Submillimeter lightcurves of Vesta

Icarus 192 (2007) 448–459 www.elsevier.com/locate/icarus

Submillimeter lightcurves of Vesta Matthew A. Chamberlain a,∗ , Amy J. Lovell b,1 , Mark V. Sykes a a Planetary Science Institute, 1700 E Fort Lowell Suite 106, Tucson, AZ 85719, USA b Agnes Scott College, 141 E. College Ave., Decatur, GA 30030, USA

Received 28 March 2007; revised 24 July 2007 Available online 12 September 2007

Abstract Thermal lightcurves of Asteroid Vesta with significant amplitude have been observed at 870 µm (345 GHz) using the MPIfR 19-channel bolometer of the Heinrich–Hertz Submillimeter Telescope. Shape and albedo are not sufficient to explain the magnitude of this variation, which we relate to global variations in thermal inertia and/or other thermophysical parameters. Vesta’s lightcurve has been observed over several epochs with the same general shape. However, there are some changes in morphology that may in part be related to viewing geometry and/or asteroid season. Inconsistent night-to-night variations exhibit the inherent difficulties in photometry at this wavelength. We are able to match the observed brightness temperatures with a relatively simple thermal model that integrates beneath the surface and assumes reasonable values of thermal inertia, loss tangent and refractive index, and without having to assume low values of emissivity in the submillimeter. High flux portions of the submillimeter lightcurve are found to correspond to regions with weak mafic bands observed in Hubble Space Telescope images. © 2007 Elsevier Inc. All rights reserved. Keywords: Asteroid Vesta; Asteroids, surfaces; Photometry; Radio observations; Infrared observations

1. Introduction Asteroid Vesta is one of the largest surviving terrestrial protoplanets from the formation of the Solar System, along with Ceres and Pallas. Visible lightcurves (e.g., Hoffmann and Geyer, 1993), disk-resolved images with adaptive optics (Drummond et al., 1998) and Hubble Space Telescope (HST) images (Binzel et al., 1997; Thomas et al., 1997a, 1997b) indicate that the surface of Vesta is heterogeneous. Unlike most asteroids, which usually exhibit double-peaked, shape-dominated lightcurves, Vesta’s albedo features result in a single-peaked lightcurve. The pole position of Vesta was initially estimated by photometric astrometry (such as Taylor et al., 1985), and has been refined with disk-resolved imaging. A shape model produced with HST images shows deviations of ∼15 km in the surface of Vesta from a (289 × 280 × 229 km) triaxial ellipsoid (note in Thomas et al., 1997b). * Corresponding author. Fax: +1 (520) 622 8060.

E-mail address: [email protected] (M.A. Chamberlain). 1 Visiting Astronomer, Arecibo, Puerto Rico.

0019-1035/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2007.08.003

We present here observations of Vesta at 870 µm (345 GHz) from several epochs from 2001 to 2004. In most of these epochs large amplitudes were observed in the thermal lightcurves. A unique set of material properties may be probed by observing thermal submillimeter radiation. The absorption length (L) of photons in the submillimeter is comparable to the e-folding depth (δd ) of the diurnal thermal wave into asteroid surfaces. At other wavelengths, e.g. mid-infrared and radio, this is not the case. Thermal radiation in the submillimeter samples the entire diurnal thermal wave, including the surface temperature and the average diurnal temperature, with a depth weighting that depends on the ratio of δd to L. Adopting the approximation for radio wave thermal radiation from terrestrial surfaces, L is typically ten wavelengths (de Pater, 1999) or ∼8.7 mm for our observations. Diurnal wave depth (δd ) is a function of thermal inertia (I ), density (ρ),√specific heat (c) and the diurnal period (P ); δd = I (ρc)−1 P /π . Using values of likely asteroid materials and the sidereal period of Vesta, ρ = 1500 kg m−1 , c = 750 J kg−1 K−1 and P = 19,230 s. I of main belt asteroids are low, ∼15 J m−2 s−0.5 K−1 (Spencer, 1990; Müller and Lagerros, 1998). Some asteroids have moderately high I ,

Submillimeter lightcurves of Vesta

449

Table 1 Aspect data associated with 12:00 UT of the first date of each epoch, negative phase angles indicate Vesta observations before opposition UT date

DSol (AU)

 (AU)

Phase (deg)

Sub-Earth lat. (deg)

Sub-solar lat. (deg)

True anomaly (deg)

Solar long. (deg)

14, 17 December 2001 4, 5 January 2003 11, 12 May 2003 2 May 2004

2.572 2.348 2.229 2.240

1.633 2.027 1.466 2.460

+8.3 −24.6 +21.1 −24.2

+11 +14 +13 −26

+14 +22 +8 −27

176 269 306 60

31 124 161 275

though these are typically smaller near-Earth asteroids, e.g., Betulia with I = 180 J m−2 s−0.5 K−1 from Harris et al. (2005). The δd for low and moderate thermal inertia materials on Vesta would be 1.0 mm and 1.2 cm, respectively, of which the former is most likely. This δd is one order of magnitude less than the estimated L, so the submillimeter radiation should primarily sample the diurnal average temperature of a surface, with a small component due to the surface temperature. Mid-infrared radiation arises from top tens of microns, well within in the diurnal thermal surface, and is independent of the properties below the surface. This mid-infrared radiation is sensitive to surface roughness which preferentially directs radiation towards the direction of illumination increasing the observed brightness temperature (Spencer, 1990; Lagerros, 1998). Thermal radiation from wavelengths much longer than the submillimeter come from depths where the temperature does not change significantly over the diurnal cycle. These longer wavelengths will, however, be sensitive to the seasonal wave and to different properties of the subsurface. Hence, submillimeter thermal radiation, which is intermediate to these other wavelengths, offers a unique probe into the properties of the surface within the skin depth of the diurnal thermal wave. A number of investigators (Webster and Johnston, 1989; Redman et al., 1992, 1998; Müller and Lagerros, 1998; Müller and Barnes, 2007) have suggested that emissivities at submillimeter and longer wavelengths are low, in the ∼0.6 range. Of these investigations, only Müller and Barnes (2007) observed variations in the thermal lightcurve that were attributed to heterogeneous material properties. Low effective emissivity values of Vesta in the submillimeter and millimeter are cited by Redman et al. (1998) as evidence that the top millimeters of the surface are transparent at these wavelengths. 2. Data 2.1. Observations Continuum observations at 870 µm (345 GHz) were made of Vesta using a MPIfR 19-channel bolometer at the Heinrich– Hertz Submillimeter Telescope (SMT). The bandpass of the bolometer is ν/ν ≈ 0.15. The effective bandpass profile is not uniform, with a peak sensitivity near 340 GHz, and it is affected by the transmission of the atmosphere. The telescope beam was typically switched by 20 with a wobbling secondary mirror to remove the sky contribution. Skydip observations were made about once an hour to measure the atmospheric optical depth and the submillimeter transmissivity of the sky as a function of elevation. These skydips allow the observed fluxes to be con-

