Expert: Philip Stahl - 11/30/2009

Question
The stars in the inner part of a galaxy move faster than the density wave/s and the stars in the outer parts move slower than the wave/s. There is a point where the stars and wave/s are moving at the same speed. Why is the spiral pattern not affected by these factors?  

Answer
Hello,

First, it is well to bear in mind that the particular density model will vary depending on the conditions for a given galaxy. I perhaps did not make clear in earlier answers there are differing density wave models, and these pertain to a number of variables, factors such as: how tightly wound the spiral is, the degree of axial symmetry of the galaxy, and the modeling assumptions - in particular the potential-gravitational fields imposed on the system which determine the locations of orbital resonance in conjunction with the equations used.

I believe in a prior answer I noted that the use of density wave model development is largely contingent on the Boltzmann eqn.:  

@f/ @t + v*grad f + F/m*@f/@t = (@f/@t)_C

where @ denotes partial derivative, and (@f/@t)_C  is the time rate of change in f (the velocity distribution function) due to *collisions*. Technically, the Boltzmann eqn. is applied to FLUIDS and for that purpose, the galaxies to which density wave approaches or models are applied are modeled firstly in the fluid format. (It is easier when dealing with an agglomeration of some 100 or 200 billion separate stars and associated orbits to think of them as comprising a "fluid" as opposed to say, 100 billion separate bodies to be treated in a 100 billion -body problem of celestial mechanics!)

The referencing of stars, their locations and movements meanwhile embodies particulate approaches that are more kinematical in nature (but often *less amenable* to consistency with the density wave approach). Orbital assumptions, declarations are not simple by any means, and merely because a source says or asserts that "The stars in the inner part of a galaxy move faster than the density wave/s and the stars in the outer parts move slower than the wave/s." this should not be take too literally without posing a lot of further questions. (And one could argue here that "apples" and "oranges" are being compared because the two entities, stars and density waves arise from differing backgrounds - kinematic-particle based and fluid mechanical, wave based.)

For example, what class is the spiral? How tightly wound? One must recognize too that an orbit that *appears* closed (e.g. elliptic) in one reference frame may not be so in another. As an example, assume the (polar) coordinates for a galactic rotating frame are given as (r, phi) with:

d(phi)/dt =  d(theta)/dt - OMEGA_p  where OMEGA_p is the angular velocity of the rotating frame. Then orbits are described by a Hamiltonian (recall the Hamiltonian adds kinetic and potential energies of the system):

H = ½(p_r^2 + p_phi^2/r^2)  + V(r) - p_phi OMEGA_p

where the p_r, p_phi are the particle momenta referred to the associated coordinates, and V(r) is the gravitational potential. The point is that H can change depending on the coordinates, and what is presented for the previous frame as H = E - J OMEGA_p (with simplification, p_phi  = J) may well be different for another frame.

Second, the question cannot really be properly answered unless a full vetting of the assumed density waves for the particular galaxy are presented. In this sense, one recognizes that a full analysis of density waves for a galaxy - call it "Barred G1"- is needed before one can say stars in a given G1 region (e.g. inner or outer) "move faster or more slowly" than the *waves* at that place. We need to know then: the physical conditions for the establishment of the desnity waves at location r1 in G1  and r10 in G1 where the r's denote radial distances from the center with r10 = 10 r1.

As noted earlier (previous answer) also: The density wave models themselves are based on the *mode* chosen for particular dynamical wave equations that can be applied to the fluid framework. (In generic dynamical terms, a "mode" is a standing wave that can be supported by a disk of given dimensions, mass. )

Lastly, whenever one brings up questions to do with density waves in galaxies, it is important to bear in mind there remain enormous stumbling blocks even when applied to the simplest models of galactic disks (e.g. "zero thickness" disks).  One of these arises from potential (V(r) - seen earlier in the particle-star orbit context) theory. Thus, the perturbed gravitational field at one location depends on the *density perturbation* at *every other location*. How will you know, ab initio, that the density perturbation at location r, phi, z, say does not accelerate the associated wave (in the fluid rest frame) to a higher velocity than any stars at the same or near location? You don't unless you investigate! What does it mean to "investigate"? It means a full bore mathematical modelling procedure to locate where all the (so-called Lindblad) orbital resonances are, which ones can speed up the star, and also where the Landau damping regions are (which can impose a retardation of the waves).

In short, what I hope to have shown is that the answer to your question is far more complex than the simple posing of it might suggest. In particular, because the posing of it implies a number of tacit assumptions that may not be applicable at all. This, of course, is the difficulty when dealing in generalities, as opposed to specific cases, examples.

If you can present a *specific galaxy example*, including with a linked photograph of it, I may be able to more definitively address your question.