Potentials Due to Axisymmetric Distributions of Matter the Oort Constants and Epicycles

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 ∇ 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 ∇ 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 ∇2Φ = 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 ∇ 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 ∇2Φ = 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 LegendréPolynomials distributions d n dΘ o LegendréPolynomials 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 Legendrépolynomials, 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) = 1∀n

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 Legendré,recently unearthed, is found in the 1820 book Album de 73 portraits-charge aquarelésdes 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 LegendréPolynomials distributions LegendréPolynomials 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 → ∞). 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 ∞ ∞ 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 ⇒ ∇ Φ = 0 has solution The Oort Constants and epicycles ∞ 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 |r − r | (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 ∞ 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’.

Ratio of Earth-Moon and Moon-Sun extra part is

GM GM R2 ⊕ / = M a3/M R3 ∼ 200 R a3 ⊕ so can treat the Moon-Sun force as a perturbation.

14 / 89 Galaxies Part II Potential from axisymmetric matter distribution Potentials due to axisymmetric matter Eclipse cycle distributions R = 6360 km – need accurate alignment. Legendr´ePolynomials ⊕ Example Moon orbit inclination to ecliptic 5o, eccentricity 0.055. Motion of the Moon Potential due to thin disk Moon’s period: d The Oort Constants: the • Siderial (relative to ﬁxed star)=27 .322 = 2π/Ω rotation of our Galaxy d • Synodical (new moon - new moon)=29 .530 = 2π/(Ω − Ω⊕) The Oort Constants and d epicycles • Anomalistic month (perigee to perigee)=27 .554 = 2π/Ωr = 2π/K The Rotation Curve of • Nodical month (node to node)=27d.212 = 2π/V our Galaxy

Spiral Galaxy Rotation To get a solar ecplise we must have Curves • Moon at “new moon” - once every synodical month • Moon passing through a node - once every nodical month • For the best eclipse, moon at perigee - once every anomalistic month The “saros” or eclipse cycle comes about because 223 synodic months = 6585.32 days 242 nodical months = 6585.36 days 239 anomalistic months = 6585.54 days ... so every 6585 days, or 18 years get a similar pattern of eclipses. 15 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter distributions

Potential due to thin disk Bessel Functions Poisson’s equation Summary of derivation of Φ for thin H << R axisymmetric disk Examples Mestel disk Exponential Disk The Oort Constants: the Let’s use cylindrical polar, (R, φ, z), coordinates rotation of our Galaxy The Oort Constants and Expect Φ ≡ Φ(R, z) and Φ(R, z) = Φ(R, −z) by symmetry. epicycles Outside disk ∇2Φ = 0. The Rotation Curve of 2 our Galaxy 1 ∂ ∂Φ ∂ Φ Spiral Galaxy Rotation ⇒ R + 2 = 0 Curves R ∂R ∂R ∂z We solve this by separation of variables, letting

Φ(R, z) = J(R)Z(z)

16 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Bessel Functions distributions 2 Potential due to thin disk 1 ∂ R ∂Φ + ∂ Φ = 0 Φ(R, z) = J(R)Z(z) Bessel Functions §R ∂R ∂R ∂z2 ¤ Poisson’s equation § ¤ Summary of derivation of Φ for thin ¦2 ¥ axisymmetric disk ¦ 1 d dJ(R¥) d Examples ⇒ Z(z) R + J(R) 2 Z(z) = 0 Mestel disk R dR dR dz Exponential Disk The Oort Constants: the 1 d dJ 1 d 2Z rotation of our Galaxy 2 ⇒ R = − 2 = −k , say The Oort Constants and JR dR dR Z dz epicycles | {z } | {z } function of z The Rotation Curve of function of R our Galaxy 2 Spiral Galaxy Rotation d Z 2 Curves ⇒ − k Z = 0 (4.4) dz2 so Z = A exp(kz) + B exp(−kz)

1 d dJ and R + k2J(R) = 0 (4.5) R dR dR

17 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Bessel Functions distributions 1 d R dJ + k2J(R) = 0 4.5 Potential due to thin disk §R dR dR ¤ Bessel Functions Poisson’s equation We would quite like Φ(R, ∞) and Φ(R, −∞) to be zero, so Summary of derivation of Φ for thin ¦ ¥ axisymmetric disk Z(z) = A exp(−k|z|) Examples Mestel disk Exponential Disk is the appropriate solution for Z(z). The Oort Constants: the The R equation (4.5) is the deﬁning equation for a Bessel function. These are the rotation of our Galaxy

The Oort Constants and analogues of sines and cosines now for cylindrical as opposed to linear problems (e.g. epicycles drum beats). The Rotation Curve of our Galaxy 2 Spiral Galaxy Rotation d y 2 Curves So while + k y = 0 has solutions sin(kz) , cos(kz) (4.6) dz2 1 d dy similarly s + k2y = 0 has solutions J (ks) , Y (ks) (4.7) s ds ds 0 0

which you can look up in e.g. Abramowitz & Stegun “Handbook of Mathematical Functions”

