Was reading about dark matter and the distribution of it throughout the galaxy. it said "For example, if rotation curves are flat this means-" what exactly does this mean?
[Physics] a flat rotation curve
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I feel that exactly the opposite should be the case; that is, dark matter halo should be inside the galaxy rather than outside.
Your feeling is entirely correct, and actually agrees with dark matter theories. Your only mistake is in thinking that the dark matter halo of those theories is only surrounding the galaxy; it's also inside the galaxy, and is usually most dense at the center of the galaxy.
A slight aside: The word "halo" is admittedly confusing here, because depictions of more modern times frequently show halos as isolated rings outside the head. The analogy would suggest that dark matter is just a ring outside the galaxy. Older (Western) art, however, showed halos as emanations of light originating behind the head, which is closer to the sense used in "dark matter halo".
To be a little more precise, dark matter halos are usually modeled by an NFW profile. This is defined by the density of dark matter $\rho$ as a function of the distance from the center of the galaxy $r$: \begin{equation} \rho(r) = \frac{\rho_0} {\frac{r}{R_s}\left( 1 + \frac{r}{R_s} \right)^2}~, \end{equation} where $R_s$ is some scale radius. As you can see, this density actually goes to infinity at $r=0$. That's okay; this is just a crude model, and the total amount of mass it describes is finite. But the point is that the density is greatest near the center of the galaxy, and gradually tapers off at larger radii.
You'll also see Einasto profiles, which have \begin{equation} \rho(r) = \rho_0\, \exp[-(r/R_s)^\alpha]~, \end{equation} where $\alpha$ is some other parameter. This one is finite at $r=0$ [in fact, $\rho(0) = \rho_0$]. And again, the density is always greatest at the center of the galaxy.
You'll sometimes even see the density profile approximated as uniform out to some radius $R_s$: \begin{equation} \rho(r) = \begin{cases} \rho_0 & r\leq R_s~, \\ 0 & r > R_s~. \end{cases} \end{equation} This is a particularly crude model, where you think of the galaxy as being basically embedded in a uniform glob of dark matter with constant density, but only out to some finite radius. The advantage of this is just that it's easier to make rough calculations with. This doesn't have greater density inside the orbits than outside, but at least it's not lower density.
Regardless of the particular model, the idea is always that there's extra matter inside the various orbits, which makes them behave exactly as you said.
Yes, dark matter has to be in motion, otherwise it would fall in to the galactic center. From the fact that galaxies are stable, we can expect the Virial Theorem to hold, i.e. that dark matter has a total kinetic energy of half the total gravitational potential of the galaxy.
Yes, indeed, a particular density distribution is required to result in the observed rotation curve; taken the other way around, measurements of galactic rotation curves are measurements of the dark matter density profile. For simplicity (which turns out to be a good approximation) let's assume a spherically symmetric distribution and equate centripetal and gravitational forces: \begin{equation} \frac{mv^2}{r}= \frac{GM(r)m}{r^2} \end{equation} with $M(r)=\int \varrho(r) 4\pi r^2 \mathrm{d}r$ the dark matter mass profile. Sanity check: Outside the mass distribution, $M(r)$ can be approximated as a point mass $M$ at $r=0$ and one recovers Kepler's law $v(r)\propto 1/\sqrt{r}$, whereas close to the center, $\varrho(r)\sim const$ and thus $v(r)\propto r$. Good. Now, to get the observed flat rotation curve $v(r)=const$ requires $M(r)\propto r$. Such a mass distribution is what you get for a isothermal sphere, often used as the simplest example of a star in ASTR101 courses. A more detailed analysis and in particular a plethora of studies on intricate n-body simulations favors a modification to that profile called the Navarro–Frenk–White profile.
The take home message is that the dark matter velocity distribution is expected to roughly follow a thermal profile, i.e. a Maxwell-Boltzmann distribution. Modifications come from cropping that at the escape velocity, and from a decade-long debate whether the cores of galactic dark matter profiles are "cored" or "cusped". So, yes, our standard models of dark matter phase space distributions correctly reproduce the observed rotation curves of galaxies. Research is ongoing to investigate feedback mechanisms between the baryonic disk and the dark matter halo, and their impact on the observed distributions and scaling relations.
I can provide some pointers if you want, but to get an idea, have a look at the homepage of the Illustris collaboration. Every galaxy you see there is actually a simulated one, so we know the simulated dark matter profile and can compare this virtual universe to the real one to further our understanding of what is going on, despite not yet having detected dark matter quanta directly. The page also lists a number of papers with details for the so inclined. Or, if you prefer plots, from this first to find though somewhat dated paper comes this example of the velocity distribution in a simulated (dark matter-only) galaxy:
(Maxwell is dot-dashed, dashed and dotted other analytical models, solid black is the simulated profile. Green is what this particular paper propagates as a model, purple an idea about the spread seen in simulations. Also note the inset: as promised, the isothermal halo is a pretty good approximation.
Finally,
Have models been investigated for different motion e.g. where the dark matter orbits at constant radius, or is stationary, or is even moving at constant speed towards the galactic nucleus, to be periodically ejected?
As mentioned dark matter can not be stationary in the galaxy's gravitational potential. The dark matter halo is not expected to condense into a disk, because (a) that would require efficient mechanism for dark matter to dissipate energy and (b) disk-only galaxies aren't stable as already some of the earliest n-body simulations confirmed. Yes, in the thermal halo, the individual dark matter quanta are expected to move on elliptical orbits. Ejection (of notable quantities) would mean evaporation of the dark matter halo which is in contrast to the observation that galaxies are around all the time (i.e. are stable)
Best Answer
When you look at an image of a galaxy like our own, you can see that most of the visible mass is concentrated in the core and that the density of stars in this core region is approximately constant:
So let's compute the expected rotation curve.
The orbital velocity of a star at a distance $r$ from the centre of the galaxy is $$ v=\sqrt{M(r)/r} $$ where $M$ is the mass of the galaxy inside the orbit $r$, and I've taken $G=1$.
Now let's pick a radius $r$ that is somewhere within the core region. Since the density $\rho$ (mass per unit area) is roughly constant in this region, the mass $M$ inside $r$ will be approximately $$ M(r)=\pi r^2\rho\qquad\mathrm{(core)} $$ And so the orbital velocity in the core region will scale as: $$ v\sim\sqrt{r^2/r}=\sqrt{r}\qquad\mathrm{(core)} $$
Now pick a radius $r$ far outside the core. Since the vast majority of the inner mass $M$ is contained within the core, then as we increase $r$ the mass $M$ will remain roughly constant. The velocity will therefore drop off as $$ v\sim 1/\sqrt{r}\qquad\mathrm{(outer)} $$
Plotting these two limiting cases ($\sqrt{r}$ in blue, $1/\sqrt{r}$ in red, and a 'hybrid' dashed):
we can see the sort of shape the rotation curve should have. However, the actual observed curve is roughly flat as $r$ increases outside of the core region:
This tells us that there is more mass than we can see (i.e. dark matter). Also, from the equation $v=\sqrt{M/r}$, we can see that this is only possible if the mass $M$ increases roughly linearly with $r$.