THE DETERMINATION OF GRAVITATIONAL POTENTIAL
DIFFERENCES FROM SATELLITE-TO-SATELLITE TRACKING
CHRISTOPHER JEKELI
Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State
University, 2070 Neil Ave, Columbus, OH 43210, U.S.A.,
e-mail: jekeli.1@osu.edu
(Received: 13 November 1998; accepted: 19 November 1999)
Abstract. A new, rigorous model is developed for the difference of gravitational potential between
two close earth-orbiting satellites in terms of measured range-rates, velocities and velocity differences, and specific forces. It is particularly suited to regional geopotential determination from a
satellite-to-satellite tracking mission. Based on energy considerations, the model specifically accounts for the time variability of the potential in inertial space, principally due to earth’s rotation.
Analysis shows the latter to be a significant (±1 m2 /s2 ) effect that overshadows by many orders of
magnitude other time dependencies caused by solar and lunar tidal potentials. Also, variations in
earth rotation with respect to terrestrial and celestial coordinate frames are inconsequential. Results
of simulations contrast the new model to the simplified linear model (relating potential difference to
range-rate) and delineate accuracy requirements in velocity vector measurements needed to supplement the range-rate measurements. The numerical analysis is oriented toward the scheduled Gravity
Recovery and Climate Experiment (GRACE) mission and shows that an accuracy in the velocity
difference vector of 2 × 10−5 m/s would be commensurate within the model to the anticipated
accuracy of 10−6 m/s in range-rate.
Key words: gravitational potential, satellite-to-satellite tracking, range-rate measurements, earth
rotation.
1. Introduction
A satellite mission dedicated to the improvement of our knowledge of the earth’s
gravitational field with a direct (in situ) measurement system has been in the proposal stages for a long time and at several agencies. Of course, gravitational field
knowledge comes also by tracking satellites from ground stations, and many longwavelength models of the field have been deduced from such data. But, these
models derive from the observations of a large collection of satellites that have been
tracked over various periods during the long history of earth-orbiting satellites,
where none of these was launched for the expressed purpose of providing a global
and detailed model of the gravitational field.
Rather, the proposed gravity mapping missions are based on one of several
related measurement concepts, including the measurement of the range between
two close earth-orbiting satellites (GRAVSAT, GRM: Keating et al., 1986), tracking a low-orbiting satellite with a system of high-orbiting satellites (Jekeli and
Celestial Mechanics and Dynamical Astronomy 75: 85–101, 1999.
c 2000 Kluwer Academic Publishers. Printed in the Netherlands.
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CHRISTOPHER JEKELI
Upadhyay, 1990), or measuring the gravitational gradients on a single low-orbiting
satellite (ARISTOTELES: Bernard and Touboul, 1989; SGGM: Morgan and Paik,
1988; GOCE (Gravity Field and Steady-State Ocean Circulation Explorer): Rummel and Sneeuw, 1997).
Such a mission now has been approved and is expected to be realized in 2001.
The Gravity Recovery and Climate Experiment (GRACE: Tapley and Reigber,
1998) is a variant of the erstwhile GRAVSAT and GRM mission concepts in which
two low-altitude satellites will track each other as they circle the earth in identical
near polar orbits. Unlike GRM, the satellites are not ‘drag-free’ and nongravitational accelerations must be measured independently using on-board accelerometers. Also, the altitude of the GRACE satellites is significantly higher (400 km) than
that proposed for GRM (160 km). Another significant departure from the previous
concept is that each satellite will carry a geodetic quality GPS receiver. The purpose
of these receivers is to aid in orbit determination, as well as provide GPS satellite
occultation measurements to model the lower atmosphere.