£ 18 / 89 ¢ ¡ Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Bessel Functions distributions

Potential due to thin disk Bessel Functions Poisson’s equation Summary of derivation of Φ for thin axisymmetric disk Note that as x → 0 J0(x) → 1 and Y0(x) → −∞. Examples Mestel disk More generally the equation Exponential Disk The Oort Constants: the 1 d dy ν2 rotation of our Galaxy 2 s + k − 2 y = 0 The Oort Constants and s ds ds s epicycles

The Rotation Curve of our Galaxy has solutions Jν (ks), Yν (ks), a family of Bessel functions characterized by the index ν.

Spiral Galaxy Rotation Curves Jν - Bessel function of the ﬁrst kind Yν - Bessel function of the second kind

19 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Bessel Functions distributions

Potential due to thin disk Bessel Functions Poisson’s equation Summary of derivation of Φ for thin axisymmetric disk Examples Mestel disk Exponential Disk The Oort Constants: the rotation of our Galaxy

The Oort Constants and epicycles

The Rotation Curve of our Galaxy

Spiral Galaxy Rotation Curves

Jν , Bessel function of the ﬁrst kind Yν , Bessel function of the second kind

20 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Bessel Functions distributions

Potential due to thin disk Bessel Functions Poisson’s equation Summary of derivation of Φ for thin axisymmetric disk Examples Mestel disk 1 d dy 2 Exponential Disk Similarly s − k y = 0 → I0(ks), K0(ks) The Oort Constants: the s ds ds rotation of our Galaxy

The Oort Constants and 2 epicycles 1 d dy 2 ν and s − k + 2 y = 0 → Iν (ks), Kν (ks) The Rotation Curve of s ds ds s our Galaxy Spiral Galaxy Rotation see Abramowitz + Stegun “Handbook of Mathematical functions” Curves These are Modiﬁed Bessel Functions.

21 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Modiﬁed Bessel Functions distributions

The Oort Constants and epicycles

The Rotation Curve of our Galaxy

Spiral Galaxy Rotation Curves

Iν , Modified Bessel function of the first Kν , Modified Bessel function of the kind second kind

22 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Bessel Functions distributions

Potential due to thin disk Bessel Functions Poisson’s equation And we can take this even further. By analogy with Fourier transforms where Summary of derivation of Φ for thin sin, cos → form the basis, we have J, Y → Hankel transforms. axisymmetric disk Examples Given a function g(r), then the Hankel transform of g is Mestel disk Exponential Disk Z ∞ The Oort Constants: the rotation of our Galaxy g˜(k) = g(r)Jν (kr)rdr 0 The Oort Constants and epicycles and the inverse transform is: The Rotation Curve of our Galaxy Z ∞ Spiral Galaxy Rotation Curves g(r) = g˜(k)Jν (kr)kdk 0 look these up in books of Hankel transforms!

23 / 89 Galaxies Part II Friedrich Bessel Potentials due to axisymmetric matter distributions

The Oort Constants and epicycles

The Rotation Curve of our Galaxy

Spiral Galaxy Rotation Curves

24 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter distributions d2Z 2 1 d dJ 2 2 − k Z = 0 4.4 R + k J(R) = 0 4.5 Potential due to thin disk §dz ¤ §R dR dR ¤ Bessel Functions Poisson’s equation Returning to the axisymmetric plane distribution, we have Summary of derivation ¦ ¥ ¦ ¥ of Φ for thin (4.4) ⇒ Z(z) = exp(−k|z|) axisymmetric disk Examples (4.5) ⇒ J(R) = J0(kr) Mestel disk Exponential Disk choose J to get Φ ﬁnite at R = 0 The Oort Constants: the rotation of our Galaxy Let k > 0 then The Oort Constants and epicycles ⇒ Φ (R, Z) = Ce−kz J (kR) z > 0 The Rotation Curve of k 0 our Galaxy

Spiral Galaxy Rotation Curves kz Ce J0(kR) z < 0

This is true ∀ k > 0, but a speciﬁc k for each Φk . P General potential → k Φk

25 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter distributions

Potential due to thin disk Bessel Functions Z ∞ Poisson’s equation −k|z| Summary of derivation ⇒ Φ(R, z) = f (k)e J0(kR)dk (4.8) of Φ for thin 0 axisymmetric disk Examples Mestel disk Here f (k) is a weighting function, corresponding to the C values in the sum. So what Exponential Disk we need to do for a particular mass distribution is ﬁnd f (k). The Oort Constants: the rotation of our Galaxy If we are going to relate it to a mass distribution, the next thing we should do is look The Oort Constants and at the z = 0 plane, i.e. the region we have neglected so far since we have taken epicycles ∇2Φ = 0 and so considered regions outside the plane. The Rotation Curve of our Galaxy Note that Φk is continuous across z = 0 but ∇Φk is not, due to |z| dependence.That Spiral Galaxy Rotation Curves is where the mass is, so that is not a surprise. 2 ⇒ ∇ Φk = 0 except at z = 0 and Φk → 0 as z, R → ∞ ⇒ satisﬁes conditions for potential from an isolated mass distribution. Still need to link with ρ (or Σ(R)) in the plane.

