Let us determine the Chandrasekhar's mass limit for white dwarfs

in steemstem •  3 months ago  (edited)

I have been doing this project from last week for a class of modeling the stellar structures. So, I thought to share with you people. So let us begin. When we model a star in equilibrium configuration we often use the equations for polytropic structures. For this approach, we need to know the equation of state that governs the star which we are modeling. As an example, we can think of the sun. In the core of the sun radiative equilibrium governs while in the convective equilibrium. So, for this kind of situations, we use the equation of polytropes where the index is for n=3 and n=1.5.

In the polytropic process the heat capacity is constant all the time, dQ=cdT. This means that heat can be transferred in a way so that the heat capacity can be kept constant which is the basic difference between the polytropic and adiabatic process.We use the politropic equation also for white dwarf stars. The interesting thing is only the core of a white dwarf star is relativistic(the relativistic velocity of electron because of enormous pressure) while the envelope is nonrelativistic. Here the electron gas is degenerate. This behavior, as a consequence of Pauli's exclusion principle, forces electrons to gradually fill energy levels. This creates the so-called degenerate pressure that balances the gravity of the outer layers which works toward the center.

We have to take account the quantum effects for the equation of states while modeling the star. For that we use the equation derived by chandarasekhar instead of the Lane-Embden equation.

Here is the central density and . Here G is gravitational constant and we get from the relativistic degenerate electron gas.

Here, are electron mass, Planck constant, chemical composition, and atomic unit mass. It can be shown that the solution for density profile shows that there can exist a mass limit in small radiuses, which we call Chandrasekhar's mass.
We can see from here that for helium nucleus which is most often the case for white dwarf and mass is 1.459 sollar mass. Until now no white is observed having more than this mass limit.

Equation which Chandrasekhar used for white dwarfs is Lane-Embden equation for index n=3. Here we are interested in the limiting case when R goes near to zero. Here the velocity of electron is ultrarelativistic.

.....(1)
Here is defined from and where is angular velocity and is central density. Here is defined from and where is angular velocity and is central density. Here we used . K is a constant which we find while we use the guess .
Equation (1) Chandrasekhar solved using perturbation calculation.
Here, is angle between normal direction to the equator of the object, is second Lorentz polynomial. can be found from boundary condition of the potential.
here, defines limit for non-rotating configuration. Function satisfy the differencial equations bellow. .
..........(2)
.......(3)
Solving (2) and (3) we can find the profile for density of non-rotating configuration.
Profile for rotating configuration, . that means is showed in the figure. In the table is given modification of mass limit depending on parameter v. Codes are done for equation (2) and (3). Where in the first python file we find the result for non-rotating polytropes and then use this result to find the result for the second one. Here is the code for equation (2) where I counted the polytrope as non-rotating.

So, the result we get from the python code is the graph below.
Now I used the result from the non-rotating code and used it to get the rotating white dwarfs. Here goes the code for python code for rotating white dwarfs.
We can get a graph from these codes. So, here it goes

n the figure is shown profile for different values of parameter v. Although angular velocity which corresponds to parameter v are small, density profile looks very different. We can also see that increasing v, increases the density of outer envelope. We can also see the validity limit of the approximation.For this perturbation approach, for v=0.1 outer envelope has almost same density as the central region which is not predicted. This corresponds to the angular velocity of rotation ( for density in the central region of When we take the results and put it in a table for velocity, angular velocity and Chandrasekhar's mass limit it looks like this.

The boundary until when we can use this perturbation method is shown by the dotted line

References

[1] Subrahmanyan Chandrasekhar. The equilibrium of distorted polytropes. i. the rotational problem. Monthly Notices of the Royal Astronomical Society, 1933
[2] Subrahmanyan Chandrasekhar and Subrahmanyan Chandrasekhar. An in- troduction to the study of stellar structure, volume 2. Courier Corporation, 1957.
[3] John P Cox. Principles of stellar structure. volume 2. applications to stars. 1968.
[4] Rudolf Kippenhahn, Alfred Weigert, and Achim Weiss. Stellar structure and evolution, volume 192. Springer, 19
Every photo, code is my original work.
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