Designer basis functions for potentials in galactic dynamics Prasenjit Saha Canadian Institute for Theoretical Astrophysics McLennan Labs, University

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1 Designer basis functions for potentials in galactic dynamics Prasenjit Saha Canadian Institute for Theoretical Astrophysics McLennan Labs, University of Toronto, 60 St. George St., Toronto M5S 1A7, Canada Summary: Existing methods for solving Poisson's equation using basis functions can be made considerably more exible, yet simpler, by using non-orthogonal basis functions. The rst part of this paper describes how such basis sets can be constructed. The second part gives explicit formulas (up to quadratures and matrix inversions) for basis functions being spherical harmonics times radial functions; two basis sets in the literature appear as special cases. To appear in Mon. Not. R. astr. Soc. 1

2 1 THE GENERAL PROBLEM Suppose we are doing a problem that requires approximate solutions of Poisson's equation. This may be a particle simulation, a gas simulation, or a normal mode analysis of a stellar dynamical system. Suppose also that we have a crude idea of the behaviour of the potential (such as asymptotic small-r and large-r forms) and further that is very smooth in particular we may want to smooth over the graininess in a simulation. Then it would be sensible to express the potential as ha `prior' form W i hpolynomial in some variable(s)i: (1:1) or as a sum of such terms. The prior form W is thus a zeroth order approximation to. This paper suggests how we can solve eciently for such forms of. 1.1 Known bases For some special W, there are elegant basis sets in the literature which allow one to solve Poisson's equation r 2 = 4 (1:2) in a simple and ecient way. These basis sets n are biorthonormal, i.e., m r 2 n d 3 x = mn : (1:3) Assuming the m are complete, we have = X n c n n ; c n = 1 4 n d 3 x: (1:4) Such basis sets are typically eigenfunctions of hsome weight functioni r 2 ; biorthogonality and completeness follow since r 2 is Hermitian. Since spherical harmonics are eigenfunctions of the angular part of r 2, one obvious form is lmp = Y lm (; ) F lp (r): (1:5) The basis sets given by Clutton-Brock (1973; hereafter CB73), Polyachenko & Shukhman (1981), and Hernquist & Ostriker (1992; hereafter HO92) are of the type (1.5), with F lp being eigenfunctions of the radial part of r 2 (apart from some weight function). For analogous examples in disc systems see Clutton-Brock (1972) and Kalnajs (1976). These bases are useful, but they are inexible in their choice of the weight function (or prior form) W ; they are also highly non-trivial to discover, so a worker wishing to tailor a basis set to a particular application has little hope of nding a new eigenfunction set with a well-matched W. Generality can be gained by dropping the requirement that the n be eigenfunctions of r 2 and simply constructing a biorthonormal set by a Gramm-Schmidt method. In Saha (1991) I adopted a basis of the form (1.5), with F lp not being eigenfunctions of anything in particular, but made biorthogonal by Gramm-Schmidt. Robijn (1992) uses a similar method, but with spheroidal harmonics rather than spherical harmonics for the angular part. 2

3 1.2 Non-biorthogonal bases In fact the biorthogonality requirement can be dropped as well. If we take any linearly independent and complete set n, we can expand = X n c n n (1:6a) where the expansion coecients c n satisfy X n M mn c n = 4 m d 3 x; where M mn = m r2 n d 3 x (1:6b) and M mn must be non-singular. Evaluating and inverting M mn needs to done only once, which will typically be a trivial overhead compared to computing R n d 3 x. The proposal of this paper is to take n of the form (1.1) and then use (1.6) to nd the expansion coecients. The polynomial factor guarantees linear independence of the n. Completeness in the strict sense does not follow, (but as illustrated later in this paper) a weaker form of completeness will hold that seems adequate for applications. 1.3 Discussion The second part of this paper is devoted to the details of the spherical-harmonic-timesradial-functions form (1.5). It turns out that the bases of CB73 and HO92 strictly speaking, linearly transformed versions of them appear as special cases. In two dimensions the problem is more dicult and are not related by a two-dimensional operator and it is not clear whether the scheme of this paper could be adapted to thin discs. For discs of nite thickness, however, bases using cylindrical coordinates seem feasible and appropriate. It is also worth mentioning that the technique suggested here is not specic to solving Poisson's equation. In particular, we could expand the momentum density p(x) from a sample of points in terms of some basis functions f n (x), thus: p = X n a n f n (1:7a) where the a n satisfy X n M mn a n = vf m d 3 x; where now M mn = f mf n d 3 x: (1:7b) Now if we combine such an expansion with a Poisson solver (1.6), the ratio 4 P m a nf n Pn c nr 2 n (1:8) gives a smoothed velocity eld. 3