26 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Poisson’s equation distributions

The Oort Constants and epicycles Use Gauss’ Theorem (≡ Poisson’s equation plus divergence theorem) to determine Σ The Rotation Curve of our Galaxy in the z = 0 plane. Spiral Galaxy Rotation Curves Over the cylinder ZZZ ZZZ ZZZ 4πGρdV = ∇2ΦdV = ∇ . (∇Φ) dV

ZZ = ∇Φ . ˆnd 2S

27 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Poisson’s equation distributions

Potential due to thin disk RRR RR 2 Bessel Functions 4πGρdV = ∇Φ . ˆnd S Poisson’s equation § ¤ Summary of derivation of Φ for thin Consider the limit in which the cylinder height → 0. Then if A is the area of an end of axisymmetric disk ¦ ¥ Examples the cylinder Mestel disk Exponential Disk The Oort Constants: the rotation of our Galaxy LHS = 4πGΣA The Oort Constants and epicycles The Rotation Curve of ! our Galaxy ∂Φ ∂Φ RHS = − × A Spiral Galaxy Rotation ∂z ∂z Curves z=0+ z=0−

∂Φ0+ Equating these ⇒ = 4πGΣ(R) ∂z 0−

28 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Poisson’s equation distributions

Potential due to thin disk Bessel Functions Poisson’s equation Summary of derivation of Φ for thin axisymmetric disk Z ∞ Z ∞ −k0+ −k0− Examples RHS = − kf (k)e J0(kR)dk − kf (k)e J0(kR)dk Mestel disk Exponential Disk 0 0 The Oort Constants: the Z ∞ Z ∞ rotation of our Galaxy = − kf (k)J0(kR)dk − kf (k)J0(kR)dk The Oort Constants and epicycles 0 0 Z ∞ The Rotation Curve of our Galaxy = −2 kf (k)J0(kR)dk Spiral Galaxy Rotation 0 Curves 1 Z ∞ ⇒ Σ(R) = − f (k)J0(kR)kdk 2πG 0

29 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Poisson’s equation distributions

Potential due to thin disk Bessel Functions Poisson’s equation Summary of derivation of Φ for thin axisymmetric disk Hence determine f (k) [and hence Φ] from inverse Hankel transform Examples Mestel disk Exponential Disk Z ∞ The Oort Constants: the f (k) = −2πG Σ(R)J0(kR)RdR rotation of our Galaxy 0 The Oort Constants and epicycles Thus the process for determining Φ from ρ in this case is Σ → f → Φ. The Rotation Curve of our Galaxy Note: For determining the circular velocity need ∂Φ , whch becomes dJ0(x) , and for Spiral Galaxy Rotation ∂R dx Curves d Bessel function J0 have dx J0(x) = −J1(x) [Example].

30 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Summary of derivation of Φ for thin axisymmetric disk distributions This has been a bit longwinded, but the steps are clear. They are: Potential due to thin disk 2 Bessel Functions 1 ∇ Φ = 0 outside disk. Solve by separation of variables. Poisson’s equation −k|z| Summary of derivation 2 Solutions of form Φk (R, z) = Ce J0(kR) ∀ k > 0 of Φ for thin 2 axisymmetric disk 3 Φk → 0 as R, z → ∞ and satisﬁes ∇ Φ = 0 Examples Mestel disk ⇒ is potential of an isolated density distribution Exponential Disk 4 General Φ can be written as The Oort Constants: the ∞ rotation of our Galaxy Z

The Oort Constants and Φ(R, z) = Φk (R, z)f (k)dk epicycles 0

The Rotation Curve of our Galaxy where f (k) is an appropriate weight function.

