PASJ: Publ. Astron. Soc. Japan 54, 1079–1089, 2002 December 25 c 2003. Astronomical Society of Japan.
Statistical Method for Deriving Spatial and Size Distributions of Sub-km Main-Belt Asteroids from Their Sky Motions Tsuko NAKAMURA National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588
[email protected]
and Fumi YOSHIDA Graduate Institute of Astronomy, National Central University, 38 Wu-chuan Li, Chuang-li, Tao-yuan, 320 Taiwan ROC
[email protected] (Received 2001 September 10; accepted 2002 September 18)
Abstract The importance of sub-kilometer-sized objects among the main-belt asteroids (MBAs) has been progressively recognized in relation to the origin of near-Earth asteroids and the formation of rubble-pile asteroids. However, exact orbit determinations of those individual objects are practically impossible, because both their discovery and subsequent follow-up observations require 8–10 m class telescopes, in which observation-time competition is severe. Therefore, we examine here instead a statistical method to deduce the spatial and size distributions of sub-km MBAs from only their apparent motion vectors on the sky. Assuming their near-opposition and near-ecliptic observations, we made simulations to estimate the accuracy of the semi-major axis (a) and inclination (I ) obtained from their sky-motion vectors. The mean errors of a and I for each asteroid were found to be about 0.15 AU and 1◦ –2◦ (for the asteroids with I < 10◦ ), respectively. Then, under a certain assumption, we calculated magnitudes and the size distribution of those computer-synthesized sub-km MBAs. Our statistical method could reproduce the slope of the cumulative size distribution for the original asteroid populations with errors of ∼ 0.05–0.1. These values are small enough for our survey purposes using the Subaru Telescope. Key words: celestial mechanics — methods: statistical — minor planets, asteroids — solar system: general 1. Introduction Collisions are a major factor dominating the origin and subsequent evolution of asteroids in the main asteroid belt. Size distributions for the main-belt asteroids (MBAs) are considered to be a direct outcome from repeated impacts between asteroids. Thanks to the Palomar–Leiden (van Houten et al. 1970), the Spacewatch (Jedicke, Metcalfe 1998) and the Sloan Digital Sky Survey (Ivezi´c et al. 2001) observations of MBAs, the size distribution for the MBAs larger than a few kilometers in diameter has well been established, and the slope for the cumulative size distribution is found to range from ∼ 1.8 to ∼ 1.3. Recently, the importance of sub-km MBAs has been progressively recognized from the following viewpoints: 1) the interrelation between near-Earth asteroids (NEAs) and MBAs, and 2) a critical size separating monolithic asteroids from “rubble piles”. Regarding the first point, NEAs are generally understood to originate from the main asteroid belt through impacts between asteroids and subsequent secular perturbations mainly associated with the Kirkwood gaps (e.g., Wisdom 1983; Morbidelli, Moons 1995). Considering this likely dynamical path from MBAs to NEAs and the fact that the majority (∼ 70%) of observed NEAs are sub-km-sized,1 it is quite natural for us to pursue any observational interrelations between NEAs and sub-km MBAs. 1
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The second point has to do with impact mechanisms between asteroids and the formation of rubble piles. Rubble piles are theoretically predicted bodies consisting of reaccumulated fragments whose relative impact velocities are less than the escape velocity from the parent asteroid (Weidenschilling 1981). The remarkably low bulk density (∼ 1.3 in cgs unit) of the asteroid Mathilde observed by the NEAR spacecraft (Veverka et al. 1999) is regarded as good evidence for the existence of rubble-piles. Recent hydrocode simulations combined with laboratory experiments (Love, Ahrens 1996; Melosh, Ryan 1997) suggested that there should be some critical diameters, near the sub-km size region, dividing rubble-pile asteroids and smaller monolithic asteroids (reality of the latter is evidenced by the existence of fast rotators among NEAs: Pravec et al. 2000; Pravec, Harris 2000). If this is the case, it may be that such a structural difference is reflected in the size distribution of sub-km MBAs. In spite of such intriguing aspects for sub-km MBAs, there had been no means, due to their faintness, to detect them with telescopes of a few meters in diameter, which were in common use a decade ago. However, the recent advent of 8–10 m class telescopes opened up a way to observe sub-km MBAs systematically. In particular, a prime-focus wide-field camera specifically designed for the Subaru Telescope (FOV: ∼ 30 × 30 ) is most suitable for survey observations of sub-km MBAs, because of its wide sky-coverage; the Subaru wide-field camera is likely to detect as many as a thousand asteroids per squaredegree, down to V ∼ 25–26.
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The orbits of individual MBAs have been calculated by traditional orbit-determination methods. This requires follow-up observations of the asteroids to have sufficient data to cover an extended period of time. Such an approach, however, is practically impossible to conduct for sub-km MBAs, because severe telescope-time competition for 8–10 m telescopes would not allow us to perform follow-up observations of individual sub-km MBAs with those telescopes. Hence, the purpose of this paper is instead to give a statistical method for deriving spatial and size distributions of subkm MBAs from their apparent sky-motion vectors, which can be measured easily and accurately from observations spanning one or two days. In section 2, we describe a methodology to calculate some of orbital elements and the relating errors for sub-km MBAs from their observed sky-motion vectors. We propose in section 3 a statistical procedure to estimate the size distribution for sub-km MBAs, which is obtained by applying the method mentioned in section 2; the result of section 3 is a quantitative substantiation of our early ideas (Nakamura 1997; Yoshida 1999). Section 4 discusses bias corrections to derive intrinsic spatial and size distributions of sub-km MBAs. In the final section, another possible approach of orbital estimation is compared with our statistical method. 2.
