparition while the asteroid’s motion was small has allowed us to estimate the phase relation’s G value of 0.4 ± 0.2. The 2000 data covering even a larger interval in solar phase angle but with the larger sky position change (therefore less reliable for an estimation of G because of a possible aspectinduced change of the asteroid’s brightness level) are consistent with the above G value estimate as well (though lower values of the uncertainty interval are preferred). We derived mean HR of 16.2 and 16.14 from the 2003 and the late October 2000 data, respectively (assuming the 2003 estimate for G of 0.4 ± 0.2 for the 2000 apparition as well), that converts to HV of 16.67 and 16.61 (both ±0.2) in the two apparitions using V − R = 0.47 ± 0.04 measured by Yu. Krugly and V. Shevchenko on 2000-10-24. Using the same G value estimate, we get mean HV = 16.56 ± 0.2 in the 1981 apparition. So, the best estimate of the mean H = HV is 16.6 ± 0.2 in all the three apparitions. Assuming pv = 0.18 for the S-type
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Fig. 15. Composite lightcurve of (3102) Krok of 2003-05-31 to 07-02 for the synodic period of 151.8 h and G = 0.35. The best-fit 4th order Fourier series is plotted as well.
classification of Krok (Binzel et al., 2004), we get an estimate of its mean diameter of 1.5 km. The apparent variations of the Krok’s synodic period during the three apparitions are correlated with its prograde PAB (as well as apparent) motion; the faster motion, the shorter synodic period. The behavior requires a retrograde direction of rotation of the asteroid. Correcting the data for a synodic–sidereal effect assuming a pole position of (260◦ , −60◦ ) in J2000—a direction close to normal to the PAB motion in the 1981 and 2000 apparitions—we found that a sidereal period about 151.6 h with an error about 0.5 h is consistent with the data of all but one of the total 45 nights. The one deviating run seen in the 1981 data (see above) could perhaps be an observational error, or it may indicate a small deviation from nearly PA rotation. We note that while the pole position cannot be estimated accurately, its retrograde nature is well established and the sidereal period estimate is accurate. Furthermore, the fact that the asteroid exhibited a very large amplitude of variation in 1981 and 2003, from aspects almost exactly 90◦ apart in the sky, indicates that the pole orientation is approximately orthogonal to those two directions. We conclude that (3102) Krok is a retrograde rotator likely in a PA spin state or nearly so, PAR = +2. If there is any tumbling motion in the asteroid, its amplitude must be small, on the order of 10% or less of the principal amplitude of variation. 3.4. Transition-range PA rotators Around log τnorm = 0, there is a transition between the range of prevailing tumblers and the range dominated by PA rotators. Asteroids with estimated log τnorm around zero contribute to the statistical definition of the transition. NPA rotators lying there were presented in Sections 3.1 and 3.2. Here we present data for asteroids in the range that exhibit a single-periodic character. Specifically, we present all of them that have estimated τ > 1 byr.
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Fig. 16. Composite lightcurve of (11398) 1998 YP11 of 1999-07-14 to 08-10 for the best fit synodic period of 38.60 h and G = 0.29.
3.4.1. (3752) Camillo This ∼ 2.6-km sized13 Apollo asteroid with log τnorm = −0.4 was observed on 14 nights during 1995-08-03 to 29 by Pravec et al. (1998). They derived a synodic period of 37.846 ± 0.012 h and found the data to be consistent with the single periodicity; ≈ 0.1-mag deviations of two of the 14 nightly lightcurves they saw have been attributed to changing observing geometry. Well fitting (to within a few 0.01 mag) overlapping data of the other runs cover about a half of the rotational period. So, we conclude that (3752) Camillo is likely a PA rotator, PAR = +2. If there is any tumbling motion in the asteroid, its amplitude is probably small, on the order of a few 0.01 mag or less. 3.4.2. (11398) 1998 YP11 We observed (11398) from Ondˇrejov on 17 nights from 1999-07-14 to 08-10. The data show a synodic period of 38.60 ± 0.03 h (the estimated synodic–sidereal effect, Psyn–sid max , is 0.06 h) and a lightcurve amplitude of about 0.25 mag. The range of solar phase angles covered was 8◦ to 27◦ , allowing us to obtain a (formally) accurate estimate of mean HR = 16.10 ± 0.03 and G = 0.29 ± 0.02 but real errors of the values may be greater due to a possible aspect-induced change of asteroid’s mean brightness as the geometry changed during the observational interval; the PAB changed by 10◦ with most of the change in the ecliptic latitude. Using V − R = 0.48 and pv = 0.18 (assumed for its S-type classification, Binzel et al., 2004), we get a mean absolute V magnitude H = 16.6 (error about 0.1 mag), an estimated mean diameter of 1.5 km and log τnorm = 0.1. A composite lightcurve is presented in Fig. 16. The data, overlapping at several phases repeatedly, are consistent with the single periodicity to within observational errors of about 0.02 mag. We conclude that (11398) 1998 YP11 is likely a PA rotator, PAR = +2. If there is any tumbling motion in the asteroid, its amplitude must be small, on the order of a few 0.01 mag or less. 13 Estimated from H = 15.41 ± 0.13 (Pravec et al., 1998) assuming
albedo pv = 0.18.
Fig. 17. Composite lightcurve of 1999 FA of 1999-03-18 and 19 for the best-fit synodic period of 10.092 h and G = 0.15 (assumed).
3.4.3. (25143) Itokawa This small Apollo asteroid (estimated effective diameter of 0.32 ± 0.03 km, see Ishiguro et al., 2003) was observed extensively from more than 10 observatories that obtained 53 nightly runs from December 2000 to September 2001. The data were collected and analyzed by Kaasalainen et al. (2003). They obtained a unique spin vector estimate with sidereal period of 12.132 h and obtained a model of its shape. Even though they found that several lightcurves did not fit well and they had to introduce time offsets to get them to fit with the other data, their solution appears safe and the reliable part of the lightcurve data fits well with the single periodicity, thus implying a PA rotation to within a few 0.01 mag. So, it is PAR = +4 object. It has log τnorm = −0.1. 3.4.4. 1999 FA This is a small, ∼ 0.24-km S type14 Apollo asteroid with log τnorm = −0.1. We observed it from Ondˇrejov on two nights of 1999-03-18 and 19 and found that it had a large amplitude (1.2 mag) lightcurve with a synodic period of 10.092 ± 0.008 h (Psyn–sid max = 0.014 h). A composite lightcurve is presented in Fig. 17. While the data overlap over only 37% of the rotational phase, the fit is excellent, showing no systematic deviations to within the data calibration errors of 0.02 mag. We therefore conclude that 1999 FA is likely a PA rotator, PAR = +2. Though this detection is somewhat less reliable than some other PAR = +2 objects mentioned in this paper, the excellent fit of the data with the single periodicity leads us to judge that if there is any tumbling motion, its amplitude is probably small, on the order of a few 0.01 mag or less. 3.4.5. 2001 VS78 We observed this Amor asteroid from Ondˇrejov on 33 nights from 2002-02-22 to 06-18. A synodic period of 14 Size estimated from H = 20.58 (ASTORB.DAT from ftp.lowell. edu/pub/elgb) assuming pv = 0.18, the S classification by Binzel and Bus, http://smass.mit.edu/.
Tumbling asteroids
Fig. 18. Composite lightcurve of 2001 VS78 of 2002-02-22 to 04-18 for the best-fit synodic period of 40.553 h and G = 0.10.
Fig. 19. Composite lightcurves of 2001 VS78 of 2002-05-02 to 22 and 05-29 to 06-18 for the best-fit synodic period of 40.553 h and G = 0.10.
40.553 h (realistic error < 0.01 h, Psyn–sid max = 0.09 h), mean absolute R magnitude HR = 15.25 ± 0.2 and G = 0.10 ± 0.1 were derived. Using V − R = 0.48 and pv = 0.18 (assumed for its S-type classification, Binzel et al., 2004), we get an estimated mean effective diameter of the asteroid of ∼ 2.2 km and log τnorm = −0.2. Composite lightcurves constructed with data covering three sub-intervals, 2002-0222 to 04-18, 05-02 to 22, and 05-29 to 06-18, fit well with the single periodicity, see composite lightcurve presented in Figs. 18 and 19. The apparent evolution of the lightcurve over the campaign is quite plausibly related to changing geometry of the observations; specifically, the phase angle decreased from about 50◦ in the February–April subinterval downto 25◦ at the end of the campaign, the apparent sky position changed by about 30◦ and the heliocentric position changed by 86◦ during the campaign. Over the shorter intervals, the lightcurve data fitted well with the single periodicity to within a few 0.01 mag (with a few notable exceptions of points around phase 0.75 in Fig. 18; see below for a suspected cause). We conclude that 2001 VS78 is likely a PA rotator, PAR = +2. If there is any tumbling motion in the body, its amplitude must be small, on the order of a few 0.01 mag or less. There is one unusual feature in the asteroid—the sharp minima occurring at phases between 0.22 to 0.28 and between 0.72 to 0.79 in Fig. 18 on five nights from 2002-03-06 to 04-09 but missing in the data covering the same phases on later dates, after 05-11. The sparse coverage of the minima does not allow us to draw a well-based conclusion. (Indeed, we have worked the asteroid on many nights but optimized
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the coverage on each night for its long-period character so that be able to work other asteroids on the same nights as well, not expecting fast, sharp changes like the ones found there at the specific phases after the campaign was over and all the data accurately calibrated and fully reduced.) Nevertheless, we consider a possibility that the sharp minima may in fact be occultation or eclipse (the latter being more likely, as the asteroid motion with respect to the Sun was more significant than its geocentric motion) events caused by a satellite of size of ∼ 0.6 diameters of the primary body orbiting it at a distance of about 4 diameters of the primary, synchronously with the primary’s rotation. An investigation of the binary hypothesis for 2001 VS78 is outside the scope of this paper and will be given in another paper on binary asteroids, but we have to keep in mind that the apparent PA rotation of 2001 VS78 (or its primary) might be due to tidal interactions in synchronized binary system and therefore irrelevant for our statistical study of solitary PA/NPA rotators, so its use in the study is conditional. 3.4.6. 2003 FG This is a small, ∼ 0.36-km sized15 Apollo asteroid with log τnorm = −0.6. We observed it on three consecutive nights of 2003-03-25.0 to 27.0 from Ondˇrejov and on one night of 2003-03-25.3 from ESO. The Ondˇrejov data were mutually linked to a same local comparison star while the ESO data are relative. The observations revealed a large amplitude (1.4 mag) lightcurve with a synodic period of 8.692 ± 0.003 h (Psyn–sid max = 0.005 h). A composite lightcurve is presented in Fig. 20. Even though the data are relatively noisy, they fit well to a singly-periodic composite, to within < 0.1 mag. We conclude that 2003 FG is likely a PA rotator, PAR = +2. Though this detection is somewhat less reliable than some other PAR = +2 objects mentioned in this paper, the good fit of the data with the single periodicity over more than half of the rotational period covered at least twice (some phases even thrice) leads us to judge that if there is any tumbling motion, its amplitude is probably small, on the order of a few 0.01 mag or less.
4. Statistics of tumbling asteroids In the previous section, we have reviewed all known tumbling asteroids with PAR codes −2. It might seem paradoxical but actually it is logical that the best detected cases (in Section 3.1) are not typical members of the current sample of tumblers. With the present state of the observational technology, the best detected tumblers must be unusual in some of their properties or circumstances—(4179) Toutatis had a very favorable close encounter that allowed it to be observed thoroughly with radar; 2002 TD60 and 2000 WL107 are fast rotators (stronger than most larger, rubble-pile asteroids, see below) that were described well with photometry of 15 Estimated from H = 19.7 (MPO 48340) assuming p = 0.18. v
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Fig. 20. Composite lightcurve of 2003 FG of 2003-03-25 to 27 for the best-fit synodic period of 8.692 h and G = 0.15 (assumed).
several nightly sessions during their favorable apparitions; and (54789) 2001 MZ7 is a “faster rotating” member of the slow tumblers population, so it is relatively large (see below for implications of the position of the asteroid in the spindiameter space on its possible younger age or higher rigidity than average). Most of the tumblers, however, are too slow rotating and/or too small for obtaining enough data to establish their second periods and they have obtained PAR = −2 (Section 3.2). With the current observational technology, studies of NPA rotators are pretty challenging and the available sample needs to be treated carefully to account for selection effects. We have established the PAR scheme in order to make useful estimates of reliability of the detections, allowing meaningful statistical studies of the population of tumbling asteroids. 4.1. Rubble-pile tumblers Figure 21 shows spin rate vs. diameter for asteroids with U 2 from the list compiled by A.W. Harris (version 2003 December 19).16 The tumblers with PAR −2 are marked there. We have also reviewed observational data for all asteroids in the range of D > 0.15 km (where rubble piles dominate; see Pravec et al., 2002) and log τnorm > −2 and established their PAR codes.17 We see a few important things from the plot. Most asteroids with estimated damping time scales (Eq. (5)) equal to the age of the Solar System or longer 16 The list is available on http://www.asu.cas.cz/~asteroid/ lcsumdat20031219.txt. For tumblers, their main periods with the strongest signals have been used as approximation of their spin periods. For (4486) Mithra where no exact period estimate was obtained, we used a value of 100 h for plotting in Fig. 21; the corresponding point is marked with an up triangle. 17 A complete list of PAR codes for asteroids we have investigated so far is available at the web page http://www.asu.cas.cz/~asteroid/tumblers. htm. In Table 1, we list tumbling asteroids mentioned in Sections 3.1 and 3.2.
Fig. 21. Asteroids’ spin rate vs. diameter plot. Tumblers with PAR −2 are marked with filled triangles. Principal axis rotators with PAR +2 in the range of D > 0.15 km and log τnorm > −2 are marked with open boxes with pluses; remaining points in the range have −1 PAR +1, i.e., they have not been reliably resolved as PA or NPA rotators.
