A. Kovacevic
Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia and Montenegro
Received 16 February 2004 / Accepted 2 September 2004
Abstract
Here I present a new attempt to determine the mass of (4) Vesta, based on its close approaches
with relatively well observed minor planets: (3057) Malaren, (5205) 1988CU7, (8331) Dawkins
and (21225) 1995GQ1. The obtained values for the mass of (4) Vesta are comparable with earlier determined ones.
Key words: celestial mechanics - astrometry
The masses for nearly 30 asteroids have been determined during the last 40 years although the history of asteroid observations is about two centuries long.
The knowledge of the masses and densities of asteroids is important in determining their composition, understanding the evolution of small bodies and of the Solar system itself, and for more precise modelling of asteroid perturbations of the high accuracy planetary ephemerides (Standish 2000).
As is well known, the method of
minor planet mass determination that considers gravitational perturbations produced by
a asteroid on other bodies (major planets, minor planets or
spacecraft) during a mutual close encounter was developed first.
However, this method is affected by significant formal errors of mass
derivation. For example, adopted masses of only five asteroids:
(1) Ceres; (2) Pallas; (4) Vesta; (11) Parthenope and (253)
Mathilde were determined with formal errors smaller than .
This may be a consequence of inhomogeneous distribution of
observations of perturbed bodies, their insufficient number and
accuracy or low gravitational effects. Nevertheless, the
majority of asteroid mass determination was based on single
asteroid close encounters.
As can be seen in Table 1, Hertz (Hertz 1966) discussed 59
observations of (197) Arete and determined the first value of an
asteroid mass. Enlarged collections of observations allowed
Hertz (1968) to recalculate and improve the mass of (4)
Vesta. Working on an extended and partly corrected set of
observations of (197) Arete, Schubart & Matson (1979)
obtained for the mass of (4) Vesta a value increased by .
Analyzing Viking lander radar ranging data, collected over
short time span (1976-1981), Standish & Hellings (1989)
obtained for the mass of (4) Vesta a greater value than the one
adopted nowadays. Later, the mass of (4) Vesta was obtained as
a solution of the JPL ephemerides DE403 (Standish et al. 1995)
and DE405 (Standish et al. 1998) where 300 asteroids
are included in the dynamical model. Hilton (1999)
determined the mass of (4) Vesta by using the largest asteroid (1) Ceres
as a perturbed body (the closest approach between these two bodies occured in 1893).
In spite of the huge number of
observations used (9354) and very careful data processing, the obtained
value is far from the adopted one and the formal error is about
of
the determined mass. An investigation of gravitational effects,
caused by mass of (4) Vesta on the motion of (1) Ceres, can
explain such a result. Using backward integration, differences in right ascension and
declination are only
and
respectively,
which are within the range of accuracy of observations. On the other hand, the gravitational influence of the mass
of Vesta calculated using forward integration (starting from the epoch JD 2 397 000.5) reached
in right ascension and
in declination at the instances of newly collected
observations.
In order to avoid problems of reliability of asteroid mass determination, Sitarski & Todorovic-Juchniewicz (1992) used the method of asteroid gravitational perturbations on the orbits of many other perturbed bodies. A simplified method was applied to mass determination of (4) Vesta by using 22 (Kuznetsov 1999) and 26 (Michalak 2000) suitable close encounters. In these cases, it could be seen that the standard deviation of the weighted mean value of the mass of (4) Vesta is strongly influenced by the number of individual determinations. Viateau & Rapaport (1997) noticed a strong perturbation of the orbit of asteroid (17) Thetis caused by two large minor planets, (4) Vesta and (11) Parthenope. These gravitational perturbations allowed Viateau & Rapaport (2001) to determine both the mass of (4) Vesta and the mass of (11) Parthenope. It was the first time that asteroid (17) Thetis had been used to determine the mass of Vesta. The authors emphasized the importance of a reliable level of accuracy of the mass of (4) Vesta, since this asteroid is the second most massive body in the main belt having a important gravitational influence on the orbits of many other solar system objects.
