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ELSEVIER Chinese Astronomy and Astrophysics 31 (2007) 288–295
CHINESEASTRONOMYAND ASTROPHYSICS
A Method of Short-arc Orbit Determinationfor the Transfer Orbit of Lunar Probes† �
LIU Lin ZHANG WeiDepartment of Astronomy, Nanjing University, Nanjing 210093
Institute of Space Environment and Astrodynamics, Nanjing University, Nanjing 210093
Abstract The short-arc orbit determination to be discussed here means amethod of initial orbit calculation without a priori information that bypassesmulti-variate iteration. It requires that the concerned dynamical problem hasan approximate analytical solution, which can represent the short arc to a cer-tain degree of accuracy. When the lunar probe enters the effective range oflunar gravitational force and approaches the moon, the dynamical problem canbe treated as a perturbed 2-body problem relative to the moon, and when theprobe is close to the earth, then as a perturbed 2-body problem relative to theearth, but for the whole transition stage, it can only be treated as a perturbedrestricted 3-body problem. The restricted 3-body problem has no analytical so-lution, even when outside the effective range of lunar gravitation and for thelarge-thrust impulsive transfer, the eccentricity of the varying elliptical orbit rel-ative to the earth is so large (greater than the Laplace limit) that an analyticalsolution can not be constructed. When the lunar gravitational perturbation istaken into consideration, we have constructed a time power series solution forthe segment of transfer orbit that takes into consideration the coaction of a non-spherical earth (including only the J2-term) and the moon, and on this basis, amethod for initial orbit calculation in the sense of perturbed 2-body problem.Numerical verification of this method indicates that it is effective as a short-arcorbit determination and that it will be useful to the ground-based measurementand control systems.
Key words: method: N-body simulation—celestial mechanics: orbit determi-nation
† Supported by National Natural Science FoundationReceived 2006–01–13; revised version 2006–04–24
� A translation of Acta Astron. Sin. Vol. 48, No. 2, pp. 220–227, 2007
0275-1062/01/$-see front matter c© 2007 Elsevier Science B. V. All rights reserved.PII:
0275-1062/07/$-see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.chinastron.2007.06.003
LIU Lin et al. / Chinese Astronomy and Astrophysics 31 (2007) 288–295 289
1. INTRODUCTION
The motion of a lunar probe, from the launch to the final target orbit, can be dividedbasically into three flying stages, each of which corresponds to a different orbit, namely, ananchor orbit near the earth, a transfer orbit in the transition stage (when the probe is flying inthe lunar-terrestrial space, after the orbit changes from the anchor orbit), and a target orbit(when the probe arrives at the neighborhood of the moon, and finally becomes a circumlunarsatellite after a further orbit change). The near-earth anchor orbit and circumlunar orbitstill belong to typical satellite orbits. This paper will discuss mainly the motion and orbitdetermination of the lunar probe, when it moves between the earth and the moon after itentered the transfer orbit ( after arriving in the neighborhood of the moon, and before thesecond change of orbit).
The motion of the probe in the transition stage can be considered as motion under thecoaction of two major celestial bodies (Earth and Moon), and for the initial motion nearthe earth, the effect of earth’s oblateness should be taken into account. The transition stagewill last about 4-5 days, during which lunar gravitation should be taken into consideration,and this is related to the problem that the probe moves under the actions of two majorcelestial bodies, with the earth taken as an oblate sphere and the moon taken as a masspoint. For this problem, we can cite neither the small-parameter power series solution usedfor studying the orbital variation of an artificial earth satellite[1−3], nor the small-parameterpower series solution used for studying the orbital variation of an artificial lunar satellite[4−5].However, we can construct a time power series solution to suit the short-arc situation, andby introducing it into the Laplace method of initial orbit determination in the sense of thecommon 2-body problem[6−8], we will give an extended Laplace method to determine ratherprecisely the probe’s orbit when it is flying to the target object (the moon), and to providethe ground-based measurement and control systems with a necessary measure during thetransition stage.
The above consideration has a practical background. For the motion of a lunar probeat the transition stage, we can now obtain the data of angle measurements with an accuracybetter than 1′′, such data should be exploited, and the method of orbit determination isgiven here particularly for this kind of data.