As shown also here with Equation (36), a simple model may be derived on
the basis of energy conservation that relates the measured range-rate between two
satellites to the gravitational potential difference. However, though widely used
to analyze the capability of a satellite-to-satellite tracking (SST) mission to determine the geopotential (Wolff, 1969; Jekeli and Rapp, 1980; Wagner, 1983; Dickey,
1997), it is hardly adequate as a model for processing actual data. In fact, this model
neglects the significant effect of earth’s rotation that causes the geopotential to vary
with time in inertial space (therefore, strictly, it is nonconservative). Furthermore,
the range-rate accounts for but a single component of the velocity vector difference
resulting from the potential difference. These deficiencies in the model are orders
of magnitude above the measurement noise level and would preclude accurate in
situ geopotential determination. It should be noted, however, that other modeling
techniques exist to determine the geopotential on global and regional bases. For
example, the range-rate or range may be expressed in terms of a spherical harmonic
series of the global geopotential (Colombo, 1984) or locally in terms of suitable
basis functions (Ilk, 1986), and the corresponding coefficients are solved using a
least-squares adjustment procedure.
The in situ model developed here is particularly suited to regional determination of the geopotential and would also be amenable to global determination
using conventional harmonic analysis techniques. It is based on an energy equation
generalized to account for the time-varying potential fields. Results of simulations
show the relationship between geopotential accuracy and, accuracies in range-rate
and velocity vector measurements associated with the GRACE mission. Clearly,
this model applies to any other SST mission to map the gravitational field of any
planet.
THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES
87
2. The Model
From energy considerations (see the Appendix), the exact relationship in inertial
space between the gravitational potential, V , and terms containing the satellite
velocity, ẋ = (ẋ1 , ẋ2 , ẋ3 ), and specific forces acting on the satellite, F̄ =
(F̄1 , F̄2 , F̄3 ), is given by (A.14) with (A.5) substituted:
Z
Z t
1 2 X t
∂V
V = |ẋ| −
F̄k ẋk dt +
(1)
dt − E0 .
2
t0
t0 ∂t
k
The first term on the right hand side is the kinetic energy and the second term
represents energy dissipation. The third term is due to the explicit time variation
of the gravitational potential in inertial space, and E0 is the energy constant of the
system.
If we measure a satellite’s velocity along its orbit, as well as the action forces
on the satellite, then (1) represents an (integral) equation that can be solved for the
potential, V . We decompose the potential as follows:
V = Vrotating earth +Vlunar tide + Vsolar tide + Vplanetary tides+
+Vsolid earth tide + Vocean tide + Vatmospheric tide +
(2)
+Vocean loading + Vatmospheric loading + Vother mass redistributions
and recognize that some parts are better known than others and most have dissimilar magnitudes and periodicities. The gravitational potential of the rotating earth
can be expressed in spherical polar coordinates in an earth-fixed coordinate frame
using spherical harmonic functions, Ȳn,m :
n ∞
kMe X X R n+1
Vrotating earth = Ve (r, θ, λ) =
Cn,m Ȳn,m (θ, λ),
(3)
R n=0 m=−n r
where r is geocentric radius, θ is co-latitude, and λ is longitude with respect to a
defined zero-meridian, kMe is the gravitational constant times earth’s total mass
(including atmosphere), R is a mean earth radius, Cn,m are coefficients that define
earth’s mass density distribution, and
cos mλ, m > 0,
¯
Ȳn,m (θ, λ) = Pn,|m| (cos θ)
(4)
sin |m|λ, m < 0,
where P̄n,m are fully normalized associated Legendre functions. The frame for the
coordinates (r, θ, λ) is fixed to the earth and it realizes the International Terrestrial
Reference System that is well defined by the International Earth Rotation Service
(IERS) (McCarthy, 1996). The coefficients, Cn,m , are assumed constant since any
temporal redistribution of mass is accounted for by the other potential components
in (2).
