Content uploaded by Vladimir A. Shefer
Author content
All content in this area was uploaded by Vladimir A. Shefer on Feb 09, 2015
Content may be subject to copyright.
0038-0946/03/3704- $25.00 © 2003
åÄIä “Nauka
/Interperiodica”
0326
Solar System Research, Vol. 37, No. 4, 2003, pp. 326–332. Translated from Astronomicheskii Vestnik, Vol. 37, No. 4, 2003, pp. 356–363.
Original Russian Text Copyright © 2003 by Shefer.
INTRODUCTION
The computation of a preliminary orbit is the first
stage in the determination of the orbit of a newly dis-
covered celestial body. This problem is usually solved
based on the minimum number of required observa-
tions made at short time intervals and ignoring the
influence of perturbing forces on the body motion. In
other words, the traditional approach is based on the
solution of the two-body problem. Along with inevita-
ble observational errors, such a model of motion dis-
torts appreciably the parameters of the preliminary
orbit. These distortions increase with the influence of
perturbations. They are especially strong in the cases
where the new object is observed when it approaches
closely a massive perturbing body (planet). It goes
without saying, that the preliminary orbit is then further
gradually improved using additional observational data
sets. The problem of constructing a sufficiently accu-
rate preliminary orbit is nevertheless quite topical.
First, such an orbit would make it possible to compute
a reliable long-term finding ephemeris to allow con-
tinuing observations. Second, it would serve as a good
initial approximation for further refinement of the orbit.
Now with the availability of modern high-precision
optoelectronic and radiotechnical positional observa-
tions, which are one to three orders of magnitude more
accurate than classical asrometric measurements, the
accuracy of the preliminary orbit obtained via the tradi-
tional approach may turn out to be much lower than that
of the reference observations. It is therefore of great
interest to develop methods that would make it possible
to construct the preliminary orbit with an accuracy
matching with the ever-increasing accuracy of the
observations employed, while preserving the advan-
tages of the traditional approach. Such methods may be
based on an analytical solution of some appropriate
intermediate orbit accounting for the principal pertur-
bations.
In this work I suggest a method, which satisfies the
above requirements. I consider the classic problem of
the determination of an orbit for a celestial body using
three observations and the corresponding time instants.
To solve this problem with the allowance for perturba-
tions, we use as a basis the scheme of the classic
Lagrange–Gauss method. Here we call this method by
the name suggested by Subbotin (1968), although in the
special literature and especially, in foreign publica-
tions, it is known as the Gauss method. In the method
that I propose here the principal part of perturbations is
allowed for via the earlier derived intermediate orbit
(Shefer 2003) based on two position vectors and the
corresponding time interval.
DESCRIPTION OF THE METHOD
Consider now the motion of a small body (an aster-
oid, comet, or a spacecraft) under the action of Newto-
nian gravitation produced by a system of point masses
(the Sun, major planets, satellites of planets), and other
forces of arbitrary nature. Here we study the motion of
the small body in the heliocentric Cartesian equatorial
coordinate system. Let us write the differential equa-
tions of motion in the general form:
(1)
where
x
is the vector of the position of the small body;
G
, the summary vector of acceleration, and dot means
differentiating with respect to time
t
.
Suppose that for each of the three time instants
,
, ( <
< )
we have a pair of observed angular
coordinates: the right ascension and declination
x
˙˙
G,=
t
1
0
t
2
0
t
3
0
t
1
0
t
2
0
t
3
0
α
i
0
δ
i
0
A Method for the Determination of an Intermediate Orbit
from Three Positions of the Small Body on the Celestial Sphere
V. A. Shefer
Research Institute of Applied Mathematics and Mechanics, Tomsk State University, Tomsk, 634050 Russia
Received February 12, 2002; in final form, June 24, 2002
Abstract
—A new method is suggested for finding the preliminary orbit from three complete measurements of
the angular coordinates of a celestial body developed by analogy with the classic Lagrange–Gauss method. The
proposed method uses the intermediate orbit that we had constructed in an earlier paper based on two position
vectors and the corresponding time interval. This intermediate orbit allows for most of the perturbations in the
motion of the body. Using the orbital motion of asteroid 1566 Icarus as an example, we compare the results
obtained by applying the classic and the new method. The comparison shows the new method to be highly effi-
cient for studying perturbed motion. It is especially efficient if applied to high-precision observational data cov-
ering short orbital arcs.