2. THE TIME POWER SERIES SOLUTION OF r AND r AND ITSSPECIFIC FORMULATION
We select the geocentric celestial coordinate system (the origin is at Earth’s center, the xy-plane is the mean equatorial plane at the epoch, the x-axis points to the corresponding meanvernal equinox), and adopt the normalized units:
⎧⎨⎩
[M ] = me (Earth’s mass)[L] = D (mean Earth-Moon distance)[T ] = (D3/Gme)1/2 ≈ 4d.3484
. (1)
In this coordinate system, if we consider only the 2-body problem by taking the earth as thecentral body, then the motion of the lunar probe corresponds to the initial value problem of
290 LIU Lin et al. / Chinese Astronomy and Astrophysics 31 (2007) 288–295
the following ordinary differential equations:{r = − r
r3
t = t0, r(t0) = r0, r(t0) = r0
. (2)
In the Laplace method of initial orbit calculation, the time (τ = t− t0) power series solutionof Eq.(2) is adopted[1−3]:
r = Fr0 + Gr0 , (3)
in which {F = F (r0, r0, τ)G = G (r0, r0, τ) . (4)
The concrete formulations accurate to the O(τ6) terms are as follows:
F = 1 − (τ2/2)u3 + (τ3/2)u5σ + (τ4/24)u5(3V 20 − 2u1 − 15u2σ
2) +
(τ5/8)u7σ(−3V 20 + 2u1 + 7u2σ
2) + (τ6/720)u7[u2σ
2(630V 20 − 420u1 − 945u2σ
2) − (22u2 − 66u1V20 + 45V 4
0 )]+ · · · , (5)
G = τ − (τ3/6)u3 + (τ4/4)u5σ + +(τ5/120)u5(9V 20 − 8u1 − 45u2σ
2) +
(τ6/24)u7σ(−6V 20 + 5u1 + 14u2σ
2) + · · · , (6)
in which σ = r0 · r0, V 20 = r2
0, un = 1/rn0 .
As shown above, to construct the time power series solution of r and r via r0 andr0 means no special requirement on the equation of motion, so it can be further extendedto construct the time power series solution of r and r under a general mechanical model.Simultaneously taking account of the effects of the J2-term of the earth oblateness and thegravitation of the third body, and adopting for this mechanical system also the geocentriccelestial coordinate system at the epoch as mentioned above, then the motion of the probecorresponds to the initial value problem of the following ordinary differential equations:⎧⎪⎨
⎪⎩r = − r
r3+ A2
(5z2
r7− 1
r5
)r − A2
2z
r5k −
2∑j=1
μ′j
(Δj
Δ3j
+r′jr′3j
)
t = t0, r = r0, r = r0
, (7)
in which A2 = 3J2a2e/2, k = (0, 0, 1)T is the unit vector in the z-direction, ae is the dimen-
sionless equatorial radius of the earth’s reference ellipsoid, Δj = rj − r′j , μ′j = (m′j/me),
m′j (j =1,2) represent the masses of the moon and sun, respectively, and r′j represent re-
spectively the position vectors of the moon and sun. The corresponding geometric relationis shown in Fig.1, where E represents the earth’s center, P , the probe, and Pj , the thirdbody (the moon or sun).
