- Email: [email protected]
Near-equilibrium dumb-bell-shaped figures for cohesionless small bodies
Icarus 265 (2016) 29–34
Contents lists available at ScienceDirect
Icarus journal homepage: www.journals.elsevier.com/icarus
Near-equilibrium dumb-bell-shaped figures for cohesionless small bodies Pascal Descamps ⇑ Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris, UMR 8028 CNRS, 77 av. Denfert-Rochereau, 75014 Paris, France
a r t i c l e
i n f o
Article history: Received 13 July 2015 Revised 4 October 2015 Accepted 10 October 2015 Available online 23 October 2015 Keywords: Asteroids Asteroids, surfaces Data reduction techniques Photometry
a b s t r a c t In a previous paper (Descamps, P. [2015]. Icarus 245, 64–79), we developed a specific method aimed to retrieve the main physical characteristics (shape, density, surface scattering properties) of highly elongated bodies from their rotational lightcurves through the use of dumb-bell-shaped equilibrium figures. The present work is a test of this method. For that purpose we introduce near-equilibrium dumbbell-shaped figures which are base dumb-bell equilibrium shapes modulated by lognormal statistics. Such synthetic irregular models are used to generate lightcurves from which our method is successfully applied. Shape statistical parameters of such near-equilibrium dumb-bell-shaped objects are in good agreement with those calculated for example for the Asteroid (216) Kleopatra from its dog-bone radar model. It may suggest that such bilobed and elongated asteroids can be approached by equilibrium figures perturbed be the interplay with a substantial internal friction modeled by a Gaussian random sphere. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction In an earlier work, dumb-bell-shaped hydrostatic equilibrium figures were proposed to describe shapes of contact-binary minor planets with a bimodal appearance (Descamps, 2015). However, small bodies in the Solar System are quite obviously not liquid but exhibit many pieces of evidence that most of them are ‘‘piles of rubble” (Davis et al., 1979), i.e. loosely consolidated aggregates of collisional fragmented material with zero tensile strength held together by mutual gravitational forces (see for example the review of Scheeres et al., 2010). One of the most convincing evidence results from high porosities measured in asteroids since the discovery of asteroid satellites which allowed density to be estimated. Significant macro-porosity sketches rubble pile models made of grains or boulders resting on each other with large voids. The fluid approach can give only overall shapes consistent with their angular momentum of rotation. Rubble piles which are much weaker than coherent structures are able to withstand shear strength due to their internal friction. This allows a much wider range of possible shapes not necessarily close to equilibrium shapes and thereby topography out of hydrostatic equilibrium can be maintained.
⇑ Fax: +33 (0)146332834. E-mail address: [email protected] http://dx.doi.org/10.1016/j.icarus.2015.10.010 0019-1035/Ó 2015 Elsevier Inc. All rights reserved.
In the present paper, near-equilibrium dumb-bell shapes are introduced as a combination of dumb-bell-shaped equilibrium figures with a Gaussian random sphere which is used to simulate the departures of real shapes from pure equilibrium figures. Although we do not know the true shape statistics of small bodies, the Gaussian hypothesis for the logarithmic radius of the perturbing sphere allows to simulate irregular shapes fully characterized by only a few statistical parameters. This paper aims at testing the reliability of our fitting and modeling protocol by dumb-bell-shaped equilibrium figures (Descamps, 2015) through simulations of noisy rotational lightcurves of pseudo-equilibrium dumb-bell shapes.
2. Near-equilibrium dumb-bell-shaped figures 2.1. Gaussian random sphere Muinonen (1998) first used lognormal statistics to modeling the irregular shapes of asteroids and cometary nuclei through the socalled Gaussian random sphere. The Gaussian sphere is fully described by only three statistical parameters: the mean radius a, the relative standard deviation r and the correlation angle C. In the limit of small standard deviations of radius and small correlation angles, the lognormal statistics reduces to the Gaussian statistics. A suitable covariance function of the logarithmic radius was devised for the generation of Gaussian spheres that closely resemble the shapes observed for asteroids. In spherical coordinates, the
30
P. Descamps / Icarus 265 (2016) 29–34
Table 1 Statistical properties of some rubble-pile asteroids (Section 2.2). ~ q
~ (°) U
~ (°) C
22 Kalliope 45 Eugenia 87 Sylvia 107 Camilla 121 Hermione 130 Elektra 216 Kleopatra 1999 KW4a 1999 KW4b 2867 Steins 4769 Castalia 25143 Itokawa
0.15 0.18 0.12 0.12 0.29 0.12 0.62 0.07 0.15 0.15 0.27 0.31
0.30 0.35 0.23 0.23 0.59 0.23 0.75 0.23 0.28 0.34 0.53 0.47
16.67 19.46 12.92 13.07 30.61 12.71 36.77 13.13 15.47 18.59 28.74 25.34
28.64 20.04 29.32 29.32 28.22 29.46 44.67 16.97 30.58 25.29 28.14 37.03
radius of a Gaussian random sphere, r(h, u) with respect to its center of mass can be fully defined by the mean radius and the covariance function of the logarithmic radius s(h, u).