4 2 HARMONIC EXPANSIONS 2.1 Formulas We assume the potential may be expanded as Then the c lmp are given by where S lm p = X lmp c lmp Y lm (; )F lp (r): (2:1) 4 pq 1 0 X q pq c lmq = S lm p (2:2) Y lm(; )F lp r 2 d 3 x (2:3a) F lp d dr r2 d dr? l(l + 1) F lq dr: (2:3b) We now do two things. First we assume a separate prior form W l for each multipole component, and factor this out, thus: pq =?1 F lp (r) = W l (r) U lp (r): (2:4) Then we change variables from r to some u having a nite domain, (say?1 to 1). This changes (2.3b) to 1 ( r 2 W l U lp r 0 W l d 2 + 2r 2 W 0 2 l + r 2? rr00 d W r 02 l + r 0h r 2 W 00 l + 2rW 0 l? l(l + 1)W li ) U lq (2:5a) where r 0 dr ; r00 d2 r 2 ; W 0 dw l l dr ; W 00 l d2 W l dr : 2 (2:5b) The cartesian components of the acceleration can be evaluated from the derivatives of thus: 10 1 x xz y r rr 2? CB C R 2 r? 0 B y r z r yz rr 2? 1 r x R 2 0 C A B sin C A (2:6) where R means p x 2 + y 2. This of course will require values of the Legendre polynomials and their derivatives, which are best got from their recursion relations. 4

5 2.2 Special cases CB73 takes W l = r l (1 + r 2 ) l+ 1 2?1 and u = r2? 1 r : (2:7) Inserting (2.7) into the form (2.5) and laboriously collecting terms yields: 1 Mpq lm = 2?(2l+1) (1? u 2 ) l+ 2 1 Ulp (1? u 2 ) d2 d? (2l + 3)u 2? 1 4 (2l + 1)(2l + 3) U lq : (2:8) HO92 take which gives: W l = pq = 2?(4l+3) 1?1 rl and u = r? 1 (1 + r) 2l+1 r + 1 : (2:9) (1? u 2 ) 2l+1 U lp (1? u 2 ) d2? 2(l + 1)(2l + 1) U lq : d? 4(l + 1)u 2 (2:10) It turns out that in both cases the eigenfunctions of the operator within the curly brackets in (2.10) and (2.8) are known functions ultraspherical polynomials and the integrals express the orthogonality relation for these eigenfunctions. Thus, if U lp is taken to be the appropriate ultraspherical polynomial then the matrix Mpq lm is diagonal. Now, for our purpose we don't need Mpq lm to be diagonal. Therefore, instead of taking U lp to be an p-th degree ultraspherical polynomial, we may take U lp = u p. This is a linearly independent basis related to the ultraspherical polynomials by a linear transformation, and will therefore give the same results for. What other choices of W l would be apt for studying (say) elliptical galaxies? One possibility would be W l = r l (1 + r)?2l ; =?1 r (1 + r)?? 1 ; (2:11a) with chosen depending on the problem at hand. The density corresponding to is = (4)?1 (1 + )r?2 (1 + r)??2 ; (2:11b) which is adjustable from a point mass ( =?1), through a Jae model ( = 0), to a Hernquist model ( = 1), and on to models with nite central density. 5

6 2.3 Completeness questions If we are trying to approximate some potential multipole moment G l (r) using our basis functions, can we make the error jg l j in our approximation arbitrarily small by including enough basis functions? Completeness in this sense does not hold, but the following weaker form of completeness does apply. Any continuous function f(u) with a nite domain can be approximated by a polynomial Q(u) in the sense that jf(u)? Q(u)j 2 (2:12) can be made arbitrarily small by making the degree of Q high enough. Since our basis consists of W l times polynomials in u, we can make jgl j 2 W 2 l dr (2:13) dr arbitrarily small. This type of argument could, of course, be applied in other coordinate systems as well. 2.4 Implementation I have written some C functions implementing the formulas of this section (and also some extra code necessary to test these). The chosen prior form is specied through user-supplied functions for r(u), u(r), dr=, W l (r), dw l =dr, and the integral in (2.5a). The code will then expand to the desired harmonic and polynomial order from a sample of masses and positions, and supply the potential and accelerations. There is also some code to t r 2 v to basis functions again with some chosen prior form and thus nd a smoothed velocity eld. (Since may become singular in some model systems, it seems better to expand r 2 v rather than v in basis functions.) Programs and documentation are available to anyone interested. ACKNOWLEDGMENTS I am grateful to James Binney, M. Slim Fayache, Lars Hernquist, Kathryn Johnston, and Scott Tremaine for discussions and correspondence. I also thank the Natural Sciences and Engineering Research Council of Canada for nancial support through an operating grant, and the Physics Department, Oxford University for their hospitality. REFERENCES Clutton-Brock, M., Astrophys. & Space Sci., 16, 101. Clutton-Brock, M., Astrophys. & Space Sci., 23, 55. Hernquist, L. & J. P. Ostriker, Astrophys. J., 386, 375. Kalnajs, A. J., Astrophys. J., 205, 745. Polyachenko, V. L. & I. G. Shukhman, Astron. h., 58, 933, English translation Sov. Astr., 25, 533. Robijn, F., Leiden preprint. Saha, P., Mon. Not. R. astr. Soc., 248,