(i.e., with log τnorm 0) are NPA rotators. Of the 12 objects with |PAR| 2 lying at or below the τ = 4.5 byr (i.e., log τnorm = 0) constant damping timescale line in Fig. 21, ten are tumblers. It is entirely consistent with the prediction by Harris (1994), see Section 1. Above the line, the situation changes rapidly; PA rotators start to prevail for log τnorm values just a few tenths less than zero. It appears that (253) Mathilde and (5645) 1990 SP with log τnorm = 0.0 are close to a transition range between the range of prevailing slower tumblers and that of faster PA rotators. (54789) 2001 MZ7 with log τnorm = −0.6 might already be somewhat unusual—it may be either younger or of a higher value of µQ than average rubble-pile asteroids of its size. (We also note that its C-classification by Dandy et al. (2003), suggests that it may have a darker albedo than we assumed, which would suggest even larger size and lower value of log τnorm about −1.0, therefore strengthening its somewhat unusual character.) Even though we expect the transition range to have a certain width due to a range of material properties as well as to varying collision ages of rubble-pile asteroids over the size range in question, the transition appears to lie between the lines of constant log τnorm = 0 and −1. The fact that the collision ages of most asteroids in the size range 0.2 to ∼ 60 km are expected to be shorter than 4.5 byr, perhaps by an order of magnitude or so, may indicate that average material properties of rubble pile asteroids are just a little bit weaker than those proposed by Harris (1994), but in any case well within the uncertainty range of his estimate given there. It would be of interest to know if the PAR +2 objects lying in the transition range actually are in PA rotation states, or if they have some residual, not fully damped tumbling motion. Even if we do not see any obvious deviations from the single periodicity in them, and therefore assign PAR values of +2 and greater, it is possible that some of them have not been damped quite to the basic rotation state and have a
Tumbling asteroids
small residual tumbling motion not apparent in the available data. A knowledge of distribution and amount of the residual energy of tumbling motion among the objects in the transition range should be useful for constraining theories of the excitation and damping of tumbling motion. A first indication for such an object may be returned by the MUSES-C (Hayabusa) mission visiting (25143) Itokawa, a small slow rotator in the transition range (Yano et al., 2004). A closer look at the exceptional case of (3102) Krok is in order. Having log τnorm of 1.8, it is the only known asteroid with PAR = +2 lying well below the log τnorm = 0 damping timescale line. Since its age cannot be longer than the age of the Solar System, its apparent single periodicity might be due to unusual physical properties or to an unusual evolutionary course leading to slow rotation. One possibility is that it might have an exceptionally low value of µQ in comparison with average rubble pile tumblers so that a much shorter damping time scale that would apply, despite its very slow spin rate (P of 151 h). The required value of µQ/T , where T is a spin excitation age of the asteroid (i.e., a time elapsed since the last significant spin excitation event), would have to be less by nearly two orders of magnitude than average for rubble-pile tumblers. Another possibility is that the mechanism that slows asteroids from initially faster spin may not induce tumbling motion, so that the tumbling motion may need to be caused by a separate mechanism, e.g., a collision after the slow spin is established. Thus not all slow rotators may have experiment a subsequent excitation into tumbling rotation. A third possibility is that its shape might be special so that tumbling motion, even if present, might not be observable in the lightcurve. For example, if it was close enough to a biaxial shape, one of the two periods would produce no detectable lightcurve signal. We conclude that even though we cannot actually resolve what is special in the exceptional case of (3102) Krok, its apparent single periodicity does not require its internal structure or evolution mechanism to be different from those of other rubble-pile tumblers. The presence of a small fraction of apparent singly-periodic objects among slow rotators dominated by tumblers may be a natural consequence of a distribution of their shapes or evolution routes. 4.2. Monolithic tumblers The smallest known NPA rotator is obviously not a rubble pile. 2000 WL107 is a superfast rotator with the periods about 0.2 h, so it clearly cannot be held together by selfgravitation only and must be a coherent body with non-zero tensile strength (even if it might be quite modest in comparison with strength of solid rocks on the earth; Pravec et al., 2002, and references therein). It has log τnorm ∼ −4, indicating that it has a value of µQ/T greater by at least four orders than average (larger) rubble pile asteroids. Because of its small size, it is likely younger by at least two orders of magnitude than the age of the Solar System, but probably not by four orders of magnitude. Thus its tumbling state
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appears to indicate a value of µQ/T greater than is typical for rubble piles, perhaps by one or two orders of magnitude. This is not unexpected for a body that is monolithic rather than a loose aggregate. It may also be that it is younger than average for its size, since a few other superfast rotators in the same size and spin rate range are not tumblers. The case of 2002 TD60 is less clear. As shown in Section 3.1, its spin rate (around instantaneous spin axis) is just such that it might be a rubble pile with zero tensile strength if its bulk density were > 3 g cm−3 . Such a density would be relatively high for a rubble pile (see Britt et al., 2002) and we also note that shapes so elongated as 2002 TD60 are generally unknown among single-period asteroids with periods shorter than 3 h—they tend to have spheroidal shapes with small amplitudes and elongations (Pravec and Harris, 2000). If 2002 TD60 was a high-density rubble pile, it would be an exceptional object, either a stony one with very little porosity (then it might be considered as being a transitional object between “true” rubble piles with high porosities and coherent “monoliths”; see Richardson et al. (2002), for discussion on possible spectrum of internal structures of asteroids), or an iron-rich, high density rubble pile. The radar observations of 2002 TD60 indicate the radar albedo about 0.25, that argues for non-metallic composition, so the latter possibility appears eliminated. The former possibility must be admitted. So, we conclude that 2002 TD60 is either a coherent body with non-zero tensile strength like those smaller superfast rotators, or one with a “transitional” internal structure of low porosity and high density even if it might have a negligible tensile strength. Its log τnorm ∼ −2 indicates a higher value of µQ/T than that of rubble pile tumblers of just a bit greater sizes. Most of the high value of µQ/T in this case may be due to the lower value of T appropriate for fewhundred meter diameter objects, but is also consistent with a greater value of µQ inferred for a near-monolithic body. The abundance of NPA rotators in the range of monoliths (D < 0.15 km) is unclear. The single known tumbler in the size range, 2000 WL107 , is about in the middle of the known sample of small monoliths both with respect to size and spin rate. Since lightcurve data for most of the other objects in the sample were not available to us—most of the objects were observed by Whiteley et al. (2002b) and their full lightcurve data still await publication—we do not know if the tumbler 2000 WL107 is a real exception with higher rigidity or younger age than other objects in its size range, or if it is perhaps just a high µQ/T tail member of only moderately wide distribution of the parameters in the population of monoliths. Future studies and, especially, careful thorough observations of small asteroids with D < 0.15 km, will be particularly interesting to shed a light on the abundance of NPA rotators among them. A future statistical study of them may lead to interesting inferences of their internal properties.
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4.3. Elongations of tumblers Observed lightcurve amplitudes of the tumblers are similar to or greater than the average amplitude of asteroids with spin rates < 6/day (which do not show the tendency to spheroidal shapes like faster-rotating rubble piles close to the barrier at ∼ 11/d; see Pravec and Harris, 2000, Fig. 6). There are a few possible explanations. First, it might be a selection effect—higher amplitude (i.e., more elongated) tumblers might be easier to recognize, as the main period and consequently also deviations from it are easier to detect unambiguously in a larger-amplitude lightcurve. Another observational effect that may contribute is that in a tumbling asteroid, a larger variation of cross-section is seen during their NPA rotation than for a PA rotator of the same shape, hence some increased total amplitude. An effect that biases against detection of low-amplitude tumblers is that more regular objects, with principal moments of inertia less different from one another have Pψ much longer than Pφ (see Samarasinha and A’Hearn, 1991), therefore much more difficult to detect. We conclude that the observational selection effects might well explain the entire apparent lack of low-amplitude tumblers, so that NPA rotation may be just as common among low-amplitude, more spheroidal asteroids as among the more extremely shaped objects.
5. Mechanisms causing slow NPA rotations Most slowly rotating asteroids larger than 0.2 km with log τnorm 0 probably have been slowed down from faster rotations (see Pravec and Harris, 2000). It seems likely that they were PA rotators earlier then when they rotated faster. The fact that we see them in NPA rotation states now when rotating slowly suggests that the mechanism (or mechanisms) that caused their slow rotations also induced their NPA rotations. Or alternatively, the slowed down asteroids might be more sensitive to excitation by other processes, e.g., by impacts, or tidal forces during planetary encounters. 5.1. YORP effect as a spin-down mechanism Rubincam (2000) proposed that the delayed re-radiation of thermal energy from a spinning body could produce a torque and alter the spin rate and orientation of small asteroids (see his paper for a brief review of the concept, going back at least 50 years to the Russian publications by Yarkovsky and by Radzievskii, and American publications by O’Keefe and Paddack). Rubincam has coined the term “YORP” (Yarkovsky–O’Keefe–Radzievskii–Paddack) for the effect. Rubincam computed torque models for several small bodies of known shape (martian satellites Phobos and Deimos, and asteroids 951 Gaspra and 243 Ida). He found that even for a variety of shapes, the rate of change of spin and/or spin orientation is similar, with scaling proportional to the inverse square of both object dimension (diameter)
and solar distance (orbital semi-major axis). Furthermore, the pattern of spin-up versus spin-down, and the evolution of spin axis orientation, is similar for all of the shapes studied. Rubincam noted that this mechanism might plausibly explain very slow rotations, including that of 253 Mathilde. ˇ Vokrouhlický and Capek (2002) carried the work further considering additional asteroid shapes and tracing spin evolution including the effect of precession of the orbit plane of the asteroid. All of this work was theoretical without a firm observational basis until Slivan et al. (2003) recently showed that the spin vectors of relatively large members of the Koronis asteroid family, bodies up to ∼ 40 km in diameter, are non-random in both orientation and spin rate and thus have likely been substantially altered over time. Vokrouhlický et al. (2003) have modeled this as a result of the YORP effect. They show that in the case of Koronis family members, initially retrograde spins should be driven to near pole-upright (180◦ obliquity), with the spin rate either spun up to periods of only a few hours, or spun down to rates asymptotically approaching zero rotation rate. Prograde rotators should be driven to spins in a particular resonance state with obliquity around 50◦ , poles aligned in longitude with respect to a particular resonance angle, and spinning with periods around 8–10 h. All members of the Koronis family that have spins and pole orientations determined fall into one or another of these three “Slivan states,” indicating convincingly that their spins have been radically altered over the age of the asteroid family, almost certainly by this radiation pressure torque. While the members of the Koronis family provide a “controlled sample” of asteroids with defined expected end states, the YORP effect itself does not depend on this particular family membership so should alter spins of all small asteroids, in varying ways depending on location in the asteroid belt. As noted above, the rate of spin alteration by YORP should be proportional to the inverse square of the asteroid diameter, all other things being equal (shape, thermal properties, distance from the Sun, etc.). In an example calculation to model possible evolution of the Koronis family member (167) Urda, a 40-km diameter slow retrograde rotator, Vokrouhlický et al. (2003) calculate a deceleration rate of ∼ 1 cy (day 109 yr)−1 , thus an asteroid this large can despin totally from an initial rate of several cycles per day, i.e., less than ∼ 6 h period over the age of the Solar system. Scaling by diameter squared, an asteroid only ∼ 1 km in diameter should evolve 1600 times faster, thus it could despin from an initial period of ∼ 4 h in only a few million years, which is far shorter than the expected collision lifetime of km-sized asteroids. At the other extreme, the largest known very slow rotator is 253 Mathilde, with a diameter of 58 km and spin period of 418 h. Scaling from the example of (167) Urda, the deceleration rate should be about half as fast, so Mathilde might plausibly slow to its present rate from an initial rotation period of around ten hours, which is not an unreasonable initial spin state. Indeed, over the entire size range from Mathilde down to the smallest slow rotators, the time of spin-down estimated in this way is short com-
Tumbling asteroids
pared to the age of the Solar System, or for objects smaller than a few tens of km diameter, shorter than the expected catastrophic collision time scale for their size. Thus YORP despinning appears to be a plausible cause for the excess of very slow rotators, which has been a mystery since the class of objects was first noticed twenty years ago. 5.2. Is YORP effect an explanation for the slow tumblers? The next question is, can YORP despinning cause tumbling among slow rotators, or is some other mechanism required to produce the non-principal axis rotation? It is noteworthy that YORP despinning has the characteristic of “sliding friction” in that the torque appears to first order not to depend on rate of spin. Thus a despinning asteroid should eventually “slide to a halt” and become spin-locked to its orbital period, as is the case for tidal friction, which leads to satellites like the Earth’s Moon having a rotation period equal to the orbit period. We do not see this with asteroids, so something is keeping the rotation excited above this end state. The tumbling itself could be the cause of shutting off further slowing (Rubincam, 2000), as the non-principal axis motion might destroy the axisymmetric re-radiation pattern to halt further deceleration. However, this does not explain the onset of tumbling in an obvious way. One possibility is collisional excitation. A difficulty with understanding asteroid spins in terms of collisional evolution has been that at the present very high impact velocities, impacts even up to the size triggering catastrophic disruption do not impart much spin change on the parent body. However for an asteroid that is barely spinning (such as Mathilde), a sub-catastrophic collision might still suffice to alter the spin by enough to set it tumbling, even if not spinning it up much from its very slow rate. Another possibility could be the interplay between the YORP evolution of spin rate and orientation and the precession of the asteroid’s orbital plane. Vokrouhlický (personal communication) has recently looked at the spin axis evolution of nearly-stopped asteroids and finds that the orientation and spin rate becomes nearly chaotic, which could perhaps induce non-principal axis spin (tumbling). Certainly it prevents spin-orbit locking such as we see as a common tidal evolution end state. Finally, for planet-crossing asteroids, tidal encounters with planets can deliver a substantial torque to the spin of an irregularly shaped body, leaving it in a state of non-principal axis rotation (Scheeres et al., 2004). This would be especially effective for a slowly rotating asteroid, so even a rather distant planetary encounter could set it tumbling without appreciably spinning it up. A first clue may be the couple asteroids with log τnorm > 0 that are not tumbling, most notably (3102) Krok. An explanation may be that a different mechanism (collisions or tidal encounters) is needed to excite tumbling from the mechanism that slows the rotation to the rate that tumbling can persist. The fact that a few slow rotators appear not to be tumbling suggests that for these, they have been slowed but by chance have not
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yet experienced an “exciting” event since become slow rotators.
Acknowledgments The work at Ondˇrejov was supported by the Grant Agency of the Czech Academy of Sciences, Grant A3003204, and by the Space Frontier Foundation within “The Watch” program. The work at Space Science Institute (A.W.H.) was supported by grant NAG5-13244 from the NASA Planetary Geology-Geophysics Program. The work at Modra was supported by the Slovak Grant Agency for Science VEGA, Grant 1/0204/03. Yu.K. and V.Sh. are grateful to V.G. Chiorny for his help during their observations of (3102) Krok. We thank David Vokrouhlický for discussing YORP mechanism and sharing his newest findings. We thank Donald Korycansky and an anonymous reviewer for their valuable reviews of the paper.
Note added in proof Our observations of 2004 FH, a ∼ 0.02-km sized Aten asteroid, on 2004-03-18 have revealed that it is a NPA rotator, PAR = −3, with periods 0.050397 ± 0.000004 and 0.037233 ± 0.000006 h. The detection of the two monolithic tumblers, 2000 WL107 and 2004 FH, in the sample of 40 asteroids with D < 0.15 km (but with only a part of them observed well to reveal their PA or NPA rotation) indicates that there is a subpopulation of monolithic NPA rotators overlapping the larger population of monolithic PA rotators in the same spin-size range.
Appendix A. Fourier series in two dimensions A function F (ψ, φ), where ψ, φ are angular variables over −π, π , can be expanded with Fourier series in the two variables (Rektorys et al., 1988): F (ψ, φ) =
∞
[aj k cos j ψ cos kφ + bj k sin j ψ cos kφ
j,k=0
+ cj k cos j ψ sin kφ + dj k sin j ψ sin kφ], (A.1) where aj k =
bj k =
cj k =
λj k π2 λj k π2 λj k π2
π π F (ψ, φ) cos j ψ cos kφ dψ dφ,
(A.2)
F (ψ, φ) sin j ψ cos kφ dψ dφ,
(A.3)
F (ψ, φ) cos j ψ sin kφ dψ dφ,
(A.4)
−π −π π π −π −π π π −π −π
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λj k dj k = 2 π
P. Pravec et al. / Icarus 173 (2005) 108–131
π π F (ψ, φ) sin j ψ sin kφ dψ dφ,
(A.5)
−π −π
δ 0 δ 0 1 j 1 k λj k = , 2 2
(A.6)
and δi0 is Kronecker’s delta. The Fourier series given by Eq. (A.1) can be rewritten into a following form that is convenient for some purposes: F (ψ, φ) = C0 +
∞ [Cj 0 cos j ψ + Sj 0 sin j ψ] j =1
+
∞ ∞ Cj k cos(j ψ + kφ) k=1 j =−∞
+ Sj k sin(j ψ + kφ) ,
(A.7)
where C0 = a00 ,
Cj 0 = aj 0 ,
Sj 0 = bj 0 ,
S0k = c0k , C0k = a0k , aj k ± dj k cj k ± b j k , S±j,k = , C±j,k = 2 2 for j, k > 0.
(A.8)
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Tumbling asteroids P. Pravec a,∗ , A.W. Harris b , P. Scheirich a , P. Kušnirák a , L. Šarounová a , C.W. Hergenrother c , S. Mottola d , M.D. Hicks e , G. Masi f,g , Yu.N. Krugly h , V.G. Shevchenko h , M.C. Nolan i , E.S. Howell i , M. Kaasalainen j , A. Galád k,a , P. Brown l , D.R. DeGraff m , J.V. Lambert n , W.R. Cooney Jr. o , S. Foglia p a Astronomical Institute, Academy of Sciences of the Czech Republic, Friˇcova 1, CZ-25165 Ondˇrejov, Czech Republic b Space Science Institute, Boulder, CO 80301, USA c Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA d DLR Institute of Space Sensor Technology and Planetary Exploration, Rutherfordstr. 2, D-12489 Berlin, Germany e Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA f Physics Department, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy g Campo Catino Observatory, Guarcino, I-03016, Italy h Institute of Astronomy of Kharkiv National University, Sumska Str. 35, Kharkiv 61022, Ukraine i Arecibo Observatory, National Astronomy and Ionosphere Center, PR 00612, USA j Department of Mathematics/Rolf Nevanlinna Institute, University of Helsinki, PO Box 4, FIN-00014, Finland k Modra Observatory, Astronomical Institute, FMFI Comenius University, Bratislava, Slovak Republic l Department of Physics and Astronomy, University of Western Ontario, London, ON N6A 3K7, Canada m Alfred University, 1 Saxon Drive, Alfred, NY 14802, USA n Air Force Maui Optical and Supercomputing Facility, Kihei, HI 96753, USA o Blackberry Observatory, 1927 Fairview Dr., Port Allen, LA 70767, USA p Serafino Zani Observatory, Lumezzane, I-25065, Italy
Received 23 March 2004; revised 16 July 2004 Available online 25 September 2004
Abstract We present both a review of earlier data and new results on non-principal axis rotators (tumblers) among asteroids. Among new tumblers found, the best data we have are for 2002 TD60 , 2000 WL107 , and (54789) 2001 MZ7 —each of them shows a lightcurve with two frequencies (full terms with linear combinations of the two frequencies are present in the lightcurve). For 2002 TD60 , we have constructed a physical model of the NPA rotation. Other recent objects which have been found to be likely tumblers based on their lightcurves that do not fit with a single periodicity are 2002 NY40 , (16067) 1999 RH27 , and (5645) 1990 SP. We have done a statistical analysis of the present sample of the population of NPA rotators. It appears that most asteroids larger than ∼ 0.4 km with estimated damping timescales (Harris, 1994, Icarus 107, 209) of 4.5 byr and longer are NPA rotators. The statistic of two short-period tumblers (D = 0.04 and 0.4 km) with non-zero tensile strength suggests that for them the quantity µQ/T , where µ is the mechanical rigidity, Q is the elastic dissipation factor, and T is a spin excitation age (i.e., a time elapsed since the last significant spin excitation event), is greater by two to four orders of magnitude than the larger, likely rubble-pile tumblers. Among observational conditions and selection effects affecting detections of NPA rotations, there is a bias against detection of low-amplitude (small elongation) tumblers. 2004 Elsevier Inc. All rights reserved. Keywords: Asteroids; Excited rotation; Photometry
* Corresponding author. Fax: +420-323-620263.