a expðsðh; uÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r2 1 l XX sðh; uÞ ¼ slm Y lm ðh; uÞ
rðh; uÞ ¼
ð1Þ
l¼0 m¼l
s(h, u) is the logarithmic radial distance, a and r are respectively the mean radius and standard deviation for the radial distance Ylm(h, u)s represents orthonormal spherical harmonics. The value of radial standard deviation r quantizes the irregularity in shape relative to a sphere (r = 0 for spherical bodies). The slms are random variables which obey normal statistics with zero means and variances depending on the shape statistics specified by the so-called covariance function of logradius. If we denote the angular distance between two directions (h1, u1) and (h2, u2) by c, the covariance function Rs(c) is related to the correlation function Cs(c) by Rs(c) = b2Cs(c), where b is the standard deviation of the logradius. The variances r2 and b2 are interrelated through r2 = exp(b2) 1. The correlation function is generally described by a series expansion of Legendre polynomials (including lower and upper bounds for the degree), the coefficients of the Legendre polynomials Cl follow a power-law dependence Cl / lm. It appears to be a true law, with m 4, for overall shapes of asteroids (Muinonen and Lagerros, 1998) and even for Saharan dust particles (Nousiainen et al., 2003). However, in the present work we make use of the Gaussian correlation function, the correlation between two radii over solid angle c is then given by
C s ðcÞ ¼ exp
1 sin c=2 2 sin2 C=2 2
!
ð2Þ
where C is the correlation angle of the Gaussian sphere, defined as pffiffiffi the angular displacement over which the correlation drops to 1= e. A small correlation angle leads to increase short-distance fluctuations on the body surface (higher number of ‘‘valleys” and ‘‘hills”).
40
Γ
=
° 20
~ = 40° Γ
0°
~Γ = 3
30
~
r~
Φ (°)
Object
~
50
20
10
0 0.0
0.2
0.4
0.6
0.8
σ~ ~ against the relative standard Fig. 1. The standard deviation of slope angle U ~ for the 12 individual shapes of ~ and the correlation angle C deviation of the radius r rubble-pile asteroids listed in Table 1. Iso-gamma lines are drawn for ~ ¼ 20 ; 30 and 40 . Most of bodies have a correlation angle close to C ~ ¼ 30 C and a standard deviation of the radius ranging from 0.1 to 0.3.
This choice is made suitable from the fact that the elongation and flatness of the figures are already included in the base dumb-bell figure (see Section 2.3). Such a correlation function was also used for the same reason for the natural extension of the Gaussian random sphere in the form of the Gaussian random ellipsoid (Muinonen and Pieniluoma, 2011). Further details regarding the description and generation of Gaussian random spheres are given in Muinonen (1998) and Muinonen and Lagerros (1998). 2.2. Statistics of some irregular shapes of small bodies The inverse problem of determining the statistical parameters from a sample shape is briefly described by Muinonen and ~, the relative standard deviation Lagerros (1998). The mean radius a ~ , and the standard deviation of slopes q ~ of an individual of radius r shape are given by the following relationships:
~ ¼ EðrÞ a 1 r~ 2 ¼ ~2 ½Eðr2 Þ EðrÞ2 a ! r 2u 1 r 2h 2 q~ ¼ E 2 þ 2 2 2 r r sin h
ð3Þ
where E((r(h, u)) is the intrinsic expectation of radius r(h, u)
EðrÞ ¼
1 4p
Z p Z 2p 0
0
rðh; uÞ sin h dh du
rh and ru are the partial derivatives of the radius. The standard devi~ ations of radius and slope can be related to the correlation angle C
Table 2 Best-fit dumb-bell shaped solutions from simulated lightcurves of near-equilibrium DB-shaped bodies for a nominal aspect angle w = 85° and a scattering parameter k = 0.24. ~ ) are shown for two values of X and three standard deviations ~;C ~; U Statistical parameters of the Gaussian random sphere (rGRS, UGRS, CGRS) and the resulting DB-shaped bodies (r of radius rGRS of the associated GRS.
X
rGRS
UGRS (°)
CGRS (°)
r~
~ (°) U
~ (°) C
Xfit
kfit
wfit (°)
Std. dev. (mag)
v2r
0.29
0.097 0.16 0.22
12.9 20.7 27.5
24.5 24.2 23.8
0.85 0.82 0.79
41.1 41.7 42.9
50.3 47.7 44.1
0.287 ± 0.002 0.287 ± 0.002 0.287 ± 0.002
0.17 ± 0.08 0.24 ± 0.10 0.23 ± 0.10
83.3 ± 2.2 83.1 ± 2.2 82.5 ± 2.2
0.021 0.039 0.045
1.1 3.8 4.6
0.36
0.12 0.18 0.24
17.0 24.9 32.7
23.4 21.8 21.5
0.43 0.37 0.35
34.3 33.5 36.1
34.9 31.2 27.2
0.348 ± 0.010 0.346 ± 0.012 0.311 ± 0.015
0.29 ± 0.08 0.18 ± 0.07 0.56 ± 0.20
82.9 ± 1.9 78.9 ± 2.7 78.8 ± 2.7
0.025 0.032 0.057
1.6 2.7 8.4
P. Descamps / Icarus 265 (2016) 29–34
31
Fig. 2. Confidence levels for six near-equilibrium DB-shaped test bodies listed in Table 2. Simulated lightcurves have been computed for an aspect angle w = 85° and a scattering parameter k = 0.24. The contours were calculated by holding the aspect angle w fixed at its optimum value while varying X and k. The small contours, drawn with solid lines, give the region of the X–k space in which there is a 68.4% probability of finding the true values of the two parameters (1r region). The large contours, shown as dashes lines, correspond to the regions 2r (95.4%) and 3r (99.7%). Best-fit solutions are given in Table 2.
