Supporting Online Material for

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1 Supporting Online Material for Farside Gravity Field of the Moon from Four-way Doppler Measurements of SELENE (Kaguya) Noriyuki Namiki,* Takahiro Iwata, Koji Matsumoto, Hideo Hanada, Hirotomo Noda, Sander Goossens, Mina Ogawa, Nobuyuki Kawano, Kazuyoshi Asari, Sei-itsu Tsuruta, Yoshiaki Ishihara, Qinghui Liu, Fuyuhiko Kikuchi, Toshiaki Ishikawa, Sho Sasaki, Chiaki Aoshima, Kosuke Kurosawa, Seiji Sugita, Tadashi Takano *To whom correspondence should be addressed. This PDF file includes: Materials and Methods Figs. S1 to S7 Tables S1 and S2 References Published 13 February 29, Science 323, 9 (29) DOI: /science

2 Materials and methods SELENE was launched on 14 September 27. Soon after SELENE arrived at the Moon on October 4, Rstar was released from Main successfully at :36 a.m. on 9 October (Universal Time). Rstar was then inserted into an elliptic orbit at an altitude ranging from 12 to 2,395 km, an inclination of 9.1, and an orbital period of 4 hours and 5 minutes. Main was placed in a near circular lunar orbit at a mean altitude of 1 km on 19 October after a series of maneuvers. Rstar is a simple spin-stabilized satellite with the shape of an octagonal prism with a spin rate of 11 rpm or 5.4-sec period. The height, diameter, and mass of spacecraft are.65 m,.99 m, and 45 kg, respectively (S1). The spacecraft has one coaxial vertical dipole antenna for both S- and X-band communication with UDSC, and two pairs of S-band patch omni-directional antennas for transmission to and reception from Main. The dipole antenna is placed as close to the cylindrical axis of symmetry as possible. Each pair of the patch antennas are placed at either upper or lower deck of the subsatellite and are used for communication between Rstar and Main. The spacecraft does not have any thruster to control its orbit or attitude. The tracking sub-system is mostly located on Rstar with one transponder on Main to establish the link. Vstar is dedicated to the satellite VLBI experiment of SELENE (S2). The main function of RSAT is to relay Doppler tracking signals between Main over the far side and a ground-based antenna. All tracking data involved are processed with the GEODYN II software (S3) that was modified specifically to analyze the SELENE four-way Doppler tracking data via a collaboration of NAOJ and NASA/Goddard Space Flight Center. Rstar s spin induces high-frequency noise that is removed from the.1 s sampling data by a digital low-pass filter. A Doppler frequency shift due to the spin is corrected in data processing. Differential VLBI data between Rstar and Vstar are not yet included. The data weights (Table S1) are larger than the post-fit residual RMS, which in general is.5 mm/s for Main and <.5 mm/s for both Rstar and Vstar. Four-way tracking data are weighted at 1 mm/s, despite large extremes in the residuals with respect to the LP-based a priori models, primarily caused by far side gravity signals not accounted for in these models. Post-fit four-way residuals with respect to SGM9d (Fig. S2) have an RMS of.52 mm/s (6.6 mm/s with respect to LP1K). Arc lengths for Main are about 12 hours, limited by unloadings of the momentum wheels onboard the satellite. Rstar arc lengths are the same, to prevent build-up of dynamical orbit errors in the fourway data. Vstar arc lengths are 2.4 days on average, depending on data coverage. The four-way data are processed in the batch estimator in a similar way as conventional tracking data: the orbit and variational partials are simultaneously integrated. The four-way data link the position of Rstar and Main together, so these orbits are determined simultaneously. Initially, the orbits of Rstar and Main are determined separately from two-way data, and after convergence, they are merged, the four-way data are added, and the orbits are re-adjusted to fit both the four-way and two-way data. The converged arc of this combination is used as the reference orbit for gravity field determination. Orbit errors from Rstar influence the residuals as well, but through analysis of the data it is estimated that a worst-case orbit error of about 2 m in Rstar s orbit has an effect of 1 mm/s on the four-way data residuals, well below the observed signal (S4, S5). Next to SELENE data, tracking data from LO I-V, A15/16ss, Clementine, LP, and some SMART-1 data are also taken into account. The extended mission data of LP are not yet included. Data weights are also summarized in Table S1. The larger weighting values are chosen to emphasize the new SE- LENE tracking data in the solutions. The arcs are all processed with full-scale modelling of the forces acting on the satellite, and of the measurement link. The initial a priori gravity field was LP75G, and the estimated model has been expanded up to degree and order 9. The DE43 ephemeris (S6) was adopted for the computation of the third-body perturbations as well as the lunar librations and coordinate system. Solar radiation pressure is modelled as a simple cannonball model. Estimated parameters, next to the gravity field, include the spacecraft state vectors, solar radiation pressure coefficients, and measurement biases per station. The process of data reduction has been iterated three times, and subsequent differences between gravity models became sufficiently small, with the only real differences in the uncovered parts of the northern far side (Fig S2).