SOLAR SYSTEM RESEARCH
Vol. 37
No. 4
2003
A METHOD FOR THE DETERMINATION 327
of the small body. Hereafter subscript
i
acquires values
1, 2, or 3 associating the given quantity with the corre-
sponding time instant. Without restricting the general-
ity of our analysis, hereafter we assume that angular
coordinates are the geocentric coordinates referred to
the instants of ephemeris time. Geocentric and helio-
centric coordinates of the small body are related to each
other via the following vector equation:
(2)
where the geocentric vector
S
i
of the position of the Sun
is determined at the observing time
,
and
L
i
is the unit
vector of the form
Here the unknown quantities are the heliocentric posi-
tion vector
x
i
and geocentric distance
ρ
i
.
Lagrange (1778) showed that in the case of short
time intervals between observations simple approxi-
mate formulas exist for heliocentric coordinates. These
formulas allow rather good first approximations to be
obtained for geocentric distances. The solution found
by Lagrange acquired a practically convenient form in
the famous treatise of Gauss (1809). Among more
recent works where the inference of this solution is
described in detail one should mention the monographs
of Dubyago (1949), Escobal (1965), Subbotin (1968),
and Herrick (1971).
We now use the formulas of the classic approach to
obtain
ρ
1
,
ρ
2
, and
ρ
3
in the first approximation. After
this stage, the observing times should be corrected to
account for the time it takes the ray of light (signal) to
cover the distance from the small body to the observer:
(3)
where
c
is the speed of light. This procedure performs
the transition from the observed time to real time.
Thus equations (2) define three position vectors
x
1
,
x
2
,
and
x
3
at time instants
t
1
,
t
2
,
and
t
3
(
t
1
<
t
2
<
t
3
)
. Their
values at this stage are approximate and require further
refinement.
Consider now the two vectors
x
1
and
x
3
correspond-
ing to the positions of the small body at times
t
1
and
t
3
,
respectively. In accordance with the theory of interme-
diate motion developed in my earlier paper (Shefer
2003), let us compute the position vectors
,
and
on the intermediate orbit relative to the fictitious attract-
ing center; the position vectors
Z
1
and
Z
3
of the ficti-
tious center, and the gravitational constants
μ
1
and
μ
3
.
To this end, we use the following formulas:
x
i
ρ
i
L
i
S
i
,–=
t
i
0
L
i
δ
i
0
α
i
0
coscos
δ
i
0
α
i
0
sincos
δ
i
0
sin
⎝⎠
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎛⎞
.=
t
i
t
i
0
1
c
---
ρ
i
,–=
q
1
*
q
3
*
(4)
where
Here (
a
·
b
) denotes the scalar product of two vec-
tors
a
and
b
. According to Shefer (2003), we obtain two
positions on the intermediate orbit in the space of para-
metric variables
u
. These vectors are defined by follow-
ing vectors:
(5)
and the corresponding instants of fictitious time:
(6)
We determine the parametric position vector
u
2
and
the corresponding instant
θ
2
of fictitious time at the
middle time instant
t
=
t
2
from the following formulas:
(7)
where
(8)
The Keplerian orbit in the parametric space is planar
and therefore vectors
u
1
,
u
2
, and
u
3
are related as fol-
lows:
(9)
where
(10)
The quantities η
12
, η
23
, and η
13
in formulas (10) are the
Gaussian ratios of the areas of the sectors of conic sec-
q
1
*
λ
1
G
1
, q
3
*
– λ
3
G
3
,–==
Z
1
x
1
q
1
*
, Z
3
– x
3
q
3
*
,–==
μ
1
λ
1
G
1
2
R
1
*
, μ
3
λ
3
G
3
2
R
3
*
,==
λ
1
G
3
2
G
1
G
1
G
3
⋅()G
3
–[]x
3
x
1
–()⋅{}
G
1
2
G
3
2
G
1
G
3
⋅()
2
–
---------------------------------------------------------------------------------------
,=
R
1
*
q
1
*
,=
λ
3
G
1
2
G
3
G
1
G
3
⋅()G
1
–[]x
3
x
1
–()⋅{}
G
1
2
G
3
2
G
1
G
3
⋅()
2
–
---------------------------------------------------------------------------------------
–,=
R
3
*
q
3
*
.=
u
1
q
1
*
, u
3
μ
3
μ
1
-----
q
3
*
==
θ
1
0, θ
3
μ
3
μ
1
-----
t
3
t
1
–().==
u
2
μ
2
μ
1
-----
q
2
*
, θ
2
μ
2
μ
1
-----
t
2
t
1
–(),==
q
2
*
x
2
Z
2
,–=
Z
2
t
3
t
2
–
t
3
t
1
–
-------------
Z
1
t
2
t
1
–
t
3
t
1
–
-------------
Z
3
,+=
μ
2
μ
1
μ
3
t
3
t
1
–()
μ
1
t
2
t
1
–()μ
3
t
3
t
2
–()+
-------------------------------------------------------
.=
u
2
d
1
u
1
d
3
u
3
,+=
d
1
η
13
θ
3
θ
2
–()
η
23
θ
3
θ
1
–()
-----------------------------
, d
3
η
13
θ
2
θ
1
–()
η
12
θ
3
θ
1
–()
-----------------------------
.==
328
SOLAR SYSTEM RESEARCH Vol. 37 No. 4 2003
SHEFER
tions and triangles constructed for the vector pairs {u
1
,
u
2
}, {u
2
, u
3
}, and {u
1
, u
3
}, respectively.