that partly characterizes the correlation function for an individual shape by
~ ¼ 2 arcsin C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 2Þ loge ð1 þ r ~ 2q
ð4Þ
~ by Furthermore, we define the slope angle U
~ ¼ arctan q ~ U
ð5Þ
~ is a random variable with The standard deviation of slopes q ~ zero means, so that U can be considered as the maximum angle
~ ¼ 0 for a spherical of repose (or slope angle) of the free surface (U surface). It is often pointed out that loosely consolidated piles of aggregated particles have slopes that are maintained at the angle of repose with respect to horizontal. Granular bodies with no tensile strength, commonly called rubble piles or gravitational aggregates, can withstand considerable shear strength. The shear
32
P. Descamps / Icarus 265 (2016) 29–34
Fig. 3. Simulated lightcurves (crosses) of near-equilibrium DB-shaped test bodies and best-fit solutions (solid lines). Fitting parameters and goodness-of-fit are listed in Table 2.
strength is due to the fact that interlocking particles are prevented from sliding over one another by the confining pressure (self-gravity in the present case of asteroids). The friction angle ~ characterizes this geometric friction due to interlocking and u rearrangement of finite-sized constituents (Nedderman, 1992). The geometric source of friction decreases with lowering density. Typically the friction angle of materials with some shear strength ~ ¼ 0 under pressure ranges from 20° to 40°, while a body with u is a pure fluid. In other words, it means that a given body with non-zero angle of friction may exhibit a large variety of permissible
shapes out of equilibrium shapes. It has been recently shown that ~ cannot overpass the friction angle the maximum angle of repose U
~ when a Mohr–Coulomb yield function of the non-cohesive mateu rial is used. However, with a Drücker–Prager law, the difference ~ u ~ ) can be as large as 10° (Modaressi and Evesque, 2001). Con(U ~, sequently, in first approximation, we may admit the slope angle U
~ of a given by (5), as a reliable estimator of the friction angle u cohesionless body. We applied the statistical parameters given by (3) to a sample of some irregular small bodies considered or known as rubble piles
P. Descamps / Icarus 265 (2016) 29–34
33
Fig. 4. Equatorial views of the near-equilibrium DB-shaped test bodies (upper views) and best-fit DB solutions (lower view) with true rendering (k) according to Table 2 results.
(Table 1). They all share a common noteworthy property: a great amount of macroscopic porosity, ranging from 20% to 40%, which is considered to characterize rubble pile asteroids. The slope angle ~ of each body for which a tridimensional shape model r(h, u) does U exist is plotted in Fig. 1 as a function of both the standard deviation ~ . All test bodies have roughly the r~ and the correlation angle C same correlation angle close to 30°. Their slope angle is ranging from 15° to 35° with a relative standard deviation of radius ranging from 0.1 to 0.3. As expected, the slope angle increases with the standard deviation of the radius. Since models in the dataset have unequal quality – due to different techniques used to recover their shapes – some of them exhibit smoother and convex surfaces which artificially lower the slope angle. Nevertheless, the obtained slope angles are in good agreement with what it is expected for the angle of friction of a granular material (Strahler, 1971). Out of the investigated bodies, only the values of the statistical parameters related to (216) Kleopatra shows striking discrepancies. It is due to its elongated and bilobed shape. The only way to reconcile statistical parameters of Kleopatra with mean values of other bodies is to consider its shape as a combination of an equilibrium figure, supposed to have a zero angle of friction, with a Gaussian random sphere endowed with common statistics. This suggests to introduce in the next section the concept of nearequilibrium figure of a rotating body in order to account of bifurcated shapes such as dumb-bell shapes with the same underlying physics which is at work in rubble pile bodies. 2.3. Near-equilibrium dumb-bell figures The radius of a near-equilibrium dumb-bell figure of a spinning body is defined as follows
rðh; uÞ ¼
re ðh; uÞ r GRS ðh; uÞ
ð6Þ
where re is the base dumb-bell equilibrium figure of a rotating fluid body perturbed by a Gaussian random sphere (GRS) of radius rGRS. The Gaussian random sphere model is used to synthesize nonequilibrium shapes. Such a shape modeling closely resembles the
one used in the case of equilibrium raindrops (Nousiainen and Muinonen, 1999). By definition, the slope angle and the friction angle of a fluid body are zero, therefore we adopt statistic parameters of the random Gaussian sphere (rGRS, CGRS, UGRS) as the true statistic parameters of this near-equilibrium figure. In other words, this makes the assumption that a near-equilibrium body is not a fluid body but has a shape resembling an equilibrium figure with internal friction modeled by the associated Gaussian random sphere. Thereby a wide variety of shapes can be derived for a given slope angle. In the present paper this formalism is used to derive shapes close to the equilibrium dumb-bell-shaped figures. Such equilibrium figures have been recently used to fit lightcurves of some small bodies (Descamps, 2015). The conclusive agreement of the model with a variety of observing techniques may suggest that real bodies have shapes close to equilibrium figures. The concept of near-equilibrium figures allows to synthesize such bodies through an intimate correlation with a concealed, elusive underlying equilibrium state.