3 Four-way Doppler measurements of SELENE Namiki et al., 28 To show the contribution of the new observations to the far side gravity field, we have also shown in Fig 3A correlations between gravity and topography, localized within a spherical cap of radius of 75 at N and 18 E by using spatio-spectral localization technique (S7). Localized SGM9d coefficients reveal a constantly high correlation with topography for degree lower than 6, and the difference with localized LP1K is evident. Uniform coverage of tracking data results in more dependence of coefficients on tracking data and less on a priori information. This is easily confirmed from contribution measure plots of SGM9d and LP1K (Fig. S4). The errors of the gravity coefficients using a data weight of 1 mm/s for the four-way data are probably over-optimistic, but the range given in Figure 3B is indicative of the actual error on the coefficients. In comparison with LP1K, the spectrum of SGM9d shows an increase in power after degree 3. This difference is likely due to the increase in far side resolution of SGM9d, because correlations between topography and gravity are higher in SGM9d than in the LP gravity models (Fig. 3A). In order to investigate the significance of our new finding that some of the far-side basins have concentrated Bouguer anomalies, we performed a simple rheological calculation. The lack of strong Bouguer anomalies over topographic rims and depressions indicates that the impact basins have no substantial post-impact deformation. This would mean that the basins have not gone through isostatic compensation via MOHO movement. Furthermore, the fact that these basins retain their high topographic relieves indicates that they have not gone through significant crustal lateral flow either. The deformation of a planetary body is generally very complicated involving elastic deformation, brittle failure, and ductile flow. Thus, a full understanding of the deformation of lunar crust and mantle subsequent to impact basin formation would require complex computational code calculations, which would be out of the scope of this study and will be explored as a separate contribution. Nevertheless, a simple rheological calculation may extract useful information on lunar deformation processes from our new observation data. Generally, the vertical distribution of the maximum stress that each depth of a planetary body can support has a peak at an intermediate depth, because both the upper part and lower part are weak (e.g., S8). The upper part cannot support a large stress because it would deform in a brittle fashion due to its low normal stress. Neither could the lower part because it would deform in a ductile fashion due to its high temperature. Then, the intermediate layer receives the highest stress and supports the majority of the load imposed on the local area. Thus, we use a simple viscosity one-layer viscous fluid model to assess the effective viscosity necessary to support the topographic relieves of impact basins without allowing isostatic compensation. In this model, we assume a uniform viscosity in the top layer and an inviscid substrate layer. This simplification may be justified because the viscosity of the mantle decreases very rapidly (i.e., exponentially) with depth. The solution of motion in such a fluid system can be given analytically and has two modes of motion (S9). One corresponds to isostatic compensation and the other to crustal lateral flow (or viscous relaxation). An example of the solution is given in Fig. S6, showing that isostatic compensation takes place much more rapidly than viscous relaxation. Here crustal density, mantle density, lunar gravity, crustal viscosity, and mantle viscosity, are assumed to be 2,9 kg/m 3, 3,4 kg/m 3, 1.62 m/s 2, 1 24 Pa s, Pa s, respectively. The fact that the Hertzsprung basin does not exhibit a strong positive Bouguer anomaly indicates that substantial isostatic compensation did not occur. If the cooling timescale of the Moon is on the order of 1 9 yr, the timescale of isostatic compensation for ~5 km of wavelength needs to be significantly larger than 1 9 yr, perhaps 1 1 yr. Such long time scales would require an effective viscosity > 1 27 Pa s for 2 km of thickness, > 5 x 1 27 Pa s for 1 km, >3 x 1 28 Pa s for 5 km. These high values of effective viscosity (> 1 27 Pa s) would require temperatures lower than 75 K for dry olivine, 85 K for dry plagioclase, and 7 K for dry diabase (Fig. S7). 2

