# The solid angle hidden in polyhedron gravitation formulations

Original Article

First Online:

Received:

Accepted:

- 312 Downloads
- 10 Citations

## Abstract

Formulas of a homogeneous polyhedron’s gravitational potential typically include two arctangent terms for every edge of every face and a special term to eliminate a possible facial singularity. However, the arctangent and singularity terms are equivalent to the face’s solid angle viewed from the field point. A face’s solid angle can be evaluated with a single arctangent, saving computation.

## Keywords

Gravitational potential Polyhedron Solid angle Prism singularity Gore## Supplementary material

## References

- Abramowitz M, Stegun IA (1964) Handbook of Mathematical Functions. National Bureau of Standards, Washington, D. C., republished by Dover, New York, 1965Google Scholar
- Barnett CT (1976) Theoretical modeling of the magnetic and gravitational fields of an arbitrarily shaped three-dimensional body. Geophysics 41(6):1353–1364CrossRefGoogle Scholar
- Binzel RP, DeMeo FE, Burt BJ, Cloutis EA, Rozitis B, Burbine TH, Campins H, Clark BE, Emery JP, Hergenrother CW, Howell ES, Lauretta DS, Nolan MC, Mansfield M, Pietrasz V, Polishook D, Scheeres DJ (2015) Spectral slope variations for OSIRIS-REx target asteroid (101955) Bennu: possible evidence for a fine-grained regolith equatorial ridge. Icarus 256:22–29CrossRefGoogle Scholar
- Blokh YI (1997) Fedor A. Sludskii, founder of Russian geophysics. Izvestiya: Physics of the Solid Earth 33(3):252–254Google Scholar
- Carvalho PCP, Cavalcanti PR (1995) Point in polyhedron testing using spherical polygons. In: Graphics Gems V, Academic Press, chap II–2, pp 42–49Google Scholar
- Cayley A (1874–1875) On the potentials of polygons and polyhedra. In: Proceedings of the London Mathematical Society, vol VI, pp 20–34, republished in The Collected Mathematical Papers, Arthur Cayley, vol IX, Cambridge 1889, Chapter 602, pp 266–280. http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABS3153
- Conway JT (2015) Analytical solution from vector potentials for the gravitational field of a general polyhedron. Celest Mech Dyn Astron 121(1):17–38. doi: 10.1007/s10569-014-9588-x CrossRefGoogle Scholar
- D’Urso MG (2013) On the evaluation of the gravity effects of polyhedral bodies and a consistent treatment of related singularities. J Geod 87(3):239–252CrossRefGoogle Scholar
- D’Urso MG (2014) Analytical computation of gravity effects for polyhedral bodies. J Geod 88(1):13–29, doi: 10.1007/s00190-013-0664-x, https://www.researchgate.net/publication/258158187_Analytical_computation_of_gravity_effects_for_polyhedral_bodies
- D’Urso MG, Russo P (2002) A new algorithm for point-in polygon test. Survey Rev 36(284):410–422CrossRefGoogle Scholar
- Eriksson F (1990) On the measure of solid angles. Mathematics Magazine 63(3):184–187, http://www.maa.org/sites/default/files/Eriksson14108673.pdf
- Göetz HJ, Lahmeyer B (1988) Application of three-dimensional modeling in gravity and magnetics. Geophysics 53(8):1096–1108. doi: 10.1190/1.1442546 CrossRefGoogle Scholar
- Golizdra GY (1981) Calculation of the gravitational field of a polyhedron. Izvestia: Physics of the Solid Earth 17(8):625–628Google Scholar
- Greenberg MD (1978) Found Appl Math. Prentice-Hall, Englewood CliffsGoogle Scholar
- Holstein H, Ketteridge B (1996) Gravimetric analysis of uniform polyhedra. Geophysics 61(2):357–364CrossRefGoogle Scholar
- Holstein H, Schürholz P, Starr AJ, Chakraborty M (1999) Comparison of gravimetric formulas for uniform polyhedra. Geophysics 64(5):1438–1446CrossRefGoogle Scholar
- Hudson RS, Ostro SJ, Jurgens RF, Rosema KD, Giorgini JD, Winker R, Rose R, Choate D, Cormier RA, Franck CR, Fry R, Howard D, Kelley D, Littlefair R, Slade MA, Benner LAM, Thomas ML, Mithell DL, Chodas PW, Yeomans DK, Scheeres DJ, Palmer P, Zaitsev A, Koyama Y, Nakamua A, Harris AW, Meshkov MN (2000) Radar observations and physical model of asteroid 6489 Golevka. Icarus 148(1):37–51, doi: 10.1006/icar.2000.6483, http://www.sciencedirect.com/science/article/pii/S0019103500964832
- Ikeda H, Kominato T, Matsuoka M, Ohnishi T, Yoshikawa M (2008) Orbit determination of Hayabusa during close proximity phase. In: 26th International Symposium on Space Technology, http://www.senkyo.co.jp/ists2008/pdf/2008-d-38.pdf
- Kellogg OD (1929) Foundations of Potential Theory. J. Springer, republished by Dover, New York, 1953Google Scholar
- Kwok YK (1991) Gravity gradient tensors due to a polyhedron with polygonal facets. Geophys Prospect 39:435–443CrossRefGoogle Scholar
- Leathem JG (1913) Volume and Surface Integrals Used in Physics, 2nd edn. Cambridge University Press, London, https://archive.org/details/volumesurfaceint00leatrich
