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Motion of an artificial satellite of a non-spherical Earth
TRANSLATIONS CONCERNED WITH THE PHYSICAL SCIENCES PUBLISHED IN
ARTIFICIAL EARTH SATELLITES U.S.S.R.
(ISKUSSTVENNYE SPUTNIKI ZEMLI) (PUBLISHED BY THE U.S.S.R. ACADEMY MOSCOW)
OF SCIENCES,
Planet.
Space
Sci. 1964. Vol.
12, pp. 705 to 717.
MOTION
Pergamon
Press Ltd.
Printed
in Northern
Ireland
OF AN ARTIFICIAL SATELLITE NON-SPHERICAL EARTH
OF A
G. V. SAMOILOVICH
Translated by H, S. H. Massey from Iskusstvennye Sputniki Zen& No. 16, p. 140 (1963)
Disturbances of the form of approximation question is investigated equations in osculating
Kepler motion of an artificial satellite of the Earth depend on the of the field of attraction of a geoid. In the present paper this by numerical integration on an electronic computer of differential elements at an interval of ten revolutions of a satellite.
1. MODELS OF THE EARTH’S EVALUATION OF THE
GRAVITATIONAL APPROXIMATIONS
FIELD
The terrestrial attraction potential can be described in the form of an infinite expansion of spherical functionso*“‘, and truncated for practical application. Increasing the number of terms improves the approximation to the gravitational field of a geoid; at the same time the coefficients become increasingly difficult to determine reliably. According to Kaula’s’3) estimates the mean square errors in the coefficients commencing at the fifth harmonic are comparable with or even greater than the values of the coefficients. Bearing this in mind, together with the ideas expressed by Zhongolovich’“), we can write the gravitational potential in the form
(1.1) and estimate the error introduced by rejecting the higher harmonics. In (1.1) we are using the notations: p-gravitational constant; cnicr d,,-constant coefficients in a spherical function of order n; P = P&sin yl); v and L-satellite’s geocentric latitude and longitude; r--focal radius of satellite; the quantity r,, is defined by the expression r0 = p/W,,, where IV, is the gravity potential on a geoid. We should first point out that for the mean square value Km = @/uir)(~*~~)zc~~~~ (averaged over a sphere) the following inequality with respect to the potential of a spherical Earth is true: 15,” 1 < 2\‘r ( (‘z,,/ q2, where
From this we conclude that, at average distances, not more than 0.4-0.05 per cent of the potential of a spherical Earth corresponds to the disturbing potential V,, (at distances of 6700 km “c r I 16000 km). The coefficients with harmonics above the second order become smaller in comparison to c, so that the term V, is greater than the subsequent ones. We can eliminate it (and also the term containing Poe) from the sum (1.1) and write down for the remainder the mean square estimate with respect to the potential of a spherical Earth (Appendix I): 1
o_,f< -r
1
( ILw) r-ci’
n=2
+ 2(i --4 Z(m--1)~~>Q!) + qe(i-1) f: Cm111 n==z
l)q2a,z)l’z
(1.2)
706
G. V. SAMOILOVICH
In this inequality Go denotes the mean square value of the anomalous potential (the anomalies are that part of the potential expansion which is not taken into account). The mean square differences a,(N) between the heights of a spheroid and the heights of the model geoid used are given by KaulaC5) (up to IZ= 32). It should be pointed out that in this case the limits in the sums of (1.2) are as follows: nr = 32, n = i + 1 = 2 since Kaula 15) allows for harmonics starting from the second sectorial one (corresponding to spherical function P&. If we calculate (1.2) for the values of r given above and different values of i, then it is clear that, assuming the Earth to be spheroidal, we shall introduce an error two orders less than the value of 17~~into the calculations. From what follows it will be clear that the disturbances of a satellite’s motion (at an interval of up to ten revolutions) in the field of a spheroid may be as much as several tens of kilometers, so the error in the position of a satellite at average distances from the Earth will not be more than 1 km in this case. When four spherical harmonics are taken into account in the potential expansion the error will be three orders less than SzO(i.e. not more than hundreds of meters in the satellite’s position). Eight harmonics in the expansion does not increase the accuracy by much, and 16 harmonics could reduce the error to 1O-4* &,, if the coefficients in this expansion could be determined with sufficient certainty. Therefore by writing the potential in the form (1.1) we are introducing errors of not more than hundreds of meters into calculations of a satellite’s position. Further simplifications of (1.1) can be achieved if we leave only the main terms in this sum (in terms of elementary spherical functions and the corresponding coefficients). Then [c,,P&sin
y)l + (c22cos 22 + cl,, sin 21)P,,(sin v) +
: 3c30P3,(sin y) +
0
3 4c4,P4,(sin y) .
0
I
(1.3)
A disturbance which is at least an order less than the term with the harmonic P4,, is produced by the terms with attached fourth-order harmonics (and also with the harmonic P,,). The terms corresponding to the harmonics P,, and P33, although they are slightly larger, are also small ; in addition their coefficients are not determined with sufficient accuracy. On the basis of (1.3) the following elementary models of the disturbing potential can be constructed : V l3 =
I/ 1 *
zz
l! Y
(3’2c 1p 20
20
(sin
y)
r.
Vc = V,* + V..* = V,* + @ co 4c40P,o(siny), r 0 r
Vn = Vi* + V3* = V,* + I”_ r” ‘(c,, cos 2J + d,, sin 2Jt)P2,(sin w), r 0r V,
=
VI*
+
V,*
=
VI*
+
lf
iL
r 0 r
3c30P30(siny),
(1.4) i
MOTION
OF AN ARTIFICIAL
SATELLITE
OF A NON-SPHERICAL
EARTH
707
In the case I’, the Earth is represented in the form of a spheroid, in case I’, in the form of an ellipsoid of rotation allowing for the second degree of compression, in case Vn in the form of a tri-axial ellipsoid and in case V, in the form of an non-symmetrical spheroid. The motion of a satellite in the field of these potentials will be investigated. 2. SYSTEM
OF
DIFFERENTIAL
EQUATIONS.
METHOD
OF
INVESTIGATION
As the evolution of an orbit can be most clearly represented by osculating elements it is convenient to solve the present problem in these parameters. Since we shall discuss orbits of different eccentricities, including circular ones, we shall use a system of differential equations with respect to p, q, k, Q, i(@ writing it down as a function of the argument u as in(‘).* The order of magnitude of the terms in the right-hand sides of the equations written as a function of the argument t is determined by the model used of the gravitational field. It should be retained when changing to a fresh argument (Appendix 11). Bearing this in mind the system of equations as functions of the argument u with an accuracy up to and including terms of the order of the Earth’s polar flattening should be written down as follows: 1 r2 dfi 1 -== [Fvi _t- e,], I du Jpp 1 + q cos u + k sin u sin i
dp
r"[Z-Ie,l,
xl
di
sin2 ~1+ +k sin 2~) cot i -
du
;
di
= (k sin2 u -
+ (k sin u -
&q sin 2~) cot i - du
I iq sin 221)cos i dS2 du
i
dQ
(q sin2 u + frk sin 2~) cos i du
+ -!_ (k + sin u) 2P dt
- 1.2 du-&P
[l + 2
cot i sin u lVi], J
P 1 + q cos u + k sin II’
?“=
e1 = 9c,,2r,4 L sin i cos3 i sin3 u, r5
0 = 9c 2r 4 1 2
20
0
T
r
. sm i cos2 i sin3 u cos u,
e 3 = 9c.20 2r0 4 L cos2 i sin3 u[sin i cos2 u r5
a(3 sin2 i sin” tl -
e4 = 9c202r04 I cos2 i sin2 II cos u[sin i sin2 43 sin2 i + 1) r5 * The notations
here and below are the same as in Ifi).
