Which Chandrasekhar Limit do I use? 1.39 or 1.44?

Question

Different sources online say that the Chandrasekhar Limit is either 1.40 or 1.39, or 1.44 solar masses. Why the discrepancy? I heard it might have to do with the composition of the white dwarf, but, correct me if I'm wrong, don't pretty much all white dwarf stars have the same composition?

Answer 1

The "classic" Chandrasekhar mass is given by $$ M_{\rm Ch} = 1.445 \left(\frac{\mu_e}{2}\right)^{-2}\ M_{\odot},$$ where $\mu_e$ is the number of mass units per electron ($\mu_e =2$ for ionised carbon, oxygen or helium; $\mu_e = 1$ for hydrogen, $\mu_e= 56/26$ for iron (56)).

This assumes a white dwarf star of uniform, ionised composition and an equation of state given by that for ideal fermions (electrons in this case). It also ignores rotation and it uses Newtonian gravity. When defined in this way, the Chandrasekhar mass is obviously composition dependent - but most white dwarfs are expected to be formed from something with $\mu_e =2$.

In practice, a white dwarf can never become this massive. There are (at least) 4 effects that can mean the de facto Chandrasekhar mass (if it is taken to mean the maximum mass of a stable white dwarf) and at least one effect that can increase it.

The effects that reduce it:

Electrostatic interactions The electrons and ions do not form an ideal Fermi gas because of Coulomb interactions. The net result is to make a white dwarf slightly more compressible and the maximum mass about 2% lower. The correction is composition dependent; it is stronger for white dwarfs made of material with a larger atomic number.

General Relativity Massive white dwarfs are strongly affected by General Relativity. White dwarfs will become unstable at a finite density when the hydrostatic equilibrium is treated with GR. This finite density is reached at about $1.38-1.39 M_{\odot}$.

Inverse beta decay At high densities the electron Fermi energy becomes high enough to initiate inverse beta decay reactions. Electrons are captured by protons in the nuclei and this renders the star unstable. For a Carbon white dwarfs this will also happen at about $1.39 M_{\odot}$, for oxygen the threshold is lower at $\sim 1.37M_{\odot}$.

Pycnonuclear reactions At high densities, even at low temperatures, fusion reactions can be initiated by quantum tunnelling. This changes the composition of the white dwarf and can change $\mu_e$ or lower the density threshold for inverse beta decay making the star unstable.

The effect that can increase the maximum mass for stability is rotation. Some authors have claimed that the limit can be increased to as high as $2.6M_{\odot}$ in certain circumstances, though these are usually referred to as Super-Chandrasekhar mass (e.g. Das & Mukhopadhyay 2013) and are not stable. A stability analysis of rotating white dwarfs in GR found that the maximum stable mass was only increased to around $1.47 M_{\odot}$ for a Carbon white dwarf (Boshkayev et al. 2012), but these would be rotating faster than any observed white dwarfs.

Answer 2

A white dwarf is a stellar remnant composed almost entirely of electron degenerate matter. Chandrasekhar determined that there is a mass limit at which electron degeneracy pressure is insufficient to prevent gravitational collapse, and the equation is

$$M_{\mathrm{limit}} = \frac{\omega_3^0 \sqrt{3\pi}}{2} \left(\frac{\hbar c}{G}\right)^{3/2} \frac{1}{(\mu_e m_\mathrm{H})^2},$$ where $\mu_e$ is the average molecular weight per electron (depending on stellar composition), $m_\mathrm{H}$ is the mass of hydrogen, and $\omega_3^0$ is a numerical constant (approx. 2.018236).

The Chandrasekhar limit $M_{CH}$ is therefore the maximum mass of a stable white dwarf. For a typical white dwarf composed mostly of carbon and oxygen, Wikipedia gives the limit as "about 1.4 $M_{\odot}$", but as the question notes, other sources give quite a broad range of values. For example, Mazzali, P. A.; Röpke, F. K.; Benetti, S.; Hillebrandt, W. (2007) use $M_{CH}$ = 1.38 $M_{\odot}$; and in this answer, Rob Jeffries gives $M_{CH}$ = 1.45 $M_{\odot}$. Rob goes on to note that:

In more recent years, the Chandrasekhar limit has evolved to colloquially mean the maximum possible mass for a white dwarf. The main corrections to the ideal case considered by Chandrasekhar are Coulomb corrections, the possible onset of inverse beta decay (electron capture) and the instability at a finite density predicted by using GR rather than Newtonian gravity. The latter probably sets the "Chandrasekhar mass" for carbon white dwarfs to be 1.38 $M_{\odot}$.

A critical factor is $\mu_e$, which depends upon the chemical composition of the star. This is important, as not all white dwarfs are the same.

When a low to medium mass main-sequence star (like the Sun) has fused all the hydrogen to helium in its core, it evolves into a red giant and starts fusing the helium into carbon and oxygen. If the star isn't massive enough to reach the billion degrees required for carbon to start fusing, the C and O accumulate in the core, and when the giant eventually sheds its outer envelope as a planetary nebula, the remnant is a "standard" white-hot C-O white dwarf.

However, a star of 8-10.5 $M_{\odot}$ is big enough to reach the core temperature required to fuse carbon. On the other hand, it won't be hot enough to fuse heavier nucleosynthesis products such as oxygen, neon and magnesium. The end result is not a C-O white dwarf but an O-Ne-Mg white dwarf.

Add to this that accreting white dwarfs are adding hydrogen and/or helium to their outer envelope, and it's clear that the composition of a white dwarf can vary considerably, and this is no doubt one of the reasons for the range of $M_{CH}$ values.

To complicate things, a surprising number of physicists continue to assume that a type Ia supernova occurs when an accreting white dwarf (for example, with a red giant binary companion) reaches the relevant Chandrasekhar limit for its chemical composition. However, the current view of astronomers is that just before the white dwarf reaches $M_{CH}$, the increased pressure and density raise the core temperature to the point where carbon can fuse. Exactly what happens next is an area of active research, but the result is a carbon detonation and runaway thermonuclear reaction that spreads through the core within seconds, releasing a massive amount of energy that completely disrupts the white dwarf.

See also:

• Rob's answer to How massive does a main sequence star need to be to go type 1 supernova, and
• both answers to Why does the Chandrasekhar limit affect white dwarfs differently?.