We now return to astrophysics, namely pertaining to the spiral galaxies, and what engenders the spiral shape. Of course, one can easily duplicate the shape using something as simple as dropping some cream in a cup of tea. One simply stirs up the black tea with a spoon, then drops several droplets of cream into it, and 'Voila!' a mini-spiral "galaxy" forms before one's eyes. Of course, things aren't quite so simple for real galaxies!

First, it's well to bear in mind that the particular density model (for a given galaxy) will vary depending on the conditions for a given galaxy. In principle then, there can be differing density wave models, and these pertain to a number of variables, factors such as: how tightly wound the spiral is (there are different grades which are assigned, e.g. Sa, Sb, Sc etc.), the degree of axial symmetry of the galaxy, and the modeling assumptions - in particular the potential-gravitational fields (V(r)) imposed on the system which determine the locations of orbital resonance in conjunction with the equations used.

The gravitational potential energy is defined according to:

V(r) = - GMm/r

where G is the Newtonian gravitational constant, m is the unit or elemental mass within the galaxy, M is associated with the central mass concentration, and r is the distance from m to M. The negative sign, as usual, indicates a

*bound system*.

Interestingly, the use of density wave model development is largely contingent on the Boltzmann eqn., which is also used (as we saw last year) in space plasma physics. Thus:

@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, i.e. between masses within the system. 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."*should not be taken 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 in a spiral galaxythat

*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, φ) with:

dφ/dt = dΘ/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_φ^2/r^2) + V(r) - p_φ OMEGA_p

where the p_r, p_φ 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_φ = J) may well be different for another frame.

Second, we see from this that the question as to why the spiral pattern is not affected by stars much further out cannot really be properly answered unless a full vetting of the assumed density waves for the particular galaxy is 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 density waves at location r1 in G1 and r10 in G1 where the r's denote radial distances from the center with r10 = 10 (r1).

Another problem (or complexity) in dealing with density wave models is the fact they are mainly 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.) More broadly, most astronomers who work in this specialist area use the term interchangeabley with

**Fourier m-component**. (And it should be understood here that one of the main tools is Fourier analysis of the waves, but alas Fourier analysis is only taught usually to those who take advanced Calculus or analysis courses).

As an example, a particular Fourier coefficient, call it a_n, applicable to a wave - may be defined:

a_n = 1/π INT (-π to π) f(x) cos mx dx

where INT denotes integral, and m is the Fourier m-component)

What types of modes can one have in these models? One is the "global" or m= 1 mode. Then there are the unstable (m= 2) modes.

Whether one mode or another appears (or is used in a spiral galaxy modelling) is critical since it may well determine at what stage a barred spiral develops, if at all. Alas, another complexity enters here since mode analysis is not simply a stand alone but also incorporates a subtle aspect called "marginal stability analysis" wherein one will solve for a quantity Q and if it is very close to 1 one has the case of marginal stability and tightly wound modes or in the case of spirals, around the Sa class. The trouble is that when one seriously incorporates any heating of the disk for whatever reason (say a massive central black hole sucking up matter and generating much radiation) then the desired values of Q are soon out of range, making it impossible for a given spiral structure to sustain itself.

Lastly, whenever one considers density waves in galaxies, it's 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) 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,φ, 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've shown is that density wave analysis as applied to galaxies is a field almost to itself in terms of being amenable to general understanding. And complexity is often compounded because the mere posing of a question to do with a particular spiral galaxy's form almsot always introduces 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.

Too bad we can't have a model based on as simple a fluid description as what happens to cream dropped into a stirred cup of tea!

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