Asteroid photometric and polarimetric phase curves: empirical modeling

Available online at www.sciencedirect.com R

Icarus 161 (2003) 34 – 46

www.elsevier.com/locate/icarus

Asteroid photometric and polarimetric phase curves: empirical interpretation S. Kaasalainen,a J. Piironen,a M. Kaasalainen,a A.W. Harris,b K. Muinonen,a,c and A. Cellinoc a

Observatory, P.O. Box 14, 00014 University of Helsinki, Finland b Jet Propulsion Laboratory, Pasadena, CA 91109, USA c Astronomical Observatory of Torino, via Osservatorio 20, I-10025 Pino Torinese, Italy Received 23 April 2002; revised 19 August 2002

Abstract A method for interpretation of asteroid phase curves, based on empirical modeling and laboratory measurements, is outlined and preliminary results are presented. A linear– exponential function is used to describe the opposition peaks and negative polarization surges of various asteroids and laboratory samples and a statistical algorithm is used in parameter estimation. The linear– exponential function describes well the phase curves, but dense phase angle coverage, particularly at small phase angles must be obtained to improve the results. Major emphasis should also be put on laboratory study: with an extensive library of laboratory measurements, a stronger connection between the phase curve properties and surface characteristics is possible. © 2003 Elsevier Science (USA). All rights reserved. Keywords: Asteroids; Meteorites; Photometry; Polarimetry

1. Introduction The opposition effect is a nonlinear brightness increase near zero phase angle viewing geometry. Observed and measured brightness peaks exist for numerous solar system objects and various laboratory samples (Muinonen et al., 2002a; Kaasalainen et al., 2001; Belskaya and Shevchenko, 2000; and references therein). Recent interpretations of the opposition effect are based on two scattering mechanisms: shadowing, which is caused by the regolith particles and rough features hiding their shadows in the backscattering direction, and coherent backscattering (Mishchenko and Dlugach, 1992; Muinonen, 1994), which is constructive interference between two electromagnetic waves, propagating in the random medium in reversed paths. The reversal of the degree of linear polarization to negative near zero phase angle is also suggested to be due to coherent backscattering. Negative linear polarization has also been observed for asteroids, rings, and satellites (Rosenbush et al., 1997 and references therein), but extensive laboratory measurements of linear polarization near zero phase angle are quite rare. Several light scattering and radiative transfer functions

have been used in the modeling of the photometric opposition peak, based on either or both of the scattering mechanisms (Bowell et al., 1989; Muinonen et al., 2002a). A scalar approximation of coherent backscattering has been studied more widely than the vector approach that has not been available until recently (Muinonen 2002, Mishchenko et al., 2000). The vector approach is essential in estimating the amplitudes of the peaks and in the physical study of the polarization opposition effect. The two important features of the opposition surge, with respect to its interpretation, are the amplitude and angular width of the opposition peak, since the shape of the opposition surge depends on the physical properties of the surface. The polarimetric properties also seem to be characteristic of the surface material, the corresponding key parameters being the amplitude and width of the polarization opposition effect. (The slope of the linear part of the phase curve is related to the darkness of the surface material (cf. Lagerkvist et al., 1990), but the main emphasis of this work is laid on the interpretation of peak parameters rather than the linear part.) To get a reliable approximation for either of the key parameters, a substantial number of data

0019-1035/03/$ – see front matter © 2003 Elsevier Science (USA). All rights reserved. PII: S 0 0 1 9 - 1 0 3 5 ( 0 2 ) 0 0 0 2 0 - 9