LIU Lin et al. / Chinese Astronomy and Astrophysics 31 (2007) 288–295 291
Fig. 1 The geometric relation among the earth, probe and the third body
Similar to the 2-body problem, we can construct the time (τ = t − t0) power seriessolution of the equation of motion. Neglecting the derivations, the result can be writtendirectly as follows:
⎧⎨⎩
x = Fx0 + Gx0 + F ′x′0 + G′x′0y = Fy0 + Gy0 + F ′y′0 + G′y′0z = Fzz0 + Gz z0 + F ′z′0 + G′z′0⎧⎨
⎩x = Fdx0 + Gdx0 + F ′dx
′0 + G′dx
′0
y = Fdy0 + Gdy0 + F ′dy′0 + G′dy
′0
z = Fdzz0 + Gdz z0 + F ′dz′0 + G′dz
′0
. (8)
Here
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
F = 1 +(τ2
/2)f2 +
(τ3
/6)f3 +
(τ4
/24
)f4 +
(τ5
/120
)f5 +
(τ6
/720
)f6 + O
(τ7
)Fz = F + A2
[(τ2
/2)(−2u5) +
(τ3
/6)(10u7σ) +
(τ4
/24
) (10u7V
20 − 6u8 − 70u9σ
2)]
G = τ +(τ3
/6)g3 +
(τ4
/24
)g4 +
(τ5
/120
)g5 +
(τ6
/720
)g6 + O
(τ7
)Gz = G + A2
[(τ3
/6)(−2u5) +
(τ4
/24
)(20u7σ)
]F ′ = μ′
[(τ2
/2)f ′2 +
(τ3
/6)f ′3 +
(τ4
/24
)f ′4 + O
(τ5
)]G′ = μ′
[(τ3
/6)g′3 +
(τ4
/24
)g′4 + O
(τ5
)],
(9)
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
Fd = τ · f2 +(τ2
/2)f3 +
(τ3
/6)f4 +
(τ4
/24
)f5 +
(τ5
/120
)f6 + O
(τ6
)Fdz = Fd + A2
[τ (−2u5) +
(τ2
/2)(10u7σ) +
(τ3
/6) (
10u7V20 − 6u8 − 70u9σ
2)]
Gd = 1 +(τ2
/2)g3 +
(τ3
/6)g4 +
(τ4
/24
)g5 +
(τ5
/120
)g6 + O
(τ6
)Gdz = Gd + A2
[(τ2
/2)(−2u5) +
(τ3
/6)(20u7σ)
]F ′d = μ′
[τ · f ′2 +
(τ2
/2)f ′3 +
(τ3
/6)f ′4 + O
(τ4
)]G′d = μ′
[(τ2
/2)g′3 +
(τ3
/6)g′4 + O
(τ4
)]. (10)
292 LIU Lin et al. / Chinese Astronomy and Astrophysics 31 (2007) 288–295
The relevant quantities in these formulae are,⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
f2 = −u3 + A2u5
(5z2
0u2 − 1) − μ′u′3
f3 = 3σu5 + A2u7
(5σ
(1 − 7z2
0u2
)+ 10z0z0
)+ μ′ (3u′5σ
′)f4 = u5
(3V 2
0 − 2u1 − 15σ2u2
)+ A2
[6u8
(4z2
0u2 − 1) − 5u7V
20
(7z2
0u2 − 1)
+35u9σ2(9z2
0u2 − 1)
+ 10u7z2 − 140u9σz0z0
]+μ′
[−u3u
′3 + 3u′5
(V ′20 − u3α
)+ 3u5 (u′3 − u13)α1 − 15u′7σ
′2 − μ′ (2u′6)]
f5 = 15u7σ[2u1 − 3V 2
0 + 7u2σ2]
f6 = u7
[u2σ
2(630V 2
0 − 420u1 − 945u2σ2) − (
22u2 − 66u1V20 + 45V 4
0
)], (11)
⎧⎪⎪⎨⎪⎪⎩
g3 = f2
g4 = 2f3
g5 = u5
[9V 2
0 − 8u1 − 45u2σ2]
g6 = 30u7σ[5u1 − 6V 2
0 + 14u2σ2] , (12)
⎧⎪⎨⎪⎩
f ′2 = u′3 − u13
f ′3 = 3 (u15σ1 − u′5σ′)
f ′4 = u3 (u13 − u′3) + 15(u′7σ
′2 − u17σ21
) − 3u′5(V ′20 − u3α
)+ 3u15V
21 + 2μ′ (u′6 − u16)
,
(13)
g′3 = f ′2, g′4 = 2f ′3 . (14)
The parameters appearing in the above are defined as follows:⎧⎪⎪⎪⎨⎪⎪⎪⎩
un = 1/rn0 , σ = r0 · r0, V 2
0 = r20 ,
u′n = 1/Δn0 , σ′ = Δ0 · Δ0, V ′20 = Δ
2
0, α = r0 · Δ0 ,
u1n = 1/r′n0 , σ1 = r′0 · r′0, V 21 = r′
2
0, α1 = r0 · r′0Δ0 = r0 − r′0, Δ0 = r0 − r′0
, (15)
where r′0 and r′0 are the values of r′j and r′j of the third body at the time t0.