l)], l]
(2.1)
G. V. SAMOILOVICH
708
The functions gi, Fii, Wi contain all the terms (starting from terms of the first order). The index i (i = B, C, D, E, F) denotes the model used for the gravitational field in accordance with (1.4). In the numerical integration of system (2.1) on the computer the constant values in the expansion of the potential were taken from referencec4). The disturbed motion is fully characterized by the quantities 6t = t - t, and Ar - the change in the radius vector* : Ar
=
=
J(x
-
.x,)~
+
(y
-
y,)”
+
(Z
-
J6r2 + r”(co? u + cos2 i sin2 u) 6Q2 + r2 sin2 u 6i2 -
z,)~
2r2 sin u cos u sin i di 8Q
The function dr = r - rn is the change in the focal radius vector modulus. For greater geometrical clarity, however, we also discuss below the disturbances of the osculating elements 6p, dq, Sk, dCl, di, the disturbances 6e, 6w (the transformation to which is made in accordance with Samoilovich’6) and the disturbances of the modulus of the orbital velocity 6V = V - V,. TABLE
1
i
w
Variant no.
p (km)
e
(degrees)
1 2 3 4 5 6 7
6996 11216 7364 14800 7364 7364 6996
0.0499 0.735 0 1.01 0 0 0.0499
0 0 0 0
(degrees) _____ 90 45 0 45 90 63.4 63.4
The disturbances of the parameters within the range 2kn < u < 2(k + 1)~ (k = 0, at the points 2krr (k = 1, . . . , 10) 1 * * 3 9) are called periodic and the disturbances qkasi-secular. In actual fact both contain both secular and long-period components. The variants of the solution shown in Figs. 1-l 1 are characterized by the values of the initial parameters given in Table 1. 6e
FIG.
1. NATURE
OF PERIODIC
ARE THE VARIANT
DISTURBANCES
NUMBERS;
OF ECCENTRICITY
THE SCALE
e.
OF de FOR VARIANT
THE
NUMBERS
ON THE CURVES
3a IS ON THE RIGHT.
* The index %” relates to parameters of motion in a normal field. The value of Ar is equal to the modulus of the vector difference (Ar = IY- r,l) unlike 6r which is equal to the difference of the moduli of the same vectors (6~ = 1~1- Ir,l).
MOTION OF AN ARTIFICIAL
SATELLITE OF A NON-SPHERICAL
EARTH
-27
u
FIG.
2.
NATURE
90
MU
OF PERIODIC DISTURBANCES OF ORBIT PARAMETERP. CURVES ARE THE VARIANT NUMBERS.
u
78U
i7R
II
THE
NUMBERS
ON THE
36P
FIG. 3. NATUREOF PERIODICDISTURBANCESOF ANGULARDISTANCEOFPERIGEE FOR ZAND 4. THESCALEFORVARIANT~ISONTHELEFT,FORVARIANT~INTHECENTRE,FORVARIANT
VARIANTS 1,
4 ON THE RIGHT.
ff FIG. 4. NATURE
90
73R
z7u
,‘,
J‘w
OF PERIODIC DISTURBANCES OF ANGULAR DISTANCE OF PERIGEE OF CIRCULAR ORBITS (FOR VARIANTS 5, 6 AND 3).
709
710
G. V. SAMOILOVICH
u-3
u -7l’//J -z41F3
-3N-3
ff
90
u
F1c.5. NATUREOFPERIODICDISTURBANCESOFFUNCTION~. THE VARIANTNUMBERS;
THE SCALE OF6qFOR
L7
FIG. 6. NATUREOFPERIODIC
THENLJMBERSONTHECURVES VARIANT 3 IS ON THE RIGHT.
I
I
I
JQ
mu
zm
-Z./PI
DISTURBANCESOFFUNCTION
3fU”
k.
ARE
1 *“
JJP
THENUMBERS
ON THE CURVES ARE
THE VARIANT NUMBERS.
4 u
90
FIG.~. NATUREOFPERIODICDISTURBANCESOF ON THE CURVES ARE THE VARIANTNUMBERS;
780
z7u
LONGITUDEOF THESCALEOFfin
ASCENDING LOOP. THE NUMBERS FORVARIANT 4 IS ONTHERIGHT.
MOTION
OF AN
ARTIFICIAL
SATELLITE
OF A NON-SPHERICAL
EARTH
FIG. 8. NATURE OF PERIODIC DISTURBANCES OF ORBITAL INCLINATION. THE NUMBERS ON THE CURVESARETHEVARIANTNUMBERS(~D-FIRSTVARIANTOFPARAME~RSELECTIONANDVARIANT D OF POTENTIALMODELSELECTION); SCALEFORVARIANTS2 AND 6 ONTHE LEFT,FORVARIANT4 IN THE CENTRE,FOR VARIANT 1DON THE RIGHT.
6t. set
‘.i
-7GJ
-2uu
u
30
76U
z7u
u
3w
FIG.~. NATUREOFPERIODICDISTURBANCESOFFUNCTION t. THENUMBERSONTHECURVESARE THEVARIANTNUMBERS,THESCALEFORVARIANT2ISONTHERIGHT,FORTHEOTHERSONTHELEFT (FOR VARIANT 4 It% = 42,850 AT u = 170”).
0 FIG. 10. NATUREOFPERIODIC
90
DISTURBANCESOF THE VARLANTNUMBERS,THESCALE
78U
z7g
FUNCTIONS. FORVARIANT
THE NUMBERSONTHE 2 ISONTHERIGHT.
CURVES ARE
711
712
Cr. V. SAMOILOVICH
26
FIG. II. NATURE OF PEKIODIC DISTURBANCES OF Ar. THE NUMBERS ON THE CURVES ARE THE VARIANTNUMBERS; THESCALEFORVARIANT 21s ONTHERIGHT(FORVARIANT 4Ar = 41,290 AT u = 170”). 3. DISTURBANCES
OF ELLIPTIC
ORBITS
GRAVITATIONAL
IN DIFFERENT
MODELS
OF
THE
FIELD
The nature of the periodic disturbances of the elements of motion of a satellite is defined in all models of the potential by the disturbance of a spheroid’s field. The orbital inclination and the longitude of the ascending loop are exceptions; with polar orbits they are disturbed only in the field of models which allow for the Earth’s tri-axiality (Fig. 8, curve 10). In these models of the gravitational field long-period components (caused by the daily rotation of the Earth) with a period of 12 hr appear in the quasi-secular disturbances of the elements. In orbits with an inclination of i = 63.4” there are none of these daily oscillations in the values of 6 and cc). The quasi-secular disturbances Sp and 6e in the field of a spheroid and an ellipsoid of rotation can be explained by appearance of long-period oscillations connected with the movement of the apside line. The periodic disturbances for different models of the gravitational field are given in Table 2 and the numerically largest quasisecular disturbances in Table 3. TABLE 2
Model
B
C
D E F ___~
np (u = 90”) (km) 1 7 -19.71 -19.68 -19.92 - 19.66 -19.83
-15.76 -15.77 -16.02 -15.74 -16.00
6e (u = 60”)
-15.54 -15.56 -15.87 -15.57 -15.88
7 x x x x x
di
Model B C D E F
cY.Q(u= 360”) 1 7 0 0 -1”*33 0 -1O.33
-13’12” -13’10” -13’21” -13’12” -13’19”
60 (N = 90’)
1 10-4 1O--4 10-k 10-d 1O-4
-13.95 -13.97 -14.30 -13.94 -14.31
___
u = 360” 1
N == 90” 1
0 0 2.89 0 2”.89
-1’56” - 1’56” -2’00” -1’56” -2’00”
63.4’.
