Our aim in this paper was explore the possibility of hitting a gold ball into orbit around various objects inthe solar system by using the orbital velocity equation. We found this to be possible around four moons,four dwarf planet candidates and a trojan of Jupiter with masses greater than 10^18kg

Eddison, Jacob

Levin, Guy

Marsden, Joe

Vigneswara, Darshan

English

University of Leicester Open Journals

Journal of Physics Special TopicsAn undergraduate physics journalP6 4 Tee Off Into SpaceJ. Eddison, G. Levin, J. Marsden and D. VigneswaraDepartment of Physics and Astronomy, University of Leicester, Leicester, LE1 7RHNovember 2, 2017AbstractOur aim in this paper is to explore the possibility of hitting a golf ball into orbit around variousobjects in our solar system by using the orbital velocity equation. We found this to be possiblearound four moons, six dwarf planet candidates and a trojan of Jupiter (See Table 1).IntroductionThe gravity and atmosphere on Earth limitsthe distance a golf ball can be hit. If instead agolfer was placed on a small moon with almostno atmosphere, could they theoretically hit a golfball so fast so that it would enter orbit?. In thispaper, we will review notable solar system ob-jects in order to identify which ones are suitableto attempt this ambitious shot.TheoryIn 1974, astronaut Alan Shepard played golfon the Moon during the Apollo 14 mission.The reduced gravitational strength and vacuumshould have massively helped Shepard but heestimated his shot only travelled approximately180 m [1]. His relatively short drive was prob-ably due to the limited movement that Shepardexperienced in his space suit.If we want to explore how fast a ball needs tobe travelling to enter orbit we need to use theorbital velocity equation. This can be derivedby setting the gravitational force acting on anobject equal to its centripetal force:Fg =GMmr2(1)Fc =mv2or(2)Where G is the gravitational constant, M is themass of the body, m is the golf ball mass, r is theorbital radius and vo is the ball’s orbital velocity.We can rearrange for vo so that:vo =√GMr(3)We can set the minimum radius of the orbitto equal the radius of a body to calculate theminimum orbital velocity. For the Moon, M =7.346×1022 kg and r = 1737m [2]. the Moon’s vois 1680ms−1. For comparison, the current worldrecord golf ball speed is only 97.1ms−1 [3].If an orbital shot on the Moon is not possi-ble, we can use equation 3 to determine vo forother objects in the solar system. An orbitalshot around a solar system object is possible solong as v0 < 97.1ms−1. This assumes that thegolfer would be able to hit the golf ball at themaximum speed despite the harsh environment,and that the shot has no inclination.ResultsA list of known solar system objects to theGigagram range of was used [2]. All planets,moons, dwarf planets or notable objects that hada lower radius than the Moon were included. Ob-jects with r < 100km were not included due totheir insignificant size and because many of theirmasses are only rough estimates. Asteroids andun-named objects were not considered which sig-nificantly simplified the final list.This meant that the orbital velocities were cal-culated for a total of 46 objects. The list consistsof 23 moons, 5 dwarf planets and 18 objects yetto be fully classified.Body r (km) M (1018 kg) vo (ms−1)Hiiaka 160 20 91.3Actaea 140 18.6 94.1Hyperion 205 5.58 42.6Sila 125 11 76.6Chariklo 124 9 69.6Chiron 116 7 63.4Hektor 113 10 60.0Ceto 112 5.4 56.7Phoebe 107 8.29 72.1Deucalion 105 3.14 44.7Bienor 104 3.15 44.9Table 1: Notable solar system objects that were foundto have a vo < 97.1ms−1.We found that an orbital golf ball shot waspossible around 4 moons, 6 dwarf planet can-didates and Hektor, an L4 Trojan of Jupiter. ATrojan is is an object that orbits at the Lagrangepoint of another gravitational body.A golfer would have more than one reason tovisit some of these objects. The dwarf planetsChariklo and Chiron both have rings, Hiiaka or-bits the egg shaped dwarf planet Haumea, whichalso has rings. The moons Hyperion and Phoebewould offer stunning views of the Gas Giants.DiscussionOne assumption we make is that the solar sys-tem objects were spherical. In reality an irreg-ular shaped object would lead to gravitationalperturbations in the golf balls orbit, which couldmake it unstable. However a ball with suffi-cient velocity would be able to complete one or-bit around all of the objects listed.Furthermore a ball hit on the surface of a bodycloser to its centre of mass would require highervelocity or a precise inclination to stop it hittingthe surface. Many low mass solar system ob-jects have irregular shapes, this was accountedfor where possible by using the maximum radiifor the calculations.Another assumption is that the golfer wouldnot experience the limited movement while wear-ing a space suit. The reason this was not con-sidered was because there was no quantifiableway to calculate it’s effect on the orbital veloc-ity. While this is a concern, many of the cal-culated orbital velocities are significantly belowthe world record speed. Furthermore, futureimprovements to space suit design could makeswinging a golf club easier.If a body had an atmosphere this would alsomean a higher orbital velocity is required. It isunderstood that the atmospheres of smaller ob-jects in the solar system are generally extremelythin so they would not effect the ball. The Moonhas an atmosphere 2 × 1014 times thinner thanEarth’s, it was assumed that small objects wouldhave similarly thin atmospheres [4].ConclusionHitting a golf ball into orbit might seem impos-sible here from Earth, but we have successfullyshown that there are a large variety of moons,dwarf planets and asteroids that such shots arepossible from. Future space exploration mightlead the sport of golf to new frontiers.References[1] http://www.pga.com/news/golf-buzz/feb-6-1971-alan-shepard-plays-golf-moon[Accessed 1/11/17][2] http://www.gutenberg.us/articles/eng/list_of_solar_system_objects_by_size [Accessed 1/11/17][3] http://www.guinnessworldrecords.com/world-records/fastest-golf-drive/ [Accessed1/11/17][4] https://futurism.com/the-atmospheres-on-our-moons/ [Accessed2/11/17]

P6_4 Tee Off Into Space

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P6_4 Tee Off Into Space

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