Spiral Galaxy Rotation 5 Use Gauss’ theorem to determine Curves 1 Z ∞ Σ(R) = − f (k)J0(kR)kdk 2πG 0 6 Hence Z ∞ f (k) = −2πG Σ(R)J0(kR)RdR 0 So given Σ, use item (6) to determine f (k), and then (5) to obtain Φ. 31 / 89 Galaxies Part II Potential due to thin disk Potentials due to axisymmetric matter Circular velocity in the thin disk distributions R ∞ −k|z| Potential due to thin disk Φ(R, z) = 0 f (k)e J0(kR)dk 4.8 Bessel Functions § ¤ Poisson’s equation The circular velocity in the plane of a plane distribution of matter is given by Summary of derivation of Φ for thin ¦ ¥ axisymmetric disk 2 Examples vC (R) ∂Φ Mestel disk = Exponential Disk R ∂R z=0 The Oort Constants: the rotation of our Galaxy and we have The Oort Constants and d d epicycles x = kR J0(kR) = k J0x = −kJ1(kR) The Rotation Curve of dR dx our Galaxy d m m Spiral Galaxy Rotation [x Jm(x)] = x Jm−1(x) Curves §dx ¤ Then since we have equation (4.8), then ¦ ¥ 2 Z ∞ vC (R) = − f (k)J1(kR)kdk R 0

32 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Mestel disk distributions

Potential due to thin disk Bessel Functions Poisson’s equation Summary of derivation of Φ for thin A Mestel disk has the surface density axisymmetric disk Σ0R0 Examples distribution Σ(R) = R Mestel disk Exponential Disk The Oort Constants: the rotation of our Galaxy

The Oort Constants and epicycles Thus

The Rotation Curve of our Galaxy Z R Z R 0 0 0 0 Spiral Galaxy Rotation M(< R) = 2πΣ(R )R dR = 2πR0Σ0 dR Curves 0 0

= 2πΣ0R0R

→ ∞ as R → ∞

33 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Mestel disk distributions

Potential due to thin disk Bessel Functions f (k) = −2πG R ∞ Σ(R)J (kR)RdR Poisson’s equation 0 0 Summary of derivation § ¤ of Φ for thin axisymmetric disk Examples ∞ ¦ ¥ Mestel disk Z Exponential Disk f (k) = −2πGΣ0R0 J0(kR)dR The Oort Constants: the 0 rotation of our Galaxy

The Oort Constants and 2πGΣ0R0 epicycles = − k The Rotation Curve of our Galaxy Spiral Galaxy Rotation From Gradshteyn and Ryzkik 6.511.1 R ∞ J (bx)dx = 1 Re(ν)>−1 Curves § 0 ν b b>0 ¤ Z ∞ ¦ −k|z| J0(kR) ¥ ⇒ Φ(R, z) = −2πGΣ0R0 e dk 0 k

34 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Mestel disk distributions Potential due to thin disk v 2 (R) R ∞ 2πGΣ0R0 C = − f (k)J (kR)kdk f (k) = − k Bessel Functions § R 0 1 ¤ § ¤ Poisson’s equation Summary of derivation 2 ∞ of Φ for thin v (R) Z ¦ ¥ axisymmetric disk ¦ c ¥ Examples and = 2πGΣ0R0 J1(kR)dk Mestel disk R 0 Exponential Disk The Oort Constants: the 2 rotation of our Galaxy ⇒ vc (R) = 2πGΣ0R0 = const

The Oort Constants and epicycles

The Rotation Curve of M(< R) = 2πΣ0R0R our Galaxy § ¤ Spiral Galaxy Rotation Note that Curves GM(R) ¦ ¥ v 2(R) = c R exactly in this case even though distribution is a disk, not spherical. 2 GM(R) More generally, ﬁnd vc ≡ R to within 10% [reasonable accuracy] for most smooth Σ distributions. Conclude that measurement of vc (R) is a good measure of mass inside R. 35 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Exponential Disk distributions Here Potential due to thin disk Bessel Functions Σ(R) = Σ0 exp [−R/Rd] Poisson’s equation Summary of derivation This has ﬁnite mass of Φ for thin axisymmetric disk Z ∞ Examples M = 2πΣ exp [−R/Rd] RdR Mestel disk 0 Exponential Disk 0 The Oort Constants: the ∞ rotation of our Galaxy Z = 2πΣ R2 e−x xdx The Oort Constants and 0 d epicycles 0 | {z } The Rotation Curve of =1 our Galaxy

Spiral Galaxy Rotation 2 Curves = 2πΣ0Rd Then Z ∞ −R/Rd f (k) = −2πGΣ0 e J0(kR)RdR 0

