Galactic Astronomy 2016

Transcription

1 10 Potential theory The study of dynamics is fundamental to any understanding of galaxies. The following few chapters provide a brief summary of the most important results. More details can be found in Binney and Tremaine (197). For the topics discussed here, Newtonian gravitation theory is su cient. Only in regions close to black holes found in the centres of galaxies are relativistic e ects important. Because galaxies contain such a large number of stars, the gravitational potential may generally be regarded as smooth and constant. A large part of the stellar dynamics problem is then the determination the potential and the orbits of stars in that potential. The exception occurs during the collapse phase, or during encounters with other galaxies, when the potential can change on a timescale comparable to the orbital period Gravitational potential The gravitational force vector can be written as the gradient of a scalar function U. This results in considerable simplification of dynamical equations. For a continuous distribution of mass pxq, Newtons law gives a force per unit mass x 1 x F pxq x 1 x 3 G px1 qd 3 x 1, (10.1) where G is Newton s constant and r pxq. (10.2) G px 1 q pxq x 1 x d3 x 1. (10.3) is the scalar gravitational potential. Di erentiating the potential and applying Gausss theorem gives Poisson s equation r 2 pxq pxq. (10.4) In empty space 0 and this becomes Laplace s equation r 2 pxq 0. (10.5) Page 63

2 10.2 Potential energy of a mass distribution The fact that the gravitational force can be written as the gradient of a potential means that it is conservative force the energy expended by a test particle moving from point A to point B does not depend on the path taken. In particular, the energy required to bring a small mass m from infinity to x is dw m m m x x x r F ds, ds, d psq ds ds, m pxq. (10.6) (which of course is negative). Therefore, to create a change pxq in density by bringing in matter from infinity requires energy W pxq pxqd 3 x. (10.7) This produces a change in the potential pxq that satisfies Poisson s equation, r 2 pxq pxq, hence W 1 r 2 d 3 x, 1 1 rr p r q r r s d 3 x, r p r q 1 2 pr r q d 3 x. (10.) The first term is a perfect divergence and so can be converted to a surface integral by Gauss s theorem. If the volume of integration extends to infinity and the mass distribution is localized, the potential goes to zero on the surface and this term vanishes. We then have W 1 G r 2 d 3 x. (10.9) To obtain a second form, apply the divergence theorem and Poisson s equation, W 1 ˇˇrp r q r 2 d 3 x, G 1 d 3 x. (10.10) 2 The Chandrasekhar potential energy tensor is defined by W j k x j B Bx k d3 x. (10.11) Page 64

3 This tensor is symmetric and its trace is the potential energy, W W j j ÿ W j j, j x r d 3 x. (10.12) For a spherically symmetric body, the potential energy tensor is diagonal and W where Mprq is the mass interior to radius r. 0 M prqrdr, (10.13) 10.3 Newton s theorems Two elementary theorems are easily proved: 1. A body inside a spherical shell of matter experiences no net gravitational force. 2. A body outside a spherical shell of matter experiences the same force as it would if all the shell s mass were concentrated into a point at its centre. These imply that the e ect of a spherically-symmetric mass distribution is the same as if the mass interior to r were concentrated into a point at the centre Thin disk For a thin axisymmetric disk, B 2 1 B pr, zq Bz2 R BR R B BR,» pr, zq. (10.14) where the second term on the right, the divergence of the radial force, is much smaller than the first. Therefore the vertical force on a particle above the disk is F z B Bz» pr, zqdz, where prq is the mass surface density of the disk.» prq, (10.15) Page 65

4 10.5 Potential-density pairs Here are some examples of potential-density pairs: Point mass M prq GM r, (10.16) pxq M 3 pxq (10.17) Homogeneous sphere of radius a and density $ & 2 G 3 p3a2 r 2 q r a prq % a3 r a # 3r r a prq 0 r a (10.1) (10.19) Plummer potential prq? GM r2 ` b, (10.20) 2 prq 3M 4 b 3 3{2 ˆ1 ` r2. (10.21) Compare this with the luminosity distribution that produces the Modified Hubble intensity profile. b 2 Kuzmin potential GM pr, zq a (10.22) R2 `pa` z q 2 pr, zq am 2 pr2 ` a 2 q 3{2 pzq prq pzq. (10.23) This potential, created by a thin disk, mimics that of a point mass M located a distance a along the axis on the opposite side of the disk. Miyamoto-Nagai potential GM pr, zq br 2 `pa`?z, (10.24) 2 ` b 2 q 2 pr, zq b2 M 4 ar 2 `pa`3? z 2 ` b 2 qpa `?z 2 ` b 2 q 2 rr 2 `pa`?z. (10.25) 2 ` b 2 q 2 s 5{2 pz 2 ` b 2 q3{2 Page 66

5 This potential can be varied continuously between the Plummer and Kuzmin potentials. It can therefore describe a range of flattened systems. Multipole expansion A general solution that gives the potential for any mass distribution is provided by an expansion in spherical harmonics and radial polynomials, Yl m p q 1 r 2l ` 1 r l`1 lm pxqx l`2 dx ` r l lm pxqx 1 l dx, (10.26) pr, q ÿ l,m lm 0 pr, qy m l p qd. (10.27) where p, q and d sinp qd d. r The dipole term (l 1) vanishes if the origin is taken at the centre of mass. The spherical harmonics, Yl m p q obey the following relations Yl m p qyl m1 1 p qd ll 1 mm 1, r2 Yl m p q 0. (10.2) Page 67