Fig. 1. Schematic orbit of Vesta with positions of epochs observed in heliocentric ecliptic coordinates. Associated aspect data may also be found in Table 1. Dotted lines indicate the lines of sight from the Earth at the times of observation. Straight solid lines indicate the positions of Vesta equinox and solstice; Vesta’s northern summer is the position to the upper left. The plane of Vesta’s orbit is inclined 7◦ from the ecliptic; the ecliptic longitude of the ascending node is 104◦ .

verted to flux at the top of the atmosphere. A typical value of zenith optical depth during these observing runs was 0.7. Fluxes were calibrated with standard millimeter and submillimeter on-sky calibration sources. Mars and Uranus were primary calibrators. Secondary calibrators used included planets Saturn and Neptune, and non-planetary calibrators Orion-A, W3(OH), 3C279 and IRC + 10216. Calibrators were observed typically every 1 to 2 h. Substantial changes in the counts-toJansky value derived with these on-sky standards were useful to indicate times of unfavorable atmospheric conditions. Submillimeter observations of Vesta reported here were made in four observing epochs from 2001 to 2004. Observations in 2001 were first reported in Barrera-Pineda et al. (2002). Table 1 contains the aspects of Vesta from these epochs. Table 1 includes two measures of the Vesta’s angular position in its orbit, the true anomaly and solar longitude, which are with respect to the position of perihelion and vernal equinox, respectively. Fig. 1 shows schematically the position of Vesta and the line of sight from Earth when the lightcurves were observed. Observations from all the epochs were folded together by plotting individual observations against the sub-Earth astero-

450

M.A. Chamberlain et al. / Icarus 192 (2007) 448–459

Fig. 2. Filtered submillimeter relative lightcurves of Vesta. Lightcurves are shown for each epoch we obtained data and have been subjected to a 30◦ boxcar filter within which a weighted mean is calculated. Submillimeter flux is plotted with respect to west longitudes using the reference longitude from the note in Thomas et al. (1997b). For comparison, relative flux is shown from a rotating ellipsoid fitted to the shape of Vesta and for a sub-observer latitude of 13◦ N. Surface temperatures of the ellipsoid are assumed to be uniform so the model lightcurve is only due to shape (lowest panel).

centric longitude. The thermal submillimeter flux observed is less sensitive to surface temperature than shorter thermal wavelengths, and hence this radiation is less sensitive to the relative position of the Sun. Therefore, we align the observations with the sub-observer longitude. We use the reference longitude, pole solution and period of Vesta from the note added in the proof of Thomas et al. (1997b). The calculation of the subEarth longitude also requires consideration of the aspect, i.e. the relative positions of the Earth and the asteroid at the time of the observation, as well as the distance and time for light to travel to the observer. The approach used is the same described by Chamberlain et al. (2007). The period of Vesta is well constrained and the uncertainty of the longitudes calculated is only ∼1◦ . The relative lightcurves from all epochs are

shown in Fig. 2. The flux shown is scaled by the average flux observed in each epoch. Dates and times (UT) from each epoch when Vesta was observed (i.e. not lighttime corrected) with a sub-Earth longitude of 0◦ are: 14 December 2001 09:51:27, 4 January 2003 11:07:48, 11 May 2003 02:34:44, and 2 May 2004 16:33:09. The period of Vesta may be used to find the other times when the 0◦ longitude was observed within a single epoch, or observing run, while the aspect is approximately the same. Table 2 shows the average observed fluxes of Vesta from these epochs, and the apparent brightness temperature (Tb ) after correcting for the apparent solid angle of the model Vesta ellipsoid (note added in Thomas et al., 1997b) for all epochs that we have reasonable absolute calibrations. For example, for

Submillimeter lightcurves of Vesta

Table 2 Vesta thermal flux and Tb at each epoch Date

Fluxav (Jy)

Tb,obs (K)

Tb,range (K)

January 2003 May 2003 May 2004

1.10 2.38 0.93

133 152 161

107–158 142–159 152–173

an effective cross-sectional diameter of 500 km, the conversion of Janskys observed at 870 µm to Kelvin is 31.25 K/(Jy AU2 ). Also in Table 2 is the range of Tb associated with the filtered lightcurve observed. 2.2. Processing The main challenge in observing thermal lightcurves in the submillimeter is the inherent difficulty obtaining accurate photometry at these wavelengths. The wavelength is not long enough for the Earth’s atmosphere to be effectively transparent; flux from the sky in the beam is typically thousands of times greater than the flux from the object being observed. The lightcurves shown above in Fig. 2 are the final, filtered lightcurves found by combining data from all rotations available in each epoch. Here we discuss the steps taken to produce these filtered lightcurves from the raw data. A discussion of the statistical significance of the lightcurves may be found in Section 3.1 following. The lightcurves in Fig. 2 are constructed using all data available from each epoch. There is no easy way to distinguish “good” from “bad” data, and the data coverage available in each epoch is limited. Purging bad data is no simple option. Errors are estimated for each data point with contributions from the uncertainties in the sky and sky-plus-object integrations, transparency of the atmosphere, and counts-to-Jansky value from observations of on-sky calibrators. Data are first projected onto the sub-observer longitude with respect to the reference longitude and pole solution in the note of Thomas et al. (1997b). The filtered lightcurves (shown in Fig. 2) are found by passing a 30◦ boxcar filter over the data and finding the weighted mean of the points within the boxcar. The weighting of each point is the inverse of the square of the uncertainty of the point, so that data points with larger error bars have little contribution to the filtered lightcurve. 30◦ was found to be wide enough to capture enough data to reduce the noise and small enough not to smooth out the features in the lightcurves we are attempting to resolve. The uncertainty in a counts-to-Jansky calibration value is estimated by comparing that value with a linear interpolation in time from other values observed nearby. The variations are typically about ten percent, but can be significantly higher in times of unstable atmospheric conditions. There is also a contribution to the uncertainty from the error in the on-off integrations of the on-sky calibrators. However, because of the relatively high fluxes of these calibrators, these integration uncertainties are much smaller than the variability in the calibration value over time. Similarly, the uncertainty of the transparency of the atmosphere is also estimated by the difference of skydip optical depth values with linear temporal interpolations from other

