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SIMPLIFICATION OF GEOPOTENTIAL PERTURBING FORCE
ACTING ON A SATELLITE
M. Eshagh1, M. Abdollahzadeh 2
1Royal Institute of Technology, Division of Geodesy, Stockholm, Sweden
2 K.N.Toosi University of Technology, Dept. of Geodesy and Geomatics, Tehran, Iran
e-mails: eshagh@kth.se, m_abdollahzadeh@sina.kntu.ac.ir
M. Najafi-Alamdari
K.N.Toosi University of Technology, Dept. of Geodesy and Geomatics, Tehran, Iran
e-mail: mnajalm@yahoo.com
ABSTRACT. One of the aspects of geopotential models is orbit integration of
satellites. The geopotential acceleration has the largest influence on a satellite with
respect to the other perturbing forces. The equation of motion of satellites is a second-
order vector differential equation. These equations are further simplified and
developed in this study based on the geopotential force. This new expression is much
simpler than the traditional one as it does not derivatives of the associated Legendre
functions and the transformations are included in the equations. The maximum degree
and order of the geopotential harmonic expansion must be selected prior to the orbit
integration purposes. The values of the maximum degree and order of these
coefficients depend directly on the satellite’s altitude. In this article, behaviour of
orbital elements of recent geopotential satellites, such as CHAMP, GRACE and
GOCE is considered with respect to the different degree and order of geopotential
coefficients. In this case, the maximum degree 116, 109 and 175 were derived for the
Earth gravitational field in short arc orbit integration of the CHAMP, GRACE and
GOCE, respectively considering millimeter level in perturbations.
Keywords: Geopotential, orbit integration, average power acceleration, orbital
elements
1. INTRODUCTION
The spherical harmonic expansion is a mathematical tool for approximating the
Earth’s gravitational field. The harmonic coefficients of this expansion can be
determined in various ways, say, by using terrestrial and/or satellite data. Only the
long wavelength structure of the Earth’s gravitational field can be determined by the
satellites, because of attenuation of the gravitational signal due to the satellite altitude.
Different satellite missions have been dedicated for such aims. The last three satellite
missions are CHAMP (Challenging Minisatellite Payload) [Reigber et al., 1999 and
2004], GRACE (Gravity Recovery and Climate Experiments) [Tapley et al., 2005]
and upcoming GOCE (Gravity Field and Steady-state Ocean Circulation Explorer)
missions [ESA, 1999, Albertella et al., 2002, Balmino et al., 1998 and 2001]. The
CHAMP mission was designed based on satellite-to-satellite tracking and analysis of
ARTIFICIAL SATELLITES, Vol. 43, No. 2 - 2008
DOI: 10.2478/v10018-009-0006-7
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the precise orbit of the satellite for recovery of the gravitational field. The GRACE
mission is a common project between US and Germany and in this technique low-low
mode of the satellite to satellite tracking (SST) are used as space gradiometry. The
GOCE mission was dedicated to be launched and in this satellite mission the space
gradiometry technique is used as well. The GOCE mission is considered to be the first
gradiometric mission although there is a concept for the GRACE satellites as a large
one-dimensional gradiometer.
Satellite orbit analysis is a well-known technique for gravitational field recovery
cf. e.g., Kaula (1966), Visser (1992), Sneeuw (1992). It is important to consider that
although the satellite gradiometry techniques [Rummel et al., 1993, Keller and Sharifi,
2005, Sharifi, 2006] are used in two last missions, the satellite’s orbit should be
determined as precise as possible so that the extracted perturbations can be analyzed
without worrying about biases in the solution. The precise orbit determination (POD)
[Su, 2000 and Wolf, 2000] can be done in different ways, such as Kinematic POD,
dynamic POD, reduced dynamic POD, etc; see e.g. Rim and Schutz (2001). In this
paper we concentrate on orbit integration [Eshagh, 2003a and 2003b] which is the
prediction part of the orbit in the dynamic POD. Numerical integration of the orbit
has some benefits with respect to analytical one [Kaula, 1966], as it is not restricted to
the mathematical models of the perturbing forces. For details of the algorithms; see
e.g. Su (2000), Wolf (2000), Eshagh (2003a and 2005) and Sharifi (2006), Eshagh and
Najafi-Alamdari (2005a, 2005b, 2006 and 2007).
The equation of motion of satellite is a second-order vector differential equation
and the satellite acceleration vector is integrated twice to obtain the vector of
velocities and positions. Integrating the equation of motion once yields a velocity
vector (and three unknowns), integrating it twice results in a position vector (six
unknown integration constant). The traditional expression for the vector differential
equation of satellite’s motion is complicated as it includes partial derivatives of the
spherical coordinates with respect to Cartesian coordinates, derivative of associated
Legendre functions (ALF) and singular terms when satellite approaches the poles
(near polar satellite). The satellite acceleration is a summation of different
gravitational and non-gravitational accelerations, but the largest perturbation is due to
the geopotential field. In this study we concentrate on the geopotential force and
simplify the vector differential equation of motion of satellite so that it excludes the
mentioned complications. This paper continues the previous studies of the authors and
also considers the behavior of the orbital elements in each degree of the geopotential
field in the recent satellite missions CHAMP, GRACE and GOCE.
