Gravitational Acceleration, Gravitational Potential, Slope, and Elevation Calculations
The SBMT includes standard plate colorings that show slope, elevation, gravitational acceleration, and gravitational potential. Users can load these from the plate coloring area in the control panel of the SBMT.
Gravitational acceleration and gravitational potential were calculated at each plate center using an in-house implementation of the algorithm of either Werner and Scheeres (1997) or Cheng (2012). In all cases, a uniform density distribution was assumed. An additional term was added to both the gravitational potential and acceleration to correct for the rotation of the small body.
Slopes were computed as the angular separation in degrees between the plate normal vector and the negative gravitational acceleration vector at each plate center.
Elevation was computed as follows. First a reference potential, R, was selected. The reference potential was chosen to be either the minimum potential across all plates, or the average potential weighted by the plate area. Typically, fast rotating asteroids (<12 hours) used the minimum potential for a reference. The elevation, E, was computed as:
where g is the magnitude of the acceleration vector at the ith plate center.
Table 1, below, lists the density, rotation rate, and reference potential used in the gravity calculations. For bodies with shape models of various resolutions, the listed reference potential is for the highest-resolution shape model.
Table 1. Assumptions for geophysical calculations.
| Near-Earth Asteroids | ||||
|---|---|---|---|---|
| Body | Shape model | Density (g/cm3) | Rotation Rate (radians/sec) | Reference Potential (J/kg) |
| 433 Eros | Gaskell, 2008 | 2.67 | 0.000331165761670640 | -53.765039959572114 (avg) |
| Thomas et al., 2001 | 2.67 | 0.000331165761670640 | -54.17362021247867 (avg) | |
| Neumann et al., 2001 | 2.67 | 0.000331165761670640 | -53.80499090631424 (avg) | |
| NAV team, 2001 | 2.67 | 0.000331165761670640 | -53.95545945308596 (avg) | |
| 25143 Itokawa | Gaskell et al., 2008 | 1.95 | 0.000143857148947075 | -0.014684684411415587 (avg) |
| Ostro et al. (2004) | 1.95 | 0.000143857148947075 | -0.01730245623407668 (avg) | |
| 101955 Bennu | Bennu (Nolan) | 1.0 | 0.000407026411379 | -0.020418417791909282 (avg) |
| OLA v20 | 1.194 | 0.00040626 | -0.02518662565534758 (min) | |
| OLA v20 PTM | 1.194 | 0.00040626 | -0.02518662565534758 (min) | |
| SPC v13 | 1.260 | 0.00040613 | -0.02653228310925632 (min) | |
| SPC v14 | 1.260 | 0.00040613 | -0.02632918411768206 (min) | |
| SPC v20 | 1.186 | 0.00040626 | -0.02511642021272778 (min) | |
| SPC v28 | 1.186 | 0.00040626 | -0.02525083554352243 (min) | |
| SPC v42 | 1.186 | 0.00040626 | -0.02526143248386209 (min) | |
| SPO v34 | 1.186 | 0.00040626 | -0.02530641097907325 (min) | |
| 4179 Toutatis | Hudson et al. (2004) | 2.1 | 9.91664347724e-06 | -0.8279371652279144 (avg) |
| Main-Belt Asteroids | ||||
| Body | Shape model | Density (g/cm3) | Rotation Rate (radians/sec) | Reference Potential (J/kg) |
| 1 Ceres | SPC | 2.16 | 1.923403740501147297e-4 | -136092.1506029048 (avg) |
| 4 Vesta | Gaskell (2013) | 3.456 | 0.00032671047399107775 | -68595.43861780204 (avg) |
| Thomas (2000) | 3.456 | 0.00032671047399107775 | -70013.7480908538 (avg) | |
| 21 Lutetia | SPC | 3.4 | 0.0002136718326725556 | -2295.0723059440766 (avg) |
| Farnham et al. (2013) | 3.4 | 0.0002136718326725556 | -2316.19578918338 (avg) | |
| 243 Ida | Thomas et al. (2000) | 2.6 | 0.000376665486656 | -186.9993041380229 (avg) |
| Stooke (2016) | 2.6 | 0.000376665486656 | -184.199526907021 (avg) | |
| 253 Mathilde | Thomas et al. (2000) | 1.3 | 4.17798760005e-06 | -238.5679818133799 (avg) |
| 951 Gaspra | Thomas et al. (2000) | 2.7 | 0.000247843113933 | -28.27656741870852 (avg) |
| Stooke (2016) | 2.7 | 0.000247843113933 | -26.195966470210653 (avg) | |
| 2867 Steins | Farnham et al. (2013) | 1.8 | 0.000288636316183 | -3.6814488593903745 (avg) |
| Comets | ||||
| Body | Shape model | Density (g/cm3) | Rotation Rate (radians/sec) | Reference Potential (J/kg) |
| 1P/Halley | Stooke (2016) | 0.6 | 3.30554782575e-05 | -3.359549744507553 (avg) |
| 9P/Tempel 1 | Farnham and Thomas (2013) | 0.47 | 4.28434E-05 | -1.37739 (avg) |
| 81P/Wild 2 | Farnham et al. (2005) | 0.6 | 0.000129283648296 | -0.6227439262399692 (avg) |
| 103P/Hartley 2 | Farnham and Thomas (2013) | 0.3 | 9.64270304969e-05 | -0.02575579466484785 (avg) |
| Moons | ||||
| Body | Shape model | Density (g/cm3) | Rotation Rate (radians/sec) | Reference Potential (J/kg) |
| Deimos | Thomas (2000) | 1.471 | 5.76043287486e-05 | -15.869706456990233 (avg) |
| Phobos | Gaskell (2011) | 1.876 | 0.00022803304110600688 | -66.90513164418878 (avg) |
| Thomas (2000) | 1.876 | 0.00022803304110600688 | -66.84279886827008 (avg) | |
References
Werner, R.A. and Scheeres, D.J., 1997, Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia, Celestial Mechanics and Dynamical Astronomy, 65, 313-344.
Cheng, A.F. et al., 2012, Efficient Calculation of Effective Potential and Gravity on Small Bodies, Asteroids, Comets, Meteors, abstract 6447.