R ∞ −αx α Gradshteyn + Ryzhik e J0 (βx) xdx = § 0 [β2+α2]3/2 ¤ 36 / 89 ¦ ¥ Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Exponential Disk distributions [ Actually they have something (6.623.2) which requires a little work: Potential due to thin disk Bessel Functions Z ∞ ν 3 Poisson’s equation −αx ν+1 2α(2β) Γ(ν + 2 ) Summary of derivation e Jν (βx)x dx = √ 3 of Φ for thin 2 2 ν+ 2 axisymmetric disk 0 π (α + β ) Examples Mestel disk √ Exponential Disk and you need to put in ν = 0, Γ(3/2) = π/2. The Oort Constants: the Then put α = 1/R and β = k.] rotation of our Galaxy d ∞ The Oort Constants and R −αx α 0 e J0 (βx) xdx = 3/2 epicycles § [β2+α2] ¤ The Rotation Curve of our Galaxy OK, so 2 ¦ ¥ Spiral Galaxy Rotation 2πGΣ0Rd Curves f (k) = − 2 3/2 [1 + (kRd ) ] Hence Z ∞ J (kR)e−k|z| Φ(R, z) = −2πGΣ R2 0 dk 0 d 3/2 0 2 1 + (kRd )

37 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Exponential Disk distributions Use Potential due to thin disk Z ∞ Jν(xy)dx 1 1 Bessel Functions = I ay K ay Poisson’s equation 2 2 1/2 ν/2 ν/2 Summary of derivation 0 (x + a ) 2 2 of Φ for thin axisymmetric disk Examples Mestel disk Exponential Disk You can do this with help from Gradshteyn + Ryzhik again, using (6.552.1) The Oort Constants: the rotation of our Galaxy d of this gives The Oort Constants and da epicycles Z ∞ The Rotation Curve of Jν (xy) dx y 1 0 1 y 0 1 1 our Galaxy −a = Iν/2( ay)Kν/2( ay) + Iν/2( ay)Kν/2( ay) 2 2 3/2 2 2 2 2 2 2 Spiral Galaxy Rotation 0 (x + a ) Curves so for ν = 0 Z ∞ J (xy) dx y 1 1 y 1 1 −a 0 = − I ( ay)K ( ay) + I ( ay)K ( ay) 3/2 0 1 1 0 0 (x 2 + a2) 2 2 2 2 2 2

38 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Exponential Disk distributions or Potential due to thin disk Z ∞ Jν (xy) dx y 1 1 1 1 Bessel Functions = I ( ay)K ( ay) − I ( ay)K ( ay) 3/2 0 1 1 0 Poisson’s equation 0 2 2 2a 2 2 2 2 Summary of derivation (x + a ) of Φ for thin axisymmetric disk Then with x = k, y = R and a = 1/R this becomes Examples d Mestel disk Exponential Disk Z ∞ Jν (kR) dk R R R R R The Oort Constants: the = I0( )K1( ) − I1( )K0( ) rotation of our Galaxy 2 3/2 2 0 (1 + (kRd) ) 2Rd 2Rd 2Rd 2Rd 2Rd The Oort Constants and epicycles or The Rotation Curve of our Galaxy R R R R Spiral Galaxy Rotation Φ(R, 0) = −πGΣ0R I0( )K1( ) − I1( )K0( ) Curves 2Rd 2Rd 2Rd 2Rd

Also you ﬁnd for the circular velocity (with y = R/2Rd ) ∂Φ v 2 = R = 4πΣ R y 2[I K − I K ] C ∂R 0 d 0 0 1 1 which is helpfully left as an example... 39 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Exponential disk distributions

The Oort Constants and epicycles

The Rotation Curve of our Galaxy

Spiral Galaxy Rotation Curves

40 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Exponential disk distributions

The Oort Constants and epicycles

The Rotation Curve of our Galaxy

Spiral Galaxy Rotation Curves

41 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Kuzmin disk distributions

The Oort Constants and epicycles

The Rotation Curve of our Galaxy

Spiral Galaxy Rotation Curves

Image credit: Joshua Barnes 42 / 89 Galaxies Part II Thin Disk: Examples Potentials due to axisymmetric matter Kuzmin disk distributions

The Oort Constants and epicycles

The Rotation Curve of our Galaxy

Spiral Galaxy Rotation Curves −GM Φ = pR2 + (a + |z|)2 Most (over)used disk model in astronomy is that credited to Miyamoto-Nagai (>1000 citations), which is a slight modiﬁcation of Kuzmin disk.

43 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter The Galaxy in the optical distributions

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

44 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter The Galaxy in the infrared distributions

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 Galactic coordinate system

45 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Suppose all stars are in circular orbits, with velocities v(R) = RΩ(R) distributions Observe a star at distance d, galactic longitude `. 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

46 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Line of sight velocity distributions

Potential due to thin disk The radial velocity, as seen from the Earth, is The Oort Constants: the rotation of our Galaxy The Oort Constants and vR = v cos α − v0 sin ` epicycles

The Rotation Curve of our Galaxy But π Spiral Galaxy Rotation sin ` sin 2 + α cos α Curves = = R R0 R0 and so vR0 v v0 vR = − v0 sin ` = − R0 sin ` R R R0

Now if we look at only nearby stars (d << R0):

R0 − R ≈ d cos `

47 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Line of sight velocity distributions

