THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES FROM SATELLITE-TO-SATELLITE TRACKING

Transcription

1 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., (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 (±1m 2 /s 2 ) 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 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: , c 2000 Kluwer Academic Publishers. Printed in the Netherlands.

2 86 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 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.

3 THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES 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 = (ẋ 1, ẋ 2, ẋ 3 ), and specific forces acting on the satellite, F = ( F 1, F 2, F 3 ), is given by (A.14) with (A.5) substituted: V = 1 2 ẋ 2 k t t 0 F k ẋ k dt + t t 0 V dt E 0. (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 E 0 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 = V rotating earth +V lunar tide + V solar tide + V planetary tides + +V solid earth tide + V ocean tide + V atmospheric tide + (2) +V ocean loading + V atmospheric loading + V other 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, Ȳ n,m : V rotating earth = V e (r,θ,λ)= km e R n n=0 m= n ( ) R n+1 C n,m r Ȳn,m (θ, λ), (3) where r is geocentric radius, θ is co-latitude, and λ is longitude with respect to a defined zero-meridian, km e is the gravitational constant times earth s total mass (including atmosphere), R is a mean earth radius, C n,m are coefficients that define earth s mass density distribution, and { cos mλ, m 0, Ȳ n,m (θ, λ) = P n, m (cos θ) sin m λ, m < 0, (4) where P n,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, C n,m, are assumed constant since any temporal redistribution of mass is accounted for by the other potential components in (2).

4 88 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 + ζ P + ζ N + θ S, (5) λ = α + α P + α N ω e t + λ S, (6) where the coordinates (ζ, α) are the co-declination and right ascension in the inertial frame of epoch J The terms θ S and λ S rotate the terrestrial pole of date to the celestial pole of date using coordinates of polar motion, ω e 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, ζ N and α N account for the nutations of the celestial pole and transform it from its true to its mean direction of date, and ζ P and ζ P describe the precession of the pole from its mean direction of date to its mean direction at a defined epoch, currently J 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 V e asafunctionof(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 V e = V e θ θ + V e λ λ, (7) where θ/ and λ/ 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 ω e. In fact, the precession rates in right ascension and in declination are less than 50 arcsec per year, or less than rad/s. Similarly the nutation rates in longitude and ecliptic obliquity are less than rad/s, and polar motion rates are less than rad/s for the main Chandler wobble. These rates are seven to eight orders or magnitude smaller than ω e = rad/s, and we may approximate V e = ω e V e α, (8) where, because of the linear relationship (6), / λ = / α. Furthermore, we assume to a similar level of approximation that ω e 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

5 THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES 89 deformation of the quasi-elastic earth, is given approximately by (Torge, 1991; Lambeck, 1988) V B (r,θ,α)= 3 ( ) ( km B r 2 ( ) ) R k 2. 4 r B r B r [ ( sin 2θ sin 2θ B cos(α α B ) + sin 2 θ sin 2 θ B cos 2(α α B ) + 3 cos 2 θ 1 ) 3 ( cos 2 θ B 1 )], (9) 3 where km B is the gravitational constant times the mass of the body, (r B,θ B,α B )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/r B 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 (r B,θ B,α B )are all functions explicitly of time due to the motion of the body with respect to the earth. However, the largest rate is in α B 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 r B,then V B V B α α B + V B θ B θ B, (10) again, because / α B = / α. Ifn B denotes the mean angular motion of the body, then the rate in co-declination varies between zero and ± sin(i)n B, 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, n M = rad/s. The sidereal year is about 365 days long, implying that the sun s mean motion is n S = rad/s. The corresponding rates in right ascension, α M and α S, 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 km B r 2 { 3.0m 2 /s 2, moon, = 4 r B r B 1.4m 2 /s 2 (11), sun. These potentials are smaller than earth s gravitational potential by seven orders of magnitude. Since the corresponding gradients compare similarly, we have ( ) ( ) VB O < 10 8 Ve O (12)

6 90 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 V e ω e α. (13) Now, since x 1 = r cos θ cos α and x 2 = r cos θ sin α, it is readily shown that ( ) V V e V e = ω e x 1 x 2. (14) x 2 x 1 Substituting (A.12) we then have ( ( V = ω e x 1 F 2 + δv ẍ 2 x 2 ) x 2 ( F 1 + δv )) ẍ 1, (15) x 1 where, from (2), V = V e + 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 ω e (x 2 ẍ 1 x 1 ẍ 2 ), (16) which yields t V dt = ω e (x 1 ẋ 2 x 2 ẋ 1 ) (17) t 0 (the constant of integration is relegated to E 0 ). As an aside, (17) can also be written as t t 0 V dt = ω e α(x1 2 + 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, ωe 2(x2 1 + 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): V = 1 2 ẋ 2 k t t 0 F k ẋ k dt ω e (x 1 ẋ 2 x 2 ẋ 2 ) E 0. (19)