E-mail address: [email protected] (P. Pravec). 0019-1035/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.07.021
Tumbling asteroids
1. Introduction The rotational motion of an asteroid is described by = Iˆω, L
(1)
is the asteroid’s angular momentum vector, Iˆ is its where L inertia tensor, and ω is its angular velocity vector. The inertia tensor is generally a symmetric tensor containing six independent components. A convenient choice of the system of coordinates in the asteroid-fixed frame gives zero non-diagonal components. The diagonal components I1 I2 I3 are then the principal moments of inertia; the axes are called the principal inertia axes. The rotational kinetic energy is generally 1 T ˆ E= ω (2) Iω 2 and particularly for the principal inertia axes choice of the system of coordinates E=
1 I1 ω12 + I2 ω22 + I3 ω32 . 2
(3)
the basic (lowFor a given angular momentum vector L, est energy) state of rotation occurs when the asteroid rotates around its principal axis of the maximum moment of inertia (I3 ), with the rotational kinetic energy of Emin = I3 ω32 /2 = L2 /(2I3 ). Unless the body has I1 = I2 = I3 (like, e.g., homogeneous sphere, or cube), it may be in an excited state of rotation with Emin < E L2 /(2I1 ). In the excited state, unless the energy is equal to L2 /(2I1 ) that corresponds to rotation around the principal axis with the lowest moment of inertia I1 , the body’s rotation is complex as described in, e.g., Kaasalainen (2001).1 We also call the non-principal axis (NPA) rotational motion “tumbling” (the word was first used by Harris, 1994, as a suitable single-word term to describe the motion). A tumbling asteroid generally does not return to a same orientation at any single period, but it shows a period of rotation around one of the two extremal principal axes, Pψ , and a quasi-period of precession of the axis around P ¯ . (The latter is rather an average period of Euler angle L, φ ¯˙ where φ¯˙ = Pψ /2 φ˙ dt/(P /2).) φ, Pφ¯ = 2π/φ, ψ 0 In a non-rigid body, a NPA rotation results in a stressstrain cycling within the body. The excess energy is dissipated in the body’s interior and the spin state evolves to lower energy states. An approximate expression for the characteristic time scale τ of damping of the excited NPA rotation has been derived by Burns and Safronov (1973) assuming a low amplitude libration τ∼
µQ , ρK32 R 2 ω3
(4)
1 A state with E = L2 /(2I ) is unstable (see, e.g., Black et al., 1999). 2 A probability to see a body with exactly the energy of L2 /(2I2 ) is infinites-
imal and it can be neglected for real cases.
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where µ is the rigidity of the material composing the asteroid, Q is the quality factor (ratio of the energy contained in the oscillation to the energy lost per cycle), ρ is the bulk density of the body, K32 is a dimensionless factor relating to the shape of the body with a value ranging from ∼ 0.01 for a nearly spherical one to ∼ 0.1 for a highly elongate or oblate one, R is the mean radius of the asteroid, and ω is the angular velocity of rotation. Harris (1994) estimated the parameters in Eq. (4) and expressed the damping timescale as τ=
P3 , C 3 D2
(5)
where P = 2π/ω is the rotation period, D is the mean diameter of the asteroid, and C is a constant of about 17 (uncertain by about a factor of 2.5) for P in hours, D in kilometers and τ in billions (109 ) of years. He found that most of the asteroids’ rotations studied at the time had damping time scales shorter than their likely ages but he found that several small slow rotators could exhibit a NPA rotation. The first identified tumbling asteroid was (4179) Toutatis, see Hudson and Ostro (1995). Since that time, a few other cases have been recognized (Harris, 1994; Mottola et al., 1995; Pravec et al., 1998; Harris et al., 1999; Benner et al., 2002). We have examined the observational evidence for, and characteristics of, the population of tumbling asteroids, revisiting earlier publications from a recent compilation by A.W. Harris and analyzing new data obtained by us in the past few years. This paper is organized as follows. Section 2 outlines methods of analysis of observational data used. In Section 3, we give a review of observational results on NPA rotators as well as on PA rotators with long damping timescales. A statistical analysis and discussion on properties of the population of NPA rotators is given in Section 4. In Section 5, we discuss mechanisms (especially the YORP effect) that could explain the population of tumbling slow rotators. We note that a reader not interested in data and technical details on individual objects may want to jump over Section 3 and go directly to Section 4 where the characteristics of the tumbling asteroids population are discussed.
2. Observational data and analysis methods Time-resolved photometric observations, also called “lightcurve observations”, contain information on rotation of the observed asteroid. An analysis of the data for a periodicity (and effects of changing geometry) can provide an estimate of the synodic rotation period of the asteroid during the particular interval. The Fourier series method developed by Harris et al. (1989) and implemented by Pravec et al. (1996, 2000a) is a powerful technique for the period derivation. It assumes that the asteroid’s lightcurve contains just a single period. In a case where the analyzed data show systematic deviations from pure periodicity that cannot be accounted for
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by effects of changing geometry, a 2-dimensional Fourier series technique (suggested by Kaasalainen, 2001) can be used to find two periods present in the lightcurve as described in Section 2.1. New photometric observations presented in Section 3 were taken with several different telescopes/stations, all equipped with CCDs. The largest contribution was with the Ondˇrejov 0.65-m telescope equipped with the SBIG ST-8 camera and the Apogee AP7p camera (since May 2001). Other telescopes are mentioned in paragraphs for individual objects. Most of the data have been taken in the Cousins R filter with absolute calibration using the standards of Landolt (1992) with absolute errors of 0.01–0.02 mag unless stated otherwise. The CCD images were reduced in standard ways as described in Pravec et al. (1996), Rabinowitz (1998), Whiteley et al. (2002a), Krugly et al. (2002). D. De Graff used an IDL program CCDPHOT written by Mark Buie to reduce his observations of 2002 NY40 . 2.1. Two-dimensional Fourier series A lightcurve of a NPA rotator is a function of Euler angles ψ and φ. In constant geometric conditions, or if small changes of the conditions can be neglected or corrected for by another method (e.g., the magnitude-phase dependence is corrected using the H –G relation), the NPA rotator lightcurve can be approximated with Fourier series in the two variables (see Appendix A). Substituting ψ = 2πt/Pψ and approximating φ with 2πt/Pφ¯ in Eq. (A.7), we get the Fourier series in the following form . F ψ(t), φ(t) = F m (t) m 2πj 2πj Cj 0 cos t + Sj 0 sin t = C0 + Pψ Pψ j =1 m m
2πj 2πk + Cj k cos t Pψ Pφ¯ k=1 j =−m 2πj 2πk + + Sj k sin (6) t , Pψ Pφ¯ +
where C0 is the mean reduced light flux, Cj k and Sj k are the Fourier coefficients of the corresponding linear combination of the two frequencies Pψ−1 , Pφ−1 ¯ , and t is the time (for arbitrary epoch). We note that in practice we cannot tell a priori which of the periods of the Fourier series fitted to the lightcurve data actually correspond to precession and rotation. We therefore use subscripts (1, 2) rather than ¯ for the derived periods; an actual assignment of the (ψ, φ) derived periods to rotation and precession depends on a subsequent physical modeling. The series is truncated at the order m, usually the highest order for which a convergent solution can be obtained. The number of Fourier coefficients of the 2-dimensional Fourier series is (2m + 1)2 — cf. the number of coefficients of 1-dimensional Fourier series of (2m + 1). Note that the number of coefficients of
the 2-dimensional Fourier series grows so large with m of only 3 or 4 that we normally cannot use high orders that would be needed to fit minor features of the lightcurve (cf. single-periodic lightcurves that sometimes contain significant signals in harmonics of orders up to 10). Thus the 2-dimensional Fourier series fitted to the complex lightcurve data may not fit smaller lightcurve features fully just because it must be truncated at the relatively low order. Nevertheless, a use of the 2-dimensional Fourier series for frequency analysis of lightcurve data of tumblers has an advantage in that it does not introduce any a priori assumptions or idealizations of shape, scattering law and albedo distribution of the body so derivation of the periods is model-independent and also much faster than any (even much simplified) physical model which would need a number of non-linear parameters.2 When describing results obtained with the method, we use both the terms “period” and “frequency” depending on context and purpose, the conversion being always fj = 1/Pj , where j is integer. A signal (amplitude) of the harmonic (jf1 + kf2 ) is expressed as 2 2 Anorm (7) j k = Cj k + Sj k /C0 . The normalization to the mean reduced light flux is because we fit the photometric data in linear units (in “luminosity”) rather than in magnitudes. For illustrative purposes, remember that in a case of a small total lightcurve amplitude, a corresponding (peak-to-mean) variation in magnitudes is . ≈ 2.5Anorm/ ln 10 = 1.09Anorm. When solving for the periods P1 and P2 , we start with an initial period estimate from the previous single-period search of the data. The period estimate may be one of the two basic periods present in the NPA rotator, or it may correspond to a strong-signal linear combination of the two basic frequencies (usually a low-integers combination). With this assumption, we can scan the periods space much more quickly, in steps, searching for one of the two basic periods in one step while keeping the other period fixed at the estimate obtained in a previous step. 2.2. Synodic–sidereal period difference and the phase angle bisector Before ascribing deviations from strict single-periodicity of a lightcurve to non-principal axis rotation, it is important to assess the possible deviations just due to changing geometry. There are well-known effects on the amplitude of variation due to changing aspect (from equatorial to polar), 2 As already noted in Kaasalainen (2001), we point out that a use of the 2-dimensional Fourier series describing a lightcurve of NPA rotator differs from a simple sum of two Fourier series as is used for lightcurves of asynchronous binaries (e.g., Pravec et al., 2000b) which do not contain the full suite of sum and difference frequency components needed to fit the tumbling case. This may help to resolve between the two hypotheses—tumbler vs. asynchronous binary—in cases where the character is not obvious.
Tumbling asteroids
and on solar phase angle, but also the rotation phase can show shifts from strict periodicity due to changing rate of motion of the viewing aspect along the direction of longitude in the asteroid’s rotation coordinates. To quantify this we use the Phase Angle Bisector (PAB), the mean direction between the heliocentric and geocentric directions to the asteroid. As discussed by Harris et al. (1984), this serves as a convenient approximation for an effective viewing direction, as if the solar phase angle were reduced to zero. We can then consider the motion of the PAB as a measure of the possible difference between the observed synodic rotation period and the presumed constant sidereal period. In the ideal case where the PAB lies in the equator of the asteroid and moves along the equator in longitude only, the synodic–siderial period difference is: d(PAB) 2 (8) P . dt When the direction of the PAB is off the equator and motion is mainly along longitude, the observed value of Psyn–sid can be greater than given above; when the motion of the PAB is mainly in latitude, it can be less, so the above expression should be considered only a dimensional estimate of the possible difference between the observed synodic period and the sidereal one. Note that if the motion of the PAB is in the same sense as the direction of asteroid rotation, then Psyn–sid is positive and the synodic period is longer than the sidereal period. If the direction of motion of the PAB is opposite the direction of rotation, then Psyn–sid is negative and the synodic period is shorter than the sidereal one. Psyn–sid =
2.3. PAR codes Analyzing lightcurve data for a large number of asteroids listed in our database, we have found a need to establish a scheme for classification of quality of the data with respect to their ability to reveal PA or NPA rotation. We have therefore established a “PAR” codes scale as follows: −4 Physical model of the NPA rotation constructed. −3 NPA rotation reliably detected with the two periods resolved. An ambiguity of the periods solution may be tolerated provided the resulting spectrum of frequencies with significant signal is the same for the different solutions. −2 NPA rotation detected based on deviations from the single periodicity but the second period not resolved. −1 NPA rotation possible, some deviations from the single periodicity are seen but not at a conclusive level. 0 Not enough data to say if it is single- or multipleperiodic. +1 PA rotation is consistent with the data but coverage poor, thus not a reliable detection. +2 PA rotation likely, or deviations from PA rotation are small, as some overlapping data fit with a single period.
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+3 PA rotation quite likely as many overlapping data covering a few cycles fit with a single period. +4 PA spin vector solution obtained. The “PAR” name of the scale was chosen as an abbreviation of “Principal Axis Rotation” quality detection codes, with positive value meaning that the data indicate a PA rotation and negative value indicating a NPA rotation. For studies of PA vs. NPA rotations in a given sample of objects, only cases with |PAR| 2 can be considered as reliable enough detections for a statistical study. Cases with PAR of +1 or −1 are not reliable detections of a PA or NPA rotations, respectively; the two codes are intended to serve just as a guide for further observations. 2.4. Physical modeling of NPA rotation Where enough data are available for a particular NPA rotator (i.e., PAR = −3), a physical model of the NPA rotation can be attempted with the available data to infer physical meaning of the two detected periods in terms of rotation and vector, and precession, to establish an orientation of the L to estimate a shape and principal moments of inertia of the asteroid. Methods of construction of physical models of tumbling asteroids from their lightcurves were developed by Kaasalainen (2001) and Kryszczy´nska et al. (1999). One of us (P. Scheirich) has developed a model of one of the best tumblers, presented in Section 3.1, based largely on the method by Kaasalainen (2001) with a few modifications. A detailed description of the model, its refinement, and an application to lightcurve data of a few other tumblers will be given in a future paper. We outline the main points of the inversion method below. The shape of an asteroid is approximated by a triaxial ellipsoid with semiaxes a, b, and c. In following, we use the notation where c is always the axis around which the ellipsoid is seen as rotating, and it is normalized to unity. The c axis precesses around L—the angular momentum vector that is constant in a case of absence of external forces. The NPA motion can be described as divided into two classes: the long-axis mode (LAM) where the asteroid’s rotation axis c is the longest one (so, a, b < 1), and the short-axis mode (SAM) where c is the shortest axis (a, b > 1). The corresponding principal moments of inertia for the axes a, b, and c are Ia , Ib , and Ic , respectively, with Ic normalized to unity. Defining the symmetric Isym = (Ia + Ib )/2 and the antisymmetric Iant = (Ia − Ib )/2 allows one to constrain their values from observed periods (see Kaasalainen, 2001) in the inverse problem. The NPA rotation motion can be described with the following eight parameters: (its magnitude L and the di• The angular momentum L rection in ecliptic coordinates λL , βL ).
