最新の「かぐや」月重力場モデルについて

松本晃治(1)、SanderGoossens (1) 、石原吉明 (1) 、劉 慶会 (2) 、菊池冬彦(1) 、岩田隆浩 (2) 、並木 則行 (3) 、野田寛大 (1) 、花田英夫 (1) 、RSAT/VRADミッションチーム

(1) 国立天文台 RISE月探査プロジェクト(2) 上海天文台(3) JAXA/ISAS 固体惑星科学研究系(4) 千葉工業大学惑星探査研究センター

SELENE gravimetry
RSAT Mission
4-way Doppler for far-side coverage
Kaguya above
the far side

Rstar

VRAD Mission
Differential VLBI
Rstar

Vstar

4-way DOPP
2-way
RARR+
4-way
DOPP

Earth
JAXA Usuda (UDSC)

differential
VLBI

EarthVLBI stations (Japan, China, Australia, Germany)Switching and Same beam VLBI Observation
Vstar

Rstar

Vstar
Separation> Beam width

Switchinginterval is 2 min.

Rstar
Separation< Beam width

Beam width
0.1 deg. (X)
0.37 deg. (S)Same beam
VLBI
Switching
VLBI
Mostof atomosphericand ionosphericfluctuation,clock offset, and instrumental delay can be canceled out.
Additionally, fluctuations whose periods are shorter thanswitching interval can be canceled out in same beam VLBI.

Status of the SELENE satellites• 2007.10.31 Start acquiring 4-way Doppler data•

2008.07.23 One of four reaction wheels of Main satellite failed (AMD interval12h→6h)• 2008.10.31 End of nominal mission•2008.12.26 Another wheel went out of order→ thruster control mode

• 2009.01.30 The last 4-way Doppler data acquisition with3 reaction wheels turned on• 2009.02.12 Rstar (relay sub-satellite) crashed into theMoon• 2009.06.10 Controlled crash of the Main satellite• 2009.06.29 The last tracking of Vstar (VLBI sub-satellite)

VRADデータ解析進捗状況
観測期間2007年11月〜2009年6月（全20ヵ月） 一月当たり約100時間◎Rstar&Vstar same beam VLBI観測データ (2009年2月まで、全16ヵ月間)
相関処理（１次処理）：完了遅延推定（２次処理）：完了 → 7ヵ月分を【SGM100h_VLBI】に使用◎Rstar&Vstar switching VLBI観測データ (2009年2月まで、全16ヵ月間)
相関処理（１次処理）：完了遅延推定（２次処理）：25％（4ヵ月分）完了◎Rstar&QSO or Vstar&QSO 相対VLBI観測データ (Vstarは2009年6月まで)
相関処理（１次処理）：完了

Data and analysis setting for
SELENE Gravity Model version h (SGM100h)
Tracking data
• SELENE：2007.10.20～2008.12.26 & 2009.01.30
Doppler + range (no VLBI data)• Historical: LO I-V, A15/16ss, Clementine,
LP nominal mission, SMART-1Setting•••••••GEODYN II, SOLVE system
Expanded up to degree and order 100
Ephemeris: DE421
A Kaula-type constraint of 3.6x10-4/n2
Solar radiation pressure model
SELENE Main: box + wing
SELENE R/Vstar and other satellites : cannonball
Mean arc length of Rstar = 2.6 days
VLBI data not included

4-way Doppler data coverageachieved during the lifetime of Rstar

White solid line indicates the boundary between the near-side and the far-sideOld and new views of far-side gravity field

LP100K

SGM100hSGM90d and SGM100h

Far-side
SGM90dNamiki et al. (2009)
based on 5-month of SELENE dataplus historical data

Near-side
SGM100h
Matsumoto et al. (submitted to JGR)
based on 14-month of SELENE dataplus historical data

Gravity anomaly errors from the full covariance matrix
LP100KMax: 95 mGal
Nearside: 25 mGal
Farside : 55 mGal

SGM100h

Far side

Near side

SGM90dMax: 75 mGal
Nearside: 20 mGal
Farside : 33 mGal

Far-side

Near-side

Max: 62 mGal
Nearside: 26 mGal
Farside : 35 mGal

Correlation between gravity and topography

RMS degree variances
A priori constraint63倍

3.1倍

SGM100h gives more than one order of magnitude smaller
formal errors with respect to LP100K for degrees 7-39.