3. Simulations Several three-dimensional geometries of Gaussian spheres were generated using a computer program based on the code developed by Muinonen and Nousiainen (2002). According to (6), nearequilibrium dumb-bell shaped figures are then generated for two values of the normalized angular velocity X = 0.29 and 0.36 and three values of the standard deviation of the radius rGRS covering the range of calculated values individual shape models (Table 1); CGRS is kept fixed to about 25°. The resulting statistical parameters ~; U ~ ) are given in Table 2. Fig. 4 illustrates the generated sample ~; C (r shapes. It is evident that a small change in rRGS can result in an increasingly deformed shape that closely resembles the shapes observed for asteroids. In the case X = 0.29, which gives a double-lobed, narrow-waisted shape, we can notice that final sta~; U ~ ) are very close to the values computed ~; C tistical parameters (r for the dog-bone radar shape model of Kleopatra (Table 1).
34
P. Descamps / Icarus 265 (2016) 29–34
To generate theoretical lightcurves, we adopted an empirical scattering model for the surfaces first proposed by Kaasalainen et al. (2001) combining, through a weight factor k, a lunar-type reflection – described by the Lommel-Seeliger law appropriate for low albedo rocky surfaces – and an icy-type law – Lambertian or diffuse reflection suitable for high-albedo surfaces with multiple scattering.
I ¼ ð1 kÞ
l0 þ k l0 l0 þ l
ð7Þ
In the above equation, l0 and l are the cosines of the angles between the surface normal and the incidence and emission directions, respectively. A pure Lambert scattering is obtained with k = 1.0; this is the case of bright bodies with high limb darkening. Conversely, it is expected that a low-albedo body, with no multiple scattering and negligible limb darkening, has a small k value. Adopting an arbitrary surface scattering parameter k = 0.24, six rotational lightcurves were obtained for a given aspect angle w of 85° (Fig. 3). A Gaussian noise with standard deviation of 0.02 mag was added to the synthetic lightcurves. The fit to the observational data is carried out by exploring systematically the interplay of all of them for affecting the lightcurve amplitude and morphology according to the method described in Descamps (2015). Fig. 2 shows the confidence levels derived for each lightcurve. The contours were calculated by holding the aspect angle fixed at its optimum value in each case which gives the smallest value of the goodness-of-fit criterion v2. Best-fit solutions for X, k, and w are given in Table 2. We can conclude that the method is able to successfully recover the initial shape, scattering and aspect parameters with a great accuracy especially in the case where X is smaller than 0.32 which guarantees a narrowed middle section of the dumb-bell-shaped body (see Fig. 2 of Descamps, 2015). 4. Conclusion We show in the present work that the method based on dumbbell-shaped equilibrium figures is reliable for dealing with lightcurves of high amplitudes (1 magnitude). Even though real
shapes are rough and irregular, the irregularities, physically associated with the level of internal friction, are essentially characterized by a short range of standard deviation of the radius and the same correlation angle. Concerning the bilobed asteroids, this result still holds true for the associated Gaussian random sphere which fully characterizes the level of internal friction. It suggests that shapes of rubble pile bodies may be governed by the rivalry between a fluid-type behavior and deformations arising from the internal friction. Acknowledgment I would like to thank Pr. Karri Muinonen for providing me with the G-Sphere code. References Davis, D.R. et al., 1979. Collisional evolution of asteroids: Populations, rotations and velocities. In: Gehrels, T. (Ed.), Asteroids. Univ. of Arizona Press, Tucson, pp. 528–557. Descamps, P., 2015. Dumb-bell-shaped equilibrium figures for fiducial contactbinary asteroids and EKBOs. Icarus 245, 64–79. Kaasalainen, M., Torppa, J., Muinonen, K., 2001. Optimization methods for asteroid lightcurve inversion – II. The complete inverse problem. Icarus 153, 37–51. Modaressi, A., Evesque, P., 2001. Is the friction angle the maximum slope of a free surface of an non cohesive material? Poudres et Grains 12, 83–102. Muinonen, K., 1998. Introducing the Gaussian shape hypothesis for asteroids and comets. Astron. Astrophys. 332, 1087–1098. Muinonen, K., Lagerros, J.S.V., 1998. Inversion of shape statistics for small Solar System bodies. Astron. Astrophys. 333, 753–761. Muinonen, K., Nousiainen, T., 2002. G-sphere, available under the GNU General Public License available from the authors. Muinonen, K., Pieniluoma, T., 2011. Light scattering by Gaussian random ellipsoid particles: First results with discrete-dipole approximation. J. Quant. Spectrosc. Radiat. Trans. 112, 1747–1752. Nedderman, R.M., 1992. Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge. Nousiainen, T., Muinonen, K., 1999. Light scattering by Gaussian, randomly, oscillating raindrops. J. Quant. Spectrosc. Radiat. Trans. 63, 643–666. Nousiainen, T., Muinonen, K., Räisänen, P., 2003. Scattering of light by large Saharan dust particles in a modified ray optics approximation. J. Geophys. Res. 108 (D1), 4025. Scheeres, D.J. et al., 2010. Scaling forces to asteroid surfaces: The role of cohesion. Icarus 210, 968–984. Strahler, A.N., 1971. The Earth Sciences. Harper and Row, New York.
