A&A 395, L17-L19 (2002)
DOI: 10.1051/0004-6361:20021444
M. Kuzmanoski - A. Kovacevic
Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Yugoslavia
Received 24 June 2002 / Accepted 2 October 2002
Abstract
From the perturbing effects of the asteroid (16) Psyche on the
motion of the asteroid (13206) 1997GC22 the mass of (16) Psyche is
determined. A close approach between these two asteroids at
,
and at a relative velocity of
,
occured in July 1974. The value of
has been found for the mass of (16) Psyche,
which yields a density of
.
These values are very different from those obtained by other
authors, but the mass is much closer to the value based on the IRAS
estimation of (16) Psyche diameter and its M-type taxonomical
classification.
Key words: celestial mechanics - minor planets, asteroids - methods: numerical
It is well known that the gravitational influences of the large main-belt asteroids in the motions of both other asteroids and some major planets, can not be neglected. The achieved accuracy in the measuring of the positions of the solar system bodies requires including of large asteroids as perturbing bodies in the dynamical model of the solar system. Thus, in the construction of DE403 (Standish 1995) and DE405 (Standish 1998) ephemerides, 300 asteroids were included, with the masses determined by different methods.
Since the first successful attempt (Hertz 1966) to determine an asteroid mass by the most frequently used dynamical method, the masses of nearly 30 asteroids have been estimated in the same way. Summary of almost all estimations of asteroid masses until 2001 is given by Krasinsky et al. (2001). Generally, the asteroids with mass determined by dynamical method belong to C or S types, with the exception of (16) Psyche, which has been classified as M type asteroid.
The first determination of the mass of (16) Psyche, based on close
encounter with asteroid (94) Aurora, was made by Viateau (2000). In
this case, the values of
for
the mass and
for the density
were calculated. However, Krasinsky et al. (2001), from close
encounters with three asteroids ((263) Dresda, (2819) Ensor, (2589) Daniel), found a weighted mean value of
.
These results are very different with respect
to the mass based on the IRAS data. Using the mean diameter of
Psyche given by IRAS
and the mean density
of
for an asteroid of M-type
(Standish 2000; Krasinsky et al. 2001), one gets an estimated value of
.
In a list of asteroid close encounters we compiled, we have found
the close encounter which occured between asteroid (16) Psyche and
asteroid (13206) 1997GC22. These two asteroids approached each
other on July 17, 1974, at the minimum distance of
.
The relative velocity at that moment was
,
while the angle of deflection of the perturbed
asteroid was
.
These kinematical parameters
revealed that this close encounter should be suitable for mass
determination of (16) Psyche. For the analysis of the motion of
(13206) 1997GC22 we used the observational data, as well as
initial orbital elements, which can be found at the public database
AstDys (see http://hamilton.dm.unipi.it/astdys).
There are 127 available observations which cover the time span
1960-2002, but with an uneven distribution. Thus, 123 postencounter
observations cover the time span 1995-2002, while only four
preencounter positions of the perturbed body were observed in 1960,
during a very short 9-day interval. During the process of
calculation of the perturbed orbit, only two of
postencounter observations were discarded. Both coordinates (right
ascension and declination) were discarded if one of them gave a
residual above .
Calculated RMS of orbital residuals
is
.
The numerical integration of the differential equations of motion
is carried out by Addams-Bashforth-Moulton predictor-corrector
method, implemented by Moshier (1992). In the dynamical model we
have included all major planets plus four largest asteroids (their
masses as used are given in Table 1), as well as (16) Psyche as a
perturbing body.
Bearing in mind that some other asteroids could perturb the motion of (13206) 1997GC22 during the interval of time considered, the close
encounters between this asteroid and all asteroids larger than
in diameter were searched.
Also, gravitational effects of
these asteroids on motion of (13206) 1997GC22 were analysed.
It was found that only perturbing effects of (1) Ceres were
in right ascension and
in declination, as a result of two
close encounters: in 1964 (at distance of
)
and 1975
(at
).
Perturbing effects of all other asteroids were found to be negligible.
The mass determination of (16) Psyche was performed by means of the
classical least-squares method, widely used by many authors.
According to this method, as it is well known, correction
of the mass of the perturbing asteroid is computed along with
the corrections
of six
osculating elements of the perturbed asteroid. These corrections
are solutions of the system of linear equations:
![]() |
(1) |
Firstly, we have computed variations of the mutual distance
between (16) Psyche and (13206) 1997GC22 for the period
1960-2002, covered by observations. During this period only one close
encounter has occured, as can be seen in Fig. 1. Note that the
distance changed very slowly around the epoch of the minimum, so
that these asteroids were at a distance smaller than
from June 25 to August 7, 1974.
![]() |
Figure 1: Mutual distance between (16) Psyche and (13206) 1997GC22 from 1960 to 2002. |
Open with DEXTER |
Calculation of the mass of (16) Psyche was done by means of
initial osculating elements of the perturbed asteroid for two
epochs: JD 2452500.5 used for backward integration and
JD 2437000.5 used for forward integration. In both
cases, the numerical procedure has been initialized by an assumed
value of
for the mass of (16)
Psyche. Integrating backwards and using Eqs. (1),
coefficients of
(partial derivates
)
at the moments of postencounter observations
are practicaly 0, as well as the coefficients computed at the
moments of preencounter observations in forward integration.
Bearing in mind the number of preencounter (4) and postencounter (121) observations, different results were expected for the mass
and its formal error. However, differences between obtained values
were negligible.
The same method has been applied on both, the Keplerian orbital
elements and the position and velocity vectors, for both above
mentioned two initial epochs. Obtained values for mass and its
formal error were identical.
Correlation coefficients between parameters are given in the Table 3 (using Keplerian orbital elements) and Table 4 (using initial position and velocity vectors).
![]() |
![]() |
The final value obtained for the mass of (16) Psyche is
.
Its mean density, based on IRAS
diameter of
,
is thus
.
As can be seen, the value obtained in
this work is much closer to the one estimated from the taxonomic
type and IRAS data,than those previously available.
This result should be quite reliable, because the perturbing
effects of (16) Psyche on the motion of asteroid (13206) 1997GC22
were very large. The differences in right ascensions and
declinations, as inferred from the two forward integrations, with
and without (16) Psyche in the dynamical model, can be as large
in right ascension (shown in Fig. 2), and
in declination (shown in Fig. 3).
In this sense, very different results for the mass of (16) Psyche, found by Viateau (2000) and Krasinsky et al. (2001), could be explained by the small corresponding effects on motions of perturbed asteroids used in their determinations.
![]() |
Figure 2: Differences of the geocentric right ascensions of the perturbed body (13206) 1997GC22, taking into account the gravitational effects of the perturbing body (16) Psyche. |
Open with DEXTER |
![]() |
Figure 3: Same as Fig. 2, but for declinations. |
Open with DEXTER |
The values for the mass
and
density of (16) Psyche
obtained from the
close encounter used in this analysis is the first successfull
attempt based on a dynamical method, leading to the conclusion
that composition of an asteroid is metallic. The mass of Psyche
based on its taxonomic
type is about
smaller.
For still better determination of the
mass and density of (16) Psyche from this close approach one needs new
observations at more oppositions and more precise determination of
the orbit of the asteroid (13206) 1997GC22. Of course, one can
expect that other asteroid close encounters with (16) Psyche will
become available in the future.
Acknowledgements
Authors would like to thank S. L. Moshier for providing computer program for numerical integration. Also, they are grateful to Z. Knezevic for his useful 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."