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

points must be measured, with a reasonable accuracy. Otherwise, interpretation of the opposition peak by means of either comparison with laboratory data or modeling does not produce conclusive results. Even with a successful fit of a scattering model or function, the resulting parameters are not unambiguous, and do not necessarily represent the investigated surface, at least with the accuracy required for reliable surface type characterization. This is almost without exception the case for datasets of only few data points, that do not span the distinguishable parts (i.e., the nonlinear part near zero phase) of the phase curve so that the features of the curve could be determined with any method of fitting. Therefore, systematic approach to data acquisition, both telescopic and laboratory, and evaluation of the goodness of the fitting procedure are essential for establishing a reliable method for remote sensing with the aid of phase curves. A simple mathematical function that reproduces the phase curves in a unique way is useful in the qualitative classification and comparison of the phase curve properties. The function has proven a useful tool in analysis, for which physical models are often too complicated and can lead to ambiguous results (see, e.g., M. Kaasalainen et al., 2001 and Mallama et al., 2002). Perhaps a wide range (even from 0° up to 180°) of scattering angles is always needed for effective physical modeling. Furthermore, many of the scattering models need a numerical approach (e.g., Monte Carlo simulation), making them more difficult to apply in a simple feature like an opposition peak. (The major advantage of the physical models is, however, that the parameters are more easily restricted into a “realistic” range.) In this paper we present a four-parameter linear– exponential function that is used to fit asteroid phase curves and polarimetric data available in the literature. The function is also applied to a set of laboratory experiments. An empirical (mathematical) model (combined with a statistical approach, i.e., search for best solution with the aid of probability distribution) can describe the phase curves better and more reliably than the complicated scattering models, and the problems described above can be avoided. 2. Empirical modeling We make use of a four-parameter linear– exponential function for semiempirical modelling of both polarimetric and photometric phase curves (Piironen et al., 2000; Kaasalainen et al., 2001; M. Kaasalainen et al., 2001; Muinonen et al., 2002b; also cf. Nelson et al., 2000):

冉 冊

f共 ␣ 兲 ⫽ a exp ⫺

␣ ⫹ b ⫹ k␣. d

(1)

For photometry, f(␣) is the relative intensity, a is the height and d the width of the opposition peak, and b is the background intensity. For polarimetry, f(␣) is the degree of linear polarization, a can be described as an amplitude coefficient, d the angular scale of the polarization surge, and

35

b is the balancing amplitude coefficient. In both cases, k is the slope of the linear part, and ␣ is the phase angle. (Note that the shape of the phase curve can be described with only three parameters here.) To investigate the goodness of the fits of Eq. (1), statistical analysis is done with the aid of the probability density p for the parameters P ⫽ (a, b, d, k) (see Muinonen et al., 2002b). The probability density for the two parameters a and d is obtained by integration with respect to b and k:

冕冕 k2

p共a, d兲 ⬀

k1

b2

dbdk 冑det A exp关⫺␹ 2/ 2兴.

(2)

b1

Here the matrix A contains the partial derivatives with respect to the four parameters. ␹2 is the obtained minimum of the least-squares fit of the parameters. The method has been applied for photometry and polarimetry of Ceres in Muinonen et al. (2002b). A preliminary application was used for satellite photometry by Kaasalainen et al. (2001).

3. Observations and measurements Both astronomical and laboratory data have been used in this analysis. The polarimetric phase curves analyzed here were obtained from the Small Bodies Node at NASA’s Planetary Data System. References for the photometric data for each asteroid are in Table I. The original data are also found in Lagerkvist et al. (2001) (see also http://www. astro.uu.se), except for 51 Nemausa, which are from the observations in 1983 by A. H., tabulated in Table II. The polarization data are analyzed as such. At the first run of the probability distribution integration, a measurement error of ⬍0.2 (% polarization) has been used. After that, the distributions were computed again, using the RMS error of the best fit as a measurement error approximation. For photometry, the magnitudes were first transformed to intensities (with I ⫽ 10(⫺0.4xm)), and a relative measurement error of 4% (a combination of approximated instrument error and the uncertainty of the absolute magnitude determination) has been applied to describe the goodness of the data. Not all the data in the databases were included in the analysis: many of the phase curves contained an insufficient number of data points for analysis, others contained no points at small phase angles, or were too noisy for the integration to converge. Some data of this kind, however, were included in the analysis as examples of too sparse datasets. The laboratory data were obtained with a device developed at the Observatory of the University of Helsinki. The experimental setup consists of a 632.8-nm He–Ne laser, polarizing plates, a beamsplitter to reach the smallest phase angles, and a CCD detector. Phase curves are analyzed for a set of meteorite rocks and aluminium oxide abrasive powder with a l-␮m mean grain radius. Four different meteorite rocks were compared: Bjurbo¨le and Buschhof (both olivine– hypersthene chondrites), Allende (carbonaceous

36

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

Table 1 Asteroid characterization: taxonomic types, G-values, albedos, diameters (Lagerkvist et al. 2001), photometric phase curve properties a⫹b ( ⫽ peak size, d ⫽ peak width), and references to the data b

