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Impact of far-side satellite tracking on gravity estimation in the SELENE project
Adv. Space Res. Vol. 23, No. 11, pp. 1809-1812,1999 Q 1999 COSPAR. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0273-I 177/99 $20.00 + 0.00
Pergamon www.elsevier.nl/locate/asr
PII: SO273-1177(99)00542-6
IMPACT OF FAR-SIDE SATELLITE TRACKING ON GRAVITY ESTIMATION IN THE SELENE PROJECT K. Matsumoto’,
K. Hekil, and D. D. Rowlands
‘Div. Earth Rotation, National Astronomical 0861, Japan 2 NASA/Goddard Space Flight Center(GSFC),
Observatory(NAO), Greenbelt,
2-12 Hoshigaoka,
Mizusawa,
Iwate, O23-
n/r0 20771, lJSA
ABSTRACT Preliminary results of numerical simulation are presented to examine the gravity estimation capability in the Japanese lunar exploration project SELENE (SELenological and ENgineering Explorer), which will be launched in 2003. One of the new characteristics of the SELENE lunar gravimetry is 4-way satellite-tosatellite Doppler tracking of a low-altitude lunar orbiter by means of a high-altitude relay satellite. It is shown that planned satellites configuration will provide a good far-side data coverage of the lunar orbiter and will improve lunar gravity field as well as far-side selenoid. 0 1999 COSPAR. Published by Elsevier Science Ltd. INTRODUCTION Many gravity potential models of the Moon have been developed mainly from 2-way Doppler tracking data of spacecrafts orbiting the Moon: in which the lunar gravity potential is usually modeled in terms of spherical harmonics as follows, V(gi,X,r)
= y
5 (F)n n=O
2 (C,,cosmX+ m=O
5’,,sinmX)
P,,(sin4)
(1)
where the expansion is given in spherical coordinates with latitude 4, longitude X, radius r; G is the universal constant of gravitation; M is the lunar mass; R is some reference radius; c,, and s,,,, are the normalized selenopotential coefficients to be determined, and p,, are the normalized associated Legendre functions of degree n and order m.. The modern lunar gravity models such as LUNGOD (Konopliv et al., 1993), GLGM-2 (Lemoine et al., 1997) and LP75G (Konopliv et al., 1998) g’lve coefficients to high degree and order up to 60, 70 and 75, respectively. In determining the gravity coefficients in a. least-squares sense, however, a kind of constraint taken from Kaula’s (1966) rule of thumb has been imposed in order to avoid numerical instabilities stemming from spatial data coverage limited to almost near-side. The Kaula-type signal constraint acts as a gravity field smoother and results in good data fit over near-side, but the far-side gravity field is almost meaningless as is clearly shown by Floberghagen (this issue). In the Japanese lunar exploration project SELENE (SELenological and ENgineering Explorer) to be launched in 2003 (Namiki et al., this issue), far-side data coverage will be greatly improved by means of high-low 4-way satellite-to-satellite Doppler tracking. In this a.rticle we show that the present situation of the lunar gravimetry may drastically change by the data coverage improvement when the SELENE mission is realized. SELENE
GRAVIMETRY
One of the two satellit,es consisting the SELENE is a 1una.r orbit,er (hereinafter referred to as LORB) and the other is a relay sub-satellite (hereinafter referred to as RSAT). LORB is a low-altitude (100 km above 1809
1810
K. Matsumotoet al.
10-10
1” 0
5 10 15 20 25 30 35 40 45 50 55 60 Degree
Figure 1. Data coverage of LORB based on 3-month simulation. The map is centered on 270’ E longitude, with the far-side of the Moon to the left of the center, and the near-side to the right.