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CHRISTOPHER JEKELI
The potential in (1) is supposed to be in the inertial frame. Hence, using (3)
requires a transformation from the fixed terrestrial to the inertial (mean celestial)
frame. It is convenient to describe this transformation in terms of co-latitude and
longitude angles
θ = ζ + 1ζP + 1ζN + 1θS ,
(5)
λ = α + 1αP + 1αN − ωe t + 1λS ,
(6)
where the coordinates (ζ, α) are the co-declination and right ascension in the inertial frame of epoch J2000.0. The terms 1θS and 1λS rotate the terrestrial pole
of date to the celestial pole of date using coordinates of polar motion, ωe is earth’s
rate of rotation and the corresponding term in (6) rotates the terrestrial into the
celestial frame about the 3-axis by the Greenwich sidereal time, 1ζN and 1αN
account for the nutations of the celestial pole and transform it from its true to
its mean direction of date, and 1ζP and 1ζP describe the precession of the pole
from its mean direction of date to its mean direction at a defined epoch, currently
J2000.0. Detailed expressions for these terms can be found in (Mueller, 1969) and
(Seidelmann, 1992). Each one is an explicit function of time, meaning that if (5)
and (6) are substituted into (3), then Ve as a function of (ζ, α) depends explicitly
on time.
These transformations sometimes are interpreted to cause time dependencies in
the harmonic coefficients. However, in the sequel the present interpretation of timedependent coordinates is preferred and more appropriate. In this way the explicit
time-derivative of the potential is given by
∂Ve
∂Ve ∂θ
∂Ve ∂λ
=
+
,
∂t
∂θ ∂t
∂λ ∂t
(7)
where ∂θ/∂t and ∂λ/∂t denote explicit time derivatives of these coordinates (now
in the inertial frame). We note that the dominant explicit time-derivative component in (5) and (6) is −ωe . In fact, the precession rates in right ascension and in
declination are less than 50 arcsec per year, or less than 8 × 10−12 rad/s. Similarly
the nutation rates in longitude and ecliptic obliquity are less than 3 × 10−12 rad/s,
and polar motion rates are less than 3 × 10−13 rad/s for the main Chandler wobble.
These rates are seven to eight orders or magnitude smaller than ωe = 7.292115 ×
10−5 rad/s, and we may approximate
∂Ve
∂Ve
= −ωe
,
∂t
∂α
(8)
where, because of the linear relationship (6), ∂/∂λ = ∂/∂α. Furthermore, we
assume to a similar level of approximation that ωe is constant.
The other potential terms in (2) may be analyzed similarly. Expressed in inertial
frame coordinates with origin at earth’s center of mass, the tidal potential of an
extra-terrestrial body, including the indirect effect arising from the consequent
THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES
89
deformation of the quasi-elastic earth, is given approximately by (Torge, 1991;
Lambeck, 1988)
5 !
R
3 kMB r 2
VB (r, θ, α) =
1 + k2
.
4 rB
rB
r
1
sin 2θ sin 2θB cos(α − αB ) + sin2 θ sin2 θB cos 2(α − αB ) + 3 cos2 θ −
3
1
cos2 θB −
,
(9)
3
where kMB is the gravitational constant times the mass of the body, (rB , θB , αB ) are
its coordinates in the inertial frame, and k2 = 0.29 is Love’s number (an empirical
number based on observation). Equation (9) treats the body as a point mass and
neglects terms with powers in r/rB greater than 2, which is adequate in the present
context for the most influential bodies, the sun and the moon. Also, it is assumed
that the elastic response to the tidal potential is instantaneous. In reality there is
a lag, which to a first approximation is constant and, therefore, presently of no
consequence.