In view of formulas (2), (4), and (5)–(8), vector rela-
tion (9) can be rewritten in the following form:
(11)
where
Let us now introduce the following designations for
the components of vectors L
i
and Q:
We now solve equations (11) for ρ
1
, ρ
2
, and ρ
3
to
obtain:
(12)
(13)
d
1
μ
1
ρ
1
L
1
μ
2
ρ
2
L
2
– d
3
μ
3
ρ
3
L
3
+ Q,=
Q d
1
μ
1
S
1
Z
1
+()=
– μ
2
S
2
Z
2
+()d
3
μ
3
S
3
Z
3
+().+
L
ix
L
iy
L
iz
,,{}L
i
, Q
x
Q
y
Q
z
,,{}Q.≡≡
ρ
1
1
d
1
μ
1
-----------
a
11
Q
x
a
12
Q
y
a
13
Q
z
++(),=
ρ
2
1
μ
2
-----
a
21
Q
x
a
22
Q
y
a
23
Q
z
++(),–=
(14)
where
ρ
3
1
d
3
μ
3
-----------
a
31
Q
x
a
32
Q
y
a
33
Q
z
++(),=
a
11
L
2y
L
3z
L
3y
L
2z
–
D
------------------------------------
, a
12
L
2z
L
3x
L
3z
L
2x
–
D
-------------------------------------
,==
a
13
L
2x
L
3y
L
3x
L
2y
–
D
-------------------------------------
,= a
21
L
1z
L
3y
L
3z
L
1y
–
D
------------------------------------
,=
a
22
L
1x
L
3z
L
3x
L
1z
–
D
-------------------------------------
,= a
23
L
1y
L
3x
L
3y
L
1x
–
D
-------------------------------------
,=
a
31
L
1y
L
2z
L
2y
L
1z
–
D
------------------------------------
,= a
32
L
1z
L
2x
L
2z
L
1x
–
D
-------------------------------------
,=
a
33
L
1x
L
2y
L
2x
L
1y
–
D
-------------------------------------
,=
DL
1x
L
2y
L
3z
L
3y
L
2z
–()=
+ L
2x
L
1z
L
3y
L
3z
L
1y
–()L
3x
L
1y
L
2z
L
2y
L
1z
–().+
Table 1. Comparison of the efficiency of the two methods for orbit determination from three geocentric angular positions
applied to the orbit of Icarus (t
2
= 1966, August 3.0 ET, ρ
2
= 1.052 AU)
t
3
– t
1
, days ρ
1
, AU ρ
3
, AU
LG3 LGS3
Δt, days Δr, AU Δv, AU/day Δt, days Δr, AU Δv, AU/day
1 1.042 1.061 1.8×10
–7
3.1×10
–5
5.6×10
–7
0.0 3.6×10
–10
8.9×10
–12
2 1.047 1.056 1.8×10
–7
3.1×10
–5
5.6×10
–7
0.0 1.0×10
–9
2.7×10
–11
4 1.033 1.071 1.7×10
–7
3.0×10
–5
5.5×10
–7
0.0 3.9×10
–9
1.0×10
–10
10 1.007 1.102 1.7×10
–7
2.9×10
–5
5.4×10
–7
0.0 2.5×10
–8
6.1×10
–10
20 0.969 1.157 1.5×10
–7
2.7×10
–5
5.2×10
–7
–4.7×10
–10
1.2×10
–7
2.1×10
–9
40 0.916 1.278 1.4×10
–7
2.4×10
–5
4.8×10
–7
–2.8×10
–9
5.2×10
–7
7.7×10
–9
80 0.912 1.540 1.2×10
–7
2.1×10
–5
4.1×10
–7
3.3×10
–9
5.7×10
–7
7.6×10
–8
160 1.165 2.015 1.2×10
–7
2.0×10
–5
3.3×10
–7
1.5×10
–8
2.7×10
–6
1.7×10
–7
Table 2. Comparison of the efficiency of the two methods for orbit determination from three geocentric angular positions
applied to the orbit of Icarus (t
2
= 1968, June 15.0 ET, ρ
2
= 0.04255 AU)
t
3
– t
1
,
days
ρ
1
,
10
–2
AU
ρ
3
,
10
–2
AU
LG3 LGS3
Δt, days Δr, AU Δv, AU/day Δt, days Δr, AU Δv, AU/day
0.0625 4.252 4.258 –7.8 × 10
–8
1.4 × 10
–5
5.4 × 10
–6
0.0 3.7 × 10
–8
1.5 × 10
–8
0.125 4.250 4.262 –7.8 × 10