Contents lists available at ScienceDirect
Icarus journal homepage: www.journals.elsevier.com/icarus
Near-equilibrium dumb-bell-shaped figures for cohesionless small bodies Pascal Descamps ⇑ Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris, UMR 8028 CNRS, 77 av. Denfert-Rochereau, 75014 Paris, France
a r t i c l e
i n f o
Article history: Received 13 July 2015 Revised 4 October 2015 Accepted 10 October 2015 Available online 23 October 2015 Keywords: Asteroids Asteroids, surfaces Data reduction techniques Photometry
a b s t r a c t In a previous paper (Descamps, P. [2015]. Icarus 245, 64–79), we developed a specific method aimed to retrieve the main physical characteristics (shape, density, surface scattering properties) of highly elongated bodies from their rotational lightcurves through the use of dumb-bell-shaped equilibrium figures. The present work is a test of this method. For that purpose we introduce near-equilibrium dumbbell-shaped figures which are base dumb-bell equilibrium shapes modulated by lognormal statistics. Such synthetic irregular models are used to generate lightcurves from which our method is successfully applied. Shape statistical parameters of such near-equilibrium dumb-bell-shaped objects are in good agreement with those calculated for example for the Asteroid (216) Kleopatra from its dog-bone radar model. It may suggest that such bilobed and elongated asteroids can be approached by equilibrium figures perturbed be the interplay with a substantial internal friction modeled by a Gaussian random sphere. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction In an earlier work, dumb-bell-shaped hydrostatic equilibrium figures were proposed to describe shapes of contact-binary minor planets with a bimodal appearance (Descamps, 2015). However, small bodies in the Solar System are quite obviously not liquid but exhibit many pieces of evidence that most of them are ‘‘piles of rubble” (Davis et al., 1979), i.e. loosely consolidated aggregates of collisional fragmented material with zero tensile strength held together by mutual gravitational forces (see for example the review of Scheeres et al., 2010). One of the most convincing evidence results from high porosities measured in asteroids since the discovery of asteroid satellites which allowed density to be estimated. Significant macro-porosity sketches rubble pile models made of grains or boulders resting on each other with large voids. The fluid approach can give only overall shapes consistent with their angular momentum of rotation. Rubble piles which are much weaker than coherent structures are able to withstand shear strength due to their internal friction. This allows a much wider range of possible shapes not necessarily close to equilibrium shapes and thereby topography out of hydrostatic equilibrium can be maintained.
⇑ Fax: +33 (0)146332834. E-mail address: [email protected] http://dx.doi.org/10.1016/j.icarus.2015.10.010 0019-1035/Ó 2015 Elsevier Inc. All rights reserved.
In the present paper, near-equilibrium dumb-bell shapes are introduced as a combination of dumb-bell-shaped equilibrium figures with a Gaussian random sphere which is used to simulate the departures of real shapes from pure equilibrium figures. Although we do not know the true shape statistics of small bodies, the Gaussian hypothesis for the logarithmic radius of the perturbing sphere allows to simulate irregular shapes fully characterized by only a few statistical parameters. This paper aims at testing the reliability of our fitting and modeling protocol by dumb-bell-shaped equilibrium figures (Descamps, 2015) through simulations of noisy rotational lightcurves of pseudo-equilibrium dumb-bell shapes.
2. Near-equilibrium dumb-bell-shaped figures 2.1. Gaussian random sphere Muinonen (1998) first used lognormal statistics to modeling the irregular shapes of asteroids and cometary nuclei through the socalled Gaussian random sphere. The Gaussian sphere is fully described by only three statistical parameters: the mean radius a, the relative standard deviation r and the correlation angle C. In the limit of small standard deviations of radius and small correlation angles, the lognormal statistics reduces to the Gaussian statistics. A suitable covariance function of the logarithmic radius was devised for the generation of Gaussian spheres that closely resemble the shapes observed for asteroids. In spherical coordinates, the
30
P. Descamps / Icarus 265 (2016) 29–34
Table 1 Statistical properties of some rubble-pile asteroids (Section 2.2). ~ q
~ (°) U
~ (°) C
22 Kalliope 45 Eugenia 87 Sylvia 107 Camilla 121 Hermione 130 Elektra 216 Kleopatra 1999 KW4a 1999 KW4b 2867 Steins 4769 Castalia 25143 Itokawa
0.15 0.18 0.12 0.12 0.29 0.12 0.62 0.07 0.15 0.15 0.27 0.31
0.30 0.35 0.23 0.23 0.59 0.23 0.75 0.23 0.28 0.34 0.53 0.47
16.67 19.46 12.92 13.07 30.61 12.71 36.77 13.13 15.47 18.59 28.74 25.34
28.64 20.04 29.32 29.32 28.22 29.46 44.67 16.97 30.58 25.29 28.14 37.03
radius of a Gaussian random sphere, r(h, u) with respect to its center of mass can be fully defined by the mean radius and the covariance function of the logarithmic radius s(h, u).
