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Some possibilities for determining the influence of earth body tides on the motion of artificial satellites
Adv. Space Res. © COSPAR, 1981.
Vol. 1, pp. 29—35. Printed in Great Britain.
0273—1177/81/0301—0029 $05.00/0
SOME POSSIBILITIES FOR DETERMINING THE INFLUENCE OF EARTH BODY TIDES ON THE MOTION OF ARTIFICIAL SATELLITES J. Koste1eck~’ Research Institute of Geodesy, Topography and Cartography, Geodetical Observatory Pecnj~,Ondiejov, Czechoslovakia ABSTRACT The paper treats some aspects of the widespread concept of a deforming potential and a more gez2eralized approach to the Dirichiet problem in defining the influence of Earth tides on a satellite’s movement. In considering theoretical,rather than practical aspeots,the solution of the Diriohiet problem is discussed for various reference surfaces in place of the Earth INTRODUCTION In the following article, we treat generally the influence of tidal potential on satellite motion. This influence has beeli studied for fifteen years approximately, and many significiant results were achieved in practical applications. With regard to increasing precision of observations and to raising claims of precision evaluation of measured data for the aims of geodynarnics, further possibilities of development of present theories and the generalisation of some considerations will be discussed. DEPORNING POTENTIAL The deforndng potential, including the influence body, is given as ( see [1] ): VT(Po) where ~V
=
GM’/~Ik(P)
is defined as
( /~)~v(
cos~p )
of non—spherical perturbing
+
l~v
(1.1)
30
J. Koste1eck~ M= GM’/~
( p/~ )m
—
{
~Ij k(P)
[1
~ ~u,k ~n,k
-
Sflk
(
a0’/~
c~’2(costp
sin c~ ) 5n,k sin kT 0
—
)
~
Z
+
?/L~
1~n,k
~n,k
(
sin
~ ~i:ik~ (P
+
~ )J }
+
0)~I~I
‘~n,kcos kT~- Sfl,k sin kT’)
~“
~o~p [k0c00
)]
)m c~’2 cosqa
~III21~n (1~’o)~(0n,k cos
kT~(1 2)
with the following meaning of the symbols: GM’ is the product of the constant of gravity and mass of perturbed body, p is the geocentric distance of P0 ,~ is the1~istanoebetween centers of gravity of perturbing and perturbed body, C~’ ( cos ) are the ultraspheric ( Gegenbauer ) polynomials; a0, ~n ~ S ~, are the lenght factor and Stokes dynamical constants of the pertur~3ing~8ay, P are the usual Legendre polynomials, c , s k are the functions o~’~tokes constants and T0 , are the ~I~erioa~’ co—ordinates of the centre of masses relative to the prime meridian and equator reap., in the equatorial system of perturbing body. The k (P0) functions represent positionally dependent Love a e].astioal parame~ers.
q.i
We suppose, that a phase delay is included in Lp , T~ , d~ values. Comparing Eqs. (1.1) and (1.2) with “classical” deforming potential neglecting the non—sphericity of the Earth ), the difference between “classical” formula and formula in Eq. (1.1) is equal to ~V. THE SATELLITE MOTION APPROACH If we want to determine the influence of the deforming potential on the satellite motion, we must solve the well known exterior Dirichlet problem in the following form: We find the harmonic function boundary VT(PO)
J
~
~
=
VT(P)
lim
with boundary condition
(
on the Earth’s (2.la)
VT(P) ;
further, lim P
—~
VT(P)
=
0
(2.lb)
00
has to be valid. If eq. (2.la) is fulfilled, then the point P is regular. We will be concerned, at present, with the existence ana uniqueness of that solution. From Eqs. (1.1) and (1.2), there follows the continuity and integrability in square of the boundary function VT(Po) because the radius vector of the Earth’s boundary is continu~ousand the ri&hthand side of Eq. (1.1) converges uniformly. Then, we can write VT(P
)
C
(
~E
)
and
)
VT(P
~E is a set of continuous functions and L2 °
where
C
( ~E 2 is a set of square L
Effects of Earth Tides on Satellites
31
integrable functions. The Earth’s surface may be characterized ( with the sufficient precision by a continuous, partly smooth function f(~) with the condition D f(~)A(~)
,
(2.2)
where the left—hand side of Eq. (2.2) is the derivative of the in
~
point in the
A
(
direction
)
A is the azimuth
f
function
for all