The coordinates (rB , θB , αB ) are all functions explicitly of time due to the motion of the body with respect to the earth. However, the largest rate is in αB since
the sun and moon, respectively, depart by at most 23.5◦ and 29◦ in declination from
the equatorial plane. If we ignore the time dependence of rB , then
∂VB
∂VB
∂VB
θ̇B ,
≈−
α̇B +
∂t
∂α
∂θB
(10)
again, because ∂/∂αB = −∂/∂α. If nB denotes the mean angular motion of the
body, then the rate in co-declination varies between zero and ± sin(i)nB , the latter
occurring when the body crosses the celestial equator, where i is the inclination of
its orbit. The length of a sidereal month is approximately 27 days, hence, for the
moon, nM = 2.7 × 10−6 rad/s. The sidereal year is about 365 days long, implying
that the sun’s mean motion is nS = 2.0 × 10−7 rad/s. The corresponding rates in
right ascension, α̇M and α̇S , have the same respective orders of magnitude. These
rates are 1 to 2 orders of magnitude less than earth’s rate of rotation.
Evaluating the first term (Doodson’s constant) in (9) for the sun and the moon,
we find with a satellite altitude of 400 km:
3.0 m2 /s2 , moon,
3 kMB r 2
=
(11)
4 rB
rB
1.4 m2 /s2 , sun.
These potentials are smaller than earth’s gravitational potential by seven orders of
magnitude. Since the corresponding gradients compare similarly, we have
∂VB
∂Ve
−8
O
< 10 O
(12)
∂t
∂t
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CHRISTOPHER JEKELI
for the principal bodies, moon and sun; the effects of other planets may be ignored.
Lambeck (1988) also gives the potential due to the ocean tides (including the
loading effect on the solid earth) and states that the amplitudes are less than 15% of
the solid earth tidal effect that is included in (9). On the basis of these magnitudes
we may safely neglect these as well as all other potentials in (2) as far as the explicit
time derivative is concerned; and we have from (8):
∂V
∂Ve
≈ −ωe
.
∂t
∂α
(13)
Now, since x1 = r cos θ cos α and x2 = r cos θ sin α, it is readily shown that
∂V
∂Ve
∂Ve
.
(14)
− x2
= −ωe x1
∂t
∂x2
∂x1
Substituting (A.12) we then have
∂V
∂δV
∂δV
,
− ẍ2 − x2 F̄1 +
− ẍ1
= ωe x1 F̄2 +
∂t
∂x2
∂x1
(15)
where, from (2), V = Ve + δV . Again, the gradients of the perturbing potential,
δV , are about seven orders of magnitude less than the acceleration of the satellite,
and in most cases so are accelerations associated with the atmospheric drag and
solar radiation pressure that constitute F̄. Neglecting these terms, we have
∂V
≈ ωe (x2 ẍ1 − x1 ẍ2 ),
∂t
which yields
Z t
∂V
dt = −ωe (x1 ẋ2 − x2 ẋ1 )
t0 ∂t
(16)
(17)
(the constant of integration is relegated to E0 ). As an aside, (17) can also be written
as
Z t
∂V
(18)
dt = −ωe α̇(x12 + x22 ).
∂t
t0
This differs from the usual ‘rotation potential’ found in textbooks on Celestial
Mechanics. The difference is that here the potential is given in the inertial frame,
whereas the rotation potential, ωe2 (x12 + x22 ) (see, e.g., Danby, 1988), applies to
the earth-fixed (rotating) frame. To distinguish our term, we call it the ‘potential
rotation’ term, since it accounts for the rotation of the potential in the inertial frame.
Finally, we arrive at the model for the potential from (1) and (17):
Z
1 2 X t
V = |ẋ| −
F̄k ẋk dt − ωe (x1 ẋ2 − x2 ẋ2 ) − E0 .
(19)
2
t0
k
THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES
91
This expresses the desired gravitational potential in terms of measured quantities,
specific force and velocity (also satellite position is required, but not to extremely
high accuracy for the potential rotation term). The model is approximate only because certain time dependencies in the gravitational potential have been neglected
according to (16). The energy dissipation is not negligible, being of approximately
the same order as the potential rotation term. However, it is ignored at present to
simplify the subsequent analysis.
