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: jekeli.1@osu.edu

revision submitted to

Celestial Mechanics and Dynamical Astronomy

11 October 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.

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

- 1 -

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

- 2 -

(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:

- 3 -

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

- 4 -

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)

- 5 -

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)

- 6 -

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

- 7 -

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

- 8 -

∂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):

- 9 -

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

- 10 -

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)

- 11 -

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

- 12 -

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

- 13 -

(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

- 14 -

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)

- 15 -

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.

- 16 -

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,

- 17 -

∆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

- 18 -

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).

- 19 -

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.

- 20 -

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

- 21 -

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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