We can say, further, that between the Earth’s surface and the surface of a sphere, there is a homeomorphisin ( ~E is the so called Jordan surface and that the inversion relative to bE has been made. Next, some characteristic of the inner problem may be transformed to the exterior problem. From the definition of the set E and boundary bE , the existence of closed conus K in every point P 0 at the boundary ~E is warranted and the relation EflK = 0 is valid. It is known from the potential theory C see for instance [4] ) that points with this characteristic are regular points and the set E is regular if and only if all bE points are regular. Last considerations say that solution of Dirichlet problem exists in the sense of (2.1), and that it is unique in consequence of the maximum principle. Remark: M.V. Keldych proved in his work [5] the existence of the generalized Dirichlet problem solution for all bounded open regions of Em A SIMPLIFICATION In the preceding paragraphs, the problems were outlined rather generally. Now, let us treat separate cases from the point of view of possible simplifications. a) k2 (rove number), E2 ( phase delay ) are constant ( for every tidal wave ), = 4ing = 0body foris n>2, a homogenous the Earth ~phere. is represented by a sphere, and the pertui This assumption is most useful in the tasks of the k 2 , computation from the analysis of the satellite motion. The ~ormulation deduced from Eq. (1.1), if i~V is neglected, leads to the solution of the Laplace equation for a sphere by spherical harmonics. The result of it has been used in practical applications ( see [6 — 12] ); one is used in all works concerned with the determination of the ocean tidal components for solid body tides reduction — see e.g. [13 — 16] ). b) k2, ~, are constant, k~= by ro’~ational ellipsoid and
=
tL
0 for n>2 , the Earth is represented perturbing bodies are spherical.
This case is partly solved in [17] . There, the estimate of non—sphericity of the Earth was made in the case of k2 solution from satellite observations and the ellipsoidal harmonics ( see [18] or [19] ) were used for the Dirichlet problem solution. The result has the form VT(P)
=
VT( u,
o, ~ )
=
~
Q~mu/a~ ~m(C~
n—Om=O
[ ~n,m ~ b~1~ E’n,m(
‘
‘
~
0
005
0
) )
cos
ma
sin
ma
+
]
(31)
32
J. Kosteleck~
where u, 0, ~. are the ellipsoidal co—ordinates of the point P , P are Legendre polynomials, a,b are the semi—axes of the rotational ellips~i~ and Q is defined by: -
~2 )XW’2
Q~(z) = 1/2 P~(z) ln
dmQ~(Z)/ d tm
[(1
+ t
)/(
1
—
) ~ )
t
(3.2) —
]
1/k Pk_l (z)P~k~ Prom (3.1), it may be seen that for the boundary value
Qn~m~~’a~= 1
~
is valid and the coefficients case.
~n m
m
are computed like in a “spherical”
‘
p,
ç,
c) k and E are geographically dependent, k( ci), x ) , E C ci), i~ the Earth is represented by a sphere, perturbing bodies have a spherical shape. This problem was solved by Kaula ( see [10] ) and also lately by Balmino, see (21, 22] . Principally, the Dirichlet problem is solved for a sphere, but the perturbing potential is more complicated thanin the a) case. The group transformations known from the quantum mechanics are used for the transformation between Keplerian elements and spherical coordinates. The result — the perturbing potential — is given by Poisson integral. The advantage is that the resulting disturbing potential is a function of the Kepler elements and that it may be successfully used in Lagrange planetary equations for description of the satellite motion. THE QUESTIONS OF THE GENERALIZED SOLUTION We can thake further generalization. We shall try to write the Dirichlet problem for the surface that approaches closely the Earth topography. We will have the positionally dependent elastic parameters phase delays
k~ and the
e~ )
,
E~(?~~,A
)
(41)
which are defined or determined for real points of Earth topography The continuity of k~ , ~ follows from the physical reality and, with respect to Eq. (1.1), the defo~ming potential is also continuous. We shall suppose, that the Earth surface may be approximated by a series of spherical harmonics see e.g. [23] ). For this assumption we will, have the Dirichiet problem for a smooth regular surface. We can designate the spherical functions approximation of the Earth~ssurface of 1 degree as bE 1 and the boundary function as VTIbEe. The cases, when bE1 is monotonously convergent to the boundary inside or outside are discerned in the1potential theory for irregular boundaries ( see § 5 in [4] ); then VT ~+VT. For regular boundaries and this is our case — this difference disappears. We turn our attention to the monotonous convergence of the series. In practice, the Earth~ssurface is approximated on the finite base of the functions by least—squares method. In this case, the ~E1 surfaces are on both sides of the boundary and the assumption of monotoneity inside or outside is not fulfilled. The difficulty can be easily removed. As we said
Effects
33
of Earth Tides on Satellites
above, the approximation was proved by least-squares method and
S~~E1
~E )2 dfl= mAn.