7 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, ρ 12, between two satellites following each in approximately the same orbit. We have ρ 12 = e T 12 x 12, wherex 12 = x 2 x 1,ande 12 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 ρ 12 = e T 12ẋ12 (20) since ė T 12 e 12 = 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 (ω e = 0), the energy Equation (19) reduces to V = 1 2 ẋ 2 E 0. (21) Taking the along-track derivative, denoted by d a, on both sides yields d a V = ẋ T d a ẋ. (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 V 2 V 1 V 12 ẋ 1 ρ 12. (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 V 12 t = ẋ T 1 ẋ ẋ 12 2 ( F 2k ẋ 2k F 1k ẋ 1k ) dt k t 0 ω e (x 121 ẋ 22 x 22 ẋ 121 x 11 ẋ x 122 x 11 ) E 012, (24)

8 92 CHRISTOPHER JEKELI Figure 1. The geometry of residual quantities referred to a reference orbit. where the first two terms derive from ẋ 2 2 ẋ 1 2 =(ẋ 2 ẋ 1 ) T (ẋ 2 +ẋ 1 ),ande 012 is a constant. Omitting the dissipative term, we write V 12 = ẋ T 1ẋ ẋ VR 12 E 012 (25) with VR 12 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). The residual quantities are, for example, V 12 = V 12 V12 0, ẋ 1 = ẋ 1 ẋ 0 1, and ρ 12 = ρ 12 ρ 12 0, where the reference potential (sans dissipative energy term) is given analogous to (25) by V 0 12 = (ẋ0 1 )T ẋ ẋ VR 0 12 E0 0 12, (26) and the residual potential difference is V 12 = ẋ T 1 ẋ12 (ẋ 0 1 )T ẋ (ẋ0 12 )T ẋ ẋt 12 ẋ 12 + V R 12 E 012. (27) Corresponding to the approximation (23), we define the approximate residual model, designated with the symbol as ˆV 12 = ẋ 0 1 ρ 12. (28)

9 THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES 93 The error in this model relative to the true model (27) is given by ɛ V 12 = ˆV 12 V 12 = (ẋ 0 2 ẋ0 1 e 12) T ẋ 12 + ( ẋ 1 ẋ 0 1 e 12) T ẋ (ẋ 1 ) T ẋ ẋ 12 2 V R 12 + E 012 = v 1 + v 2 + v 3 + v 4 V R 12 + E 012, (29) which is readily derived using ρ 12 = ρ 12 ρ12 0 = et 12ẋ12 (e 0 12 )T ẋ 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 ẋ 0 1 =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 m 2 /s 2 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 (n max = 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 180 n=0 n m= n ( ) R n+1 C nmŷnm(θ, λ). (30) r 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 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 m 2 /s 2. 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

10 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 (ω e = 0, for this case, only). In this case, the terms, v 1 and v 2, 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 ω e = 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 v 1 and v 2 for the case depicted in Figure 3.

11 THE DETERMINATION OF GRAVITATIONAL POTENTIAL DIFFERENCES 95 Figure 5. Model error terms v 3, v 4, V R 12 for the case depicted in Figure 3. caused by the second term, v 2, 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 v 3 has the same order of magnitude as the error due to range-rate measurement error (10 6 m/s), andv 4 is practically negligible, but the potential rotation term, V R 12, on the order of ±1m 2 /s 2, 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 V 12 = ẋ 0 1 ρ 12 v 1 v 2 v 3 v 4 + V R 12 E 012. (31) It requires also measurements of velocity vectors and their intersatellite differences. The constant, E 012, 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

12 96 CHRISTOPHER JEKELI Figure 6. Range-rate and velocity accuracy requirements for potential difference determination according to (31). potential difference, δ V 12, and accuracies in range-rate (δ ρ 12 ), absolute (δẋ 1 ), and intersatellite (δẋ 12 ) velocity measurements. The principal term affected by errors in absolute velocity is v 2, while the velocity difference error affects mostly the potential rotation term. Computation of these two terms also requires accurate absolute (for V R 12 ) and relative (for v 2 ) 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 m 2 /s 2 (corresponding to an accuracy of 1 cm in geoid differences) requires accuracies in range-rate, velocity, and position as follows: δ ρ 12 = m/s, δẋ 1 = m/s, δẋ 12 = m/s δx 1 = 7m, δx 12 = m. (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.

13 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 m 2 /s 2 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 = Q i (A.1) dt q i q i

14 98 CHRISTOPHER JEKELI where {q i, q i } are generalized coordinates, T is the kinetic energy of the particle, and Q i is a component of the generalized force Q i = j F j x j q i, (A.2) F j being the j th force acting on the particle and expressed in inertial Cartesian coordinates {x k }=x=x(q i ), 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 x k = q k. (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 = 1 2 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, 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.

15 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, dt (A.8) 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),andH is the Hamiltonian of the motion only if F = 0. We have dh dt = k H dx k x k dt + k H dẋ k ẋ k dt + H. (A.10) Noting the dependencies of T and V on ẋ k, x k,andt, this simplifies to dh = V ẋ k + dt x k k k From (A.6) and (A.8), dv = dẋ k x k dt F k, dt dẋ k dẋ k dt V. andfrom(a.5),dt/dẋ k =ẋ k. Substituting these into (A.11) yields dh = F k ẋ k V dt. k Integrating both sides and using (A.9), we obtain T V = t t V F k ẋ k dt dt + E 0, k 0 t 0 (A.11) (A.12) (A.13) (A.14) where E 0 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 ( F = 0), then (A.14) expresses the energy conservation law.

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