1. Widescreen Test Pattern (16:9)
Aspect Ratio Test
(Should appear circular)
4x3
16x9

2. Spiral Structure in Galaxies
12th March 2010
Suprit Singh (Talk for the course: Galactic Dynamics)

3. Overview Wave Mechanics of Differentially rotating disks
Kinematic Density Waves
Dispersion Relation
Local Stability of Disks
Spiral Structure Basics
Lin-Shu Hypothesis
Geometry
The Winding Problem
Pattern Speed

4. Spiral Structure: Basics

5. The Lin-Shu Hypothesis
Lin and Shu suggested that the spiral arms can be thought of in terms of density waves: compressions and rarefactions in the distribution of stars.
Coupling this with Lindblad's ideas and the hypothesis that spiral patterns are long lived lead to the Lin-Shu hypothesis that spiral structure is just a stationary density wave.
Unfortunately, Lin & Shu were only half right: spiral patterns are density waves, but they're definately not stationary!

6. Geometry
We can characterize spiral structure in terms of rotational symmetry. If I(R,f) is the observed intensity distribution in a disk then if
I(R, f+2p/m) = I(R,f)
(i.e. a rotation by 2p/m radians leaves the galaxy looking the same) then the galaxy (and spiral pattern) is said to have m-fold symmetry and m arms (m > 0).
Most galaxies have m = 2 and the predominance of two-armed galaxies is something that any good theory of spiral structure should be able to explain.

7. The Winding Problem I
The pitch angle at some radius R is defined as the angle between a tangent to the spiral arm and the circle R = constant.
It's useful to define a function describing a mathematical curve which runs along the center of an arm.
If we have m arms we can write this as
The function f(R; t) is known as the shape function and allows us to define a radial wavenumber

8. The Winding Problem II The pitch angle is then
and for galaxy with flat rotation curve
W(R)R = 200km/s at R = 5 kpc and after 10 Gyr the pitch angle would be This is much smaller than any observed galaxy and is known as the winding problem.

9. The Pattern Speed
In the Lin-Shu hypothesis, the sprial arms are a density wave pattern that rotates rigidly. We can then always move to a rotating frame with some angular frequency Wp in which the pattern remains stationary. This pattern speed is not the same as the rotational frequency of the disk.
The radius at which Wp = W (R) is known as the corotation radius. At smaller radii, W (R) > Wp .
Observationally, dust lanes are seen to lie on the inside of arms as defined by bright stars. If this reflects a time lag between the point of maximum compression of gas and the formation of stars it suggests that gas is flowing into arms from the inside. Since most arms are trailing this implies that the gas is rotating faster than the spiral pattern. Therefore, spiral patterns (at least those in grand design spirals) must be inside corotation.
Measuring the pattern speed is, in general, difficult (its a pattern, not a physical object) and relies on the assumption that there is in fact a well-defined pattern speed.

10. Wave Mechanics of Differentially rotating disksII

11. Kinematic Density Waves
Any particle orbitting in an axissymmetric galaxy will execute a periodic orbit with some well defined period Tr. During this time, the azimuthal angle will increase by some amount Df (not necessarily equal to 2p).
The corresponding radial and aziumthal frequencies are Wr = 2p/Tr and Wf = 2p/Tf
We can consider the orbit in a frame which rotates with frequency Wp. In this frame, the azimuthal position of the particle if fp=f- Wpt then in one radial period Dfp = Df - Wp Tr . Choose Wp such that orbit is closed.

12. The Lin-Shu Theory

13. Tight Winding Approximation

14. Potential of the Spiral Pattern I

15. Potential of the Spiral Pattern II

16. Dispersion Relation I

17. Dispersion Relation II

18. Dispersion Relation III

19. Dispersion Relation IV

20. Dispersion Relation V

21. Dispersion Relation VI

22. Dispersion Relation VII (??) Not yet!
Lets first see why WKB works?
Dispersion Relation VII (??) Not yet!
Pattern rotates at constant speed: it is a growing mode of oscillations
The waves propagate in a part of the galaxy bordered by resonances and/or turning-points which deflect waves.
That part acts as a resonant cavity for waves. Waves grow in the stellar disk but saturate due to the transfer of wave energy to gas disk.
Waves grow between the Inner Lindblad Resonance and Corotational Res.
CR region acts as an amplifier of waves due to over-reflection
Density waves may exist
in this ‘resonant cavity’
Two ILRs, one inner and one outer One OLR.
Condition for ILR

23. Dispersion Relation VII (Finally!!)

24. Local Stability of Disks (Toomre)

25. That’s all Folks. Thanks for your kind attention*
References :
Binney and Tremaine. Galactic Dynamics 2nd Edition
Bertin and Lin. Spiral Structure in Galaxies
My friend Google for Images/Videos
That’s all Folks. Thanks for your kind attention*
*after a ‘Galactic’ amount of spiraling 