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THE DETERMINATION OF GRAVITATIONAL POTENTIAL
DIFFERENCES FROM SATELLITE-TO-SATELLITE TRACKING
Christopher Jekeli
Department of Civil and Environmental Engineering and
Geodetic ScienceThe Ohio State University
2070 Neil Ave.Columbus, OH 43210
e-mail: [email protected]
revision submitted to
Celestial Mechanics and Dynamical Astronomy
11 October 1999
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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.
Keywords: 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
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several agencies. Of course, gravitational field knowledge comes
also by tracking satellites from
ground stations, and many long-wavelength 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 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. GRACE, the
Gravity Recovery And Climate Experiment (Tapley and Reigber,
1998) is a variant of the erstwhile
GRAVSAT and GRM mission concepts in that 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 non-gravitational 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 (23), 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
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(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 non-conservative).
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.
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, x = x1, x2, x3 , and specific
forces acting on the satellite, F = F1, F2, F3 , is given by
(A.14) with (A.5) substituted:
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V = 12 x
2– Fk xk dt
t0
t
Σk
+ ∂V∂t dtt0
t
– E0 (1)
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
+ Vocean loading + Vatmospheric loading + Vother mass
redistributions
(2)
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, Yn,m :
Vrotating Earth ≡ Ve(r,θ,λ) =
kMeR Σn = 0
∞ Rr
n + 1Cn,m Yn,m(θ,λ)Σm = – n
n(3)
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
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Yn,m(θ,λ) = Pn, m (cosθ)cosmλ, m ≥ 0
sin m λ, m < 0(4)
where Pn,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).
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 + ∆ζP + ∆ζN + ∆θS (5)
λ = α + ∆αP + ∆αN – ωet + ∆λS (6)
where the coordinates (ζ,α) are the co-declination and right
ascension in the inertial frame of
epoch J2000.0. The terms ∆θS and ∆λS 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 frame into the celestial about the
3-axis by the Greenwich sidereal time;
∆ζN and ∆αN account for the nutations of the celestial pole and
transform it from its true to its
mean direction of date; and ∆ζP and ∆αP 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)
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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 time-dependent coordinates is
preferred and more appropriate. In this way the explicit
time-derivative of the potential is given by
∂Ve∂t =
∂Ve∂θ
∂θ∂t +
∂Ve∂λ
∂λ∂t (7)
where ∂θ ∂t∂θ ∂t and ∂λ ∂t∂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∂t = – ωe
∂Ve∂α (8)
where, because of the linear relationship (6), ∂ ∂λ∂ ∂λ = ∂ ∂α∂
∂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
deformation of the quasi-elastic Earth, is
given approximately by (Torge, 1991; Lambeck, 1988)
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VB(r,θ,α) =
34
kMBrB
rrB
21 + k 2
Rr
5⋅
sin2θ sin2θB cos(α – αB) + sin2θ sin2θB cos2(α – αB) + 3 cos2θ
–13 cos
2θB –13
(9)
where kMB is the gravitational constant times the mass of the
body, (rB,θB,αB) are its
coordinates in the inertial frame, and k 2 = 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 rBr 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∂t ≈ –
∂VB∂α αB +
∂VB∂θB
θB (10)
again, because ∂ ∂αB∂ ∂αB = – ∂ ∂α∂ ∂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, αM and αS , have the same respective orders
of magnitude. These rates are 1
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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:
34
kMBrB
rrB
2= 3.0 m
2/s2 , moon1.4 m2/s2 , sun
(11)
These potentials are smaller than Earth’s gravitational
potential by seven orders of magnitude.
Since the corresponding gradients compare similarly, we have
O
∂VB∂t < 10
–8 O∂Ve∂t (12)
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∂t ≈ – ωe
∂Ve∂α (13)
Now, since x1 = r cosθ cosα and x2 = r cosθ sinα , it is readily
shown that
∂V∂t = – ωe x1
∂Ve∂x2
– x2∂Ve∂x1
(14)
Substituting (A.12) we then have
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∂V∂t = ωe x1 F2 +
∂δV∂x2
– x2 – x2 F1 +∂δV∂x1
– 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 F .