3. Satellite-to-Satellite Tracking
SST, for example, as proposed for the GRACE mission, constitutes the very precise
measurement of the range, ρ12 , between two satellites following each in approximately the same orbit. We have ρ12 = eT12 x12 , where x12 = x2 − x1 , and e12 is the
unit vector identifying the direction to the second satellite from the first. Then, the
range-rate, being derived from the measured range, is the projection of the velocity
difference between the satellites onto the line joining them
ρ̇12 = eT12 ẋ12
(20)
since ėT12 e12 = 0. We treat the range-rate as the measurement, noting that it is only
a component of the velocity difference.
For satellites in drag-free orbits (F̄ = 0) and a static gravitational field (ωe = 0),
the energy Equation (19) reduces to
1
V = |ẋ|2 − E0 .
2
Taking the along-track derivative, denoted by da , on both sides yields
da V = ẋT da ẋ.
(21)
(22)
If the two satellite are close then the left side may be interpreted as the difference
in gravitational potential between the satellites and the along-track differential
velocity as the range-rate, thus
V2 − V1 ≡ V12 ≈ |ẋ1 |ρ̇12 .
(23)
This relates the measurements directly to potential differences along the orbit. It is
the model assumed in the analyses by Wolff (1969), Fischell and Pisacane (1978),
Rummel (1980), Jekeli and Rapp (1980), Wagner (1983), and Dickey (1997), among
others.
Up to the approximations discussed in connection with (19), the correct expression is given by
XZ t
1
T
2
V12 = ẋ1 ẋ12 + |ẋ12 | −
(F̄2k ẋ2k − F̄1k ẋ1k ) dt −
2
t0
k
−ωe (x121 ẋ22 − x22 ẋ121 − x11 ẋ122 + x122 x11 ) − E012 ,
(24)
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CHRISTOPHER JEKELI
Figure 1. The geometry of residual quantities referred to a reference orbit.
where the first two terms derive from |ẋ2 |2 − |ẋ1 |2 = (ẋ2 − ẋ1 )T (ẋ2 + ẋ1 ), and E012
is a constant. Omitting the dissipative term, we write
V12 = ẋT1 ẋ12 + 12 |ẋ12 |2 + V R12 − E012
(25)
with V R12 denoting the difference in potential rotation terms.
It is customary to introduce a known reference potential that accounts for the
longest wavelengths of the signal. We denote all quantities referring to such a reference field by the superscript ‘0’, and by definition, it and all associated quantities,
in particular the corresponding orbital reference ephemerides of both satellites,
can be computed without error. The reference field may be a potential with just
the central and second zonal harmonic terms, or it may be a low-degree spherical
harmonic expansion of the potential, say, complete to degree and order 10. For
the present purposes, a harmonic expansion complete to degree and order 2 will
suffice to provide a reasonably quantitative illustration. The residual to any of the
reference quantities is denoted with the prefix ‘1’.
It must be emphasized that a residual quantity is the difference between a quantity that refers to the actual orbit and a quantity that refers to a reference orbit. That
is, the only common coordinate between the two is time, and not position. Figure 1
illustrates this situation. It is assumed that there is a point in time when the two
orbits are tangent (i.e., their Keplerian elements coincide).
0
The residual quantities are, for example, 1V12 = V12 − V12
, 1ẋ1 = ẋ1 − ẋ01 ,
0
and 1ρ̇12 = ρ̇12 − ρ̇12 , where the reference potential (sans dissipative energy term)
is given analogous to (25) by
0
0
V12
= (ẋ01 )T ẋ012 + 12 |ẋ012 |2 + V R12
− E0012 ,
(26)
and the residual potential difference is
1V12 = ẋT1 ẋ12 − (ẋ01 )T ẋ012 + (ẋ012 )T 1ẋ12 + 12 1ẋT12 1ẋ12 + 1V R12 − 1E012 . (27)
Corresponding to the approximation (23), we define the approximate residual model,
designated with the symbol ‘∧ ’ as
1V̂12 = |ẋ01 |1ρ̇12 .
