What will happen if a white dwarf star has mass higher than the Chandrasekhar limit, i.e. 1.4 times the mass of the Sun?
[Physics] What happens to a white dwarf star if it has mass higher than the Chandrashekhar limit
astrophysicsstarsstellar-evolution
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You are a little confused in your stellar evolution model. After the ignition of hydrogen fusion in the core of a star, it will next progress to helium fusion, then to carbon/oxygen fusion via the triple-alpha process (I've skipped a lot of steps and details there, if you want the details you can look at either Hansen & Kawaler's Stellar Interiors text or Dina Prialnik's Introduction to Stellar Structure text). What happens next is mass-dependent (using $M_\odot\simeq2\cdot10^{33}$ g and the mass of the star as $M_\star$):
- $M_\star\gtrsim 8M_\odot$
- able to continue fusion in the core
- will later blow up in core-collapse supernova events, producing either a neutron star or a black hole (mass-dependent) after forming iron in the core
- $M_\star\in(\sim0.5,\,\sim8)M_\odot$
- unable to continue fusion in the core due to insufficient temperatures
- will proceed into the planetary nebula phase (which has nothing to do with forming planets, but it's discoverer, William Herschel, thought that it was a planetary system forming)
- these stars form the white dwarfs that the Chandrasekhar limit applies to
- $M_\star\lesssim0.5M_\odot$
- unable to produce helium in the core (insufficient temperatures)
- expected to continue burning hydrogen for $t_{burn}>t_{age\,of\,universe}$
Thus, not every star produces iron in the core; this only applies to stars with mass $\gtrsim8M_\odot$.
The Chandrasekhar limit arises from comparing the gravitational forces to an $n=3$ polytrope (see this nice tool from Dr Bradley Meyer at Clemson University on polytropes)--polytropes basically mean $P=k\rho^{\gamma}$ where $P$ is the pressure, $k$ some constant, $\rho$ the mass density and $\gamma$ the adiabatic index.
That is, in order to find the limit, you need to use the hydrostatic pressure, $$ 4\pi r^3P=\frac32\frac{GM^2}{r}\tag{1} $$ and insert the pressure of the polytrope of index $n=3$ (requires numerically solving the Lane-Emden equation) and then solving (1) for the mass, $M$. If you've done it correctly, you'll find $M_{ch}=1.44M_\odot$.
The gravitational binding energy is the sum of the gravitational potential energy, $\Omega$, and the total internal kinetic energy, $U$.
If you calculate $\Omega + U$ for a star governed solely by ideal ultra-relativistic electron degeneracy pressure, the net binding energy is zero. This corresponds to the "traditional" Chandrasekhar limit for infinite density and zero radius, which occurs at a mass of $1.44 M_{\odot}$ for a carbon or oxygen white dwarf.
In truth, this situation does not occur in nature. There are a number of small corrections to the equation of state - e.g. electrostatic interactions, but more importantly there are at least two reasons why the white dwarf would become unstable at a mass lower than the canonical Chandrasekhar mass and at finite radius. (i) Neutronisation (aka electron capture) may occur, leading to the removal of degenerate electrons and instability; (ii) If one uses the appropriate Tolman-Oppenheimer-Volkhoff (TOV) general relativistic expression for hydrostatic equilibrium, then the WD becomes unstable (for a carbon WD) at around $1.397M_{\odot}$ and at a small, but finite radius of about 1000 km (see Mathew & Nandy 2017 ).
An approximation of $\Omega \sim -1.5GM^2/R$ (which is valid for a gas governed by relativistic degeneracy pressure - i.e.for a $n=3$ polytrope, http://www.astro.princeton.edu/~gk/A403/polytrop.pdf) gives $\Omega= -6\times 10^{44}$ J.
It is somewhat harder to calculate $U$ on the back of an envelope - you really need to integrate a numerical model in spherical shells, solving the TOV hydrostatic equilibrium equation in GR. However, here goes. Let's get an estimate by using the energy density of gas at the average density of the WD ($6.6\times10^{11}$ kg/m$^3$).
For a carbon gas at this density, the Fermi momentum is $p_F=1.9\times10^{-21}$ kg m/s and the relativity parameter, $p_F/m_e c \simeq 7$. Then, approximating this as ultra-relativistic, the average kinetic energy per electron is $(3/4)p_{F}c$ and the kinetic energy density $u=8.4\times10^{25}$ kg/m$^3$. Multiplying by the stellar volume gives $U=3.5\times10^{44}$ J.
Hence binding energy $\Omega + U = -2.5\times10^{44}$ J.
This is five times the $-5\times 10^{43}$ J quoted in the references you dug out. This could easily be due to my crude approximations in the calculations of $\Omega$ and $U$ (subtracting one big, uncertain number from another), but I also note that in your references they talk about a central density of $2\times10^{12}$ kg/m$^3$, whereas the central density of a WD at the GR Chandrasekhar limit is actually $2.35\times10^{13}$ kg/m$^3$. So I guess their WD is also factor of 2-3 bigger and so their GPE is a factor of 2-3 smaller because of this.
I am puzzled by where this central density comes from (if indeed that's what it is) and would appreciate any comments on this (rather than a downvote).
Footnote:
The OP raises the question of rotation. This might change things. Boshkayev et al. (2013) finds a GR Chandrasekhar limit of 1.386$M_{\odot}$ for non-rotating WDs and a central density of $2.12\times10^{13}$ kg/m$^3$ (consistent with what I use above). The rotating models (shown in Fig.2) show that a WD with 1.38$M_{\odot}$ and central density of 2-3$\times10^{12}$ kg/m$^3$ is possible, but these should be stable - the Chandrasekhar limit is increased by rotation and occurs at lower central densities in these cases, but always I think $\geq 7\times10^{12}$ kg/m$^3$.
Further footnote
After correspondence with one of the authors of the original SN Type 1A progenitor papers, it turns out they are using WD structures that do not use GR in the calculation. Hence the lower central densities at a given mass.
Best Answer
According to Wikipedia
The Chandrasekhar limit is the maximum mass of a stable white dwarf star. The limit was first published by Wilhelm Anderson and E. C. Stoner, and was named after Subrahmanyan Chandrasekhar, the Indian-American astrophysicist who improved upon the accuracy of the calculation in 1930, at the age of 19.
White dwarfs with masses greater than the limit undergo further gravitational collapse, evolving into a different type of stellar remnant, such as a neutron star or black hole.
Also read this nice lecture.