A&A 370, 311-319 (2001)
Astronomical Institute, Faculty of Mathematics, Physics and Informatics,
Comenius University, Bratislava,
Received 17 July 2000 / Accepted 31 January 2001
The first 9511 numbered asteroids are studied in terms of their mutual closest approaches and encounter velocities during the period from November 6, 1967, to September 13, 2023. Several large asteroids (diameter and above) were (will be) encountered by smaller counterparts within a distance of . Thus, they are possible candidates for mass determination by the astrometrical method. Similarly, the search for effective perturbers is extended to even smaller asteroids for the much closer separation distance of and below. Only the simplified method for evaluation of observable effects on a perturbed body is used. Asteroid masses alone are not computed here. But a stronger criterion to reveal pairs for this purpose in comparison to some specially devoted papers should compensate for the difference and act as a reliable test. The best candidates for mass determination at present are asteroids (1), (2), (4), (10), (11), (24), (52) and (65). This list may be extended by at least (29) in the next 5 years and by many others in the next two decades. Several other strong perturbers from the last three decades are not included in the list, while there is still only a limited number of (or no) precise and reliable observations of perturbed asteroids before a close encounter. It seems that a perturbation by (10) is at least as effective as that by (2) and could be included in asteroid orbit determination in the future. Except for their bulk density determinations (knowing the size), the masses of perturbers could occasionally be used to improve the precision of the computed orbit for perturbed large-numbered and unnumbered asteroids as well.
Key words: minor planets, asteroids - astrometry - ephemerides
Asteroid observations have a long history. Larger telescopes, more sensitive detectors and different techniques were used to increase the number of known bodies. Due to observational effects large and close asteroids were discovered at first. Most of the known asteroids orbit the Sun at a distance of 2-4 AU - in the main belt.
The orbits of asteroids are usually computed by numerical integration. This takes into account the perturbations by the eight planets (Mercury to Neptune) and, in addition, by the three largest minor planets (1) Ceres, (2) Pallas and (4) Vesta from the main belt of asteroids (Marsden 1995). The last three of these perturbers contain about half the mass of the belt. The Earth-Moon system may be considered as one body with the sum of their masses or, in the case of Earth-approaching asteroids, perturbations by both bodies are taken separatedly.
It is known that the collisions of known objects are extremely rare even in the main belt. Thus, it is expected that sufficiently high accuracy of orbit solutions for a long time period should be reached for numbered asteroids. However, from time to time interasteroidal close encounters occur. Then, some of the larger asteroids may cause deflection from the ephemeris position of their smaller counterparts. Knowledge of the encounter geometry enables us to compute the masses (and densities, if the size and shape is also evaluated) of new perturbers while the perturbed body residuals (observed minus computed positions) should decrease. Precise astrometric positions of the perturbing and perturbed bodies are needed over several years before and after their close encounter. The observations from the period closer to the encounter are helpful, as well.
The astrometrical method was used not only for the mass determination of the three largest main-belt asteroids mentioned above. Many times it was used even for the smaller ones (see e.g. Hoffmann 1989; Kuzmanoski & Knezevic 1994; Viateau 2000 and references therein). A deflection as small as from the ephemeris position of the perturbed body can be measured several (say 10-20) years after an encounter by optical telescopes from the ground. Subarcsecond measurements by meridian circles (Viateau & Rapaport 1997) and astrometric satellites like Hipparcos (Bange 1998) are the most precise and valuable - they reduce the time needed for deflection measurements.
Mass is an important physical characteristic. The rapid increase in new asteroid discoveries and the improvement (in precision) of poorly determined orbits in recent years challenge as to determine it for many objects. The first step is to provide a database of suitable close encounters. Such searching was recently done by Hilton et al. (1996) in quite a large set of asteroids. They integrated 4583 orbits for 57 years. The subject of this paper is to recognize the best candidates from the main belt for asteroid mass determination in a larger database.