261 Prymno 24 Themis 144 Vibilia 146 Lucina 211 Isolda 165 Loreley 51 Nemausa 44 Nysa 64 Angelina 419 Aurelia 335 Roberta 1 Ceres 19 Fortuna 16 Psyche 22 Kalliope 69 Hesperia 110 Lydia 1862 Apollo 6 Hebe 20 Massalia 29 Amphitrite

B C C C C CD CU E E F FP G G M M M M Q S S S

0.11 0.19 0.06 0.05 0.06 0.08 0.09 0.55 0.48 0.05 0.06 0.11 0.10 0.12 0.14 0.14 0.18 0.09 0.27 0.21 0.18

0.10 0.01 0.06 0.05 0.06 0.07 0.09 0.49 0.01 0.04 0.05 0.10 0.01 0.10 0.12 0.12 0.17 0.25 0.19 0.16

52 510 146 136 147 160 152 73 389 132 93 910 498 263 186 142 88 8 191 150 219

a⫹b b 1.40 1.37 1.34 1.29 1.34 1.20 1.48 1.30 1.18 1.17 1.63 1.47 1.30 1.52 1.46 1.37 1.43 3.70 1.48 1.41 1.47

33 Polyhymnia

S

0.33

0.01

259

1.51

1.10

60 Echo 133 Cyrene

S SR

0.25 0.26

0.15 0.21

61 69

1.54 1.42

2.3 1.4

Asteroid

Type

G

Albedo

D (km)

d (°)

Reference

1.1 3.5 1.7 2.1 2.6 1.7 1.5 1.05 4.1 1.25 2.1 3.7 2.6 3.1 2.7 2.1 2.3 26.0 2.4 2.05 3.10

Harris et al. 1989b Harris et al. 1989 Harris et al. 1989c Harris et al. 1989c Harris et al. 1989c Harris et al. 1992 See Table 2 Harris et al. 1989b Harris et al. 1989b Harris et al. 1989c Harris et al. 1992 Tedesco et al. 1983 Lupishko et al. 1981 Lupishko et al. 1982 Sealtriti et al. 1978 Poutanen et al. 1985 Taylor et al. 1971 Harris et al. 1987 Gehrels & Taylor 1977 Gehrels 1956 Tedesco et al. 1981, Lupishko et al. 1981 Zappala et al. 1982, Harris et al. 1989c Harris et al. 1989b Harris et al. 1989c

Note. The error estimations for each parameter can be best approximated graphically from the plotted distributions.

chondrite), and Ochansk (olivine– bronzite chondrite). The device and the samples are described in more detail in Kaasalainen et al. (2002).

4. Results 4.1. Feasibility The computed probability densities for the data analyzed (Figs. 1–13) are descriptive of the number and angular distribution of data points, as well as their accuracy. For Table 2 Photometric data for 51 Nemausa Phase angle (°)

V0 magnitude

20.96 17.97 17.69 8.00 1.22 0.83 0.46 8.67 9.11

8.367 8.287 8.277 7.894 7.492 7.447 7.407 7.937 7.960

data that have not too many gaps in phase angle, and contain data at small phase, the distributions are closer to a focused rounded (Gaussian-like) shape. Many of the phase curves analyzed here, however, lack data points in significant ranges of phase angle, which is the major contributor to the shape of the probability distributions and the obtained parameter values. It is not easy to get an unambiguous mathematical determination for a good dataset. Qualitatively speaking, there must be a number of data points near zero phase angle, reasonably many points in the curved region and the linear part of the phase curve, no large gaps, etc. All these features have their own effect in the goodness of the fit. As an example, three different phase curves are shown for 29 Amphitrite in Fig. 1. The first one has been taken from Lagerkvist et al. (2001), and corrected for geometry and shape using the rotation and shape model in M. Kaasalainen et al., 2002 (see also M. Kaasalainen et al., 2001 and references therein). The second one is from Tedesco and Sather (1981), and the third dataset is a combination of the second one and another one by Lupishko et al. (1981). These three have produced somewhat different results: the peak sizes ((a ⫹ b)/b) are 1.69, 1.51, and 1.47, and widths (d) are 4.6°, 2.8°, and 3.1°, respectively. It must be noted that the first two sets do not have data points near zero