Orbital Elements and Their Accuracy Deduced from Sky-Motion Vectors
2.1. Method for Deriving Semi-Major Axis and Inclination Textbooks on orbit determination teach us that, in principle, the right ascension (α) and declination (δ) of an asteroid observed at three different epochs of time can determine its six Kepler orbital elements and two sets of (α, δ) can provide an circular orbit. However, given the observation period covering one or two days and the current level of errors in positional measurements, this traditional approach never gives reliable orbital elements (see section 5). This is mainly because the eccentricity of an asteroid can never be known from short-arc observations. If so, it could be cleverer to devise a simpler approach which can give errors comparable to a traditional orbitdetermination method, since we must handle many asteroids in our proposed survey observations. Accurately measurable quantities from one- or two-day observations are only the linear motion vector of an asteroid projected on the sky, namely the total daily motion m (arcmind−1 ) and its position angle P (◦ ), or motion vector components l ≡ dλ/dt and b ≡ dβ/dt (arcmind−1 ) in ecliptic longitude (λ) √ and latitude (β) at a mean epoch of observations (m = l 2 + b2 near the ecliptic plane). Hence, in this section, we look for two-body-kinematic relations connecting (l, b) with (a, I ) under some specific orbital and observational conditions. We assume here that survey observations of sub-km MBAs are conducted in a small field of view, near opposition and near the ecliptic. The reason is that 1) the relation connecting (l, b) with (a, I ) becomes simplest under this condition, 2) the apparent motions and brightness of asteroids are maximal at opposition, and 3) asteroids with arbitrary orbital inclinations (I ) pass through the ecliptic plane. Bowell et al. (1990) already gave such a relation between (l, b) and (a, I ) under the assumption of a circular orbit; we
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note that Watanabe (1953) had also obtained a less-refined similar formula in 1953 in a tabular form. Those relations are most suitably applied to the skies within ecliptic longitudes of several degrees from the exact opposition, because the derivation of the formula assumes that an asteroid is exactly at opposition (see Appendix). However, in practical observations, it often happens that we are forced to observe a sky much more separated from opposition in longitude (elongation angle: E), caused by telescopic scheduling, or to avoid interference from the Milky Way at an assigned observation date. Moreover, if we attempt to measure phase curves of sub-km MBAs, for instance, observations at large Es are inevitable. Therefore, in Appendix of this paper, we rederive a relation as an extension of the formula by Bowell et al. (1990), because they gave only the final results in their paper. Our new formula is valid up to E ∼ 15◦ –20◦ , and tends to the one by Bowell et al. (1990) when E → 0. We emphasize again that the formula in Appendix was obtained assuming that asteroids have circular orbits, and there is no other reasonable choice because we can know nothing about the eccentricity for short-arc observations. Since in this paper we are mainly interested in nearopposition survey observations of sub-km MBAs, we calculated the a and I of asteroids in the following simulations using the formula (15) and (16) in Appendix, which are equivalent to the one given by Bowell et al. (1990). Hereafter, for simplicity, the a and I obtained by equations (15) and (16) are called Bowell’s orbit. 2.2. Error Estimates of Orbital Elements What is most important in applying equations (15) and (16) to observed motion vectors is to confirm that the a and I averaged over many asteroids should have no systematic deviations from the true a and I of the asteroids in a statistical sense; each a and I can sometimes be considerably different from the corresponding true values since eccentricities are neglected in equations (15) and (16). Without systematic errors in a and I , we expect that the global spatial and size distributions of asteroids as a whole can be correctly deduced from the observed a and I . To check the above requirement, we conducted Monte Carlo simulations of wide-field CCD survey observations by generating orbits for hypothetical asteroids in a computer. First, we constructed model populations of orbital elements mimicking the main asteroid belt. Angular elements (except for I ) were assumed to be uniformly distributed over (0◦ , 360◦ ). By picking up randomly a set of orbital elements corresponding to an asteroid one by one from a model distribution, our program tested whether or not the asteroid was located in an observational window centered at opposition of the ecliptic plane. The size of the observation window was taken to be 5◦ × 4◦ . This window size was selected as such a compromise that a single run of the simulation would give the necessary output number (∼ 5000) within a reasonable computer time. When the asteroid was found to be within the window, its input orbital elements including a and I were recorded in an observation list along with its calculated (l, b) values. We repeated this process until the number (n) of in-window asteroids reached n = 1000, 3000, and 5000. Throughout, we used a two-body ephemeris generator that could calculate rigorous
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Fig. 1. Correlation plot for simulation-generated a vs. a obtained from Bowell’s orbit. a, I , and e are uniformly distributed over 2.7 ± 0.6 AU, 15◦ ± 15◦ , and 0.2 ± 0.2, respectively, for the model population of MBAs. Angular elements are also assumed to have uniform distributions over (0◦ ,360◦ ). The observation window of 5◦ × 4◦ was adopted in this simulation.