—
(4.2)
hods. Thus, with the probability r~
r~.
—
~ = 99 .7% 3
~
is a standard deviation. If we define the new surface
is valid, where as =
±3 ~
rE
(4.4)
1
and if we take from real values of rE a convergent series, then we have according to Eq. (4.4) ) the inside ~ or outside ) monotonously convergent series of ~E1 . Let us not forget that we have in reality a finite basis of the functions only and the specific d1 is limited by the choice of the points representing the Earth’s surface ( in other words, by smoothing the Eart1i~ssurface ) and by degree 1 of the surface harmonics. We can, at present, solve the Dirichlet problem. The analytical shape of the solution surface
may not be expected
because
the deforming
potential
(1 .1)
and the
bE are defined in a very complicated way. Therefore, let us turn our attention to the approximative solution. For a further solution, the time dependence of deforming potential 1~ should • Because be emphasized. the deforining This potential fact will is ( according the assumption°) be described by to VT(P,t) , vT(P ,t) oon~inuous on CE U bE and the boundary ~E is also continuous, E is regular and bE is a Jordan surface, the function VT(P,t) may be harmonically approximated at P E CE. This fact is the consequence of Keldych—Lavrentiev theorem ( see [4] , chap. V, § 5 ). This means, in practice, that a sphere with radius R may be chosen (precisely, a system of spheres) and the harmonic coefficients C , S be determined by the condition
I VT (ç, 4’, i~, t
)
-
(
~
R/~ )~‘~
4),
7~,t
)
=
mm.
may
(4.5)
where Y is a spherical function and VT is approximated by VT! bE on the best outside representation of the Eart1~usurface. 1 Using the norm of the L 2 space, the problem requires to find the minimum of the convex functional [v~( ~ t ) ~ C R/~ )rl Y(cp7~ ~ 2 d S , (4.6)
5
)J
n
where bE is the Earth~sürface; the described method is the known leastsquares method. Let us take
~, 4’,
,~,
t
)
~ (
R/
)n Y~(4’,
~,
t
)
(4.7)
and compare VT and K for an outer point P Let the standard deviation of the least—squares method being equal to d , K being sufficiently smooth As a consequence of the maximum principle, we may use for estimate of the result - the expressiOn ~P
(
IK(P, t)
-
VT(P,
t)1s
3 d
)
=
99.7%
,
(4.8)
34
J. Kosteleckj~
supposing the Gaussian decomposition of the “residuals”. The estimation of the C , S and R parameters is not unique; therefore, an optimal solution should be searched. Let us introduce one variant, which is convenient fox’ the practical determination of the deforming potential in outer space. The name of this method is “the representation by point masses”. In the case of disturbing potential, we can write VT(P$ t)
=
S1/I(X—P)I dJ~~t
‘
p
where is the Lebesgue measure and integration goes over the Earth surface. The fundamental problem of this method is the determination of the measure • The representation of the gravity field by t~buried~tmasses was used in the Balmino a work ( see [24] ) and in detail in the work of Holota (25] Let us introduce the system of submerged, simply continuous, smooth areas 1~k+1 with the property of E ~ il.~. On every boundary bflk we can ~ determine the density ~ for which the functional ak
t)J
bET
bE
—
~
1/J(x
—
i)j dpk(Y, t)J2 d s(X)
,
(4.10)
is minimalized C Y bQ~ ). We take the system of the spheres in our case and we pass on to the expression n
~ m~ 6Y~ , where 6Y~ is the Dirac measure concentrated in • To the Eq. (4.10), the physi~alconditions must be added. In our case ~here are: ~
m~= 0
,
ui~ Y~ = 0
(4.11)
.