In the next section, we present the traditional expression of satellite’s equation of
motion. In Section 3 we derive a new expression for equation of motion and Section 4
presents an alternative non-singular expression. Section 5 deals with the system of
equation to be integrated. In Section 6 the average acceleration power of the
gravitational vector is considered to determine maximum degree of geopotential
expansion needed for orbit integration. In Section 7 a numerical study on the orbital
elements are presented and the paper is ended with conclusions in Section 8.
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2. TRADITIONAL EXPRESSION OF EQUATION OF MOTION OF
SATELLITE
The gravitational field of the Earth can be approximated by a truncated series of
spherical harmonics [Heiskanen and Moritz, 1967]:
max 1
0
cos
n
Nn
nm m nm
nmn
GM R
UP uQ m P
Rr
OT
§·
¨¸
©¹
¦¦ ,
rRt (1)
where, GM is the geocentric constant, R is the radius of the mean sphere of the Earth,
nm
u is the co-sine and sine geopotential coefficient when m d 0 and m > 0,
respectively. nm
P is the fully-normalized ALF of degree n and order m, r,
O
,
T
are
the spherical Earth-fixed coordinate of the satellite at point P and
cos 0
sin 0
m
mm
Qm
mm
O
OO
d
°
®!
°
¯
(2)
Transformations of these triple parameters r,
O
,
T
to the Cartesian quasi-inertial
coordinates of x, y, and z are:
222
rxyz , (3a)
2 sinarc z r
TS
, (3b)
tanarc y x
O
4. ( 3c)
where 4 is Greenwich apparent sidereal time. The partial derivatives of these
curvilinear and Cartesian coordinates is:
ii
rr rrww , (4a)
12
22 2
ii
i
xy zrr zr
r
T
wªº
ww
¬¼
w, (4b)
>@
1
22
ii
i
xy x y r yx x r
r
O
w wwuww
w, ( 4c)
where, i
r, i=1, 2, 3 stand for x, y and z, respectively. Now the derivatives of potential
U(P) with respect to spherical coordinates must be derived. These derivatives are
well-known for details see, e.g. Parrot (1989), Santos (1994) and Hwang and Lin,
(1998). According to Hwang and Lin (1998) the satellite acceleration can also be
expressed in the quasi-inertial frame using the following transformation (neglecting
precession, nutation and polar motion but for actual orbit analysis one cannot neglect
these effects):
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22 22
22 22
22
0sin
xxz yUP
rrx y x y r
xP
UP
yyz x
yP rr
rx y x y
zP UP
xy
zr
rr
T
TO
ªº
ªº
w
«»
«»
«»
w
ªº «»
«»
«» «»
w
«»
«» «»
«»
w
«» «»
«»
¬¼ «»
w
«»
«»
«»
w
¬¼
¬¼
(5)
The traditional expressions for the elements of gravitational vector in geocentric
Earth-fixed frame are:
max 2
2
0
1cos
n
Nn
nm m nm
nmn
UP GM R
n uQmP
rR r
OT
w§·
¨¸
w©¹
¦¦ (6a)
max 2
2
0
cos
n
Nnnm
nm m
nmn
P
UP GM R uQ m
rR r
T
O
TT
w
w§·
¨¸
ww
©¹
¦¦ (6b)
max 2
2
0
cos
sin sin
n
Nnnm
nm m
nmn
P
UP GM R mu Q m
rRr
T
O
TO T
w§·
¨¸
w©¹
¦¦ . (6c)
Non-singular expressions for the elements of the gravitational vector are given by
Eshagh (2008a). Equation (5) is the differential equation of satellite’s motion. This
equation is involved with derivatives of the curvilinear coordinates. Therefore the
second-order derivatives of the gravitational potential will have a complicated form
for each element of the acceleration vector. In the following section an attempt is
made to simplify this differential equation further.
3. NEW EXPRESSIONS FOR EQUATION OF MOTION OF
SATELLITE
As it was shown in previous section, to carry out the integration steps of satellite orbit
in the quasi-inertial frame while the geopotential model (e.g., EGM96, Lemoine et al.,
1996) as the source of geopotential is given in an Earth fixed (non-inertial) curvilinear
coordinates system, one has to transfer the satellite position from the inertial frame to
the Earth fixed frame, for the acceleration computation out of the Earth gravitational
model. This requires the Cartesian and Curvilinear coordinates of the satellite to be
computed in both the quasi-inertial and the Earth fixed frames.
