Why does a star beyond a certain mass limit (Chandrasekhar limit) only become a black hole?

Question

Why does a star beyond a certain mass limit (Chandrasekhar limit) only become a black hole?

A star is first made of hydrogen, it undergoes nuclear fussion reaction combining hydrogen into helium and releasing a large amount of energy. This process continues until the star is made up of an iron core, since iron has the largest value of binding energy per nucleon. After this, if the mass of the star is above the value of Chandrasekhar limit it becomes a black hole. What is the reason for this and why is a certain mass threshold required?

Answer 1

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 $M_{ch}=1.44M_\odot$.

Answer 2

The Chandrasekhar mass is not the dividing line between those stellar remnants that will become black holes and those that will become something else.

A compact, cold white dwarf (i.e. one supported by electron degeneracy pressure) may become unstable and collapse at close to the value of $M_{Ch}=1.44(\mu_e/2)^{-2}M_{\odot}$, where $\mu_{e}$ is the number of mass units per free electron ($\mu_e=2$ for Carbon or Oxygen) and derived using simple Newtonian mechanics. [In fact the Chandrasekhar mass is likely lower because of (i) electrostatic Coulomb corrections to the equation of state; (ii) inverse beta decay inducing instability and/or (iii) General Relativistic instability at finite density]. Anyhow, it's probably between 1.3 and 1.4 solar masses for a carbon/oxygen WD. If a white dwarf gained more mass than this it would either explode as a type Ia supernova or collapse to form a stable neutron star (e.g. Fryer et al. 1999); and certainly would not form a black hole.

The scenario described in the question is of a star forming an iron core. In this case $\mu_e =56/26$ and $M_{Ch}$ calculated from ideal electron degeneracy pressure is more like 1.24$M_{\odot}$ and reduced even further to 1.06$M_{\odot}$ by inverse beta decay instability. (e.g. Boshkayev et al. 2018).

If the core exceeds this value it will collapse, but that does not mean a black hole will form. The most likely outcome, at least for progenitor masses $<20-30M_{\odot}$ may be the formation of a neutron star supported by the strong nuclear repulsion between closely-packed neutrons. The dividing line between those objects that become black holes and those that become neutron stars is highly uncertain and may depend strongly on other factors like rotation.