Then we find an appropriate sphere from the point of view of Eqs. (4.10) and (4.11). The value of the deforming potential in the point PECE is then determined from the formula ~
t)
=
~
1/ j
(
—
P
)
m~
(
.
(4.12)
For a maximalistic estimate of the error, the analogy of (4.8) may be used. CONCLUSION of all what was told above, the contribution of z~V from Eq. (1.2) and of the non—sphericity of the Earth will be very small. The Lagrange planetary equations can be hardly analytically integrated duo •to complicated formulas, in case of numerical inte~raticn, the values of the potential roust he computed ( from Eqs. (4.7) and (4.12) ) individually for every integrating point. In other words, the functional must be minimalized again within the intervals equal to one integration step. In spite
Further difficulties are due to the fact, that the values of Love constants k , ( for which we suppose continuc~sapproximation ) are known in isolated jàints only. In this case, the task can not be “correct” and an operator of the regularisation should be introduced C see e.g. [26] ). In the future, it will be necessary to specify the solution of the discussed methods and the difference between generalized and classical solutions must
,
Effects of Earth Tides on Satellites be estimated. Then, it will be possible to adopt this difference to the motion of real geodynamical satellites. It may happen that on the basis of further practical analysis some opinions stated here will be alternated and corrected. At last, the author would like to express his thanks to his colleague Dr.P. Holota for his advices and remarks. REFERENCES 1. 2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
M. Bur~a,Bull. Astron. Inst. Czechosl. 30, 159 (1979). W.M. Kaula, Rev. Geoph. Space Phys. 2, 661 (1964). 1. Kozai, Pubi. Astron. Soc. Japan 17, 395 (1965). N.S. Landkov, Foundations of Modern Potential Theory, Springer, Berlin, 1972. M.V. Keldych, Dokl. Akad. Nauk. USSR 32, 308 (1941). Y. Kozai, Publ. Astron. Soc. Japan 20, 24 (1968). R.R. Newton, J. Geoph. Research 70, 5983 (1965). R.J. Anderle, Tech. Rep. TR-2889, U.S. Navy Weapons Lab., Dahlgreen, Va, 1971. B.C. Douglas, S.M. Klosko, J.G. Marsh and R.G. Williamson, NASA Rep. X—553—72—475, 1972. D.E. Smith, P.J. Dunn and R. Kolenkiewicz, Nature 2A4, 498 (1973). K. Lainbeck, A. Cazenave and G. Balmino, in: The Use of Artificial Satellites for Geodesy and Geodyrtamics, Tech. Univ. of Athens, 1973. J. Kosteleck~,in: Problémy sou~asnégravimet~, Brno, 1979, p. 339 (in Czech). A. Cazenave, 8. Daillet and K. Lambeck, Phil. Trans. Roy. Soc. London Ser. A 284, 595 (1977). C.C. Goad and B.C. Douglas, J. Geoph. Research 82, 898 (1977). T.L. Felsentreger, J.G. Marsch and R.G. Williamson, J. Geoph. Research 84. No B4, 1837 (1978). S. Daillet, Etude des Mardes oceanigues par lesperturbations orbitales des satellites artificiels, GRGS CNES, Dr thesis (1977). 3. Kosteleck~7,Bull, Astron. Inst. Czechosl. 28, 232 (1977). E.W. Hobson, The Theory of ~hericpl and Ellipsoidal Harmonics, Cambridge University Press, 1931. W.A. Heiskanen and H. Moritz, Physical Geodesy, Freeman and Company, San Prancisco, 1967. W.M. Kaula, The Astronomical. Journal, 1108 (1969). G. Balmino, in: Bull. GRGS, 13, 1974. G. Balmino, Studia geoph. et geod. 22, 107, (1978). 0. Balmino, K. Lambeck and W.M. Kaula, J. Geoph. Research 78, (1973). 0. Balmino, in: The Use of Artificial Satellites for Geodesy, Amer. Geophys. Union, Washington, D.C., 1972. 6w Ziemi 18, Warszawa, P. Holota, in: Observacie Sztucznych Satelit 1979, p. 159. A.N~Tikhonov and B.J. Arsenin, Metody reshenija nekorektnych zadach, Moscow, 1974.
