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The Gravity Field of a Cube Physics International, 2012, 3 (2), 50-57 ISSN: 1948-9803 ©2012 Science Publication doi:10.3844/pisp.2012.50.57 Published Online 3 (2) 2012 (http://www.thescipub.com/pi.toc) The Gravity Field of a Cube 1James Michael Chappell, 2Mark John Chappell, 1Azhar Iqbal and 1Derek Abbott 1School of Electrical and Electronic Engineering, Faculty of Engineering, Computer and Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia 2Applied Cognitive Neuroscience Research Unit, Griffith Health Institute, Griffith University, Queensland, 4111, Australia ABSTRACT We calculate the Newtonian gravitational potential and field of a cubic, homogeneous asteroid and we apply it to the orbit of possible satellites. Large astronomical objects such as stars or planets, naturally tend to form spherical shapes due to the dominance of the gravitational forces, but as a thought experiment, we consider the properties of a planet in the form of a perfect cube. We investigate the formation of stable orbits around such cubic objects, for the case of a static as well as a rotating cube employing the method of Poincare sections. The calculation of the gravitational field around non-spherical objects has a significant role in space missions to investigate asteroid belt objects that require calculating orbits around a large non-spherical mass. The calculation of such non-spherical fields also has relevance in identifying deposits or beds of ores inside the Earth, by measuring gravitational anomalies. Keywords: Gravitational Field, Cube, Satellite Orbits, Poincare Sections 1. INTRODUCTION 1885). The appearance of cubic planets in our universe may appear extremely improbable, however if gravity ∞ Much work has now been carried out in calculating acted over the L = max i=1…3|xi| metric rather than the the gravitational field around cuboid objects and so we usual Euclidean metric then cubic planets would naturally investigate in this study, the nature of the gravitational form under the force of gravity. field and satellite orbits around a hypothetical cubic The gravitational field of a cube was apparently first Earth-size planet. Plato, in the Timaeus, links what he calculated in (MacMillan, 1958) and later extended to considers the four basic elements, fire, air, water and cuboid masses (Nagy, 1966; Mufti, 2006a), including earth, with the four regular solids, using the tetrahedron varying density (Garcia-Abdeslem, 2005; Hansen, to represent fire, the octahedron to represent air, the 1999) and also for general polyhedra (Paul, 1974; icosahedron to represent water and the cube to represent Coggon, 1976), as well as for a range of simpler planar earth. Taking Plato’s idea literally, we might imagine the objects such as straight line segments as well as disks and Earth in the form of a perfect cube. We can then calculate annular shapes (Riaguas et al ., 1999; 2001; Azevedo the gravitational field around this object, the process being and Ontaneda, 2007; Azevedo et al ., 2005; simplified by the simple endpoints needed when Fukushima, 2010; Alberti and Vidal, 2007; Blesa, integrating over a right rectangular prism, assuming 2005; Gutierrez-Romero et al ., 2004; Palacian et al ., Newton’s classical law of gravitation. The equipotentials 2006; Najid and Elourabi, 2012; Najid et al ., 2012; of the field will give us, for example, the shape of lakes Iorio, 2007; 2012). These results were found to have that would form on the face of such a cube. The concept of application in the calculation of the gravitational cubical planets is in fact not new, as in 1884 the Swiss anomalies on the Earth (Mufti, 2006b) and the astronomer Arndt claimed to have discovered a cubical slowdown of the Earth’s rotation rate due to tidal drag planet orbiting beyond the orbit of Neptune, though (Celnikier, 1990). Further investigations explored the unfortunately the discovery was never verified (Vankirk, nature of the satellite orbits that would form around a Corresponding Author: James Michael Chappell, School of Electrical and Electronic Engineering, Faculty of Engineering, Computer and Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia Science Publications 50 PI James Michael Chappell et al. / Physics International 3 (2) (2012) 50-57 static cube (Liu et al ., 2011a; Werner, 1994), as well as for and with the integral over x, we achieve our final rotating cubes (Liu et al ., 2011b) using simulations and result: the method of Poincare sections (Liu et al ., 2011a; et al Scheeres ., 2000). These preliminary investigations V(X,Y,Z) =- Gρ of orbits around cubic objects are of significance as they can L-X D-Z R-Y be considered the first step towards an analysis of orbits x2 yz yz ln() x + r - arctan + xz ln around more general shaped bodies. These