Closed form expressions for the gravitational inner multipole moments of homogeneous elementary solids

Transcription

1 Closed form epressions for the gravitational inner multipole moments of homogeneous elementar solids Julian Stirling 1,2, and Stephan Schlamminger 1 1 National Institute of Standards and Technolog, 1 Bureau Drive, Gaithersburg, MD 2899, USA 2 Joint Quantum Institute, Universit of Marland, College Park, MD 2742, USA arxiv: v1 [gr-qc] 5 Jul 217 Jul 7, 217 Abstract Perhaps the most powerful method for deriving the Newtonian gravitational interaction between two masses is the multipole epansion. Once inner multipoles are calculated for a particular shape this shape can be rotated, translated, and even converted to an outer multipole with well established methods. The most difficult stage of the multipole epansion is generating the initial inner multipole moments without resorting to three dimensional numerical integration of comple functions. Previous work has produced epressions for the low degree inner multipoles for certain elementar solids. This work goes further b presenting closed form epressions for all degrees and orders. A combination of these solids, combined with the aforementioned multipole transformations, can be used to model the comple structures often used in precision gravitation eperiments. 1 Introduction In the field of precision gravitational measurements, the measurements and its associated analsis are often onl half of the battle in producing a result. The other half comes from computing the theoretical Newtonian gravitational interaction for comparison. Computation of gravitational fields, forces, and torques can be accomplished b calculating setuple integrals over the volumes of mass pairs, and summing for all pairs of source and test masses. Even with advanced methods to reduce these setuple integrals to quadruple integrals [1, 2], for certain elementar solids, this is etremel computationall intensive, especiall considering that for man measurements this needs to be entirel recalculated for multiple source mass positions. More efficient methods are available for sstems with favourable smmetries [3, 4]. An elegant method to compute gravitational interactions is to epand the problem in terms of regular solid harmonics r l i Y lm (θ i, φ i ), or more precisel its comple conjugate r l i Ylm (θ i, φ i ), of the masses closest to the origin of the chosen coordinate sstem and the irregular solid harmonics r (l+1) o Y lm (θ o, φ o ), of the masses furthest from this origin. Where r i := (r i, θ i, φ i ), and r o := (r o, θ o, φ o ) are vectors to positions inside the inner and outer masses respectivel. Triple integrals of ρ(r i )r l i Ylm (θ i, φ i ) over the volumes of the inner masses are referred to as the inner multipoles q lm, julian.stirling@nist.gov 1

2 where ρ(r) is the mass densit. Whereas triple integrals of ρ(r o )r o (l+1) Y lm (θ o, φ o ) over the volumes of the outer masses are referred to as the outer multipoles Q lm. The convergence condition for this epansion is that r i < r o for all positions integrated over. The gravitational potential energ of the sstem can be calculated as V = G l l= m= l 1 2l + 1 q lmq lm. (1) At first glance, an infinite sum over pairs of triple integrals is not necessaril a significant advance over a brute force calculation of the setuple integrals. The power of the multipole epansion becomes apparent when considering comple eperiments with multiple source and test masses. The multipole moments can be calculated for each individual mass just once and then used with all masses it interacts with, as other masses change no recalculation is needed. Also multipole moments can easil undergo translations [5] and rotations [6, 7]. As such, when calculating the effect of a mass moving, ver few new calculations are needed. Furthermore, outer multipoles can be computed from inner multipoles of the same shape [8]. Utilising the multipole transformations, the onl other calculations needed are the inner multipole moment of each mass at an arbitrar location, which is eas to calculate. Forces[9] and torques[1] can also be directl calculated from these multipoles. Efficient calculation of inner multipole moments is, as such, of great value. Low-degree (l 5) inner multipole moments have been calculated individuall for each order (m) for a number of elementar solids [11]. For higher degrees, however, either numerical methods must be emploed or each order must be calculated eplicitl. In this work we develop closed form solutions for the inner multipole moments a number of solids. These, combined with the multipole transformations, can be used to calculate gravitational interactions between comple apparatus to an required accurac with relative ease. 2 Closed forms epressions for inner multipoles For calculating inner multipoles is is helpful to write the regular solid harmonics in the clindrical coordinate sstem. From Eqn in Ref [12], the solid harmonics are given in Cartesian coordinates. It is trivial to convert this form into clindrical coordinates r l Y lm (θ, φ) = ( 1) m e imφ k ( 1) k r c 2k+m z l 2k m 2 2k+m (m + k)!k!(l m 2k)!, (2) where k is summed over all values where each factorial is non-negative. Here we are careful with our notation such that r and r c are the radial distances in the spherical and clindrical coordinate sstems respectivel, φ is the azimuthal angle for both coordinate sstems, θ is the spherical polar angle, and z is vertical position. For simplicit, we will calculate all closed forms for m. The inner multipole moments for negative m can easil be calculated with the following identit q l( m) = ( 1) m q lm. (3) 2