Potential due to thin disk v If we expand R around R = R0: The Oort Constants: the rotation of our Galaxy v v0 d v The Oort Constants and = + (R − R0) epicycles R R0 dR R R0 The Rotation Curve of our Galaxy So the radial velocity Spiral Galaxy Rotation Curves d v

vR ≈ −R0 d sin ` cos ` dR R R0 Since sin 2` = 2 sin ` cos `, we have

vR = Ad sin 2` v v0 " # vR = − R0 sin ` R R0 R d v 1 v dv § ¤ 0 0 A = − = − 2 dR R R0 2 R0 dR R ¦ R0 − R ≈ d cos ` ¥ 0 £ the Oort constant A ¢ ¡ 48 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Line of sight velocity distributions

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

vR = Ad sin 2`

" # R d v 1 v dv 0 0 A = − = − 2 dR R R0 2 R dR 0 R0

vR Then if we measure d as a function of `, then (in principle) we would know A. 49 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Tangential velocity distributions

Potential due to thin disk Similarly we can measure the tangential velocity The Oort Constants: the (from proper motions) rotation of our Galaxy

The Oort Constants and epicycles vT = v sin α − v0 cos ` The Rotation Curve of our Galaxy Note that R sin α = R0 cos ` − d, and so Spiral Galaxy Rotation Curves v v = [R cos ` − d] − v cos ` T R 0 0 i.e. v v0 v vT = − R0 cos ` − d R R0 R

For small d we have R0 − R ≈ d cos ` and v v d v 0 = + (R − R0) (∗) R R0 dR R R0

50 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Tangential velocity distributions

Potential due to thin disk

The Oort Constants: the rotation of our Galaxy v v0 v vT = − R0 cos ` − d The Oort Constants and § R R0 R ¤ epicycles

The Rotation Curve of v v0 d v our Galaxy −¦ = (R − R0) (*) ¥ §R R0 dR R R0 ¤ Spiral Galaxy Rotation Curves ¦ R0 − R ≈ d cos ` ¥ £ ¢ ¡ d v v 2 vT ≈ −R0 d cos ` − d dR R R0 R

51 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Tangential velocity distributions v v0 d v = + (R − R0) (*) Potential due to thin disk §R R0 dR R R0 ¤ The Oort Constants: the v 2 1 rotation of our Galaxy We can remove the term by using the expression (*), and cos ` = (1 + cos 2`) to R ¦ 2 ¥ The Oort Constants and obtain epicycles The Rotation Curve of d v v our Galaxy 2 vT ≈ −R0 d cos ` − d Spiral Galaxy Rotation dR R R0 R Curves ! ! 2 1 v0 dv 1 v0 dv d ≈ − d cos 2` − + d + O( ) 2 R dR 2 R dR R2 0 R0 0 R0 0

Now deﬁne vT = d(A cos 2` + B) i.e. " # " # 1 v0 dv 1 v0 dv B = − + A = − 2 R dR 2 R dR 0 R0 0 R0

52 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter distributions " # 1 v0 dv R0 d v 1 dΩ Potential due to thin disk A = − = − = − R0 2 R dR 2 dR R R0 2 dR The Oort Constants: the 0 R0 R0 rotation of our Galaxy The Oort Constants and " # epicycles 1 v0 dv 1 dΩ B = − + = − Ω + R0 The Rotation Curve of 2 R0 dR 2 dR our Galaxy R0 R0

Spiral Galaxy Rotation Curves • A measures the shear in the disk i.e. deviation from solid body rotation. For solid dΩ body rotation dR = 0, so A = 0. • B measures vorticity i.e. tendency for material in the disk to circulate due to diﬀerential rotation. Then note that Ω0 = v0/R0 = A − B and

dv = −(A + B) dR R0

53 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Measurement distributions

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

54 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Measurement distributions

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

Not easy: distances, peculiar velocities, proper motions

55 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter distributions −1 −1 −1 −1 Potential due to thin disk Best estimates are A = 14.5 km s kpc , B = −12 km s kpc . The Oort Constants: the −1 −1 −1 rotation of our Galaxy ⇒ ΩR0 = 26 km s kpc , so with R0 = 8.5 kpc v0 = 220 km s The Oort Constants and 8 epicycles ⇒ Rotation period= 2.4 × 10 years. The Rotation Curve of p −1 −1 our Galaxy Epicyclic frequency K0 = −4B(A − B) = 36 ± 10 km s kpc Spiral Galaxy Rotation Curves K0/Ω0 = 1.3 ± 0.2.