The mentioned reasons imply that new asteroid mass determinations (especially based on new recorded close encounters) are needed.
Table 1: Values for the mass of (4) Vesta determined by other authors.
The list of close encounters suitable for mass determination of 17 large main belt asteroids was published by Galad & Gray (2002). There are five strongly perturbed asteroids ((17) Thetis, (3057) Malaren, (5205) 1988CU7, (8331) Dawkins and (21225) 1995GQ1) due to gravitational influence of the mass of (4) Vesta, which were relatively well observed before and after a close encounter. Due to the fact that a close encounter with (17) Thetis had been successfully used for mass determination of (4) Vesta (Viateau & Rapaport 2001), it is not included in this study.
Geometrical and kinematical parameters (Table 2) as well as expected gravitational effects (Fig. 1) revealed the potential high efficiency of these close approaches. The influence of the mass of (4) Vesta on the trajectory of perturbed bodies has been determined by two separate computations of the orbits of perturbed bodies. The first one took into account perturbation by the adopted mass of (4) Vesta, whereas the second did not.
Bearing in mind that some other minor planets could perturb the motion of the chosen perturbed asteroids, the 9 largest asteroids have been included in the dynamical model, as well as all major planets. The mass values of perturbing asteroids used are given in Table 3. The gravitational influence of the perturbed asteroids on the perturber is negligible due to their small diameters.
Table 2:
Geometrical and kinematical parameters of close
encounters:
is minimum distance,
is relative velocity and
is angle of deflection of perturbed asteroid.
Table 3: Masses of perturbing minor planets.
The numerical integration of differential equations of motion of perturbed bodies is carried out by Addams-Bashforth-Moulton predictor-corrector method (Moshier 1992). The initial osculating orbital elements for the epoch JD 2 452 600.5 were taken from the Edward Bowell database (http://www.lowell.edu/users/elgb/).
In order to analyze the motion of perturbed asteroids, sets of
observational data were downloaded from the public database
AstDys (http://hamilton.dm.unipi.it/astdys). The classical
least-squares method was applied for the
mass determination of (4) Vesta. Corrections of
six orbital elements of the perturbed asteroid and correction of
the mass of (4) Vesta are computed simultaneously.
According to this method, the system of linear equations could be expressed in the matrix space as:
![]() |
(1) |
During the process of mass determination, it was noticed that, integrating backward and using Eq. (1),
partial derivatives
of the coordinates (right ascension and declination) of the perturbed body with respect to the perturbing mass
(coefficients of
),
calculated at the instants of postencounter observations are practically zero and vice versa, coefficients computed at
the moments of preencounter
observations in forward integration. Having in mind that the number of preencounter observational equations is 10 times
less than the
number of postencounter observational equations in each case, different results were expected for the mass and
its formal error. However, this did not affect the obtained results for the mass of (4) Vesta and corresponding
formal errors.
![]() |
Figure 1: Differences of the geocentric right ascensions of perturbed bodies caused by (4) Vesta ( left) and mutual distances between perturbed asteroids and (4) Vesta ( right). Differences of the geocentric right ascensions of perturbed bodies calculated at the instances of available sets of observations are denoted by asterisks. Pairs of plots from the top to the bottom correspond to close encounters with: (3057) Malaren, (5205) 1988CU7, (8331) Dawkins and (21225) 1995GQ1, respectively. |
Open with DEXTER |
Table 4: The distribution of observations of perturbed asteroids. In the columns N1, N2, N3, N4, N5 are given number of observations, number of discarded preenc. observations, number of used preenc. observations, number of discarded postenc. observations and number of used postenc. observations. T1 and T2 are time intervals covered by preencounter and postencounter observations. The standard deviations of orbital residuals are given in the last column.