3. APPLICATION OF THE TIME POWER SERIES SOLUTION— THEEXTENDED LAPLACE METHOD FOR ORBIT DETERMINATION
The short-arc orbit determination can be made as well for the probe moving in its transitionorbit, the basic equations can be solved by using the above-mentioned time power series, theprinciple of the orbit determination is similar to that of the short-arc orbit determinationfor the perturbed motion of satellites[9−11], and likewise applicable to both angular data andpositional data (including data of simultaneous range and angle measurements)
3.1 Angular DataThe angular data (α, δ) come from the Deep Space Network. Here we will not discuss
the details of the method and the accuracy of the data acquisition. Let (λ, μ, ν) and (X, Y, Z)be, respectively, the 3 components of the unit vector L along the directions of (α, δ) andthe 3 components of the coordinate vector R of the observing station, then we have
LIU Lin et al. / Chinese Astronomy and Astrophysics 31 (2007) 288–295 293
L =
⎛⎝ λ
μν
⎞⎠ =
⎛⎝ cos δ cosα
cos δ sinαsin δ
⎞⎠ , (16)
R =
⎛⎝ X
YZ
⎞⎠ . (17)
The relevant basic equations for orbit determination are:⎧⎨⎩
(Fν)x0 − (Fzλ)z0 + (Gν)x0 − (Gzλ)z0 = (νX − λZ) − Δ1(r′0, r′0)
(Fν)y0 − (Fzμ)z0 + (Gν)y0 − (Gzμ)z0 = (νY − μZ) − Δ2(r′0, r′0)
(Fμ)x0 − (Fλ)y0 + (Gμ)x0 − (Gλ)y0 = (μX − λY ) − Δ3(r′0, r′0)
, (18)
in which ⎧⎨⎩
Δ1(r′0, r′0) = (F ′ν)x′0 − (F ′λ)z′0 + (G′ν)x′0 − (G′λ)z′0Δ2(r′0, r′0) = (F ′ν)y′0 − (F ′μ)z′0 + (G′ν)y′0 − (G′μ)z′0Δ3(r′0, r′0) = (F ′μ)x′0 − (F ′λ)y′0 + (G′μ)x′0 − (G′λ)y′0
. (19)
3.2 Positional Data r(x, y, z)For observed positional data, the basic equations for orbit determination can be ob-
tained directly by a transformation of the series solution,⎧⎨⎩
Fx0 + Gx0 = x − (F ′x′0 + G′x′0)Fy0 + Gy0 = y − (F ′y′0 + G′y′0)Fzz0 + Gz z0 = z − (F ′z′0 + G′z′0)
. (20)
The process of orbit determination is: first, determine r0 and r0 at the epoch time t0,then transform them to the orbital elements. The derivation of r0 and r0 is an iterativeprocess, including the solution of the basic equations of Eq.(18) or Eq.(20), and the initialvalues of the iteration are selected as follows:{
F (0) = 1, F(0)z = 1, F ′(0) = 0
G(0) = τ, G(0)z = τ, G′(0) = 0
. (21)
This selection does not require any initial information, it is obviously different from theshort-arc precision orbit determination in the usual sense, the latter relates to multivariateiteration and has a more strict requirement on the selection of initial values. So, the ground-based measurement and control systems should be provided with the orbit determinationmeasure described in this paper. The above-mentioned method suits as well multi-data,thus the statistical information of the multi-data can be exploited to improve the accuracyof the actual orbit determination[12].
294 LIU Lin et al. / Chinese Astronomy and Astrophysics 31 (2007) 288–295
4. NUMERICAL VERIFICATION
4.1 Verification of the Accuracy of the Time Power Series SolutionBased on the time power series solution mentioned above, we can extrapolate the po-
sition and velocity vectors r0 and r0 at the time t0 to the position and velocity vectors r
and r at the time t, and then compare the latter with the result of high-precision numericalcalculations, so checking the accuracy of the time power series solution. Considering thecharacter of the time power series solution and the requirement of the short-arc orbit deter-mination, the value of τ(t − t0) can not be very large, i.e., in the process of verification theseries solution is extrapolated segment by segment, and in order to meet the requirement ofthe short-arc initial orbit determination, τ is taken to be 30min. At the initial time, 2008Jan 10, 0h, the selected initial orbital elements are:
a = 160450 km, e = 0.95886569, i = 9◦.0, Ω = 328◦.0, ω = 212◦.0, M = 0◦.0 .
The units used in the calculations are the normalized units listed before, and the integrationis made for one-half of the period of the large-eccentricity elliptical motion at the transitionstage, and the obtained results are listed in Table 1. In this Table, “a” stands for the resultof the numerical solution, “b”, the result of the series solution, and ω = Ω +ω has the samemeaning as in the text below.