Figure 2. Anticipated coefficient sigma degree variances; triangles, from LORB solution with near-side data only; squares, from LORB plus RSAT solution with near-side data only; pluses, from LORB only solution with far-side data included; circles, from LORB plus RSAT solution with far-side data included; solid line, a Kaula-type constraint applied to the former two solutions.
the Moon) satellite in a circular orbit with inclination of 95 ’ (or 90”; not determined in Oct. 1998). RSAT is a high-altitude (perilune height of 100 km and apolune height of 2400 km) spin-stabilized satellite in an eccentric orbit with the same inclination as LORB. RSAT is equipped with a communication instrument which relays Doppler signal to/from LORB, which realizes direct tracking of LORB in the far-side. The orbit of LORB will be perturbed by attitude control maneuvers every 18 hours, while no artificial disturbance will be made to RSAT orbit. Since gravity acceleration is inversely proportional to T~+~, LORB at 100 km altitude is more sensitive to high-degree gravity fields than RSAT. RSAT is, on the other hand, more appropriate than LORB to determine low-degree gravity fields because of attenuation of high-degree gravity signal on its relatively high mean altitude. Unperturbed long arc-length of RSAT is also preferable to determine long-wavelength gravity potentials. Consequently, the combination of gravity solutions from the two satellites will result in a fine lunar gravity field model over a wide range of wavelengths. In addition to the deployment of the two satellites and 4-way Doppler measurement, the SELENE gravimetry also includes three dimensional tracking of LORB (Heki et al., this issue) and that of RSAT (Namiki et al., this issue) by differential Very-Long-Baseline-Interferometry. SIMULATION
SETTING
AND RESULTS
GEODYN II (Pavlis et al., 1997) and SOLVE (Ullman, 1994) programs are used for the simulation and the error analysis. GEODYN II is now being modified so that it can handle interplanetary 4-way Doppler measurement but it has not been completed yet. In the present analysis, however, we make use of the simulated 2-way Doppler measurements of LORB in the far-side as if LORB was virtually observed through the Moon by omitting occultation checking part in the GEODYN II code. The far-side Doppler data are, of course, retained only if the satellites geometry allows the a-way communication link. The,following results are thus not for the actual 4-way Doppler situation, because sensitive direction in a 4-way case is different from the line-of-sight direction in a 2-way case. It will give us, however, an insight into relation between the data coverage and the accuracy of gravity field recovery. The present preliminary simulation was done for 3-month observation period, though the planned mission length is 1.2 years. The data interval is assumed to be 30 seconds and the Doppler observation accuracy is presumed to be 0.2 mm/set to be acquired at UDSC (Usuda Deep Space Center). Shown in Figure 1 is the date coverage of the LORB based on the orbit configuration and the analysis conditions described above. The Doppler data are simulated by using a priori lunar gravity field model LUNGOD (Konopliv et al., 1993), from which the lunar gravity filed is estimated up to degree and order 60. The arc length is set to 1 day for LORB and 90 days for RSAT.
1811
Far-SideSatelJiteTracking In order to see error spectrum 0, which is defined as
of the recovered
a, = [(2n + 1)-l
gravity
field we calculate
2 {,2(&J 7IZ=Q
coefficient
sigma degree variance
+ 02(s.,))]1’2
(2)
are the error of the gravity potential coefficients. Plotted in Figure 2 is the where a(C,,) and a(&,) coefficient sigma degree variances for the following 4 cases: A. The LORB solution with conventional near-side 2-way Doppler data only (triangles), B. The combined solution of LORB plus RSAT with conventional near-side 2-way Doppler data only (squares), C. The LORB solution with far-side Doppler data included (pluses), D. The combined solution of LORB plus RSAT with far-si_de Doppler data included (circles). For the cases A and B a Kaula-type signal constraint of {C,,,S,,} N 0 f 15 x 10m5/n2 is applied. In the cases A and B, the gravity coefficients higher than degree lo-15 are strongly affected by the constraint. RSAT contribution is significant for low-degree fields (n < lo), as expected in the section SELENE GRAVIMETRY. In the case C, on the other hand, gravity coefficients of degrees 5 5 R. 5 50 are greatly improved without any constraint in comparison with the cases A and B. This suggests the importance of far-side data coverage on spherical harmonic approach to lunar gravity field recovery. RSAT, in the case D, still shows significant contribution in determining very low degree (n 5 4) gravity coefficients. Since the accuracy of the lunar moments of inertia is currently dependent on the errors in 52 and C22 (Williams et al., 1996), RSAT contribution will be of particular importance to constrain the radius or density of the lunar core which can be deduced from the lunar moment of inertia. Selenoid height error are calculated by propagating a full error variance-covariance matrix of the gravity coefficients following Haagmans and Gelderen (1991). Figure 3 shows the selenoid height error which is anticipated for the case A and Figure 4 is the same but for the case C. Large selenoid error exceeding 30 m in Figure 3 indicates that even if a Kaula-type signal constraint is applied to the solution, the lunar far-side gravity filed is poorly determined from conventional near-side 2-way Doppler observation only. By adding the far-side Doppler data, however, the selenoid height errors in the far-side are significantly reduced down to below 10 m (Figure 4). SUMMARY We have presented Preliminary simulation results with regard to the lunar gravity estimation experiment in the coming SELENE project. It is shown that sufficient data coverage over the lunar far-side will be realized
0
5
Selenoidheighterror : LORB
Selenoid height error : LOREI
Near-side data only
Near-side + far-side data
IO
15
20
25
30
35
(meters)
Figure 3. Anticipated selenoid height error for LORB solution with conventional near-side 2-way Doppler data only. Unit is in meters. A Kaula-type constraint is applied to the solution in order to avoid numerical instability.