3 2.1 Inner multipoles of a clinder From smmetr we can see that the inner multipoles q lm of a homogeneous clinder of densit ρ requires m = due to rotational azimuthal smmetr and for l to be even from vertical smmetr. Using Eqn. 2 and integrating over the volume of the clinder with radius R and height H centred on the origin (See Figure 1(a) and (f)) 2l + 1 q l = ρ r l Yl(θ, φ) dv c = ρ V c l! = M 2l + 1 l/2 l! 2 l ( 1) k 2 2k k!k!(l 2k)! l/2 H/2 H/2 R 2π r c 2k z l 2k r c dφ dr c dz ( 1) k R 2k H l 2k k!(k + 1)!(l 2k + 1)!, (4) for l even, where M is the mass of the clinder. This result is consistent with that derived b Lockerbie, Veraskin, and Xu [13] and has the useful form of being the mass of the object multiplied b a geometrical factor. For efficient programming this equation can easil be written as a simple recursion relation: l/2 2l + 1 q l = M S(l, k), (5) where S(, ) = 1, (6) (l + 1)H2 S(l + 2, ) = S(l, ), 4(l + 3) (7) (l 2k + 1)(l 2k) R 2 S(l, k + 1) = S(l, k). (k + 1)(k + 2) H2 (8) 2.2 Inner multipoles of an annular section A more generalised case for the clinder is an annular section with inner radius R i and outer radius R o which etends over the azimuthal angular range from φ c φ h to φ c + φ h (See Figure 1(b) and (f)). The z integral can be solved separatel (A.1). The integral to solve is then R o φ c+φ h R i φ c φ h e imφ r 2k+m c r c dφ dr c = 2 ( R 2k+m+2 o R ) 2k+m+2 i 2k + m + 2 { e imφc sin(mφ h) m for m φ h for m =. (9) From A.1 we know that from vertical smmetr that (l m) must be even. We then combine the above result with Eqn. 22, and the other terms for q lm in front of the integral. To write the multipole as the mass multiplied b a geometric factor we need to factor out the volume 3

4 (a) (b) (c) R a R i R o d R (d) b a a (e) (f) H z - plane Figure 1: (a) (e) Cross sections of clinder, annular section, isosceles triangular prism, cuboid, and N-sided regular polgonal prism respectivel. (f) Side view for all aforementioned prisms. φ h (R 2 o R 2 i )H, e imφc sinc(mφ h ) ( 1) k+m H l 2k m 2 l 1 k!(m + k)!(l m 2k + 1)!(2k + m + 2) ( 2k+m+2 2k+m+2 ) Ro R i R 2 2, for (l m) even, and m, (1) o R i where we note that using the sinc function removes the need for separate cases for m = and m. This equation can be shown to be consistent with the results given in Adelberger et al. [11]. 2.3 Inner multipoles of an isosceles triangular prism Here we define an isosceles triangle using the same geometr as the annular section ecept with onl one radius R, with φ h < π (See Figure 1(c) and (f)). Using the solution for the z-integral for 2 a prism (A.1), the remaining integrals to solve are R cos φ h cos(φ φc) φ c+φ h φ c φ h e imφ r 2k+m c r c dφ dr c = (R cos φ h) 2k+m+2 2k + m + 2 φ c+φ h φ c φ h e imφ dφ, (11) cos 2k+m+2 (φ φ c ) 4

