# Understanding the Chandrasekhar limit for white dwarfs and its relation with supernovas

So if I understand correctly, the Chandrasekhar limit ($$\sim 1.4 \ M_{\odot}$$) is the maximum mass that a white dwarf can have. Beyond this mass, the degeneracy pressure of the electrons can no longer sustain gravity, and the star collapses. However, I personally find difficulties in applying this limit to generic stars and to make a connection with supernovas.

The only situation that I perfectly understand is when a white dwarf of small mass is already formed, and then it starts accreting mass from a companion star. When the mass of the white dwarf goes above $$1.4 \ M_{\odot}$$, a supernovae of type Ia is formed.

But can we apply the Chandrasekhar to other generic stars that have still not become white dwarfs? For example, knowing that our sun has $$1 \ M_{\odot}$$, can we ensure it will become a white dwarf? I think in this case another limit should be applied (of around $$\sim 8 \ M_{\odot}$$), but I don't understand its relation with the Chandrasekhar limit.

Above I described a situation in which a white dwarf is the stage before a supernova. But are there other situations in which a white dwarf is tha stage after a supernova? If yes, which types of supernova?

Your understanding is more-or-less right about the Chandrasekhar mass. The maximum mass is likely to be a little lower than $$1.4 M_{\odot}$$ because collapse or explosion may be triggered by either General Relativistic instabilities, inverse beta decay or by pyconuclear reactions, that all commence when the (central) density reaches $$\sim 3\times 10^{13}$$ kg/m$$^3$$ in a Carbon/Oxygen white dwarf, corresponding to a mass of about $$1.38 M_{\odot}$$ (e.g. Rotondo et al. 2011 and https://physics.stackexchange.com/a/345296/43351 ).

A Type Ia supernova might be triggered if a white dwarf close to this limit accretes more mass from a companion, or by the merger of two white dwarfs. The trigger for the explosion could be pyconuclear reactions between carbon nuclei in a dense crystalline lattice at the centre of the white dwarf, or it might be caused by the ignition of helium (from the accreted material), which can occur at lower densities nearer to the surface (see https://astronomy.stackexchange.com/a/14747/2531).

There is a very non-linear relationship (see plot below) between the initial mass of a star and the white dwarf it will eventually become. For instance it is thought that the Sun will leave behind a $$\sim 0.5 M_{\odot}$$ white dwarf, but a $$1.1M_{\odot}$$ white dwarf will have had a progenitor of $$\sim 8M_{\odot}$$. The white dwarf is essentially the ashes from the nuclear-burning core of the star, which can form a relatively small fraction of the initial mass. This "ash" never gets hot enough to burn, because electron degeneracy pressure halts any further contraction.

The big difference in mass of the progenitor and the white dwarf it leaves behind is due to mass loss, mainly in the red giant phase and the asymptotic giant branch phase, due to dusty, radiation-driven winds. These wind expel the majority of mass above the degenerate core. White dwarfs that are very close to the Chandrasekhar mass cannot be produced by normal stellar evolution, without interference or mass transfer from a binary companion.

Stars up to $$8M_{\odot}$$ will probably leave behind a degenerate core of carbon and oxygen. It is possible that slightly heavier stars may be able to leave degenerate cores of Neon or Magnesium, without burning further towards iron. Most researchers agree that beyond $$10M_{\odot}$$ that the most likely final outcome will be a core of iron-peak elements that collapses yielding a Type II supernova. All of the above mass limits may be slightly dependent on the initial chemical composition of the star.

The outcome of a Type Ia supernova is the complete destruction of the white dwarf, since the energy released is larger than the gravitational binding energy of the white dwarf (see https://physics.stackexchange.com/a/346092/43351). Type II supernova might leave behind a remnant neutron star or black hole. There are no supernovae which leave behind a white dwarf remnant.

Edit: As Peter Erwin suggests, one-way of thinking about the above processes is (roughly) that a white dwarf gets left behind if the mass of the core is less than the Chandrasekhar mass for its composition. This is satisfied for $$\leq 8M_{\odot}$$ stars, since the core mass is $$<1.2M_{\odot}$$, which is comfortably below the Chandrasekhar mass for degenerate C, O, Mg or Ne.

For higher mass stars, the core mass will be higher, but the Chandrasekhar mass for iron is lower, about $$1.2M_{\odot}$$. Therefore a white dwarf cannot be the final outcome for such stars.