4 Four-way Doppler measurements of SELENE Namiki et al., 28 In contrast, even such high viscosity would allow a much larger impact basin, such as South-Pole Aitken, to compensate isostatically within 1 Gy (Figure S6). This is consistent with high positive Bouguer anomalies observed over the South-Pole Aitken basin (Fig. 5). References S1. T. Iwata, M. Takahashi, N. Namiki, H. Hanada, N. Kawano, K. Heki, K. Matsumoto, T. Takano, J. Geod. Soc. Japan, 47, 558 (21). S2. H. Hanada, T. Iwata, N. Namiki, N. Kawano, K. Asari, T. Ishikawa, F. Kikuchi, Q. Liu, K. Matsumoto, H. Noda, S. Tsuruta, S. Goossens, K. Iwadate, O. Kameya, Y. Tamura, X. Hong, J. Ping, Y. Aili, S. Ellingsen, W. Schlüter, Adv. Space Res., 42, 341 (28). S3. D. E. Pavlis, S. G. Poulose, C. Deng, J. J. McCarthy, GEODYN II System Documentation (Contractor Rep, SGT-Inc., Greenbelt, MD, 27). S4. S. Goossens and K. Matsumoto, Adv. Space Res., 4, 43 (27). S5. K. Matsumoto, H. Hanada, N. Namiki, T. Iwata, S. Goossens, S. Tsuruta, N. Kawano, D. D. Rowlands, Adv. Space Res., 42, 331 (28). S6. E. M. Standish, X. X. Newhall, J. G. Williams, W. M. Folkner, JPL Planetary and Lunar ephemerides DE43/LE43, JPL Interoffice memorandum, (1995). S7. M. A. Wieczorek and F. J. Simons, Geophys. J. Int., 162, 655 (25). S8. S. H. Kirby and A. K. Kronenberg, Rev. Geophys. 25, 1219, S9. S. C. Solomon, R. P. Comer, J. W. Head, J. Geophys., Res., 87, 3975 (1982). S1 A. S. Konopliv, S. W. Asmar, E. Carranza, W. L. Sjogren, D. N. Yuan, Icarus, 15, 1 (21). 3

5 Four-way Doppler measurements of SELENE Namiki et al., 28 Table S1. Summary of tracking data processing in the SGM9d Mission (Station) Doppler/Range Number of observations Weight * Far side SELENE four-way (UDSC) Doppler 25,5 1 mm/s SELENE Main two-way (GN) Doppler 785,638 2 mm/s (UDSC) Range 43,895 5 m SELENE Rstar two-way (UDSC) Doppler 55,935 1 mm/s Near side Range 52,176 5 m SELENE Vstar two-way (UDSC) Doppler 19,88 1 mm/s Range 14,64 5 m LO I to V Doppler 4.5 mm/s A15/16ss Doppler 4.5 mm/s Clementine Doppler 3 mm/s Range 3,861,375 6 m Clementine (Pomonkey) Doppler 1 mm/s LP nominal mission Doppler 2 mm/s Range 4 m SMART-1 Doppler 1 mm/s *, Data weights for the historical data have been deliberately chosen higher, in order to emphasize the new Kaguya data. Four-way data are further emphasized to make up for the different amounts of tracking data over near and far side. 4