451

nearby values. The uncertainty of atmospheric transparency is corrected for the elevation of the object. Filtered and raw lightcurve data from all epochs are shown in Fig. 3 as four pairs of panels. The filtered data shown in panel A of each pair are the same data shown in Fig. 2, though here they are not scaled by the average flux. These filtered lightcurves in Fig. 3 also show error bars that include the uncertainty in the weighted means and the spread of data values within the 30◦ boxcars. Panel B of each pair shows the raw data, and uncertainties, that were used to generate the filtered lightcurves, as described above. Lightcurves were best sampled in January 2003 and May 2003. Data are sparse from December 2001 and May 2004, so the filtered lightcurves only partially cover the rotation of Vesta at these epochs. Due to uncertainty in the absolute calibration in December 2001, lightcurves from this epoch are relative only. 3. Analysis 3.1. Lightcurve significance While there are differences between the submillimeter lightcurves from each epoch in Fig. 2, there is a general consistent pattern. Submillimeter flux is relatively high near the sub-observer longitude 90◦ W, and flux is low in the region 180–270◦ W. Given the noisy data that has gone into the submillimeter lightcurves presented here, it is reasonable to ask how real the broad features in the lightcurve are. Two approaches are taken here to demonstrate that the lightcurve variations are real and not just artifacts of noise: (1) comparing the amplitude of the lightcurve to the uncertainty of the observations, and (2) comparing the shape of the lightcurve across different epochs. The significance of the variation in the lightcurve may be seen in the size of the amplitude relative to the uncertainty in the weighted means in the filtered lightcurves. The lightcurves for each epoch in Fig. 2 are the observed lightcurves that include the effects due to shape and heterogeneities in material properties of Vesta. To compare the observed lightcurves with a lightcurve due to the shape only, a model lightcurve is included in Fig. 2 based on the cross section of a triaxial ellipsoid observed at 13◦ N. The ellipsoid is based on the shape from the note in Thomas et al. (1997b) with radii of 289, 280 and 229 km and the long axis at 150/330◦ W. To calculate the model flux in Fig. 2, the ellipsoid surface is assumed to have a uniform temperature, so the relative change in flux is due to change in cross-sectional area only. The amplitude of the model lightcurve indicates that changes in the observed flux are much greater than those that would be associated with shape only. In the lightcurve shown in panel A of the January 2003 data in Fig. 3, the difference between the low and high values is several times greater than the combined uncertainty in these values. This indicates that the variation is significant to at least several σ and the chance that the variation is only due to error is a fraction of a percent. In panel A of the May 2003 lightcurve, this difference is only a few times greater than the uncertainty of the individual points. It is disappointing that the May 2003 data has

452

M.A. Chamberlain et al. / Icarus 192 (2007) 448–459

Fig. 3. Vesta submillimeter lightcurves from each epoch in Table 1. Four pairs of panels are shown, panel A of each pair shows the filtered lightcurve derived from data in panel B, with uncertainties; see text for details. Symbols are used in panel B to differentiate data from different dates within an epoch, when applicable.

poor repeatability between the two days of observing (panel B of the May 2003 data, Fig. 3). Poor repeatability reduces our confidence in the shape of the filtered May 2003 lightcurve. While there may be some uncertainty in the May 2003 lightcurve, there is a positive correlation with the January 2003 lightcurve. Fig. 4 shows the correlation between the lightcurves of these epochs in which complete submillimeter lightcurves of Vesta were observed. The correlation coefficient between these two lightcurves is relatively high, R = 0.698, which for 12 independent points means the relationship between the two lightcurves is significant with a confidence level of more than 95%. This correlation suggests that features in the lightcurves are real and correspond to some region(s) on Vesta’s surface rotating in and out of view. The data from the December 2001 and May 2004 epochs were insufficient for complete lightcurves. However, the partial lightcurve from December 2001 does suggest a minimum

Fig. 4. Correlation between filtered lightcurves from January and May 2003; R = 0.698.

Submillimeter lightcurves of Vesta

453

at about the same position as the minima in January and May 2003. Also, the amplitude suggested by this partial lightcurve from December 2001 is about the same size as the amplitude observed in January 2003. The May 2004 data appears to have a small maximum in the same position as other epochs. 3.2. Modeling We have employed a simple “billiard-ball” thermophysical model to test effects of various material properties on the submillimeter thermal flux of Vesta. This billiard-ball model has finite thermal inertia and a smooth surface, so it does not include effects such as self-radiation or shadowing. These “rough-surface” effects are important in modeling mid-infrared radiation (Spencer, 1990; Lagerros, 1998). However, it has yet to be demonstrated how significant these effects are at longer wavelengths, such as 870 µm, where photons are emitted from further below the surface. Thermal models are run to find the temperature of the surface and subsurface as a function of time, latitude and depth. Inputs to the thermal model include thermal inertia, albedo, orbit and obliquity. This temperature solution is then used to find the disk integrated Tb from a model spherical asteroid as viewed from a particular direction and set of subsurface properties. Tb is independent of asteroid size. The thermal model used to find the temperature for any location on the asteroid surface uses a semi-implicit, Crank– Nicholson 1-D diffusion scheme. Temperatures are associated with the midpoints of a series of layers corresponding to depths below the asteroid surface. The lower boundary of the model is made deeper than any significant temperature variation and is insulated. The top boundary is a layer with zero thickness, the temperature of which is the surface temperature. The thickness of the 2nd topmost layer is 0.2 times the skin depth of the diurnal temperature cycle. Each subsequent layer is 1.2 times the one above. This model is based on one applied to martian ground ice in Chamberlain and Boynton (2007), which may be referred to for more details. The surface temperature is found by a balance of insolation, thermal radiation and thermal conduction. Properties important to surface temperatures are albedo, emissivity and thermal conductivity (which is related to thermal inertia). The time step of the model is 1/100th of the diurnal cycle. Temperature cycles are solved for latitudes 5◦ apart. The model converges to a solution so that the temperature of each depth, when averaged over a full season, is the same. The model can solve either the diurnal temperature cycle at a single position of the asteroid in its orbit, or solve the full seasonal cycle (diurnal cycles are still resolved in this mode). When solving seasonal cycles, important orbit properties are semi-major axis, eccentricity, obliquity and the positions of equinoxes/solstices. A sample of the thermal model output is shown as a set of temperature profiles in Fig. 5. This figure shows both seasonal and diurnal changes in temperatures with depth. The latitude for the temperature profiles shown is 40◦ N; other model inputs are typical property values, see below and Table 3. Properties are assumed to be uniform with depth in this model. Depth is