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

37

Fig. 1. Probability densities (arbitrary units, normalized to unity) in the same xy-scale (top) and photometric phase curves (bottom) for 29 Amphitrite, compiled from different light curves with different correction. The linear– exponential function with parameters corresponding to the best fit is plotted with the data.

phase angle, contrary to the third one, which has produced a reasonably good probability distribution. A similar comparison was carried out for two datasets of 22 Kalliope (Fig. 2; the data are from Lagerkvist et al. (2001) and Scaltriti et

al. (1978)). In this case, there are more points near zero phase, and the results agree better for these two, (a ⫹ b)/b being 1.42 and 1.46, and d 3.2° and 2.7°, respectively. This example shows that one must be careful with further con-

Fig. 2. Probability densities in the same xy-scale (top) and photometric phase curves (bottom) for 22 Kalliope, compiled from different light curves with different correction. The linear– exponential function with parameters corresponding to the best fit is plotted with the data.

38

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

Fig. 3. Probability densities (arbitrary units, normalized to unity) in the same xy-scale (top) and photometric phase curves (bottom) for simulated phase curves of 50 (left and middle) and 20 (right) data points. For samples 2 and 3, a majority of points have been concentrated on the first 2 degrees of phase angle. The linear– exponential function with parameters corresponding to the best fit is plotted with the data.

clusions or classifications when there are no data points near zero phase. Consistent correction for the viewing and illumination geometry (and the object’s macroscopic shape) also affects the final shape of the opposition peak.

The required accuracy and number of data points has been studied with three sets of simulated phase curves, where the intensities have been computed with the parameters a ⫽ 0.001, d ⫽ 1.5, b ⫽ 0.004, k ⫽ ⫺1.2 ⫻ 10⫺4 for

Fig. 4. Probability densities in the same xy-scale (top) and photometric phase curves (bottom) for 133 Cyrene (left), SR-type, and 6 Hebe (right), S-type. The linear-exponential function with parameters corresponding to the best fit is plotted with the data.

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

39

Fig. 5. As in Fig. 4 for 16 Psyche and 110 Lydia, both M-type.

randomly generated phase angles ␣, plotted in Fig. 3. A 2% random uncertainty has been introduced to the phase curves, and the probability distributions have been computed sim-

ilarly as for the asteroid and meteorite samples. The first phase curve had 50 points randomly distributed over the range from 0° to 10°. A gap near zero phase has made the

Fig. 6. As in Fig. 4 for 69 Hesperia (M-type) and 44 Nysa (E-type).

40

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

Fig. 7. As in Fig. 4 for 24 Themis (C) and 261 Prymno (B-type).

result clearly worse than that of the phase curve 2, for which the majority of the points are within the first two degrees. The third sample has only 20 points, but the result is better

than that for the first sample (with 50 points), since again the majority of the data points lie in the 2° range. The peak sizes ((a ⫹ b)/b) are 1.32, 1.25, and 1.24, the original value being

Fig. 8. As in Fig. 4 for 1 Ceres (G-type) and aluminum oxide, the distributions plotted in different xy-scales.

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

41

Fig. 9. As in Fig. 4 for Allende and Buschhof.

1.25 exactly. The peak widths (d) are 1.0°, 1.10°, and 1.16°, respectively. The peak sizes for the two datasets (2 and 3) with good coverage near zero phase have been quite well

reproduced, but even a better coverage or accuracy in data would be recommended for a more accurate widths of the peaks.

Fig. 10. As in Fig. 4 for Bjurbo¨ le and Ochansk.

42

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

Fig. 11. Probability densities in the same xy-scale (top) and polarization phase curves (bottom) for 18 Melpomene (left) and 8 Flora (right), both S-type. The linear– exponential function with parameters corresponding to the best fit is plotted with the data.

Fig. 12. As in Fig. 11 for combined S-type data and 3 Juno (also S-type), the distributions plotted at different y-scale. This is a good example of how the number and accuracy of the data points affect the shape of the distribution.

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

43

Fig. 13. As in Fig. 11 for combined M-type data and 55 Pandora (also M-type).