(28)
THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES
93
The error in this model relative to the true model (27) is given by
1V12 = 1V̂12 − 1V12
= (ẋ02 − |ẋ01 |e12 )T 1ẋ12 + (1ẋ1 − |ẋ01 |1e12 )T ẋ012 +
+(ẋ1 )T 1ẋ12 + 12 |1ẋ12 |2 − 1V R12 + 1E012
= v1 + v2 + v3 + v4 − 1V R12 + 1E012 ,
(29)
0
which is readily derived using 1ρ̇12 = ρ̇12 − ρ̇12
= eT12 ẋ12 − (e012 )T ẋ012 . Equation
(28) also provides an approximate relationship for the error in potential difference
resulting from an error in the satellite-to-satellite range-rate measurement. Since
the velocity magnitude is approximately |ẋ01 | = 7700 m/s, a standard deviation in
the range-rate measurement of 10−6 m/s (to be expected for the GRACE mission) is
equivalent to a standard deviation of about 0.008 m2 /s2 in the potential difference.
4. A Simulation
To quantify the terms in the error of the potential difference model (28), the orbits
of two satellites were generated on the basis of the high-degree (nmax = 360)
spherical harmonic model of the geopotential, EGM96 (Lemoine et al., 1998), but
only up to degree and order 180
V (r, θ, λ) =
n
180
kM X X R n+1
Cnm Ŷnm (θ, λ).
R n=0 m=−n r
(30)
This model was substituted into (A.6) (with F̄ = 0) and Equation (A.8) was integrated by the Adams–Cowell multistep predictor–corrector algorithm yielding the
ephemeris (x and ẋ) of each satellite at one-second intervals. The accuracy of the
numerical integration of (A.8) was checked by comparing the potential difference
obtained from (25) to the original difference on the basis of (30) – the disagreement
over a single revolution was near the limit of the computational precision.
Other parameters of the two orbits include an initial altitude of 400 km above
the earth’s mean radius, an initial eccentricity of zero, and an initial inclination to
the equator of 87◦ . Hence they are near-polar orbits. The initial orbital elements
of the two satellites were chosen so that their separation was about 200 km and
the two orbital paths never deviated from each other by more than 60 m, mostly
in the radial direction. The orbital integration was limited to slightly more than
a single revolution of the satellite pair (about 6000 s). Also, a pair of reference
orbits was generated using a potential field complete to degree and order 2. The
resulting residual potential difference between the two satellites was on the order
of ±30 m2 /s2 .
This signal and the error in the model (28) are both shown in Figure 2 for the
special case of identical orbits for the two satellites, meaning that the gravitational
94
CHRISTOPHER JEKELI
Figure 2. Comparison of true residual potential difference to model (28) (no earth rotation, identical
orbits).
Figure 3. Comparison of true residual potential difference to model (28) (earth rotation, unequal
orbits differing by less than 60 m).
potential was assumed to be static (ωe = 0, for this case, only). In this case, the
terms, v1 and v2 , on the right side of (29) nearly cancel, and the model error is three
orders of magnitude smaller than the signal. However, when the orbits are only
similar (within 60 m, and ωe 6 = 0), the model error is as large as the signal itself
(Figure 3), but has a very long-wavelength (once-per-revolution) structure that is
Figure 4. Model error terms v1 and v2 for the case depicted in Figure 3.
THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES
95
Figure 5. Model error terms v3 , v4 , 1V R12 for the case depicted in Figure 3.
caused by the second term, v2 , in (29), as seen in Figure 4. Thus, the stratagem
of using the along-track derivative to develop the model is rather sensitive to the
radial similarity of the orbits.
Figure 5 shows the other model errors associated with the simple model (28).
Term v3 has the same order of magnitude as the error due to range-rate measurement error (10−6 m/s), and v4 is practically negligible, but the potential rotation
term, 1V R12 , on the order of ±1 m2 /s2 , is significant. Therefore, the accuracy
of the model (28) is not consistent with a measurement accuracy of 10−6 m/s.