The close encounters that occurred between the second largest perturber in the main belt and (3057) Malaren, (8331) Dawkins, (5205) 1988CU7 and (21225) 1995GQ1 have some common characteristics. The relative velocities are particularly low (Table 2) implying that these close encounters are long lasting. The evolution of the distance between (4) Vesta and the perturbed asteroids over time is given in Fig. 1. As can be seen from Table 2, the minimum distances between (4) Vesta and four perturbed asteroids are small. These close encounters of (4) Vesta are also characterized by the almost 3 year long
Besides the geometry of the mutual approaches, the quality of
asteroid mass determination depends on the number and
distribution of the observations. The time
spans covered by pre and postencounter observations of
perturbed asteroids (3057) Malaren and (5205) 1988CU7 are the
largest, as it can be seen from Table 4.
Inspection of Table 5 shows that the best comparable masses of (4)
Vesta with the adopted one have been calculated from its close
approaches with these two mentioned asteroids.
Prior to a more detailed analysis of the sensitivity of individual
solutions for the mass of (4) Vesta to the observation distribution and precision,
a global solution of the mass of Vesta was obtained:
.
According to the method proposed by Sitarski & Todorovic-Juchniewicz (1992) the set of 25
normal equations
corresponding to 2194 observational equations of four perturbed asteroids was created. Compared to the previous mass
determinations
of (4) Vesta (Table 1), it could be noticed that
the formal error thus obtained is the smallest one.
![]() |
Figure 2: Differences of the geocentric right ascensions of (17) Thetis (curve on lowest position), (67) Asia (curve on medium position) and (197) Arete (curve on highest position) caused by (4) Vesta. |
Open with DEXTER |
As already mentioned, the closest encounter occurred with (5205)
1988CU7. Almost 24 years had elapsed from the first observation of (5205) 1988CU7
to the long lasting closest approach with (4) Vesta.
During this period gravitational effects reached
in right ascension (Fig. 1).
Considering the time span of equal length relative to the close approach of the most effective single close encounters
that occurred with (17) Thetis (Viateau & Rapaport 1997) and
(67) Asia (Michalak 2000), it was noticed that they reached
and
in right
ascension, respectively. In the case of the latest close
encounter with (197) Arete corresponding gravitational effects
are less than
in right ascension within
the same time interval (Fig. 2).
Table 5:
Results of the mass determinations for (4)
Vesta. Mean densities based on a IRAS diameter of
are given
in the third column. In the last column are presented correlation coefficients between the mass
of Vesta and the mean motions of the perturbed asteroids.
Similar distribution characteristics of observations is
seen in the case of the close approach with (8331) Dawkins.
There are only five preencounter observations made
during the 12 day interval,
whereas there are 345 postencounter observations covering a 13 year period.
These five preencounter observations were made in May 1982,
only six years before the close approach.
The maximum variation in right ascension of
(8331) Dawkins due to the gravitational influence of
(4) Vesta is
within the six year
period before the closest approach. Considering the equal
time span prior to the closest encounters of the most effective
close approaches with (4) Vesta, it was noticed that greater
gravitational effects were seen on the orbit of (17)
Thetis (
)
(Fig. 2).
On the other hand, the gravitational effects calculated
by using backward integration (Fig. 1) are almost two times
lower than those calculated by forward integration (JD 2 434 500). Keeping this in mind, the mass
of (4) Vesta was determined by using both forward and backward integration.
There was no significant difference between obtained values.
The short time span of preencounter observations too near to the epoch of close approach, unequal
length of time spans covered by pre and postencounter observations and gravitational effect asymmetry influenced
the derived mass of (4) Vesta. Also, the high sensitivity of the obtained correction
of the mass of Vesta to the preencounter observations as well as to the newly collected postencounter
observations was noticed.
Collecting new observations provides a extension of the time interval
covered by postencounter observations in such a way that the
mass of (4) Vesta could be successfully determined.
For example, the value of
for the mass of (4) Vesta
is calculated by using 281 postencounter observations collected from 1990 to 2002.
