ISSN 0884�5913, Kinematics and Physics of Celestial Bodies, 2013, Vol. 29, No. 5, pp. 237–242. © Allerton Press, Inc., 2013.Original Russian Text © A.V. Morozhenko, A.P. Vidmachenko, 2013, published in Kinematika i Fizika Nebesnykh Tel, 2013, Vol. 29, No. 5, pp. 49–56.

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The linear sizes of the majority of asteroids are estimated from the data on their magnitude m(α) andalbedo A(α) (usually in the V photometric band). In our days, the error in determining the asteroid’s mag�nitude amounts to several hundredths of the magnitude. Because of this, the reliability of an estimate ofthe radius is completely determined by the error in the albedo. The latter is obtained with the polarimetric[10, 13, 17–19] and thermal (radiation) [4, 14, 15] methods. In the first one, the geometric albedo A(0) isestimated, while the bolometric albedo Ai is estimated in the second one. The latter is assumed to be prac�tically equal to the spherical albedo As in the V band. As for the accuracy of these two methods, it was foundthat the sizes they yield can differ by 30% and more [6], which simulated work on the improving of theboth methods.

In our opinion, the minimal difference that was achieved between the size values obtained with thesemethods does not indicate high reliability of these estimates yet. Because of this, below, we will attempt tofind the ways to increase the accuracy of both methods. For this, the sources of errors in the polarimetricand radiation methods will be analyzed. Since the error in determining m(α) is less than several percent,the main sources of errors should be searched for in the estimates of albedo.

POLARIMETRIC METHOD

In the 1960s, Ken Knight et al. [10] and Widorn et al. [18] independently suggested that the albedo ofasteroids be determined from the relation between the slope of the linear portion of the positive branch ofpolarization in the inversion point h and the reflectance c, which was obtained from the investigations ofthe terrestrial samples and the features in the lunar disk. For asteroids, this relation, logh = logc – βρ, canbe rewritten as

(1)

The initially assumed values of B = –1.78 and C = 1 [15, 16] was later changed to B = –1.73 and C =0.92 [19] or C = 0.98 [13]. Even from the pioneering paper [19], it follows that the estimates of A(0) for39 asteroids recalculated from the B and G bands to the V band sometimes produce the values differing bytens of percents. This gave us the idea to calculate the integral polarimetric albedos of the disks of themoon, Mercury, and Mars (with the values of B = –1.73 and C = 0.98) and to compare them with the val�ues of A'(0) and A''(0) obtained from the measurements with ignoring and accounting for the oppositioneffect, respectively.

A 0( )log B C h.log–=

On the Accuracy of Indirect Methods for Estimating the Sizes of Asteroids

A. V. Morozhenko and A. P. VidmachenkoMain Astronomical Observatory, National Academy of Sciences of Ukraine,

ul. Akademika Zabolotnogo 27, Kiev, 03680 UkraineReceived July 31, 2012

Abstract—We have shown that the least reliable data source for estimating the albedo of asteroids isthe maximal value of the degree of negative polarization. To increase the accuracy of the method thatuses the data on the slope of the positive branch of polarization, the values of the approximating coef�ficient should be selected in accordance with a specified type of asteroids. The similar situation is inthe shadow method, where the value of the phase integral q should be selected in accordance with eachof the types. Moreover, the estimates obtained by both methods will be more reliable if the phasedependences of brightness that are characteristic of a specified type of asteroids, including the rangeof the opposition effect, are used in transforming from A(0) to A(α). The modeling performed with theIrvine�Yanovitskii modification of the shadow model of Hapke showed that the values of the phasecoefficient β (10° ≤ α ≤ 20°) and q are, respectively, in the ranges of 0.016–0.030 and 0.6–1.0 for theE�type asteroids, 0.026–0.033 and 0.42–0.52 for the M�type asteroids, and 0.031–0.039 and 0.42–0.52 for the C�type asteroids.