–8
1.3 × 10
–5
5.4 × 10
–6
4.7 × 10
–10
9.1 × 10
–8
3.7 × 10
–8
0.25 4.248 4.272 –7.8 × 10
–8
1.3 × 10
–5
5.4 × 10
–6
1.9 × 10
–9
3.5 × 10
–7
1.4 × 10
–7
0.5 4.252 4.300 –7.8 × 10
–8
1.4 × 10
–5
5.5 × 10
–6
7.9 × 10
–9
1.4 × 10
–6
5.6 × 10
–7
1.0 4.293 4.387 –7.9 × 10
–8
1.4 × 10
–5
5.5 × 10
–6
3.1 × 10
–8
5.3 × 10
–6
2.1 × 10
–6
2.0 4.495 4.673 –8.3 × 10
–8
1.4 × 10
–5
5.5 × 10
–6
1.1 × 10
–7
1.9 × 10
–5
7.1 × 10
–6
4.0 5.306 5.605 –1.0 × 10
–7
1.7 × 10
–5
5.8 × 10
–6
2.9 × 10
–7
5.1 × 10
–5
1.6 × 10
–5
10.0 9.340 9.764 –2.7 × 10
–7
4.8 × 10
–5
9.1 × 10
–6
8.1 × 10
–7
1.4 × 10
–4
2.6 × 10
–5
SOLAR SYSTEM RESEARCH Vol. 37 No. 4 2003
A METHOD FOR THE DETERMINATION 329
Formulas (12)–(14) allow ρ
i
and then x
i
, t
i
to be
refined using relations (2) and (3). The iterations are
repeated until for some number n + 1 (n ≥ 1) the condi-
tion
is satisfied. Here ε
i
is the preset tolerance.
The proposed iterative scheme thus generalizes the
classical Gauss scheme (Dubyago 1949; Escobal 1965;
Subbotin 1968, and Herrick 1971) applied after the first
approximations for geocentric distances ρ
1
, ρ
2
, and ρ
3
had been obtained.
The cases where all three observed positions of the
small body lie on the ecliptic are insolvable for our
algorithm like they are for the classic method. In these
cases, like in all cases where D is equal or close to zero,
the orbit should be computed using four observations
(Dubyago 1949; Subbotin 1968).
Based on the two finally refined position vectors x
1
and x
3
and the corresponding refined time instants t
1
and t
3
, we determine the velocity vector on the
intermediate orbit using the algorithm developed in the
previous paper (Shefer 2003). Vectors = x
1
, and
determine the motion parameters for the sought-for
orbit at time = t
1
.
NUMERICAL SIMULATIONS
We developed a Fortran-90 code called LGS3
implementing the above algorithm. We compared the
efficiency of the new method with that of the classic
Lagrange–Gauss method (program LG3). The formulas
of the classic method can be found, in an easily pro-
grammable form, e.g., in Escobal (1965).
ρ
i
()
n 1+
ρ
i
()
n
– ε
i
<
x
˙
1
*
x
1
*
x
˙
1
*
t
1
*
To test our programs, we applied them to the motion
of asteroid 1566 Icarus. We analyzed the motion of
Icarus with the allowance for the perturbations due to
all nine major planets and the Moon using the coordi-
nate and velocity ephemerides DE200/LE200. We used
the fifteenth-order Everhart algorithm (Everhart 1985)
for numerical integration of the equations of motion
and performed orbital computations on a Pentium-
II/350 MHz PC with a computer precision of ε = 1.11 ×
10
−16
.