x x x x x
10-4 IO-4 10-4 10-4 10-a
7
-2”10’33” -2”10’12” -2”09’53” -2”10’08” -2”09’07”
- 1’22’38” - l”22’37” - l”22’26” -1”22’33” --1”22’31”
6t (u = 360”) l
-5.62 -5.65 -6.02 -5.62 -6.05
N.B. The numbers in the second line are numbers of variants. i =
1
(=I
Ilrln~ax,
7 -9.23 -9.25 -9.65 -9.24 -9.66
First variant:
1
9.266 9.303 9.688 9.235 9.745 i = 90”;
km 7 25.293 25.230 25.570 25.293 25.507
seventh variant:
MOTION
OF AN
ARTIFICIAL
SATELLITE
&&ax (RI) 1
B 0.6 C 0.65 D 219 E 53 F 198 ----_------~_--_.-
I
0 0 274 0 274
6 7 75 79 10s
x X x X X
lo-’ 10-1 10-G lo--B 10-a _---
713
p&m
iSelmax
7
EARTH
3
TABLE
Model
OF A NON-SPHERICAL
7
1
0 0 725 x lo+ 0 725 x IO-’
226’12” 2”25’41* 2’26’50” 226’18” 2”25’26”
7
1
0 0 19” 0 29” 2”47 9” 0 /I 2”.47 1 __--_~~____._.____
2”12’03” 2”11’42” 2”12’24” 2”12’04” 212’03”
j8t lmax (set) Model B c D E F
1 ‘di’max 0 0 44” 0 44”
7
1
7
21” 0 21”
56.2 56.5 58.2 56.1 58.5
92-4 92.5 94.7 92.4 94.8
0.288 0.280 0.500 0.840 1.032
253 252 254 253 253
N.B. The numbers in the second line are numbers of variants. First variant: i = 90”; seventh variant: i = 63.4”; quasi-secular disturbances in this table are examined at an interval of 10 revolutions.
4. DISTURBANCES DIFFERENT
OF
CIRCULAR
MODELS
OF
AND THE
HYPERBOLIC*
GRAVITATIONAL
ORBITS
IN
FIELD
The trajectory of a satellite moving in a non-central field and moving in a circle initially will have an osculating orbit (ellipse) at any time which is infinitely close to the initial value (i.e. at u -+ 0). For the succeeding period the osculating motion remains elliptic within the range 2kn- < u < 2(k- + 1)~. At points 2kn the osculating orbit in a model B field ceases to be elliptic. The perigee of the osculating orbit at the point in time u -+ 0 is formed at the point cu = r/2. At the point 2kn the function 60(u) is broken (Fig. 4): its value on the left is 6w = 3n/2 and on the right 60 = 7r/2 (in the model B field). From Fig. 4 it is clear that the apside line of polar orbits, no matter what the size of the orbit, makes two and a half revolutions (at the same average velocity) in the time of one draconic period (Fig. 4). Just as with elliptic orbits the nature of the periodic disturbances of circular and hyperbolic orbits for all models of the potential is determined by the disturbances in the model B field, and with the exception of 8~11and 6e (with circular orbits Se > 0 always and with hyperbolic orbits de < 0), is the same as in elliptic orbits (Figs. 1-I 1). The effect of the various models of the potential on the periodic disturbances of a circular and hyperbolic orbit is shown in Table 4. It should be noted that the values of the function Ar calculated for motion on hyperbolic orbits in model B and F fields differ considerably (the same is the case with the values of &). This indicates the necessity of knowing as precisely as possible the gravitational potential in the case of hyperbolic motions. The disturbances 6~ calculated in different models of the gravitational field differ most from each other in the case of circular orbits when u is close to 360”. This can be seen from * Here and subsequently orbits with e > 1 are considered as hyperbolic orbits. .&
714
G. V. SAMOILOVICH TABLE 4*
8cu
de
Model B C D E F
6; u = 180”
4; u = 170”
1.16 1.15 1.21 1.15 1.21
8.20 8.22 8.48 8.20 8.47
x x x x x
1O-3 10-s 10-s 10-Z 10-S
Model
z D E F
x x x x x
10m4 lo-* 10m4 lo-” 1O-4
4; b, = 170”
6; u=360”
179”58’33” 179”58’01” 175”52’03” 179”51’59” 175”46’30”
2’28” 2’28” 2’29” 2’28” 2’29”
-7.74 -7.74 -7.98 -7.74 -7.98
&(km) 6; u = 360 4; II = 170”
25.49 25.26 25.59 25.55 25.52
St(sec)
6; u = 180”
413,400 412,900 423,400 412,700 423.000
4; u = 170” -428,500 -428,300 -439,100 -428,300 -438,700
6 v(m/sec) 6; u = 180” 4; I( = 170” 8.45 3.31 8.46 3.31 8.89 3.32 8.46 3.29 8.91 3.31
* The second line gives the numbers of the variants and the values of u. Sixth variant: p = 7364 km, e = 0, i = 63.4”; fourth variant: p = 14800 km, e = 1.01, i = 45”. The disturbances&i for the 6th variant are measured from the initial (meaning u + 0) value of w = 90”. The argument I( = 170” for the 4th variant corresponds to r M loo km.
the data given below (i = 63*4”, end of first revolution): Field model Angular distance of perigee co (U = 360”)
B
C
D
E
F
270”
270”
187” 52’ 50”
344” 7’ 6”
186” 35’ 43”
The difference between the quasi-secular disturbances of the focal parameter of circular orbits and the corresponding disturbances for elliptic orbits in model D and F fields consists of their being positive and less in magnitude. At an interval of 10 revolutions they are equal to 6p,,, = 71.3 m (for model D) and 71.5 (for model F). The maximum quasi-secular disturbances 6e (models D and F) are 7.7 x 10e5. In view of the uncertainty in o at points u = 2krr, the quasi-secular disturbance 6w cannot be determined. The disturbances dt and Ar at u = 2~ are shown in Table 4 (they change linearly depending on the number of revolutions). 5. DISTURBANCES
DUE
TO GRAVITATIONAL
ANOMALIES
Since the term of the gravitational potential containing the harmonic PZOhas the basic disturbing effect the field of a spheroid may be taken as the normal field. In this case the effect of the square of the flattening and lack of symmetry of the hemispheres and the tri-axiality of the Earth must be due to gravitational anomalies (in accordance with (1.3)). The periodic disturbances due to the second degree of flattening and non-symmetry of the hemispheres are characterized by the appearance of higher harmonics. As an example Fig. 12 gives the periodic disturbances Sp due to gravitational anomalies. The effects of the above two anomalies on all the parameters have the same nature. In particular ISpI < 50 m, 16el I 5 x IOV, Ar I 70 m, l&l I 0.03 sec. The effect of the tri-axiality is an order greater and leads to the appearance of harmonics similar to the basic ones (caused by the term V,,). All these anomalies lead to periodic disturbances Ar of a magnitude not greater than 500 m and l&l < 0.45 sec. The maximum quasi-secular disturbances caused by the tri-axiality are given by I 8p I c 280 m, 16el I7.5 x 10P5, 16~1 I 25”, 16Qj I 20”, j&l i 22”, l&l i 2.5 set,
MOTTIQN OF AN
ARTIFICIAL
47
SATELLITE
JU
@‘/J
OF A NON-SPHERICAL
z7u
EARTH
715
LL da?”