astrometric positions from the orbital elements of asteroids. Then, for those pseudo-observations (l, b) of asteroids, they were converted to (a , I ) using equations (15) and (16). Figure 1 represents an example of a correlation plot (n = 5000) between the true a generated in this simulation and the a calculated by equations (15) and (16). Similarly, figure 2 shows a I vs. I relation corresponding to figure 1. We define here (a − a ) and (I − I ) as errors by Bowell’s orbit. From figure 1, one can see that there is no appreciable systematic deviations between a and a . For other model populations of asteroids in which distributions of a, I , and e were changed in reasonable ranges (see table 3), we obtained a vs. a correlations similar to figure 1. The trend in figure 1 is shown in table 1 quantitatively. The mean values and standard deviations (SD) of (a − a ) are given for the inner-, mid- and outer-belt regions of asteroids. In this simulation, about 30 (0.6%) out of n = 5000 showed apparently unrealistic orbits whose a -values were larger than 4 AU. Such abnormal cases seem to take place, for instance, when asteroids with a large e happened to be near their aphelion. As understandable from equation (15), those asteroids also seem to show a tendency to possess unusually large inclinations, since a large a generally means a small l (or distant asteroids show slow motions). Although those unusual asteroids are removed from the calculation of table 1, it has very little effects on the whole distribution of asteroids because of their small number. We see from table 1 that the systematic deviations expressed as the means of (a − a ) are negligibly small compared with the width of the main asteroid belt (∼ 1.2 AU) for all of the three zones. The values of the standard deviation also indicate that we can easily identify at least which zone of the belt individual asteroids
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Fig. 2. Correlation diagram for simulation-generated I vs. I calculated from Bowell’s orbit. As for orbital distributions in the model population of asteroids, see the caption of figure 1. It is obvious that for high-inclination asteroids, the deviation of I from the original I is considerable. Table 1. Errors of the semi-major axis obtained from Bowell’s orbit.
Zone
Range (AU)
Mean (a − a )
SD (a − a )
Inner-belt Mid-belt Outer-belt
2.1 < a < 2.5 2.5 < a < 2.9 2.9 < a < 3.3
0.042 0.023 0.019
0.17 0.15 0.15
belong to; this is enough for our present purpose. Here, we comment on the effects of the sample number and window size on the (a − a ) errors. For simulations with n = 1000 and 3000, and for smaller window sizes than the one adopted in figure 1, all of the mean and SD of (a − a ) showed values closer to zero than those for n = 5000. This is because the distribution of points, like in figure 1 for n = 1000 and 3000, is less scattered, and narrower window sizes are closer to the ideal observational condition treated in Appendix. Hence, we can expect smaller orbital errors than the values given in table 1 in realistic Subaru survey observations, where the obtainable sample number could be less than a few thousand and the surveyable sky area would be smaller than several square degrees in an observational run. We also stress that the values of the standard deviation in table 1 refer to a single asteroid. In an actual analysis of observations of sub-km MBAs, a semi-major axis distribution is composed from many asteroids, so that the reliability of the resulting overall a-distribution of asteroids is expected to be much higher than that of a for individual asteroids. Next, we consider figure 2 regarding the inclination. For the same reason as mentioned in the paragraphs with respect to a , only the case of n = 5000 is considered here. One can notice at a glance that the correlation between I (abscissa)
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Zone Range (deg) Mean (I − I ) SD (I − I ) Low-incl. 0 < I < 10 0.35 1.59 Medium-incl. 10 < I < 20 2.0 5.1 High-incl. 20 < I < 30 6.5 6.1
and I (ordinate) becomes worse as I increases, especially for I > 10◦ . Bowell et al. (1990) also pointed out a similar behavior of errors in the I -estimate, as a function of I . Such a tendency in I -errors is summarized quantitatively in table 2, as done for a in table 1. Since the mean values of (I − I ) for the two zones 10◦ < I < 20◦ and 20◦ < I < 30◦ are comparable to the corresponding SDs, those systematic errors must be taken into account first to estimate correct inclinations. In calculating table 2, several tens (approximately 1%) out of n = 5000 asteroids whose inclinations were found to be I > 40◦ were omitted. However, this should have no appreciable effects on the whole statistics due to the scarcity of such unrealistic asteroids. Because the error trend seen in table 2 is likely to be inherent for Bowell’s orbit, we discuss in the last section another approach that may give a reduced error of inclination. In short, from the above discussions, we conclude that Bowell’s orbit can be satisfactorily used in a statistical sense for the purpose of our survey observations of sub-km MBAs. Another point to notice is that positional measurement errors of asteroids on CCD chips have substantially no effects on the determination of a and I , so long as the sky motions of the asteroids are obtained from observations covering 1–2 hr. 3.
Statistical Estimates of Size Distribution for Sub-km Asteroids
3.1. Procedure for Calculating Size Distributions In this section, we first calculate the H -magnitude for asteroids whose a and I have already been estimated in section 2. If the albedo (p) of an asteroid is known or assumed, its diameter (D) can be obtained from an equation (Bowell et al. 1989) which is a modified version of the formula by Bowell and Lumme (1979): log D = 3.1295 − 0.5 log p − 0.2H.