At the first stage our task was to estimate the accu-
racy of the two methods. To this end, we used the nom-
inal trajectory of the asteroid computed by Shefer
(2003). We computed the pairs of geocentric coordi-
nates , and times corresponding to the helio-
centric positions x
i
on the nominal trajectory at the pre-
viously chosen ephemeris times t
i
(i = 1, 2, 3). We then
used the parameters thus determined as the input data
for the programs. Tables 1 and 2 list the results of the
computations performed using programs LG3 and
LGS3. The epochs t
1
, t
2
, and t
3
in these tables coincide
with the epochs t
1
, t
0
, and t
2
in Tables 1 and 2 of Shefer
(2003), respectively; ρ
i
= |x
i
+ S
i
| is ith geocentric distance
in AU; Δt = ( – t
1
) (days); Δr = | – x
1
| (AU/day), and
Δv = | – | (AU/day) are the errors of the time,
position and velocity vectors at epoch t
1
, respectively.
Here is the vector of velocity on the nominal trajec-
tory at time t
1
. In the time intervals considered in Table
1 the asteroid was in the aphelion part of its orbit. The
time intervals in Table 2 either include the time of the
1968 closest approach between Icarus and the Earth or
are close to this time. Like in our previous paper
(Shefer 2003), it is evident from these tables that for the
methods to be applied efficiently, the reference time
α
i
0
δ
i
0
t
i
0
t
1
*
x
1
*
x
˙
1
*
x
˙
1
x
˙
1
Table 3. Representation of the topocentric angular positions of Icarus based on the parameters of the preliminary orbit (An-
glo-Australian Observatory, CCD observations
†
)
Number of
observation
Date of observation
(UT)
LG3 LGS3
ρ, AU
cosδΔα, arcsec Δδ, arcsec cosδΔα, arcsec Δδ, arcsec
1 1995 06 05.73931 3.3633 1.5656 2.6628 1.1838 0.677
2 1995 06 08.80079 2.1070 0.7630 1.6153 0.5039 0.668
3 1995 06 19.65645 –0.3259 0.2594 –0.3433 0.2504 0.654
4 1995 06 20.69873 –0.1836 0.0933 –0.1836 0.0933 0.655
5* 1995 06 20.70184 0.0000 0.0000 0.0000 0.0000 0.655
6* 1995 07 06.66727 0.3014 –0.0393 0.0013 0.0031 0.712
7 1995 07 06.67157 0.6374 –0.1096 0.3371 –0.0671 0.712
8* 1995 07 20.53618 0.8079 –0.2670 0.0217 0.0096 0.835
9 1995 07 20.53849 0.3102 –0.3845 –0.4760 –0.1079 0.835
10 1996 06 07.77481 1124.6230 1186.2498 627.5461 647.0063 0.115
11 1996 06 10.73748 730.3780 1100.7082 419.0094 604.9206 0.102
†
Observations published in Minor Planet Circulars nos. 25371, 25478, 27370, and 27495.
330
SOLAR SYSTEM RESEARCH Vol. 37 No. 4 2003
SHEFER
interval should be reduced in the case of strong pertur-
bations, which take place in the case where the asteroid
passes the neighborhood of the Earth–Moon system.
The tabular data show that in the classic Lagrange–
Gauss method the errors of all the parameters men-
tioned above remain virtually constant independently
of the chosen length t
3
– t
1
of the reference time inter-
val. By contrast, in our algorithm the methodical errors
decrease rapidly with decreasing length of the refer-
ence interval. As is evident from the first rows of the
tables, the errors decrease approximately as (t
3
– t
1
)
2
.
Such a behavior of the errors has a rigorous theoretical
explanation if we recall that the degree of approxima-
tion of the real motion by our intermediate orbit is two
orders higher than by the Keplerian orbit (Shefer 2003).
At |Δt| < ε × 10
6
days we obtained zero Δt values,
because in our computations the Julian dates were rep-
resented with nine decimal digits. It follows from the
tables that our intermediate orbit is substantially (by
three to five orders in the first rows of the tables) more
accurate than the Keplerian orbit computed using the
classic method.
In practice, we have to deal with measurements that
are inevitably burdened with errors. Therefore, the next
stage of our computations involved real observational
data. Our theory of the motion of Icarus (Shefer 1984)
is based on photographic position observations per-
formed in 1949–1977 with rms errors of 1.6 arcsec. The
experiment involving the use of the observations of
Icarus made in the first year after its discovery (1949)
showed that the accuracy of traditional optical mea-
surements is insufficient to demonstrate the efficiency
of the new approach. More accurate optical CCD obser-
vations of Icarus are performed since 1992. We could
obtain a total of 92 CCD observations of Icarus per-
formed up to 1996 inclusive and published in the Minor
Planet Circulars. By comparing the theory with these
observations we estimated the mean error of CCD mea-
surements to be 0.6 arcsec. In order to better preserve
the information contained in the observations, it is
desirable to perform computations with the precision
higher by one order of magnitude than that of the obser-
vations. The linear error corresponding to 0.6 arcsec at
the geocentric distance of 1 AU implies, according to
Table 1, that the best time interval between the extreme
observing times t
1
and t
3
should be equal to one month.