a expðsðh; uÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ r2 1 l XX sðh; uÞ ¼ slm Y lm ðh; uÞ
rðh; uÞ ¼
ð1Þ
l¼0 m¼l
s(h, u) is the logarithmic radial distance, a and r are respectively the mean radius and standard deviation for the radial distance Ylm(h, u)s represents orthonormal spherical harmonics. The value of radial standard deviation r quantizes the irregularity in shape relative to a sphere (r = 0 for spherical bodies). The slms are random variables which obey normal statistics with zero means and variances depending on the shape statistics specified by the so-called covariance function of logradius. If we denote the angular distance between two directions (h1, u1) and (h2, u2) by c, the covariance function Rs(c) is related to the correlation function Cs(c) by Rs(c) = b2Cs(c), where b is the standard deviation of the logradius. The variances r2 and b2 are interrelated through r2 = exp(b2) 1. The correlation function is generally described by a series expansion of Legendre polynomials (including lower and upper bounds for the degree), the coefficients of the Legendre polynomials Cl follow a power-law dependence Cl / lm. It appears to be a true law, with m 4, for overall shapes of asteroids (Muinonen and Lagerros, 1998) and even for Saharan dust particles (Nousiainen et al., 2003). However, in the present work we make use of the Gaussian correlation function, the correlation between two radii over solid angle c is then given by
C s ðcÞ ¼ exp
1 sin c=2 2 sin2 C=2 2
!
ð2Þ
where C is the correlation angle of the Gaussian sphere, defined as pffiffiffi the angular displacement over which the correlation drops to 1= e. A small correlation angle leads to increase short-distance fluctuations on the body surface (higher number of ‘‘valleys” and ‘‘hills”).
40
Γ
=
° 20
~ = 40° Γ
0°
~Γ = 3
30
~
r~
Φ (°)
Object
~
50
20
10
0 0.0
0.2
0.4
0.6
0.8
σ~ ~ against the relative standard Fig. 1. The standard deviation of slope angle U ~ for the 12 individual shapes of ~ and the correlation angle C deviation of the radius r rubble-pile asteroids listed in Table 1. Iso-gamma lines are drawn for ~ ¼ 20 ; 30 and 40 . Most of bodies have a correlation angle close to C ~ ¼ 30 C and a standard deviation of the radius ranging from 0.1 to 0.3.
This choice is made suitable from the fact that the elongation and flatness of the figures are already included in the base dumb-bell figure (see Section 2.3). Such a correlation function was also used for the same reason for the natural extension of the Gaussian random sphere in the form of the Gaussian random ellipsoid (Muinonen and Pieniluoma, 2011). Further details regarding the description and generation of Gaussian random spheres are given in Muinonen (1998) and Muinonen and Lagerros (1998). 2.2. Statistics of some irregular shapes of small bodies The inverse problem of determining the statistical parameters from a sample shape is briefly described by Muinonen and ~, the relative standard deviation Lagerros (1998). The mean radius a ~ , and the standard deviation of slopes q ~ of an individual of radius r shape are given by the following relationships:
~ ¼ EðrÞ a 1 r~ 2 ¼ ~2 ½Eðr2 Þ EðrÞ2 a ! r 2u 1 r 2h 2 q~ ¼ E 2 þ 2 2 2 r r sin h
ð3Þ
where E((r(h, u)) is the intrinsic expectation of radius r(h, u)
EðrÞ ¼
1 4p
Z p Z 2p 0
0
rðh; uÞ sin h dh du
rh and ru are the partial derivatives of the radius. The standard devi~ ations of radius and slope can be related to the correlation angle C
Table 2 Best-fit dumb-bell shaped solutions from simulated lightcurves of near-equilibrium DB-shaped bodies for a nominal aspect angle w = 85° and a scattering parameter k = 0.24. ~ ) are shown for two values of X and three standard deviations ~;C ~; U Statistical parameters of the Gaussian random sphere (rGRS, UGRS, CGRS) and the resulting DB-shaped bodies (r of radius rGRS of the associated GRS.
X
rGRS
UGRS (°)
CGRS (°)
r~
~ (°) U
~ (°) C
Xfit
kfit
wfit (°)
Std. dev. (mag)
v2r
0.29
0.097 0.16 0.22
12.9 20.7 27.5
24.5 24.2 23.8
0.85 0.82 0.79
41.1 41.7 42.9
50.3 47.7 44.1
0.287 ± 0.002 0.287 ± 0.002 0.287 ± 0.002
0.17 ± 0.08 0.24 ± 0.10 0.23 ± 0.10
83.3 ± 2.2 83.1 ± 2.2 82.5 ± 2.2
0.021 0.039 0.045
1.1 3.8 4.6
0.36
0.12 0.18 0.24
17.0 24.9 32.7
23.4 21.8 21.5
0.43 0.37 0.35
34.3 33.5 36.1
34.9 31.2 27.2
0.348 ± 0.010 0.346 ± 0.012 0.311 ± 0.015
0.29 ± 0.08 0.18 ± 0.07 0.56 ± 0.20
82.9 ± 1.9 78.9 ± 2.7 78.8 ± 2.7
0.025 0.032 0.057
1.6 2.7 8.4
P. Descamps / Icarus 265 (2016) 29–34
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Fig. 2. Confidence levels for six near-equilibrium DB-shaped test bodies listed in Table 2. Simulated lightcurves have been computed for an aspect angle w = 85° and a scattering parameter k = 0.24. The contours were calculated by holding the aspect angle w fixed at its optimum value while varying X and k. The small contours, drawn with solid lines, give the region of the X–k space in which there is a 68.4% probability of finding the true values of the two parameters (1r region). The large contours, shown as dashes lines, correspond to the regions 2r (95.4%) and 3r (99.7%). Best-fit solutions are given in Table 2.