Similar to the previous section we can write the following transformation for the
satellite’s accelerations in a quasi-inertial frame, but this time we change the
transformation matrix from Cartesian to curvilinear form:
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sin cos cos cos sin
sin sin sin cos cos
cos sin 0
sin
UP
r
xP
UP
yP r
zP UP
r
TO O T O
TO O T O T
TT
TO
ªº
w
«»
w
ªº «»
ªº
«» «»
w
«»
«» «»
«»
w
«» «»
«»
¬¼
¬¼ «»
w
«»
w
¬¼
(7)
where
O O4 as defined previously. By inserting the components of the
gravitational vector Eqs. (6a)-(6c) into the Eq. (7) and after simplifications we obtain :
max 2
2
0
cos sin
n
Nn
nm m nm m nm
nmn
GM R
xP u Q m F Q m G
Rr
OOTO OT
§· ªº
¨¸ ¬¼
©¹
¦¦
(8a)
max 2
2
0
sin cos
n
Nn
nm m nm m nm
nmn
GM R
yP u Q m F Q m G
Rr
OOTO OT
§· ªº
¨¸ ¬¼
©¹
¦¦
(8b)
max 2
2
0
1 cos sin
n
Nnnm
nm m nm
nmn
P
GM R
zP uQ m n P
Rr
OTT
T
ªº
w
§·
«»
¨¸ w
©¹ «»
¬¼
¦¦
(8c)
where
1sin cos nm
nm nm
P
FnP
TTT
T
w
w (9a)
1, 1 1, 1
sin
nm
nm nm nm
nm nm
P
GmfPgP
TT
0mz (9b)
0
1,0
221
1
2221
m
nm
m
n
m
fnmnm
mn
G
G
(9c)
0
1,0
221
1
2221
m
nm
m
n
m
gnmnm
mn
G
G
(9d)
Equations (9b)-(9d) were given by Eshagh (2008a, Proposition 3), nm
fand nm
gare
derived based on normalizing Lemma 5 using Eq. (11), which will be presented later.
It should be mentioned that the order m is a part of nm
f, nm
g,
nm
F
T
and
nm
G
T
.
Now an attempt is made to simplify these coefficients further. For presenting our
mathematical derivations we define the following lemmas.
Lemma 1 [Ilk 1983, Eq. (Z. 1.43)]:
1, 1,
11
sin 21 21
nm
nm nm
Pnnm nnm
PP
nn
TT
w
w
Lemma 2 [Ilk 1983, Eq. (Z.1.37)]:
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1, 1,
1
cos 21 21
nm n m n m
nm nm
PP P
nn
T
.
Note: these relations only hold for non-normalized ALF.
Proposition 1:
1,
21
1cos sin 1 1 23
nm
nm nm
Pn
nP nmnm P
n
TT
T
w
w
Proof. By multiplying Lemma 2 by (n+1) and adding to Lemma 1 we obtain :
1,
1 cos sin 1
nm
nm n m
P
nP nmP
TT
T
w
w, (10)
by considering normalization factors
0
()!
221 !
nm nm
m
nm
PP
nnm
G
. (11)
By normalizing Eq. (10) based on Eq. (11) and after further simplifications
Proposition 1 is proved.
Corollary 1 The satellite acceleration in z-direction in the quasi-inertial frame is:
max 2
1,
2
0
21
11
23
n
Nn
nm m nm
nmn
GM R n
zP uQ m n m n m P
Rr n
O
§·
¨¸
©¹
¦¦
Here, the result follows by considering and re-substituting the results of Proposition 1
into Eq. (8c).
For simplifying the co-latitude dependent parts of Eqs. (8a) and (8b), we start our
mathematical derivations by the following lemmas.
Lemma 3 [Ilk 1983, Eq. (Z.1.44)]:
,1 ,1
11
2
nm
nm nm
Pnmnm P P
T
wªº
¬¼
w
Lemma 4 [Ilk 1983, Eq. (Z.1.40)]:
1, 1 1, 1
11
sin 21 21
nm n m n m
PP P
nn
T
Lemma 5 [Ilk 1983, Eq. (Z.1.42)]:
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1, 1 1, 1
11
sin 2
nm
nm nm
Pnmnm P P
m
T
ªº
¬¼
.
Note: all relations hold for non-normalized ALF.
