Missing matter problem and galactic kinematics This is a re-wording of an earlier problem I posted.
The standard example of the Missing Mass Problem comes from the rotational profiles of galaxies. By counting up the visible matter, we extrapolate a mass profile for a galaxy. We then apply Kepler's laws (the enclosed mass of a stable orbit can be modeled as a point mass) to calculate the expected velocity:
$$
f(v) = \sqrt{\frac{GM}{r}}
$$
Where G is the gravitational constant, M is the enclosed mass of an elliptical orbit, r is the radius of the orbit. But this formula assumes a closed, elliptical orbit. I'm sure the data exists, but I haven't been able to find it. How do we know that the Earth, for instance, is not falling towards or away from the center of the Milky Way? That is, when we apply the rules of Keplar's orbits, what information do we have that the orbits of the observed galactic bodies describe a closed ellipse and not a spiral in or out (which would change the amount of missing mass considerably)?
 A: The local standard of rest is the restframe circular orbit of a star at the position of the Sun in the azimuthally averaged Galactic potential. The Sun moves with 3d Galactic velocity coordinates of (11, 12, 7) km/s with respect to the LSR (Schonrich et al. 2009. i.e The Sun has a velocity component (wrt the LSR) of 11 km/s towards the Galactic centre (it of course has a tangential velocity of $>200$ km/s).
Sjouwerman et al. (1998) measure the line of sight velocities of 229 Maser sources towards the Galactic centre finding a mean velocity wrt the LST of $4 \pm 5$ km/s.
Reid et al. (2007) find a mean line of sight velocity, with respect to the LSR, for masers even closer to the Galactic centre as $-22 \pm 28$ km/s.
Li et al. (2010) measure velocities for 20 masers source within 2pc projected distance of the Galactic centre finding a mean velocity wrt to the LSR of $5 \pm 11$ km/s.
i.e. The Sun travels at no more than $\sim 10$ km/s radially with respect to the Galactic centre.
The tangential velocity of the Sun can be fixed with respect to the Galactic centre by observing the proper motion of Sag A* (Reid & Brunthaler 2004), which they find is almost entirely along the Galactic plane (i.e almost no vertical motion of the Sun wrt the Galactic plane). Assuming a distance of $8.0\pm 0.5$ kpc to the centre (for which there is a variety of evidence), the tangential velocity of the Sun is $241 \pm 15$ km/s, translating to a LSR tangential velocity of $236\pm 15$ km/s wrt the Galactic centre.
The Sun's motion wrt the Galactic centre is therefore almost entirely tangential
and in the absence of anything but a nearly axially symmetric Galactic potential, the Sun executes a nearly circular orbit, with epicycles in the radial and vertical directions. The  speed of the Sun's orbit and the amplitude and period of its epicycles depend on the size and shape of the Galactic potential.
I suppose you could hypothesise that the Sun was at the apogee of a highly elliptical, or otherwise non-circular orbit, but then the fact that it has very similar kinematics to 99 per cent of the nearby stars means they too would have to be on highly elliptical orbits with similar apogees (as they are moving in the same potential). But why would this be? Why should stars born over billions of years in different parts of the Galaxy have organised themselves to align their semi-major axes? By far the simplest explanation is that the orbits are close to circular and that is why the solar peculiar motion is small wrt to most stars - but large wrt halo (Population II) stars, which do have highly elliptical orbits and little circular motion.
EDIT: Part of the premise of this question is incorrect, since stars do not execute Keplerian orbits in the potential of a disk galaxy (see Why don't stars have Keplerian orbits? ). Keplerian orbits apply either to cases where objects orbit a much larger, point-like mass, or are orbiting in a spherically symmetric mass distribution where Newton's shell theorem can be applied. Neither of these are true in detail for the Milky Way and serious research work does not make these assumptions unless it is shown to be reasonable (e.g. for objects at large distances from the centre of the Galaxy).
As it says in the Introduction of the classic "Galactic Dynamics" by Binney & Tremaine 2nd ed. (2008) - " "The simplest approximate dynamical description of the Galaxy is obtained by assuming that its mass distribution is spherical. Let the mass interior to radius r be M(r). From Newton’s theorems the gravitational acceleration at radius r is equal to that of a point whose mass is the same as the total mass interior to r; thus the inward acceleration is $GM(r)/r^2$, where the gravitational constant $G = 6.674\times 10^{-11}\ m^{3} kg^{-1} s^{-2}$. The central or centripetal acceleration required to hold a body in a circular orbit with speed $v_0$ is $v_0^{2}/r$. Thus the mass interior to the solar radius $R_0$ in this crude model is $M(R_0)=v_0^{2}R_0/G$. The approximation that the mass distribution is spherical is reasonable for the dark halo, but not for the flat stellar disk."
The closed/not-closed issue I don't understand. Galactic orbits are not closed at all in the sense that orbital paths do not repeat, they undergo epicycles because the potential is not that of a point mass. A "spiral" orbit would imply that energy was being dissipated somehow (or added if the spiral were outward) - but the motion of stars in the galactic potential is essentially collisionless, there is no reason that this should happen.
A: A rotation curve is an attempt to measure the mass enclosed as a function of radius based on the fact that:
$$v_{\rm circular} = \sqrt{\frac{GM(<r)}{r}}$$
As you correctly point out this depends critically on the assumption that the orbits are, in fact, circular. The motion of individual stars or bits of gas aren't that interesting in the context of the "missing mass" problem, but rather the collective behaviour of all the matter at a given radius. In a rotationally dominated system (e.g. classic spiral galaxy), you would typically measure some bulk rotation that is circular at every radius (think of a differentially rotating disk), and some velocity dispersion at each radius, which would account for motions like inward or outward radial migrations of some stars/gas, or motion that is not in the plane of the disk. As long as the system is rotationally dominated, as in the rotational component of the motion dominates the dispersion component, then it is straightforward to model the total mass profile. You just pay attention to the bulk rotation and ignore the other components when doing your modelling.
Things get more complicated if the system is dispersion dominated. Elliptical/spheroidal systems are still not too difficult to model, but things like dwarf irregulars that are not rotationally dominated can be a real mess. The usual term is "non-circular motions". It is still usually possible to get a total mass profile that is reliable by fitting a "tilted ring model". It is in these small systems that the presence of dark matter is most obvious since the dark matter mass exceeds the mass of luminous matter within any radius (for larger systems the luminous matter typically dominates the dark matter within the extent of the luminous galaxy).
Incidentally, the current version of the "missing mass" problem is that there seems to be some dark matter missing from the central regions of some galaxies. This is usually referred to as the "cusp-core problem" these days. This is distinct from the old problem, which was "things seem to rotate too fast given the amount of mass we think we can see". Rather, this is "the standard model for the dark matter distribution (based on standard LCDM cosmology) seems to overpredict the amount of dark matter in the central regions of some galaxies". I've just written an entire journal article on the topic, which I will try to remember to link as soon as is reasonable (e.g. a couple of months once it's refereed and so on).