6 Four-way Doppler measurements of SELENE Namiki et al., 28 Table S2. Root mean square of fit values over the near side for LP1K and SGM9d for LP tracking data during the nominal mission. Date of arc RMS fit (mm/s) SGM9d LP1K July

7 Four-way Doppler measurements of SELENE Namiki et al., 28 S-band up-link for four-way Doppler/two-way Doppler and range S-band down-link for two-way Doppler and range Rstar S-band forward-link for four-way Doppler S-band return-link for four-way Doppler JAXA UDSC-64 m X-band down-link for four-way Doppler Moon Main Earth JAXA GN Two-way Doppler and range Fig. S1. Schematic diagram of the four-way Doppler measurements 6

8 Four-way Doppler measurements of SELENE Namiki et al., 28 Fig. S Map of mean residuals of four-way Doppler data in SGM9d with respect to LP1K gravity model. Residuals are plotted on the location of Main at the time of observation. Units are mm/s. The lunar near side is on the right side of the figure, and the far side is on the left. 7

9 Four-way Doppler measurements of SELENE (A) SGM9d -35 (B) -35 Namiki et al., LP165P SGM9d LP165P

10 Four-way Doppler measurements of SELENE Namiki et al., 28 Fig. S3. A comparison of SGM9d and previous gravity model, LP1K (S1) for the (A) near side, and (B) far side. The free-air gravity map from SGM9d is on the left side of the figure, and that from LP1K is on the right. Color scales are the same as for Fig. 2. 9

11 Four-way Doppler measurements of SELENE Namiki et al., 28 Harmonic Order (A)

12 Four-way Doppler measurements of SELENE Namiki et al., 28 Harmonic Order (B) Harmonic Degree Fig. S4. Contribution measure of the spherical harmonic coefficients of (A) SGM9d and (B) LP1K. Contribution measure value is a fraction of the observation in determination of the individual gravity model coefficients. Zero value indicates that the coefficient is determined by a priori constraint, not observation. 11

13 Four-way Doppler measurements of SELENE Namiki et al., 28 (A)

14 Four-way Doppler measurements of SELENE Namiki et al., 28 (B)

15 Four-way Doppler measurements of SELENE Namiki et al., 28 (C)

16 Four-way Doppler measurements of SELENE Namiki et al., 28 (D)

17 Four-way Doppler measurements of SELENE Namiki et al., 28 (E)

18 Four-way Doppler measurements of SELENE Namiki et al., 28 (F) Fig. S5. Close-up views of free-air and Bouguer gravity anomalies of primary mascon, Type I, and II basins. (A) Free-air and (B) Bouguer gravity anomalies of the Crisium basin (17.5 N, 58.5 E), (C) free-air and (D) Bouguer gravity anomalies of the Korolev basin (4.5 S, 157 W), and (E) free-air and (F) Bouguer gravity anomalies of the Moscoviense basin (25 N, 147 E). The contour lines are in 1 mgal intervals. Each map covers ±2 around a center of basin in latitude and longitude. 17

19 Four-way Doppler measurements of SELENE Namiki et al., 28 Fig. S6. Flow timescale of a uniform viscous layer for different layer thicknesses as a function of wavelength λ of topographic undulation. The time scales for both isostatic compensation and topographic viscous relaxation are shown. HSPG and SPA indicate the diameters of Hertzsprung and South- Pole Aitken basins, respectively. It is noted that Hertzsprung basin does not exhibit a strong positive Bouguer anomaly, a sign of isostatic compensation, but South-Pole Aitken basin does. The viscosity of 1 24 Pa s is used in this calculation. When the viscosity is 1 times larger, all the curves are moved translationally by a factor of 1 upward. 18

20 Four-way Doppler measurements of SELENE Namiki et al., 28 Figure S7. Flow laws for different minerals. Effective viscosity σ/ε is shown as a function of temperature for dry anorthosite, dry olivine, dry diabase, and wet anorthosite. Here σ and ε are differential stress and strain rate, respectively. The strain rate is assumed to be very small (3 x 1-18 s -1 ) in this figure. 19