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• φ0 , θ0 , ψ0 —Euler angles describing the orientation of the ellipsoid at epoch t0 . We use the angles in the so-called x-convention (see, e.g., Samarasinha and A’Hearn, 1991). The Z-axis of the inertial frame is par and the XZ-plane contains a vector directed to allel to L vernal equinox. • Ia , Ib —principal moments of inertia (Ic normalized to unity). The NPA motion consists of the precession of the c axis (the instantaneous tilt between the two vectors around L is denoted θ ) with period Pφ , and the rotation of the body around the c axis with period Pψ . Since the two periods can be estimated from the lightcurve (though ambiguously in some cases)—see Section 2.1—we transform the above set of parameters to another one: λL , βL , φ0 , ψ0 , Pψ , Pφ , Isym , Iant . Furthermore, the ellipsoid axes (and therefore Isym and Iant , too) can be guessed from lightcurve amplitudes, so the number of initially unconstrained parameters shrinks to four. The transformation between the two sets of parameters and integration of Euler equations can be found in Kaasalainen (2001). The brightness of the ellipsoid as seen by observer can be computed in two ways: (1) At low phase angles, the brightness can be approximated with the apparent sky cross-section of the ellipsoid, evaluated using a formula from (Kryszczy´nska et al., 1999). (2) At higher phase angles, the brightness is computed numerically. The ellipsoid is represented with a number of triangular facets and the sum of individual contributions of all visible facets is computed. Several photometric laws can be used but the simple Lommel–Seeliger law is usually sufficient for ellipsoidal approximation of the real shape. The inverse problem is defined as the minimization of the function (oi − mi )2 , χ 2 (λL , βL , φ0 , ψ0 , Pψ , Pφ , Isym , Iant ) = i
where oi and mi are observed and modeled magnitudes at the epoch i, respectively. Initial values for Pψ , Pφ , Isym , and Iant are obtained from the lightcurve analysis, the remaining parameters (λL , βL , φ0 , ψ0 ) are sampled on an equidistant grid. Starting from each point of the grid, the local minimum is found using Nelder and Mead Simplex Algorithm (Buchanan and Turner, 1992). For data sets containing observations at large phase angle, an amplitude-phase correction coefficient m (see Zappalà et al., 1990) can be included into the model and solved for simultaneously with the other parameters.
3. Individual asteroids—PA vs. NPA rotators In this section, we present both a review of previous results and our new analyses of data for NPA rotators with PAR −2 (Sections 3.1 and 3.2) as well as for PA rotators with PAR +2 with long damping timescales (Section 3.3). In Table 1, a summary of the data for the NPA rotators is given. In order to facilitate a comparison with the prediction, in the table as well as in the text we give values of the logarithm of the damping time scale estimated from Eq. (5) normalized to the age of the Solar System, i.e., log τnorm = log(τ/4.5 byr). 3.1. Tumblers—the best cases 3.1.1. (4179) Toutatis—the best known case This S-type Apollo asteroid made repeated close approaches to the Earth in 1992, 1996, and 2000. It was extensively observed by both radar and photometric observers in December 1992–January 1993. Hudson and Ostro (1995) analyzed the 1992 radar data. The asteroid was observed from Goldstone from 1992-12-02 to 12-18 and from Arecibo on 1992-12-19. They established a model of the asteroid’s shape, spin state, and ratios of the principal moments of inertia. The motion was recognized as LAM (see Section 2.4), with the rotation period Pψ = 5.41 day and the long-axis precession period Pφ = 7.35 day. They
Table 1 Tumbling asteroids Object
PAR
4179 Toutatis 2002 TD60 2000 WL107 54789 2001 MZ7 2002 NY40 253 Mathilde 3288 Seleucus 3691 Bede 4486 Mithra 5645 1990 SP 13651 1997 BR 16064 1999 RH27 53319 1999 JM8
−4 −3 to −4 −3 −3 −2 to −3 −2 to −3 −2 −2 −2 −2 −2 −2 −2
a Ambiguity, see text.
log τnorm 1.2 −2.2 −3.9 −0.6 0.5 0.0 0.6 1.2 >0 0.0 0.5 1.6 0.4
P1 (h)
P2 (h)
A (mag)
Strongest f ’s
176 2.851 0.1609 37.57a 19.98 418 ∼ 75 227 (days) 30.39 33.64 178.5 (136)
130 6.783 0.2188 52.79a (18.43) (250)
1.2 1.4 1.1 1.4 1.3 0.5 1.0 0.5
2f1 , f2 , (f1 − 2f2 ), 2(f1 + f2 ) 2(f1 + f2 ), 2f1 , 2f2 f1 , 2f2 2f1 , 2f2 2f1
0.7 1.2 0.7 0.7
Tumbling asteroids
found dimensions along the axes corresponding to principal moments of inertia of 1.92, 2.40, and 4.60 km, and a di in the ecliptic rection of the angular momentum vector L coordinates of (180◦, −52◦ ). Spencer et al. (1995) made photometric observations from 25 observatories on 40 nights from 1992-12-08 to 1993-01-28 at phase angles of 0.2◦ to 121◦ using photoelectric photometers as well as CCDs. They found that the data contain periods of ∼ 7.4 and ∼ 3.1 day, using an algorithm assuming additive components for the two periods. From the radar model by Hudson and Ostro, it is seen that the main photometric period of 7.4 day is the precession period and that the second period of 3.1 day corresponds to a linear combination of the two frequencies of the asteroid’s NPA rotation. They estimated a mean absolute magnitude H of 15.3 and a slope parameter G of 0.10 ± 0.10. Hudson and Ostro (1998) used the radar-derived model to simulate the 1992–1993 lightcurve data of Spencer et al. and showed a very good correspondence. Using the lightcurve data that have a much longer time base than the radar observations, they further improved their model and also obtained Hapke photometric parameters of the asteroid. The 1992–1993 lightcurve data were simulated also by Kryszczy´nska et al. (1999), using dynamical parameters derived by Hudson and Ostro (1995). They noted that the lightcurve of Toutatis is dominated by the precession and the superposition of precession and rotation. The strongest signal of their synthetic lightcurve was in 2fφ , the next strong signal was in fφ and 2(fφ + fψ ); that exactly corresponds with the periods found by Spencer et al. considering that they saw the dominance of the second harmonics in their two lightcurve components. The same data were revisited also by Mueller et al. (2002) using the WindowCLEAN algorithm (Roberts et al., 1987) for obtaining characteristic frequencies of the lightcurve. They found four significant peaks in the spectrum, in decreasing order of power, of 2fφ , fφ , 2(fφ + fψ ), and f4 = 0.029 day−1 , where the last one corresponds to a curvature of the envelope of the lightcurve maxima so it does not correspond to a real frequency of the asteroid’s NPA rotation. The first three frequencies concur well with the previous results by Spencer et al. (1995) and Kryszczy´nska et al. (1999). We revisited the data by Spencer et al. with the Fourier series method. A goal of the exercise was to learn how the method works on the photometric data for the very longperiod tumbler. Results presented in the following two paragraphs showed that there is not a unique solution, nevertheless, a resulting solution spectrum corresponds well to the results found by the earlier studies. We analyzed the data from 1992-12-20.4 to 1993-01-27.9 taken at solar phase angles < 42◦ on the outbound trajectory after the close approach on 1992 December 8; this choice was due to the method’s requirement to limit synodic effects (∼ 1 h or less) as well as to keep the lightcurve amplitude change with phase angle small. A single-period
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. search showed a period of 175 h (= 7.3 day) dominating the lightcurve. An analysis using the 2-dimensional Fourier series of the 2nd order as described in Section 2.1 confirmed the main period of 175 h and provided a few possible solu. tions for the second period: 52.2, 61.2, 74.5 h = 3.1 day, 94.4 . h, and 130 h = 5.4 day. Two of the periods, 61.2 and 94.4 h, appear to be aliases. Checking signals in individual harmonics of the two solutions, we found that there was actually relatively little signal in both the first and the second harmonics for each of the two periods themselves. The significance of the 94.4 h period was because its second harmonic corresponds to a linear combination of the two other periods: . 2/94.4 h = (1/175 h + 2/130 h). Similarly, the significance of the 61.2 h period was because its second harmonic corresponds to another linear combination of the two periods: . 2/61.2 h = (3/175 h + 2/130 h). While we could not rule out the two solutions of 61.2 and 94.4 h for the second period on that basis only, the analysis suggested that they might be spurious. We were not able to resolve between the other three solutions, 52.2, 74.5, and 130 h, for the second period. Nevertheless, the first two solutions, 52.2 h and 74.5 h, are linear combinations of the frequencies corresponding to the periods 175 and 130 h, (2f1 + f2 ) and (f1 + f2 ), so all the three solutions correspond to the same frequency spectrum. With more abundant data allowing a higher order fit of the 2-dimensional Fourier series, we might be able to resolve between the multiple solutions, as we can in a few cases with better data presented below. From the available data, we would conclude that Toutatis is a NPA rotator and would give it PAR = −2 tending to −3. The real solution (according to the radar model) with the periods P1 = 175 h and P2 = 130 h is in agreement with the precession and the rotation periods by Hudson and Ostro (1995). It has most signal in 2f1 , the next strong signals in f2 , (f1 − 2f2 ), and 2(f1 + f2 ). 3.1.2. 2002 TD60 This Amor asteroid had a favorable apparition in November and December 2002. We observed it with the 0.65-m telescope from Ondˇrejov on 6 nights from 2002-11-07.1 to 12-11.9; with the 1.2-m telescope from Elginfield Observatory on 3 nights of 11-08.3, 12-03.2 and 07.4; with the Catalina 1.54-m telescope on two nights of 11-07.5 and 08.5; with the 0.6-m telescope from Modra on two nights of 12-08.0 and 08.9; and with the 0.8-m telescope from Campo Catino Observatory (observers G. Masi, C. Belmonte, and F. Mallia) on one night of 11-11.0. The asteroid was also imaged with the Arecibo radar on 4 nights from 2002-11-20 to 23. All but two of the photometric runs were done in R filter; the two exceptions are the two runs from Catalina which were done unfiltered. The Ondˇrejov observations were calibrated on the absolute magnitude scale within 0.01–0.03 mag, the other data are relative, their magnitude zero points have been adjusted in the analysis. We have found that the lightcurve is two-periodic. Its periods are P1 = 2.8513 ± 0.0001 h and P2 = 6.783 ± 0.002 h;
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Fig. 1. Lightcurve data of 2002 TD60 of 2002-11-07.1 to 14.1 folded with the period of 2.85119 h and G = 0.55. The best-fit full two-dimensional 3rd order Fourier series with the periods of 2.85119 and 6.7841 h is plotted as well.
Fig. 2. Lightcurve data of 2002 TD60 of 2002-11-24.0, 12-01.9 and 03.2 folded with the period of 2.8513 h and G = 0.55. The best-fit full two-dimensional 3rd order Fourier series with the periods of 2.8513 and 6.7805 h is plotted as well.
these values are means of estimates of the periods from three subsets described below. Since the asteroid’s lightcurve showed an apparent change of amplitude due to changing solar phase as well as aspect, we divided the data into three subsets: 2002-11-07.1 to 14.1 (7 nights, 756 points), 11-24.0 to 12-03.2 (3 nights, 428 points) and 12-07.4 to 11.9 (4 nights, 730 points). The subsets contain data over solar phase angles 27◦ –18◦ , 5◦ –10◦ , and 15◦ –20◦, respectively. The asteroid’s sky position (L, B) in the middle of each of the three intervals was (70◦ , −14◦ ), (65◦ , +5◦ ), and (62◦ , +14◦ ), respectively. The asteroid’s heliocentric vector moved at an almost constant rate of 0.9◦ day−1 in the ecliptic longitude while it stayed within 2◦ of Bh = 0◦ throughout the campaign. A full two-dimensional Fourier series of the 3rd order were fitted to the data with a satisfactory result, see Figs. 1–3. While we found that the 4th order would further improve the fits to the large-amplitude, low noise data, we have found that the dataset is just slightly insufficient in ex-
Fig. 3. Lightcurve data of 2002 TD60 of 2002-12-07.4 to 11.9 folded with the period of 2.8516 h and G = 0.55. The best-fit full two-dimensional 3rd order Fourier series with the periods of 2.8516 and 6.784 h is plotted as well.
tent for the 4th order to be used—experiments showed that it would slightly overfit the available data. While it would not affect the period solution, we stayed with the third order. The periods estimated from the three subsets were P1 = 2.8512, 2.8513, and 2.8516 h (errors 0.0001, 0.0001, and 0.0002 h), respectively, and P2 = 6.784, 6.781, and 6.784 h (errors 0.001 h), respectively. We saw that the period estimates were consistent to within a few times their 1-σ errors, so we took their means as the adopted periods solution mentioned above. The strongest signal is in 2(f1 + f2 ) in all the three data subsets (Anorm = 0.41, 0.33, and 0.31, respectively). The next two frequencies with consistently strong signal in all the three subsets are 2f1 (Anorm = 0.27, 0.28, and 0.20, respectively) and 2f2 (Anorm = 0.14, 0.19, and 0.16, respectively). At higher solar phase angles—in the subsets 1 and 3—there was also a high signal in (f1 + f2 ) while it was weak in the low solar phase subset No. 2 (Anorm = 0.32, 0.04, and 0.16 in the three subsets, respectively). As will become obvious with a physical model below, it is a shape effect equivalent to that seen in ordinary PA rotators—odd harmonics caused by shape become smaller at low solar phases. There were significant albeit weaker signals in a number of other linear combinations of the basic frequencies. We tested also a possibility that a true period present in the asteroid is that of 2.0075 h (equivalent to the frequency (f1 + f2 ), in the 2nd harmonic of which there is seen the highest signal). We found that it provided consistently poorer fits to the data in each of the subset. For example, in the 2nd subset, the sum of square residuals increased by 40% when f2 was replaced with (f1 + f2 ). A detailed check revealed that some significant-signal linear combinations of the basic frequencies f1 and f2 of the solution mentioned in the above paragraphs were missing in the tested solution with the period of 2.0075 h, so we concluded that it was not a real period present in the asteroid, rather it really was a linear addition of the two basic frequencies of the true solution mentioned above.
Tumbling asteroids
The best estimates of the mean absolute R magnitude HR and the slope parameter G are 19.41 ± 0.08 and 0.55 ± 0.1, respectively (errors are realistic). Having the basic lightcurve periods estimated with the two-dimensional Fourier series, we went on to construct a physical model of the NPA rotation using the method developed by P. Scheirich as outlined in Section 2.4. Considering that the strongest signal seen in the lightcurve is in 2(f1 + f2 ), we started with an assumption that it is LAM and that Pψ ≡ P2 and Pφ ≡ P1 . (We also made another modeling run starting with an assumption of SAM and Pψ ≡ P2 and Pφ ≡ 1/(f1 + f2 ); getting no satisfactory fit.) Starting values of axial ratios of the fitted ellipsoid were estimated using observed lightcurve amplitudes. The full amplitude of 1.4 mag at the phase angle 7◦ , corrected to 1.2 mag at 0◦ (see Zappalà et al., 1990) suggested that the body is quite elongated with a/c (the longest-to-shortest axial ratio) about 3. The largest difference between local maxima of the lightcurve was 0.9 mag (0.8 mag after correction to zero phase), suggesting a/b ∼ 2.1. It gave starting values of Isym = 0.591 and Iant = −0.321. For starting value of the amplitude-phase coefficient m, we used 0.02. We divided the optimization process into four subsequent steps, starting from simulating the low solar phase subset No. 2 and adding two next lightcurves adjacent to the beginning and the end of current subset on each step. At a first step we approximated the brightness of the ellipsoid with its sky cross-section and scanned λL , βL , φ0 , and ψ0 over their full intervals, while other parameters (Pψ , Pφ , Isym , Iant , m) were kept fixed at the starting values. From each point of the grid, all the nine parameters were iterated to reach a minimum of χ 2 using the simplex algorithm. At other steps we used the more realistic model using a triaxial ellipsoid approximated with 2292 triangular facets and the Lommel– Seeliger scattering law and a minimum of χ 2 in all parameters was found again using the simplex algorithm from each starting point. In each step, we did not use the whole set of local minima obtained in the previous step, just the deepest ones that appeared to encompass all possible solutions. (Tests with some of the other, higher minima indicated much poorer fits as supposed.) The allowed range of parameters shrank from one step to the following one. In the end we repeated the very last step with the whole dataset once again, now with the semiaxes a, b of the ellipsoid not coupled with the moments of inertia Isym , Iant but treated as free parameters (with all the other parameters varied, too). It is an approach to account for some effects of non-ellipsoidal shape and/or inhomogeneous density distribution. While the semiaxes a and b describe the “photometric” ellipsoid, the moments of inertia define a “dynamical” ellipsoid, which can be described with dynamical semiaxes
∓2Iant + 1 adyn , bdyn = (9) . 2Isym − 1
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Table 2 Parameters of the prograde and retrograde solutions for 2002 TD60 Parameter λL (◦ ) βL (◦ ) φ0 (◦ ) ψ0 (◦ ) Pψ (h) Pφ (h) Isym Iant m a b adyn bdyn
Best solution Prograde
Retrograde
155 32 333 317 6.7872 2.8503 1.78 −0.12 0.021 0.64 0.36 0.70 0.54
334 −31 177 317 6.7871 2.8515 1.80 −0.14 0.022 0.65 0.37 0.70 0.53
The “dynamical” semiaxes adyn and bdyn were derived from Isym and Iant . The angles φ0 and ψ0 are for the epoch JD 2452585.51766 (= 2002 Nov. 07.01766 UT), LT corrected.
