Why aren't all spinning galaxies shaped as discs as my young mind would expect? I understand how the innermost parts of a galaxy spin faster than the outer parts, and that could explain why some galaxies are more spiraled than others based on age. Though, this doesn't explain how the arms came into existence in the first place. Might it have something to do with an imperfect distribution of mass, and therefore an imperfect distribution of gravity, causing a split in the disc from which point gravity, centripetal force, and inertia could take over? Or does something happen earlier in a galaxy's life?
Answer
user6972's answer is great, but I thought I'd add a somewhat more technical footnote. If the mathematics are lost on you, skip to the end where I give a simple physical interpretation.
The dispersion relation for a differentially rotating fluid disk (i.e. the rotation frequency changes with radius, as opposed to a uniformly rotating disk) is:
$(\omega-m\Omega)^2 = \kappa^2-2\pi G\Sigma|k| + v_s^2k^2$
- $\omega$ is the angular frequency of a perturbing wave
- $m$ is an integer $\geq 0$ and describes the rotational symmetry of the disk (so $m=2$ for a bar structure, for instance)
- $\Omega$ is the rotation frequency of the disk
- $\kappa$ is the epicyclic frequency of the perturbation
- $\Sigma$ is the surface density of the disk (mass per unit area)
- $k$ is the wavenumber of the perturbation
- $v_s$ is the sound speed in the fluid
This may be a bit intimidating, but as I'll show in a minute it has a nice simple physical interpretation.
First though, a couple words on the assumptions that go into that dispersion relation (the full derivation is in Binney & Tremaine's Galactic Dynamics 2$^\mathrm{nd}$ Edition... it's quite involved so I won't try and outline it here).
- The disk is approximated as being two-dimensional (infinitely thin).
- Perturbations to the disk are small.
- The "tight-winding" approximation, or "short wavelength" approximation - very roughly speaking, the derivation fails if the spiral arms are not tightly wound. This is actually analogous to the WKB approximation.
- The sound speed $v_s$ is much less than the rotation speed $\Omega R$.
So, are these approximations reasonable? Checking typical disk galaxies it turns out that they are (as long as we're not talking about colliding galaxies or anything like that, which would lead to large perturbations). Besides, the idea with this analysis is not to get a nice clean result showing the theory of spiral arm formation, but rather convince ourselves that a disk is naturally unstable under certain conditions and will "want" to form spiral arms (and gain insight into what drives the instability), and we can check later that they do in fact form with simulations such as those mentioned by user6972.
Ok so with a dispersion relation based on some reasonable assumptions, we can do the usual stability analysis, requiring $\omega^2>0$ for stability. This gives:
$\mathrm{stable~if~}\dfrac{v_s\kappa}{\pi G\Sigma} > 1$
The analysis for a disk made of stars instead of a fluid disk (in reality a galaxy is a disk composed of a mixture of stars and gas) is very similar but with a couple of extra gory details... the result is nice, though:
$\mathrm{stable~if~}\dfrac{\sigma_R\kappa}{3.36G\Sigma}>1$
where $\sigma_R$ is the radial velocity dispersion of the stars in the disk; this is a measure of the distribution of radial velocities and can be thought of somewhat like a sound speed, in a certain sense it carries information about how fast the stars react to carry an impulse. This is a reasonably famous result called "Toomre's stability criterion".
Ok so now for the simple physical interpretation of the stability criteria. First, I should point out that $v_s$ (sound speed), $\sigma_R$ (velocity dispersion) and $\kappa$ (epicyclic frequency) are all similar quantities; they describe the ability of a system to respond to a disturbance. If I poke one side of a cloud of gas, the other side only finds out about it through pressure as fast as the speed of sound (or velocity dispersion/epicyclic frequency) can carry the message.
Now imagine I have a rotating disk of gas with nice smooth properties and I squeeze a little piece of gas (or group of stars) a bit. Two things happen - the squeezed piece of gas will "push back" outwards since I've increased the pressure, but I've also caused a slight increase in density, which will exert a bit of extra gravitational force. It turns out the gravitational force is proportional to $G\Sigma$, and the pressure force is proportional to $v_s\kappa$ (or $\sigma_R\kappa$). The same argument applies in reverse if I stretch the gas/stars a bit - the pressure drops, but so does the gravitational force. So the interpretation of the stability criteria above is that if, when I squeeze a bit of gas (or stars) a little bit, if the increase in pressure is sufficient to balance the increase in gravity, the gas will un-squeeze itself; it is stable. On the other hand, if gravity wins out against pressure, the disk is unstable and collapses locally.
Ok, so how does this lead to spiral arms? Well, you can show that spirals are a natural structure to form under this sort of instability with the parameters of a typical galaxy (depending on the details, a bar is also a possibility). It's a lot of work, though, and I'm not sure it brings a lot more insight - at this point, in my opinion, it's time to switch to simulations and see that yes, indeed, spirals seem to form because of this instability.
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