Proposition 2:
1, 1 1, 1 1, 1 1, 1
cos 1 sin
nm
nm nm nm nm
nm nm nm nm nm
PnPaPbPcPdP
TT
T
w
w
where
0
1,0
21
222121
m
nm
m
nmnm
nm
a
nn
G
G
0
1,0
21
222121
m
nm
m
nmnm
nm
b
nn
G
G
0
1,0
212
1
222123
m
nm
m
nm nm
nm
c
nn
G
G
0
1,0
212
31
222123
m
nm
m
nm nm
nm
d
nn
G
G
Proof. We have to derive a relation for cos nm
P
TT
w
w. In this case let us differentiate
Lemma 2 with respect to T (co-latitude)
1, 1,
1
sin cos 21 21
nm nm nm
nm
PPP
nm nm
Pnn
TT
TT T
www
ww w
(12)
considering Lemma 3 for the derivatives of the unnormalized ALF and inserting them
in Eq. (12) we finally obtain
1, 1 1, 1
sin cos 1
22 1
nm
nm nm nm
Pnm
PnmnmPP
n
TT
T
wªº
¬¼
w
1, 1 1, 1
112
22 1 nm nm
nm nm nm P P
n
ªº
¬¼
. (13)
by taking sin nm
P
T
into the right hand side of Eq. (13) and considering Lemma 4 we
obtain:
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1, 1 1, 1
12
cos 22 1 22 1
nm
nm nm
Pnmnm nm nm
PP
nn
TT
w
w
1, 1 1, 1
112 1
22 1 22 1
nm nm
nm nm nm nm
PP
nn
. (14)
Multiplying Lemma 4 by –(n+1) and adding to Eq. (14) we have
1, 1 1, 1
1
cos 1 sin 22 1 22 1
nm
nm nm nm
Pnmnm nm nm
nP P P
nn
TT
T
w
w
1, 1 1, 1
112 31
22 1 22 1
nm nm
nm nm nm nm
PP
nn
. (15)
The proposition is proved by normalizing Eq. (15) and considering Eq. (11) as the
normalizing factor.
Corollary 2 The satellite acceleration in x- and y-directions of the quasi-inertial
frame are:
max 2
1, 1 1, 1
2
0
cos
n
Nn
nm m nm nm
nm nm
nmn
GM R
xP u Q m a P b P
Rr
OO
§· ª
¨¸ ¬
©¹
¦¦
1, 1 1, 1 1, 1 1, 1
sin
nm nm m nm nm
nm nm nm nm
cP dP Q m f P g P
OO
º
¼
max 2
1, 1 1, 1
2
0
sin
n
Nn
nm m nm nm
nm nm
nmn
GM R
yP u Q m a P b P
Rr
OO
§· ª
¨¸ ¬
©¹
¦¦
1, 1 1, 1 1, 1 1, 1
cos
nm nm m nm n m
nm nm nm nm
cP dP Q m f P gP
OO
º
¼
The corollary follows by inserting Proposition 2 into Eqs. (8a) and (8b) and
considering Eqs. (9a) and (9b). It should be emphasized that nm
fand nm
gare derived
based on normalizing Lemma 5 using Eq. (11), see e.g. Eshagh (2008a, Proposition
3).
Note: For the zonal terms of satellite accelerations presented in Corollary 2,
0
nm
G
T
, i.e. the second term in both equations of Corollary 2 will vanish.
We have presented the elements of the satellite acceleration vector in Corollaries 1
and 2. It is obvious that the presented formulas are very simple to use as there is
neither first- and/or second-order derivatives of the ALF nor singular terms. Also the
formulation inherently includes necessary transformation. The satellite’s acceleration
in x- and y-directions, which were presented in Corollary 2 can also be further
simplified. In the following section we present alternative formulas for these
accelerations.
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4. ALTERNATIVE NON-SINGULAR EXPRESSIONS FOR EQUATION OF
MOTION OF SATELLITE
In the previous section we obtained non-singular expressions for the satellite’s
accelerations in a quasi-inertial frame. We can also further simplify the relations
presented in Corollary 2 considering some trigonometric simplifications. The result of
these simplifications is:
max 2
2
0
11
2
n
Nn
nm m m
nmn
GM R m
xP u Q m Q m
Rr m
OO
ªº
§·
4 4
«»
¨¸
©¹ «»
¬¼
¦¦
nm nm
m
FG
m
TT
§·
¨¸
¨¸
©¹
, for 0mz, (16a)
max 2
00
2
0
cos
n
N
nn
n
GM R
xP u F
Rr
OT
§·
¨¸
©¹
¦
, for m=0, (16b)
max 2
2
0
11
2
n
Nn
nm m m
nmn
GM R m
yP u Q m Q m
Rr m
OO
ªº
§·
4 4
«»
¨¸
©¹ «»
¬¼
¦¦
nm nm
m
FG
m
TT
§·
¨¸
¨¸
©¹
, for 0mz (16c)
max 2
00
2
0
sin
n
N
nn
n
GM R
yP u F
Rr
OT
§·
¨¸
©¹
¦
, for m=0. (16d)
It should be mentioned again that
0
nm
G
T
for the zonal terms and Eqs. (16a)-(16d)
can further be simplified. We have added mmin our formulation in Eq. (16b) to
keep the right sign for
1
m
Qm
O
4
, when m is negative a minus sign (-)
appears by multiplication of -1 to this coefficient, and when it becomes positive, a
plus sign (+) appears instead.
Equations (16a) and (16d) contain two terms
nm nm
FG
TT
and
nm nm
FG
TT
, when m < 0 and m > 0, respectively, and they are simplified in
the following propositions.
Proposition 3:
1, 1 1, 1 1, 1 1, 1
cos 1 sin sin
nm nm
nm nm nm nm
nm nm nm nm nm
PP
nPmaPbPcPdP
TT
TT
wcccc
w
where
0
1,0
21
121
222121
m
nm
m
nmnm
m
anm n
mnn
G
G
ªº
c
«»
«»
¬¼
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0
1,0
21
121
222121
m
nm
m
nmnm
m
bnm n
mnn
G
G
ªº
c
«»
«»
¬¼
0
1,0
212
1
222123
m
nm
m
nm nm
nm
c
nn
G
G
c
0
1,0
212
31
222123
m
nm
m
nm nm
nm
d
nn
G
G
c
.