5 which is solved in A.2. Factoring out the mass M = ρhr 2 cos 2 φ h tan φ h, e imφc 2 l 1 ( 1) k+m H l 2k m (R cos φ h ) 2k+m (m + k)!k!(l m 2k + 1)!(2k + m + 2) ( ) k ( ) m k tan ( 1) p 2j+2p φ h, for (l m) even, and m, (12) 2p j 2j + 2p + 1 where m 2 denotes rounding m down to the nearest integer. Calling the base of the triangle 2 a = 2R sin φ h and the shortest line to the base d = R cos φ h, a more simple form is: ( ) k m ( 1) p 2p ( ) k j e imφc 2 l 1 1 2j + 2p + 1 ( a 2d 2.4 Inner multipoles of a cuboid ( 1) k+m H l 2k m d 2k+m (m + k)!k!(l m 2k + 1)!(2k + m + 2) ) 2j+2p, for (l m) even, and m. A cuboid can be described as a sum of two pairs of isosceles triangular prisms. Defining a cuboid of height (z-ais) H to be consistent with the above prisms, the other other two sides a and b are defined such that when φ c =, a is parallel to the -ais and b is parallel to the -ais (See Figure 1(d) and (f)). B smmetr we can see that the each pair of isosceles triangles are offset b an angle π therefore m must alwas be even for a nonzero multipole. As with all prisms centred in z, (l m) must be even, and therefore l is also even. The inner multipoles for a cuboid are thus: ( 1) m/2 e imφc ( 1) k H l 2k m (m + k)!k!2 l+2k+m (l m 2k + 1)!(2k + m + 2) m/2 ( ) k ( ) m k a ( 1) p 2k+m 2j 2p b 2j+2p + b 2k+m 2j 2p a 2j+2p, 2p j 2j + 2p + 1 (13) for both m and l even, and m. (14) 2.5 Inner multipoles of an N-sided regular polgonal prism Consider an N-sided regular polgonal prism, with height H with its centre of figure at the origin. The angle between the right-most side (in the -plane) and the -ais is φ c. The side length of the polgon is a (See Figure 1(e) and (f)). The inner multipole moments can easil be calculated b combining the results for N identical isosceles triangular prisms each rotated b an angle 2π N with respect to the last. B smmetr, the angular term for the N prisms add to N if m is a 5

6 (a) (b) z R P - plane Figure 2: (a) Base of azimuthal section of a cone. (b) Side view of cone. multiple of N, or else it vanishes, hence the moment is simpl (again non-zero for (l m) is even): e imφc ( 1) k+m H l 2k m a 2k+m ( ) m ( 1) p (m + k)!k!2 l+2k+m 1 (l m 2k + 1)!(2k + m + 2) 2p k ( ) ( ) k tan 2j+2p 2k m π N for (l m) even, and m =, N, 2N,.... (15) j 2j + 2p Inner multipoles of an azimuthal section of a cone Consider a cone with a base centred at the origin with a radius R, the ape of the cone is on the z-ais with z = P. The cone is defined in the azimuthal angular range from φ c φ h to φ c + φ h (See Figure 2). The azimuthal integral for the inner multipoles was alread solved in Section 2.2. The radial and z integrals are: P R zr P r c 2k+m+1 z l 2k m dr c dz = = (R/P )2k+m+2 2k + m + 2 P (P z) 2k+m+2 z l 2k m dz (16) (2k + m + 1)!(l 2k m)! R 2k+l+2 P l 2k m+1. (17) (l + 3)! We can therefore write the inner multipole moments as q lm = 3M (l + 3)!e imφc sinc(mφ h ) ( 1) k+m (2k + m + 1)!R 2k+m P l 2k m, for m. (18) 2 2k+m 1 (m + k)!k! 6

7 Figure 3: Top view of two overlapping holes which can be modelled as two clindrical sections in the angular range without overlap, plus two isosceles triangular prisms. 3 Discussion Care must be taken, however, when performing numerical calculations. First man programming languages define sinc() as sin(π) rather than sin(). Also, for large degree multipole moments, π numerical rounding errors become significant as the sum over k has terms with alternating sign which individuall can be man orders of magnitude larger than the final result. As a rule of thumb we find that for l 5 quadruple-precision floats should be used for calculations requiring precision better than 1 part in 1 6. Using quadruple-precision floating point operations we have found results are still accurate beond double-precision for l > 1. This can be checked on an individual basis b comparing the ratio of the magnitude of largest term in the sum over k and the sum itself to the numerical precision of the data tpe used. For eample, we can write an inner multipole moment of an object as q lm = A( ) k ( 1) k B k ( ), (19) where A and B are functions of the variables needed to describe the object. We can then estimate the relative error in our numerical calculations as q lm q lm ma(b k( )) k ( 1)k B k ( ) P B k, (2) where P Bk is the numerical precision of the floating point data tpe used to store B k and its sum (P Bk 1 16 for double-precision and P Bk 1 34 for quadruple-precision). This estimate assumes all other sources of numerical error are negligible. 4 Conclusion We have derived the inner multipole moments for a number of homogeneous elementar solids in terms of their mass multiplied b a geometrical factor. Using the translation and rotation equations for multipoles, a number of more comple but commonl occurring shapes can be modelled. For eample, overlapping clindrical holes can be modelled as two clindrical sections in the angular range without overlap, plus two isosceles triangular prisms, all with negative mass (see Figure 3); an irregular polgon prism can be modelled as a combination of isosceles triangular prisms; or a truncated cone can be modelled as one cone subtracted from another. The equations provided are 7