We found two solutions that mirror each other. The resulting parameters are summarized in Table 2. (Errors are likely in the last digits, dominated by systematic errors of the idealized model.) We note that the fitted model has a rather large r.m.s. residual of 0.18 mag (for both solutions); it is particularly apparent as imperfections of the fit to the lightcurve maxima and minima that we deem is caused by a real shape differing from the idealized ellipsoidal model. Note also that relative differences between the “dynamical” and the “photometric” semiaxes of the ellipsoid, (adyn − a)/a and (bdyn − b)/b, are 8 and 40%, respectively. The best-solution fitted curves to four sample sessions are presented in Fig. 4. We note that the notation of semiaxes used in this paragraph (see Section 2.4) differs from the conventional notation where a b c. The estimated axial ratios in the conventional notation are a/b = 1.5 and a/c = 2.7. The tilt angle θ oscillates between 53◦ and 61◦ for both solutions. It is illustrative to look at why the strongest signal was in 2(fψ + fφ ). The lightcurve showed local maxima close to the times when the angle between the asteroid’s shortest axis and the line of sight reached its extrema (minimal or maximal values). This occurred with an average period of 1.0 h (with variations of ±0.1 h), corresponding to the frequency 2(fψ + fφ ). Thus, the strongest signal in the lightcurve was due to an apparent “rotation” of the shortest axis around L. This was due to the fact that the L vector was approximately was directed to the obperpendicular to the line of sight. If L server and the asteroid was at zero solar phase, the lightcurve would be dominated by 2fψ (rotational frequency). We considered if the asteroid may be a rubble pile with zero tensile strength or a coherent body. We examined the total acceleration (gravitational plus centrifugal) of the modeled ellipsoid body. It revealed that the body must have a non-zero tensile strength if the bulk density is less than ∼ 3 g cm−3 . We fixed lengths of the ellipsoid semiaxes at values 310, 200, and 110 m (for their estimation using radar data, see below), and computed the projection of the accel-
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(a)
(b)
(c) Fig. 4. Lightcurve data of 2002 TD60 of (a) 2002-11-14.1, (b) 11-24.0, (c) 12-01.9, and (d) 12-08.9, respectively. Curve is the best-fit synthetic lightcurve of the prograde solution of the numerical model. The other, mirror solution (retrograde) gives a curve indistinguishable from the plotted one.
min of 2002 TD for bulk Fig. 5. Extremal modeled surface acceleration atime 60 density ρ of (a) 2.0, (b) 3.0, and (c) 3.5 g cm−3 . Only negative values, which correspond to the acceleration vector in the direction outward from min . the surface, are shown as equidistant contours from zero down to asurf
eration vector to a normal to the surface, a = a · (− n), at each facet over a time span T = 1 month. Then we esmin (t) = min timated a time-resolved minimum asurf r∈S a(t, r) (where t is time, r is a position vector of the facet, and S is the whole surface of the ellipsoid) across the surface min (t) for densities 2, 3, for each time point. The values of asurf and 3.5 g cm−3 oscillated in intervals −3.1 to −6.4, −2.4 to +1.4, and −0.4 to +3.6 × 10−5 m s−2 , respectively, with period Pψ /2. We also estimated a surface-resolved minimum min ( r ) = mint ∈T a(t, r) (for each facet separately) across atime min ( r ) are prethe examined time span T . The values of atime sented in Fig. 5. Asteroid 2002 TD60 was observed by CW (continuous wave) radar at Arecibo Observatory November 20–23, 2002. A monochromatic, circularly polarized radio wave was transmitted to the asteroid. The received signal has components in both opposite-sense circular (OC) and same-sense circular (SC) polarization as that transmitted. As the asteroid rotates, one limb is moving towards the observer, and one is moving away, relative to the center. This gives 2Vlimb overall velocity change across the body, which produces a Doppler
shift in the received echo. There is an additional factor of two from the change of reference frame between the asteroid and the receiver. An observer on the asteroid would see the arriving signal already Doppler-shifted, and an additional shift occurs upon reflection. This results in a total bandwidth given by: ν =
4Vlimb , λT X
(10)
where Vlimb is the limb velocity in the line of sight, and λT X is the transmitted wavelength (12.5945 cm at Arecibo).3 See Ostro (1993) for a description of the principles of radar observations. Measured values of the radar bandwidth are summarized in Table 3. The best fit of the photometric model with the radar data was obtained for the longest semiaxis value of 310 m (for the both solutions). Dimensions of the other two 3 In a case of PA rotator, we get V limb = 2π r cos δ/P where r is the asteroid’s equatorial radius, δ is the sub-Earth latitude at the time of observation, and P is the rotation period.
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Table 3 Radar observations of 2002 TD60 Date & midtime UT
Res. (Hz)
ν (Hz)
2002-11-20, 04:33 2002-11-21, 04:30 2002-11-22, 04:14 2002-11-23, 04:15
0.1 0.1 0.1 0.1
4.8 4.7 5.3 6.0
semiaxes of the ellipsoid, computed using the estimated axial ratios (Table 2), are 200 and 110 m. Uncertainties of the estimates are on the order of a few tens meters. A geometric albedo pv of 0.12 was estimated by combining the absolutely calibrated R data of 2002-11-24.0 and 12-01.9 converted to H using G = 0.55 (see above) and assuming V − R = 0.45 (about average for NEAs) with the photometric model with the absolute dimensions derived above. An error of the albedo estimate is ∼ 0.05. 3.1.3. 2000 WL107 This ∼ 0.04-km sized4 Apollo asteroid made a close approach to 0.0171 AU to the Earth on 2000 December 3. The asteroid was favorably placed for photometric observations for a few days around the closest approach. It was observed by S. Mottola on two nights of 2000-12-02.0 and 03.9 with the 1.2-m telescope from Calar Alto (total of 255 relative R points, integration times 15 s); by C. Hergenrother with the 1.54-m telescope from Steward Observatory’s Catalina Station on two nights of 2000-12-03.5 and 04.5 (147 unfiltered points, plus additional 18, 16, and 14 points in the ECAS v, x, and b filters, respectively, on the first night, all with integration times 30 s); and by M. Hicks with 0.6-m telescope from Table Mountain Observatory on the night of 2000-12-03.2 in the Johnson–Cousins BVRI system (10, 3, 2, and 2 points in R, I , V , and B, respectively). We have found that the lightcurve is two-periodic and derived its periods P1 = 0.160916 h and P2 = 0.218834 h (both with formal errors of 0.000002 h). The full twodimensional Fourier series of the 4th order has been fitted to the data with a very satisfactory result, see Fig. 6. The strongest signal is in f1 and 2f2 with Anorm = 0.24 and 0.20, respectively. The second strongest couple with significant signal is f2 and 2f1 , both with Anorm = 0.06. There are several other linear combinations of the basic frequencies with significant signal; those with Anorm = 0.03 are (f1 + f2 ), (2f1 − f2 ), (f1 − 2f2 ), and (3f1 − 2f2 ). The normalized r.m.s. residual of the fit is 0.034. In addition to the usually tested things,5 we have also checked the possibility that the true period present in the asteroid was P1 = 0.32183 h, twice the P1 value of the above solution. In that case, the strongest signal would be in 2f1 4 Size estimated from our derived absolute magnitude H = 24.44 and assuming albedo pv = 0.18. 5 At minimum, we check the 2nd harmonics and whether some of the derived frequencies might not be actually a linear addition or difference of true frequencies of the asteroid.
Fig. 6. Lightcurve data of 2000 WL107 of 2000-12-02.0 to 04.5 folded with the period of 0.160916 h and G = 0.15 (assumed). The best-fit full two-dimensional 4th order Fourier series with the periods of 0.160916 and 0.218834 h and residuals of the fitted points are plotted as well.
(equivalent to f1 above) harmonic. We consider that possibility unlikely; the normalized r.m.s. residual would be 0.041 and, even more importantly, there would be insignificant signal in all linear combinations with odd multiples of f1 . Since this asteroid is a superfast rotator, we need to check that the finite integration times do not lead to significant smoothing of the lightcurve. Following the analysis method of Pravec et al. (2000a), the integration times of 30 s should cause a smoothing of the lightcurve by a factor of 0.990 in the fastest variation with 2f2 . Thus, systematic errors in the fit due to the smoothing are a few times less than the r.m.s. residual, therefore they can be neglected. Using the two-dimensional Fourier series fit, we derived the color indices from the ECAS photometry b − v = 0.20 and v − x = −0.02. Thanks to the fact that 14 to 18 measurements have been taken in each of the bands on 2000-12-03.5, uncertainties due to imperfections of the fit (there likely remain some systematic errors on the order of a few 0.01 mag, see also below) mostly canceled out and errors of the above color indices are < 0.02 mag. We derived also color indices in the Johnson–Cousins system from the Hicks measurements on 2000-12-03.2: V − R = 0.50, B − R = 1.42, and R − I = 0.33 but their errors are estimated about 0.06 mag due to the low number of observations (see above). The ECAS color indices suggest a SQ classification. A combination of the V and v points further suggests a rather steep phase relation, lower G values than the value of 0.15 we assumed throughout the analysis.6 For G = 0.15 ± 0.2, we get the mean absolute v magnitude of 24.44 ± 0.1. 6 Since just one of the sessions used in the period analysis given above was taken in the V band, the periods solution is not sensitive to the assumed value of G.
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The observational geometry changed significantly during the campaign; the asteroid’s solar phase angle spanned a range 2◦ –21◦ and it moved in the sky by 40◦ at almost constant rate of 16◦ day−1 and in almost constant position angle of about 100◦ . We checked the data for effects that might be related to the changing geometry. We have found that while the character of the brightness variation remained the same throughout the apparition, its amplitude varied somewhat with the solar phase—an amplitude-phase effect well known from ordinary asteroids. The fact that the asteroid’s lightcurve behavior showed no apparent change with the changing sky position might indicate that our aspect to the vector did not change much during the appariasteroid’s L tion. Since the asteroid’s apparent sky motion was at the almost constant rate and direction, it also suggests that a unique, unambiguous physical model of the asteroid’s NPA rotation might not be constructed from the available data; at least two mirror solutions like in the case of 2002 TD60 are to be expected. Nevertheless, the complex two-periodic character of the asteroid is reliably detected. We conclude that 2000 WL107 is a NPA rotator, PAR = −3. 3.1.4. (54789) 2001 MZ7 We observed this ∼ 3-km sized7 Amor asteroid on 47 nights from 2003-02-12 to 06-08. The R observations were calibrated with errors of 0.01–0.02 mag for all but two nights of 06-02 and 04 that have calibration errors of 0.025– 0.03 mag. We have found that the lightcurve is two-periodic, and derived its periods P1 = 37.57 h and P2 = 52.79 h. The full two-dimensional Fourier series of the 4th order has been fitted to the data with a satisfactory result, see Fig. 7. The strongest signal is in 2f1 with Anorm = 0.37. The second strongest signal is in 2f2 with Anorm = 0.11. There are several other linear combinations of the basic frequencies with significant signal; those with Anorm = 0.05–0.07 are f1 , 3f1 , f2 , (f1 − 2f2 ), 2(f1 − 2f2 ), (3f1 − 2f2 ), (2f1 − 3f2 ), 2(f1 − f2 ), 3(f1 − f2 ), 4(f1 − f2 ). The normalized r.m.s. residual is 0.023. The solution, however, is not unique. There exist a few other solutions, combinations of two of the four periods, 37.57, 52.79, 88.74, and 130.3 h, that provide a similarly good fit as the above solution. (Note that the two last periods correspond to (f1 − 2f2 ) and (f1 − f2 ) mentioned in the previous paragraph. Generally, all the frequencies with significant signal mentioned in the previous paragraph are present also in the other solutions, just for different linear combination of their basic frequencies.) A visual check of the resulting Fourier series expansions for the other possible solutions showed a less satisfactory behavior—implausibly deep minima—at some times not covered by the observations. This suggests that the ambiguity is actually due to insufficiency of the dataset—if some additional times were covered (primarily during daytime in Europe where all the 7 Estimated from H = 15.1 (MPO 39543) assuming p = 0.18. v
Fig. 7. Lightcurve data of (54789) 2001 MZ7 of 2003-02-12 to 06-08 folded with the period of 37.57 h and G = 0.15 (assumed). The points are means of (on average) three measurements taken in quick succession. Corresponding pieces of the best-fit full two-dimensional 4th order Fourier series with the periods of 37.57 and 52.79 h are plotted as well.
data have been obtained), we would probably resolve between the solutions. It is interesting that the fit is so good (whichever of the solutions is the true one) despite changes of the observing geometry. During the campaign, the asteroid’s solar phase angle decreased from 47◦ to 33◦ , its apparent sky position moved back and forth over an arc of 29◦ , and its heliocentric vector moved by 73◦ . We checked the data for effects that might be related to the changing geometry but found nothing prominent—the amplitude and character of the brightness variation remained about the same throughout the apparition.8 This might indicate that the asteroid’s motion was such that the effective viewing aspect, that is the “latitude” of the of the Earth and Sun vectors with respect to the vector L asteroid, did not change much during the apparition. It is likely that some smaller changes of the lightcurve caused by the changing geometry occurred and that they have been absorbed into the fitted Fourier series leading to some systematic errors in the above periods. We estimate this uncertainty to be of the order of 0.1 h for each of the periods. Nevertheless, the complex two-periodic character of the asteroid is reliably detected. We conclude that (54789) 2001 MZ7 is a NPA rotator, PAR = −3, even though the ambiguity between the few periods mentioned above needs to be resolved in future. 3.2. Tumblers with PAR = −2 3.2.1. (253) Mathilde—the largest tumbler Mathilde is the largest ultraslow rotator.9 Mottola et al. (1995) observed it on 49 nights from 1995-02-03 to 06-01. 8 In particular, a variation of G around the assumed value of 0.15 by ±0.2 had an insignificant effect on the solution periods or the Anorm values given above; they varied by < 0.01 h and by ∼ 0.01, respectively. 9 Mean diameter estimate of 58 km by Tedesco et al. (1992).
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Fig. 8. Lightcurve data of (253) Mathilde of 1995-02-03 to 06-01 folded with the period of 418 h and G = 0.12. The curve is the best fit single-period Fourier series, showing that the lightcurve data contain significant deviations from the pure periodicity.
Fig. 9. Lightcurve data of (253) Mathilde of 1995-02-03 to 06-01 folded with the period of 419 h and G = 0.07. Corresponding pieces of the best-fit two-dimensional Fourier series with the periods of 419 and 250 h are plotted as well.
They found that the lightcurve data had a period of 417.7 h but they showed significant deviations from pure periodicity and inferred that Mathilde is in a non-principal axis rotation state. They also estimated the mean absolute magnitude H = 10.28 ± 0.03 and G = 0.12 ± 0.06. We have downloaded the data by Mottola et al. archived in the Asteroid Photometric Catalogue. As the first step, we reproduced the Mottola et al.’s result. We have found the best fit for a period of 418 ± 1 h and G of 0.12 (± a few 0.01). The mean absolute magnitude derived is H = 10.28 (error of a few 0.01). The composite lightcurve for the period is presented in Fig. 8. Having confirmed the Mottola et al.’s result of about 418 h and the significant deviations indicating the NPA rotation of Mathilde, we analyzed the data for a two-periodic nature. We have found that there is present another period of 250 h that significantly improves the fit—the sum of square residuals decreased 5-times. The best-fit full twodimensional Fourier series with the periods of 419 and 250 h is presented in Fig. 9. Uncertainty of both periods appears to be about 1 h. We note that the solution is not secure enough to rate a PAR code of −3, so we assign a PAR code of −2 tending to −3. Nevertheless, the solution with the two periods appears likely and plausible, with strong linear combinations of the basic frequencies (in order of decreasing signal strength) of 2f1 , f1 , (2f1 − f2 ), 2(f2 − f1 ), (f2 − f1 ), (f2 + f1 ), 2f2 , and f2 . The estimated mean absolute magnitude H and the slope parameter G are 10.24 and 0.07, respectively, both with errors of a few 0.01.
lightcurve is similar to that of (13651) 1997 BR (see below), both showing a high-amplitude (1.0–1.2 mag) main frequency component with an additional frequency causing deviations with amplitude on the order of ≈ 0.1 mag. We assign to (3288) Seleucus a PAR = −2.