DOI: 10.3103/S0884591313050061

DYNAMICS AND PHYSICS OF BODIES OF THE SOLAR SYSTEM

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Moon

Since the observations [5] revealed the asymmetry in the phase dependence of polarization for thephase angles +α and –α, we present below the values of h and A(0) averaged for both branches in the Uand B bands:

U: h = 0.220, A(0) = 0.082, A'(0) = 0.077 [11], A''(0) = 0.112 [2];

B: h = 0.175, A(0) = 0.103, A'(0) = 0.094 [11], A''(0) = 0.120 [2].

Mercury

The planet demonstrates approximately the same slopes at λ = 520 and 580 nm (h = 0.155) [7]. Theirvalue corresponds to A(0) = 0.118, while A'(0) = 0.094 and A''(0) = 0.122 in the V band.

Mars

We analyzed the observations carried out during the period of high transparency of the atmosphere, inMarch–April 1971 [1]. We used the data obtained in the spectral ranges of λ = 620 and 720 nm, where theinfluence of the Martian atmosphere is negligible. The values of A(0) were estimated for the longitudeswith the highest and lowest values of the polarization degree, which corresponded to the maximal andminimal values of albedo (Table 1). The maximal and minimal values of A(0) were obtained from thereducing of the data for the “averaged” Mars [9] according to the longitudinal dependence of brightness[3] (ignoring the opposition effect). The opposition effect was taken into account according to the param�eters of the surface layer [2] obtained with the Irvine�Yanovitskii modification of the shadow model byHapke. Moreover, Table 1 contains the values of the coefficient C that provide the agreement between thetheoretical and observed polarimetric albedos.

Thus, for the moon and Mercury, the use of expression (1) with B = –1.73 and C = 0.98 results in therelatively small disagreement (within 20%) between the calculated and observed values of A(0). At thesame time, for Mars, the difference may reach a factor of two. Since the inversion point of the dependenceP(α) is, as a rule, at α > 15°, the interval of the phase angles with positive values of polarization is very lim�

Table 1. Estimates of the spectral albedo A(0) for the longitude with the maximal and minimal values of albedo

Albedo h A(0) A'(0) C A"(0) C

λ = 620 nm

Amax 0.115 0.155 0.279 1.25 0.311 1.30

Amin 0.148 0.121 0.221 1.30 0.247 1.35

λ = 720 nm

Amax 0.0952 0.187 0.310 1.20 0.344 1.24

Amin 0.127 0.141 0.254 1.27 0.282 1.32

P, %

–0.6

–1.0

–1.4α, deg 105 15

Branch of negative polarization for one of the features on the lunar surface at λ =325 (points) and 840 nm (circles) [8].

KINEMATICS AND PHYSICS OF CELESTIAL BODIES Vol. 29 No. 5 2013

ON THE ACCURACY OF INDIRECT METHODS FOR ESTIMATING 239

ited (or even absent) for many asteroids. Because of this, for the asteroids that are observed only in therange of the negative branch of polarization, the empirical expression [19] is used

(2)

where Pmin is the maximal value of negative polarization. Unfortunately, the reliability of such estimates iseven lower, as evidenced by the application of this method to the moon, where the value of Pmin is practi�cally independent of the wavelength λ (see Figure). Due to this, the geometric albedo of the moon shouldnot demonstrate any spectral dependence either, while it actually increases approximately by a factor of2.5 in the wavelength range from 325 to 840 nm [11].

THERMAL METHOD

Since the thermal emission of the solar system bodies is determined by the amount of the solar radia�tion they absorb mainly at the wavelengths less than 3 μm, D. Allen [4] suggested that the bolometricalbedo of asteroids should be estimated simultaneously with the temperature from the observational dataon the thermal flux in several wavelengths (for example, 10 and 20 μm). However, in the study [4], it wasshown that the error in the estimate of the bolometric albedo may reach 30%, which is caused by theuncertainty in the spectral dependence of the emissivity of the material and the phase dependence ofintensity of thermal radiation. Moreover, the substantially larger errors are related to the uncertainty in thevalue of the phase integral q that is used for transforming from the bolometric (or spherical in the V band)albedo to the value of A(0). For different atmosphereless celestial bodies, the phase integral may differ bya factor of two or more.