Neglecting these terms, we have
∂V∂t ≈ ωe x2 x1 – x1 x2 (16)
which yields
∂V∂t dt
t0
t
= – ωe x1 x2 – x2 x1 (17)
(the constant of integration is relegated to E0 ). As an aside,
(17) can also be written as
∂V∂t dt
t0
t
= – ωe α x12 + x2
2 (18)
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 + x2
2 (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):
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V = 12 x
2 – Fk xk dtt0
t
Σk
– ωe x1 x2 – x2 x1 – E0 (19)
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
Satellite-to-satellite tracking, 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 = e12T
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:
ρ12 = e12T
x12 (20)
since e12T
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 ( F = 0 ) and a static
gravitational field ( ωe = 0 ), the energy
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equation (19) reduces to
V = 12 x2 – E0 (21)
Taking the along-track derivative, denoted by da , on both sides
yields
daV = xT
dax (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 ≈ x1 ρ12 (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
V12 = x1
Tx12 +
12 x12
2 – F2k x2k – F1k x1k dtt0
t
Σk
– ωe x121 x22 – x22 x121 – x11 x122 + x122 x11 – E012
(24)
where the first two terms derive from x22 – x1
2 = x2 – x1T
x2 + x1 , and E012 is a
constant. Omitting the dissipative term, we write
V12 = x1T
x12 +12 x12
2 + VR12 – E012 (25)
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with VR12 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 “ ∆ ”.
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).
Reference orbitTrue Orbit
V2
V1
V02
•
•••
V01
Figure 1: The geometry of residual quantities referred to a
reference orbit.
The residual quantities are, for example, ∆V12 = V12 – V120
, ∆x1 = x1 – x10
, and
∆ρ12 = ρ12 – ρ120
, where the reference potential (sans dissipative energy term)
is given analogous
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to (25) by:
V12
0 = x10 T
x120
+ 12 x120 2
+ VR120 – E0
0
12(26)
and the residual potential difference is
∆V12 = x1T
x12 – x10 T
x120
+ x120 T ∆x12 +
12
∆x12T ∆x12 + ∆VR12 – ∆E012 (27)
Corresponding to the approximation (23), we define the
approximate residual model,
designated with the symbol “^” as
∆V12 = x10 ∆ρ12 (28)
The error in this model relative to the true model (27) is given
by
ε∆V12 = ∆V12 – ∆V12
= x20
– x10
e12T
∆x12 + ∆x1 – x10 ∆e12
T
x120
+ x1T ∆x12 +
12
∆x122
– ∆VR12 + ∆E012
= ν1 + ν2 + ν3 + ν4 – ∆VR12 + ∆E012
(29)
which is readily derived using ∆ρ12 = ρ12 – ρ120
= e12T
x12 – e120 T
x120
. Equation (28) also
provides an approximate relationship between the error in
potential difference resulting from an
error in the satellite-to-satellite range-rate measurement.
Since the velocity magnitude is
approximately x10
= 7700 m/s , a standard deviation in the range-rate measurement
of 10–6 m/s
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(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) = kM
R Σn = 0180 R
rn + 1
Cnm Ynm(θ,λ)Σm = –nn
(30)
This model was substituted into (A.6) (with F = 0 ) and equation
(A.8) was integrated by the
Adams-Cowell multistep predictor-corrector algorithm yielding
the ephemeris (x and x ) 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
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identical orbits for the two satellites, meaning that the
gravitational potential was assumed to be
static ( ωe = 0 , for this case, only). In this case, the terms,
ν1 and ν2 , 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 ≠ 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 caused by
the second term, ν2 , 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.