that partly characterizes the correlation function for an individual shape by
~ ¼ 2 arcsin C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ 2Þ loge ð1 þ r ~ 2q
ð4Þ
~ by Furthermore, we define the slope angle U
~ ¼ arctan q ~ U
ð5Þ
~ is a random variable with The standard deviation of slopes q ~ zero means, so that U can be considered as the maximum angle
~ ¼ 0 for a spherical of repose (or slope angle) of the free surface (U surface). It is often pointed out that loosely consolidated piles of aggregated particles have slopes that are maintained at the angle of repose with respect to horizontal. Granular bodies with no tensile strength, commonly called rubble piles or gravitational aggregates, can withstand considerable shear strength. The shear
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P. Descamps / Icarus 265 (2016) 29–34
Fig. 3. Simulated lightcurves (crosses) of near-equilibrium DB-shaped test bodies and best-fit solutions (solid lines). Fitting parameters and goodness-of-fit are listed in Table 2.
strength is due to the fact that interlocking particles are prevented from sliding over one another by the confining pressure (self-gravity in the present case of asteroids). The friction angle ~ characterizes this geometric friction due to interlocking and u rearrangement of finite-sized constituents (Nedderman, 1992). The geometric source of friction decreases with lowering density. Typically the friction angle of materials with some shear strength ~ ¼ 0 under pressure ranges from 20° to 40°, while a body with u is a pure fluid. In other words, it means that a given body with non-zero angle of friction may exhibit a large variety of permissible
shapes out of equilibrium shapes. It has been recently shown that ~ cannot overpass the friction angle the maximum angle of repose U
~ when a Mohr–Coulomb yield function of the non-cohesive mateu rial is used. However, with a Drücker–Prager law, the difference ~ u ~ ) can be as large as 10° (Modaressi and Evesque, 2001). Con(U ~, sequently, in first approximation, we may admit the slope angle U
~ of a given by (5), as a reliable estimator of the friction angle u cohesionless body. We applied the statistical parameters given by (3) to a sample of some irregular small bodies considered or known as rubble piles
P. Descamps / Icarus 265 (2016) 29–34
33
Fig. 4. Equatorial views of the near-equilibrium DB-shaped test bodies (upper views) and best-fit DB solutions (lower view) with true rendering (k) according to Table 2 results.
(Table 1). They all share a common noteworthy property: a great amount of macroscopic porosity, ranging from 20% to 40%, which is considered to characterize rubble pile asteroids. The slope angle ~ of each body for which a tridimensional shape model r(h, u) does U exist is plotted in Fig. 1 as a function of both the standard deviation ~ . All test bodies have roughly the r~ and the correlation angle C same correlation angle close to 30°. Their slope angle is ranging from 15° to 35° with a relative standard deviation of radius ranging from 0.1 to 0.3. As expected, the slope angle increases with the standard deviation of the radius. Since models in the dataset have unequal quality – due to different techniques used to recover their shapes – some of them exhibit smoother and convex surfaces which artificially lower the slope angle. Nevertheless, the obtained slope angles are in good agreement with what it is expected for the angle of friction of a granular material (Strahler, 1971). Out of the investigated bodies, only the values of the statistical parameters related to (216) Kleopatra shows striking discrepancies. It is due to its elongated and bilobed shape. The only way to reconcile statistical parameters of Kleopatra with mean values of other bodies is to consider its shape as a combination of an equilibrium figure, supposed to have a zero angle of friction, with a Gaussian random sphere endowed with common statistics. This suggests to introduce in the next section the concept of nearequilibrium figure of a rotating body in order to account of bifurcated shapes such as dumb-bell shapes with the same underlying physics which is at work in rubble pile bodies. 2.3. Near-equilibrium dumb-bell figures The radius of a near-equilibrium dumb-bell figure of a spinning body is defined as follows
rðh; uÞ ¼
re ðh; uÞ r GRS ðh; uÞ
ð6Þ
where re is the base dumb-bell equilibrium figure of a rotating fluid body perturbed by a Gaussian random sphere (GRS) of radius rGRS. The Gaussian random sphere model is used to synthesize nonequilibrium shapes. Such a shape modeling closely resembles the
one used in the case of equilibrium raindrops (Nousiainen and Muinonen, 1999). By definition, the slope angle and the friction angle of a fluid body are zero, therefore we adopt statistic parameters of the random Gaussian sphere (rGRS, CGRS, UGRS) as the true statistic parameters of this near-equilibrium figure. In other words, this makes the assumption that a near-equilibrium body is not a fluid body but has a shape resembling an equilibrium figure with internal friction modeled by the associated Gaussian random sphere. Thereby a wide variety of shapes can be derived for a given slope angle. In the present paper this formalism is used to derive shapes close to the equilibrium dumb-bell-shaped figures. Such equilibrium figures have been recently used to fit lightcurves of some small bodies (Descamps, 2015). The conclusive agreement of the model with a variety of observing techniques may suggest that real bodies have shapes close to equilibrium figures. The concept of near-equilibrium figures allows to synthesize such bodies through an intimate correlation with a concealed, elusive underlying equilibrium state.