Proof. By considering Eq. (15) and Lemma 5 we obtain after simplification:
1, 1
1
cos 1 sin sin 2 2 1
nm nm
nm nm
PP
nmnm nm m
nPm P
nm
TT
TT
w
ªº
«»
w
«»
¬¼
1, 1 1, 1
112
1
22 1 22 1
nm nm
nm nm nm
nm mPP
nm n
ªº
«»
«»
¬¼
1, 1
31
22 1 nm
nm P
n
(17)
and the proposition is proved by normalizing Eq. (17) using Eq. (11).
Proposition 4:
1, 1 1, 1 1, 1 1, 1
cos 1 sin sin
nm nm
nm nm nm nm
nm nm nm nm nm
PP
nPmaPbPcPdP
TT
TT
wcc cc c c
w
where
0
1,0
21
121
222121
m
nm
m
nmnm
m
anm n
mnn
G
G
ªº
cc
«»
«»
¬¼
0
1,0
21
121
222121
m
nm
m
nmnm
m
bnm n
mnn
G
G
ªº
cc
«»
«»
¬¼
Proof. Again by considering Eq. (15) and Lemma 5 and after simplification we
obtain:
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1, 1
1
cos 1 sin sin 2 2 1
nm nm
nm nm
PP
nmnm nm m
nPm P
nm
TT
TT
w
ªº
«»
w
«»
¬¼
1, 1 1 , 1
112
1
22 1 22 1
nm nm
nm nm nm
nm mPP
nm n
ªº
«»
«»
¬¼
1, 1
31
22 1 nm
nm P
n
(18)
and the proposition is proved by normalizing Eq. (18) using Eq. (11).
Note: For m=0 terms the third term in the first square bracket of the nm
ac,nm
bc,nm
acc
and nm
bcc coefficients presented in Propositions 3 and 4 vanish. In other words,
000nnn
aaa
cc c
and 000nnn
bbb
cc c
.
Corollary 3 The satellite acceleration in x- and y-directions of the quasi-inertial
frame based on Eqs. (1
6
a) and (1
6
b) and Propositions 3 and 4 are:
max 2
2
0
11
2
n
Nn
nm m m
nmn
GM R m
xP u Q m Q m
Rr m
OO
ªº
§·
4 4
«»
¨¸
©¹ «»
¬¼
¦¦
1, 1 1, 1 1, 1 1, 1
nm nm nm nm
nm nm nm nm
mm
aP b P c P d P
mm
§·
cc c c
¨¸
¨¸
©¹
, 0mz
max 2
2
0
11
2
n
Nn
nm m m
nmn
GM R m
yP u Q m Q m
Rr m
OO
ªº
§·
4 4
«»
¨¸
©¹ «»
¬¼
¦¦
1, 1 1, 1 1, 1 1, 1
nm nm nm nm
nm nm nm nm
mm
aP b P c P d P
mm
§·
cc c c
¨¸
¨¸
©¹
, 0mz.
The advantages of our new expressions are to exclude singular terms, derivatives of
the ALF and simplicity because of having same type of the coordinates for the same
point at satellite level. All the coefficients of the ALF
(nm
ac,nm
bc,nm
cc,nm
dc,nm
acc ,nm
bcc ,nm
ccc and nm
dcc ) are constant and do not change by satellite’s
position. The ALF are needed to be computed once for each position of the satellite.
5 NUMERICAL ORBIT INTEGRATION
Equation (5) shows the three differential equations to be solved. Numerical
integration is the simplest and most efficient technique for the solution. The Runge-
Kutta [Babolian and Maleknejad, 1994, Eshagh 2003a, Eshagh and Najafi Alamdari,
2006] is one of the well-known single step methods of numerical integration. Adams-
Bashforth and Adams-Moulton [Babolian and Maleknejad, 1994], also Störmer-
Cowell [Santos, 1994] algorithms are two well-known methods of multi-step
integration. Step-variable methods can also be used for integration; see Eshagh
(2005). The orbit integration in different frames is given by Eshagh (2009b). Some
details for the benefits of the reader about orbit integration are given in Appendix A.
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6. AVERAGE POWER OF ACCELERATION AT SATELLITE
ALTITUDE
Depending on the satellite’s altitude, the maximum degree ma x
N, in the spherical
harmonic expansion is considered as a cut-off degree. This is due to the factor
2n
Rr in these equations that attenuates the magnitude of the satellite acceleration
and due to the asymptotic decrease in nm
ucoefficients. The average power of the
satellite’s acceleration was investigated by Hwang and Lin (1998). This power can be
written in terms of the degree variance of the gravitational field as [Hwang and Lin,
1998]:
22
2
2
2
12 1
n
kn
knm
nmn
GM R
Pnnu
RR
§·§·
¨¸¨¸
cc
©¹©¹
¦¦
, (19)
where Rc is approximately equal to the Earth's mean radius plus the satellite mean
altitude. With specific error tolerance the cutoff degree can be determined by
comparing the power of acceleration up to the cutoff degree and the “total” power of
acceleration which can be obtained by an expansion to a very high degree max
N.