3.2.2. (3288) Seleucus This 2.2-km diameter (see Harris, 1998) Amor asteroid was observed by Debehogne et al. (1983) on the night 198203-13, by M. Carlsson on 1982-03-21, both from ESO, and by Harris et al. (1999) on seven nights from 1982-04-22 to 30. Harris et al. analyzed the complete dataset and found a best fit period of ∼ 75 h but showed that the data do not fit with any simple periodicity. They concluded that (3288) Seleucus was not simply periodic, but is a tumbling asteroid. We note that the complex character of the Seleucus
3.2.3. (3691) Bede This relatively large (6-km mean diameter, assuming a dark albedo of 0.06 suggested by its Xc classification by Binzel et al., 2004) Amor asteroid was observed by Pravec et al. (1998) on 27 nights from 1995-12-28 to 1996-03-21. They have found a synodic period of 226.8 ± 0.7 h (formal error). The data show a few deviations from single periodicity that cannot be accounted for by changing geometric conditions, which were minor to moderate at the relevant times. We conclude that (3691) Bede is likely a NPA rotator, PAR = −2. 3.2.4. (4486) Mithra This ≈ 2-km sized10 Apollo asteroid was observed by Ostro et al. (2000) with the Arecibo and Goldstone radars on 8 days from 2000-07-22 to 08-09. They revealed a doublelobed object, “apparently more severly bifurcated than any other near-Earth asteroid imaged to date.” The bandwidth of the echoes were consistently very narrow, “implying some combination of very slow rotation (evident from the barely noticeable variation in the appearance of images over several hours) and a radar line of sight not far from the apparent spin vector at any time during the experiment; the radarobserved sky arc was only about 35◦ .” They further mentioned that “No simple periodicity in the day-to-day image sequence is evident, so non-principal-axis rotation is suggested. The alignment of the two lobes is almost parallel to the projected, apparent, instantaneous spin vector in some images but almost perpendicular to it in others, providing additional evidence for a very unusual spin state.” Therefore, we consider (4486) Mithra to be another slow rotator 10 Estimated from H = 15.6 (MPO 38159) assuming p = 0.2. v
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Fig. 10. Lightcurve data of (5645) 1990 SP of 2002-03-12 to 05-07 folded with the period of 30.39 h and G = 0.0. Individual points plotted are means of (on average) six measurements taken in quick succession. The fitted 2nd-order Fourier series to the April–May data is plotted to facilitate visual comparison of the data behavior at different rotational phases, see text.
in a NPA rotation state, PAR = −2. Its main period appears to be on the order of days that places it to the range of log τnorm > 0. 3.2.5. (5645) 1990 SP We observed this ∼ 1-km large11 Apollo asteroid on two nights of 2002-03-12 and 14 and on 9 nights from 2002-0406 to 05-07. The first two nights data were taken at a large phase angle of about 54◦ and at a higher ecliptic latitude of the PAB of +24◦ ; we used them for a check of consistency of the results obtained using the later nine nights data taken at the moderate phase angles of 11◦ to 24◦ and at lower latitudes of the PAB of +4◦ to +15◦ . Analyzing the later nine nights data, we found the best fit for a period of 30.39 ± 0.02 h, a mean absolute R magnitude of HR = 16.75 ± 0.2 and G = 0.0 ± 0.1. There exists another period solution of 44.5 h (i.e., ∼ 1.5× the above solution) that is formally not ruled out but is implausible as it would require a very asymmetric lightcurve with maxima differing by 0.4 mag at the moderate solar phase angles. The two March nights data support the shorter period solution as well. A composite of the lightcurve data folded with the adopted solution is presented in Fig. 10. The data do not fit together well with a single periodicity. Particularly significant is the difference by ∼ 0.3 mag between the data of 2002-04-08 and 18 around phase of 0.15 in Fig. 10 while the calibration errors were only 0.02 mag. Noticeable is also the poor fit of the data of the nights of 2002-04-06 and 10 that would require a quite steep jump by . > 0.2 mag over just 0.02 = 7◦ in rotational phase (at phase of 0.7 in Fig. 10) if the asteroid were single periodic. We therefore conclude that (5645) 1990 SP is likely a NPA rotator, PAR = −2. 3.2.6. (13651) 1997 BR This 0.8-km sized (assuming pv = 0.18 based on its Sclassification by Binzel et al., 2004) Apollo asteroid was ob11 Estimated from H = 17.2 ± 0.2 assuming p = 0.2. v
Fig. 11. Lightcurve data of (16064) 1999 RH27 of 1999-10-31 to 200002-12 folded with the period of 178.5 h and G = −0.15. Individual points plotted are means of (on average) four measurements taken in quick succession.
served by Pravec et al. (1998) on 20 nights from 1997-02-01 to 03-07. They derived the best-fit period of 33.644 ± 0.009 h (formal error) but found that the data did not fit well, the r.m.s. residual of the fit was 2- to 3-times greater than the typical errors of the points. The largest discrepancies were apparent around phases of the lightcurve maxima, making them of quite different shapes on different dates. The reader may note also the apparent similarity between the character of the complex lightcurve of 1997 BR with that of (3288) Seleucus, see above. We conclude that (13651) 1997 BR is a NPA rotator, PAR = −2. 3.2.7. (16064) 1999 RH27 We observed this Amor asteroid on 33 nights from Ondˇrejov and one night from Kharkiv from 1999-10-31 to 200002-12. The data were calibrated with errors of 0.01 mag. They fit best with a synodic period of 178.5 h, mean HR = 16.05 and G = −0.15 (realistic errors are in the last digits; Psyn–sid max = 3 h). Assuming V − R = 0.36 and pv = 0.06 based on its C-classification (Binzel et al., 2001), we estimate a mean diameter of ∼ 2.8 km. A composite lightcurve is presented in Fig. 11. The fit is far from perfect. It may be, however, caused in large part by the significant change of observing geometry—the phase angle increased from 12◦ to 30◦ and the PAB moved from 43◦ to 113◦ in ecliptic longitude at an almost constant rate of 0.7◦ day−1 during the observational interval. A few points/runs, however, deviate by more than 0.1 mag from other points at nearby rotational phases taken within relatively short time intervals (a few weeks) while the observational geometry did not change so much. The significance of the deviations with relation to a possible NPA motion of the asteroid is strengthened by the fact that the lightcurve behavior—apart from the deviations—was overally regular, consistent with an aspect not far from equator-on during the whole apparition. For a PA rotator in such conditions, we would expect a smoother fit, with lightcurve amplitude increasing due to the amplitude-phase effect. So, we conclude that the apparent deviations indicate that (16064) 1999 RH27 is likely a NPA rotator, PAR = −2.
Tumbling asteroids
3.2.8. (53319) 1999 JM8 This large Apollo asteroid was observed extensively with the Goldstone and Arecibo radars over 18 days from 199907-18 to 08-09 (Benner et al., 2002). They found it to be a slow-rotator with an asymmetric, irregular shape with typical overall dimension within 20% of 7 km, in a NPA spin state with a dominant periodicity of ∼ 7 days. They further noted that “images obtained between July 31 and August 9 show apparent regular rotation features from day to day, suggesting that the rotation state is not far from principal axis rotation.” We observed the asteroid on 20 sessions total from 1999-07-03 to 21; 15 nights from Ondˇrejov (mostly by L. Šarounová), 2 nights by Yu. Krugly and V. Shevchenko from Kharkiv, 2 nights by S. Mottola and F. Lahulla from Calar Alto, and 1 night by M. Hicks from Table Mountain Observatory. All the data were calibrated with errors 0.01–0.02 mag. On 1999-07-18 Hicks measured the color indices B − R = 1.08 ± 0.04, V − R = 0.45 ± 0.035, and R − I = 0.40 ± 0.05; the modest red slope shown by the indices is consistent with the X-classification of 1999 JM8 (Binzel et al., 2004). The photometric data best fit with a period of 136 h and G = −0.09 but significant deviations from a simple periodicity are apparent, see the composite lightcurve in Fig. 12. While the deviations are likely partly caused by changing observing geometry during the interval covered—the phase angle increased from 85◦ to 121◦ and the PAB moved by 29◦ —some of them cannot be due to changing geometry and therefore indicate that the rotation is not single-periodic. Particularly, the two points of 1999-07-14 and 15 are brighter by 0.3–0.5 mag than points taken at similar rotational phases in earlier as well as later cycles; the behavior cannot be explained by the evolution of geometry of a singly-periodic object. Another feature, the steep rise seen in the lightcurve of 1999-07-19 in the lightcurve maximum also suggests a presence of another frequency contributing to the behavior seen (cf. lightcurves of other tumblers with similar behavior mentioned in this section). We cannot derive the second
Fig. 12. Lightcurve data of (53319) 1999 JM8 of 1999-07-03 to 21 folded with the period of 136 h and G = −0.09. Individual points plotted are means of (on average) six measurements taken in quick succession, their errors are always less than the calibration errors of 0.01–0.02 mag. The fitted 2nd-order Fourier series is plotted to facilitate visual comparison of the data behavior at different rotational phases, see text.
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period of the asteroid’s NPA motion, and we also cannot be sure whether the detected period of 136 h does not correspond to some linear combination of real frequencies of the asteroid’s NPA motion. Nevertheless an excited rotation state is indicated by the lightcurve data as well as from the radar observations. We conclude that (53319) 1999 JM8 is a NPA rotator, PAR = −2. 3.2.9. 2002 NY40 This Apollo asteroid made a close approach to 0.0035 AU to the Earth on 2002 August 18. It was favorably placed for photometric observations on its inbound trajectory during a few weeks before the closest approach, an extensive photometric campaign was organized. We observed it in 21 sessions from 2002-07-31.3 to 0816.4. Most of the photometry was done in R filter, and 16 of the 21 sessions were calibrated in the standard Cousins R system. For one (08-11.3 by W. Cooney) of the five uncalibrated sessions, a zero point of the magnitude scale was derived from an overlap with another session (by D. De Graff). The other four uncalibrated sessions were retained in the analyzed dataset as they provide additional useful information on the lightcurve shape at the particular times, and we derived their magnitude scale zero points to provide a good fit. The data were obtained by D. De Graff with 0.82-m telescope from Stull Observatory (9 calibrated sessions 2002-07-31.3 to 08-12.2; calibrations done using the list of secondary standards “Loneos Reference Star Catalog” by B. Skiff12); by L. Šarounová and P. Pravec with the 0.65-m telescope from Ondˇrejov (6 calibrated sessions 0803.0 to 14.9); by M. Hicks with 0.6-m telescope from Table Mountain Observatory (one calibrated session 08-07.4; also measured V − R = 0.423 ± 0.010); by J. Lambert with the US Air Force 0.36-m telescope from AMOS Remote Maui Experimental site (2 unfiltered sessions of 08-13.4 and 16.4); by S. Foglia, C. Cremaschini, W. Marinello, M. Micheli and S. Zubani with 0.4-m telescope from Lumezzane Observatory (one uncalibrated V session 08-08.0); by W. Cooney with 0.3-m telescope from Blackberry Observatory (one unfiltered session 08-11.3, but see above); and by G. Masi with the 0.8-m telescope from Campo Catino Observatory (one uncalibrated R session 08-13.9). There are available three additional sessions of 08-17.3 (calibrated R session by M. Hicks), 08-17.4 (unfiltered session by J. Lambert), and 08-18.1 (uncalibrated R session by J. Licandro et al., personal communication). We checked them for consistency with results reported below but did not use them directly in the analysis as observational geometric conditions changed rapidly within one day around the closest approach that would require introducing additional parameters into the solution (actually requiring a pole solution) while the added information was not enough to support the increase of the number of free parameters. Therefore 12 Available on ftp://ftp.lowell.edu/pub/bas/starcats/loneos.phot.
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we left the three additional sessions for the future modeling. We found that the main period of the lightcurve is P1 = 19.98 ± 0.01 h (realistic error). A composite lightcurve presented in Fig. 13 shows significant deviations from single periodicity on the order of 0.1 mag. Changing geometric conditions cannot account for the deviations; the observational geometry changed slowly during the interval when it was on the inbound trajectory (2002-07-31.3 to 08-16.4)— the asteroid’s solar phase angle was in a range of 11◦ –16◦ , its apparent sky position changed by 8◦ (but only by 4◦ over the 20 sessions before 08-15), and its heliocentric vector moved at an almost constant rate of 0.8◦ day−1 in the ecliptic longitude while it stayed within 1◦ of Bh = +1◦ throughout the interval. The deviations from cycle to cycle must be mainly due to the asteroid itself, and indicate that 2002 NY40 is a NPA rotator. A search for the second period in the data did not yield a unique result but a period of P2 = 18.43 ± 0.01 h appears likely. No higher than the 2nd order of the full twodimensional Fourier series could be used—an attempt to use the third order showed that the data are not sufficient to describe the lower-amplitude deviations reliably to the higher order. So, the resulting fit shown in Fig. 13 necessarily has significant residuals on the order of several 0.01 mag. Most of the signal is in 2f1 , Anorm = 0.38. The next significant linear combinations of the two basic frequencies are f2 , 2f2 , (f1 + f2 ), 2(f1 − f2 ), (f1 + 2f2 ), and (f1 − 2f2 ), with Anorm = 0.03 to 0.06. The solution is still not sufficient for a PAR code of −3, so we give it PAR of −2 tending to −3. The lightcurve shows also another interesting feature; there are apparent sharp bends in the data covering the secondary minimum around phase 0.5 in Fig. 13, creating a “plateau” at the bottom of the minimum. It was well covered at times around 2002-08-11.4, 13.9, and 16.4, and with a poorer coverage also around 08-12.2. The behavior suggests the presence of large non-convex features in the asteroid, consistent with the radar images (see below). We estimated a mean absolute R magnitude of HR = 18.81 ± 0.2 assuming G = 0.15 ± 0.2. Since V − R =
Fig. 13. Lightcurve data of 2002 NY40 of 2002-07-31.3 to 08-16.4 folded with the period of 19.98 h and G = 0.15 (assumed).