• Awesome explanation, thanks! Apr 11, 2020 at 0:36
• It might be useful to add that the <i>reason</i> for core collapse supernovae is that the degenerate core of the massive star exceeds the Chandrasekhar limit. Apr 12, 2020 at 21:31
• Also, I don't think pair-instability supernovae have any relation to the Chandrasekhar limit. Apr 12, 2020 at 21:31
• Who mentioned pair instability supernovae @PeterErwin ? Apr 12, 2020 at 21:53
• The question seemed to be about "supernovae" in general, not just type Ia (e.g., "can we apply the Chandrasekhar to other generic stars that have still not become white dwarfs?"). I'm just suggesting that for the sake of completeness, one could point out that there are, in principle, some supernovae that don't involve white dwarfs, and where the Chandrasekhar limit isn't relevant. Apr 12, 2020 at 21:57

For a star of initially greater than about 11 solar masses, once fusion is complete there remains sufficient mass, more than 1.39 solar masses (the Chandrasekhar limit), that electron degeneracy pressure is not enough to prevent the continued collapse of the star due to gravity. The iron core undergoes catastrophic collapse in which electrons combine with positrons to produce neutrons, together with a massive surge of high energy neutrinos (this has been detected). Some of the neutrinos interact with infalling matter, fragmenting nuclei and producing protons and electrons. A runaway explosion is instigated in the infalling matter, again creating a supernova. Numerous further interactions create heavier-than-iron material, including radioactive elements up to (and probably beyond) uranium. The remains of the core forms a neutron star, supported by neutron degeneracy pressure.

At the end of fusion for a star of initial mass more than about 40 solar masses the core is more than about 3 solar masses (the Tolman-Oppenheimer-Volkoff limit), large enough that a black hole may form. Since the angular momentum of the core is contained in a radius of only about 10 km, neutron stars may acquire huge rotational velocities when they can be observed as pulsars.

Supernovae are classified according to their spectra. Type II supernovae have hydrogen in their spectra, while type I do not. The majority of supernovae form from stars with initially less than about 40 solar masses, and form from red supergiants which still have hydrogen in the envelope. Type Ib and type Ic supernova form from Wolf-Rayet stars, where Ib means that helium is present and Ic means that it is not. Types II, Ib and Ic supernovae are observed in spiral galaxies where active star formation takes place but not in ellipticals.

Type Ia supernovae are observed in all galaxies, and do not arise directly from the core collapse of a large star. A normal type Ia supernova occurs when a carbon- oxygen white dwarf which has been acquiring mass from a companion reaches a mass of 1.38 solar masses (just below the Chandrasekhar limit) where carbon is ignited in a runaway explosion called carbon detonation. In a much rarer event, an oxygen-neon-magnesium white dwarf can reach the Chandrasekhar limit, 1.39 solar masses, and the collapse of the core ignites oxygen. Because normal type Ia supernovae always explode at precisely the same point they have extremely uniform luminosity, which makes them suitable as “standard candles”, whose distance can be estimated from observed magnitude. On average a supernova should occur in the Milky Way about every fifty years, but they are likely to be obscured by dust in the Galactic disc. The last Milky Way supernova clearly seen by the naked eye was Kepler’s Supernova, which occurred in 1604, no further than 6 kiloparsecs or about 20 000 light years from Earth, and was brighter than all planets except Venus. Remnants of more recent supernovae have also been found. Depending upon type, a supernova could affect the Earth biosphere from a distance of 3 000 light years. A type II supernova nearer than about 25 light years could destroy half the ozone layer.

• Thanks a lot! This explanation is very useful. I'm guessing that stars with initial masses above 11 solar masses will lead to supernovas, after which a neutron star or black hole is produced, right? Apr 10, 2020 at 17:02
• That's right. At the end of fusion for a star of initial mass more than about 40 solar masses the core is more than about 3 solar masses (the Tolman-Oppenheimer-Volkoff limit), large enough that a black hole may form. Apr 10, 2020 at 17:06
• "A white dwarf is always the stage before a supernova." -- You've skipped over Type Ib and Type II supernovae, which don't don't come from white dwarfs. (Also: pair-instability supernovae?) Apr 12, 2020 at 21:26
• Garrgh, Thanks @PeterErwin, I must have had a senior moment. I pasted the wrong passage from my notes. Apr 13, 2020 at 6:53