Galaxy Disks: rotation and epicyclic motion

Transcription

1 Galaxy Disks: otation and epicyclic motion 1. Last time, we discussed how you measue the mass of an elliptical galaxy. You measue the width of the line and apply the adial Jeans equation, making some assumptions about anisotopy. 2. Galaxy disks ae much easie: they ae mostly odeed, with vey little andomness. Once again, you use a spectogaph. inceasing λ Hβ OIII NII H NII 6563 an absobtion line, maybe spectoscopic slit Ionized gas HII egions and diuse (ambient) stalight. big bulge systems looks vey much like an isothemal sphee flat! v c -½ (non-kepleian) small bulge systems most of the light the Milky Way is vey typical v c 220km/s (a) As you might expect, v c inceases as L inceases. Empiical Tully-Fishe elation: L v 4 c. Can be used as standad candles. (b) The coesponding Fabe-Jackson elation fo ellipticals is L σ Of couse, disks ae not innitely cool thee's a velocity dispesion tenso with compenents σ 2, σ 2 θθ, and σ2 zz in pola coodinates. 1

2 (a) Now you know how to integate an obit numeically, but it tuns out that if you assume that the deviations fom ciculaity ae small, you can teat the poblem analytically. A petubation appoach. (b) Let's go to a coodinate system in which a paticle in cicula motion would be at est. y petubed position x 0 Ω 0 t θ Let Ω() = v c() = o + xˆx + yŷ x o y o (θ Ω o t) 4. Now you know that in an inetial fame, the equation of motion is given by = Φ (a) In the absence of a potential, = constant. (b) This is not so in an acceleating fame. In a fame otating with an angula velocity of Ω o, = Φ 2Ω o v }{{} Ω o ( Ω o ) }{{} Coiolisfoce centifugalfoce The Coiolis foce explains why stoms otate counteclockwise in the nothen hemisphee. The centifugal foce explains why the Eath bulges at the equato. (c) Both of these foces ae ctitious, in the sense that we've chosen a non-intetial coodinate system. (d) This is developed in section 7.2 of Symon's Classical Mechanics and in appendix 1.D.3 of Binney & Temain. 5. We will substitute ou expession fo into this equation, but st notice that Φ = v2 c = Ω2 Ω 2 o o + Ω 2 }{{ odx } Ω 2 o 2 dω +2 o Ω o d x ˆx

3 whee I've expanded the poduct in a 2D Taylo seies aound o. One vecto equation: ẍˆx + ÿŷ Ω 2 dω oˆx 2 o Ω o d xˆx 2Ω oẋŷ + 2Ω o ẏˆx +Ω 2 }{{} o 2 scala equations: ˆx: ẍ 2Ω o ẏ = 4Ω o A o x Hee, Oot's A is dened as ( A o ) dω 2 d ŷ: ÿ + 2Ω o ẋ = 0 o Coiolis Anyone cae to suggest a solution? Small motion about an equilibium position. Ty: These give: and x = x o sin(κt + φ) y = y o cos(κt + φ) κ 2 x o (sin κt + φ) + 2y o Ω o sin(κt + φ) = 4Ω o A o x o sin(κt + φ) κ 2 y o cos(κt + φ) + 2κx o cos(κt + φ) = 0 y o = 2Ω o x o κ κ 2 = 4Ω o (Ω o A o ) 6. What does motion look like? Thee ae paametic equations fo an ellipse in catesian coodinates. Motion is said to be etogade. κt=π/2 Ω κt=0 κt=π x y guiding cente 3

4 7. This may be somewhat familia. Keple, when he intoduced ellipses as the obits of the obits of the planets, was tying to avoid the use of epicycles. Hee we ae taking what looks like a giant step backwads and eintoducing them! (a) This is called the epicyclic appoximation and κ is called the epicyclic fequency. 8. Case I: Galaxy v c is constant; Ω() = v c A = d 2 d v c = 1 v c 2 = 1 2 Ω κ 2 = 4Ω(Ω 1 Ω) = 2Ω2 2 κ = 2Ω (a) Leads to an obit that isn't closed a osette. (b) The epicycle is complete befoe the guiding cente completes an obit. This is what you found. (c) yo x o = 2 a bit squashed. 9. Case II: Keple ( GM Ω = 3 ) 1/2 A() = (GM) 1/2 = 3 2 3/2 4 Ω κ 2 = 4Ω(Ω A) = Ω 2 κ = Ω (a) Hee, one epicycle pe guiding obit; obits close on themselves. (b) yo x o = 2Ω κ = 2 faily at: that's whee Hippachus went wong. You can't completely avoid ellipses! 4

5 10. If you aveage ove an obit, you have < ẏ 2 > obit < ẋ 2 = y2 o > obit x 2 = 4Ω2 o κ 2 guiding cente tails guiding cente leads But that's not the whole stoy. When we measue velocity dispesion at a point, we'e aveaging ove stas. ( 1 A ) = < σ2 yy > stas x 2 o Ω < σxx 2 > stas yo 2 = < σ2 θθ > stas < σ 2 = κ2 > stas 4Ω 2 (This comes fom second moments of collisionless Boltzmann) Case I: < σ 2 θθ > stas < σ 2 > stas = 1 2 We measue a lage σ in the adial diection, then in the tangential diection. 5