We selected all CCD observations performed in 1995–
1996 at the Anglo-Australian Observatory (Siding
Spring, Australia) and then chose a number of reference
observations among the 1995 measurements. Asterisks
in Table 3 indicate the running numbers of these refer-
ence observations. These observations are most uni-
Table 4. Representation of the topocentric angular positions of Icarus based on the parameters of the preliminary orbit (Pal-
omar Observatory, fictitious observations, standard error 0.1 arcsec)
Number of
observation
Date of observation
†
(UT)
LG3 LGS3
ρ, AU
cosδΔα, arcsec Δδ, arcsec cosδΔα, arcsec Δδ, arcsec
1* 1949 06 27.23889 0.0000 0.0000 0.0000 0.0000 0.269
2 1949 06 29.23056 –0.1445 0.0036 –0.0342 0.0139 0.300
3 1949 07 01.31562 –0.1957 0.1757 –0.0323 0.1831 0.333
4* 1949 07 13.20660 –0.3385 0.0600 –0.0390 –0.0009 0.530
5 1949 07 14.21701 –0.5473 0.0838 –0.2266 0.0166 0.548
6 1949 07 23.19583 –0.6824 0.2670 –0.0527 0.1431 0.705
7* 1949 07 24.18750 –0.7608 0.1289 –0.0839 –0.0016 0.723
8 1950 08 03.19167 –266.0205 175.5590 118.1862 –77.5207 0.807
9 1950 08 04.19167 –257.8409 169.2788 114.2377 –74.9299 0.824
10 1950 08 05.20903 –249.6622 163.2368 110.5441 –72.2655 0.842
†
Dates correspond to real observations of Icarus (Minor Planet Circular no. 270; Harvard Announcement Cards nos. 1017 and 1096).
Table 5. Representation of the topocentric angular positions
of Icarus based on the parameters of the preliminary orbit
(Palomar Observatory, fictitious observations, standard error
0.01 arcsec)
Number
of obser-
vation
LG3 LGS3
cosδΔα,
arcsec
Δδ, arcsec
cosδΔα,
arcsec
Δδ, arcsec
1* 0.0000 0.0000 0.0000 0.0000
2 –0.0843 0.0118 0.0260 0.0221
3 –0.1250 0.0033 0.0385 0.0106
4* –0.3385 0.0600 –0.0390 –0.0009
5 –0.3613 0.0643 –0.0406 –0.0028
6 –0.6901 0.1257 –0.0605 0.0018
7* –0.7608 0.1289 –0.0839 –0.0016
8 –349.3618 230.2291 34.6122 –22.6981
9 –338.3479 222.1790 33.5038 –21.8803
10 –327.5658 214.2526 32.4196 –21.1038
SOLAR SYSTEM RESEARCH Vol. 37 No. 4 2003
A METHOD FOR THE DETERMINATION 331
formly distributed over the one-month interval and fit
best the nominal trajectory. We constructed the prelim-
inary orbits based on the selected reference observa-
tions. We then used the parameters of these orbits deter-
mined at the epoch of the first reference observation to
represent the selected observations with full allowance
for all perturbations (via numerical integration of the
equations of motion). Table 3 gives the results of the
representation of all observations inside the reference
time interval and immediately adjacent to it. In the
remaining cases we left only one observation for every
night. As is evident from the table, our approach repre-
sents the third reference observation 35 times better
than in the classic one. This is a good indicator of the
high quality of the orbit constructed using our method.
The observations not used for the construction of orbits
are, on the average, 1.5 times better represented by our
initial motion parameters than by those inferred using
the classical Lagrange–Gauss method. This conclusion
also applies to the observations performed in 1996,
which are separated from those made in 1995 by almost
a complete revolution of the asteroid about the Sun.