∆V12
time [s]
m2 /
s2
∆V12 – ∆V12 × 103
Figure 2: Comparison of true residual potential difference to
model (41) (no Earth
rotation, identical orbits)
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time [s]
m2 /
s2
∆V12∆V12
^
Figure 3: Comparison of true residual potential difference to
model (41) (Earth rotation,
unequal orbits differing by less than 60 m)
Figure 5 shows the other model errors associated with the simple
model (28). Term ν3 has
the same order of magnitude as the error due to range-rate
measurement error ( 10–6 m/s ), and ν4
is practically negligible; but the potential rotation term,
∆VR12 , 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.
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ν2
ν1 x 10
time [s]
m2 /
s2
Figure 4: Model error terms ν1 and ν2 for the case depicted in
Figure 3.
ν4 x 104
ν3 x 103
Potential Rotation Term
time [s]
m2 /
s2
Figure 5: Model error terms ν3 , ν4 , and ∆VR12 for the case
depicted in Figure 3.
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:
∆V12 = x10 ∆ρ12 – ν1 – ν2 – ν3 – ν4 + ∆VR12 – ∆E012 (31)
It requires also measurements of velocity vectors and their
intersatellite differences. The constant,
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∆E012 , 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
potential difference, δ∆V12 , and
accuracies in range-rate ( δρ12 ), absolute ( δx1 ), and
intersatellite ( δx12 ) velocity measurements.
The principal term affected by errors in absolute velocity is ν2
; while the velocity difference error
affects mostly the potential rotation term.
Computation of these two terms also requires accurate absolute
(for ∆VR12 ) and relative (for
ν2 ) 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:
δρ12 = 1×10–5 m/s , δx1 = 5×10
–4 m/s , δx12 = 2×10–4 m/s
δx1 = 7 m , δx12 = 1×10–2 m(32)
The vector position requirements are easily satisfied with GPS,
while the velocity vector
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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.
δ∆V12 [m2/s2]
[m/s
]
δx1 δx12
δρ12
Figure 6: Range-rate and velocity accuracy requirements for
potential difference
determination according to (31).
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δ∆V12 [m2/s2]
[m]
δx1
δx12
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 satellite-to-
satellite tracking 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.
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APPENDIX
From classical mechanics (Goldstein, 1950), Lagrange’s equation
for the motion of a particle is
given by
ddt
∂T∂qi
– ∂T∂qi= Qi (A.1)
where {qi,qi} are generalized coordinates, T is the kinetic
energy of the particle, and Qi is a
component of the generalized force:
Qi = F j ⋅
∂x j∂qi
Σj
(A.2)
F j being the jth 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
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-
respect to space. Under these premises, the kinetic energy is
given by
T = 12 x2
(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,
F , 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 + F (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.
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):
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-
ddt
x = F (A.8)
where x 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,x,t) , and H is the Hamiltonian of the
motion only if F = 0 .
We have
dHdt
= ∂H∂xkdxkdtΣk +
∂H∂xk
dxkdtΣk +
∂H∂t (A.10)
Noting the dependencies of T and V on xk , xk , and t, this
simplifies to
dHdt
= – ∂V∂xkxkΣk +
dTdxk
dxkdtΣk –
∂V∂t (A.11)
From (A.6) and (A.8),
∂V∂xk
=dxkdt
– Fk (A.12)
and from (A.5), dT dxkdT dxk = xk . Substituting these into
(A.11) yields
dHdt
= Fk xkΣk –∂V∂t (A.13)
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-
Integrating both sides and using (A.9), we obtain:
T – V = Fk xk dt
t0
t
Σk
– ∂V∂t dtt0
t
+ E0 (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 non-gravitational
forces are absent ( F = 0 ), then (A.14)
expresses the energy conservation law.
Acknowledgments: The author is grateful to the reviewers for
their valuable comments. Thiswork was supported by a grant from the
University of Texas, Austin, Contract No. UTA98-0223,under a
primary contract with NASA.
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