October 19: The Chandrasekhar Limit
Meaning of Equation
If the mass of a white dwarf exceeds 1.4 solar masses, it will collapse under its own gravity. Thus 1.4 solar masses is the maximum mass of a stable white dwarf star.
Today is Subrahmanyan Chandrasekhar’s 108th birth anniversary and so I decided to dedicate today’s equation to this legend. Today, I’ll shed light on the famous equation named after him, the Chandrasekhar limit. This article will also give you a glimpse of the importance of statistical mechanics in astrophysics. As usual, let us start from scratch.
To arrive at the Chandrasekhar limit, we first need to answer two important questions: How do stars evolve and how does Pauli’s exclusion principle save the star. We begin with the first question: How do stars evolve? I will explain it in the simplest and briefest possible way.
A star is a hot ball of plasma. There is a core region in the star that hosts nuclear fusion reaction. A star spends 90% of its life fusing the most basic nuclear fusion reaction: hydrogen to helium in its core. Such a star is known as the main sequence star, an example is our Sun. The main characteristic of such a star is that it is in perfect hydrostatic equilibrium. Getting complicated? Consider this: The star is massive. So massive that it starts collapsing under its own gravity. But what stops the inward gravitational collapse is the outward (gas) pressure of the core nuclear reaction. So the inward gravitational collapse is perfectly balanced by the outward gas pressure and such a star is said to be in hydrostatic equilibrium.
When all the hydrogen is converted into helium, next element in the chain, carbon, forms. The temperature required for the hydrogen fusion was about 15 million K and for helium fusion to carbon is about 100 million K. One day, again, all the helium burns out into carbon and what is left is an inert carbon core. The temperature required to fuse carbon is whooping 500 million K. Small to mid sized stars do not have the potential to host a full scale carbon fusion. That is all we need to know about stellar evolution to understand Chandrasekhar limit. We now move on to the next question.
Now in the absence of the core reaction, gravity gains the upper hand and starts collapsing the star. This collapse starts increasing the density of the core region. Thus the mean separation between the constituent particles decreases and becomes of the order of de-Broglie wavelength (ignore this wavelength concept if its difficult to understand. Just learn that the density increases and separation b/w particles decreases). Such a system of high density is known as degenerate system. Now who will the save the star?
Answer is electrons! These little sub-atomic particles hate being crushed. They are fermions and obey Pauli’s exclusion principle. Thus no more than two electrons (one with spin up and other spin down) can occupy the same quantum state. So as gravity tries to crush the star, all the available lower energy states start getting filled. (See Fermi Energy to understand it deeply). Now since other electrons cannot occupy the already filled lower energy states, they have to fill the higher energy states. Lower energy state electrons will say, “No! You cannot occupy this state. We have already occupied it. You need to go to higher energy levels. We will exert an outward pressure if you try to occupy this state.” This pressure is known as the electron degeneracy pressure. Thus, in a highly degenerate system, the electrons with highest energy states have incredible amount of velocity associated with them (because they are at higher energy levels and hence kinetic energy is high). So high, that now relativistic effects come into picture. (For complete mathematics of this concept, see this).
In a stable white dwarf star, the inward gravitational collapse is balanced by this electron degeneracy pressure. But if the mass of the star becomes more than 1.4 solar masses, even the electron degeneracy pressure will break down. The electrons will then combine with protons and form neutrons and thus, a neutron star. This mass limit, below which the white dwarf star is stable, is known as the Chandrasekhar limit.
You see, how important statistical mechanics is to astrophysics. It is one of the most important tool with which as astrophysicist decodes the universe.