The previous experiment showed that the use of
high-quality CCD observations by itself combined with
the new approach gives appreciable effect. Modern
optoelectronic and, especially, radiotechnical measure-
ment tools make it possible to obtain much lower
errors. To demonstrate the potential of our method
when applied to high-precision observations, we per-
formed the following, concluding experiment. We used
all the first observations of Icarus made in 1949 at Pal-
omar Observatory (Palomar Mountain, California,
USA) and three observations made at the same obser-
vatory a year later. We left all observing times
unchanged and substituted the real angular observa-
tions by fictitious measurements with the mean errors
of 0.1, 0.01, and 0.001 arcsec. Tables 4–6 list the results
of the representation of fictitious observations in terms
of the constructed orbital parameters. The observing
dates are listed in Table 4. Asterisks again indicate the
running numbers of the reference observations. We
considered all possible variants of the choice of refer-
ence observations among those made in 1949. A com-
parison of the results of computations for these variants
showed that with observations having rms errors of 0.1
and 0.01 arcsec the best reference time interval is that
defined by the extreme observing times. With observa-
tions having a mean error of 0.001 arcsec, the best
orbits are obtained using a somewhat shorter reference
interval given by the times of the third and seventh
observations. Thus, if the accuracy of the observations
of Icarus at the time of its discovery were equal to about
0.1, 0.01, or 0.001 arcsec, the ephemeris of the asteroid
computed for 1950 based on the parameters of our
intermediate orbit would be more accurate than that
obtained using the results of the classic Lagrange–
Gauss method by factors 2, 10, and 100, respectively. It
follows from the tabulated data that with observations
with errors less than 1 arcsec the use of the classic
method becomes formal, because the methodical error
in these cases exceeds the errors mentioned above.
CONCLUSIONS
In this paper we proposed a new method for the
determination of the preliminary orbit using three posi-
tions of the small body on the celestial sphere and the
corresponding time instants. We developed this method
in line with the underlying schema of the classic
Lagrange–Gauss method as far as our approach
allowed. The principal and fundamental deviation from
the classic scheme is that instead of the unperturbed
Keplerian orbit we construct an orbit that allows for
most of the perturbations in the motion of the celestial
body. We achieve it by using our earlier derived inter-
mediate perturbed orbit based on two position vectors
and the corresponding time interval.
The numerical examples analyzed in this paper lead
us to conclude that the new method may be successfully
used for studying the motion of small bodies of the
Solar System. If the boundary conditions of orbit deter-
mination are set in the form of angular positions exclu-
sively, the errors of the methods based on the solution
of the unperturbed two-body problem become fixed and
do not decrease with the decreasing length of the refer-
ence arc of the trajectory. In this case, the methodical
errors of our approach decrease proportionally to the
squared length of the reference time interval. This fact
allows such a reference arc to be selected that the accu-
racy of the intermediate orbit would always match that
of the reference observations that determine this arc.
The higher is the accuracy of the employed observa-
tions spanning a sufficiently short arc, the higher is the
accuracy of the approximation of the real motion by our
orbit. In this respect the proposed method differs favor-
Table 6. Representation of the topocentric angular positions
of Icarus based on the parameters of the preliminary orbit
(Palomar Observatory, fictitious observations, standard error
0.001 arcsec)
Number
of obser-
vation
LG3 LGS3
cosδΔα,
arcsec
Δδ, arcsec
cosδΔα,
arcsec
Δδ, arcsec
1 0.2247 0.0397 –0.0143 –0.0029
2 0.0813 0.0096 –0.0039 –0.0024
3* 0.0000 0.0000 0.0000 0.0000
4* –0.2055 0.0314 –0.0069 –0.0003
5 –0.2290 0.0365 –0.0083 0.0000
6 –0.5441 0.0833 –0.0197 –0.0008
7* –0.5915 0.0890 –0.0216 –0.0010
8 –369.9424 243.7191 –3.2225 2.1382
9 –358.2582 235.1698 –3.1241 2.0647
10 –346.8235 226.7798 –3.0302 1.9924
332
SOLAR SYSTEM RESEARCH Vol. 37 No. 4 2003
SHEFER
ably from the traditional method. It should, however, be
borne in mind, whatever the method applied, that the
reference arc should not be too short for the observa-
tional errors to affect appreciably the accuracy of orbit
determination.
Of course, the necessity to use the formulas of per-
turbed motion for determining the parameters of the
intermediate orbit makes the proposed method some-
what more complex; however, its schema preserves the
simplicity of the initial classic algorithm. All this makes
our method a highly efficient tool, which allows reli-
able parameters of the perturbed motion to be obtained
already at the stage of computing the preliminary orbit.
The approach described here can be successfully
used to construct the methods based on other sets
observing data different from the classic one.