The source of errors that is common for the both methods is the uncertainty in the phase function F(α),which is used for transforming from A(0) to A(α), i.e., to the phase angle at which the magnitude m(α) wasobserved. The latter, in turn, is used for estimating the effective size of an asteroid. It is worth noting thatthis uncertainty causes much smaller errors than the uncertainty in the phase integral does.

Alog –1.311gPmin 0.9– 0.1,±=

Table 2. Single�scattering phase functions for the moon, Mars, and C�type asteroids (a modified lunar one)

α, deg Moon Mars C�type asteroids

0 3.054 1.533 3.054

1 3.011 1.525 3.003

2 2.969 1.516 2.952

3 2.896 1.508 2.901

4 2.858 1.499 2.850

5 2.841 1.491 2.798

10 2.643 1.450 2.566

20 2.334 1.305 2.145

30 2.019 1.200 1.927

40 1.740 1.148 1.763

50 1.534 0.988 1.586

60 1.371 0.932 1.422

70 1.208 0.855 1.264

80 1.069 0.819 1.061

90 0.905 0.828 0.968

100 0.761 0.835 0.826

110 0.616 0.862 0.694

120 0.485 0.899 0.558

130 0.417 0.985 0.492

140 0.381 1.010 0.313

150 0.341 1.118 0.255

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Table 3. Values of Δm(α) for the Martian single�scattering phase function

α, deg

ω = 0.1 ω = 0.2 ω = 0.4 ω = 0.8

g

0.4 0.6 0.9 0.4 0.6 0.9 0.4 0.6 0.9 0.4 0.6 0.9

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0.14 0.09 0.06 0.13 0.08 0.05 0.11 0.07 0.04 0.06 0.04 0.02

2 0.23 0.16 0.11 0.21 0.14 0.10 0.18 0.12 0.08 0.11 0.07 0.04

3 0.30 0.21 0.15 0.28 0.20 0.14 0.25 0.17 0.12 0.14 0.10 0.07

4 0.36 0.25 0.19 0.34 0.25 0.18 0.29 0.21 0.15 0.17 0.13 0.09

5 0.41 0.31 0.23 0.38 0.29 0.21 0.33 0.25 0.19 0.19 0.15 0.11

10 0.57 0.48 0.39 0.54 0.45 0.37 0.47 0.39 0.32 0.27 0.24 0.20

20 0.84 0.77 0.69 0.79 0.73 0.66 0.69 0.64 0.58 0.43 0.41 0.38

30 1.04 0.99 0.93 0.99 0.95 0.89 0.88 0.84 0.79 0.58 0.56 0.54

40 1.21 1.18 1.13 1.16 1.12 1.08 1.03 1.01 0.98 0.71 0.70 0.69

50 1.50 1.47 1.44 1.43 1.41 1.38 1.29 1.27 1.25 0.93 0.92 0.91

60 1.71 1.69 1.67 1.64 1.62 1.61 1.49 1.47 1.46 1.11 1.10 1.10

70 1.95 1.95 1.94 1.89 1.88 1.87 1.73 1.72 1.71 1.25 1.33 1.33

80 2.19 2.19 2.19 2.12 2.11 2.11 1.96 1.95 1.95 1.56 1.56 1.56

90 2.39 2.39 2.40 2.32 2.32 2.33 2.17 2.17 2.17 1.79 1.79 1.79

100 2.63 2.64 2.66 2.56 2.57 2.58 2.41 2.42 2.43 2.06 2.06 2.06

q 0.59 0.61 0.62 0.62 0.64 0.66 0.71 0.72 0.73 0.97 0.98 0.98

A(0) 0.038 0.076 0.151 0.330

Table 4. Values of Δm(α) for the lunar single�scattering phase function

α, deg

ω = 0.1 ω = 0.2 ω = 0.3 ω = 0.5

g

0.4 0.6 0.9 0.4 0.6 0.9 0.4 0.6 0.9 0.4 0.6 0.9

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0.15 0.10 0.07 0.14 0.09 0.06 0.13 0.09 0.06 0.12 0.08 0.06