Fig. 5. Temperature profiles as a function of season and time of day. Temperature profiles are shown for 5 days at solar longitudes (Ls ) indicated. 5 instantaneous profiles are shown for 2 of these days. Profiles from the other 3 days are only shown below the diurnal wave to reduce clutter of profiles near the surface. Temperature profiles here are calculated for a latitude of 40◦ N with typical Vesta properties.

shown on a log scale in order to demonstrate both the diurnal and seasonal thermal waves. Using the guide that thermal radiation is emitted from a depth of 10 wavelengths (de Pater, 1999), 870 µm photons will probe depths of ∼8.7 mm which is below the diurnal thermal wave in Fig. 5 and will sample the diurnal average temperature. Shorter photon absorption lengths result in a greater weighting of surface temperatures in the submillimeter thermal radiation. Increasing the thermal inertia of surface materials increases the depth of diurnal thermal wave propagation and also increases the weighting of surface temperatures. Wavelengths of at least several centimeters would be required to start probing the annual average temperature at this location. Unlike visual and near-infrared observations, long-wavelength thermal emission arises from beneath the surface of the asteroid and is a function of several material properties. At the wavelengths of our observations (much longer than the peak in thermal radiation of Vesta), the thermal flux density (Iν ) is well approximated with the equation, Iν = 2kT /λ2 , where k is Boltzmann’s constant, T is temperature and λ is the wavelength. The theoretical submillimeter Tb from a single surface element is found by integrating with depth, z, and weighting the temperatures of the subsurface with a term sensitive to the absorption length. Below is the equation we use, which is a simplified form of the equation from Keihm and Langseth (1975). 

 Tb (λ, θe ) = 1 − R(θe ) sec θi

∞

κλ T (z)e−κλ z sec θi dz.

(1)

0

Tb is the brightness temperature observed of the surface element and θe and θi are emission and incidence angles, respectively. R is the Fresnel reflection coefficient, κ is the attenuation coefficient which is the inverse of the absorption length, assuming that κ is purely absorptive with no scattering. Our simplification

454

M.A. Chamberlain et al. / Icarus 192 (2007) 448–459

Table 3 Thermal modeling and Tb calculation properties Property

Symbol

Value

Source

Geometric albedo Slope parameter Phase integral Bond albedo Thermal inertia (J m−2 s−0.5 K−1 ) Density (kg m−3 )

p G q A I ρ

0.42 0.32 0.48 0.20 15 1750

Specific heat (J kg−1 K−1 ) Thermal conductivity (J m−1 K−1 ) Sidereal rotation (days) Diurnal skin depth (m) Seasonal skin depth (m) Emissivity Dielectric constant (real) Loss tangent

c k Psid. δd δs e Kr tan 

750 1.71 × 10−4 0.22258874 0.00192 0.148 0.9 3.0 0.0067

Tedesco et al. (2004) Tedesco et al. (2004) Function from Bowell et al. (1989) A = pq Spencer (1990) √ √ ρ = 3200 ln((1 + R)/(1 − R)), R = radar reflectivity (Mitchell et al., 1996) Typical rock value k = I 2 /(ρc) Drummond et al. (1998) √ δ = I (ρc)−1 P /π , P = 19,230 s P = 1.145 × 108 s Typical values used (e.g., Spencer, 1990) Kr = 1.87ρ/1000 (Ostro et al., 1999) log10 (tan ) = 0.44(ρ/1000) − 2.943 (Ostro et al., 1999)

is that κ is independent of depth and temperature. κ is found by √ 1 + tan2  4πn −1 κ= (2) λ 2 from Ostro et al. (1999), where n is the refractive index (the square root of the real part of the dielectric constant, Kr ) and tan  is the loss tangent. The apparent Tb of the “ball” for a particular viewing aspect is found by dividing the surface into elements corresponding to the latitude and time resolution of the thermal model. The thermal intensity of each element in the direction of the observer is found by integrating with depth, as in Eq. (1), using temperature profiles determined with the thermal model. These intensities are then integrated over the visible disk with a weighting by the solid angle of the element towards the observer to get a diskaveraged Tb . The typical properties assumed in our thermal model and used to calculate Vesta Tb are listed in Table 3. The Bond albedo value (A) is based on the geometric albedo (p) and G slope parameter from IRAS (Tedesco et al., 2004). The equation to convert the G value to a phase integral (q) is q = G × 0.604 + 0.29 (Bowell et al., 1989), which is used to find the Bond albedo (A = pq). The thermal inertia value adopted is based on the value Spencer (1990) found for Ceres, which is also consistent with the estimate made of Vesta’s thermal inertia by Müller and Lagerros (1998). The estimates of the diurnal and seasonal thermal skin depth assume that thermal properties are uniform with depth. This is unlikely to be strictly true because density and thermal conductivity were found to increase at depths greater than a centimeter below the surface of the Moon (e.g., Keihm and Langseth, 1975). Note that this change is below the diurnal thermal wave and will not change the weighting of the surface temperature sampled by submillimeter radiation. The value for density, 1750 kg m−3 , is based on the radar reflectivity and relations used by Mitchell et al. (1996); this value for the surface material is significantly less than the bulk density of Vesta, 3400 kg m−3 . The radar reflectivity relation to density is likely to be different for Vesta than for Ceres and

Pallas, as used by Mitchell et al. (1996), due to the effect of multiple scattering in the radar echo. However, the approach is still useful in order to get an approximation. This density value is then used with relations in Ostro et al. (1999) to estimate values for Kr and tan  of Vesta’s surface. These relations are based on laboratory measurements at frequencies that are orders of magnitude lower than our observations. However, electric properties of candidate asteroid materials have not been measured in the laboratory at frequencies appropriate to submillimeter or millimeter astronomy and these relations are the closest available. For typical values of Kr used, the reflection coefficient when viewing a surface normal (θe = 0◦ ) is 0.067. Hence, the emissivity (esubmm ) is 0.933. Emissivity decreases as θe increases, however our esubmm does not approach values quoted in the literature (∼0.6; Webster and Johnston, 1989; Müller and Lagerros, 1998; Redman et al., 1998) until θe is almost parallel to the surface. This discrepancy will be discussed later. The billiard ball thermal model is run with typical properties and other property combinations to test their influence on the observed submillimeter Tb , as described below. The main objectives of these models are to: (1) match the typical Tb , (2) see how material properties may modify the thermal lightcurve as a function of solar phase angle, and (3) see what seasonal variations in Tb may be expected. 3.2.1. Typical brightness temperature Using the material properties listed in Table 3, the rotation pole solution of the proof note in Thomas et al. (1997b), and the viewing aspects of Vesta during the January 2003 and May 2003 epochs we calculated billiard ball Tb of 144.2 and 144.5 K. These values are within the range of Tb observed (see Table 2). Model Tb may be modified by varying the values of material properties. For example, Fig. 6 shows effects of thermal inertia, tan  and dielectric constant. In this figure, the heliocentric distance and sub-solar latitude are 2.347 AU and 22◦ , which was the position of Vesta in January 2003. The Tb were calculated for an observer in the same position as the

Submillimeter lightcurves of Vesta

455

Fig. 7. Change in shape of submillimeter thermal lightcurve with solar phase angles of 30◦ before and after opposition. To generate these model lightcurves, the hemisphere facing the observer at a rotational phase of 0.5 is given a tan  of 0.09, while the other hemisphere has a value of 0.0067.

using a spherical asteroid. The end of the following section has a discussion of the effects of ellipsoidal shapes on the model results.