This means that range-rates cannot be used to full advantage to measure potential
differences, unless supplemented by velocity vector measurements.
The more accurate model for the determination of potential differences (again,
omitting the dissipative energy term), given that range-rates are the primary measurements, is obtained from (29) and (28) as
1V12 = |ẋ01 |1ρ̇12 − v1 − v2 − v3 − v4 + 1V R12 − 1E012 .
(31)
It requires also measurements of velocity vectors and their intersatellite differences.
The constant, 1E012 , is either obtained from known initial conditions, or determined empirically as a bias from a sufficiently long sequence of data. To measure
a satellite’s velocity generally requires extensive ground tracking to determine its
ephemeris. However, if the two satellites are equipped with Global Positioning
system (GPS) receivers (as in the case of the GRACE satellites), then their relative
velocities can be measured in situ using standard baseline determination procedures developed for terrestrial kinematic applications where the current accuracy is
estimated to be about 1 cm/s. In space, the accuracy would be significantly better
since the signals transmitted from the GPS satellites are unaffected by tropospheric
delays. Also, if the clock errors of the GPS satellites are known, then the absolute
velocity of either satellite can be determined quite accurately (in fact, GPS will be
used for precise orbit determination of GRACE).
Nevertheless, the accuracy requirements are rather demanding when measuring
velocities and velocity differences associated with the potential difference determination according to (31). Figure 6 shows the relationship between the accuracy in
96
CHRISTOPHER JEKELI
Figure 6. Range-rate and velocity accuracy requirements for potential difference determination
according to (31).
potential difference, δ1V12 , and accuracies in range-rate (δ ρ̇12), absolute (δ ẋ1 ),
and intersatellite (δ ẋ12 ) velocity measurements. The principal term affected by
errors in absolute velocity is v2 , while the velocity difference error affects mostly
the potential rotation term.
Computation of these two terms also requires accurate absolute (for 1V R12 )
and relative (for v2 ) position vector measurements. Figure 7 shows the corresponding relationships to the potential difference accuracies. For example, determination
of the potential difference along the satellite trajectory to an accuracy of 0.1 m2 /s2
(corresponding to an accuracy of 1 cm in geoid differences) requires accuracies in
range-rate, velocity, and position as follows:
δ ρ̇12 = 1 × 10−5 m/s,
δ ẋ1 = 5 × 10−4 m/s,
δx1 = 7 m,
δx12 = 1 × 10−2 m.
δ ẋ12 = 2 × 10−4 m/s
(32)
The vector position requirements are easily satisfied with GPS, while the velocity
vector requirements are just beyond current demonstrated GPS capability, but not
outside the realm of feasibility. Note that the anticipated order-of-magnitude higher
accuracy in range-rate for GRACE would be advantageous only with commensurate improvements in velocity and position accuracies.
THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES
97
Figure 7. Position accuracy requirements for potential difference determination according to (31).
5. Summary
An accurate model for the gravitational potential difference was developed for the
SST system concept. The model relates potential difference to in situ measurements
of velocity (consisting of range-rate, relative and absolute velocity vectors), position, and specific force. In particular, the model includes the time dependencies
of the gravitational potential in inertial space, dominated for practical purposes
by earth’s constant rotation rate. Moreover, the model also differs from models
usually used by terms that depend on the velocity difference vector. Simulations
show that the accuracy of this velocity difference is allowed to be about one order
of magnitude poorer than the range-rate accuracy. They also show that the potential
rotation term is significant at the level of 1 m2 /s2 for satellites in near-polar orbits
with 400 km altitude.