To conclude, a reasonable dataset for modelling should contain at least a few points per degree at small phase angles, spanning the exponential and curved part of the phase curve so that, e.g., the width and the height of the peak can be unambiguously determined. At larger phase angles the coverage need not be as dense as near zero, but they must enable the determination of the slope of the linear part. If an asteroid cannot be observed closer to opposition than 1°, other methods of surface study are perhaps more suitable. 4.2. The probability distributions The probability densities and phase curves for the best of the asteroid data and the laboratory samples are plotted in Figs. 2–13. The best fit parameters are tabulated in Tables I, III, and IV along with Tholen taxonomic classes, albedos, G

values, and diameters (Lagerkvist et al., 2001; Bowell et al., 1989). The phase curves corresponding to the best parameters all fit well to the observations, indicating that the phase curves can be reproduced with the linear– exponential function. Figures 2–10 show the photometric data and distributions. Generally, the M and S type asteroids tend to have slightly larger peaks than do the C (B) types. The small peak size, however, is in many cases explained rather by insufficient data, which tends to flatten out the fitted peaks and gives rise to the great variation in the peak widths. Plotting the peak sizes and widths versus albedo did not show any distinct functional behaviour for the asteroids investigated here. For the laboratory measurements, a general trend can be pointed out (Table III): the white aluminum oxide powder has produced a very sharp peak, and the black Allende chunk a very wide peak. Obtaining a few more data points

Table 3 Laboratory sample characteristics (Kaasalainen et al. 2002) Sample

Type

Surface characteristics

a⫹b b

d (°)

Allende Bjurbo¨ le Buschhof Ochansk

CV3 LL4 L6 H4

Black chunk, rough Sawed piece, medium gray Light gray chunk Dark gray chunk

1.39 1.29 1.19 1.32

1.5 1.35 0.65 1.10

White, 1 ␮m grain size

1.18

0.925

Al2O3

44

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

Table 4 Asteroid characterization as in table I (Lagerkvist et al. 2001) and polarimetric phase curve properties (a ⫽ amplitude, d ⫽ width) Asteroid 24 Themis 511 Davida 139 Juewa 44 Nysa 64 Angelina 1 Ceres 16 Psyche 55 Pandora M Combined 3 Juno 7 Iris 8 Flora 18 Melpomene 20 Massalia 30 Urania 39 Laetitia 40 Harmonia 192 Nausikaa 433 Eros S Combined

Tax. type

G

Albedo

D (km)

a (%-units)

d (°)

C C CP E E G M M

0.19 0.05 0.06 0.55 0.48 0.11 0.12 0.30

0.01 0.05 0.05 0.49 0.01 0.10 0.10 0.32

510 336 162 73 389 910 263 67

S S S S S S S S S S

0.24 0.28 0.24 0.22 0.21 0.17 0.29 0.24 0.23 0.46

0.22 0.21 0.22 0.22 0.19 0.13 0.29 0.20 0.21 0.01

243 202 141 147 150 104 158 110 107 78

6.0 9.2 8.0 0.8 5.2 6.6 6.0 2.0 2.45 2.25 3.5 2.6 3.1 1.85 3.6 3.1 3.5 3.1 2.6 2.05

5.9 8.2 10.0 3.7 1.9 7.5 1.8 5.0 5.8 3.5 10.6 8.7 8.75 5.5 5.2 10.4 10.3 6.5 11.4 5.8

would in many cases lead to more accurate classification of the peak size parameter a, which also depends crucially on the y-axis intersection with the linear part (⫽b). It must also be noted that the peak sizes computed in this way are always approximate, and have uses only in qualitative interpretation. The laboratory results presented here are preliminary. Extensive measurements are needed to search systematically for relations between, e.g., peak size and grain size/ porosity/albedo of the sample (see, e.g., Nelson et al., 2000, where the opposition effect for simulated regoliths is studied as a function of sample grain size), which can then be applied in the interpretation of asteroid phase curves. It must be considered though, that direct comparison between the disk-integrated and disk-resolved (laboratory) phase curves is seldom possible, since the former include shadowing effects at several roughness length scales not present in the latter (M. Kaasalainen et al., 2001). The macroscopic shape, however, does not have a strong effect on the peak shape at small phase angles. For the polarimetric data, the amplitude a is somewhat lower for the S-type asteroids, whereas the C- and E-groups show larger amplitudes, but no strong conclusions should be made. The polarization data are yet more defective near zero phase, and therefore estimates of peak widths are not feasible. There is a great need for polarimetry of all asteroids at small phase angles. No comparisons between photometric and polarimetric parameters of single objects have been made at this point, this being not justified with the quality of present data. Since most of the computed distributions are not focused and rounded shapes (see, e.g., 55 Pandora in Fig. 13), any standard method of error determination for the fitted param-