2 0.25 0.18 0.13 0.24 0.17 0.12 0.23 0.16 0.11 0.22 0.14 0.11

3 0.35 0.26 0.19 0.33 0.25 0.18 0.31 0.23 0.18 0.30 0.22 0.17

4 0.41 0.32 0.24 0.40 0.30 0.23 0.38 0.29 0.22 0.36 0.27 0.21

5 0.46 0.36 0.28 0.44 0.35 0.27 0.42 0.33 0.26 0.40 0.31 0.24

10 0.68 0.58 0.49 0.65 0.56 0.47 0.62 0.54 0.45 0.59 0.51 0.43

20 0.96 0.89 0.82 0.93 0.86 0.79 0.89 0.83 0.76 0.85 0.79 0.73

30 1.24 1.19 1.14 1.20 1.15 1.09 1.15 1.11 1.05 1.10 1.06 1.01

40 1.52 1.48 1.45 1.47 1.44 1.39 1.42 1.38 1.35 1.36 1.34 1.30

50 1.78 1.76 1.72 1.73 1.70 1.68 1.67 1.65 1.62 1.61 1.58 1.56

60 2.05 2.03 2.01 1.99 1.97 1.96 1.92 1.90 1.89 1.85 1.84 1.82

70 2.34 2.33 2.32 2.28 2.27 2.26 2.21 2.19 2.19 2.13 2.12 2.11

80 2.66 2.65 2.65 2.58 2.58 2.58 2.50 2.50 2.50 2.42 2.41 2.41

90 3.04 3.04 3.05 2.96 2.95 2.95 2.86 2.86 2.87 2.77 2.77 2.78

100 3.46 3.46 3.48 3.36 3.36 3.38 3.26 3.26 3.28 3.16 3.16 3.17

q 0.42 0.43 0.45 0.44 0.46 0.46 0.46 0.48 0.49 0.49 0.50 0.52

A(0) 0.075 0.147 0.217 0.284

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ON THE ACCURACY OF INDIRECT METHODS FOR ESTIMATING 241

Thus, it is presently impossible to specify even an approximate error in the estimate of the size of thisor that asteroid. This circumstance should be taken into account when the data on their sizes are used(e.g., when estimating the density of the material of asteroids and, therefore, their probable nature andstructure).

Since the main source of the uncertainty in the calculated “thermal” albedo is the uncertainty in thephase integral, we will present below the calculated changes in the magnitude Δm(α), the geometricalbedo A(0), and the phase integral q caused by different values of the porosity g and the single scatteringalbedo ω for several types of asteroids. The calculations were performed with the use of the data from thepaper [2], where it is shown that the phase dependence of brightness of asteroids of E, M, S, and C typescan be satisfactorily described with the Irvine�Yanovitskii modification of the shadow model of Hapke.Here, the unknowns are the single�scattering phase function χ(α), the parameters g and ω, and the trans�parency coefficient κ, whose value is connected with ω by relation κ = (1 – ω)0.25. It has turned out thata good agreement between the observed phase dependences of brightness and the calculations for theasteroids of different types can be reached for the single�scattering phase functions presented in Table 2.For the E type, it is the Martian single�scattering phase function, it is the lunar one for M and S types, andit is the modified lunar one for the C type.