15
20
(meters)
Figure 4. Same as Figure 3, but for LORB solution with far-side Doppler data included. No constraint is applied to the solution.
1812
K. Matsumoto et 01.
by high-low satellite-to-satellite 4-way Doppler measurement with planned satellites configuration. The farside data result in significant improvement of lunar gravity field over a wide range of wavelengths as well as improvement of the far-side selenoid. LORB and RSAT contribution to the gravity field solution are complementary and RSAT is expected to contribute to improve low degree gravity coefficients such as Jz and Czz. Although the results shown here might be too optimistic because simulated situation is idealized one (e.g., solar radiation pressure and effect of spin of RSAT are not taken into account, and actual 4-way Doppler measurement is not simulated), a new high-precision lunar gravity model will come with more isotropic tracking data coverage provided by the SELENE project. ACKNOWLEDGMENTS GEODYN II/SOLVE programs were introduced to NAO under a joint research contract with NASA/GSFC and we thank Ben Chao for his continuous help. The authors also would like to thank Rune Floberghagen of Delft Institute for Earth-Oriented Space Research for providing the error propagation software which was used to calculate selenoid height error. REFERENCES Floberghagen, R., J. Bouman, R. Koop, and P. Visser, Regularisation of lunar gravity field solutions, Adv. Space Res., this issue. Haagmans, R.H.N. and M. Gelderen, Error variances-covariances of GEM-Tl: Their characteristics and implications in geoid computation, J. Geophys. Res., 96, 20011-20022 (1991). Heki, K., K. Matsumoto, and R. Floberghagen, Three-dimensional tracking of a lunar satellite with differential very-long-baseline-interferometry, Adu. Space Res., this issue. Kaula, W.M., Theory of satellite geodesy, Blaisdell Pub. Co., Waltham, Mass. (1966). Konopliv, A.S., W.L. Sjogren, R.N. Wimberly, R.A. Cook, and A. Vijayaraghavan, A high-resolution lunar gravity field and predicted orbit behavior, AAS paper 93-622 in Proc. of the AAS/AIAA Astrodynamics Specialist Conf., August 16-19, Victoria B.C., Canada (1993). Konopliv, A.S., A.B. Binder, L.L. Hood, A.B. Kucinskas, W.L. Sjogren, and J.G. Williams, Improved gravity field of the Moon from Lunar Prospector, Science, 281, 1476-1480 (1998). Lemoine, F.G., D.E. Smith, M.T. Zuber, G.A. Neumann, and D.D. Rowlands, A 70th degree lunar gravity model (GLGM-2) from Clementine and other tracking data, J. Geophys. Res., 102, 16339-16359 (1997). Namiki, N., H. Hanada, T. Tsubokawa, N. Kawano, M. Ooe et al., Selenodetic experiments of SELENE: Relay subsatellite, differential VLBI, and laser altimeter, Adw. Space Res., this issue. Pavlis, D.E., D. Moore, S. Luo, J.J. McCarthy, and S.B. Luthcke, GEODYN Operations Manual, 5 Volumes, Hughes/STX, prepared for NASA Goddard Space Flight Center, Greenbelt, Maryland (1997). Ullman, R.E., SOLVE Program: Mathematical formulation and guide to user input, Hughes/STX contractor report, contract NAS5-31760, NASA Goddard Space Flight Center, Greenbelt, Maryland (1994). Williams, J.G., X X Newhall and J.O. Dickey, Lunar moments, tides, orientation, and coordinate frames, Planet. Space Ski., 44, 1077-1080 (1996).