The asteroid orbital elements used in this paper were taken from the CD Guide 7.0 (Gray 1998). In this software the orbital elements for various epochs are included due to the need of rapid drawing of asteroids in star charts for any time in the recent past and near future. Only a little programming is needed to convert them from a compressed form to a readable format.
The initial osculating elements for epoch October 14, 1998 (JD 2451100.5) were based on observations and were taken from Bowell et al. (1994). All perturbations mentioned above were taken into account. However, orbital elements for the other epochs derived on the CD are adjusted for planetary perturbations only. Asteroidal perturbations were not included at all. The asteroid positions between adjacent epochs can be computed as a two-body problem at first approximation. Thus, it is possible to study the whole sample of asteroids for mutual approaches without an additional numerical integration. But we should bear in mind that the computations of orbital elements excluding the perturbations by large asteroids is a source of inaccuracy in the estimation of the close encounter's effect on these asteroids. The actual positions of asteroids may differ from their computed ones far from the initial epoch. These can be checked by examining observations from the past. According to the CD's manual this comparison was done for some asteroids, and the difference in position was less than 1 arcsec.
By the release of the CD (October 1998) there were 9511 numbered asteroids and thousands of multiopposition and singleopposition unnumbered asteroids known. In this paper only numbered asteroids were studied during the period from November 6, 1967 (JD 2439800.5) to September 13, 2023 (JD 2460200.5). In general, their orbits are much better determined in comparison to multiopposition unnumbered asteroids. Only (719) Albert was lost at the time of the CD's release. Increasing the number of items (unnumbered asteroids) and choosing a longer time period would affect not only the time needed to process the data but also the reliability of data.
According to Carpino & Knezevic (1996)
the magnitude of the change in the mean motion of the perturbed body depends on several quantities during the encounter.
Neglecting the orbital eccentricity of the scattered asteroid, they obtained
To find out possible candidates for mass determinaton, the mass of the
perturber should be assessed from asteroid diameter D and the evaluated
value of its bulk density ,
as it is not known in advance.
Three quantities (D, r and v) out of six that characterize may differ by several orders of magnitude (in different pairs).
An auxiliary P parameter can be defined by them so that
P is a measure of .
In this paper the search for possible candidates for mass determination
is focused on the search for largest values of P.
It is computed from the equation
It must be noted that neither the P parameter nor are valid outside the sphere of action of the perturber. But the computation of the P parameter is just an approximation allowing comparisons between the encounters.
The following steps were taken to gain and process the data in chronological order:
As can be seen in Tables 1 and 2, the lowest values of the P parameter
for pairs considered to be significant for the mass determination are as
small as 1
These need at least some comment.
Hilton et al. (1996) searched for convenient close approaches
using a scattering formula. But according to Carpino &
Knezevic (1996) the use of only angular deflection of
the trajectory produced by the close approach ()
may lead to
overestimating the importance of low-velocity encounters, since, in fact,
we are interested in the change in relative velocity
Contrary to ,
the P parameter directly represents
could better serve as a measure of observable effects on a
The main results of this paper are summarized in Tables 3 and 4. These contain pairs with the largest values of their Pparameter encountered within . Only pairs with were chosen. These are higher than most of the P values in pairs suspected to use for mass determination according to both Hoffmann (1989) or Hilton et al. (1996), respectively. In fact, these references served to set a low limit for the values of parameter P due to an unknown needed. As this paper is not focused on (1), (2), and (4) perturbations, the comparison of the results is restricted. For example, only one pair from the former (and much shorter) list - (65) Cybele and (526) Jena - really passed the chosen criterion.
Table 5 contains additional pairs with encountered at a mutual distance of . Pairs with (1) and (4) were removed. Otherwise, this table would contain 107 extra items with (1) and 98 with (4). The largest P value would be 133 here, but it is mentioned in the third part of the Table 2 - since it was recognized by Hilton et al. (1996). Only six pairs were left - one pair with (2) and five with (10). These are really strong perturbators, but much less effective than (1) and (4) are, if one judge by the numbers of pairs as indicators of effectiveness of encounters. Although (2) is large, its orbit is highly inclined to the ecliptic. It means that effective encounters are rare and even v is large.