eters can not be considered suitable for the purpose; i.e., it is hard to find a single error estimate that would adequately describe all the distributions presented here. Any standard error determination produces errors that can be larger than the iteration step in the computation of the probability distribution, forcing one to round off the best parameter values obtained, reducing the accuracy of the parameter estimation procedure. The deviations of each parameter can be best approximated graphically from the plotted distributions. 4.3. Combined probability distributions The combined probability distributions were investigated in order to find out their possible uses for taxonomic purposes. Figures 12 and 13 show the probability distributions for the combined polarimetric data for S- and M-type asteroids. For photometric data, the joint distributions for Sand M-type asteroids are shown as products of the individually computed distributions in Fig. 14. For this, the computations were repeated for each asteroid, and the data were normalized to b ⫽ 1 in each case. The multiplied distribution for photometric data of four M-type asteroids is centered near the same point as the corresponding product for five S-type asteroids (see Fig. 14), indicating that no differences could be distinguished between the two taxonomic classes with the aid of combined distributions. (We also computed a multiplied distribution for four C-type asteroids, but that did not bring out any significant difference.) No clear connection between the model parameters and taxonomic classes can be made at this point. A larger number of good quality datasets are needed to determine whether the asteroids of similar taxonomic

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

45

Fig. 14. Left: the multiplied probability distribution for the photometric data of five S-type as teroids: 29 Amphitrite, 60 Echo, 6 Hebe, 20 Massalia, and 33 Polyhymnia. Right: the product of the individual distributions for four M-type asteroids: 69 Hesperia, 22 Kalliope, 110 Lydia, and 16 Psyche. All distributions were first normalized to unity.

type produce any common and distinguishable phase curve properties. It must be noted, however, that the taxonomy for, e.g., M type asteroids is not based on composition only, but on spectra (and albedo) (Belskaya and Shevchenko, 2000), and the diversity in composition and surface properties is likely to lead into diversity in the phase curve properties of the asteroids in the same class. Therefore, taxonomic classification of asteroids with the aid of phase curves is not relevant, at least at this point.

5. Conclusions and future work We introduce a new viewpoint for the study of the opposition effect. Thus far, most of the phase curve study is phased on complicated multiparameter modelling, leading to multiple solutions, and even at its best, to solutions that might not describe the actual properties of the object. Furthermore, most of the observed phase curves published so far have been made for various objects and samples, lacking any systematic approach. This is why strong connections with most of the physical parameters and phase curve properties have not yet been made. We emphasize that a fruitful study of phase curves must be based on systematic laboratory measurements and empirical phase curve modeling. In this way, it should be possible to establish a better understood connection between laboratory and asteroid phase curves. In other words, we should be able to answer (at least for classification purposes) the basic question “what would a typical phase curve of an asteroid covered with this material look like?” and its inverse “what is the phase curve of the material that covers this asteroid?” An extensive library of laboratory measurements should then enable us to answer the final question “what are the physical characteristics of the material that covers this asteroid?” The phase curves can be described in the linear– exponential form. The use of probability distributions allows the search for a unique set of parameters. The data needs to have a dense phase coverage for the fitting parameters to be accurate. For some of the phase curves analyzed here, the

addition of even a few data points would crucially improve the fits. For this reason, a table is provided for some asteroids with information of the phase angle regions of required data (see Table V). In the near future, we plan to fill the gaps in polarimetric phase curves using the five-color photopolarimeter attached to the 2.5-m CASLEO telescope in Argentina. Despite the fact that an observed disk-integrated phase curve has lost much of the information available in the laboratory, we know that the phase behavior still holds important clues to the nature of the surface material, particularly since the disk-integration transform is relatively straightforward to represent (cf. M. Kaasalainen et al., 2001). Not all the surface physical parameters can perhaps be solved from one single feature, but they can be characterized in a classifying sense rather than as direct numerical values. This is precisely why we believe that the empirical approach is a robust one, and also more straightforward than the use of uncertain physical scattering models. Once the quality of data is improved, the method presented can provide new insights to the remote sensing of asteroids.

