Galaxies Part II Potentials due to axisymmetric matter distributions Potential due to thin disk The Oort Constants: the Stellar Dynamics and Structure of Galaxies rotation of our Galaxy Potentials due to axisymmetric distributions of matter The Oort Constants and epicycles The Rotation Curve of our Galaxy Spiral Galaxy Rotation Vasily Belokurov Curves [email protected] Institute of Astronomy Lent Term 2016 1 / 89 Galaxies Part II OutlineI Potentials due to axisymmetric matter 1 Potentials due to axisymmetric matter distributions distributions Legendr´ePolynomials Potential due to thin disk The Oort Constants: the Example rotation of our Galaxy Motion of the Moon The Oort Constants and epicycles 2 Potential due to thin disk The Rotation Curve of our Galaxy Bessel Functions Spiral Galaxy Rotation Poisson's equation Curves Summary of derivation of Φ for thin axisymmetric disk Examples Mestel disk Exponential Disk 3 The Oort Constants: the rotation of our Galaxy 4 The Oort Constants and epicycles 5 The Rotation Curve of our Galaxy 6 Spiral Galaxy Rotation Curves 2 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to 2 axisymmetric matter Outside matter, ρ = 0 , so r Φ = 0. distributions Legendr´ePolynomials Axisymmetric, so independent of spherical polar φ, Example Motion of the Moon Φ = Φ(r; θ) only Potential due to thin disk The Oort Constants: the rotation of our Galaxy The Oort Constants and epicycles The Rotation Curve of our Galaxy Spiral Galaxy Rotation Curves 3 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to 2 axisymmetric matter Outside matter, ρ = 0 , so r Φ = 0. distributions Legendr´ePolynomials Axisymmetric, so independent of spherical polar φ, Example Motion of the Moon Φ = Φ(r; θ) only Potential due to thin disk The Oort Constants: the Then r2Φ = 0 becomes rotation of our Galaxy The Oort Constants and 1 @ @Φ 1 @ @Φ epicycles 2 2 r + 2 sin θ = 0 : The Rotation Curve of r @r @r r sin θ @θ @θ our Galaxy Spiral Galaxy Rotation If we assume the function Φ(r; θ) is separable, ie Curves Φ(r; θ) = R(r) × Θ(θ) then, as substituting and dividing by RΘ, 1 d dR 1 d dΘ r 2 + sin θ = 0 R dr dr Θ sin θ dθ dθ i.e. each term has to be constant (and one = - the other) 4 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to 2 axisymmetric matter Outside matter, ρ = 0 , so r Φ = 0. distributions Legendr´ePolynomials Axisymmetric, so independent of spherical polar φ, Example Motion of the Moon Φ = Φ(r; θ) only Potential due to thin disk The Oort Constants: the Then r2Φ = 0 becomes rotation of our Galaxy The Oort Constants and 1 @ @Φ 1 @ @Φ epicycles 2 2 r + 2 sin θ = 0 : The Rotation Curve of r @r @r r sin θ @θ @θ our Galaxy Spiral Galaxy Rotation If we assume the function Φ(r; θ) is separable, ie Curves Φ(r; θ) = R(r) × Θ(θ) then, as substituting and dividing by RΘ, 1 d dR 1 d dΘ r 2 + sin θ = 0 R dr dr Θ sin θ dθ dθ i.e. each term has to be constant (and one = - the other) 4 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter distributions Let Legendr´ePolynomials Example 1 d dR Motion of the Moon r 2 = n(n + 1) (say) (4.1) Potential due to thin disk R dr dr The Oort Constants: the rotation of our Galaxy 1 d dΘ The Oort Constants and then sin θ = −n(n + 1)Θ(θ) (4.2) epicycles sin θ dθ dθ The Rotation Curve of our Galaxy Set µ = cos θ (−1 ≤ µ ≤ 1) Spiral Galaxy Rotation Curves d dΘ (1 − µ2) + n(n + 1)Θ = 0 (4.3) dµ dµ since dµ = − sin θdθ { Legendres equation. £ Want Θ(θ) to be finite at θ = 0; π (µ = ±1). And the only way this¢ can happen is if ¡ n = 0; 1; 2 . - a non-negative integer 5 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter distributions Let Legendr´ePolynomials Example 1 d dR Motion of the Moon r 2 = n(n + 1) (say) (4.1) Potential due to thin disk R dr dr The Oort Constants: the rotation of our Galaxy 1 d dΘ The Oort Constants and then sin θ = −n(n + 1)Θ(θ) (4.2) epicycles sin θ dθ dθ The Rotation Curve of our Galaxy Set µ = cos θ (−1 ≤ µ ≤ 1) Spiral Galaxy Rotation Curves d dΘ (1 − µ2) + n(n + 1)Θ = 0 (4.3) dµ dµ since dµ = − sin θdθ { Legendres equation. £ Want Θ(θ) to be finite at θ = 0; π (µ = ±1). And the only way this¢ can happen is if ¡ n = 0; 1; 2 . - a non-negative integer 5 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Legendr´ePolynomials distributions d n dΘ o Legendr´ePolynomials 2 dµ (1 − µ ) dµ + n(n + 1)Θ = 0 4.3 Example § ¤ Motion of the Moon Solution of (4.1) is Potential due to thin disk ¦ ¥ The Oort Constants: the rotation of our Galaxy n B R(r) = Ar + n+1 The Oort Constants and r epicycles The Rotation Curve of And (4.3) is the defining differential equation for Legendr´epolynomials, so we have our Galaxy Spiral Galaxy Rotation Curves and Θ(θ) = CPn(cos θ) for a given n Presumably you've seen Legendr´epolynomials before: P0(x) = 1 Pn(1) = 18n P1(x) = x 1 2 