P has a large uncertainty (of the order of 0.1-1 ) mainly due to the uncertainty in diameter D (and albedo). The uncertainty of the separation distance r is mostly not so crucial until an extremely close encounter occurs (e.g. in a distance below 0.0001 ). In such cases the uncertainty in P may exceed even . Only two such pairs were significant in that manner (from step 5). In 2014 (1961) will be encountered by (5569) at a distance of and (445) by (1764) at a distance of , respectively. The uncertainties are more than and actual values of P may not pass the chosen limit. The sizes of these perturbers are far below . The diameter of (1961) may be overestimated due to extremely low albedo of 0.019 used here. An additional uncertainty for P comes from the computed asteroid orbit alone that affects the encounter geometry. In general, this one increases with the time separation from the 1998 (encounters both back to the past and into the future), as the asteroid perturbations are not taking into account from this starting year of integration. For example, the orbital elements used for (17) in 1968, when it encountered (11), were not precise due to a strong perturbation by (4) in 1996 ( ). One should determine its orbit from precise observations before 1996 only or take the perturbation by (4) into account, but this was realized by Viateau & Rapaport (1997), who used the (11)-(17) encounter for mass determination. In this special case even (17) as a large asteroid contributes to the total P. Its actual value in Table 3 should be . (These asteroids met each other even in 1997, but related P was below .)
The largest P ( ) was computed for the pair (1) and (5303) in 1996 (Table 3). Neglecting the perturbation by (1) the ephemeris position of (5303) would be more than off from its real position after 3.5 years! Such a large perturbation alone may improve the mass determination of the largest asteroid. Improvements can be reached from approaches of others, as well. Similarly, the mass determination of (4) Vesta can be improved from several large perturbations. The masses of smaller preliminary candidates could be specially computed including other quantities that may cause deflection from the ephemeris. However, the most crucial element for mass determination of smaller perturbers is the time period covered by reliable and precise observations.
The P parameter may be used to evaluate the deflection from the ephemeris under some assumptions only, namely the perturbers's density and energy efficiency. Assuming the density of the perturber is , the semimajor axis of perturbed body , and the energy efficiency , the correction factor of 0.0065 may be used to convert the values between the parameter and . This means, that the perturbed body deflection of could be revealed after two to three decades (in the main belt), if . Smaller P may be used only if precise observations are available during much longer time period (for low-numbered asteroids) or astrometric measurements reached subarcsecond precision during several years. The latter are relatively new and still do not cover the period of more than a decade for most objects.
The last column of Tables 3-5 contains the year from which astrometric positions for both members of a given pair are known. It helps to recognize if that pair can be used for mass determination. But it is not always a reliable indicator due to the low precision of some measurements (mostly the very earliest) in the past and the long intervals between them. Perturbers that are emphasized in Tables 3 and 5 have convenient conditions for mass determination at present. Several asteroids (perturbers) were not emphasized due to poor (e.g. less than 10 observations, positions above precision) or no observations of related perturbed bodies before an encounter. For example, (511) was encountered by (7191) in 1969, but the latter has only 4 observations before the approach, although 2 of them are from 1949 - 20 years to the encounter. The first astrometric observations for the perturbed bodies by (121) and (704) in 1982 and 1997 come from only 12 and 13 years to the encounter, respectively. Almost all pairs in Table 4 could probably be used for mass determination in the future. The only exception is probably (104)-(9049), as the latter will have 13 years period of observations before an encounter with in 2004. On the other hand, the development of milliarcsecond astrometry may enlarge the number of observations that are very accurate. The limit put on the P parameter ( ) in Table 4 may become too conservative then.