(1)
Since the absolute magnitude (H ) of an asteroid near opposition is given by H = V − 5 log r(r − 1) − δV ,
(2)
where V and δV stand for the apparent magnitude and light variation in V -band, respectively, we need the heliocentric distance r of this asteroid to calculate the H -magnitude. However, because of the two-body formula r = a(1 − e2 )/(1 + ecosω) near the ecliptic, its r can never been estimated without knowledge of this asteroid’s e and ω (argument of perihelion), whose determination is impossible from observations covering 1–2 d (see Appendix). When we think about many asteroids as a whole having similar a and e values instead of a single asteroid, however, the
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situation becomes more tractable. Upon averaging r over various e cos ω for those asteroids, the mean r is expected to approach a. Then, it would be reasonable to assume that r ∼ = a (or ∼ e = 0) for each asteroid from the beginning, because this brings about the same result, as an ensemble of asteroids. Actually, there is no other choice for estimating r. It should be noted, in this sample-averaging, that the light variation (δV ) of asteroids would also be averaged out, since they would be at various phases of the light curves. This seemingly crude expectation must, of course, be ascertained by simulating whether or not the above assumption r ∼ = a can give the correct size distribution. Before discussing the results of the simulations, however, it would be appropriate to review here the relation between the size and H -magnitude distributions of asteroids. It is well-known that a cumulative number distribution for MBAs brighter than magnitude H is expressed as log N (< H ) = C − αH.
(3)
Let us call α the slope for log N vs. H plot. If we rewrite equation (3) with the help of equation (1), it results in N (> D) = c · D −γ ,
(4)
where N (> D) is the cumulative number of asteroids larger than diameter D (corresponding to H ), and γ , the power-law index (or the slope for log N vs. log D plot), is connected with α by the equation γ = 5α.
(5)
The value of ∼ 1.8 mentioned above for the Palomar–Leiden and Spacewatch surveys corresponds to the slope γ . We now describe the simulations in order to understand how faithfully our above proposed method can reproduce the originally hypothesized size distribution for sub-km MBAs. Below, variables without primes generally indicate input parameters and quantities, and the same variables with primes represent those obtained as output from the simulation. The simulation procedure consisted of the following processes: I. Make a power-law random-number generator. This program outputs H -magnitudes as random numbers whose relative frequency obeys a power-law with a specified power-law index γ (or α). The output H covered a wide range with a H -bin step of 0.25. II. Combine the output from process I with orbital elements mimicking the distribution of orbits in the main asteroid belt. Angular elements other than I are changed randomly between (0◦ ,360◦ ), as before. The data set of a H -magnitude and orbital elements consists of an asteroid as input in this simulation. III. 1) Set up a survey window centered at opposition and the ecliptic, 2) calculate the position of the asteroid from process II on a specified date using the two-body ephemeris generator mentioned above, and 3) examine whether or not the asteroid is in the observational window and brighter than a specified limiting magnitude Vlim . The window size is taken to be 5◦ × 4◦ , because of the reason mentioned in section 2. Vlim is set to be 25, assuming the Subaru observations.
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IV. If the asteroid passes the test of process III, pick it up, and calculate its sky motion vectors (l, b) in the ecliptic coordinate system.
V. 1) Estimate a and I of the asteroid from process IV using Bowell’s orbit, equations (15) and (16), 2) calculate r assuming r = a , and obtain the size (D) of the asteroid, with the help of equations (1) and (2) for an appropriate albedo, and 3) store the relating data on the asteroid into an observation list file. An averaged albedo for S-type and C-type was adopted here.
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Table 3. Orbital-element ranges for three model populations of sub-km MBAs.
Case
a (AU)
I (deg)
e
A B C
2.6 ± 0.5 2.7 ± 0.6 2.6 ± 0.6
10 ± 10 15 ± 15 7.5 ± 5
0.2 ± 0.1 0.2 ± 0.2 0.175 ± 0.08
VI. Repeat processes I–V until the number of in-window asteroids attains n = 5000 in typical simulations. VII. 1) Produce a cumulative size distribution from the collected samples in process VI, 2) make a least-squares fitting to obtain its H -slope (α ) and power-law index (γ ), and 3) compare the γ (the slope of the original size distribution) with the γ (the slope reproduced from this simulation). Although we also included in some computer runs light variations (δV ) with an amplitude of ∼ ±1.0 mag, we did not find their overall effects on the α - or γ -slopes; this is probably because the simulated range of H is much wider than that for δV . Hence, δV in equation (2) was not taken into account in the following simulations. 3.2. Errors of Slope Estimates In the following part, we examine based on simulations how measurements of the slope in size distributions are affected by (i) different possible orbital populations of sub-km MBAs, (ii) different sample numbers (n), and (iii) different slope values (α or γ ). (i) Dependence on the orbital populations We simulated size distributions with a nominal value of α = 0.35 (γ = 1.75), for three assumed model populations of sub-km MBAs whose orbital characteristics are given in table 3. The reason of this approach is that the orbital distributions for sub-km MBAs are totally unknown in advance of survey observations. In table 3, Case A is intended to be nearest to the global distributions of the existing small asteroids.2 Case B is a more widened version for each element, considering that the spatial distribution for sub-km MBAs is possibly more scattered than that for km-sized asteroids. Case C reflects the modal values and FWHMs of a-, I -, and e-distributions for existing small MBAs, so that the distribution-widths in the three elements are narrowest among the three cases. We anticipate that reality exists somewhere among the three cases, A–C, for sub-km MBAs. Figure 3 gives the simulated size distribution for Case B as an example; two other cases showed a similar behavior. The abscissa is the H -magnitude and the ordinate is the logarithm of the cumulative number N (H ). Small filled circles represent the cumulative H -distribution generated as input data, crosses stand for the distribution obtained from a of simulated asteroids using Bowell’s orbit and equation (2), and for those dis2
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Fig. 3. Comparison of computer-generated size (H -) distributions with those estimated by Bowell’s orbit. The former and the latter distributions are expressed by small filled circles and crosses, respectively. This graph is for Case-B asteroid population given in table 3. A straight line in the upper part is drawn just to show the nominal slope of α = 0.35 (γ = 1.75).