Fig. 6. Changes in the model submillimeter disk-averaged Tb as √ a function of thermal inertia (I ), loss tangent (tan ) and refractive index (n = Kr ).

Earth in January 2003. Surface and subsurface properties of the model are set to values in Table 3, except for the property being evaluated. 3.2.2. Changes with phase angle We calculated model lightcurves at different solar phase angle to help understand the differences in lightcurve amplitudes between May 2003 and January 2003. A billiard ball model was constructed with material properties that vary with longitude, and a model lightcurve was produced by rotating the ball. Fig. 7 shows apparent Tb calculated as the billiard ball rotates for two different viewing directions. The sub-observer latitude for both directions is 0◦ , and the sub-observer longitude corresponding to positions 30◦ either side of the asteroid midday meridian. “−30◦ ” is used to indicate the observer position before opposition and with a view that includes the evening terminator. The heliocentric distance and sub-solar latitude are again the same as the position of Vesta in the January 2003 epoch. For the example shown, one hemisphere is given a different value for tan  (0.09 instead of 0.0067), and this hemisphere modifies the Tb observed as it rotates in and out of view. This anomalous hemisphere is facing the observer at a rotational phase of 0.5 in Fig. 7. Note the decrease in amplitude and change in shape due to the change in solar phase angle. While this model does not explain the change in the amplitude of the lightcurves between January 2003 and May 2003, it does illustrate that substantial amplitude changes with solar phase angle are plausible. This model lightcurve was generated

3.2.3. Changes with asteroid season Thermal models were run to demonstrate how Tb can vary as a function of Vesta’s position in orbit, or its “seasons.” In Figs. 8 and 9, apparent Tb is plotted against solar longitude (Ls ) which is the angular position of a planet or asteroid in its orbit with respect to its seasons. Ls = 0◦ corresponds to the position of northern vernal equinox. For Vesta, perihelion and aphelion correspond to Ls values of 215 and 35◦ , respectively, shown schematically in Fig. 1. The results in Figs. 8 and 9 are calculated with a thermal model that solves the seasonal and diurnal cycles. The viewing aspects of the apparent Tb in Fig. 8 are for an observer at zero solar phase, i.e. viewing Vesta from the same direction as solar illumination. A series of Tb are calculated corresponding to a range of tan  values. Large tan  values have shorter absorption lengths. For a tan  of 0.01 and a refractive index of 1.7, the absorption length is 8.1 mm for a 870 µm photon. The same thermal model results and same series of tan  are used to calculate Tb in Fig. 9. However, in this figure the viewing aspects are adjusted to match those of each observing epoch for which we have with reasonable absolute calibrations. The Ls is used to position the results from each epoch; there is a different sub-observer latitude and solar phase angle with each epoch which also affect the Tb observed. Also included in Fig. 9 are the average Tb observed from each epoch with bars to indicate the range associated with each filtered lightcurve. Fig. 9 shows that there is a greater variation in the Tb observed than can be produced with the models developed so far. In Fig. 8 at low values of tan , thermal radiation samples deeper into the seasonal wave where there is less temperature variation over the seasons. At intermediate tan  values, radiation samples the diurnal wave and two effects become apparent. Firstly, the Tb becomes sensitive to heliocentric distance; Vesta starts to appear warmer when closer to the Sun. Recall that as-

456

M.A. Chamberlain et al. / Icarus 192 (2007) 448–459

Fig. 8. Response of Tb to Vesta’s seasons for different values of tan . The Tb shown are for an observer at zero solar phase angle.

Fig. 9. Vesta Tb as a function of tan  for an observer with the same viewing aspect as the indicated observing epochs. Observed Tb are shown; arrows indicate the range of Tb observed over the filtered lightcurve. Lines are used to link model results between different epochs and do not represent systematic variations as a function of solar longitude.

teroid surfaces have very low thermal inertias and temperatures on the surface are close to equilibrium temperatures which are independent of properties below the surface. However, these temperatures do not propagate far into the surface. Secondly, Tb varies several degrees seasonally due to the sub-solar latitude. When viewing from zero solar phase angle, submillimeter Tb is higher at high absolute values of sub-solar latitudes. At the limit of when the sub-solar point is at the pole, then the Tb would be the same as derived for a non-rotating model when the temperature profile is constant and equal to the equilibrium temperature. At high tan  values shown in Fig. 8, the heliocentric distance effect (which is single peaked with solar longitude) increases, dominating the sub-solar latitude effect (double peaked). Tb also increases and approaches equilibrium temperatures as tan  increases. The model results that have been presented in this and previous sections show Tb of a spherical asteroid. These results are used to show the effects of various surface material properties