Appendix
From classical mechanics (Goldstein, 1950), Lagrange’s equation for the motion
of a particle is given by
d ∂T
∂T
−
= Qi
(A.1)
dt ∂ q̇i
∂qi
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CHRISTOPHER JEKELI
where {qi , q̇i } are generalized coordinates, T is the kinetic energy of the particle,
and Qi is a component of the generalized force
Qi =
X
Fj ·
j
∂xj
,
∂qi
(A.2)
Fj being the j th force acting on the particle and expressed in inertial Cartesian
coordinates
{xk } = x = x(qi ),
k = 1, 2, 3.
(A.3)
The application at hand is the motion of a satellite in orbit around earth (or
any other planet). As such the motion is unconstrained in terms of the coordinates
and the system is trivially holonomic. It is simplest in this case to specialize the
generalized coordinates to Cartesian coordinates
xk = qk .
(A.4)
The coordinate frame is assumed to be inertial in the sense of being fixed to earth’s
center of mass (it is in free fall in the gravitational fields of the sun, moon, and other
planets) and not rotating with respect to space. Under these premises, the kinetic
energy is given by
T = 12 |ẋ|2 ,
(A.5)
with the further assumption that the satellite has unit mass. The forces acting on the
satellite are divided into kinematic forces (Martin, 1988) due to the gravitational
fields, V , and action forces, F̄, caused variously by atmospheric drag, solar radiation pressure, albedo (earth-reflected solar radiation), occasional thrusting of the
satellite as part of orbital maintenance, and a host of other minor effects, such as
electrostatic and electromagnetic interactions and thermal radiation (Seeber, 1993).
We write for the total force
F = ∇V + F̄.
(A.6)
The total gravitational potential, V , comprises the potentials of all masses in the
universe and it is a function of position in the inertial frame and of time, but not of
velocity
V = V (x, t).
(A.7)
We use the sign convention for the potential that is common in geodesy and geophysics. The temporal dependence arises from earth’s rotation (also, not constant);
the moon’s, sun’s, and planets’ motion relative to the earth, and the change in
potential due to solid earth tides, atmospheric and ocean tides, their loading effects,
and other terrestrial mass redistributions of secular (e.g., post-glacial rebound) and
periodic type.
THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES
99
Lagrange’s equation derives from the principle of virtual work and ultimately
is based on Newton’s second law of motion to which one returns upon substituting
(A.5) and (A.4) into (A.1) and (A.2)
d
(ẋ) = F,
(A.8)
dt
where ẋ is also linear momentum (for unit mass). However, equations like (A.1)
expressing energy relationships are more suited to our purpose since they treat
position and momentum as distinct coordinates (states) of the system. Along this
line, define
H = T − V.
(A.9)
We note that H = H (x, ẋ, t), and H is the Hamiltonian of the motion only if
F̄ = 0.
We have
X ∂H dxk X ∂H dẋk
dH
∂H
=
+
+
.
(A.10)
dt
∂xk dt
∂ ẋk dt
∂t
k
k
Noting the dependencies of T and V on ẋk , xk , and t, this simplifies to
X dT dẋk
X ∂V
dH
∂V
ẋk +
=−
−
.
dt
∂x
d
ẋ
dt
∂t
k
k
k
k
(A.11)
From (A.6) and (A.8),
dV
dẋk
=
− F̄k ,
∂xk
dt
and from (A.5), dT /dẋk = ẋk . Substituting these into (A.11) yields
X
dH
∂V
F̄k ẋk −
=
.
dt
∂t
k
Integrating both sides and using (A.9), we obtain
Z t
XZ t
∂V
T −V =
F̄k ẋk dt −
dt + E0 ,
0
t0 ∂t
k
(A.12)
(A.13)
(A.14)
where E0 is the constant of integration. If the gravitational potential is static in
inertial space (principally, no earth rotation) and if the nongravitational forces are
absent (F̄ = 0), then (A.14) expresses the energy conservation law.
100
CHRISTOPHER JEKELI
Acknowledgements
The author is grateful to the reviewers for their valuable comments. This work was
supported by a grant from the University of Texas, Austin, Contract No. UTA980223, under a primary contract with NASA.
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