tributions straight lines corresponding to equation (3) are fitted by least-squares. For a comparison we also drew the nominal slope of α = 0.35 (γ = 1.75) as the upper straight line in figure 3. This is because this slope never exactly coincides with the slope for the input distribution, expressed by the small filled circles, due to errors in the numerical approximations (see process I in subsection 3.1). As in section 2, we removed in the statistics any unusual asteroids of about 1% with a ² 4 AU, which is likely to be due to an inevitable drawback of Bowell’s orbit. One can see from figure 3 that an agreement between the input H -distribution and that obtained from Bowell’s orbit is fairly good, which is also the case for two other populations (A and C) of hypothetical sub-km MBAs. We analyze the situation quantitatively in table 4. Table 4 summarizes the least-squares-fitted slope values for the input size distribution and the simulated distribution by Bowell’s orbit. For clarity hereafter, we express the H -slope for the input distribution (filled circles in figure 3) as α, and that for the simulated distribution (crosses) as α . From α and α , γ and γ are calculated using equation (5). ∆γ (≡ γ − γ ) given in table 4 defines the slope error in the simulations. ∆γ
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Table 4. Generated and Bowell’s slopes for three model populations of sub-km MBAs.
Case A B C
α
α
γ
γ
∆γ
0.335 0.343 1.675 1.715 −0.040 0.352 0.329 1.760 1.645 0.115 0.352 0.349 1.760 1.745 0.015
∆γ − ∆γ −0.070 0.085 −0.015
Note. ∆γ stands for the value (γ − γ ).
represents the mean of ∆γ ; namely 0.030. Considering that ∆γ and ∆γ − ∆γ correspond, respectively, to the systematic and random errors between the original power-law index and that obtained by Bowell’s orbit, one can understand from table 4 that, if a correction of the systematic error (∆γ ) is applied, the simulation can reproduce the original power-law index to an error level of less than 0.10 (or equivalently, less than 0.02 in α-slope). (ii) Sample-number dependence Next, in order to see how the slope determination depends on the sample number, we calculated the cumulative size distributions for n = 1000, 3000, and 5000. They are shown in figure 4, in which the small filled circles represent the initial distributions generated in a computer, the crosses show the output distributions obtained via Bowell’s orbit, and the straight lines are fitted curves to the output data by least squares. The input population of MBAs is the same as Case B in table 3. In this simulation, the window size is taken to be 2◦ × 2◦ , which is close to a coverable sky area by Subaru’s one-night observations. Table 5 summarizes the fitted slope values corresponding to figure 4. Note again that computer-generated α-slopes are not exactly equal to the initially specified value (α = 0.35 or γ = 1.35), because of the numerical approximation in the process I of subsection 3.1. One can see from the table that the samplenumber dependence of α is negligibly small in the range of 1000 < n < 5000, since the original slopes are reproduced in this simulation with errors less than 0.01–0.02 in α (or 0.05– 0.1 in γ ). Hence we expect that such an accuracy in slope determination, as indicated in table 5, can be realized in actual one-night Subaru observations. (iii) Dependence on γ -values In (i) and (ii), simulations were made by specifying the slope as γ = 1.75 (or α = 0.35) beforehand. In actual survey observations, however, the γ for sub-km MBAs is unknown, but is the one to be pursued. Therefore, it is necessary to examine by simulations such systematic and random errors of γ as shown in table 4, for a wide range of γ . Figure 5 shows a diagram connecting several values of γ obtained from Bowell’s orbit (abscissa) with the corresponding correction (ordinate). The corrections are defined here by ∆γ = γ − γ , so that the addition of ∆γ to the observed γ brings about the true slope γ (see table 4). Actually, each value of the correction (filled circle) was calculated as a mean (∆γ ) averaged over the three model populations (Case A through Case C), and the attached error bar is the standard deviation for the three cases. From figure 5, we see that Bowell’s slope (γ ) to be obtained
Fig. 4. Comparison of computer-generated size (H -) distributions with those estimated by Bowell’s orbit, as a function of sample number (n). Small filled circles and crosses represent the former and the latter distributions, respectively. Curves for n = 1000, 3000, and 5000 are shown. A straight line fitted to output data by least squares is drawn in each panel. The observation window was taken to be 2◦ × 2◦ in this case.
from observations positively correlates with the correction (∆γ ). Therefore, by fitting a linear relation to that trend, we can remove at least the systematic error of γ . Then, the uncontrollable errors that are left to us are deviations of the filled
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Table 5. Slope-value dependence on the sample number.