Currently equal to 2160 is the highest possible degree but at satellite level such high
degree geopotential model does not make sense. In order to determine the highest
useful degree one may compute the relative power up to degree k
max
k
k
N
P
P
U
, (20)
we should have 1
k
U
| or
14
110
k
U
for the k to be the cut-off degree.
Based on the average acceleration power of the satellite, Eq. (19), and the criterion
mentioned above (Eq. 20), the cut-off degree of the spherical harmonic expansion of
the gravitational field for integrating the orbit of the CHAMP, GRACE and GOCE
satellites are obtained 154, 137 and 261, respectively.
7. NUMERICAL ESTIMATION OF SATELLITE ORBIT
The perturbations of the orbital elements are enlarged by decreasing the satellite’s
elevation. The approximate magnitude of these orbital elements for the recent satellite
missions are presented in Table 1.
Table 1. Orbital characteristics of the recent satellite missions, CHAMP, GRACE and
upcoming GOCE
Orbital
elements CHAMP GRACE GOCE
a (metre) 6823287 6882043 6628281
e <0.004 <0.005 <0.001
i (deg.) 87.3 89.05 96.6
:(deg.) 144 74.51 0
Z(deg.) 257 68.35 0
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The GRACE twin satellites have higher altitudes than the others, its orbit is more
polar as its inclination is closer to 90 deg. The GOCE satellite has the most circular
and inclined orbit. The current orbital elements of the CHAMP and the GRACE are
available on http://www.gfz-potsdam.de/pb1/op/champ/ and
http://www.csr.utexas.edu/grace/, respectively. Also the initial position and velocity
of the GOCE were received by authors from ESA (European Space Agency) (Plank,
personal communication) and converted to the orbital elements.
In order to study the behaviour of orbital elements due to different harmonics we
need to integrate Corollaries 1 and either 2 or 3 numerically. Among the various
techniques of solving the vector system of differential equations, the well-known 4-th
order Runge-Kutta integrator was used in this study because of its simplicity.
The computational speed on the orbit integration process is directly related to the
computation of the satellite accelerations due to different resolutions of the
gravitational field. A double summation can be summed up using two loops in
computer programming, but it is not the best way, in computational point of view by
increasing maximum degree of the spherical harmonic expansion computational time
is increased too. In such cases, vectorization techniques are preferred. Sharifi (2006)
and Eshagh (2009a) have also used this technique for global synthesis and analysis of
the Earth gravitational field successfully.
We consider one day revolution of the CHAMP, GRACE and GOCE satellites to
compare the behaviour of their orbital elements at each degree of the gravitational
field. At first step maximum degree max
N of the spherical harmonic coefficients of
the Earth gravitational field were determined for each satellite using average
accelerations power presented in Section 5. The integration performs max
N times for
computing the satellite state vector in each step. The state vector is converted to the
orbital elements and the orbital elements due to the central (spherical) gravitational
field are subtracted to obtain the perturbations. The following figures show the
behaviour of orbital elements for each satellite with respect to different harmonic
degree of gravitational field. In these figures we consider the maximum absolute
value of the geopotential perturbation for a specific degree along the satellite motion.
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Fig.1. Absolute maximum value of geopotential perturbation in one day revolution of
satellites, (a) perturbation in semi-major axis, (b) perturbation in eccentricity, (c)
inclinations, (d) right ascension of ascending node, (e) perigee argument, (f) mean
anomaly
Figure 1(a) shows the semi-major axis of the orbital ellipses of the recent satellite
missions. The behaviour of the semi-major axis of the CHAMP and GRACE satellites
are more or less the same as would be expected because of having close altitudes.
However, in Fig. 1(a) one can see that the perturbation of these two satellites differs
in some degrees. The perturbations of the CHAMP and the GRACE satellites in semi-
major axis are nearly in the same order up to degree 50 but they differ in higher
degrees. Because of lower altitude of the GOCE than the other satellites the semi-
major axis of this satellite is more perturbed. It is perturbed more or less in the same
order as the other satellites up to degree 30 but it differs. In order to see more details
we can present the perturbations of the maximum degree on the orbital elements. A
Perturbation in semi-major axis
(m)
Perturbation in eccentricity
020 40 60 80 100 120 140 160 180 200
10
-14
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
GRACE
CHAMP
GOCE
020 40 60 80 100 120 140 160 180 200
10
-7
10
-5
10
-3
10
-1
10
1
10
3
10
5
10
7
GRACE
CHAMP
GOCE
Degree
(a)
Degree
(b)
Degree
(c)
Degree
(d)
Perturbation in inclination (deg.)