0.423 ± 0.010, the mean absolute V magnitude is H = 19.23 ± 0.2. An absolute magnitude for the maximum of the lightcurve is brighter by 0.3 mag while that for the flat minimum (with the “plateau”) is fainter by 0.5 mag than the mean H . Rivkin et al. (2003) found a striking match between their 0.3–4 µm spectrum of 2002 NY40 and spectra of LL6 ordinary chondrites. They also measured thermal flux of the asteroid and estimate a geometric albedo of 0.25 ± 0.03, assuming a non-rotating model with no thermal inertia. Müller et al. (2004) estimated an albedo of 0.34 ± 0.06 with a nearEarth asteroid thermal model and a thermophysical model. Using an average of the two estimates and our mean absolute magnitude estimate gives an effective diameter of the asteroid of 0.35 ± 0.07 km (combined error of the H and the albedo estimates). Radar images show that this is an elongated object which has two fairly spherical lobes, with diameters 0.33 and 0.24 km. The maximum dimension must be at least 0.61 km, which makes the effective diameter 0.36 ± 0.07 km (Howell et al., personal communication), which agrees well with our estimate. 3.3. Principal axis rotators with long τ 3.3.1. (3102) Krok—a singly-periodic ultraslow rotator Krok is an exceptional object. Though its log τnorm is nearly 2, i.e., its estimated damping timescale is longer by two orders of magnitude than the age of the Solar System (even more compared to its expected collision lifetime), its lightcurve appears to be singly-periodic, or nearly so, showing variations consistent with the synodic–sidereal rotation period variation of a PA retrograde rotator. Harris et al. (1992) derived a synodic period of 147.8 h from their observations taken during eight nights from 198108-27 to 09-04. We re-analyzed their data and found a synodic period of 148 ± 1 h (realistic error estimate). A lower limit of the amplitude is 1.0 mag (at phase angle of 24◦ ). The only apparent deviation from single periodicity is on the two nights of 1981-08-29 and 09-04, where the magnitude levels at the same rotation phase differ by ∼ 0.1 mag, or equivalently, the rotation phase is displaced by a couple hours, when plotted at the derived period (the period is more tightly constrained by overlapping data on other nights). We observed the asteroid in two recent apparitions; on 22 nights (19 from Ondˇrejov and 3 from Kharkiv Observatory) from 2000-07-26 to 10-29 and on 15 nights from Ondˇrejov from 2003-05-31 to 07-02. Our analysis of the first 14 nights of the 2000 apparition, taken during 2000-07-26 to 09-08, revealed a synodic period of 149.4 ± 0.1 h (formal error; real error a few times greater) and amplitude of 1.6 mag at phase angles in a range 42◦ to 55◦ . All the 14 nights data are consistent with the single periodicity. Adding the data of two and six nights taken one and two lunations later, from 2000-09-30 to 10-06 and from 2000-10-21 to 29, respectively, are consistent with the same general lightcurve but are displaced in rotation phase from
Tumbling asteroids
Fig. 14. Composite lightcurve of (3102) Krok of 2000-07-26 to 10-29 for the synodic period of 149.4 h and G = 0.35. Points of eight nights of 2000-09-30 to 10-29 show a shift in rotation phase due to a change of the synodic period with the decreasing rate of motion of the asteroid, see text.
that predicted with a constant synodic period. The shift is consistent in magnitude with an expected synodic–sidereal period difference from the PAB motion, and is in a sense indicating retrograde rotation, i.e., the synodic period increased while the asteroid PAB motion decreased (see also below). A lower amplitude of 0.7 mag was seen at the lower phase angle of 12◦ of the six nights observations in the last decade of October. A composite lightcurve of the 2000 data is presented in Fig. 14. The 2003 data revealed a synodic period of 151.8 ± 0.3 h (formal error; real error a couple times greater) and a lightcurve amplitude of 1.3 mag at the phase angles from 17◦ to 37◦ . Again, all the 15 nights data are consistent with the single periodicity. The motion of the asteroid (both apparent as well as PAB) during this apparition was several times slower than during the two previous apparitions so the synodic period of the 2003 apparition is close to a sidereal rotation period of the asteroid. A composite lightcurve of the 2003 data is presented in Fig. 15. The 20◦ phase angle interval covered during the 2003 apparition while the asteroid’s motion was small has allowed us to estimate the phase relation’s G value of 0.4 ± 0.2. The 2000 data covering even a larger interval in solar phase angle but with the larger sky position change (therefore less reliable for an estimation of G because of a possible aspectinduced change of the asteroid’s brightness level) are consistent with the above G value estimate as well (though lower values of the uncertainty interval are preferred). We derived mean HR of 16.2 and 16.14 from the 2003 and the late October 2000 data, respectively (assuming the 2003 estimate for G of 0.4 ± 0.2 for the 2000 apparition as well), that converts to HV of 16.67 and 16.61 (both ±0.2) in the two apparitions using V − R = 0.47 ± 0.04 measured by Yu. Krugly and V. Shevchenko on 2000-10-24. Using the same G value estimate, we get mean HV = 16.56 ± 0.2 in the 1981 apparition. So, the best estimate of the mean H = HV is 16.6 ± 0.2 in all the three apparitions. Assuming pv = 0.18 for the S-type
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Fig. 15. Composite lightcurve of (3102) Krok of 2003-05-31 to 07-02 for the synodic period of 151.8 h and G = 0.35. The best-fit 4th order Fourier series is plotted as well.
classification of Krok (Binzel et al., 2004), we get an estimate of its mean diameter of 1.5 km. The apparent variations of the Krok’s synodic period during the three apparitions are correlated with its prograde PAB (as well as apparent) motion; the faster motion, the shorter synodic period. The behavior requires a retrograde direction of rotation of the asteroid. Correcting the data for a synodic–sidereal effect assuming a pole position of (260◦ , −60◦ ) in J2000—a direction close to normal to the PAB motion in the 1981 and 2000 apparitions—we found that a sidereal period about 151.6 h with an error about 0.5 h is consistent with the data of all but one of the total 45 nights. The one deviating run seen in the 1981 data (see above) could perhaps be an observational error, or it may indicate a small deviation from nearly PA rotation. We note that while the pole position cannot be estimated accurately, its retrograde nature is well established and the sidereal period estimate is accurate. Furthermore, the fact that the asteroid exhibited a very large amplitude of variation in 1981 and 2003, from aspects almost exactly 90◦ apart in the sky, indicates that the pole orientation is approximately orthogonal to those two directions. We conclude that (3102) Krok is a retrograde rotator likely in a PA spin state or nearly so, PAR = +2. If there is any tumbling motion in the asteroid, its amplitude must be small, on the order of 10% or less of the principal amplitude of variation. 3.4. Transition-range PA rotators Around log τnorm = 0, there is a transition between the range of prevailing tumblers and the range dominated by PA rotators. Asteroids with estimated log τnorm around zero contribute to the statistical definition of the transition. NPA rotators lying there were presented in Sections 3.1 and 3.2. Here we present data for asteroids in the range that exhibit a single-periodic character. Specifically, we present all of them that have estimated τ > 1 byr.
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Fig. 16. Composite lightcurve of (11398) 1998 YP11 of 1999-07-14 to 08-10 for the best fit synodic period of 38.60 h and G = 0.29.
3.4.1. (3752) Camillo This ∼ 2.6-km sized13 Apollo asteroid with log τnorm = −0.4 was observed on 14 nights during 1995-08-03 to 29 by Pravec et al. (1998). They derived a synodic period of 37.846 ± 0.012 h and found the data to be consistent with the single periodicity; ≈ 0.1-mag deviations of two of the 14 nightly lightcurves they saw have been attributed to changing observing geometry. Well fitting (to within a few 0.01 mag) overlapping data of the other runs cover about a half of the rotational period. So, we conclude that (3752) Camillo is likely a PA rotator, PAR = +2. If there is any tumbling motion in the asteroid, its amplitude is probably small, on the order of a few 0.01 mag or less. 3.4.2. (11398) 1998 YP11 We observed (11398) from Ondˇrejov on 17 nights from 1999-07-14 to 08-10. The data show a synodic period of 38.60 ± 0.03 h (the estimated synodic–sidereal effect, Psyn–sid max , is 0.06 h) and a lightcurve amplitude of about 0.25 mag. The range of solar phase angles covered was 8◦ to 27◦ , allowing us to obtain a (formally) accurate estimate of mean HR = 16.10 ± 0.03 and G = 0.29 ± 0.02 but real errors of the values may be greater due to a possible aspect-induced change of asteroid’s mean brightness as the geometry changed during the observational interval; the PAB changed by 10◦ with most of the change in the ecliptic latitude. Using V − R = 0.48 and pv = 0.18 (assumed for its S-type classification, Binzel et al., 2004), we get a mean absolute V magnitude H = 16.6 (error about 0.1 mag), an estimated mean diameter of 1.5 km and log τnorm = 0.1. A composite lightcurve is presented in Fig. 16. The data, overlapping at several phases repeatedly, are consistent with the single periodicity to within observational errors of about 0.02 mag. We conclude that (11398) 1998 YP11 is likely a PA rotator, PAR = +2. If there is any tumbling motion in the asteroid, its amplitude must be small, on the order of a few 0.01 mag or less. 13 Estimated from H = 15.41 ± 0.13 (Pravec et al., 1998) assuming
albedo pv = 0.18.
Fig. 17. Composite lightcurve of 1999 FA of 1999-03-18 and 19 for the best-fit synodic period of 10.092 h and G = 0.15 (assumed).
3.4.3. (25143) Itokawa This small Apollo asteroid (estimated effective diameter of 0.32 ± 0.03 km, see Ishiguro et al., 2003) was observed extensively from more than 10 observatories that obtained 53 nightly runs from December 2000 to September 2001. The data were collected and analyzed by Kaasalainen et al. (2003). They obtained a unique spin vector estimate with sidereal period of 12.132 h and obtained a model of its shape. Even though they found that several lightcurves did not fit well and they had to introduce time offsets to get them to fit with the other data, their solution appears safe and the reliable part of the lightcurve data fits well with the single periodicity, thus implying a PA rotation to within a few 0.01 mag. So, it is PAR = +4 object. It has log τnorm = −0.1. 3.4.4. 1999 FA This is a small, ∼ 0.24-km S type14 Apollo asteroid with log τnorm = −0.1. We observed it from Ondˇrejov on two nights of 1999-03-18 and 19 and found that it had a large amplitude (1.2 mag) lightcurve with a synodic period of 10.092 ± 0.008 h (Psyn–sid max = 0.014 h). A composite lightcurve is presented in Fig. 17. While the data overlap over only 37% of the rotational phase, the fit is excellent, showing no systematic deviations to within the data calibration errors of 0.02 mag. We therefore conclude that 1999 FA is likely a PA rotator, PAR = +2. Though this detection is somewhat less reliable than some other PAR = +2 objects mentioned in this paper, the excellent fit of the data with the single periodicity leads us to judge that if there is any tumbling motion, its amplitude is probably small, on the order of a few 0.01 mag or less. 3.4.5. 2001 VS78 We observed this Amor asteroid from Ondˇrejov on 33 nights from 2002-02-22 to 06-18. A synodic period of 14 Size estimated from H = 20.58 (ASTORB.DAT from ftp.lowell. edu/pub/elgb) assuming pv = 0.18, the S classification by Binzel and Bus, http://smass.mit.edu/.
Tumbling asteroids
Fig. 18. Composite lightcurve of 2001 VS78 of 2002-02-22 to 04-18 for the best-fit synodic period of 40.553 h and G = 0.10.
Fig. 19. Composite lightcurves of 2001 VS78 of 2002-05-02 to 22 and 05-29 to 06-18 for the best-fit synodic period of 40.553 h and G = 0.10.
40.553 h (realistic error < 0.01 h, Psyn–sid max = 0.09 h), mean absolute R magnitude HR = 15.25 ± 0.2 and G = 0.10 ± 0.1 were derived. Using V − R = 0.48 and pv = 0.18 (assumed for its S-type classification, Binzel et al., 2004), we get an estimated mean effective diameter of the asteroid of ∼ 2.2 km and log τnorm = −0.2. Composite lightcurves constructed with data covering three sub-intervals, 2002-0222 to 04-18, 05-02 to 22, and 05-29 to 06-18, fit well with the single periodicity, see composite lightcurve presented in Figs. 18 and 19. The apparent evolution of the lightcurve over the campaign is quite plausibly related to changing geometry of the observations; specifically, the phase angle decreased from about 50◦ in the February–April subinterval downto 25◦ at the end of the campaign, the apparent sky position changed by about 30◦ and the heliocentric position changed by 86◦ during the campaign. Over the shorter intervals, the lightcurve data fitted well with the single periodicity to within a few 0.01 mag (with a few notable exceptions of points around phase 0.75 in Fig. 18; see below for a suspected cause). We conclude that 2001 VS78 is likely a PA rotator, PAR = +2. If there is any tumbling motion in the body, its amplitude must be small, on the order of a few 0.01 mag or less. There is one unusual feature in the asteroid—the sharp minima occurring at phases between 0.22 to 0.28 and between 0.72 to 0.79 in Fig. 18 on five nights from 2002-03-06 to 04-09 but missing in the data covering the same phases on later dates, after 05-11. The sparse coverage of the minima does not allow us to draw a well-based conclusion. (Indeed, we have worked the asteroid on many nights but optimized
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the coverage on each night for its long-period character so that be able to work other asteroids on the same nights as well, not expecting fast, sharp changes like the ones found there at the specific phases after the campaign was over and all the data accurately calibrated and fully reduced.) Nevertheless, we consider a possibility that the sharp minima may in fact be occultation or eclipse (the latter being more likely, as the asteroid motion with respect to the Sun was more significant than its geocentric motion) events caused by a satellite of size of ∼ 0.6 diameters of the primary body orbiting it at a distance of about 4 diameters of the primary, synchronously with the primary’s rotation. An investigation of the binary hypothesis for 2001 VS78 is outside the scope of this paper and will be given in another paper on binary asteroids, but we have to keep in mind that the apparent PA rotation of 2001 VS78 (or its primary) might be due to tidal interactions in synchronized binary system and therefore irrelevant for our statistical study of solitary PA/NPA rotators, so its use in the study is conditional. 3.4.6. 2003 FG This is a small, ∼ 0.36-km sized15 Apollo asteroid with log τnorm = −0.6. We observed it on three consecutive nights of 2003-03-25.0 to 27.0 from Ondˇrejov and on one night of 2003-03-25.3 from ESO. The Ondˇrejov data were mutually linked to a same local comparison star while the ESO data are relative. The observations revealed a large amplitude (1.4 mag) lightcurve with a synodic period of 8.692 ± 0.003 h (Psyn–sid max = 0.005 h). A composite lightcurve is presented in Fig. 20. Even though the data are relatively noisy, they fit well to a singly-periodic composite, to within < 0.1 mag. We conclude that 2003 FG is likely a PA rotator, PAR = +2. Though this detection is somewhat less reliable than some other PAR = +2 objects mentioned in this paper, the good fit of the data with the single periodicity over more than half of the rotational period covered at least twice (some phases even thrice) leads us to judge that if there is any tumbling motion, its amplitude is probably small, on the order of a few 0.01 mag or less.
4. Statistics of tumbling asteroids In the previous section, we have reviewed all known tumbling asteroids with PAR codes −2. It might seem paradoxical but actually it is logical that the best detected cases (in Section 3.1) are not typical members of the current sample of tumblers. With the present state of the observational technology, the best detected tumblers must be unusual in some of their properties or circumstances—(4179) Toutatis had a very favorable close encounter that allowed it to be observed thoroughly with radar; 2002 TD60 and 2000 WL107 are fast rotators (stronger than most larger, rubble-pile asteroids, see below) that were described well with photometry of 15 Estimated from H = 19.7 (MPO 48340) assuming p = 0.18. v
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Fig. 20. Composite lightcurve of 2003 FG of 2003-03-25 to 27 for the best-fit synodic period of 8.692 h and G = 0.15 (assumed).
several nightly sessions during their favorable apparitions; and (54789) 2001 MZ7 is a “faster rotating” member of the slow tumblers population, so it is relatively large (see below for implications of the position of the asteroid in the spindiameter space on its possible younger age or higher rigidity than average). Most of the tumblers, however, are too slow rotating and/or too small for obtaining enough data to establish their second periods and they have obtained PAR = −2 (Section 3.2). With the current observational technology, studies of NPA rotators are pretty challenging and the available sample needs to be treated carefully to account for selection effects. We have established the PAR scheme in order to make useful estimates of reliability of the detections, allowing meaningful statistical studies of the population of tumbling asteroids. 4.1. Rubble-pile tumblers Figure 21 shows spin rate vs. diameter for asteroids with U 2 from the list compiled by A.W. Harris (version 2003 December 19).16 The tumblers with PAR −2 are marked there. We have also reviewed observational data for all asteroids in the range of D > 0.15 km (where rubble piles dominate; see Pravec et al., 2002) and log τnorm > −2 and established their PAR codes.17 We see a few important things from the plot. Most asteroids with estimated damping time scales (Eq. (5)) equal to the age of the Solar System or longer 16 The list is available on http://www.asu.cas.cz/~asteroid/ lcsumdat20031219.txt. For tumblers, their main periods with the strongest signals have been used as approximation of their spin periods. For (4486) Mithra where no exact period estimate was obtained, we used a value of 100 h for plotting in Fig. 21; the corresponding point is marked with an up triangle. 17 A complete list of PAR codes for asteroids we have investigated so far is available at the web page http://www.asu.cas.cz/~asteroid/tumblers. htm. In Table 1, we list tumbling asteroids mentioned in Sections 3.1 and 3.2.
Fig. 21. Asteroids’ spin rate vs. diameter plot. Tumblers with PAR −2 are marked with filled triangles. Principal axis rotators with PAR +2 in the range of D > 0.15 km and log τnorm > −2 are marked with open boxes with pluses; remaining points in the range have −1 PAR +1, i.e., they have not been reliably resolved as PA or NPA rotators.