For each of the single�scattering phase functions (Tables 3–5), the calculations were fulfilled with threeprobable values of the porosity parameter g (0.4, 0.6, and 0.9) and four values of ω. The minimal value ofω was assumed to be 0.1, since the correction for the multiple scattering in the range of α ≤ 100° hardlyproduces an effect on the shape of the phase function and, therefore, on the value of the phase integral ifsmaller values of ω is used. When selecting the largest ω, we limited ourselves with the values that provideA(0) to be close to 0.3. It follows from the tables that the values of the phase coefficient β = (m(10°) –m(20°))/10° and q are, respectively, within the ranges of 0.016–0.030 and 0.6–1.0 for the E�type asteroids,0.026–0.033 and 0.42–0.52 for the M� and S�type asteroids, and 0.031–0.039 and 0.42–0.52 for the C�type asteroids.

According to the data [12], 17 of 18 asteroids approaching the earth are characterized by the value ofβ = 0.020–0.038, and only one (887 Alinda) is characterized by β = 0.044.

Table 5. Values of Δm(α) for the modified lunar single�scattering phase function

α, deg

ω = 0.1 ω = 0.2 ω = 0.3 ω = 0.4

g

0.4 0.6 0.9 0.4 0.6 0.9 0.4 0.6 0.9 0.4 0.6 0.9

0 0 0 0 0 0 0 0 0 0 0 0 0

1 0.15 0.10 0.07 0.14 0.10 0.07 0.14 0.09 0.06 0.13 0.08 0.06

2 0.27 0.16 0.13 0.24 0.17 0.13 0.23 0.15 0.12 0.22 0.15 0.11

3 0.34 0.26 0.19 0.33 0.24 0.18 0.31 0.23 0.17 0.30 0.22 0.15

4 0.41 0.32 0.24 0.40 0.30 0.24 0.38 0.29 0.22 0.36 0.28 0.21

5 0.48 0.38 0.30 0.46 0.36 0.28 0.44 0.35 0.27 0.42 0.33 0.26

10 0.71 0.61 0.52 0.68 0.59 0.50 0.65 0.57 0.48 0.62 0.54 0.46

20 1.05 0.98 0.91 1.01 0.95 0.88 0.98 0.92 0.85 0.93 0.88 0.81

30 1.29 1.24 1.18 1.22 1.20 1.14 1.20 1.15 1.10 1.15 1.11 1.06

40 1.50 1.47 1.42 1.46 1.42 1.38 1.40 1.37 1.33 1.35 1.32 1.28

50 1.75 1.72 1.69 1.70 1.67 1.64 1.64 1.61 1.59 1.58 1.55 1.53

60 2.01 1.99 1.97 1.95 1.93 1.92 1.89 1.87 1.86 1.82 1.80 1.79

70 2.30 2.28 2.28 2.23 2.22 2.21 2.16 2.15 2.14 2.09 2.08 2.07

80 2.66 2.66 2.66 2.59 2.58 2.58 2.51 2.50 2.51 2.43 2.42 2.42

90 2.97 2.97 2.98 2.89 2.89 2.90 2.80 2.80 2.81 2.71 2.71 2.72

100 3.37 3.38 3.40 3.29 3.29 3.31 3.19 3.19 3.20 3.09 3.09 3.10

q 0.42 0.43 0.45 0.45 0.45 0.46 0.46 0.48 0.49 0.49 0.51 0.52

A(0) 0.075 0.147 0.217 0.284

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In conclusion, note that the less reliable estimates of albedo are those obtained from the data on themaximal degree of negative polarization (Eq. (2)).

The slope of the branch of positive polarization can yield the reliable results only in the case when thevalue of the coefficient C in Eq. (1) is selected in accordance with a specified type of the asteroid.

The same situation is with the thermal method, because the main source of the uncertainty here is theuncertainty in the phase integral, whose values are different for different types of asteroids. The commonproblem for both methods is a choice of the phase function, including the behavior in the range of theopposition effect.

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Translated by E. Petrova