On the Gravitational Fields of Maclaurin Spheroid Models of Rotating Fluid Planets

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ON THE GRAVITATIONAL FIELDS OF MACLAURIN SPHEROID MODELS OF ROTATING FLUID PLANETS

Dali Kong1, Keke Zhang1, and Gerald Schubert2 1 Center for Geophysical and Astrophysical Fluid Dynamics and Department of Mathematical Sciences, University of Exeter, Exeter EX4 4QF, UK; [email protected], [email protected] 2 Department of Earth and Space Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA, USA; [email protected] Received 2012 October 15; accepted 2012 December 14; published 2013 January 28

ABSTRACT Hubbard recently derived an important iterative equation for calculating the gravitational coefficients of a Maclaurin spheroid that does not require an expansion in a small distortion parameter. We show that this iterative equation, which is based on an incomplete solution of the Poisson equation, diverges when the distortion parameter is not sufficiently small. We derive a new iterative equation that is based on a complete solution of the Poisson equation and, hence, always converges when calculating the gravitational coefficients of a Maclaurin spheroid. Key words: gravitation – hydrodynamics – planets and satellites: general – planets and satellites: interiors Online-only material: color figure

1. INTRODUCTION potential Vg which are a solution of the Poisson equation, 2 An oblate Maclaurin spheroid model of a rapidly rotating ∇ Vg =−4πGρ, (3) fluid planet or star is described by the equation must remain constant on the bounding surface of the Maclaurin x2 y2 z2 a spheroid. In order to determine the shape (i.e., the size of α + + = 1orr =˜r(μ) = , (1) or l) and the external gravity field of a Maclaurin spheroid, an a2 a2 b2 1+l2μ2 expression for the gravitational potential Vg is needed in the external domain b r a. Hubbard’s (2012) iterative method where (x,y,z) are Cartesian coordinates with z along the axis of is based on the expression for the total potential U in the form rotation, (r, θ, φ) are spherical polar coordinates with μ = cos θ, and a and b (a>b) are the equatorial and polar radii of U(r, μ) = V (r, μ)+V (r, μ) g c the Maclaurin spheroid, respectively. The distortion parameter ∞ 2k due to the effect of rotation is given by either αor l defined GM a = 1 − J2kP2k(μ) = − = 2 − 2 r r as α (a b)/a with 0 <α<1orl (a b )/b k=1 with 0

1 The Astrophysical Journal, 764:67 (4pp), 2013 February 10 Kong, Zhang, & Schubert the multipole moments J2k for k = 1, 2, 3,...,kmax, where k is the truncation parameter in expansions (4) and (7). This max ∞ ω is because the summation k=1 in Equation (4)or(7)must kmax (r, μ) be replaced with k=1 in any practical computation. For a convergent expansion, however, the solution of the problem, A e.g., U(r, μ), would become independent of kmax for sufﬁciently b (III) large kmax.

3. A NEW ITERATIVE EQUATION (I) (II) We show that, although Equations (4) and (7) do not explicitly a O require an expansion in a small distortion parameter, the small- distortion condition is still implicitly required in the iterative equation (7) derived by Hubbard (2012). This is because the (IV) expansion of the gravitational potential Vg used in Equation (4) B or (7) is based on an incomplete solution of the Poisson equation (3) in the external domain b r a and, hence, diverges when the distortion parameter is not sufficiently small. We derive a new iterative equation that is based on a complete Figure 1. Sketch of different integration domains in a Maclaurin spheroid in solution of the Poisson equation (3) and, thus, always converges which (r, μ) represents an external√ point in the domain b r a, the point A 2 2 when calculating the gravitational coefficients of Maclaurin has coordinates (r, μ = μr = a − r /(rl)), and the point B has coordinates spheroids. (r, μ =−μr ). We begin by expressing the gravitational potential (A color version of this figure is available in the online journal.) Vg(r, μ, φ), a solution of the Poisson equation (3), in the form

ρ(r,μ)dV separately because they have different forms of solution for the V (r, μ, φ) = G , (8) Poisson equation (3). g |r − r| V Consider ﬁrst the gravitational potential Vg in the region r>a for which only the expansion (11) is needed and, hence, the where denotes the volume integration over the Maclaurin V analysis is simple. It is straightforward, after making use of spheroid, r = (r, μ, φ) is the position vector located in the Equation (11), to show that exterior of the Maclaurin spheroid, r = (r,μ,φ) denotes the position vector of the density ρ(r ,μ,φ) within the interior of +1 r˜ ∞ l the Maclaurin spheroid, and (r ) Vg(r, μ) = ρG Pl(μ)Pl(μ ) − rl+1 1 0 l=0 ∞ l 1 4π (r )l = × (r )2 dr dμ when r a. (13) |r − r| 2l +1 rl+1 l=0 m=−l Carrying out the integration in the radial direction and noting × Y m(μ, φ)Y m(μ ,φ) when r>r, (9) l l that 1 M = (4πρ/3) [r˜(μ)]3 dμ, ∞ l l 0 1 = 4π r |r − r| 2l +1 (r)l+1 we obtain l=0 m=−l ∞ × m m 2k Yl (μ, φ)Yl (μ ,φ) when rr, (11) | − | l+1 l l the gravitational potential V (r, μ) in this domain is much r r = r g l 0 more complicated than Equation (14). Let r = (r, μ)bean ∞ l external point in the domain b r a which is displayed 1 r in Figure 1. There always exist two circles representing the = Pl(μ)Pl(μ ) when rsphere of the radius and the bounding surface r˜ of the Maclaurin spheroid. It is sufﬁcient, because of We have to consider the gravitational potential Vg(r, μ)inthe the axisymmetry property, to illustrate various regions of the two different external domains, marked by r>aand b