P2(x) = 2 3x − 1 6 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Legendr´ePolynomialsmials distributions Legendr´ePolynomials Example Motion of the Moon Potential due to thin disk The Oort Constants: the rotation of our Galaxy The Oort Constants and epicycles The Rotation Curve of our Galaxy Spiral Galaxy Rotation Curves 7 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Legendr´ePolynomials distributions Legendr´ePolynomials Recurrence relation: Example Motion of the Moon Potential due to thin disk (n + 1)Pn+1(x) = (2n + 1)xPn(x) − nPn−1(x) : The Oort Constants: the rotation of our Galaxy Orthogonal polynomials on [−1; 1] i.e. The Oort Constants and epicycles 1 The Rotation Curve of Z our Galaxy Pn(x)Pm(x) = 0 if n 6= m Spiral Galaxy Rotation −1 Curves 2 = if n = m 2n + 1 n 1 d n P (x) = x 2 − 1 : n 2nn! dx n Pn(x) is oscillatory: Pn(x) has n zeros in [-1,1] (so higher n ) more oscillations) ) Any function on [-1,1] can be reproduced as a sum of these (cf Fourier series). 8 / 89 Galaxies Part II Legendr´eand Fourier Potentials due to axisymmetric matter distributions Legendr´ePolynomials Example Motion of the Moon Potential due to thin disk The Oort Constants: the rotation of our Galaxy The Oort Constants and epicycles The Rotation Curve of our Galaxy Spiral Galaxy Rotation Curves The only known portrait of Legendr´e,recently unearthed, is found in the 1820 book Album de 73 portraits-charge aquarel´esdes membres de l'Institut, a book of caricatures of seventy-three famous mathematicians by the French artist Julien-Leopold Boilly 9 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Legendr´ePolynomials distributions Legendr´ePolynomials Note then for general solution outside axisymmetric body Example Motion of the Moon Potential due to thin disk B R(r) = n+1 The Oort Constants: the r rotation of our Galaxy The Oort Constants and (want Φ finite as r ! 1). epicycles And if the body is symmetric about θ = π=2 plane, then only need even n, since these The Rotation Curve of our Galaxy are symmetric about µ = 0 and the odd n ones are antisymmetric. Spiral Galaxy Rotation Curves 1 1 X BnPn(cos θ) X B2k P2k (cos θ) Φ(r; θ) = = r n+1 r 2k+1 n=0 k=0 GM J 1 J 35 15 3 = − + 2 3 cos2 θ − 1 + 4 cos4 θ − cos2 θ + ::::: r r 3 2 r 5 8 4 8 If the body is nearly spherical, treat J terms as perturbations. 10 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Example distributions Legendr´ePolynomials Potential due to a ring of matter Example Motion of the Moon Mass M , radius a (in full generality) Potential due to thin disk The Oort Constants: the 2 rotation of our Galaxy Axisymmetric ) no φ dependence ) r Φ = 0 has solution The Oort Constants and epicycles 1 X Bn The Rotation Curve of Φ = A r n + P (cos θ) our Galaxy n r n+1 n Spiral Galaxy Rotation n=0 Curves So if we need to work out Φ somewhere useful to get the coefficients An and Bn. Trick: let's try the easiest place - on the z axis. At height z on the z-axis Z ρd 3r0 GM Φ = −G 0 = − 1 jr − r j (a2 + z2) 2 " usual sum 11 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Example distributions For small z, Legendr´ePolynomials Example 1 Motion of the Moon − GM z2 2 Potential due to thin disk Φ = − 1 + 2 The Oort Constants: the a a rotation of our Galaxy The Oort Constants and GM z2 3 z4 epicycles = − 1 − 2 + 4 − · · · The Rotation Curve of a 2a 8 a our Galaxy Spiral Galaxy Rotation But on axis also θ = 0; cos θ = 1; r = z so Curves 1 X Bn Φ(r) = A zn + there n zn+1 n=0 Match terms, and so for small r < a GM r 2 3 Φ = − 1 − P (cos θ) + r 4P (cos θ) ··· a 2a2 2 8a4 4 12 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Example distributions Legendr´ePolynomials n 2 o Example GM r 3 4 Φ = − a 1 − 2 P2(cos θ) + 4 r P4(cos θ) ··· Motion of the Moon § 2a 8a ¤ Potential due to thin disk The Oort Constants: the ¦ ¥ rotation of our Galaxy The Oort Constants and P0(cos θ) = 1 epicycles 1 The Rotation Curve of P (cos θ) = (3 cos2 θ − 1) our Galaxy 2 2 Spiral Galaxy Rotation 1 Curves P (cos θ) = (35 cos4 θ − 30 cos2 θ + 3) 4 8 So if you want the potential close to the plane: z θ ∼ π=2 r −! R; and r cos θ = z or cos θ = R GM 1 ) Φ(R; Z) ' − 1 − 3z2 − R2 + ::: a 4a2 13 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Motion of the Moon distributions Legendr´ePolynomials Can apply this to the motion of the Moon due to Example Motion of the Moon the time average of the Sun (axisymmetric part of Potential due to thin disk the potential) The Oort Constants: the rotation of our Galaxy GM⊕ GM 1 2 2 The Oort Constants and Φ = − − 1 − 2 3z − R epicycles R a 4a The Rotation Curve of our Galaxy 8 11 R = R⊕M = 4 × 10 m a = R ⊕ = 1:5 × 10 m Spiral Galaxy Rotation 24 30 Curves M⊕ = 6 × 10 kg M = 2 × 10 kg Mass of moon is about 1=81th that of the Earth, so it is the 'test particle'.