n
Generated α
Simulated α
1000 3000 5000
0.329 0.361 0.386
0.349 0.365 0.382
circles from the fitted line, namely random errors of γ , and one can see from figure 5 that those deviations are smaller than 0.05–0.10. Hence, we conclude that the statistical method described in section 2 can estimate the size distribution of subkm MBAs with an error of γ better than 0.05–0.10. In other words, if there is a difference in γ s between multi-km and subkm MBAs by an amount of 0.05–0.10, our proposed statistical method can easily distinguish the difference. 4. Bias Corrections Since our planned survey observations will be conducted in a narrow field of view near opposition and near the ecliptic, only covering a time span of one or two days, bias corrections specific to such observational conditions must be taken into account in order to derive intrinsic spatial and size distributions of sub-km MBAs. Although observational and orbital biases in asteroid surveys were already investigated by several authors (e.g., Benedix et al. 1992; Jedicke 1996; Jedicke, Metcalfe 1998; Tancredi 1998), all of their discussions were based on orbits obtained by traditional orbit-determination methods. There, it was assumed that the six Keplerian orbital elements for each asteroid are known or measured. However, in our approach, only a and I are estimated statistically. Hence, there are still good reasons for us to reexamine the orbital bias in our proposed method from a different viewpoint. In this paper, we distinguish two kinds of biases, that is, the absolute bias and the relative bias. An absolute bias correction is necessary when estimating the absolute number of asteroids, for instance, in a specific orbital region and brighter than some limiting magnitude. To make such an estimate, however, detailed information on the photometric characteristics of the telescopic and CCD camera system used for observations and the meteorological sky conditions are required, in addition to information on the orbital configurations mentioned below; the meteorological information can be known for the first time after actual observations. Hence, here we restrict ourselves exclusively to a discussion of the relative bias. The relative bias concerns only a relative number of asteroids normalized with the asteroid number in a specified orbital region. The relative bias correction is still very useful to estimate the intrinsic distribution profile of asteroids, though a constant leading to an absolute distribution remains unknown. In this section, we calculate the relative biases for a and I , which are major bias-affected components. Figure 6 represents the a- or distance-bias, normalized with the number of objects located at sufficiently remote distances. The relative bias in the ordinate was actually defined here as the number ratio between near-ecliptic objects with r ∼ 6 AU and those with r = a (AU). The observational field of view was
Fig. 5. Corrections to be applied to the power-law index (γ ) calculated by Bowell’s orbit, as a function of γ . The γ is called Bowell’s slope here for simplicity. The straight line is a least-squares-fitted curve to the data points shown as filled circles. The error bars for each point were estimated from variations among the three cases in table 3.
taken, as before, to be 5◦ × 4◦ centered at opposition and near the ecliptic. We changed the angular elements at random in (0◦ , 360◦ ). The relative biases were calculated for three kinds of orbits, namely circular orbits, slightly elongated orbits, and elliptic orbits whose eccentricity distribution is similar to that for existing asteroids. Figure 6 can be understood as follows. For example, let a circular orbit with a ∼ 6 AU and that with a = 3 AU contain the same number of asteroids. Then, figure 6 tells us that about 80% of them with a = 3 AU are observed in the above observation window. Similarly, one can see that only 60% of elliptic orbits with a = 3 AU are detected in the same window; in general, elliptic orbits are found to be more biased than nearcircular orbits. Therefore, such biases must be corrected to get the true number/distribution of asteroids. The cause of the abias can easily be interpreted at least qualitatively. This bias is due to a perspective effect that, when observed from the Earth’s orbit, a fixed observational window contains a greater number of objects uniformly distributed in the main-asteroid belt at more remote distances. Next, we examine the I -bias. In figure 7 are calculated the relative biases normalized with the number of near-ecliptic asteroids, for the inner-, mid- and outer-belt asteroids. The location and size of the observation window were the same as in figure 6. Orbital elements other than a are taken from a distribution similar to that for existing MBAs. We emphasize that the number of high-inclination asteroids observed in a small window near opposition and near the ecliptic are heavily biased; for example, in a specified observation window (the same as in figure 6), less than 10% of asteroids with I > 10◦ are detected, compared with the number of near-ecliptic asteroids. We would therefore say that the I -bias correction is of primary importance in our planned survey observations. In practical
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Fig. 6. Relative biases in a for near-ecliptic asteroids as a function of a. The relative bias is defined here to be the number ratio between near-ecliptic asteroids with r ∼ 6 AU and those with r = a (AU). The observation window of 5◦ × 4◦ was assumed, centered at near opposition and near the ecliptic plane. The relative bias curves are calculated for circular, near-circular, and elliptic orbits.
bias-analysis, we must calculate the relative bias corrections as a function of both a and I , but this is beyond the scope of our present paper. 5.
Discussion and Summary
5.1. Comparison with Short-Arc Orbit Determination In section 2, we estimated the likely errors in a and I obtained from our statistical method based on Bowell’s orbit. In particular, it was shown that, for orbits with I > 10◦ , the I error for individual asteroids is as large as 5◦ –6◦ . This tendency seems to be inevitable, as long as Bowell’s orbit is used, Hence, in this section, we investigate the possibility of another approach. In past issues of Minor Planet Circulars (published by the Minor Planet Center of IAU), there are many reports on the initial elliptic orbits of newly discovered asteroids, which were calculated from positions covering only a few days using a traditional orbit-determination method. Most of them became “numbered” asteroids with an elapse of time of about 10 yr, through repeated follow-up opposition observations. Therefore, for those asteroids, we can confirm how accurate their initial orbits are by comparing them with the definitive orbits of the corresponding numbered asteroids (notice that the orbital effects due to the precessional movement of the equatorial coordinate system and from planetary perturbations are much smaller than the errors in question here). We expect that the results given here will be useful to assess the orbits obtained if a traditional orbit determination were applied to the Subaru data which cover one day or so. For this purpose, we picked up nearly 90 asteroids that fulfill the conditions mentioned in the previous paragraph, taken from the MPCs for
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Fig. 7. Relative biases in I for the main-belt asteroids as a function of I . The relative bias here is also calculated as the number ratio between near-ecliptic asteroids and asteroids with the inclination I , for the same window as in figure 6. Three curves are for the inner-, midand outer-main asteroid belt.