Table 5 Asteroids with moderately good parameter estimates but gaps in data (phase angle regions of lacking data points are given) Photometry

␣ ⬍ 1.5° 8° ⬍ ␣ ⬍ 13° ␣ ⬍ 0.9°, 3.5° ⬍ ␣ ⬍ 8° 2° ⬍ ␣ ⬍ 8° ␣ ⬍ 2°, 3.5° ⬍ ␣ ⬍ 6.5° 2.5° ⬍ ␣ ⬍ 7.5°

16 Psyche 24 Themis 29 Amphitrite 44 Nysa 110 Lydia 133 Cyrene Polarimetry 1 Ceres 8 Flora 18 Melpomene 40 Harmonia 511 Davida

␣ ␣ ␣ ␣ ␣

⬍ ⬍ ⬍ ⬍ ⬍

1°, 1.5 ⬍ ␣ ⬍ 4.5° 4°, 4.5° ⬍ ␣ ⬍ 9° 3° 4° 4.5°, 5° ⬍ ␣ ⬍ 8°

46

S. Kaasalainen et al. / Icarus 161 (2003) 34 – 46

Acknowledgments The reviews by Ellen Howell and Beth Ellen Clark helped improve the article significantly.

References Belskaya, I.N., Shevchenko, V.G., 2000. Opposition effect of asteroids. Icarus 147, 94 –105. Bowell, E., Hapke, B., Domingue, D., Lumme, K., Peltoniemi, J., Harris, A.W., 1989. Application of photometric models to asteroids. Binzel, R.P., Gehrels, T., Matthews, M.S. (Eds.), In Asteroids II. Univ. of Arizona Press, Tucson, pp. 524 –556. Gehrels, T., 1956. Photometric studies of asteroids. V. The light-curve and phase function of 20 Massalia. Astrophys. J. 123, 331–338. Gehrels, T., Taylor, R.C., 1977. Minor planets and related objects. XXII. Phase functions for (6) Hebe. Astron. J. 82, 229 –237. Harris, A.W., Young, J.W., Goguen, J., Hammel, H.B., Hahn, G., Tedesco, E.F., Tholen, D.J., 1987. Photoelectric lightcurves of the asteroid 1862 Apollo. Icarus 70, 246 –256. Harris, A.W., Young, J.W., Bowell, E., Martin, L.J., Millis, R.L., Poutanen, M., Scaltriti, F., Zappala, V., Schober, H.-J., Debehogne, H., Zeigler, K.W., 1989. Photoelectric observations of Asteroids 3, 24, 60, 261, and 863. Icarus 77, 171–186. Harris, A.W., Young, J.W., Contreiras, L., Dockweiler, T., Belkora, L., Salo, H., Harris, W.D., Bowell, E., Poutanen, M., Binzel, R.P., Tholen, D.J., Wang, S., 1989b. Phase relations of high albedo asteroids: 44 Nysa and 64 Angelina. Icarus 81, 365–374. Harris, A.W., Young, J.W., 1989c. Asteroid lightcurve observations from 1979 –1981. Icarus 81, 314 –364. Harris, A.W., Young, J.W., Dockweiler, T., Gibson, J., Poutanen, M., Bowell, E., 1992. Asteroid lightcurve observations from 1981. Icarus 95, 115–147. Kaasalainen, M., Torppa, J., Mulnonen, K., 2001. Optimization methods for asteroid light curve inversion. II. The complete inverse problem. Icarus 153, 37–51. Kaasalainen, M., Torppa, J., Piironen, J., 2002. Models of twenty asteroids from photometric data. Icarus 159, 369 –395. Kaasalainen, S., Muinonen, K., Piironen, J., 2001. Comparative study on opposition effect of icy solar system objects. J. Quant. Spectrosc. Radiat. Transfer 70, 529 –543. Kaasalainen, S., Piironen, J., Muinonen, K., Karttunen, H., Peltoniemi, J., 2002. Laboratory experiments on backscattering from regolith samples. Appl. Opt. 41, 4416 – 4420. Lagerkvist, C.-I., Piironen, J., Erikson, A., 2001. Asteroid Photometric Catalogue. University of Uppsala, Sweden. Lagerkvist, C.-I., Magnusson, P., 1990. Analysis of asteroid lightcurves. II. Absolute magnitudes and slope parameters in a generalized HG-system. Astron. Astrophys. Suppl. Ser. 86, 119 –165. Lupishko, D.F., Tupieva, F.A., Velichko, F.P., Kiselev, N.N., Chernova, G.P., 1981. UBV photometry of the asteroids 19 Fortuna and 29 Amphitrite. Astron. Vestn. 15, 25–31 Eng. trans. in Solar System Astron. 81, 19 –24, 1981.