2 The Astrophysical Journal, 764:67 (4pp), 2013 February 10 Kong, Zhang, & Schubert circles in a meridional plane√ are marked by the point A whose be written as = 2 − 2 coordinates are (r, μr a r √/(rl)) and by the point B ∞ μr r˜(μ ) 2k =− 2 − 2 r whose coordinates are (r, μr a r /(rl)). It follows 4πρG P (μ)P (μ )(r )2 dr dμ 2k+1 2k 2k that, in contrast to the domain r>a, the gravitational potential 0 r (r ) k=0 Vg(r, μ)inb

3 The Astrophysical Journal, 764:67 (4pp), 2013 February 10 Kong, Zhang, & Schubert

2.5 is illustrated in Figure 2 for a Maclaurin spheroid with α = 0.35, showing the total potential U(r,˜ cos θ) on the bounding 2 surface of the Maclaurin spheroid as a function of θ.Inthe case of a convergent expansion, we anticipate that the scaled 1.5 total potential U(r,˜ cos θ)/(GM/a) on the bounding surface of the Maclaurin spheroid not only remains constant (which = ) 1 is U(a,0)/(GM/a) 1.2622) but also is independent of the truncation parameter kmax. Using three different truncations 0.5 kmax = 15, 20, 25 in our Equations (17) and (18), our calculation GM/a ( shows that the total scaled potential U(r,˜ cos θ)/(GM/a), which U/ 0 is depicted in Figure 2 on the solid line, remains constant and independent of the truncation parameter kmax. The scaled total −0.5 potential U(r,˜ cos θ)/(GM/a) calculated using Equations (4) kmax =15, 20, 25 and (7) given by Hubbard (2012) becomes strongly divergent in kmax =15 −1 k =20 the polar regions where the departure from spherical geometry max = ˜ kmax =25 is the largest. At kmax 15, the value of U(r,cos θ)/(GM/a) −1.5 oscillates slightly in the polar regions with a small amplitude 0 0.1 0.2 0.3 0.4 0.5 θ |U(b, 1)/(GM/a) − 1.26|≈0.26 which is shown by the dashed = line in Figure 2. The amplitude of oscillation increases to Figure 2. Scaled total potential U for a Maclaurin spheroid with α 0.35 | − |≈ = is plotted on its bounding surface as a function of θ. Three solutions using U(b, 1)/(GM/a) 1.25 2.3atkmax 25 which is shown three different truncation parameters kmax = 15, 20, 25 are calculated. The by the circled line in Figure 2. When kmax increases further, horizontal solid line represents the solutions obtained from Equations (17) the amplitude of divergent oscillation becomes too large to = = and (18)forkmax 15, 20, 25 while other lines (the dashed line for kmax 15, be shown in the figure. In fact, the divergent behavior of the dotted line kmax = 20, and the circled line for kmax = 25) are calculated from Equations (4)and(7). Equations (4) and (7) should be expected because they are based on an incomplete solution of the Poisson equation (see 4. DISCUSSION a relevant discussion in Section 38 of Zharkov & Trubitsyn 1978). However, our new iterative equation (18)—that is based It is important to point out that when the distortion parameter on a complete solution of the Poisson equation and contains α or l is sufficiently small, our Equations (17) and (18) approach the four extra terms—always converges for calculating the Equations (4) and (7) given by Hubbard (2012). This is because gravitational fields of any Maclaurin spheroid at any practically large kmax. K0 → 0,K2k → 0,N0 → 0,N2 → 0,N2k → 0as α → 0. K.Z. is supported by UK NERC, STFC, and Leverhulme In other words, the iterative equation (7) derived by Hubbard grants and G.S. is supported by the National Science Foundation (2012) mathematically represents the small flattening limit of under grant NSF AST-0909206. our iterative equation (18). Our calculation shows that when the distortion parameter α is not sufficiently small (α 0.29), REFERENCES Hubbard’s (2012) expansions (4) and (7) diverge and, hence, cannot produce physically meaningful solutions. Hubbard, W. B. 2012, ApJL, 756, L15 Lamb, H. 1932, Hydrodynamics (Cambridge: Cambridge Univ. Press) Comparison of the convergent behavior of Equations (17) Zharkov, V. N., & Trubitsyn, V. P. 1978, Physics of Planetary Interiors (Tucson, and (18) with the divergent behavior of Equations (4) and (7) AZ: Pachart)