∼ 1985–1995. Because we are interested in how the number of observations (n) and the observational time-span (∆t) affect the accuracy of the initial orbits, we selected only those asteroids with 1 d < ∆t < 6 d and 3 < n < 11. Figure 8 shows ∆a, ∆e, and ∆I as a function of the observation arc (∆t), where ∆a, ∆e, and ∆I are defined by the absolute values of the initial orbit minus the definitive orbit. For each panel, an exponential curve was fitted empirically by a least-squares method. We can see a weak correlation for each element, though the scattering of the data points is fairly large. These errors in ∆a, ∆e, and ∆I are considered, except for poor knowledge on eccentricity, to be mainly caused by positional errors of star catalogs used in astrometric reduction, and hence will not be drastically reduced for the present. Given the empirical curves also usable to our purposes, we attempt to estimate the orbital errors of our planned survey observations when a traditional orbit-determination is applied. Assuming one-night observations with the Subaru Telescope, the coverable observation-arc would be 0.5 d at longest. Then, as for ∆a, one can see from figure 8 that ∆a is about 0.2 AU at ∆t = 0.5 d. This value is comparable to that estimated statistically using Bowell’s orbit in table 1 of section 2. Similarly, we see that the mean ∆e error at ∆t = 0.5 d is ∼ 0.1. However, since the error ∆ω associated with e is found to be as large as 90◦ –100◦ and the r is calculated with e cos ω, the e-errors inferred from figure 8 are far from being useful for our purpose to improve the r-estimate. Next, let us examine the I -error. One can read from figure 8 the ∆I at ∆t = 0.5 d to be 2.5◦ –3◦ . The values seem to be roughly half the statistically-estimated I -errors given in table 2. If the difference is substantial, it might be better to perform a traditional orbit-determination rather than the statistical approach described in section 2, as far as high-inclination
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Subaru observations (Yoshida et al. 2001, 2002). Although we had to make a crude assumption on the heliocentric distance in reducing the data for the former testing run because of a very short time coverage (< 1 hr), we could apply the statistical procedure described in this paper very satisfactorily to the latter survey data. As a result, both observations could give a mutually consistent new outcome that the slope of the size distribution for sub-km MBAs has considerably lower values (γ ∼ = 1.0–1.3) than that for multi-km MBAs, and this slope was found to be a reasonable extension for the SDSS slope (Ivezi´c et al. 2001). The result also indicates that, according to figure 5, the systematic correction for the slope is almost unnecessary because of the proximity of the γ to unity. 5.3. Summary • We proposed a statistical method to estimate the semimajor axis and inclination of asteroids for which only sky motion vectors are measured, and extensively tested the method by Monte Carlo simulations. • The mean errors of semi-major axis and inclination for individual asteroids were found to be about 0.15 AU and 1◦ –2◦ (for I < 10◦ ), respectively. • It was shown that the slope of cumulative size distribution of asteroids can be determined with errors of ∼ 0.05–0.1. We thank the students, Yusuke Sato of Tokyo Gakugei University who provided us with the data used in figure 8, and Budi Dermawan of the University of Tokyo and Bandung Institute of Technology who kindly combined the text and figures in the final manuscript. Appendix
Fig. 8. Short-arc orbital element errors of asteroids estimated by a traditional orbit-determination method. The data are taken from the MPCs during 1985–1995. Vertical error bars stand for the standard deviation for the data points.
asteroids are concerned. Thus, this is a target for future investigations. Regarding the number of data points, there were no correlations of n at all with the errors in any orbital elements in the range of 3 < n < 11. This suggests that, in a short-arc orbit determination, the length of time-span of observations is important, and even circular orbits obtained from only two data points can sometimes be useful to obtain a better estimate of I . 5.2. Lessons Learned from an Actual Survey Observation After submission of the original version of this paper, we could obtain results from a testing run and a survey run of
Let us assume that survey observations of asteroids be made in a very narrow sky-field, near the ecliptic plane. In figure 9, S stands for the Sun, T the Earth, B an asteroid in opposition, and A another asteroid which is off opposition by the B–S–A angle (E), respectively. The orbit of the Earth is assumed to be circular. Let the velocity vectors of the Earth and asteroid A be ˙ y, ˙ z˙ ) respectively. Also, SA = r = x 2 + y 2 + z 2 V0 and v = (x, and ST = a0 (1 AU). Then, the ecliptic components (l, b) of the apparent motion vector for asteroid A are approximated with a sufficient accuracy for E º 15◦ –20◦ by l = (y˙ − V0 · cos E)/TA
(6)
and b = z˙ /TA, in which TA = r 2 + a02 − 2ra0 cos E.
