Widescreen Test Pattern (16:9)

Aspect Ratio Test

(Should appear circular)

16x9

4x3

SPIRAL STRUCTURE IN GALAXIES

Suprit Singh (Talk for the course: Galactic Dynamics)12th March 2010

Overview• Spiral Structure Basics• Lin-Shu Hypothesis• Geometry• The Winding Problem• Pattern Speed

• Wave Mechanics of Differentially rotating disks• Kinematic Density Waves • Dispersion Relation• Local Stability of Disks

Spiral Structure: BasicsI

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!

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.

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

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 0.14. This is much smaller than any observed galaxy and is known as the winding problem.

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.

Wave Mechanics of Differentially rotating disksII

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.

The Lin-Shu Theory

Tight Winding Approximation

Potential of the Spiral Pattern I

Potential of the Spiral Pattern II

Dispersion Relation I

Dispersion Relation II

Dispersion Relation III

Dispersion Relation IV

Dispersion Relation V

Dispersion Relation VI

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.

mor

m

p

p

/

)(

Condition for ILR

Lets first see why WKB works?

Dispersion Relation VII (Finally!!)

Local Stability of Disks (Toomre)

*after a ‘Galactic’ amount of spiraling

That’s all Folks. Thanks for your kind attention*

References :

1. Binney and Tremaine. Galactic Dynamics 2nd Edition

2. Bertin and Lin. Spiral Structure in Galaxies3. My friend Google for Images/Videos