35
Vol. 1, pp. 29—35. Printed in Great Britain.
0273—1177/81/0301—0029 $05.00/0
SOME POSSIBILITIES FOR DETERMINING THE INFLUENCE OF EARTH BODY TIDES ON THE MOTION OF ARTIFICIAL SATELLITES J. Koste1eck~’ Research Institute of Geodesy, Topography and Cartography, Geodetical Observatory Pecnj~,Ondiejov, Czechoslovakia ABSTRACT The paper treats some aspects of the widespread concept of a deforming potential and a more gez2eralized approach to the Dirichiet problem in defining the influence of Earth tides on a satellite’s movement. In considering theoretical,rather than practical aspeots,the solution of the Diriohiet problem is discussed for various reference surfaces in place of the Earth INTRODUCTION In the following article, we treat generally the influence of tidal potential on satellite motion. This influence has beeli studied for fifteen years approximately, and many significiant results were achieved in practical applications. With regard to increasing precision of observations and to raising claims of precision evaluation of measured data for the aims of geodynarnics, further possibilities of development of present theories and the generalisation of some considerations will be discussed. DEPORNING POTENTIAL The deforndng potential, including the influence body, is given as ( see [1] ): VT(Po) where ~V
=
GM’/~Ik(P)
is defined as
( /~)~v(
cos~p )
of non—spherical perturbing
+
l~v
(1.1)
30
J. Koste1eck~ M= GM’/~
( p/~ )m
—
{
~Ij k(P)
[1
~ ~u,k ~n,k
-
Sflk
(
a0’/~
c~’2(costp
sin c~ ) 5n,k sin kT 0
—
)
~
Z
+
?/L~
1~n,k
~n,k
(
sin
~ ~i:ik~ (P
+
~ )J }
+
0)~I~I
‘~n,kcos kT~- Sfl,k sin kT’)
~“
~o~p [k0c00
)]
)m c~’2 cosqa
~III21~n (1~’o)~(0n,k cos
kT~(1 2)
with the following meaning of the symbols: GM’ is the product of the constant of gravity and mass of perturbed body, p is the geocentric distance of P0 ,~ is the1~istanoebetween centers of gravity of perturbing and perturbed body, C~’ ( cos ) are the ultraspheric ( Gegenbauer ) polynomials; a0, ~n ~ S ~, are the lenght factor and Stokes dynamical constants of the pertur~3ing~8ay, P are the usual Legendre polynomials, c , s k are the functions o~’~tokes constants and T0 , are the ~I~erioa~’ co—ordinates of the centre of masses relative to the prime meridian and equator reap., in the equatorial system of perturbing body. The k (P0) functions represent positionally dependent Love a e].astioal parame~ers.
q.i
We suppose, that a phase delay is included in Lp , T~ , d~ values. Comparing Eqs. (1.1) and (1.2) with “classical” deforming potential neglecting the non—sphericity of the Earth ), the difference between “classical” formula and formula in Eq. (1.1) is equal to ~V. THE SATELLITE MOTION APPROACH If we want to determine the influence of the deforming potential on the satellite motion, we must solve the well known exterior Dirichlet problem in the following form: We find the harmonic function boundary VT(PO)
J
~
~
=
VT(P)
lim
with boundary condition
(
on the Earth’s (2.la)
VT(P) ;
further, lim P
—~
VT(P)
=
0
(2.lb)
00
has to be valid. If eq. (2.la) is fulfilled, then the point P is regular. We will be concerned, at present, with the existence ana uniqueness of that solution. From Eqs. (1.1) and (1.2), there follows the continuity and integrability in square of the boundary function VT(Po) because the radius vector of the Earth’s boundary is continu~ousand the ri&hthand side of Eq. (1.1) converges uniformly. Then, we can write VT(P
)
C
(
~E
)
and
)
VT(P
~E is a set of continuous functions and L2 °
where
C
( ~E 2 is a set of square L
Effects of Earth Tides on Satellites
31
integrable functions. The Earth’s surface may be characterized ( with the sufficient precision by a continuous, partly smooth function f(~) with the condition D f(~)A(~)
,
(2.2)
where the left—hand side of Eq. (2.2) is the derivative of the in
~
point in the
A
(
direction
)
A is the azimuth
f
function
for all