Lupishko, D.F., Belskaya, I.N., Tupieva, F.A., Chernova, G.P., 1982. UBV photometry of the M-type asteroids 16 Psyche and 22 Kalliope. Astron. Vesin. 16, 101–108. Eng. trans. in Solar System Res. 16, 75– 80, 1982. Mallama, A., Wang, D., Howard, R.A., 2002. Photometry of Mercury from SOHO/LASCO and Earth. Icarus 155, 253–264. Mishchenko, M.I., Diugach, J.M., 1992. Can weak localization of photons explain the opposition effect of Saturn’s rings? Mon. Not. R. Astron. Soc. 254, 15–18. Mischcenko, M.I., Luck, J.-M., Nieuwenhuizen, T.M., 2000. Full angular profile of the coherent polarization opposition effect. J. Opt. Soc. Am. A 5, 888 – 891. Muinonen, K., 1994. Coherent backscattering by Solar System dust particles. In Asteroids, Comets, Meteors, 1993 (A. Milani, M. Di. Martino, and A. Cellino, Eds.), IAU Symposium 160, pp. 271–296. Kluwer Academic, Dordrecht. Muinonen, K., 2002. Coherent backscattering by absorbing and scattering media. 6th Conference on Electromagnetic and Light Scattering by Nonspherical Particles (Bo Gustafson, Ludmilla Kolokolova, and Gorden Videen, Eds.), pp. 223–226. Adelphi, MD. Muinonen, K., Shkuratov, Yu.G., Ovcharenko, A., Piironen, J., Stankevich, D., Miloslavskaya, O., Kaasalainen, S., Josset, J.-L., 2002a. The SMART-1 AMIE experiment: Implication to the lunar opposition effect. Planet. Space Sci., in press. Mulnonen, K., Piironen, J., Kaasalainen, S., Cellino, A., 2002b. Asteroid photometric and polarimetric phase curves: Empirical modelling. Mem. Soc. Astron. Ital. 73, 716 –721. Nelson, R.M., Hapke, B.W., Smythe, W.D., Spilker, L.J., 2000. The opposition effect in simulated planetary regoliths. Reflectance and circular polarization ratio change at small phase angle. Icarus 147, 545–558. Piironen, J., Muinonen, K., Kera¨ nen, S., Karttunen, H., Peltoniemi, J., 2000. Backscattering of light by snow: Field measurements. Verstraete, Michel M., Menenti, Massimo, Peltoniemi, Jouni (Eds.), In Advances in Global Change Research, Volume 4. Observing Land from Space: Science Customers and Technology. Kluwer Academic, Dordrecht, pp. 219 –228. Poutanen, M., Bowell, E., Martin, L.J., Thompson, D.T., 1985. Photoelectric photometry of Asteroid 69 Hesperia. Astron. Astrophys. Suppl. Ser. 61, 291–297. Rosenbush, V.K., Avramchuk, V.V., Rosenbush, A.E., Mishchenko, M.I., 1997. Polarization properties of the Galilean satellites of Jupiter: Observations and preliminary analysis. Astrophys. J. 487, 402– 414. Sealtriti, F., Zappala, V., Stanzel, R., 1978. Lightcurves, phase function and pole of the Asteroid 22 Kalliope. Icarus 34, 93–98. Tedesco, E.F., Sather, R.E., 1981. Minor planets and related objects. XXIX. Asteroid 29 Amphitrite. Astron. J. 86, 1553–1558. Tedesco, E.F., Taylor, R.C., Drummond, J., Harwood, D., Nickoloff, I., Scaltriti, F., Schober, H.J., Zappala, V., 1983. Worldwide photometry and lightcurve observations of 1 Ceres during the 1975–1976 apparition. Icarus 54, 23–29. Taylor, B.C., Gehrels, T., Silvester, A.B., 1971. Minor planets and related objects. VI. Asteroid (110) Lydia. Astron. J. 76, 141–146. Zappala, V., Scaltriti, F., Lagerkvist, C.-I., Rickman, H., Harris, A.W., 1982. Photoelectric photometry of Asteroids 33 Polyhymnia and 386 Siegena. Icarus 52, 196 –201.