3. Simulations Several three-dimensional geometries of Gaussian spheres were generated using a computer program based on the code developed by Muinonen and Nousiainen (2002). According to (6), nearequilibrium dumb-bell shaped figures are then generated for two values of the normalized angular velocity X = 0.29 and 0.36 and three values of the standard deviation of the radius rGRS covering the range of calculated values individual shape models (Table 1); CGRS is kept fixed to about 25°. The resulting statistical parameters ~; U ~ ) are given in Table 2. Fig. 4 illustrates the generated sample ~; C (r shapes. It is evident that a small change in rRGS can result in an increasingly deformed shape that closely resembles the shapes observed for asteroids. In the case X = 0.29, which gives a double-lobed, narrow-waisted shape, we can notice that final sta~; U ~ ) are very close to the values computed ~; C tistical parameters (r for the dog-bone radar shape model of Kleopatra (Table 1).
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P. Descamps / Icarus 265 (2016) 29–34
To generate theoretical lightcurves, we adopted an empirical scattering model for the surfaces first proposed by Kaasalainen et al. (2001) combining, through a weight factor k, a lunar-type reflection – described by the Lommel-Seeliger law appropriate for low albedo rocky surfaces – and an icy-type law – Lambertian or diffuse reflection suitable for high-albedo surfaces with multiple scattering.
I ¼ ð1 kÞ
l0 þ k l0 l0 þ l
ð7Þ
In the above equation, l0 and l are the cosines of the angles between the surface normal and the incidence and emission directions, respectively. A pure Lambert scattering is obtained with k = 1.0; this is the case of bright bodies with high limb darkening. Conversely, it is expected that a low-albedo body, with no multiple scattering and negligible limb darkening, has a small k value. Adopting an arbitrary surface scattering parameter k = 0.24, six rotational lightcurves were obtained for a given aspect angle w of 85° (Fig. 3). A Gaussian noise with standard deviation of 0.02 mag was added to the synthetic lightcurves. The fit to the observational data is carried out by exploring systematically the interplay of all of them for affecting the lightcurve amplitude and morphology according to the method described in Descamps (2015). Fig. 2 shows the confidence levels derived for each lightcurve. The contours were calculated by holding the aspect angle fixed at its optimum value in each case which gives the smallest value of the goodness-of-fit criterion v2. Best-fit solutions for X, k, and w are given in Table 2. We can conclude that the method is able to successfully recover the initial shape, scattering and aspect parameters with a great accuracy especially in the case where X is smaller than 0.32 which guarantees a narrowed middle section of the dumb-bell-shaped body (see Fig. 2 of Descamps, 2015). 4. Conclusion We show in the present work that the method based on dumbbell-shaped equilibrium figures is reliable for dealing with lightcurves of high amplitudes (1 magnitude). Even though real
shapes are rough and irregular, the irregularities, physically associated with the level of internal friction, are essentially characterized by a short range of standard deviation of the radius and the same correlation angle. Concerning the bilobed asteroids, this result still holds true for the associated Gaussian random sphere which fully characterizes the level of internal friction. It suggests that shapes of rubble pile bodies may be governed by the rivalry between a fluid-type behavior and deformations arising from the internal friction. Acknowledgment I would like to thank Pr. Karri Muinonen for providing me with the G-Sphere code. References Davis, D.R. et al., 1979. Collisional evolution of asteroids: Populations, rotations and velocities. In: Gehrels, T. (Ed.), Asteroids. Univ. of Arizona Press, Tucson, pp. 528–557. Descamps, P., 2015. Dumb-bell-shaped equilibrium figures for fiducial contactbinary asteroids and EKBOs. Icarus 245, 64–79. Kaasalainen, M., Torppa, J., Muinonen, K., 2001. Optimization methods for asteroid lightcurve inversion – II. The complete inverse problem. Icarus 153, 37–51. Modaressi, A., Evesque, P., 2001. Is the friction angle the maximum slope of a free surface of an non cohesive material? Poudres et Grains 12, 83–102. Muinonen, K., 1998. Introducing the Gaussian shape hypothesis for asteroids and comets. Astron. Astrophys. 332, 1087–1098. Muinonen, K., Lagerros, J.S.V., 1998. Inversion of shape statistics for small Solar System bodies. Astron. Astrophys. 333, 753–761. Muinonen, K., Nousiainen, T., 2002. G-sphere, available under the GNU General Public License available from the authors. Muinonen, K., Pieniluoma, T., 2011. Light scattering by Gaussian random ellipsoid particles: First results with discrete-dipole approximation. J. Quant. Spectrosc. Radiat. Trans. 112, 1747–1752. Nedderman, R.M., 1992. Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge. Nousiainen, T., Muinonen, K., 1999. Light scattering by Gaussian, randomly, oscillating raindrops. J. Quant. Spectrosc. Radiat. Trans. 63, 643–666. Nousiainen, T., Muinonen, K., Räisänen, P., 2003. Scattering of light by large Saharan dust particles in a modified ray optics approximation. J. Geophys. Res. 108 (D1), 4025. Scheeres, D.J. et al., 2010. Scaling forces to asteroid surfaces: The role of cohesion. Icarus 210, 968–984. Strahler, A.N., 1971. The Earth Sciences. Harper and Row, New York.