表側裏側

Imbrium盆地

Hertzsprung
盆地

Free-airgravity
Anomaly

Free-airgravity
Anomaly

Bouguergravityanomaly

Bouguergravityanomaly

topography

topography

Elliptical signature of the South Pole-Aitken Basin

Bouguer gravity anomaly around SPA

SGM100h_VLBI 【VLBI data included】
Data
•
••
•

S-band same-beam VLBI data.
Differential phase delays are determined for January, March,April, May, June, July, and August 2008 (7 months).
Doubly differenced 1-way range (DDOR).
Expected accuracy of differential phase delay = 3.44 ps(Kikuchi et al., Radio Science, 2009) → 1 mm.

Setting•••••

Cannon ball model for solar radiation pressure (R/Vstar).
R/Vstar mean arc length = 13.6 days (15 arcs).
Fit the orbits of 3 satellites to 2-way range, 2-way Doppler, 4way Doppler, and VLBI simultaneously.
DDOR data weight = 1 cm.
RMS VLBI residual: 1.4 cm with SGM100h as a priori→ 0.9 cm (post-fit, i.e., with SGM100h_VLBI)

Same-beam VLBI residuals

数cmの振幅をもつ長周期変動が見られる。
太陽輻射圧モデル誤差に起因か？

Contribution of VLBI data

5倍
1.6倍

Ratio of degree 2 sigma variance
SGM100h / SGM100h_VLBI = 1.6
LP100K / SGM100h_VLBI = 5.0

Contribution of VLBI data- orbit overlap statistics 245m

• Mean arc length = 13.6 days , overlapping duration = 1 day.
• In general, overlap position differences decrease by adding VLBI data toconventional 2-way Doppler and range data.
• The orbit consistency is further improved, in more than half the cases,
with the VLBI-included model SGM100h_VLBI.

k2 estimates

Quoted sigma’s for k2 are ten times the formal error (except for LLR).
LP99: Tracking data of the low-altitude extended mission of Lunar Prospector.

MOI (Moment of inertia) estimates【Setup】[SGM90d, Namiki et al., 2009] k2 value: 0.023295
J2 coefficient (unnormalised): 203.514682×10-6
C22 coefficient (unnormalised): 22.401900 ×10-6[SGM100h, Matsumoto et al., 2009(In preparation)] k2 value: 0.024046
J2 coefficient (unnormalised): 203.538961 ×10-6
C22 coefficient (unnormalised): 22.416703 ×10-6[SGM100h_VLBI] k2 value: 0.024029
J2 coefficient (unnormalised): 203.433343 ×10-6
C22 coefficient (unnormalised): 22.429570 ×10-6
Libration parameters (Konopliv et al.,1998):
β = 631.486 ×10-6 +/- 0.09 ×10-6 , γ= 227.871 ×10-6 +/- 0.09 ×10-6【Result】
C/MR^2 = 0.3932+/- 0.0002 [LP165P, Konopliv et al., 2001]
C/MR^2 = 0.393348714 +/- 0.000118 [SGM90d]
C/MR^2 = 0.393479418 +/- 0.000116 [SGM100h]
C/MR^2 = 0.393444435 +/- 0.000096 [SGM100h_VLBI]
Errors for C/MR^2 are based on five times the formal errors of the gravity field.

Core density estimates【Setup/Assumption】
Internal structure : 3 layer (crust, mantle, core)(The following parameters are fixed.)
Density : Crust:2850, Mantle:3360 [kg/m3]
Radius of the Moon : 1737.15 [km]GM = 4902.8003 [km3/s2] [LP165P]
Crustal thickness : 53 [km]GM = 4902.8010 [km3/s2] [SGM90d]
Core radius : 450 [km]GM = 4902.8021 [km3/s2] [SGM100h]
GM = 4902.8009 [km3/s2] [SGM100h_VLBI]【Results in this presentation】
Core density = 3781 〜 6650 [kg/m3], err = ±28% [LP165P, Konopliv et al., 2001]
Core density = 5434 〜 7131 [kg/m3], err = ±14% [SGM90d]
Core density = 6387 〜 8054 [kg/m3], err = ±12% [SGM100h]Core density = 6275 〜 7662 [kg/m3], err = ±10% [SGM100h_VLBI]
4700 [kg/m3] : FeS
5300 [kg/m3] : Eutectic Fe-FeS
7400 [kg/m3] : Liquid Fe
7700 [kg/m3] : solid Fe8100 [kg/m3] : γ-Fe
[Kuskov, 1995] など

Summary• Historical tracking data + SELENE range &Doppler data → SGM100h model• Far side gravity errors are drastically reduced.
• k2 estimates still diverge to ～11% depending ondata used(*).
• VLBI data improve the accuracy of the low
degree coefficient and orbit consistency(*)• MOI and core density of the Moon is estimated.
The error is reduced from 30% to 10%
compared with LP165P.
(*)The processing is still in progress.

長期にわたるサポートに感謝いたします。