020 40 60 80 100 120 140 160 180 200
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
GRACE
CHAMP
GOCE
Perturbation in right ascension of
ascending node (deg.)
020 40 60 80 100 120
10
-8
10
-6
10
-4
10
-2
10
0
10
2
GRACE
CHAMP
GOCE
020 40 60 80 100 120 140 160 180 200
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
10
4
GRACE
CHAMP
GOCE
Perturbation in perigee
argument(deg.)
Perturbation in mean anomaly (deg)
020 40 60 80 100 120 140 160 180 200
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
10
2
GRACE
CHAMP
GOCE
Degree
(e)
Degree
(f)
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maximum degree could be considered for specific level of perturbation in each
element. The millimeter level has been considered in the perturbation of semi-major
axis and 1 arc-second for the angular elements. The maximum degree is 116 to see
perturbations in millimeter level in semi-major axis of the CHAMP; this value is 109
and 175 for the GRACE and GOCE, respectively. The inclinations of the satellites are
presented in Fig. 1(c). The figure shows the small perturbations in the inclination.
They are in the same order for the lower degrees up to 20 and decreasing to 1 arc-
second. In Fig. 1(d) the large perturbations in the right ascension of ascending node
are decreasing fast. They decrease to 1 arc-second up to degree 40 for the GOCE
while it is about 20 for other satellites. The interesting matter is to see small
perturbations on the satellite’s orbit eccentricity, and this is why for dynamic orbit
determination and analytical solution of the Earth gravitational field the satellite’s
orbit is assumed circular. It expresses that perturbation in eccentricity does not play an
important role in orbit integration and recovery of the geopotential coefficients as the
orbits are more or less circular. The behaviour of the perigee argument and mean
anomaly is larger in the GOCE than the other satellites as it is expected. The
perturbations in perigee arguments and mean anomaly are up to same degree 83 for
the CHAMP and GRACE while these are considerably perturbed (twice larger) in
GOCE.
In comparison with average power of accelerations, 154, 137 and 261 in these
satellites missions, we can say that these values are too optimistic and theoretical, and
the above numbers discussed are also too pessimistic for the maximum degree of
perturbations. We expect that these differences are due to the selection of short arc
orbit in this study. As it is known now, the maximum degree of the spherical
harmonic expansion of the geopotential considered to be determined by the orbital
analysis of CHAMP satellite is about 119 which does not confirm with the average
power and the number that we obtained. The gravitational field extracted from the
GRACE mission is expanded up to degree and order 150; but it should be kept in
mind that satellite gradiometric data of the GRACE mission helped the solution to
derive higher degrees and orders. Perturbation analysis of the satellite’s orbit at
altitude of the GRACE satellites cannot yield such resolution for the gravitational
field from space. The maximum degree obtained using the average power of the
acceleration for the GOCE is 261 but in this numerical study we obtain 175 based on
short arc orbit consideration.
8. CONCLUSIONS
The newly presented expression for satellite acceleration computation in this paper is
very simple for programming as it is not involved with the associated Legendre
function derivatives and singular terms. However it should be mentioned that the
singularity happens only for those satellites whose orbits inclined towards the poles.
Since the GOCE orbit is away from the poles, the singularity is not needed to be
considered. The coefficients of the associated Legendre functions in new expressions
are constants and do not change by the position of a satellite and they should be
computed once for all the integration process. The associated Legendre functions
should be generated up to only two more harmonic degrees and orders in the new
expression than in the traditional method. Another achievement is the capability of
investigating perturbations of a satellite orbit parameters due to a single harmonic
coefficient of the geopotential field during one day revolution of satellite. The average
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acceleration power was considered for the recent satellite’s missions and the
maximum degree of the spherical harmonic expansion are obtained 154, 137 and 261,
for the CHAMP, GRACE and GOCE, respectively. The maximum absolute values of
the perturbations of the orbital elements with respect to each degree of geopotential
were computed by frequent integrating the satellites’ orbit and comparing with
Keplerian orbit. The perturbations were visualized in some figures presenting the
behaviour of the orbital elements clearly. As we expected the perturbations of the
CHAMP and GRACE satellite are similar and the GOCE orbit is perturbed
considerably larger than the others. The maximum degrees of the spherical harmonic
expansion of the orbit integration procedure was obtained based on a predefined
amount of the perturbations (millimeter level in this study). This maximum values are
116, 109 and 175 for the CHAMP, GRACE and GOCE satellites, respectively, which
contradicts with the degrees obtained from average acceleration power because we
have considered one day revolution of satellite in our investigations.
