Astr 5465 Mar. 29, 2018 Galactic Dynamics I: Disks

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1 Galati Dynamis Overview Astr 5465 Mar. 29, 2018 Subjet is omplex but we will hit the highlights Our goal is to develop an appreiation of the subjet whih we an use to interpret observational data See Binney Tremaine and referenes therein for a full treatment of the subjet Distribution of mass determines gravitational aeleration on eah star Aeleration at eah 3d point and 3d veloity onstitutes 6 parameters in phase spae whih define the motion of eah star at one point in time We need to integrate these over time to determine stellar trajetory (orbit) Following stars via an N-body simulation is urrently impossible Instead let s onsider analyti models Distribution of stars reflets their orbital trajetories over a long time interval. So we assume a steady state for now Disk galaxies Key onepts inlude Cirular motion Deviations from irular motion Resonanes Density Waves Instabilities 1

2 Consider the properties of stellar orbits in the disk of the Milky Way Orbits are approximately irular but not preisely so let s see what information is available Jan Oort parameterized stellar orbits in the following way: Let the radial and tangential veloites of a star be: V r = Arsin2l V t = Br + Ar os2l where A and B are the Oort onstants given by: A = 1 " V 2 R dv dr B = 1 " V 2 R + dv dr = 1 2 R " dω dr = 1 2 R " dω dr + 2Ω R A is a measure of the loal radial gradient of the irular veloity, the shear B is a measure of the loal vortiity (url) Hipparos yields: A = km/s/kp, B = km/s/kp Combining them gives some insight into loal stellar orbits:! A + B = dv = +2.4 km/s/kp (flat rotation urve) " dr! A B = V = Ω( ) = 27.2 km/s/kp (P = 2π / Ω( ) = 230 Myr " R R o When ombined with a measure of = 8kp: V ( ) = 218 ( /8 kp) km/s 2

3 Consider small perturbations from irular orbits The veloity will be almost unhanged as star is perturbed radially but the small hange is signifiant Result is an elliptial orbit with a > R Angular momentum must be onserved: As r inreases, V must derease and vie versa Stars perturbed initially outward will fall behind those on irular orbit F grav > F ent so stars moves bak in Stars perturbed initially inward will lead those in irular orbit F grav < F ent so star moves bak out The yle repeats and so elliptial orbit an be modeled as an epiyle entered on the guiding (irular) orbit. We define and angular veloity for the epiyle k and it is retrograde. For a Keplerian potential (orbit about point mass) we have: W g = k g and so the orbit is a losed ellipse This is not true in general and so the orbits are not losed Unless we onsider a rotating frame with W = W g 1/2 k g then orbits are losed ellipses entered on galaxy If we have a phase shift of these orbits with radius we see a spiral-like pattern similar to that seen at the right. Now we have a dynamial method for produing spiral arms 3

4 Dynamial basis for epiyli motion of disk stars Consider the potential of a flattened, axisymmetri disk: F(R,z). Sine angular momentum is onserved (no azimuthal fores): r = Φ(R, z) and L z = R 2 φ = onst. For ylindrial oordinates (R, φ, z): R R φ 2 = Φ Φ and z = z with d dt (L z ) = 0 Note the entrifugal aeleration term. Consider z motions about the plane: Φ z ( = 0 sine the disk is ontinuous and symmetri about z = 0 If we expand the z-fore for small z (linear terms): z = Φ ( z z 2 Φ ( z 2 = z 2 Φ ( z 2 = ν 2 z where ν 2 = 2 Φ ( z 2 This is the equation of motion for symple harmoni motion with frequeny ν and the solution is: z(t) = Z os(νt +ψ 0 ) For the Milky Way near the sun, ν 2 = 4πGρ 0 or ν 0.096Myr 1 So the vertial osillation period (2π /ν) 6.5 x 10 7 yr 1/3 Ω 4

5 Now onsider the radial motions about the irular guiding (referene) orbit: Φ ( = V 2 2 = Ω g For non-irular orbits the equation of motion is: R = R φ 2 Φ We an also write this interms of the angular momentum: R = Φ eff where Φ eff = Φ(R, z)+ L 2 z 2R sine L = 2 z φ R2 We an plot Φ eff to ilustrate the sharp rise at small r and the slow rise at large R. The minimum ours at the guiding orbit: Φ eff ( = 0 = Φ φ g (R 2 = Φ ( V g For some small perturbation x onsider the potential at R = R + x : R = x = Φ eff ( x 2 2 Φ eff ( 2 = x 2 2 Φ eff = κ 2 x ( One again this is the equation of simple harmoni motion about : x(t) = X os(κt +φ 0 ) where κ 2 = 2 Φ eff 2 ( = φ (( + 3L 2 z R g R or: 4 g κ 2 = R dω2 dr + 4Ω2 ( 5

6 Now onsider the azimuthal motions about the irular guiding (referene) orbit: Sine: L z = R 2 Ω g = onst., hanges in R yield hanges in Ω: Ω = φ = L z R = L z 2 ( + x) L z 1 2x 2 2 R g R ) = Ω 2x g 1 g ( R ) g ( Integrating yields: φ(t) = Ω g t 2Ω X g κr ) sin(κt +φ ) 0 g ( Thus, φ follows the guiding orbit with small amplitude SHM suprposed. If we let y be the azimuthal perturbations: Near the Sun we predit: κ 2 0 = 4B(A B) = 4BΩ 0 where κ 0 = 37 km/s/kp and Ω 0 = A B = 27 km/s/kp This orresponds to κ 0 / Ω (stars make 1.3 yles per orbit) and sine κ 0 / 2Ω then epiyles have radial/azimuthal 0.7 The observed veloity ellipsoid at = is: σ R /σ φ = κ 0 / 2Ω in good agreement, but: σ R /σ φ = 2Ω 0 /κ beause there are more stars at smaller R This results in about 1 kp exursions in R. Similarly, for z: σ z 30 km/s with ν Myr 1 with exursions of about 300 p. y(t) = 2Ω g κ X sin(κt +φ 0 ) and so the frequeny κ is that same as in x (radial diretion) but out of phase by 90 o. Together: x(t) = X os(κt) y(t) = 2Ω g κ X os(κt) assuming φ 0 = 0 Some properties are: elliptial epiyle with radial/azimuthal = κ /2Ω retrograde epiyli motion For a Keplerian potential: κ = Ω (losed ellipses) For flat rotation urve: κ = 2Ω For solid body rotation: κ = 2Ω (losed oval orbits) 6