We can say, further, that between the Earth’s surface and the surface of a sphere, there is a homeomorphisin ( ~E is the so called Jordan surface and that the inversion relative to bE has been made. Next, some characteristic of the inner problem may be transformed to the exterior problem. From the definition of the set E and boundary bE , the existence of closed conus K in every point P 0 at the boundary ~E is warranted and the relation EflK = 0 is valid. It is known from the potential theory C see for instance [4] ) that points with this characteristic are regular points and the set E is regular if and only if all bE points are regular. Last considerations say that solution of Dirichlet problem exists in the sense of (2.1), and that it is unique in consequence of the maximum principle. Remark: M.V. Keldych proved in his work [5] the existence of the generalized Dirichlet problem solution for all bounded open regions of Em A SIMPLIFICATION In the preceding paragraphs, the problems were outlined rather generally. Now, let us treat separate cases from the point of view of possible simplifications. a) k2 (rove number), E2 ( phase delay ) are constant ( for every tidal wave ), = 4ing = 0body foris n>2, a homogenous the Earth ~phere. is represented by a sphere, and the pertui This assumption is most useful in the tasks of the k 2 , computation from the analysis of the satellite motion. The ~ormulation deduced from Eq. (1.1), if i~V is neglected, leads to the solution of the Laplace equation for a sphere by spherical harmonics. The result of it has been used in practical applications ( see [6 — 12] ); one is used in all works concerned with the determination of the ocean tidal components for solid body tides reduction — see e.g. [13 — 16] ). b) k2, ~, are constant, k~= by ro’~ational ellipsoid and
=
tL
0 for n>2 , the Earth is represented perturbing bodies are spherical.
This case is partly solved in [17] . There, the estimate of non—sphericity of the Earth was made in the case of k2 solution from satellite observations and the ellipsoidal harmonics ( see [18] or [19] ) were used for the Dirichlet problem solution. The result has the form VT(P)
=
VT( u,
o, ~ )
=
~
Q~mu/a~ ~m(C~
n—Om=O
[ ~n,m ~ b~1~ E’n,m(
‘
‘
~
0
005
0
) )
cos
ma
sin
ma
+
]
(31)
32
J. Kosteleck~
where u, 0, ~. are the ellipsoidal co—ordinates of the point P , P are Legendre polynomials, a,b are the semi—axes of the rotational ellips~i~ and Q is defined by: -
~2 )XW’2
Q~(z) = 1/2 P~(z) ln
dmQ~(Z)/ d tm
[(1
+ t
)/(
1
—
) ~ )
t
(3.2) —
]
1/k Pk_l (z)P~k~ Prom (3.1), it may be seen that for the boundary value
Qn~m~~’a~= 1
~
is valid and the coefficients case.
~n m
m
are computed like in a “spherical”
‘
p,
ç,
c) k and E are geographically dependent, k( ci), x ) , E C ci), i~ the Earth is represented by a sphere, perturbing bodies have a spherical shape. This problem was solved by Kaula ( see [10] ) and also lately by Balmino, see (21, 22] . Principally, the Dirichlet problem is solved for a sphere, but the perturbing potential is more complicated thanin the a) case. The group transformations known from the quantum mechanics are used for the transformation between Keplerian elements and spherical coordinates. The result — the perturbing potential — is given by Poisson integral. The advantage is that the resulting disturbing potential is a function of the Kepler elements and that it may be successfully used in Lagrange planetary equations for description of the satellite motion. THE QUESTIONS OF THE GENERALIZED SOLUTION We can thake further generalization. We shall try to write the Dirichlet problem for the surface that approaches closely the Earth topography. We will have the positionally dependent elastic parameters phase delays
k~ and the
e~ )
,
E~(?~~,A
)
(41)
which are defined or determined for real points of Earth topography The continuity of k~ , ~ follows from the physical reality and, with respect to Eq. (1.1), the defo~ming potential is also continuous. We shall suppose, that the Earth surface may be approximated by a series of spherical harmonics see e.g. [23] ). For this assumption we will, have the Dirichiet problem for a smooth regular surface. We can designate the spherical functions approximation of the Earth~ssurface of 1 degree as bE 1 and the boundary function as VTIbEe. The cases, when bE1 is monotonously convergent to the boundary inside or outside are discerned in the1potential theory for irregular boundaries ( see § 5 in [4] ); then VT ~+VT. For regular boundaries and this is our case — this difference disappears. We turn our attention to the monotonous convergence of the series. In practice, the Earth~ssurface is approximated on the finite base of the functions by least—squares method. In this case, the ~E1 surfaces are on both sides of the boundary and the assumption of monotoneity inside or outside is not fulfilled. The difficulty can be easily removed. As we said
Effects
33
of Earth Tides on Satellites
above, the approximation was proved by least-squares method and
S~~E1
~E )2 dfl= mAn.