FKL 12. EFFECTOF ~~v~TATI~~AL
ANOMALIES ON DISTURBANCES OF THEFOCAL PARAMETERS: CI-EFFECTOFTI~ESQUAREOFT~EPOLARTLATIENING; &EFFECTOFTHE EARTH’STRI-AXIALITY; c--EFFECT OF NON-SYMMETRY OF THE HEMISPHERES; ~-TOTAL EFFECT OF ANOMALIES; THE SCALE FOR THE DASHED
LINES (h AND d}ARE ONTHE
RIGHT.
Ar i 280 m, The flattening has an effect chiefly on the quasi-secular motions of the apside with the disturbances due to the tci-axiality. line and the loop line. They are comparable The nun-symmetry of the hemispheres has the greatest effect on the eccentricity (in this case reaching a value of the order of lo-?) and on Ar. On the 10th revolution the quasi-secular disturbance Ar due to the non-symmetry is 500 m and exceeds the disturbance due to tri-axiality, The total effect of all the anomalies (including the periodic and quasi-secular parts) does not exceed 1.5 km for Ar (in the case of orbits with i = 63-4” we have Ar I 800 m) over the fO revolutions, whilst /&I < 1.2 see over 3 revolutions and j&i 2 29 set over the 10 revolutions, On circular orbits the non-symmetry and tri-axiahty of the Earth cause a strong change in the periodic disturbances of the apside line at the end of the draconic period. The change in the value of the disturbance 6w may be as much as $0” when each of the three anomalies occurs, On circular orbits with an inclination close to 63.4” there is a break in the function 6w due to the effect of tri-axiality at the points u = 140 and 220’. As the A large disturbance (about IOU0 m) appears inclination decreases this feature disappears. in the function SP on circular orbits at the end of the period; this can be explained by the large value of the long-period oscillation (due to the Earth’s rotation). The nature of the change of the short-period disturbances due to the action of the various gravitational anomalies can be seen in Table 5. The total effect on the anomalies listed on the general (periodic and quasi-secular)
Type of anomnly
A ~mslc
-
---ERect of square of flattening Effect of hemispheres’ nonsymmetry Effect of Earth’s tri-axiality Total effect of anomalies
Hyperbolic orbits e -I: l*Ol,p = 14,800 z/ = 170”
Circular orbits p = 7364 km
Cm)
prJmax W.3
300
0
300 400 700
0 0.4 0.4
200 10500
220 -
10100
6170
G. V. SAMOILOVICH
716
disturbances of circular orbits does not exceed the following < 4000 m, 16t 1 < 3.5 sec. The above analysis shows that the Earth can be considered of problems which are of practical importance.
values
over 10 orbits:
as a spheroid
/k
in a number
REFERENCES 1. 2. 3. 4. 5. 6. 7.
G. N. DUBOSHIN,Theory of Attraction (in Russian). Fizmatgiz (1961). M. F. SUBBOTIN,Course in Celestial Mechanics (in Russian). Vol. III. Gostekhizdat (1949). W. M. KAULA, J. Geophys. Res. 66, 1799 (1961). I. D. ZHONGOLOVICH,Byull. Astr. Teor. Inst. 6, 505 (1957). W. M. KAULA, Army Map Service Technical Report N 24 (1959). G. V. SAMOILOVICH,Isk. Sput. Zemli No. 16, p, 136 (I 963). D. YE. OKHOTSIMSKII, T. M. ENEYEVand G. P. TARATYNOVA, Usp. Fiz. Nauk 63, No. la, 33 (1957). APPENDIX
After eliminating
the terms containing
I
P,, and Pz,, from (1.1) we obtain
(cz2 cos 22 + d,, sin 21)P,, cos kl + d,, sin ki)P,,c or
y&&f
0 3
(1.1)
1
n-1 N,
r2 n=e r
Here N, = rOPn are the height deviations of a geoid corresponding to the Legendre polynomial P, ; only the sectorial harmonic formed part of N,. From this we can obtain the expression for the mean square value of Vzd:
which, after certain
manipulations,
allows us to write down the estimate:
The quantity i + 1 = n denotes the number of the lowest spherical harmonic present in the anomaly potential, whilst m denotes the number of the highest harmonic for which allowance can be made in this expansion. If the anomalous potential is written in form (1.1) then we should put n = 2 in the lower limit of the first sum and allow for only the sectorial harmonic in the estimate of the second order harmonics. The mean square deviations gn = gn(N) are given below (quoted from Kaulac5)): n
gn2, m2
n
un2, m2
n
un2, wP
n
308 460 140 28 41 3-3 19.6 14.6
10
7.8 7.6 2.4 4.4 5.8 4-8 I.1 2.0
18 19 20 21 22 23 24 25
2.8 1.3 O-8 l-5 1.0 0.8 0.9 0.7
26 27 28 29 30 31 32
11 12 13 14 1.5 16 17
ls,2, t?23
o-7 0.3 0.5 o-2 o-1 o-05 o-09
MOTION
OF AN ARTIFICIAL
The mean square difference whilst
SATELLITE
OF A NON-SPHERICAL
of the heights of a geoid and a spheroid
:a,”
EARTH
717
is about 1075 WZ~(~),
= 1062m2
n-2
Therefore the mean difference of the heights of a geoid and a shape represented by 32 2 as a function of n it can be seen that harmonics is about 3*6m. From the data given for a7&
i.e. the remaining The estimate
harmonics (from 5 to 32) have on the whole the same effect as one quarter. simplified at q I O-7 and i < 2. given for aAfi2 can be considerably APPENDIX
Let us write the equations Q
in the osculating
= Qi(qlj . . . 395; U)zj,
IL
elements
(i=l,...,
(ql, . . . , q5) in the form 5;j
= 1, 2, 3)
_& can be taken as g, F or p. More strictly we should write the right-hand sides form of the sum: Qil(ql, . . . , q5: u) s” + Qz2(q1, . . . , q5; u) F + Qia(ql, . . . , q5; This, however, complicates the arguments without really changing the essentials. Since Ii = & + _&, where the indices 1 and n denote the highest order of the 21 present in the cirresplfnding functions, then when we change to the argument u the ential equations can be written in the form: d&A_ - zQi(ql, du JPP
. . . ,q5; 4[$
f
Fi(41, f . * ,q5;
in the u)@. terms differ-
Io,h~I;jl.
The indices below the functions z and @ denote the highest order to the terms present in these functions. System (2.2) was obtained on the basis of these equations. I should like to thank L. P. Pellinen for his useful discussions and V. N. Lavrik who programmed and solved the equations on a computer.