on thermal fluxes, and are not being fitted to data at this time. By using a spherical model, results shown do not include effects due to shape. For Vesta, these shape effects are minimal. Shape effects would become more significant for asteroids that are less spherical or for asteroids with higher obliquities than Vesta. Ellipsoidal effects modify the lightcurves and seasonal changes observed in thermal flux. Vesta rotates about its short c axis, so the lightcurve is sensitive to the difference in the a and b axes. Seasonal changes in thermal flux are sensitive to the oblateness. To quantify the scale of shape effects for Vesta, we calculated the thermal response of a non-rotating ellipsoid when viewed/illuminated from different directions. The model albedo is 0, emissivity 1, heliocentric distance 2.4 AU and the shape is from the note in Thomas et al. (1997b), 289 × 280 × 229 km. This is similar to the non-rotating model used by Redman et al. (1998), only here the shape is an ellipsoid. Note that submillimeter Tb of non-rotating bodies are higher because of higher temperatures at depth, hence changes in Tb are larger. The ellipsoidal shape of an asteroid affects the thermal flux observed in two ways, firstly by changing the cross-sectional area observed and secondly the degree of ‘flattening’ on the observed/illuminated surface, only the later affects the Tb . In the case of a thermal lightcurve (viewed and illuminated over the equator) the amplitude due to Vesta’s shape is 3.9%, of which 3.2% is due to cross sectional area and 0.6% due a change in thermal intensity (corresponding to a 1.3 K increase in Tb ). The thermal flux from an oblate spheroid with Vesta’s dimensions, when illuminated/viewed from 30◦ above or below the equator, would be increased by 7.7% relative to an equatorial view. This would be associated with an increase in cross-sectional area by 6.5%, and in thermal intensity of 1.0% (corresponding to a 2.3 K increase in Tb ). Note that the model results in figures shown are in Tb , which are demonstrated here to be relatively insensitive to Vesta’s shape. However, for asteroids with more irregular shapes or higher obliquities, shape effects would need to be considered in the treatment of Tb as well as observed thermal fluxes. 4. Discussion Tb is sensitive to many properties, as illustrated in Fig. 6. No single set of properties has yet been found to match the Tb from all epochs. The fit between some epochs may be improved selectively. Comparing the Tb observed to model results in Fig. 9, it appears that a high tan  value in the range of 0.02–0.05 would be sufficient to match the Tb in the epochs May 2003 and May 2004, though clearly this model would not reproduce the Tb from January 2003. While it may be attractive to dismiss the January 2003 in favor of the two consistent epochs, we do not consider this a prudent thing to do. The Tb for the January 2003 epoch is the average of many integrations over two days, including several independent calibration values. On this basis, the January 2003 should be more reliable than the May 2004 which is the average of a single set of integrations. Likewise, we consider the Tb from May 2003 to be a reliable value.

Submillimeter lightcurves of Vesta

The increase in Tb from January 2003 to May 2003 is evidence of a surface with a low tan . High tan  values would decrease the Tb in May 2003 due to the higher contribution of surface temperatures to the submillimeter Tb and that the cooler morning terminator was viewed in the May 2003. We note that the change in the Tb from January 2003 to May 2003 in models is less than the change in the observed Tb suggesting that there are components still to be included in the models. Properties such as surface roughness, which has been important in modeling mid-infrared observations (Spencer, 1990), may resolve these discrepancies. These ideas will be incorporated into future work. At this time we refrain from matching model parameters to data while there is still considerable uncertainty in the data. We are, however, encouraged that the initial estimates of the parameters produced model Tb that were within the range of Tb observed. None of the models yet presented offer a comprehensive explanation of the submillimeter thermal radiation of Vesta. However, these models do demonstrate some processes that contribute to the observed variations. We have been able to reproduce the observed thermal flux densities at 870 µm, to within a few percent, with reasonable values of thermal inertia, tan  and Kr in a relatively simple thermal model. We do not need to use esubmm values as low as 0.6 that have been cited in the literature (see discussion below). For comparison, the esubmm associated with the Kr we use is 0.93. We define the esubmm value as the efficiency with which a surface is able to emit thermal radiation at this wavelength. Note that esubmm does not have to be related to the mid-infrared emissivity which is important to the surface temperature. Thermal fluxes in the submillimeter are much lower than in the mid-infrared from asteroid surfaces, so variations in esubmm do not affect the energy balance, or temperature, of the surface. Previous workers have usually recognized that cooler material below the asteroid surface may contribute to, and reduce, the thermal flux observed at wavelengths ∼1 mm. However, emissivity is still usually calculated as a ratio of the flux observed to the flux calculated with only surface temperatures over the asteroid. Emissivity and models have been used a number of different ways in the literature, and in some cases, emissivity is treated as an observable quantity. Redman et al. (1998) compared the Tb observed to that of a non-rotating model, which is the one model in which the temperatures at depth are the same as surface temperatures. Redman et al. found that the “effective emissivity” varied from 1.4 to 0.3 from the mid-infrared to the centimeter wavelengths. Webster and Johnston (1989) and Altenhoff et al. (1994) find emissivity values by comparing Tb observed to those obtained with a rapid-rotating model. It is not clear why temperatures at depth should be expected, a priori, to be those of rapid-rotator, as typically this is the model applied to bodies with very rapid rotation, high thermal inertia, or a high thermal parameter (as defined by Spencer et al., 1989). Temperatures at depth may be expected to be the average diurnal temperature of the surface, which for low thermal inertia surfaces that are typical of asteroids, can be much lower than average temperatures of rapid rotators. However, rapid rotator values were found to work well for Webster and Johnston (1989) and Altenhoff et al. (1994) who find esubmm ∼ 1 at

457

λ ∼ 1 mm, and esubmm ∼ 0.8 at λ of a few centimeters. Müller and Lagerros (1998) use a detailed thermal model that includes the observed shapes of asteroids, surface roughness and thermal inertia, and is used to find thermal fluxes from the mid-infrared out to the millimeter just with surface temperatures. Note that the absorption length of photons in the infrared is much shorter, if the L = 10λ guide is used. Hence, infrared photons are only emitted from depths well within δd , so these only sample the surface temperature and these models reproduce the fluxes observed with reasonable values of emissivity (∼0.9). However, at millimeter wavelengths, low esubmm of ∼0.6 are required to match the observed Tb with the approach of Müller and Lagerros (1998). The difference with our model is that it integrates with depth to calculate thermal radiation, as has been applied to lunar thermal radiation at radio wavelengths (e.g., Keihm and Langseth, 1975). This integration samples through to below the diurnal thermal wave where the physical temperatures are cooler than on the surface when observing the illuminated side of the asteroid. It is possible to determine which approach of calculating submillimeter and millimeter thermal fluxes is most appropriate by observing the sensitivity of thermal radiation to phase angle. If thermal radiation is only from the surface and modified by the surface emissivity, there would a large variation in flux as an observer views an asteroid from different angles with a maximum flux when observing near zero phase. In contrast, if thermal radiation is from depth, the variation in flux with observer direction will be much smaller because a substantial portion of the thermal flux comes from below the diurnal thermal wave. There is another lightcurve at a wavelength ∼1 mm in the literature; Redman et al. (1992) show a 1-mm lightcurve by combining 0.8 and 1.1 mm data. While the data is noisy, Redman et al. (1992) fit two Fourier components to these data with magnitudes that are different from zero by 3.4σ . The amplitude of the lightcurve was ∼10% and this was stated to be consistent with a lightcurve induced by shape only. However, the shape model used at the time had a larger difference between the a and b axes relative to the most recent shape model derived with HST images (note added in Thomas et al., 1997b) or adaptive optics (Drummond et al., 1998), even a 10% amplitude is now inconsistent with a shape-induced lightcurve. The Redman et al. (1992) lightcurve has a smaller amplitude than most of the lightcurves presented here. This may just be due to the position of Vesta. Redman et al. (1992) data were taken in September 1989, when the true anomaly of Vesta was 42◦ , to which our May 2004 data is the closest (Table 1 and Fig. 1). May 2004 is also the epoch in which we observed the smallest lightcurve amplitude. Unfortunately, our data coverage was least in this epoch and little more can be said with confidence. At this position, Vesta’s southern hemisphere is observed while our other epochs observe the northern hemisphere (see sub-observer and sub-solar latitudes in Table 1). It appears that the southern hemisphere of Vesta is more longitudinally homogeneous than the north, and hence amplitude of the lightcurve is not as great when viewing the south. There may also be properties in the south that increase the Tb here, either larger tan  or