Now, using a widened set of postencounter observations (340) collected until
October 2003, the value of
is obtained. Adding eight observations collected in October and November 2003 led to
for the mass of Vesta. Also, if it is possible to
find preencounter observations that were made far enough
from the close approach, a more reliable mass determination of (4) Vesta could be
expected.
As mentioned before, the total number and high
accuracy of new observations could affect the mass determination
of (4) Vesta based on this close encounter.
In the case of the mutual approach with (21225) 1995GQ1, the
maximum variation of the right ascension, as a consequence of gravitational perturbation of its orbit
due to (4) Vesta, is
after backward integration whereas it is three times greater after
forward integration (Fig. 1). The gravitational effects recorded in the preencounter time interval
are greater than those seen within the equal time spans of closest
approaches with (67) Asia and (197) Arete, whereas in the case
of (17) Thetis the gravitational influence is almost equal (Fig. 2).
As in the previous cases, calculated values
of the mass of (4) Vesta by using backward and forward integration (JD 2 434 300.5)
do not show significant discrepancies. As mentioned in the case of the close encounter with (8331) Dawkins,
the correction of the mass is sensitive to the preencounter observations used as well as to
the newly collected postencounter observations.
Therefore, the result obtained by using the set of
postencounter observations collected until June 2002 was
,
whereas after adding 72 observations collected until
the end of November 2003 the obtained value of the
mass of (4) Vesta is
.
It is possible that the uneven
distribution of observations and differences in gravitational effects affect the obtained
value of the mass of (4) Vesta (see Table 4). 100 postencounter observations
cover the time span 1995-2003, while only 4 preencounter positions of
the perturbed body were observed in 1972 during a very short 1 day interval. The maximum
variation of right ascension of
(21225) 1995GQ1 was
during the period when preencounter observations were made.
The available set of observations
of (21225) 1995GQ1 covers only seven oppositions, i.e. too small an arc of the orbit. Much more reliable calculation
of corrections of elements of the perturbed orbit and perturbing mass could be expected after a widening of the postencounter
set of observations. All previous attempts of determination of the mass of (4) Vesta
were based on close approaches with asteroids numbered
less than 5000.
Three of the four considered perturbed asteroids are relatively faint objects. Despite this, the obtained results are in agreement with previous determinations of the mass of Vesta and are of comparable precision. Using faint objects will provide an enlargement of the number of close encounters and perturbed asteroids suitable for asteroid mass determination. Concerning the adopted value of the mass of (4) Vesta the best results are obtained from the close encounters with (3057) Malaren and (5205) 1988CU7 (see Table 5), where strongly expressed gravitational effects, as well as distribution of observations on the post-encounter and pre-encouter part of the orbit, are well balanced.
However, the values of the mass of (4) Vesta obtained from its close approaches with (8331) Dawkins and (21225) 1995GQ1 (Table 5) are influenced by very different gravitational effects on the pre and postencounter part of orbit as well as by different length of time spans covered by preencounter and postencounter sets of observations and their number. In spite of the fact that these two results are too far from the adopted value, it can be seen that these deviations are comparable with deviations of the masses of Vesta obtained by some other authors using single close encounters.
Having in mind that gravitational effects are strong in all four cases of close encounters, we could expect that collection of new observations of (21225) 1995GQ1 and (8331) Dawkins will provide better postencounter coverage in such a way that a more reliable mass of (4) Vesta could be determined within the next few years.
The planned launch of the Dawn Discovery Mission (Russel et al. 2002) emphasizes the importance of new attempts of mass determination of the two largest perturbers in the main belt.
Acknowledgements
The author would like to thank M. Kuzmanoski, O. Atanackovic-Vukmanovic and D. Mirkovic for help. Also, the author is grateful to the referee for his/her invaluable suggestions and remarks. This research was supported by the Ministry of Science, Technologies and Development of Serbia through No. 1238 "Positions and motion of minor bodies of the solar system."