ARTIFICIAL EARTH SATELLITES U.S.S.R.
(ISKUSSTVENNYE SPUTNIKI ZEMLI) (PUBLISHED BY THE U.S.S.R. ACADEMY MOSCOW)
OF SCIENCES,
Planet.
Space
Sci. 1964. Vol.
12, pp. 705 to 717.
MOTION
Pergamon
Press Ltd.
Printed
in Northern
Ireland
OF AN ARTIFICIAL SATELLITE NON-SPHERICAL EARTH
OF A
G. V. SAMOILOVICH
Translated by H, S. H. Massey from Iskusstvennye Sputniki Zen& No. 16, p. 140 (1963)
Disturbances of the form of approximation question is investigated equations in osculating
Kepler motion of an artificial satellite of the Earth depend on the of the field of attraction of a geoid. In the present paper this by numerical integration on an electronic computer of differential elements at an interval of ten revolutions of a satellite.
1. MODELS OF THE EARTH’S EVALUATION OF THE
GRAVITATIONAL APPROXIMATIONS
FIELD
The terrestrial attraction potential can be described in the form of an infinite expansion of spherical functionso*“‘, and truncated for practical application. Increasing the number of terms improves the approximation to the gravitational field of a geoid; at the same time the coefficients become increasingly difficult to determine reliably. According to Kaula’s’3) estimates the mean square errors in the coefficients commencing at the fifth harmonic are comparable with or even greater than the values of the coefficients. Bearing this in mind, together with the ideas expressed by Zhongolovich’“), we can write the gravitational potential in the form
(1.1) and estimate the error introduced by rejecting the higher harmonics. In (1.1) we are using the notations: p-gravitational constant; cnicr d,,-constant coefficients in a spherical function of order n; P = P&sin yl); v and L-satellite’s geocentric latitude and longitude; r--focal radius of satellite; the quantity r,, is defined by the expression r0 = p/W,,, where IV, is the gravity potential on a geoid. We should first point out that for the mean square value Km = @/uir)(~*~~)zc~~~~ (averaged over a sphere) the following inequality with respect to the potential of a spherical Earth is true: 15,” 1 < 2\‘r ( (‘z,,/ q2, where
From this we conclude that, at average distances, not more than 0.4-0.05 per cent of the potential of a spherical Earth corresponds to the disturbing potential V,, (at distances of 6700 km “c r I 16000 km). The coefficients with harmonics above the second order become smaller in comparison to c, so that the term V, is greater than the subsequent ones. We can eliminate it (and also the term containing Poe) from the sum (1.1) and write down for the remainder the mean square estimate with respect to the potential of a spherical Earth (Appendix I): 1
o_,f< -r
1
( ILw) r-ci’
n=2
+ 2(i --4 Z(m--1)~~>Q!) + qe(i-1) f: Cm111 n==z
l)q2a,z)l’z
(1.2)
706
G. V. SAMOILOVICH
In this inequality Go denotes the mean square value of the anomalous potential (the anomalies are that part of the potential expansion which is not taken into account). The mean square differences a,(N) between the heights of a spheroid and the heights of the model geoid used are given by KaulaC5) (up to IZ= 32). It should be pointed out that in this case the limits in the sums of (1.2) are as follows: nr = 32, n = i + 1 = 2 since Kaula 15) allows for harmonics starting from the second sectorial one (corresponding to spherical function P&. If we calculate (1.2) for the values of r given above and different values of i, then it is clear that, assuming the Earth to be spheroidal, we shall introduce an error two orders less than the value of 17~~into the calculations. From what follows it will be clear that the disturbances of a satellite’s motion (at an interval of up to ten revolutions) in the field of a spheroid may be as much as several tens of kilometers, so the error in the position of a satellite at average distances from the Earth will not be more than 1 km in this case. When four spherical harmonics are taken into account in the potential expansion the error will be three orders less than SzO(i.e. not more than hundreds of meters in the satellite’s position). Eight harmonics in the expansion does not increase the accuracy by much, and 16 harmonics could reduce the error to 1O-4* &,, if the coefficients in this expansion could be determined with sufficient certainty. Therefore by writing the potential in the form (1.1) we are introducing errors of not more than hundreds of meters into calculations of a satellite’s position. Further simplifications of (1.1) can be achieved if we leave only the main terms in this sum (in terms of elementary spherical functions and the corresponding coefficients). Then [c,,P&sin
y)l + (c22cos 22 + cl,, sin 21)P,,(sin v) +
: 3c30P3,(sin y) +
0
3 4c4,P4,(sin y) .
0
I
(1.3)
A disturbance which is at least an order less than the term with the harmonic P4,, is produced by the terms with attached fourth-order harmonics (and also with the harmonic P,,). The terms corresponding to the harmonics P,, and P33, although they are slightly larger, are also small ; in addition their coefficients are not determined with sufficient accuracy. On the basis of (1.3) the following elementary models of the disturbing potential can be constructed : V l3 =
I/ 1 *
zz
l! Y
(3’2c 1p 20
20
(sin
y)
r.
Vc = V,* + V..* = V,* + @ co 4c40P,o(siny), r 0 r
Vn = Vi* + V3* = V,* + I”_ r” ‘(c,, cos 2J + d,, sin 2Jt)P2,(sin w), r 0r V,
=
VI*
+
V,*
=
VI*
+
lf
iL
r 0 r
3c30P30(siny),
(1.4) i
MOTION
OF AN ARTIFICIAL
SATELLITE
OF A NON-SPHERICAL
EARTH
707
In the case I’, the Earth is represented in the form of a spheroid, in case I’, in the form of an ellipsoid of rotation allowing for the second degree of compression, in case Vn in the form of a tri-axial ellipsoid and in case V, in the form of an non-symmetrical spheroid. The motion of a satellite in the field of these potentials will be investigated. 2. SYSTEM
OF
DIFFERENTIAL
EQUATIONS.
METHOD
OF
INVESTIGATION
As the evolution of an orbit can be most clearly represented by osculating elements it is convenient to solve the present problem in these parameters. Since we shall discuss orbits of different eccentricities, including circular ones, we shall use a system of differential equations with respect to p, q, k, Q, i(@ writing it down as a function of the argument u as in(‘).* The order of magnitude of the terms in the right-hand sides of the equations written as a function of the argument t is determined by the model used of the gravitational field. It should be retained when changing to a fresh argument (Appendix 11). Bearing this in mind the system of equations as functions of the argument u with an accuracy up to and including terms of the order of the Earth’s polar flattening should be written down as follows: 1 r2 dfi 1 -== [Fvi _t- e,], I du Jpp 1 + q cos u + k sin u sin i
dp
r"[Z-Ie,l,
xl
di
sin2 ~1+ +k sin 2~) cot i -
du
;
di
= (k sin2 u -
+ (k sin u -
&q sin 2~) cot i - du
I iq sin 221)cos i dS2 du
i
dQ
(q sin2 u + frk sin 2~) cos i du
+ -!_ (k + sin u) 2P dt
- 1.2 du-&P
[l + 2
cot i sin u lVi], J
P 1 + q cos u + k sin II’
?“=
e1 = 9c,,2r,4 L sin i cos3 i sin3 u, r5
0 = 9c 2r 4 1 2
20
0
T
r
. sm i cos2 i sin3 u cos u,
e 3 = 9c.20 2r0 4 L cos2 i sin3 u[sin i cos2 u r5
a(3 sin2 i sin” tl -
e4 = 9c202r04 I cos2 i sin2 II cos u[sin i sin2 43 sin2 i + 1) r5 * The notations
here and below are the same as in Ifi).