The Tables 3-5 cannot be viewed as a complete list of pairs with , even excluding those with (1) or (4). Several large asteroids may be larger than assumed here (from taxonomic class or assumed albedo), or some dark asteroids with absolute magnitude slightly above 8 (the limit in step 6) are so large that if encountered very slowly by a counterpart ( ) at a distance just above (the limit in step 5) P could exceed a given threshold.
|Assumed albedo of 0.04 due to P type.|
|Large uncertainty due to r <<.|
|(10)||Hygiea||(1965)||van de Kamp||2000/10/15||0.02141||3.296||6.3||1927|
The parameter P is defined to reveal the effective perturbers for some asteroids. It is connected to the change in relative velocity and mean motion of the perturbed body. These quantities are sensitive to the mass of the perturbing body and to the closeness and mutual velocity of the perturbed body. Asteroid masses alone are not computed here.
There are several pairs with a large value of P among the first 9511 numbered asteroids. Larger members of these are good candidates for mass determination by the astrometric method, if precise measurements (for both) are known long before their mutual encounter. Unlike computations that use only angular deflection of the perturbed body trajectory as a criterion in searching for the effective perturbators, the P parameter should not overestimate low-velocity encounters.
Special attention has been paid to finding smaller perturbers in addition to the three largest ones (from the main belt) that are usually taken into account for orbit determination. Thus, the search for pairs is limited mainly to mutual encounter distance of . It is found that at present the masses of asteroids (1) Ceres, (2) Pallas, (4) Vesta, (11) Parthenope, (24) Themis, (52) Europa, and (65) Cybele could be best evaluated from encounters by other asteroids within such a distance. In Table 3 they are in bold type in pairs that could be used for this purpose. Astrometric observations of related perturbed bodies are available for several years (or decades) before an encounter. In the next 5 years at least (10) Hygiea and (29) Amphitrite could be added to the list due to recent encounters with smaller asteroids in 1998 and 1994, respectively. Table 4 summarizes pairs, the majority of which can be used for mass determination during the next decades.
Asteroid approaches to the fifteen largest main belt asteroids were studied up to a distance of . It was found that except for (1), (2) and (4), even (10) as a fourth largest asteroid is an effective pertuber above limit. Moreover, its mass could be determined at present from one such distant encounter (emphasized in Table 5), in contrast to recent encounters by several asteroids within a distance of (Table 3) whose preencounter positions are probably not sufficient to reveal deflection from ephemeris yet. In the total number of significant encounters it can displace even (2), since the latter has a highly inclined orbit! Thus, its mass could be included in orbit determination in the future as a standard as well (at least for the outer part of the main belt).
In addition to these candidates, the list of large perturbers will rapidly increase due to new discoveries (by the end of 2000 the number of numbered asteroids doubled), due to longer period covered by observation (recent discoveries scarcely have observations back to the 1970s, but observations from the 1990s are evenly covered for lot of asteroids which allow to rely on observations and make the orbit more precise) and due to observations of better precision (astrometric satellites, meridian circles, radar), respectively. The chosen selection process should guarantee to list (nearly) all pairs of . This large value of P enable to reveal possible candidates for mass determination quickly. Within 3 decades the deflection from the ephemeris of perturbed asteroid can be >1 . Some significant perturbers can't be used for mass determination at present while preencounter astrometric observations of perturbed bodies are missing. However, they may be recorded on older plates, e.g., from the sky surveys from the 1960s. In the near future, encounters with smaller values of P may become of great interest. This could be done next - e.g., after new version of CD GUIDE is released (encounters with smaller P values are far from completeness at this point - only some of them are available by request).
Multiopposition asteroids become numbered after a reliable orbit is computed. Standard orbit determination may not be sufficient to cover all precise observations with small residuals in extremely rare occasions over a very long time period (several decades). There are two possible ways to decrease the orbit uncertainty.
This work was supported by the Slovak Grant Agency for Science, VEGA, (grant No. 1/7157/20). The author would like to thank Bill Gray from Project Pluto, Bowdoinham, ME, USA, for his kind advice and allowing me to use orbital elements from the CD GUIDE 7.0.