(7)
(8)
We note that these expressions are valid only for E º 15◦ –20◦ , since the formulation here is not an exact one. ˙ y, ˙ z˙ ) by the orbital What we must do next is to express (x, elements of asteroid A. In order to make the mathematical expressions simpler, we take advantage of the orbital configurations specific to our planned survey observations.
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If, therefore, we measure (l,b) in the unit of V0 /a0 (= 1.991× 10−7 s−1 = 59. 14 d−1 ) and r, a and TA are measured by a0 (1 AU), the above two equations on (l, b) are expressed as √ (1 + ε) cos I / p = TA · l + cos E (11) and
√ (1 + ε) sin I / p = TA · b,
(12) ∼ where ε ≡ e cos ω with f + ω = 0. If the eccentricity and ω for each observed asteroid are given, equations (11) and (12) can be solved to obtain a and I in principle. However, the eccentricity can never been known from short-arc observations themselves. It is also noted from equations (11) and (12) that no knowledge of affects estimates of a and I most seriously when ω ∼ 0◦ or 180◦ . One practical way to tackle the problem is to assume that e ∼ = 0 (eccentricities for the majority of MBAs are less than ∼ 0.2–0.3); justification for such a compromise comes from an expectation that the r averaged over many asteroids with various ε’s will look like the case of e ∼ 0. Some manipulation of equations (11) and (12) with the assumption that e ∼ = 0 results in Fig. 9. Orbital geometry of asteroid observations near the ecliptic.
First, by taking the SA-line as x-axis, the nodal longitude (Ω) of asteroids detected in this observational field of view can be limited to Ω ∼ = 0◦ or 180◦ (see figure 9). Second, from the condition that asteroid A should be in the ecliptic plane (that is, β ∼ = 0), we have the relation f + ω = 0◦ or 180◦ , depending on whether the asteroid is at the ascending node or descending node, where f and ω represent the true anomaly and the argument of perihelion. Then, knowledge of the two-body problem teaches us that y˙ and z˙ can be expressed as y˙ = ±r f˙ cos I and z˙ = ±r f˙ sinI , respectively (the plus sign applies for f + ω = 0◦ ; the minus sign for f + ω = 180◦ ). Hereafter, we discuss only the case of f + ω = 0◦ for√simplicity. Considering that V0 = GM/a0 , r = p/(1 + e cos f ), and √ r f˙ = GM/p · (1 + e cos f ), where p = a(1 − e2 ), e is the eccentricity, M the solar mass, and G the gravitational constant, equations (6) and (7) can be rewritten as √ (1 + e cos f ) · cos I / p/a0 − cos E l = (V0 /a0 ) · (9) TA/a0 and b = (V0 /a0 ) ·
√ (1 + e cos f ) · sin I / p/a0 . TA/a0
(10)
a(a 2 + 1 − 2a cos E)(l 2 + b2 ) + 2al cos E a 2 + 1 − 2a cos E + a cos2 E − 1 = 0. (13) This could be solved iteratively, starting from such an initial guess of a that equation (14) gives, since E would be less than 10◦ –20◦ in practical applications. When E → 0, the equation tends to a quadratic equation, a(a − 1)(l 2 + b2 ) + 2al + 1 = 0.
(14)
By solving equation (14) and combining equations (11) and (12), we have the following final expressions (15) and (16) for the semi-major axis and inclination (for E ∼ 0◦ ). Note that in (15) and (16) the left-side terms are expressed with primes, to indicate that they are values obtained from an approximate formulation: a = 1/(2m2 ) · m2 − 2l ± (2l − m2 )2 − 4m2 (15) and tan I =
|b| , l + 1/(a − 1)
(16)
where m2 = l 2 + b2 . This solution corresponds to the asteroid located at opposition (point B) in figure 9, and is equivalent to that given in Bowell et al. (1990).
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Bowell, E., Skiff, B. A., Wasserman, L. H., & Russell, K. S. 1990, in Asteroids Comets Meteors 1990, ed. C. I. Lagerkvist, H. Rickman, & B. A. Lindblad (Uppsala: Uppsala Univ.), 19 ˇ Tabachnik, S., Rafikov, R., Lupton, R. H., Quinn, T., Ivezi´c, Z., Hammergren, M., Eyer, L., Chu, J., et al. 2001, AJ, 122, 2749 Jedicke, R. 1996, AJ, 111, 970 Jedicke, R., & Metcalfe, T. S. 1998, Icarus, 131, 245 Love, S. G., & Ahrens, T. J. 1996, Icarus, 124, 141
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Veverka, J., Thomas, P., Harch, A., Clark, B., Bell, J. F., III, Carcich, B., Joseph, J., Murchie, S., et al. 1999, Icarus, 140, 3 Watanabe, T. 1953, Journ. of the Univ. of Mercantile Marine, No. 4, Ser. A, 195 (in Japanese) Weidenschilling, S. J. 1981, Icarus, 46, 124 Wisdom, J. 1983, Icarus, 56, 51 Yoshida, F. 1999, Master Thesis, Fukuoka Univ. of Education, p.96 (in Japanese) Yoshida, F., Nakamura, T., Fuse, T., Komiyama, Y., Yagi, M., Miyazaki, S., Okamura, S., Ouchi, M., & Miyazaki, M. 2001, PASJ, 53, L13 Yoshida, F., Nakamura, T., Watanabe, J., Kinoshita, D., Yamamota, N., & Fuse, T. 2002, PASJ, submitted
