Feng & Gallo have published a series of extremely similar papers, all of which essentially claim that they have "discovered" a major flaw in the way (some) astrophysicists think about rotation curves. Instead of assuming spherical symmetry, they try to solve for the mass distribution, using a rotation curve, without assuming spherical symmetry, instead adopting a planar geometry with cylindrical symmetry.
Of course they do have a point; statements that the flat rotation curve can be compared with a Keplerian prediction (that assume spherical symmetry, or that all the mass is concentrated at the centre) are overly simplistic. So far so good, but they then go on to claim that their analysis is compatible with the total stellar mass of galaxies and that dark matter is not required.
So, in their planar model (and obviously this is open to criticism too) they invert rotation curves to obtain a radially dependent surface density distribution that drops pseudo-exponentially.
Problem 1: They concede (e.g. in Feng & Gallo 2011) that "the surface mass density decreases toward the galactic periphery at a slower rate than that of the luminosity density. In other words, the mass-to-light ratio in a disk galaxy is not a constant". This is an understatement! They find exponential scale lengths for the mass that are around twice (or more for some galaxies) the luminosity scalelengths, so this implies a huge, unexplained increase in the average mass to light ratio of the stellar population with radius. For the Milky way they give a luminosity scalelength of 2.5 kpc and a mass scale length of 4.5 kpc, so the $M/L$ ratio goes as $\exp[0.18r]$, with radius in kpc (e.g. increases by a factor 4 between 2 kpc and 10 kpc). They argue this may be due to the neglect in their model of the galactic bulge, but completely fail to explain how this could affect the mass-to-light ratio in such an extreme way.
Problem 2: In their model they derive a surface mass density of the disk in the solar vicinity as between 150-200 $M_{\odot}/pc^2$. Most ($\sim 90$%) of the stars in the solar neighbourhood are "thin disk" stars, with an exponential scale height of between $z_0= 100-200$pc. If we assume the density distribution is exponential with height above the plane and that the Sun is near the plane (it is actually about 20pc above the plane, but this makes little difference), a total surface mass density of $\sigma = 200M_{\odot}/pc^2$ implies a local volume mass density of $\rho \simeq \sigma/2z_0$, which is of order $0.5-1 M_{\odot}/pc^3$ for the considered range of possible scale heights. The total mass density in the Galactic disk near the Sun, derived from the dynamics of stars observed by Hipparcos, is actually only $0.076 \pm 0.015 M_{\odot}/pc^3$ (Creze et al. 1998), which falls short of Feng & Gallo's requirements by an order of magnitude. (This does not bother the cold, non-baryonic dark matter model because the additional (dark) mass is not concentrated in the plane of the Galaxy).
Problem 3: For the most truncated discs that they consider, with an edge at $r=15$ kpc, the total Galaxy mass is $1.1\times10^{11}\ M_{\odot}$ (again from Feng & Gallo 2011 ). The claim is then that this "is in very good agreement with the Milky Way star counts of 100 billion (Sparke & Gallagher 2007)". I would not agree. Assuming "stars" covers the full stellar mass range, then I wouldn't dissent from the 100 billion number; but the average stellar mass is about $0.2\ M_{\odot}$ (e.g. Chabrier 2003), so this implies $\sim 5$ times as much mass as there is in stars (i.e. essentially the same objection as problem 2, but now integrated over the Galaxy). Gas might close this gap a little, white dwarfs/brown dwarfs make minor/negligible contributions, but we still end up requiring some "dark" component that dominates the mass, even if not as extreme as the pseudo-spherical dark matter halo models. Even if a factor of 5 additional baryonic dark matter (gas, molecular material, lost golf balls) were found this still leaves the problem of points 1 and 2 - why does this dark matter not follow the luminous matter and why does it not betray its existence in the kinematics of objects perpendicular to the disc.
Problem 4: Feng & Gallo do not include any discussion or consideration of the more extended populations of the Milky Way. In particular they do not consider the motions of distant globular clusters, halo stars or satellite galaxies of the Milky Way, which can be at 100-200 kpc from the Galactic centre (e.g. Bhattachargee et al. 2014). At these distances, any mass associated with the luminous matter in the disk at $r \leq 15$ kpc can be well approximated using the Keplerian assumption. Proper consideration of these seems to suggest a much larger minimum mass for the Milky way independently of any assumptions about its distribution, though perhaps not in the inner (luminous) regions where dark matter appears not to be dominant and which is where F&G's analysis takes place. i.e the factor of 5-10 "missing" mass referred to above may be quite consistent with what others say about the total disk mass and the required dark matter within 15kpc of the Galactic centre (e.g. Kafle et al. 2014). To put it another way, the dynamics of these very distant objects require a large amount of mass in a spherical Milky Way halo, way more than the luminous matter and way more even than derived by Feng & Gallo. For instance, Kafle et al. model the mass (properly, using the Jeans equation) as a spheroidal bulge, a disk and a spherical (dark) halo using the velocity dispersions of halo stars out to 150 kpc. They find the total Galaxy mass is $\sim 10^{12} M_{\odot}$ and about 80-90% is in the spherical dark halo. Yet this dark halo makes almost no contribution to the mass density in the disk near the Sun.
Problem 5: (And to be fair I do think this is beyond the scope of what Feng & Gallo are doing) Feng & Gallo treat this problem in isolation without considering how their rival ideas might impact on all the other observations that non-baryonic dark matter was brought in to solve. Namely, the dynamics of galaxies in clusters, lensing by clusters, the CMB ripples, structure formation and primordial nucleosynthesis abundances to state the obvious ones. A new paradigm needs to do at least as well as the old one in order to be considered competitive.
