furthermore, given that there is a tight inverse mass-radius relation, it seems that the density should be well constrained and not allowed to vary much.

It is not a problem that the densities for stability against e- captures are higher. There is a simple explanation for the difference you observe, and it's neither an oversight nor GR. The central densities we have chosen are appropriate for Carbon ignition, which is what matters. The Chandrasekhar-limit or the stability of WDs is only indirectly related to ignition, which occurs when Carbon ignites, and NOT when the WD becomes gravitationally unstable. The references for the appropriate densities where pycnonuclear reactions lead to C12 ignition are given in the paper. I quote text from the introduction:

"The rate of the screened 12C–12C fusion reaction is still quite uncertain (e.g. Itoh et al. 2003; Gasques et al. 2005, 2007; Jiang et al. 2007). The central density at the time of ignition, however, depends only mildly on the exact value of this reaction rate (Cooper, Steiner & Brown 2009; Iapichino & Lesaffre 2010). More important is the initial mass and the accretion and cooling history of the WD, which determines the thermodynamic state of the interior. This results in a range of possible central densities at ignition, from less than 2 × 10^9 to over 5 × 10^9 g cm−3 (Lesaffre et al. 2006)."

Would you then say that the central densities associated with Carbon ignition and gravitational collapse are not the same? While Carbon ingnition is a much more complicated feature to simulate numerically, the Lane-Emden equation for white dwarfs could be used to investigate gravitational instability.

I think I understand. You are saying that what matters is the density at which ignition takes place, not the exact mass/radius parameters of the WD?

However I am still puzzled as to how you get a 1.38Msun WD with densities of only 2-5E9 g/cc? In the more recent Fink et al 2014, you say "we adopt M_Ch [Chandrasekhar mass] CO WDs in hydrostatic equilibrium, with a central density of 2.9E9 g/cc".

It's not a temperature effect, because even at T~1E9K, at these densities the degeneracy parameter is ~100. (see www.geogebratube.org/material/show/id/28528 ), so they are effectively cold. 

The Lesaffre paper refers only to the Eggleton/Pols code for solving the stellar structure equations, but these papers do not use the TOV general relativistic hydrostatic equilibrium equation.

So it seems that the initial conditions do neglect GR and that although I see that it is the densities that are most important here, I think these densities are reached in a WD of slightly lower mass (probably only a few hundredths) than you assume, but certainly less than M_Ch.

"Would you then say that the central densities associated with Carbon ignition and gravitational collapse are not the same?"

Yes, that's exactly what I'm saying.

At densities we're concerned with GR is negligible and would be a secondary effect even to Coulomb corrections. And Jeffries is right that the WDs are essentially cold. I think the problem may be one of incorrect use of Chandrasekhar-mass WD in the literature. Strictly speaking, a Mch WD cannot exist, since the Chandraskhar mass is the limiting mass. All non-rotating WDs therefore must have a lower mass. When we say "Chandrasekhar-mass white dwarf", we always mean "near Chandraskhar-mass white dwarf". Why don't you try it out and integrate a hydrostatic profile using TOV. Pick a central density and see what mass you get. The effect will certainly be small compared to metallicity effects (e.g. how much Ne22), Coulomb corrections to the equation of state (it's a plasma) or even non-zero temperature.

I don't have to do it - there are lots of publications that do it already!

For example Boshkayev et al. (2013)  http://adsabs.harvard.edu/abs/2013ApJ...762..117B

This says that non-rotating C WDs collapse due to GR at a density of 2.12e10 g/cc and a mass of 1.386 solar masses. This threshold is lower than that due to inverse beta decay stability (which would occur at densities of 4e10 g/cc). This is what most people would call the "Chandrasekhar limit". However, the same paper also points out the Pyconuclear reactions might get going at lower densities and result in instability. They say at densities >1e10 g/cc, but whatever, that isn't the detail I was trying to understand - I guess these things are uncertain so let's go with the (2-5)e9 g/cc suggested by Lesaffre.

So then I think the relevant plot in the Boshkayev paper is the left hand panel of Fig.2, which shows mass versus central density. The solid line shows the relation for cold, non-rotating C WDs. As you say I think the adoption of GR makes only a small difference. The mass of a WD at (2-5)E9 g/cc is 1.35-1.37 solar masses. This can be compared with the classic Hamada & Salpeter (1961,  ApJ, 134, 683, Table 1), which gives values for Newtonian WDs with similar central densities as 1.37-1.395 solar masses.

I think you are correct that this effect is smaller (actually I think it's comparable) than electrostatic corrections, but everyone includes those right? I don't think finite temperature effects would make much difference at these kinds of densities. If you add Ne22 to a C WD, doesn't it enhance the discrepancy? An increase in mass per electron results in a less massive WD for the same density.

Good, I now feel that my question is answered. Your initial models do neglect GR and that is why you have 1.39 solar mass WDs with those central densities. So, as long as you don't care that the total mass of the WD is out by a few percent at a given central density, I guess it doesn't really matter?

Thanks for linking the reference. This is very interesting and I wasn't aware of those papers. It seems like that GR may be a bigger effect than I had thought. However, after skimming the paper it seems to me that the different approach to the Coulomb correction may actually cause the biggest difference. In particular, I think that the large difference between the masses of the WDs as a function of composition (He4, C12, O16, or Fe56) are due to Coulomb corrections (please correct me if I'm wrong). I will try to understand this better. However, we recently had a death in the immediate family and it will take some time before I get back to work. I agree that adding Ne22 should lower the mass for a given central density. How to deal with electrostatic corrections in multi-component plasmas would be a whole new discussion.

Ivo, thanks for the interesting reply. 

If I look at the left hand panel of Fig.5 in the Boshkayev paper, this shows the influence of composition. Looking at the lower (non-rotating) boundary of the stability region you can see there is only a very small difference in the mass vs central density relation between C12 and O16 WDs. There is of course big difference for Fe56, but this mainly because of the larger mu_e. The differences in Chandrasekhar mass are driven by the different mechanism for triggering instability. For C12 it is a GR instability, but for O16 it is inverse beta instability at a lower density (ignoring pyconuclear reactions in both cases).

Fig.7 shows the effects of GR by computing structures using the (unrealistic) Chandrasekhar eqn. of state for mu_e=2 (no electrostatic correction) in Newtonian and GR. You can see that at a given density the GR mass is lower and perhaps 2-3 hundredths lower at densities of 1e10 g/cc.

However, the critical test would be a GR calculation for their new relativistic RFMT EOS vs a GR calculation for the Hamada & Salpeter EOS and they don't show that.

Of more help is Fig.5 right hand panel in Rotondo et al. (2011) http://arxiv.org/abs/1012.0154; who show several mass vs central density plots for C12 WDs. They compare the RFMT + GR case with the Hamada & Salpeter + Newtonian results. This shows that the M vs density relations are very close until densities of 1e9 g/cc, where they diverge. This suggests that the electrostatic corrections are close to identical at high densities. Indeed, Fig.2 of this paper shows that the ratio of the pressures predicted by these two equations of state is ~unity for densities above 1e5 g/cc; so I infer that the difference in the mass vs central density curves at high densities must be *almost entirely* due to GR.