—
(4.2)
hods. Thus, with the probability r~
r~.
—
~ = 99 .7% 3
~
is a standard deviation. If we define the new surface
is valid, where as =
±3 ~
rE
(4.4)
1
and if we take from real values of rE a convergent series, then we have according to Eq. (4.4) ) the inside ~ or outside ) monotonously convergent series of ~E1 . Let us not forget that we have in reality a finite basis of the functions only and the specific d1 is limited by the choice of the points representing the Earth’s surface ( in other words, by smoothing the Eart1i~ssurface ) and by degree 1 of the surface harmonics. We can, at present, solve the Dirichlet problem. The analytical shape of the solution surface
may not be expected
because
the deforming
potential
(1 .1)
and the
bE are defined in a very complicated way. Therefore, let us turn our attention to the approximative solution. For a further solution, the time dependence of deforming potential 1~ should • Because be emphasized. the deforining This potential fact will is ( according the assumption°) be described by to VT(P,t) , vT(P ,t) oon~inuous on CE U bE and the boundary ~E is also continuous, E is regular and bE is a Jordan surface, the function VT(P,t) may be harmonically approximated at P E CE. This fact is the consequence of Keldych—Lavrentiev theorem ( see [4] , chap. V, § 5 ). This means, in practice, that a sphere with radius R may be chosen (precisely, a system of spheres) and the harmonic coefficients C , S be determined by the condition
I VT (ç, 4’, i~, t
)
-
(
~
R/~ )~‘~
4),
7~,t
)
=
mm.
may
(4.5)
where Y is a spherical function and VT is approximated by VT! bE on the best outside representation of the Eart1~usurface. 1 Using the norm of the L 2 space, the problem requires to find the minimum of the convex functional [v~( ~ t ) ~ C R/~ )rl Y(cp7~ ~ 2 d S , (4.6)
5
)J
n
where bE is the Earth~sürface; the described method is the known leastsquares method. Let us take
~, 4’,
,~,
t
)
~ (
R/
)n Y~(4’,
~,
t
)
(4.7)
and compare VT and K for an outer point P Let the standard deviation of the least—squares method being equal to d , K being sufficiently smooth As a consequence of the maximum principle, we may use for estimate of the result - the expressiOn ~P
(
IK(P, t)
-
VT(P,
t)1s
3 d
)
=
99.7%
,
(4.8)
34
J. Kosteleckj~
supposing the Gaussian decomposition of the “residuals”. The estimation of the C , S and R parameters is not unique; therefore, an optimal solution should be searched. Let us introduce one variant, which is convenient fox’ the practical determination of the deforming potential in outer space. The name of this method is “the representation by point masses”. In the case of disturbing potential, we can write VT(P$ t)
=
S1/I(X—P)I dJ~~t
‘
p
where is the Lebesgue measure and integration goes over the Earth surface. The fundamental problem of this method is the determination of the measure • The representation of the gravity field by t~buried~tmasses was used in the Balmino a work ( see [24] ) and in detail in the work of Holota (25] Let us introduce the system of submerged, simply continuous, smooth areas 1~k+1 with the property of E ~ il.~. On every boundary bflk we can ~ determine the density ~ for which the functional ak
t)J
bET
bE
—
~
1/J(x
—
i)j dpk(Y, t)J2 d s(X)
,
(4.10)
is minimalized C Y bQ~ ). We take the system of the spheres in our case and we pass on to the expression n
~ m~ 6Y~ , where 6Y~ is the Dirac measure concentrated in • To the Eq. (4.10), the physi~alconditions must be added. In our case ~here are: ~
m~= 0
,
ui~ Y~ = 0
(4.11)
.