l)], l]
(2.1)
G. V. SAMOILOVICH
708
The functions gi, Fii, Wi contain all the terms (starting from terms of the first order). The index i (i = B, C, D, E, F) denotes the model used for the gravitational field in accordance with (1.4). In the numerical integration of system (2.1) on the computer the constant values in the expansion of the potential were taken from referencec4). The disturbed motion is fully characterized by the quantities 6t = t - t, and Ar - the change in the radius vector* : Ar
=
=
J(x
-
.x,)~
+
(y
-
y,)”
+
(Z
-
J6r2 + r”(co? u + cos2 i sin2 u) 6Q2 + r2 sin2 u 6i2 -
z,)~
2r2 sin u cos u sin i di 8Q
The function dr = r - rn is the change in the focal radius vector modulus. For greater geometrical clarity, however, we also discuss below the disturbances of the osculating elements 6p, dq, Sk, dCl, di, the disturbances 6e, 6w (the transformation to which is made in accordance with Samoilovich’6) and the disturbances of the modulus of the orbital velocity 6V = V - V,. TABLE
1
i
w
Variant no.
p (km)
e
(degrees)
1 2 3 4 5 6 7
6996 11216 7364 14800 7364 7364 6996
0.0499 0.735 0 1.01 0 0 0.0499
0 0 0 0
(degrees) _____ 90 45 0 45 90 63.4 63.4
The disturbances of the parameters within the range 2kn < u < 2(k + 1)~ (k = 0, at the points 2krr (k = 1, . . . , 10) 1 * * 3 9) are called periodic and the disturbances qkasi-secular. In actual fact both contain both secular and long-period components. The variants of the solution shown in Figs. 1-l 1 are characterized by the values of the initial parameters given in Table 1. 6e
FIG.
1. NATURE
OF PERIODIC
ARE THE VARIANT
DISTURBANCES
NUMBERS;
OF ECCENTRICITY
THE SCALE
e.
OF de FOR VARIANT
THE
NUMBERS
ON THE CURVES
3a IS ON THE RIGHT.
* The index %” relates to parameters of motion in a normal field. The value of Ar is equal to the modulus of the vector difference (Ar = IY- r,l) unlike 6r which is equal to the difference of the moduli of the same vectors (6~ = 1~1- Ir,l).
MOTION OF AN ARTIFICIAL
SATELLITE OF A NON-SPHERICAL
EARTH
-27
u
FIG.
2.
NATURE
90
MU
OF PERIODIC DISTURBANCES OF ORBIT PARAMETERP. CURVES ARE THE VARIANT NUMBERS.
u
78U
i7R
II
THE
NUMBERS
ON THE
36P
FIG. 3. NATUREOF PERIODICDISTURBANCESOF ANGULARDISTANCEOFPERIGEE FOR ZAND 4. THESCALEFORVARIANT~ISONTHELEFT,FORVARIANT~INTHECENTRE,FORVARIANT
VARIANTS 1,
4 ON THE RIGHT.
ff FIG. 4. NATURE
90
73R
z7u
,‘,
J‘w
OF PERIODIC DISTURBANCES OF ANGULAR DISTANCE OF PERIGEE OF CIRCULAR ORBITS (FOR VARIANTS 5, 6 AND 3).
709
710
G. V. SAMOILOVICH
u-3
u -7l’//J -z41F3
-3N-3
ff
90
u
F1c.5. NATUREOFPERIODICDISTURBANCESOFFUNCTION~. THE VARIANTNUMBERS;
THE SCALE OF6qFOR
L7
FIG. 6. NATUREOFPERIODIC
THENLJMBERSONTHECURVES VARIANT 3 IS ON THE RIGHT.
I
I
I
JQ
mu
zm
-Z./PI
DISTURBANCESOFFUNCTION
3fU”
k.
ARE
1 *“
JJP
THENUMBERS
ON THE CURVES ARE
THE VARIANT NUMBERS.
4 u
90
FIG.~. NATUREOFPERIODICDISTURBANCESOF ON THE CURVES ARE THE VARIANTNUMBERS;
780
z7u
LONGITUDEOF THESCALEOFfin
ASCENDING LOOP. THE NUMBERS FORVARIANT 4 IS ONTHERIGHT.
MOTION
OF AN
ARTIFICIAL
SATELLITE
OF A NON-SPHERICAL
EARTH
FIG. 8. NATURE OF PERIODIC DISTURBANCES OF ORBITAL INCLINATION. THE NUMBERS ON THE CURVESARETHEVARIANTNUMBERS(~D-FIRSTVARIANTOFPARAME~RSELECTIONANDVARIANT D OF POTENTIALMODELSELECTION); SCALEFORVARIANTS2 AND 6 ONTHE LEFT,FORVARIANT4 IN THE CENTRE,FOR VARIANT 1DON THE RIGHT.
6t. set
‘.i
-7GJ
-2uu
u
30
76U
z7u
u
3w
FIG.~. NATUREOFPERIODICDISTURBANCESOFFUNCTION t. THENUMBERSONTHECURVESARE THEVARIANTNUMBERS,THESCALEFORVARIANT2ISONTHERIGHT,FORTHEOTHERSONTHELEFT (FOR VARIANT 4 It% = 42,850 AT u = 170”).
0 FIG. 10. NATUREOFPERIODIC
90
DISTURBANCESOF THE VARLANTNUMBERS,THESCALE
78U
z7g
FUNCTIONS. FORVARIANT
THE NUMBERSONTHE 2 ISONTHERIGHT.
CURVES ARE
711
712
Cr. V. SAMOILOVICH
26
FIG. II. NATURE OF PEKIODIC DISTURBANCES OF Ar. THE NUMBERS ON THE CURVES ARE THE VARIANTNUMBERS; THESCALEFORVARIANT 21s ONTHERIGHT(FORVARIANT 4Ar = 41,290 AT u = 170”). 3. DISTURBANCES
OF ELLIPTIC
ORBITS
GRAVITATIONAL
IN DIFFERENT
MODELS
OF
THE
FIELD
The nature of the periodic disturbances of the elements of motion of a satellite is defined in all models of the potential by the disturbance of a spheroid’s field. The orbital inclination and the longitude of the ascending loop are exceptions; with polar orbits they are disturbed only in the field of models which allow for the Earth’s tri-axiality (Fig. 8, curve 10). In these models of the gravitational field long-period components (caused by the daily rotation of the Earth) with a period of 12 hr appear in the quasi-secular disturbances of the elements. In orbits with an inclination of i = 63.4” there are none of these daily oscillations in the values of 6 and cc). The quasi-secular disturbances Sp and 6e in the field of a spheroid and an ellipsoid of rotation can be explained by appearance of long-period oscillations connected with the movement of the apside line. The periodic disturbances for different models of the gravitational field are given in Table 2 and the numerically largest quasisecular disturbances in Table 3. TABLE 2
Model
B
C
D E F ___~
np (u = 90”) (km) 1 7 -19.71 -19.68 -19.92 - 19.66 -19.83
-15.76 -15.77 -16.02 -15.74 -16.00
6e (u = 60”)
-15.54 -15.56 -15.87 -15.57 -15.88
7 x x x x x
di
Model B C D E F
cY.Q(u= 360”) 1 7 0 0 -1”*33 0 -1O.33
-13’12” -13’10” -13’21” -13’12” -13’19”
60 (N = 90’)
1 10-4 1O--4 10-k 10-d 1O-4
-13.95 -13.97 -14.30 -13.94 -14.31
___
u = 360” 1
N == 90” 1
0 0 2.89 0 2”.89
-1’56” - 1’56” -2’00” -1’56” -2’00”
63.4’.