2 K 2 = R dΩ + 4Ω2 § dR ¤ See¦ examples sheet 3 ¥ £ The Sun makes 1.3 oscillations¢ in the radial direc- ¡ tion in the time it takes to complete the orbit around the Galactic centre

56 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter 2 distributions ν2 = ∂ Φ Ω2 = 1 ∂Φ ∂z2 R ∂R Potential due to thin disk § ¤ § ¤

The Oort Constants: the rotation of our Galaxy Poisson’s equation in¦ cylindrical coordinates:¥ ¦ ¥ The Oort Constants and epicycles 1 ∂ ∂Φ ∂2Φ The Rotation Curve of 4πGρ = R + our Galaxy R ∂R ∂R ∂z2 Spiral Galaxy Rotation Curves 1 dv 2 ≈ c + ν2 R dR If we could measure V, the vertical frequency, we could estimate the mass density in the Galactic plane, but we can’t. But, if we assume that the matter distribution was spherical then mass internal to the Sun: 2 v0 GM ≈ 2 R0 R0 R v 2 ⇒ M ≈ 0 0 ≈ 1011 M G 57 / 89 Galaxies Part II The Oort Constants: the rotation of our Galaxy Potentials due to axisymmetric matter Jan Oort, 1930s distributions

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

58 / 89 Galaxies Part II The Oort Constants and epicycles Potentials due to axisymmetric matter distributions

Potential due to thin disk The Oort Constants: the A star on an orbit with the guiding center radius Rg : rotation of our Galaxy

The Oort Constants and epicycles x ≡ R − Rg The Rotation Curve of our Galaxy 2 The angular speed of the circular orbit Ωg = Lz /Rg Spiral Galaxy Rotation Curves −2 ˙ pφ Lz x φ = 2 = 2 1 + R Rg Rg 2x ' Ωg 1 − Rg

59 / 89 Galaxies Part II The Oort Constants and epicycles Potentials due to axisymmetric matter 2x φ˙ ' Ωg 1 − distributions § Rg ¤ Potential due to thin disk If we could measure vφ(R0) − vc (R0) and vR for a group of stars, each with its own The Oort Constants: the ¦ ¥ rotation of our Galaxy guiding center, is there a way to obtain the relevant frequencies? The Oort Constants and epicycles The Rotation Curve of ˙ ˙ our Galaxy vφ(R0) − vc (R0) = R0(φ − Ω0) = R0(φ − Ωg + Ωg − Ω0) Spiral Galaxy Rotation " # Curves dΩ ' R (φ˙ − Ω ) − x 0 g dR Rg

2Ω dΩ v (R ) − v (R ) ' −R x + φ 0 c 0 0 R dR Rg 2 Evaluate at R0 and dropping terms of order of x dΩ v (R ) − v (R ) ' −x 2Ω + R φ 0 c 0 dR R0

60 / 89 Galaxies Part II The Oort Constants and epicycles Potentials due to axisymmetric matter The Oort’s constants deﬁnition: distributions Potential due to thin disk 1 vc dvc 1 dΩ The Oort Constants: the A(R) ≡ − = − R rotation of our Galaxy 2 R dR 2 dR The Oort Constants and epicycles 1 vc dvc 1 dΩ The Rotation Curve of B(R) ≡ − + = − Ω + R our Galaxy 2 R dR 2 dR

Spiral Galaxy Rotation Curves Therefore, dΩ v (R ) − v (R ) ' −x 2Ω + R = 2Bx φ 0 c 0 dR R0 Epicyclic radial motion solution:

x(t) = X cos(Kt + α)

Hence, averaging over the phases of stars near the Sun

2 2 2 [vφ − vc (R0)] = 2B X

61 / 89 Galaxies Part II The Oort Constants and epicycles Potentials due to axisymmetric matter The Oort’s constants deﬁnition: distributions Potential due to thin disk 1 vc dvc 1 dΩ The Oort Constants: the A(R) ≡ − = − R rotation of our Galaxy 2 R dR 2 dR The Oort Constants and epicycles 1 vc dvc 1 dΩ The Rotation Curve of B(R) ≡ − + = − Ω + R our Galaxy 2 R dR 2 dR

Spiral Galaxy Rotation Curves Therefore, dΩ v (R ) − v (R ) ' −x 2Ω + R = 2Bx φ 0 c 0 dR R0 Epicyclic radial motion solution:

x(t) = X cos(Kt + α)

Hence, averaging over the phases of stars near the Sun

2 2 2 [vφ − vc (R0)] = 2B X

61 / 89 Galaxies Part II The Oort Constants and epicycles Potentials due to axisymmetric matter distributions

Potential due to thin disk

The Oort Constants: the rotation of our Galaxy 2 2 2 The Oort Constants and [vφ − vc (R0)] = 2B X epicycles § ¤

The Rotation Curve of Similarly, for the radial velocity our Galaxy ¦ ¥ 2 Spiral Galaxy Rotation v 2 = −2B(A − B)X Curves R Taking ratio: 2 2 [vφ − vc (R0)] −B B K0 ' = − = 2 2 A − B Ω0 4Ω vR 0