investigations, 2 xr for example, have application to future missions to 2 (3) y xz investigate asteroids, where orbits need to be calculated ()y + r - arctan + xy ln() z + r 2 yr around rotating non-spherical objects (Liu et al ., 2011a; 2 Werner and Scheeres, 1996; Michalodimitrakis and z xy - arctan et al et al 2 zr z=-D-Z Bozis, 1985; Liu ., 2011b; Scheeres ., 1998; y=-B-Y X=-L-X 1996; Yu and Baoyin, 2012a; 2012b; 2012c). In this study we firstly produce a calculation for the using x = x, x = y, x = z and D = L,D = B,D = D we field of a perfect cube or rectangular solid of uniform 1 2 3 1 2 3 can write Equation 4: density, with which we then investigate the formation of stable orbits around such objects using the method of Dj− X j Poincare sections, before concluding with some light- v ln xi + 1 3 3 () hearted remarks regarding the nature of life on a xi VX,Y,Z= − G ρ (4) ()1 2 3 ∑ ∑ 2 hypothetical cubic planet. j1= i1 = xi v − arctan 2 2 xi r 1.1. Analysis xj=− D j − X j 1.1.1. Gravitational Potential of a Cube For the gravitational potential of a right rectangular We first seek the Newtonian gravitational potential of prism mass, where v = x 1x2x3. The pairs of log and a rectangular solid of uniform density ρ with Newton's 1 arctan terms combine to produce the expected falloff universal gravitational constant G. We suppose the r rectangular solid has a length 2L, breadth 2B and depth in gravitational potential at large distances. For example, 2D, oriented along the x, y, z axes respectively, centered for a 2×2×2 m 3 cube, with Gρ = 1, as x → ∞, yz ln (x+r)- on the origin, then we have the potential Equation 1: x2 yz 8 arctan → − , as expected for a point source. 2 xr r V( X,Y,Z ) = In order to reveal the equipotentials around this D B L dx′ dy ′ dz ′ −G ρ object, we can imagine a very low density fluid being ∫−DBL ∫ − ∫ − 2 2 2 ()()()Xx−′ +− Yy ′ +− Zz ′ added to a face of a cube and by plotting Equation (3) at a constant potential, we find a lake as shown in Fig. 1 . A DZ− BY − LX − dxdydz (1) = −G ρ perfect circle is shown for comparison and we can see ∫zDZyBYxLX=−− ∫ =−− ∫ =−− 2 2 2 x+ y + x how the water is pulled up towards the corners due to the LX− BY − D− Z extra mass present there. =−ρG In z +++ x2 y 2 z 2 dydx, ∫x=−− LX ∫ y =− BY − () z=− D − Z The surface of the lake, for someone boating on the surface, will be approximately spherical in shape, as it is where, we made the substitution, x = x'-X, y = y'-Y and z for a large lake upon the Earth, as is also shown in Fig. = z'-Z and completed the integral over the z coordinate. 1, found by plotting Equation (3) at a constant potential Next, integrating over the y variable and using in three dimensions. 2 2 2 r= x + y + z , we find Equation 2: 1.2. Gravitational Field L− X We calculate the gravitational field vectors from V=−ρ G dx[[yln()() z ++ r zln y + r ∫x=− L − X Equation 5: D− Z B− Y y yz (2) −y + xarctan − xarctan r ∂V ∂ V ∂ V gV=−∇=− e1 + e 2 + e 3 (5) x xr z=− D − Z y=− B − Y ∂x ∂ y ∂ z Science Publications 51 PI James Michael Chappell et al. / Physics International 3 (2) (2012) 50-57 Fig. 1. A lake formed on the surface of a cube. As expected, the edge of the lake is ‘pulled up’ towards the corners, due to the extra mass present there, but forming a nearly spherical surface ∂v greater gravitational force of attraction over the The field vector in the x direction g x =− , can be ∂x corners of the cube and hence an orbiting satellite will deduced from Equation (2), before the last integral is suffer significant perturbation in these areas, refer calculated and hence by the fundamental theorem of Fig. 3 . A cube has three symmetry planes parallel to calculus we find Equation 6: its faces, six symmetry planes in the diagonal plane and four asymmetry planes that contain the regular D− X hexagonal cross section. It has been shown (Liu et al ., 3 yz j j gx =ρ G∑ yln()() z ++ r z ln y +− r x arctan (6) 2011a), that periodic orbits can form in all three cases, j= 1 xr xj=− D j − X j although only planar-type orbits form when orbiting around the axis of symmetry, that is parallel to the and from symmetry we can also easily deduce the field faces or diagonally across the corners. strengths in the y and z directions, giving the field If we take the planar satellite orbits, around the r 3 faces and the centre of the edges, we produce the strength vector g= ∑ i1= g i e i , where Equation 7: orbital Equation 8: gi = D− X && && 3 b b x= g,yx = g y (8) v (7) Gρ xjk ln()() x ++ r x kj ln x +− r x i arctan 2 ∑ x r b= 1 i x=− D − X b b b Solving this equation numerically for the specific case, of a satellite orbiting a cube with a side length equal to the for distinct i, j, k, where e i are the unit vectors for the x, y, z coordinate system.
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