Pergamon www.elsevier.nl/locate/asr
PII: SO273-1177(99)00542-6
IMPACT OF FAR-SIDE SATELLITE TRACKING ON GRAVITY ESTIMATION IN THE SELENE PROJECT K. Matsumoto’,
K. Hekil, and D. D. Rowlands
‘Div. Earth Rotation, National Astronomical 0861, Japan 2 NASA/Goddard Space Flight Center(GSFC),
Observatory(NAO), Greenbelt,
2-12 Hoshigaoka,
Mizusawa,
Iwate, O23-
n/r0 20771, lJSA
ABSTRACT Preliminary results of numerical simulation are presented to examine the gravity estimation capability in the Japanese lunar exploration project SELENE (SELenological and ENgineering Explorer), which will be launched in 2003. One of the new characteristics of the SELENE lunar gravimetry is 4-way satellite-tosatellite Doppler tracking of a low-altitude lunar orbiter by means of a high-altitude relay satellite. It is shown that planned satellites configuration will provide a good far-side data coverage of the lunar orbiter and will improve lunar gravity field as well as far-side selenoid. 0 1999 COSPAR. Published by Elsevier Science Ltd. INTRODUCTION Many gravity potential models of the Moon have been developed mainly from 2-way Doppler tracking data of spacecrafts orbiting the Moon: in which the lunar gravity potential is usually modeled in terms of spherical harmonics as follows, V(gi,X,r)
= y
5 (F)n n=O
2 (C,,cosmX+ m=O
5’,,sinmX)
P,,(sin4)
(1)
where the expansion is given in spherical coordinates with latitude 4, longitude X, radius r; G is the universal constant of gravitation; M is the lunar mass; R is some reference radius; c,, and s,,,, are the normalized selenopotential coefficients to be determined, and p,, are the normalized associated Legendre functions of degree n and order m.. The modern lunar gravity models such as LUNGOD (Konopliv et al., 1993), GLGM-2 (Lemoine et al., 1997) and LP75G (Konopliv et al., 1998) g’lve coefficients to high degree and order up to 60, 70 and 75, respectively. In determining the gravity coefficients in a. least-squares sense, however, a kind of constraint taken from Kaula’s (1966) rule of thumb has been imposed in order to avoid numerical instabilities stemming from spatial data coverage limited to almost near-side. The Kaula-type signal constraint acts as a gravity field smoother and results in good data fit over near-side, but the far-side gravity field is almost meaningless as is clearly shown by Floberghagen (this issue). In the Japanese lunar exploration project SELENE (SELenological and ENgineering Explorer) to be launched in 2003 (Namiki et al., this issue), far-side data coverage will be greatly improved by means of high-low 4-way satellite-to-satellite Doppler tracking. In this a.rticle we show that the present situation of the lunar gravimetry may drastically change by the data coverage improvement when the SELENE mission is realized. SELENE
GRAVIMETRY
One of the two satellit,es consisting the SELENE is a 1una.r orbit,er (hereinafter referred to as LORB) and the other is a relay sub-satellite (hereinafter referred to as RSAT). LORB is a low-altitude (100 km above 1809
1810
K. Matsumotoet al.
10-10
1” 0
5 10 15 20 25 30 35 40 45 50 55 60 Degree
Figure 1. Data coverage of LORB based on 3-month simulation. The map is centered on 270’ E longitude, with the far-side of the Moon to the left of the center, and the near-side to the right.
Figure 2. Anticipated coefficient sigma degree variances; triangles, from LORB solution with near-side data only; squares, from LORB plus RSAT solution with near-side data only; pluses, from LORB only solution with far-side data included; circles, from LORB plus RSAT solution with far-side data included; solid line, a Kaula-type constraint applied to the former two solutions.