8 relativel simple to code to allow multipole calculations of Newtonian gravitational interactions between comple structures to an desired degree. 5 References References [1] Y. T. Chen and A. Cook, Gravitational eperiments in the laborator (Cambridge Universit Press, Cambridge, 1993), p [2] J. Stirling, A vector field approach to calculating gravitational forces, New Journal of Phsics In Press. [3] H. S. Cohl and J. E. Tohline, The Astrophsical Journal 527, 86 (1999). [4] J. P. Selvaggi, S. Salon, and M. V. K. Chari, Classical and Quantum Gravit 25, 1513 (28). [5] C. D Urso and E. Adelberger, Phsical Review D 55, 797 (1997). [6] J. D. Jackson, Classical Electrodnamics (John Wile & sons, Inc, New York, 1999), chap. 9. [7] D. A. Varshalovich, A. N. Moskalev, and V. K. V K Khersonskii, Quantum theor of angular momentum (World scientific, New Jerse, 1988), chap. 5. [8] C. Trenkel and C. C. Speake, Phsical Review D 6, 1751 (1999). [9] J. Stirling, Phs. Rev. D 95, (217). [1] R. Newman, M. Bantel, E. Berg, and W. Cross, Philosophical Transactions of the Roal Societ A: Mathematical, Phsical and Engineering Sciences 372, (214). [11] E. G. Adelberger, N. A. Collins, and C. D. Hole, Classical and Quantum Gravit 23, 125 (26). [12] W. J. Thompson, Angular Momentum (John Wile & sons, Inc, New York, 1994), chap. 4. [13] N. A. Lockerbie, A. V. Veraskin, and X. Xu, Classical and Quantum Gravit 1, 2419 (1993). A Integrals A.1 z-integral for prisms For prismatic solids, the z-integral can be solved separatel from the other two coordinates: [ H/2 ( H ) l 2k m+1 ( 1 f lmk (H) := z l 2k m 2 H ) ] l 2k m+1 2 dz = 2 2k+m (l m 2k)! 2 2k+m (l m 2k)!(l 2k m + 1). (21) H/2 8

9 The integral vanishes if (l m) is odd, therefore: f lmk (H) = A.2 Integral used for flat sides H l 2k m+1 2 l (l m 2k + 1)!, (22) for (l + m) even, and zero otherwise. To integrate the radial coordinate over a flat edge the following integral must be solved: g km (φ c, φ h ) := φ c+φ h φ c φ h Substituting φ i = φ φ c for smmetr, then e imφc φ h φ h e imφ i cos 2k+m+2 φ i dφ i = e imφc e imφ dφ. (23) cos 2k+m+2 (φ φ c ) φ h φ h cos(mφ i ) cos 2k+m+2 φ i dφ i, (24) where the imaginar part of the integral is odd and therefore evaluates to zero. But for m ( ) m cos(mφ i ) = ( 1) p cos m 2p φ i sin 2p φ i. (25) 2p Substituting this into Eqn. 24 gives ( m g km (φ c, φ h ) = e imφc ( 1) p 2p The integral in the sum can be rewritten as ) φ h φ h tan 2p φ i (cos 2 φ i ) k+1 dφ i. (26) φ h φ h (1 + tan 2 φ i ) k tan 2p φ i cos 2 φ i dφ i, (27) using the identit 1 + tan 2 φ i = 1 cos 2 φ i. If we substitute = tan φ i, then tan φ h tan φ h (1 + 2 ) k 2p d = 2 and therefore b substituting Eqn. 28 into Eqn. 26, we conclude that φ c+φ h φ c φ h e imφ dφ = cos 2k+m+2 2e imφc (φ φ c ) k ( ) k tan 2j+2p+1 φ h j 2j + 2p + 1, (28) ( ) k m ( 1) p 2p ( ) k tan 2j+2p+1 φ h j 2j + 2p + 1, for m. (29) 9