Then we find an appropriate sphere from the point of view of Eqs. (4.10) and (4.11). The value of the deforming potential in the point PECE is then determined from the formula ~
t)
=
~
1/ j
(
—
P
)
m~
(
.
(4.12)
For a maximalistic estimate of the error, the analogy of (4.8) may be used. CONCLUSION of all what was told above, the contribution of z~V from Eq. (1.2) and of the non—sphericity of the Earth will be very small. The Lagrange planetary equations can be hardly analytically integrated duo •to complicated formulas, in case of numerical inte~raticn, the values of the potential roust he computed ( from Eqs. (4.7) and (4.12) ) individually for every integrating point. In other words, the functional must be minimalized again within the intervals equal to one integration step. In spite
Further difficulties are due to the fact, that the values of Love constants k , ( for which we suppose continuc~sapproximation ) are known in isolated jàints only. In this case, the task can not be “correct” and an operator of the regularisation should be introduced C see e.g. [26] ). In the future, it will be necessary to specify the solution of the discussed methods and the difference between generalized and classical solutions must
,
Effects of Earth Tides on Satellites be estimated. Then, it will be possible to adopt this difference to the motion of real geodynamical satellites. It may happen that on the basis of further practical analysis some opinions stated here will be alternated and corrected. At last, the author would like to express his thanks to his colleague Dr.P. Holota for his advices and remarks. REFERENCES 1. 2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
M. Bur~a,Bull. Astron. Inst. Czechosl. 30, 159 (1979). W.M. Kaula, Rev. Geoph. Space Phys. 2, 661 (1964). 1. Kozai, Pubi. Astron. Soc. Japan 17, 395 (1965). N.S. Landkov, Foundations of Modern Potential Theory, Springer, Berlin, 1972. M.V. Keldych, Dokl. Akad. Nauk. USSR 32, 308 (1941). Y. Kozai, Publ. Astron. Soc. Japan 20, 24 (1968). R.R. Newton, J. Geoph. Research 70, 5983 (1965). R.J. Anderle, Tech. Rep. TR-2889, U.S. Navy Weapons Lab., Dahlgreen, Va, 1971. B.C. Douglas, S.M. Klosko, J.G. Marsh and R.G. Williamson, NASA Rep. X—553—72—475, 1972. D.E. Smith, P.J. Dunn and R. Kolenkiewicz, Nature 2A4, 498 (1973). K. Lainbeck, A. Cazenave and G. Balmino, in: The Use of Artificial Satellites for Geodesy and Geodyrtamics, Tech. Univ. of Athens, 1973. J. Kosteleck~,in: Problémy sou~asnégravimet~, Brno, 1979, p. 339 (in Czech). A. Cazenave, 8. Daillet and K. Lambeck, Phil. Trans. Roy. Soc. London Ser. A 284, 595 (1977). C.C. Goad and B.C. Douglas, J. Geoph. Research 82, 898 (1977). T.L. Felsentreger, J.G. Marsch and R.G. Williamson, J. Geoph. Research 84. No B4, 1837 (1978). S. Daillet, Etude des Mardes oceanigues par lesperturbations orbitales des satellites artificiels, GRGS CNES, Dr thesis (1977). 3. Kosteleck~7,Bull, Astron. Inst. Czechosl. 28, 232 (1977). E.W. Hobson, The Theory of ~hericpl and Ellipsoidal Harmonics, Cambridge University Press, 1931. W.A. Heiskanen and H. Moritz, Physical Geodesy, Freeman and Company, San Prancisco, 1967. W.M. Kaula, The Astronomical. Journal, 1108 (1969). G. Balmino, in: Bull. GRGS, 13, 1974. G. Balmino, Studia geoph. et geod. 22, 107, (1978). 0. Balmino, K. Lambeck and W.M. Kaula, J. Geoph. Research 78, (1973). 0. Balmino, in: The Use of Artificial Satellites for Geodesy, Amer. Geophys. Union, Washington, D.C., 1972. 6w Ziemi 18, Warszawa, P. Holota, in: Observacie Sztucznych Satelit 1979, p. 159. A.N~Tikhonov and B.J. Arsenin, Metody reshenija nekorektnych zadach, Moscow, 1974.
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