the Moon) satellite in a circular orbit with inclination of 95 ’ (or 90”; not determined in Oct. 1998). RSAT is a high-altitude (perilune height of 100 km and apolune height of 2400 km) spin-stabilized satellite in an eccentric orbit with the same inclination as LORB. RSAT is equipped with a communication instrument which relays Doppler signal to/from LORB, which realizes direct tracking of LORB in the far-side. The orbit of LORB will be perturbed by attitude control maneuvers every 18 hours, while no artificial disturbance will be made to RSAT orbit. Since gravity acceleration is inversely proportional to T~+~, LORB at 100 km altitude is more sensitive to high-degree gravity fields than RSAT. RSAT is, on the other hand, more appropriate than LORB to determine low-degree gravity fields because of attenuation of high-degree gravity signal on its relatively high mean altitude. Unperturbed long arc-length of RSAT is also preferable to determine long-wavelength gravity potentials. Consequently, the combination of gravity solutions from the two satellites will result in a fine lunar gravity field model over a wide range of wavelengths. In addition to the deployment of the two satellites and 4-way Doppler measurement, the SELENE gravimetry also includes three dimensional tracking of LORB (Heki et al., this issue) and that of RSAT (Namiki et al., this issue) by differential Very-Long-Baseline-Interferometry. SIMULATION
SETTING
AND RESULTS
GEODYN II (Pavlis et al., 1997) and SOLVE (Ullman, 1994) programs are used for the simulation and the error analysis. GEODYN II is now being modified so that it can handle interplanetary 4-way Doppler measurement but it has not been completed yet. In the present analysis, however, we make use of the simulated 2-way Doppler measurements of LORB in the far-side as if LORB was virtually observed through the Moon by omitting occultation checking part in the GEODYN II code. The far-side Doppler data are, of course, retained only if the satellites geometry allows the a-way communication link. The,following results are thus not for the actual 4-way Doppler situation, because sensitive direction in a 4-way case is different from the line-of-sight direction in a 2-way case. It will give us, however, an insight into relation between the data coverage and the accuracy of gravity field recovery. The present preliminary simulation was done for 3-month observation period, though the planned mission length is 1.2 years. The data interval is assumed to be 30 seconds and the Doppler observation accuracy is presumed to be 0.2 mm/set to be acquired at UDSC (Usuda Deep Space Center). Shown in Figure 1 is the date coverage of the LORB based on the orbit configuration and the analysis conditions described above. The Doppler data are simulated by using a priori lunar gravity field model LUNGOD (Konopliv et al., 1993), from which the lunar gravity filed is estimated up to degree and order 60. The arc length is set to 1 day for LORB and 90 days for RSAT.
1811
Far-SideSatelJiteTracking In order to see error spectrum 0, which is defined as
of the recovered
a, = [(2n + 1)-l
gravity
field we calculate
2 {,2(&J 7IZ=Q
coefficient
sigma degree variance
+ 02(s.,))]1’2
(2)
are the error of the gravity potential coefficients. Plotted in Figure 2 is the where a(C,,) and a(&,) coefficient sigma degree variances for the following 4 cases: A. The LORB solution with conventional near-side 2-way Doppler data only (triangles), B. The combined solution of LORB plus RSAT with conventional near-side 2-way Doppler data only (squares), C. The LORB solution with far-side Doppler data included (pluses), D. The combined solution of LORB plus RSAT with far-si_de Doppler data included (circles). For the cases A and B a Kaula-type signal constraint of {C,,,S,,} N 0 f 15 x 10m5/n2 is applied. In the cases A and B, the gravity coefficients higher than degree lo-15 are strongly affected by the constraint. RSAT contribution is significant for low-degree fields (n < lo), as expected in the section SELENE GRAVIMETRY. In the case C, on the other hand, gravity coefficients of degrees 5 5 R. 5 50 are greatly improved without any constraint in comparison with the cases A and B. This suggests the importance of far-side data coverage on spherical harmonic approach to lunar gravity field recovery. RSAT, in the case D, still shows significant contribution in determining very low degree (n 5 4) gravity coefficients. Since the accuracy of the lunar moments of inertia is currently dependent on the errors in 52 and C22 (Williams et al., 1996), RSAT contribution will be of particular importance to constrain the radius or density of the lunar core which can be deduced from the lunar moment of inertia. Selenoid height error are calculated by propagating a full error variance-covariance matrix of the gravity coefficients following Haagmans and Gelderen (1991). Figure 3 shows the selenoid height error which is anticipated for the case A and Figure 4 is the same but for the case C. Large selenoid error exceeding 30 m in Figure 3 indicates that even if a Kaula-type signal constraint is applied to the solution, the lunar far-side gravity filed is poorly determined from conventional near-side 2-way Doppler observation only. By adding the far-side Doppler data, however, the selenoid height errors in the far-side are significantly reduced down to below 10 m (Figure 4). SUMMARY We have presented Preliminary simulation results with regard to the lunar gravity estimation experiment in the coming SELENE project. It is shown that sufficient data coverage over the lunar far-side will be realized
0
5
Selenoidheighterror : LORB
Selenoid height error : LOREI
Near-side data only
Near-side + far-side data
IO
15
20
25
30
35
(meters)
Figure 3. Anticipated selenoid height error for LORB solution with conventional near-side 2-way Doppler data only. Unit is in meters. A Kaula-type constraint is applied to the solution in order to avoid numerical instability.