458

M.A. Chamberlain et al. / Icarus 192 (2007) 448–459

thermal inertia, relative to the north and this is part of the reason for the high Tb observed. These anomalous properties could be associated with the large impact structure identified in the south (Thomas et al., 1997a). The quoted Tb at 1 mm in Redman et al. (1992) is cooler than our results (∼140 K compared to our 161 K in May 2004). Part of this difference may be attributed to the difference in the size of Vesta assumed, Redman et al. (1992) used a diameter of 520 km which gives an area 1.5% larger than the area when observing the Vesta ellipsoid from a direction 17◦ below the equator (the direction of the Redman et al. observation). This would increase the Redman et al. (1992) by 2 K. The Redman et al. Tb is also cooler due to the different aspect. The phase angle of the Redman et al. (1992) data was +25◦ (viewing the morning terminator) and from a direction 17◦ S, relative to a phase of −24◦ (evening terminator) and 26◦ S during our May 2004 observations. The Redman et al. aspect appeared 5 K cooler when Tb was calculated with our billiard ball model from these two aspects with typical properties (Table 3). This difference due to aspect could be enhanced by increasing the tan , which increases the weighting of surface temperatures in the Tb , though then other properties would need to be modified. We defer this to a later time when better data may be available. Also, different properties over the southern impact feature or details of the Vesta shape may explain the Tb difference. These effects have yet to be explored. Disk-resolved images from the HST allow us to see if there are any features in the northern hemisphere that would explain the pronounced lightcurve amplitudes when viewing Vesta’s north. While there are features with lower albedo on the surface of Vesta (Binzel et al., 1997), these appear to be centered on the equator and would then be observed from all aspects. The HST false color ‘geological map’ shows an anomalous region close to the low albedo region but centered 10 to 20◦ north of the equator. Mafic absorption bands near 1 µm are shallower in this region. This region may also be associated with anomalous thermophysical properties to produce the thermal lightcurves observed that have larger amplitudes when observing Vesta from the north. Data in thermal lightcurves presented here do not resolve the disk of Vesta, but data are most sensitive to the material properties of the sub-observer latitude. The longitude range of this weak mafic band region extends from 60 to 140◦ W, which coincides with position of the maximum in the thermal lightcurve, as observed in most epochs. There is also an association between the submillimeter and visible lightcurves. Two visible lightcurves are shown in Fig. 10, which were chosen to sample a range of sub-observer and sub-solar latitudes comparable to our submillimeter observations. Data in the lightcurves are from the photometric catalogue Lagerkvist et al. (1984) and the references in the caption. The visible lightcurves are plotted with respect to the sub-phase bisector longitude, which is the direction between the Sun and the observer from the asteroid, as defined by Harris et al. (2001). Note that unlike most asteroids, whose lightcurves are dominated by shape, Vesta’s lightcurve is single peaked due to albedo features on the surface. For a time, it was uncertain whether Vesta had a 5- or 10-h period. Taylor (1973), who was

Fig. 10. Sample visible lightcurves of Vesta from Taylor (1973) (A) and Taylor et al. (1985) (B). The second date in the title is the epoch of the observations. The sub-solar and sub-observer latitudes were 18◦ N and 9◦ N respectively for (A), and 27◦ S and 25◦ S for (B). The different symbols in each panel represent data from different nights.

the source for panel A of Fig. 10, argued that the period was 10 h. However, the data to support this 10-h period and two distinct maxima were from a ‘southern hemisphere aspect’ while data with equatorial aspects, like the data we use, did not show such distinct maxima. The minimum in the lightcurve corresponds to the position of the low albedo features identified in HST images and the maximum in the submillimeter thermal lightcurve. While low albedo regions have warmer temperatures due to increased absorption of insolation, albedo alone is clearly insufficient to explain the changes in thermal flux observed. The 10% variation in albedo (0.40 to 0.44) corresponds to a 6.7% change in the absorption of sunlight. Hence the change in equilibrium temperature, and submillimeter flux √ density, is 1.6% ( 4 1.067). Like in the submillimeter, the amplitude of the visible lightcurves is greater when the sub-solar and sub-observer latitudes are ∼10◦ N with respect to the amplitudes when these latitudes are ∼27◦ S. 5. Conclusions While there is still considerable uncertainty in the Vesta submillimeter data, there is enough data to resolve a thermal lightcurve with an amplitude that is clearly inconsistent with shape or albedo influences alone. We propose that the observed high-amplitude thermal lightcurve is due primarily to variations in thermophysical properties across the asteroid surface. Evidence that supports the reality of the lightcurve amplitude includes the size of the amplitude with respect to the uncertainty in the data points and also the similarity in the lightcurves be-