x x x x x
10-4 IO-4 10-4 10-4 10-a
7
-2”10’33” -2”10’12” -2”09’53” -2”10’08” -2”09’07”
- 1’22’38” - l”22’37” - l”22’26” -1”22’33” --1”22’31”
6t (u = 360”) l
-5.62 -5.65 -6.02 -5.62 -6.05
N.B. The numbers in the second line are numbers of variants. i =
1
(=I
Ilrln~ax,
7 -9.23 -9.25 -9.65 -9.24 -9.66
First variant:
1
9.266 9.303 9.688 9.235 9.745 i = 90”;
km 7 25.293 25.230 25.570 25.293 25.507
seventh variant:
MOTION
OF AN
ARTIFICIAL
SATELLITE
&&ax (RI) 1
B 0.6 C 0.65 D 219 E 53 F 198 ----_------~_--_.-
I
0 0 274 0 274
6 7 75 79 10s
x X x X X
lo-’ 10-1 10-G lo--B 10-a _---
713
p&m
iSelmax
7
EARTH
3
TABLE
Model
OF A NON-SPHERICAL
7
1
0 0 725 x lo+ 0 725 x IO-’
226’12” 2”25’41* 2’26’50” 226’18” 2”25’26”
7
1
0 0 19” 0 29” 2”47 9” 0 /I 2”.47 1 __--_~~____._.____
2”12’03” 2”11’42” 2”12’24” 2”12’04” 212’03”
j8t lmax (set) Model B c D E F
1 ‘di’max 0 0 44” 0 44”
7
1
7
21” 0 21”
56.2 56.5 58.2 56.1 58.5
92-4 92.5 94.7 92.4 94.8
0.288 0.280 0.500 0.840 1.032
253 252 254 253 253
N.B. The numbers in the second line are numbers of variants. First variant: i = 90”; seventh variant: i = 63.4”; quasi-secular disturbances in this table are examined at an interval of 10 revolutions.
4. DISTURBANCES DIFFERENT
OF
CIRCULAR
MODELS
OF
AND THE
HYPERBOLIC*
GRAVITATIONAL
ORBITS
IN
FIELD
The trajectory of a satellite moving in a non-central field and moving in a circle initially will have an osculating orbit (ellipse) at any time which is infinitely close to the initial value (i.e. at u -+ 0). For the succeeding period the osculating motion remains elliptic within the range 2kn- < u < 2(k- + 1)~. At points 2kn the osculating orbit in a model B field ceases to be elliptic. The perigee of the osculating orbit at the point in time u -+ 0 is formed at the point cu = r/2. At the point 2kn the function 60(u) is broken (Fig. 4): its value on the left is 6w = 3n/2 and on the right 60 = 7r/2 (in the model B field). From Fig. 4 it is clear that the apside line of polar orbits, no matter what the size of the orbit, makes two and a half revolutions (at the same average velocity) in the time of one draconic period (Fig. 4). Just as with elliptic orbits the nature of the periodic disturbances of circular and hyperbolic orbits for all models of the potential is determined by the disturbances in the model B field, and with the exception of 8~11and 6e (with circular orbits Se > 0 always and with hyperbolic orbits de < 0), is the same as in elliptic orbits (Figs. 1-I 1). The effect of the various models of the potential on the periodic disturbances of a circular and hyperbolic orbit is shown in Table 4. It should be noted that the values of the function Ar calculated for motion on hyperbolic orbits in model B and F fields differ considerably (the same is the case with the values of &). This indicates the necessity of knowing as precisely as possible the gravitational potential in the case of hyperbolic motions. The disturbances 6~ calculated in different models of the gravitational field differ most from each other in the case of circular orbits when u is close to 360”. This can be seen from * Here and subsequently orbits with e > 1 are considered as hyperbolic orbits. .&
714
G. V. SAMOILOVICH TABLE 4*
8cu
de
Model B C D E F
6; u = 180”
4; u = 170”
1.16 1.15 1.21 1.15 1.21
8.20 8.22 8.48 8.20 8.47
x x x x x
1O-3 10-s 10-s 10-Z 10-S
Model
z D E F
x x x x x
10m4 lo-* 10m4 lo-” 1O-4
4; b, = 170”
6; u=360”
179”58’33” 179”58’01” 175”52’03” 179”51’59” 175”46’30”
2’28” 2’28” 2’29” 2’28” 2’29”
-7.74 -7.74 -7.98 -7.74 -7.98
&(km) 6; u = 360 4; II = 170”
25.49 25.26 25.59 25.55 25.52
St(sec)
6; u = 180”
413,400 412,900 423,400 412,700 423.000
4; u = 170” -428,500 -428,300 -439,100 -428,300 -438,700
6 v(m/sec) 6; u = 180” 4; I( = 170” 8.45 3.31 8.46 3.31 8.89 3.32 8.46 3.29 8.91 3.31
* The second line gives the numbers of the variants and the values of u. Sixth variant: p = 7364 km, e = 0, i = 63.4”; fourth variant: p = 14800 km, e = 1.01, i = 45”. The disturbances&i for the 6th variant are measured from the initial (meaning u + 0) value of w = 90”. The argument I( = 170” for the 4th variant corresponds to r M loo km.
the data given below (i = 63*4”, end of first revolution): Field model Angular distance of perigee co (U = 360”)
B
C
D
E
F
270”
270”
187” 52’ 50”
344” 7’ 6”
186” 35’ 43”
The difference between the quasi-secular disturbances of the focal parameter of circular orbits and the corresponding disturbances for elliptic orbits in model D and F fields consists of their being positive and less in magnitude. At an interval of 10 revolutions they are equal to 6p,,, = 71.3 m (for model D) and 71.5 (for model F). The maximum quasi-secular disturbances 6e (models D and F) are 7.7 x 10e5. In view of the uncertainty in o at points u = 2krr, the quasi-secular disturbance 6w cannot be determined. The disturbances dt and Ar at u = 2~ are shown in Table 4 (they change linearly depending on the number of revolutions). 5. DISTURBANCES
DUE
TO GRAVITATIONAL
ANOMALIES
Since the term of the gravitational potential containing the harmonic PZOhas the basic disturbing effect the field of a spheroid may be taken as the normal field. In this case the effect of the square of the flattening and lack of symmetry of the hemispheres and the tri-axiality of the Earth must be due to gravitational anomalies (in accordance with (1.3)). The periodic disturbances due to the second degree of flattening and non-symmetry of the hemispheres are characterized by the appearance of higher harmonics. As an example Fig. 12 gives the periodic disturbances Sp due to gravitational anomalies. The effects of the above two anomalies on all the parameters have the same nature. In particular ISpI < 50 m, 16el I 5 x IOV, Ar I 70 m, l&l I 0.03 sec. The effect of the tri-axiality is an order greater and leads to the appearance of harmonics similar to the basic ones (caused by the term V,,). All these anomalies lead to periodic disturbances Ar of a magnitude not greater than 500 m and l&l < 0.45 sec. The maximum quasi-secular disturbances caused by the tri-axiality are given by I 8p I c 280 m, 16el I7.5 x 10P5, 16~1 I 25”, 16Qj I 20”, j&l i 22”, l&l i 2.5 set,
MOTTIQN OF AN
ARTIFICIAL
47
SATELLITE
JU
@‘/J
OF A NON-SPHERICAL
z7u
EARTH
715
LL da?”