Note in conclusion that the formulation of the prob-
lem of orbit determination from the minimum number
of observations with the allowance for perturbations is
by no means a new one. Thus, Gibbs (1888) derived a
formula relating the coordinates of a celestial body at
three consecutive time instants to the second derivatives
of the coordinates with respect to time at the same time
instants, by expanding the coordinates and their second
derivatives into Taylor series in time. This makes it pos-
sible to use not only the Keplerian terms in the right-
hand sides of the equations of motion (1), but also
allows additional perturbing accelerations to be
included in the formula derived. Numerov (Noumeroff)
(1923a, 1923c) was, apparently, the first to use this pos-
sibility in practice. His method of orbit determination
from three observations is based just on such a general-
ized variant of the Gibbs formula. The main formula of
the Numerov method for the determination of the pre-
liminary orbit is based on expanding the coordinates
into a series in time up to the forth-order terms inclu-
sive. As for our method, it is based on closed expres-
sions, which in the absence of perturbations yield the
exact orbit. This fact alone speaks for our approach, at
least in the case of weakly perturbed motions, which
are those most commonly found in the Solar System.
Numerov proposes a method for further refinement of
the preliminary orbit by including higher-order terms in
the Gibbs formula. This method is based on the formula
of Numerov’s own method of numerical integration of
equations of motion in special coordinates. However, it
is a method for improving the preliminary orbit. We do
not address this stage of the construction of the final
orbit in this paper. A good example of the use of
Numerov’s method can be found in the work of its dis-
ciple Postoev (1926). While giving the credits to the
pioneering works of B.V.Numerov, we, however, do not
aim here at a detailed comparison of the two methods.
Such a comparison, and, especially, numerical simula-
tions aimed at comparing the efficiency of the two
methods, may become a subject of a separate study.
ACKNOWLEDGMENTS
I am deeply grateful to prof. Yu.V. Batrakov for a
number of useful comments and advice in connection
with this work. This study was supported by the Minis-
try of Science, Industry, and Technology of the Russian
Federation (State Contract no. 40.022.1.1.1108).
REFERENCES
Lagrange, J.L. Sur le problème de la dètermination des
orbites des comètes, d’apres trois observations, Nouv.
Mem. Acad. Roy. Sci. et Belles-Lettres, Berlin, 1778.
Noumeroff, B. Mèthode Nouvelle De La Dètermination Des
Orbites Et Le Calcul Des Éphèmerides En Tenant
Compte Des Perturbations, Tr. Glav. Ross. Astrofiz.
Observ, 1923, vol. 2, pp. 188–288.
Dubyago, A.D., Opredelenie orbit (Orbit Determination),
Moscow: Gos. Izd. Tekh. Teor. Lit., 1949.
Escobal, P.R., Methods of Orbit Determination, New York:
Wiley, 1965. Translated under the title Metody oprede-
leniya orbit, Moscow: Mir, 1970.
Everhart, E., An Efficient Integrator That Uses Gauss-Radau
Spacings, Dynamics of Comets: Their Origin and Evolu-
tion (Proc. Of the 83rd IAU Colloq., Rome, 1984),
Carusi, A. and Valsecchi, G.B., Eds., Dordrecht: Reidel,
1985, pp. 185–202.
Gauss, C.F., Theoria Motus Corporum Coelestium in Sec-
tionibus Conicis Solem Ambientium, Hamburg, 1809.
Translated under the title Teoriya dvizheniya nebesnykh tel,
obrashchayushchikhsya vokrug Solntsa po konicheskim
secheniyam, Moscow, 1859.
Gibbs, J.W., On the Determination of Elliptic Orbits From
Three Complete Observations, Mem. Nat. Acad. Sci.,
1888, vol. 4, pp. 81–104.
Herrick, S., Astrodynamics, London: Van Nostrand Rein-
hold, 1971, vol. 1. Translated under the title Astrodi-
namika, Moscow: Mir, 1977, vol. 2.
Numerov, B.V., New Method of Orbit Determination and
Ephemeris Calculations Taking into Account Perturba-
tions, Tr. Astron. Otd. Petrogr. Univ., 1923a, vol. 4,
pp. 29–39.
Postoev, A.I., Application of Numeroff Heliocentric Coordi-
nates Variations Method to Determination of Orbit of
Planet Emma (283), Russ. Astron. Zh., 1926, vol. 3,
no. 2, pp. 103–127.
Shefer, V.A., Study of the Motion of the Unusual Minor
Planet Ikar Motion, Byull. ITA, Akad. Nauk SSSR, 1984,
vol. 15, pp. 347–349.
Shefer, V.A., Determination of an Intermediate Perturbed
Orbit from Two Position Vectors, Astron. Vestn., 2003,
vol. 37, no. 3, pp. 265–272.
Subbotin, M.F., Vvedenie v Teoreticheskuyu Astronomiyu
(Introduction to Theoretical Astronomy), Moscow:
Nauka, 1968.