Table 1 Numerical verification of the time power series solution
T (min) a (km) e i (◦) Ω (◦) ω (◦) M + ω(◦)
1575a 156821.341 0.95816700 8.888532 327.726694 212.284792 226.561829b 156819.857 0.95816660 8.888567 327.727018 212.284497 226.562519
3150a 156836.995 0.95966530 8.359897 326.195815 213.634681 272.778807b 156835.508 0.95966490 8.359941 326.196190 213.634339 272.780163
4725a 156771.791 0.96366377 6.880192 319.095847 220.477043 318.403883b 156770.298 0.96366331 6.880286 319.096641 220.476296 318.405923
6300a 156434.293 0.97188704 4.364110 267.793248 271.681194 0.989091b 156432.762 0.97188621 4.364106 267.802419 271.672107 0.992649
4.2 The Result of Short-arc Orbit DeterminationAdopting again the same initial time and the initial orbital elements of the transfer
orbit as in the previous section, and taking the angular data for an illustrative example, theorbit is determined by using a group of simulated observational data (each group consistsof 50 observations at time intervals of 1min, i.e., over a 50min-span), and the coordinatesof the observing station is selected to be
(6372.4124 km, 118◦.820916, 31◦.893611) .
The result is shown in Table 2. Here, t0 is the epoch of the orbit determination, the ob-servational data are distributed in the preceding and following 25min. “A” indicates theresult obtained by using a complete force model for the extrapolation, “B” — the resultobtained with the simulative data without additional errors and in the way of 2-body prob-lem, “C” — the result obtained by using a complete force model and using the simulativedata in addition to small errors of size about 10−8 , “D” — the result obtained by using
LIU Lin et al. / Chinese Astronomy and Astrophysics 31 (2007) 288–295 295
the simulative data in addition to errors of size about 10−6, and in order to approach thereal situations, these errors are not purely random. Here, the 10−8 and 10−6 errors are allin units of radians (relative to the Earth’ center).
Table 2 Numerical verification of the short-arc orbit determination
t0(min) a (km) e i (◦) Ω(◦) ω(◦) M + ω(◦)
700
A 156772.885 0.9579274 8.97521 327.88653 212.16823 204.52762B 156995.190 0.9582140 8.95750 327.85395 212.17780 204.38883C 156772.805 0.9579272 8.97522 327.88655 212.16823 204.52770D 156774.451 0.9578618 8.97996 327.89556 212.16858 204.55371
The results of numerical verification illustrate that the time power series solution itselfhas a rather high accuracy, and that it is obviously suitable for the short-arc orbit deter-mination of the probe at its transition stage. When the probe is sufficiently far from theearth (for the example in this paper, when the probe has reached 20 Earth’s equatorial radiifrom the earth), the effect of the lunar gravitational perturbation is obvious, the order ofmagnitude of the perturbation exceeds apparently 10−4: to make orbit determination byusing a 2-body model under such circumstances would cause rather large errors, the result“B” has proved this point, so, the effect of the moon has to be taken into account simultane-ously. The method given in this paper accommodates itself to data of angular measurementsonly, or combined range & angular measurements, or positional measurements only, but atoo short arc segment of measurements is inadvisable (depending on the specific measure-ments), for then the nature of the short-arc orbit determination will cause loss of fidelity inthe result of the orbit determination, especially in the orbital semi-major axis a. When aprobe is launched into its transition stage, the present method will provide the ground-basedmeasurement and control systems with a means of short-arc orbit determination.
References
1 Kozai Y., AJ, 1959, 64, 367
2 Liu Lin, ChA&A, 1977, 1, 63
3 Liu Lin, AcASn, 1975, 16, 65
4 Liu Lin, Wang Jia-song, ChA&A, 1998, 22, 328
5 Liu Lin, Wang Jia-song, AcASn, 1998, 39, 81
6 Taff L. G., Celestial Mechanics, John Wiley & Sons, 1985
7 Liu Lin, Orbit Mechanics of Artificial Earth Satellite, Beijing: Higher Education Publishing House,
1992
8 Liu Lin, Orbit Theory of Spacecraft, Beijing: National Defence Industry Press, 2000
9 Liu Lin, Wang Xin, ChA&A, 2003, 27, 335
10 Liu Lin, Wang Xin, AcASn, 2003, 44, 175
11 Liu Lin, Wang Jian-feng, Journal of Spacecraft TT&C Technology, 2004, 23, 41
12 Jia Pei-zhang, Wu Lian-da, AcASn, 1997, 38, 353