Submillimeter lightcurves of Vesta

tween epochs (primarily between January 2003 and May 2003). There is also evidence that the submillimeter lightcurve shape and amplitude is a function of solar phase angle and/or asteroid season. We used simple thermal models to explore effects of some thermophysical properties. We find that we can reproduce the Tb observed using reasonable values of thermal inertia, tan  and Kr without having to impose a low value of emissivity at these wavelengths. High-amplitude submillimeter lightcurves can be produced with regions of different properties on the surface of Vesta. Submillimeter thermal lightcurves also change in non-intuitive ways, as shown in Fig. 7. Regions on Vesta’s surface with different properties respond differently to phase angles and/or seasons. This is certainly worthy of further investigation, ideally with more lightcurve observations to characterize these effects. The constraints to models of Vesta submillimeter radiation are limited by the restricted observational coverage and noisy data. Since the flux response with phase and season vary for different properties, more observations sampling various phases and seasons could potentially constrain material properties. In addition, since thermal emission originates from approximately a few to ten wavelengths beneath the surface, simultaneous multi-wavelength observations can constrain the properties with depth. Ideally, simultaneous, well-calibrated multi-wavelength observations (e.g. mid-infrared to centimeter) would be conducted over a full rotation, at several phase angles, and even over different asteroid seasons, in order to constrain the distribution of materials across the surface. Best seasonal coverage with ground-based observatories will require using facilities in both northern and southern hemispheres. This research would also benefit from laboratory measurements of Kr and tan  of candidate asteroid materials at these wavelengths/ frequencies. Acknowledgments This work was supported by the NASA Planetary Astronomy Program, Grant NNG05GF37G (PI MVS). We would like to thank the staff and telescope operators at the Heinrich–Hertz Submillimeter Telescope for assistance obtaining the data presented here. Thanks also to R.O. Redman and J.D. Drummond for helpful reviews that improved the manuscript. References Altenhoff, W.J., Johnston, K.J., Stumpff, P., Webster, W.J., 1994. Millimeterwavelength observations of minor planets. Astron. Astrophys. 287, 641– 646. Barrera-Pineda, P.S., Lovell, A.J., Schloerb, F.P., Carrasco, L., 2002. Variability of the thermal emission of Vesta at 870 µm. Bull. Am. Astron. Soc. 34, 859. Binzel, R.P., Gaffey, M.J., Thomas, P.C., Zellner, B.H., Storrs, A.D., Wells, E.N., 1997. Geologic mapping of Vesta from 1994 Hubble Space Telescope images. Icarus 128, 95–103. Bowell, E., Hapke, B., Domingue, D., Lumme, K., Peltoniemi, J., Harris, A.W., 1989. Application of photometric models to asteroids. In: Binzel, R.P.,

459

Gehrels, T., Matthews, M.S. (Eds.), Asteroids II. Univ. of Arizona Press, Tucson, AZ, pp. 524–556. Chamberlain, M.A., Boynton, W.V., 2007. Response of martian ground ice to orbit-induced climate change. J. Geophys. Res. 112, doi:10.1029/ 2006JE002801. E06009. Chamberlain, M.A., Sykes, M.V., Esquerdo, G.A., 2007. Ceres lightcurve analysis—Period determination. Icarus 188, 451–456. de Pater, I., 1999. The Solar System at radio wavelengths. In: Weissman, P.R., McFadden, L.A., Johnson, T.V. (Eds.), Encyclopedia of the Solar System. Academic Press, San Diego, CA, pp. 524–556. Drummond, J.D., Fugate, R.Q., Christou, J.C., Hege, E.K., 1998. Full Adaptive Optics images of Asteroids Ceres and Vesta; rotational poles and triaxial ellipsoid dimensions. Icarus 132, 80–99. Harris, A.W., Carlsson, M., Young, J.W., Lagerkvist, C.I., 2001. The lightcurve and phase relation of the Asteroid 133 Cyrene. Icarus 58, 377–382. Harris, A.W., Mueller, M., Delbó, M., Bus, S.J., 2005. The surface properties of small asteroids: Peculiar Betulia—A case study. Icarus 179, 95–108. Hoffmann, M., Geyer, E.H., 1993. Spots on 4-Vesta and 7-Iris—Large areas or little patches. Astron. Astrophys. Suppl. 101, 621–627. Keihm, S.J., Langseth, M.G., 1975. Lunar microwave brightness temperature observations reevaluated in the light of Apollo program findings. Icarus 24, 211–230. Lagerkvist, C.I., Piironen, J., Carlsson, A., 1984. Asteroid Photometric Catalogue, fifth update. Uppsala Univ. Press, Uppsala. Lagerros, J.S.V., 1998. Thermal physics of asteroids. IV. Thermal infrared beaming. Astron. Astrophys. 332, 1123–1132. Mitchell, D.L., Ostro, S.J., Hudson, R.S., Rosema, K.D., Campbell, D.B., Velez, R., Chandler, J.F., Shapiro, I.I., Giorgini, J.D., Yeomans, D.K., 1996. Radar observations of Asteroids 1 Ceres, 2 Pallas, and 4 Vesta. Icarus 124, 113–133. Müller, T.G., Barnes, P.J., 2007. 3.2 mm lightcurve observations of (4) Vesta and (9) Metis with the Australia Telescope Compact Array. Astron. Astrophys. 467, 737–747. Müller, T.G., Lagerros, J.S.V., 1998. Asteroids as far-infrared photometric standards for ISOPHOT. Astron. Astrophys. 338, 340–352. Ostro, S.J., Hudson, R.S., Rosema, K.D., Giorgini, J.D., Jurgens, R.F., Yeomans, D.K., Chodas, P.W., Winkler, R., Rose, R., Choate, D., Cormier, R.A., Kelley, D., Littlefair, R., Benner, L.A.M., Thomas, M.L., Slade, M.A., 1999. Asteroid 4179 Toutatis: 1996 Radar observations. Icarus 137, 122–139. Redman, R.O., Feldman, P.A., Matthews, H.E., Halliday, I., Creutzberg, F., 1992. Millimeter and submillimeter observations of the Asteroid 4 Vesta. Astron. J. 104, 405–411. Redman, R.O., Feldman, P.A., Matthews, H.E., 1998. High-quality photometry of asteroids at millimeter and submillimeter wavelengths. Astron. J. 116, 1478–1490. Spencer, J.R., 1990. A rough-surface thermophysical model for airless planets. Icarus 83, 27–38. Spencer, J.R., Lebofsky, L.A., Sykes, M.V., 1989. Systematic biases in radiometric diameter determinations. Icarus 78, 337–354. Taylor, R.C., 1973. Minor planets and related objects. XIV. Asteroid (4) Vesta. Astron. J. 78, 1131–1139. Taylor, R.C., Tapia, S., Tedesco, E.F., 1985. The rotation period and pole orientation of Asteroid 4 Vesta. Icarus 62, 298–304. Tedesco, E.F., Noah, P.V., Noah, M., Price, S.D., 2004. IRAS Minor Planet Survey. IRAS-A-FPA-3-RDR-IMPS-V6.0. NASA Planetary Data System. Thomas, P.C., Binzel, R.P., Gaffey, M.J., Storrs, A.D., Wells, E.N., Zellner, B.H., 1997a. Impact excavation on Asteroid 4 Vesta: Hubble Space Telescope results. Science 277, 1492–1495. Thomas, P.C., Binzel, R.P., Gaffey, M.J., Zellner, B.H., Storrs, A.D., Wells, E., 1997b. Vesta: Spin pole, size, and shape from HST images. Icarus 128, 88–94. Webster Jr., W.J., Johnston, K.J., 1989. On the wavelength dependence of apparent emissivity of asteroid microwave emissions—Ceres and Vesta. Publ. Astron. Soc. Pacific 101, 122–125.