APPENDIX A
Schematically we can write the system of differential equation as:
xx (A1a)
yy (A1b)
zz (A1c)
4,,,xfxyzt '
(A1d)
5,,,yfxyzt '
(A1e)
6,,,zfxyzt '
(A1f)
where, x
, y
and z
are the satellite’s velocities in the quasi-inertial frame. The
functions 4
f, 5
f are the formulas presented in either Corollary 2 or 3 and 6
f is
Corollary 1. These equations can be solved by numerical integration algorithms. In
this study we use 4-th order Runge-Kutta algorithm for solving the equation of motion
of the satellite:
111213141
(22 )/6
ii
xxkk kk
(A2a)
112223242
(22 )/6
ii
yykkkk
(A2b)
113233343
(22 )/6
ii
zzkk kk
(A2c)
114243444
(22 )/6
ii
xxk k kk
(A2d)
115253545
(22 )/6
ii
yykkkk
(A2e)
116263646
(22 )/6
ii
zzk k kk
(A2f)
The coefficients of jk
k, j=1, 2, 3, 4 and k=1,2,…, 6 are presented. i is the epoch
number.
Generally, a higher order differential equations than first, is converted to a system of
first order differential equations for the numerical solution; see Appendix B. In orbit
integration the numerical solution of this system of differential equation yields the
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position and velocity of the satellite simultaneously in the next epoch of integration.
For more detail see Appendix B.
The coefficient of 4-th order Runge-Kutta algorithm for solving the system of
differential equation of satellite motion Eqs. (A1a)-(A1f)
11
12
13
14 4
15 5
16 6
(, ,)
(, ,)
(, ,) ,
ii
ii
ii
iii i
iii i
iii i
kxt
kyt
kzt
kfxyzt
kfxyzt
kfxyzt
'
'
'
'
'
'
,
21 14
22 15
23 16
24 4 11 12 13
25 5 11 12 13
26 6 11 12 13
2
2
2
2, 2, 2
2, 2, 2
2, 2, 2
ii
ii
ii
iii i
iii i
iii i
kxk t
kyk t
kzk t
k fxk yk zk t
k fxk yk zk t
k fxk yk zk t
'
'
'
'
'
'
31 24
32 25
33 26
34 4 21 22 23
35 5 21 22 23
36 6 21 22 23
2
2
2
2, 2, 2
2, 2, 2
2, 2, 2
ii
ii
ii
iii i
iii i
iii i
kxk t
kyk t
kzk t
kfxk yk zk t
kfxk yk zk t
k fxk yk zk t
'
'
'
'
'
'
,
41 34
42 35
43 36
44 4 31 32 33
45 5 31 32 33
46 6 31 32 33
,,
,,
,,
ii
ii
ii
iii i
iii i
iii i
kxkt
kykt
kzkt
k fxkykzk t
k fxkykzk t
kfxkykzk t
'
'
'
'
'
'
i
x , i
y, i
z, i
x
, i
y
and i
z
are the three components of the position and velocity at
reference epoch i, respectively. i
t'is the integration step size.
APPENDIX B
The solution of the higher order differential equation leads to a system of first order
differential equations and this system of differential equations can numerically be
solved. For an m-order differential equation we write
y=y(t), a t bdd (B.1)
is the solution of the following m-order differential equation
() ( 1)
() (,,, ,)
mm
ytfyy y t
c
" (B.2)
with the following initial values
01
()yt
D
, 02
()yt
D
c (1)
0
,, ()
m
m
yt
D
". (B.3)
A system of first order differential equations can be written:
12
() (), () (), ,yt v t y t v t
c
!(1)
() ()
m
m
ytvt
(B.4)
and
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1
2
2
3
(1)
() ( 1)
12
(,,, ,)
(, , , ,)
m
mm
m
m
dv dy v
dt dt
dv dy v
dt dt
dv dy yfyyyt
dt dt
fvv v t
°
°c
°
°
°
®
°
°c
°
°
°
¯
#
"
"
(B.5)
is the system of m first order differential equations with the following initial values
10 0 1
20 0 2
(1)
00
() ()
() ()
() ()
m
mm
vt yt
vt yt
vt y t
D
D
D
c
# (B.6)
The numerical solution of the system of differential equations by for example Runge-
Kutta is
()
nn
yyt , and ()
nn
zzt , (B.7)
(,,)yfyzt
c , and (,,)zgyzt
c , (B.8)
and coefficients of the Runge-Kutta algorithm are:
1(,,)
iii i
kfyzt t '
1(,,)
iii i
mgyzt t '
211
(2, 2, 2)
ii iii
k
fy
kzm tt t ''
211
2, 2, 2
ii iii
mgyk zm x t t ''
322
(2, 2,2)
iiiii
kfyk zm t t t ''
322
2, 2, 2
iiiii
mgyk zm t t t ''
433
(, , )
iiiii
kfykzmt tt ''
433
,,
iiiii
mgykzmt t t ''
(B.9)
and the differential equations are numerical solved by the following relations
11234
22 /6
ii
yykkkk
(B.10)
11234
22 /6
ii
zzmmmm
. (B.11)
Acknowledgments. The authors would like to thank Prof. Lars E. Sjöberg for his
valuable comments on the draft version of this paper. The first author appreciates the
Swedish National Space Board for the financial support, project no. 63/07:1.
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64
Received: 2008-11-27,
Reviewed: 2009-03-05,
Accepted: 2009-03-09.
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