62 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter 21cm line of neutral hydrogen distributions

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

frequency 1420.4058 MHz, wavelength 21 cm, probability 1 in 107 years

63 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter 21cm line of neutral hydrogen distributions

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

64 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter 21cm line of neutral hydrogen distributions

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

65 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter Rotation curve from 21cm line distributions

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

66 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter Rotation curve from 21cm line distributions

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

67 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter Terminal velocity distributions Potential due to thin disk Gas moves on circular orbits - use 21cm HI emission, and measure Doppler shift given `. The Oort Constants: the rotation of our Galaxy We have The Oort Constants and v v0 epicycles vR = − R0 sin ` R R0 The Rotation Curve of our Galaxy If Ω(R) = v decreases monotonically with R, the we can look at the 21cm line proﬁle Spiral Galaxy Rotation R Curves and pick the maximum Doppler shift as the one corresponding to the smallest circle along the line of sight at Galactic longitude `. This occurs when R = R0 sin ` Therefore vR (max) = v(R) − v0 sin ` and hence v(R) = vR (max) + v0 sin ` Note that this works only for −90o < ` < 90o, since a line of sight for 90o < ` < 270o is never a tangent to a circular orbit.

68 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter Non-circular motion distributions

Potential due to thin disk

The Oort Constants: the rotation of our Galaxy

The Oort Constants and epicycles

The Rotation Curve of our Galaxy • Axisymmertic expansion Spiral Galaxy Rotation Curves • Oval distortion (elliptical orbits) • Spiral structure • Random motions

69 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter Axisymmetric expansion distributions

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

70 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter Oval distortion and spiral structure distributions

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

71 / 89 Galaxies Part II The Rotation Curve of our Galaxy Potentials due to axisymmetric matter Rotation curve from 21cm line distributions

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

72 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter distributions

Potential due to thin disk • The Oort Constants: the spirals contain disks of gas. In a symmetric potential settles down to rotate on rotation of our Galaxy circular orbits in centrifugal balance The Oort Constants and epicycles • gas collides with itself and shocks, therefore paths of gas motion rarely intersect, The Rotation Curve of unlike stars our Galaxy Spiral Galaxy Rotation • if the potential is spherically symmetric then the measure of the circular velocity Curves vc (R) is a good measure of the mass M(R) contained within radius R

v 2 dΦ GM(R) c = ' R dR R2 or v 2R M(< R) ' c G

73 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 253 distributions

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

74 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter We can obtain vC (R) by observing the Doppler shift of a spectral line. For the gas in distributions the spiral’s disk, our options are: Potential due to thin disk The Oort Constants: the • HI - radio - neutral gas, largest range in radius rotation of our Galaxy The Oort Constants and • Hα - optical - warm gas, inner regions epicycles

The Rotation Curve of • CO - mm - molecular gas, very inner regions our Galaxy

Spiral Galaxy Rotation Problems: Curves • Instrumental beam smearing: each pointing has data from a range of radii, need to deconvolve • Intrinsic a) absorption: especially if galaxy is close to edge on position b) ﬁnite thickness of gas layer Spiral arms cause kinks in apparent rotation curve, because • gas motion is not exactly circular • arms provide non-axisymmetric potential • arms contribute to optical extinction

75 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter We can obtain vC (R) by observing the Doppler shift of a spectral line. For the gas in distributions the spiral’s disk, our options are: Potential due to thin disk The Oort Constants: the • HI - radio - neutral gas, largest range in radius rotation of our Galaxy The Oort Constants and • Hα - optical - warm gas, inner regions epicycles

The Rotation Curve of • CO - mm - molecular gas, very inner regions our Galaxy

75 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 253 rotation curve distributions

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

76 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 3079 distributions

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

77 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 3079 rotation curve from PV diagram distributions

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

78 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 4565 distributions

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

79 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 6946 distributions

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

80 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 1808 distributions

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

81 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter Rotation curves of nearby spirals distributions

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

82 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter IC 342 distributions

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

83 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 5907 distributions

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

84 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter NGC 5907’s secret distributions

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

85 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter Vera Rubin, 1980 distributions

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

“...This form for the rotation curves implies that the mass is not centrally condensed, but that signiﬁcant mass is located at large R. The integral mass is increasing at least as fast as R. The mass is not converging to a limiting mass at the edge of the optical image. The conclusion is inescapable that non-luminous matter exists beyond the

optical galaxy...” 86 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter The similarity of the rotation curves distributions

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

87 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter The dynamical components distributions

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

88 / 89 Galaxies Part II Spiral Galaxy Rotation Curves Potentials due to axisymmetric matter Tully-Fisher relation distributions

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

89 / 89