15
20
(meters)
Figure 4. Same as Figure 3, but for LORB solution with far-side Doppler data included. No constraint is applied to the solution.
1812
K. Matsumoto et 01.
by high-low satellite-to-satellite 4-way Doppler measurement with planned satellites configuration. The farside data result in significant improvement of lunar gravity field over a wide range of wavelengths as well as improvement of the far-side selenoid. LORB and RSAT contribution to the gravity field solution are complementary and RSAT is expected to contribute to improve low degree gravity coefficients such as Jz and Czz. Although the results shown here might be too optimistic because simulated situation is idealized one (e.g., solar radiation pressure and effect of spin of RSAT are not taken into account, and actual 4-way Doppler measurement is not simulated), a new high-precision lunar gravity model will come with more isotropic tracking data coverage provided by the SELENE project. ACKNOWLEDGMENTS GEODYN II/SOLVE programs were introduced to NAO under a joint research contract with NASA/GSFC and we thank Ben Chao for his continuous help. The authors also would like to thank Rune Floberghagen of Delft Institute for Earth-Oriented Space Research for providing the error propagation software which was used to calculate selenoid height error. REFERENCES Floberghagen, R., J. Bouman, R. Koop, and P. Visser, Regularisation of lunar gravity field solutions, Adv. Space Res., this issue. Haagmans, R.H.N. and M. Gelderen, Error variances-covariances of GEM-Tl: Their characteristics and implications in geoid computation, J. Geophys. Res., 96, 20011-20022 (1991). Heki, K., K. Matsumoto, and R. Floberghagen, Three-dimensional tracking of a lunar satellite with differential very-long-baseline-interferometry, Adu. Space Res., this issue. Kaula, W.M., Theory of satellite geodesy, Blaisdell Pub. Co., Waltham, Mass. (1966). Konopliv, A.S., W.L. Sjogren, R.N. Wimberly, R.A. Cook, and A. Vijayaraghavan, A high-resolution lunar gravity field and predicted orbit behavior, AAS paper 93-622 in Proc. of the AAS/AIAA Astrodynamics Specialist Conf., August 16-19, Victoria B.C., Canada (1993). Konopliv, A.S., A.B. Binder, L.L. Hood, A.B. Kucinskas, W.L. Sjogren, and J.G. Williams, Improved gravity field of the Moon from Lunar Prospector, Science, 281, 1476-1480 (1998). Lemoine, F.G., D.E. Smith, M.T. Zuber, G.A. Neumann, and D.D. Rowlands, A 70th degree lunar gravity model (GLGM-2) from Clementine and other tracking data, J. Geophys. Res., 102, 16339-16359 (1997). Namiki, N., H. Hanada, T. Tsubokawa, N. Kawano, M. Ooe et al., Selenodetic experiments of SELENE: Relay subsatellite, differential VLBI, and laser altimeter, Adw. Space Res., this issue. Pavlis, D.E., D. Moore, S. Luo, J.J. McCarthy, and S.B. Luthcke, GEODYN Operations Manual, 5 Volumes, Hughes/STX, prepared for NASA Goddard Space Flight Center, Greenbelt, Maryland (1997). Ullman, R.E., SOLVE Program: Mathematical formulation and guide to user input, Hughes/STX contractor report, contract NAS5-31760, NASA Goddard Space Flight Center, Greenbelt, Maryland (1994). Williams, J.G., X X Newhall and J.O. Dickey, Lunar moments, tides, orientation, and coordinate frames, Planet. Space Ski., 44, 1077-1080 (1996).