- Macmillan WD (1930) The Theory of the Potential. McGraw Hill, republished by Dover, New York, 1958Google Scholar
- Mehler FG (1866) Über die Anziehung eines homogenen Polyeders. Journal für die reine und angewandte Mathematik LXVI:375–381Google Scholar
- Mertens F (1868) Bestimmung des Potentials eines homogenen Polyeders. Journal für die reine und angewandte Mathematik LXIX:286–288Google Scholar
- Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74(7):552–560, doi: 10.1007/s001900000116, URL:http://link.springer.com/article/10.1007%2Fs001900000116
- Okabe M (1979) Analytical expressions for gravity anomalies due to homogeneous polyhedral bodies and translations into magnetic anomalies. Geophysics 44(4):730–741CrossRefGoogle Scholar
- Paul MK (1974) The gravity effect of a homogeneous polyhedron for three-dimensional interpretation. Pure Appl Geophys 112(III):553–561CrossRefGoogle Scholar
- Petrović S (1996) Determination of the potential of homogeneous polyhedral bodies using line integrals. J Geod 71:44–52CrossRefGoogle Scholar
- Plouff D (1976) Gravity and magnetic fields of polygonal prisms and application to magnetic terrain corrections. Geophysics 41(4):727–741CrossRefGoogle Scholar
- Pohánka V (1988) Optimum expression for computation of the gravity field of a homogeneous polyhedral body. Geophys Prospect 36:733–751CrossRefGoogle Scholar
- Richardson JE, Melosh HJ (2006) Modeling the ballistic behavior of solid ejecta from the Deep Impact cratering event. In: 37th Lunar and Planetary Science Conference, Lunar and Planetary Institute, Houston, http://www.lpi.usra.edu/meetings/lpsc2006/pdf/1836.pdf
- Rossi A, Marzari F, Farinella P (1999) Orbital evolution around irregular bodies. Earth Planets Space 51:1173–1180CrossRefGoogle Scholar
- Scheeres DJ, Durda DD, Geissler PE (2002) The fate of asteroid ejecta. In: Bottke WF Jr, Cellino A, Paolicchi P, Binzel RP (eds) Asteroids III. The University of Arizona, Tucson, pp 527–544Google Scholar
- Scheeres DJ, Miller JK, Yeomans DK (2003) The orbital dynamics environment of 433 Eros: a case study for future asteroid missions. InterPlanetary Network (IPN) Progress Report 42-152, http://ipnpr.jpl.nasa.gov/progress_report/42-152/152F.pdf
- Selby SM, Girling B (eds) (1965) Standard Math Tables. The Chemical Rubber Company, ClevelandGoogle Scholar
- Silva AA, Prado AFBdA, Winter OC (2011) Study of trajectories around a non-spherical body. In: Proceedings of the 10th WSEAS international conference on system science and simulation in engineering, World Scientific and Engineering Academy and Society (WSEAS), pp 42–47Google Scholar
- Sludskii FA (1863) Ob’yklonenii otvesnykh linii (On the deflection of plumb lines). Master’s thesis, Moscow, Univ. TipografiyaGoogle Scholar
- Spiegel MR (1959) Schaum’s Outline Series: Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis. McGraw-Hill, New YorkGoogle Scholar
- Strakhov VN, Lapina MI (1990) Direct gravimetric and magnetometric problems for homogeneous polyhedrons. Geophys J 8(6):740–756Google Scholar
- Strakhov VN, Lapina MI, Yefimov AB (1986a) A solution to forward problems in gravity and magnetism with new analytical expressions for the field elements of standard approximating bodies. I. Izvestiya, Earth Sciences 22(6):471–482Google Scholar
- Strakhov VN, Lapina MI, Yefimov AB (1986b) Solution of direct gravity and magnetic problems with new analytical expressions of the field of typical approximating bodies. II. Izvestiya, Earth Sciences 22(7):566–577Google Scholar
- Todhunter I (1886) Spherical Trigonometry, 5th edn. Macmillan and Co., http://www.gutenberg.org/ebooks/19770
- Tsoulis D (2012) Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics 77(2):F1–F11. doi: 10.1190/geo2010-0334.1 CrossRefGoogle Scholar
- Tsoulis D, Petrović S (2001) On the singularities of the gravity field of a homogeneous polyhedral body. Geophysics 66(2):535–539CrossRefGoogle Scholar
- van Oosterom A, Strackee J (1983) The solid angle of a plane triangle. IEEE Trans Biomed Eng 30(2):125–126. doi: 10.1109/TBME.1983.325207 CrossRefGoogle Scholar
- Waldvogel J (1979) The newtonian potential of homogeneous polyhedra. J Appl Math Phys (ZAMP) 30:388–398CrossRefGoogle Scholar
- Werner RA (1994) The gravitational potential of a homogeneous polyhedron, or, don’t cut corners. Celest Mech Dyn Astron 59(3):253–278. doi: 10.1007/BF00692875
- Werner RA, Scheeres DJ (1997) Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celest Mech Dyn Astron 65(3):313–344. doi: 10.1007/BF00053511 CrossRefGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2016