(i.e., with log τnorm 0) are NPA rotators. Of the 12 objects with |PAR| 2 lying at or below the τ = 4.5 byr (i.e., log τnorm = 0) constant damping timescale line in Fig. 21, ten are tumblers. It is entirely consistent with the prediction by Harris (1994), see Section 1. Above the line, the situation changes rapidly; PA rotators start to prevail for log τnorm values just a few tenths less than zero. It appears that (253) Mathilde and (5645) 1990 SP with log τnorm = 0.0 are close to a transition range between the range of prevailing slower tumblers and that of faster PA rotators. (54789) 2001 MZ7 with log τnorm = −0.6 might already be somewhat unusual—it may be either younger or of a higher value of µQ than average rubble-pile asteroids of its size. (We also note that its C-classification by Dandy et al. (2003), suggests that it may have a darker albedo than we assumed, which would suggest even larger size and lower value of log τnorm about −1.0, therefore strengthening its somewhat unusual character.) Even though we expect the transition range to have a certain width due to a range of material properties as well as to varying collision ages of rubble-pile asteroids over the size range in question, the transition appears to lie between the lines of constant log τnorm = 0 and −1. The fact that the collision ages of most asteroids in the size range 0.2 to ∼ 60 km are expected to be shorter than 4.5 byr, perhaps by an order of magnitude or so, may indicate that average material properties of rubble pile asteroids are just a little bit weaker than those proposed by Harris (1994), but in any case well within the uncertainty range of his estimate given there. It would be of interest to know if the PAR +2 objects lying in the transition range actually are in PA rotation states, or if they have some residual, not fully damped tumbling motion. Even if we do not see any obvious deviations from the single periodicity in them, and therefore assign PAR values of +2 and greater, it is possible that some of them have not been damped quite to the basic rotation state and have a
Tumbling asteroids
small residual tumbling motion not apparent in the available data. A knowledge of distribution and amount of the residual energy of tumbling motion among the objects in the transition range should be useful for constraining theories of the excitation and damping of tumbling motion. A first indication for such an object may be returned by the MUSES-C (Hayabusa) mission visiting (25143) Itokawa, a small slow rotator in the transition range (Yano et al., 2004). A closer look at the exceptional case of (3102) Krok is in order. Having log τnorm of 1.8, it is the only known asteroid with PAR = +2 lying well below the log τnorm = 0 damping timescale line. Since its age cannot be longer than the age of the Solar System, its apparent single periodicity might be due to unusual physical properties or to an unusual evolutionary course leading to slow rotation. One possibility is that it might have an exceptionally low value of µQ in comparison with average rubble pile tumblers so that a much shorter damping time scale that would apply, despite its very slow spin rate (P of 151 h). The required value of µQ/T , where T is a spin excitation age of the asteroid (i.e., a time elapsed since the last significant spin excitation event), would have to be less by nearly two orders of magnitude than average for rubble-pile tumblers. Another possibility is that the mechanism that slows asteroids from initially faster spin may not induce tumbling motion, so that the tumbling motion may need to be caused by a separate mechanism, e.g., a collision after the slow spin is established. Thus not all slow rotators may have experiment a subsequent excitation into tumbling rotation. A third possibility is that its shape might be special so that tumbling motion, even if present, might not be observable in the lightcurve. For example, if it was close enough to a biaxial shape, one of the two periods would produce no detectable lightcurve signal. We conclude that even though we cannot actually resolve what is special in the exceptional case of (3102) Krok, its apparent single periodicity does not require its internal structure or evolution mechanism to be different from those of other rubble-pile tumblers. The presence of a small fraction of apparent singly-periodic objects among slow rotators dominated by tumblers may be a natural consequence of a distribution of their shapes or evolution routes. 4.2. Monolithic tumblers The smallest known NPA rotator is obviously not a rubble pile. 2000 WL107 is a superfast rotator with the periods about 0.2 h, so it clearly cannot be held together by selfgravitation only and must be a coherent body with non-zero tensile strength (even if it might be quite modest in comparison with strength of solid rocks on the earth; Pravec et al., 2002, and references therein). It has log τnorm ∼ −4, indicating that it has a value of µQ/T greater by at least four orders than average (larger) rubble pile asteroids. Because of its small size, it is likely younger by at least two orders of magnitude than the age of the Solar System, but probably not by four orders of magnitude. Thus its tumbling state
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appears to indicate a value of µQ/T greater than is typical for rubble piles, perhaps by one or two orders of magnitude. This is not unexpected for a body that is monolithic rather than a loose aggregate. It may also be that it is younger than average for its size, since a few other superfast rotators in the same size and spin rate range are not tumblers. The case of 2002 TD60 is less clear. As shown in Section 3.1, its spin rate (around instantaneous spin axis) is just such that it might be a rubble pile with zero tensile strength if its bulk density were > 3 g cm−3 . Such a density would be relatively high for a rubble pile (see Britt et al., 2002) and we also note that shapes so elongated as 2002 TD60 are generally unknown among single-period asteroids with periods shorter than 3 h—they tend to have spheroidal shapes with small amplitudes and elongations (Pravec and Harris, 2000). If 2002 TD60 was a high-density rubble pile, it would be an exceptional object, either a stony one with very little porosity (then it might be considered as being a transitional object between “true” rubble piles with high porosities and coherent “monoliths”; see Richardson et al. (2002), for discussion on possible spectrum of internal structures of asteroids), or an iron-rich, high density rubble pile. The radar observations of 2002 TD60 indicate the radar albedo about 0.25, that argues for non-metallic composition, so the latter possibility appears eliminated. The former possibility must be admitted. So, we conclude that 2002 TD60 is either a coherent body with non-zero tensile strength like those smaller superfast rotators, or one with a “transitional” internal structure of low porosity and high density even if it might have a negligible tensile strength. Its log τnorm ∼ −2 indicates a higher value of µQ/T than that of rubble pile tumblers of just a bit greater sizes. Most of the high value of µQ/T in this case may be due to the lower value of T appropriate for fewhundred meter diameter objects, but is also consistent with a greater value of µQ inferred for a near-monolithic body. The abundance of NPA rotators in the range of monoliths (D < 0.15 km) is unclear. The single known tumbler in the size range, 2000 WL107 , is about in the middle of the known sample of small monoliths both with respect to size and spin rate. Since lightcurve data for most of the other objects in the sample were not available to us—most of the objects were observed by Whiteley et al. (2002b) and their full lightcurve data still await publication—we do not know if the tumbler 2000 WL107 is a real exception with higher rigidity or younger age than other objects in its size range, or if it is perhaps just a high µQ/T tail member of only moderately wide distribution of the parameters in the population of monoliths. Future studies and, especially, careful thorough observations of small asteroids with D < 0.15 km, will be particularly interesting to shed a light on the abundance of NPA rotators among them. A future statistical study of them may lead to interesting inferences of their internal properties.
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4.3. Elongations of tumblers Observed lightcurve amplitudes of the tumblers are similar to or greater than the average amplitude of asteroids with spin rates < 6/day (which do not show the tendency to spheroidal shapes like faster-rotating rubble piles close to the barrier at ∼ 11/d; see Pravec and Harris, 2000, Fig. 6). There are a few possible explanations. First, it might be a selection effect—higher amplitude (i.e., more elongated) tumblers might be easier to recognize, as the main period and consequently also deviations from it are easier to detect unambiguously in a larger-amplitude lightcurve. Another observational effect that may contribute is that in a tumbling asteroid, a larger variation of cross-section is seen during their NPA rotation than for a PA rotator of the same shape, hence some increased total amplitude. An effect that biases against detection of low-amplitude tumblers is that more regular objects, with principal moments of inertia less different from one another have Pψ much longer than Pφ (see Samarasinha and A’Hearn, 1991), therefore much more difficult to detect. We conclude that the observational selection effects might well explain the entire apparent lack of low-amplitude tumblers, so that NPA rotation may be just as common among low-amplitude, more spheroidal asteroids as among the more extremely shaped objects.
5. Mechanisms causing slow NPA rotations Most slowly rotating asteroids larger than 0.2 km with log τnorm 0 probably have been slowed down from faster rotations (see Pravec and Harris, 2000). It seems likely that they were PA rotators earlier then when they rotated faster. The fact that we see them in NPA rotation states now when rotating slowly suggests that the mechanism (or mechanisms) that caused their slow rotations also induced their NPA rotations. Or alternatively, the slowed down asteroids might be more sensitive to excitation by other processes, e.g., by impacts, or tidal forces during planetary encounters. 5.1. YORP effect as a spin-down mechanism Rubincam (2000) proposed that the delayed re-radiation of thermal energy from a spinning body could produce a torque and alter the spin rate and orientation of small asteroids (see his paper for a brief review of the concept, going back at least 50 years to the Russian publications by Yarkovsky and by Radzievskii, and American publications by O’Keefe and Paddack). Rubincam has coined the term “YORP” (Yarkovsky–O’Keefe–Radzievskii–Paddack) for the effect. Rubincam computed torque models for several small bodies of known shape (martian satellites Phobos and Deimos, and asteroids 951 Gaspra and 243 Ida). He found that even for a variety of shapes, the rate of change of spin and/or spin orientation is similar, with scaling proportional to the inverse square of both object dimension (diameter)
and solar distance (orbital semi-major axis). Furthermore, the pattern of spin-up versus spin-down, and the evolution of spin axis orientation, is similar for all of the shapes studied. Rubincam noted that this mechanism might plausibly explain very slow rotations, including that of 253 Mathilde. ˇ Vokrouhlický and Capek (2002) carried the work further considering additional asteroid shapes and tracing spin evolution including the effect of precession of the orbit plane of the asteroid. All of this work was theoretical without a firm observational basis until Slivan et al. (2003) recently showed that the spin vectors of relatively large members of the Koronis asteroid family, bodies up to ∼ 40 km in diameter, are non-random in both orientation and spin rate and thus have likely been substantially altered over time. Vokrouhlický et al. (2003) have modeled this as a result of the YORP effect. They show that in the case of Koronis family members, initially retrograde spins should be driven to near pole-upright (180◦ obliquity), with the spin rate either spun up to periods of only a few hours, or spun down to rates asymptotically approaching zero rotation rate. Prograde rotators should be driven to spins in a particular resonance state with obliquity around 50◦ , poles aligned in longitude with respect to a particular resonance angle, and spinning with periods around 8–10 h. All members of the Koronis family that have spins and pole orientations determined fall into one or another of these three “Slivan states,” indicating convincingly that their spins have been radically altered over the age of the asteroid family, almost certainly by this radiation pressure torque. While the members of the Koronis family provide a “controlled sample” of asteroids with defined expected end states, the YORP effect itself does not depend on this particular family membership so should alter spins of all small asteroids, in varying ways depending on location in the asteroid belt. As noted above, the rate of spin alteration by YORP should be proportional to the inverse square of the asteroid diameter, all other things being equal (shape, thermal properties, distance from the Sun, etc.). In an example calculation to model possible evolution of the Koronis family member (167) Urda, a 40-km diameter slow retrograde rotator, Vokrouhlický et al. (2003) calculate a deceleration rate of ∼ 1 cy (day 109 yr)−1 , thus an asteroid this large can despin totally from an initial rate of several cycles per day, i.e., less than ∼ 6 h period over the age of the Solar system. Scaling by diameter squared, an asteroid only ∼ 1 km in diameter should evolve 1600 times faster, thus it could despin from an initial period of ∼ 4 h in only a few million years, which is far shorter than the expected collision lifetime of km-sized asteroids. At the other extreme, the largest known very slow rotator is 253 Mathilde, with a diameter of 58 km and spin period of 418 h. Scaling from the example of (167) Urda, the deceleration rate should be about half as fast, so Mathilde might plausibly slow to its present rate from an initial rotation period of around ten hours, which is not an unreasonable initial spin state. Indeed, over the entire size range from Mathilde down to the smallest slow rotators, the time of spin-down estimated in this way is short com-
Tumbling asteroids
pared to the age of the Solar System, or for objects smaller than a few tens of km diameter, shorter than the expected catastrophic collision time scale for their size. Thus YORP despinning appears to be a plausible cause for the excess of very slow rotators, which has been a mystery since the class of objects was first noticed twenty years ago. 5.2. Is YORP effect an explanation for the slow tumblers? The next question is, can YORP despinning cause tumbling among slow rotators, or is some other mechanism required to produce the non-principal axis rotation? It is noteworthy that YORP despinning has the characteristic of “sliding friction” in that the torque appears to first order not to depend on rate of spin. Thus a despinning asteroid should eventually “slide to a halt” and become spin-locked to its orbital period, as is the case for tidal friction, which leads to satellites like the Earth’s Moon having a rotation period equal to the orbit period. We do not see this with asteroids, so something is keeping the rotation excited above this end state. The tumbling itself could be the cause of shutting off further slowing (Rubincam, 2000), as the non-principal axis motion might destroy the axisymmetric re-radiation pattern to halt further deceleration. However, this does not explain the onset of tumbling in an obvious way. One possibility is collisional excitation. A difficulty with understanding asteroid spins in terms of collisional evolution has been that at the present very high impact velocities, impacts even up to the size triggering catastrophic disruption do not impart much spin change on the parent body. However for an asteroid that is barely spinning (such as Mathilde), a sub-catastrophic collision might still suffice to alter the spin by enough to set it tumbling, even if not spinning it up much from its very slow rate. Another possibility could be the interplay between the YORP evolution of spin rate and orientation and the precession of the asteroid’s orbital plane. Vokrouhlický (personal communication) has recently looked at the spin axis evolution of nearly-stopped asteroids and finds that the orientation and spin rate becomes nearly chaotic, which could perhaps induce non-principal axis spin (tumbling). Certainly it prevents spin-orbit locking such as we see as a common tidal evolution end state. Finally, for planet-crossing asteroids, tidal encounters with planets can deliver a substantial torque to the spin of an irregularly shaped body, leaving it in a state of non-principal axis rotation (Scheeres et al., 2004). This would be especially effective for a slowly rotating asteroid, so even a rather distant planetary encounter could set it tumbling without appreciably spinning it up. A first clue may be the couple asteroids with log τnorm > 0 that are not tumbling, most notably (3102) Krok. An explanation may be that a different mechanism (collisions or tidal encounters) is needed to excite tumbling from the mechanism that slows the rotation to the rate that tumbling can persist. The fact that a few slow rotators appear not to be tumbling suggests that for these, they have been slowed but by chance have not
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yet experienced an “exciting” event since become slow rotators.
Acknowledgments The work at Ondˇrejov was supported by the Grant Agency of the Czech Academy of Sciences, Grant A3003204, and by the Space Frontier Foundation within “The Watch” program. The work at Space Science Institute (A.W.H.) was supported by grant NAG5-13244 from the NASA Planetary Geology-Geophysics Program. The work at Modra was supported by the Slovak Grant Agency for Science VEGA, Grant 1/0204/03. Yu.K. and V.Sh. are grateful to V.G. Chiorny for his help during their observations of (3102) Krok. We thank David Vokrouhlický for discussing YORP mechanism and sharing his newest findings. We thank Donald Korycansky and an anonymous reviewer for their valuable reviews of the paper.
Note added in proof Our observations of 2004 FH, a ∼ 0.02-km sized Aten asteroid, on 2004-03-18 have revealed that it is a NPA rotator, PAR = −3, with periods 0.050397 ± 0.000004 and 0.037233 ± 0.000006 h. The detection of the two monolithic tumblers, 2000 WL107 and 2004 FH, in the sample of 40 asteroids with D < 0.15 km (but with only a part of them observed well to reveal their PA or NPA rotation) indicates that there is a subpopulation of monolithic NPA rotators overlapping the larger population of monolithic PA rotators in the same spin-size range.
Appendix A. Fourier series in two dimensions A function F (ψ, φ), where ψ, φ are angular variables over −π, π , can be expanded with Fourier series in the two variables (Rektorys et al., 1988): F (ψ, φ) =
∞
[aj k cos j ψ cos kφ + bj k sin j ψ cos kφ
j,k=0
+ cj k cos j ψ sin kφ + dj k sin j ψ sin kφ], (A.1) where aj k =
bj k =
cj k =
λj k π2 λj k π2 λj k π2
π π F (ψ, φ) cos j ψ cos kφ dψ dφ,
(A.2)
F (ψ, φ) sin j ψ cos kφ dψ dφ,
(A.3)
F (ψ, φ) cos j ψ sin kφ dψ dφ,
(A.4)
−π −π π π −π −π π π −π −π
130
λj k dj k = 2 π
P. Pravec et al. / Icarus 173 (2005) 108–131
π π F (ψ, φ) sin j ψ sin kφ dψ dφ,
(A.5)
−π −π
δ 0 δ 0 1 j 1 k λj k = , 2 2
(A.6)
and δi0 is Kronecker’s delta. The Fourier series given by Eq. (A.1) can be rewritten into a following form that is convenient for some purposes: F (ψ, φ) = C0 +
∞ [Cj 0 cos j ψ + Sj 0 sin j ψ] j =1
+
∞ ∞ Cj k cos(j ψ + kφ) k=1 j =−∞
+ Sj k sin(j ψ + kφ) ,
(A.7)
where C0 = a00 ,
Cj 0 = aj 0 ,
Sj 0 = bj 0 ,
S0k = c0k , C0k = a0k , aj k ± dj k cj k ± b j k , S±j,k = , C±j,k = 2 2 for j, k > 0.
(A.8)
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