FKL 12. EFFECTOF ~~v~TATI~~AL
ANOMALIES ON DISTURBANCES OF THEFOCAL PARAMETERS: CI-EFFECTOFTI~ESQUAREOFT~EPOLARTLATIENING; &EFFECTOFTHE EARTH’STRI-AXIALITY; c--EFFECT OF NON-SYMMETRY OF THE HEMISPHERES; ~-TOTAL EFFECT OF ANOMALIES; THE SCALE FOR THE DASHED
LINES (h AND d}ARE ONTHE
RIGHT.
Ar i 280 m, The flattening has an effect chiefly on the quasi-secular motions of the apside with the disturbances due to the tci-axiality. line and the loop line. They are comparable The nun-symmetry of the hemispheres has the greatest effect on the eccentricity (in this case reaching a value of the order of lo-?) and on Ar. On the 10th revolution the quasi-secular disturbance Ar due to the non-symmetry is 500 m and exceeds the disturbance due to tri-axiality, The total effect of all the anomalies (including the periodic and quasi-secular parts) does not exceed 1.5 km for Ar (in the case of orbits with i = 63-4” we have Ar I 800 m) over the fO revolutions, whilst /&I < 1.2 see over 3 revolutions and j&i 2 29 set over the 10 revolutions, On circular orbits the non-symmetry and tri-axiahty of the Earth cause a strong change in the periodic disturbances of the apside line at the end of the draconic period. The change in the value of the disturbance 6w may be as much as $0” when each of the three anomalies occurs, On circular orbits with an inclination close to 63.4” there is a break in the function 6w due to the effect of tri-axiality at the points u = 140 and 220’. As the A large disturbance (about IOU0 m) appears inclination decreases this feature disappears. in the function SP on circular orbits at the end of the period; this can be explained by the large value of the long-period oscillation (due to the Earth’s rotation). The nature of the change of the short-period disturbances due to the action of the various gravitational anomalies can be seen in Table 5. The total effect on the anomalies listed on the general (periodic and quasi-secular)
Type of anomnly
A ~mslc
-
---ERect of square of flattening Effect of hemispheres’ nonsymmetry Effect of Earth’s tri-axiality Total effect of anomalies
Hyperbolic orbits e -I: l*Ol,p = 14,800 z/ = 170”
Circular orbits p = 7364 km
Cm)
prJmax W.3
300
0
300 400 700
0 0.4 0.4
200 10500
220 -
10100
6170
G. V. SAMOILOVICH
716
disturbances of circular orbits does not exceed the following < 4000 m, 16t 1 < 3.5 sec. The above analysis shows that the Earth can be considered of problems which are of practical importance.
values
over 10 orbits:
as a spheroid
/k
in a number
REFERENCES 1. 2. 3. 4. 5. 6. 7.
G. N. DUBOSHIN,Theory of Attraction (in Russian). Fizmatgiz (1961). M. F. SUBBOTIN,Course in Celestial Mechanics (in Russian). Vol. III. Gostekhizdat (1949). W. M. KAULA, J. Geophys. Res. 66, 1799 (1961). I. D. ZHONGOLOVICH,Byull. Astr. Teor. Inst. 6, 505 (1957). W. M. KAULA, Army Map Service Technical Report N 24 (1959). G. V. SAMOILOVICH,Isk. Sput. Zemli No. 16, p, 136 (I 963). D. YE. OKHOTSIMSKII, T. M. ENEYEVand G. P. TARATYNOVA, Usp. Fiz. Nauk 63, No. la, 33 (1957). APPENDIX
After eliminating
the terms containing
I
P,, and Pz,, from (1.1) we obtain
(cz2 cos 22 + d,, sin 21)P,, cos kl + d,, sin ki)P,,c or
y&&f
0 3
(1.1)
1
n-1 N,
r2 n=e r
Here N, = rOPn are the height deviations of a geoid corresponding to the Legendre polynomial P, ; only the sectorial harmonic formed part of N,. From this we can obtain the expression for the mean square value of Vzd:
which, after certain
manipulations,
allows us to write down the estimate:
The quantity i + 1 = n denotes the number of the lowest spherical harmonic present in the anomaly potential, whilst m denotes the number of the highest harmonic for which allowance can be made in this expansion. If the anomalous potential is written in form (1.1) then we should put n = 2 in the lower limit of the first sum and allow for only the sectorial harmonic in the estimate of the second order harmonics. The mean square deviations gn = gn(N) are given below (quoted from Kaulac5)): n
gn2, m2
n
un2, m2
n
un2, wP
n
308 460 140 28 41 3-3 19.6 14.6
10
7.8 7.6 2.4 4.4 5.8 4-8 I.1 2.0
18 19 20 21 22 23 24 25
2.8 1.3 O-8 l-5 1.0 0.8 0.9 0.7
26 27 28 29 30 31 32
11 12 13 14 1.5 16 17
ls,2, t?23
o-7 0.3 0.5 o-2 o-1 o-05 o-09
MOTION
OF AN ARTIFICIAL
The mean square difference whilst
SATELLITE
OF A NON-SPHERICAL
of the heights of a geoid and a spheroid
:a,”
EARTH
717
is about 1075 WZ~(~),
= 1062m2
n-2
Therefore the mean difference of the heights of a geoid and a shape represented by 32 2 as a function of n it can be seen that harmonics is about 3*6m. From the data given for a7&
i.e. the remaining The estimate
harmonics (from 5 to 32) have on the whole the same effect as one quarter. simplified at q I O-7 and i < 2. given for aAfi2 can be considerably APPENDIX
Let us write the equations Q
in the osculating
= Qi(qlj . . . 395; U)zj,
IL
elements
(i=l,...,
(ql, . . . , q5) in the form 5;j
= 1, 2, 3)
_& can be taken as g, F or p. More strictly we should write the right-hand sides form of the sum: Qil(ql, . . . , q5: u) s” + Qz2(q1, . . . , q5; u) F + Qia(ql, . . . , q5; This, however, complicates the arguments without really changing the essentials. Since Ii = & + _&, where the indices 1 and n denote the highest order of the 21 present in the cirresplfnding functions, then when we change to the argument u the ential equations can be written in the form: d&A_ - zQi(ql, du JPP
. . . ,q5; 4[$
f
Fi(41, f . * ,q5;
in the u)@. terms differ-
Io,h~I;jl.
The indices below the functions z and @ denote the highest order to the terms present in these functions. System (2.2) was obtained on the basis of these equations. I should like to thank L. P. Pellinen for his useful discussions and V. N. Lavrik who programmed and solved the equations on a computer.