Stellar Evolution- White Dwarfs

8/22/2019 Stellar Evolution- White Dwarfs

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STELLAR EVOLUTION: WHITE DWARFS

Sirius A (the larger star) and

Sirius B (the small pinprick on

the lower left), part of a binary

star system of a revolutionary

period of 50 years, 8.6 lightyears away from Earth.

IMAGE COURTESY: NASA/ESA

WITH AN EMPHASIS ON CHANDRASHEKARS LIMIT AND THE LANEEMDEN

EQUATIONS

Submitted by

C.R. Aditya Narayan

Towards CBCS Project

requirements

Guide: Dr. C Venkateswaran

External Guide: Dr. V Devanathan


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Subrahmanyan Chandrasekhar(19 Oct. 1910 - 21 Aug. 1995),

without whom white dwarfs knew no limit.IMAGE COURTESY: AIP


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White Dwarfs: Introduction

A white dwarf is an electrondegenerate dwarf star or

stellar remnant that is comparable in mass to that of our

Sun and in size to that of Earth.

This makes white dwarfs one of the densest bodies known

to mankind.

They are the most common stellar remnants, usually

formed at the end of a star s lifetime whose mass is not

sufficient enough to form a heavier neutron star or a black

hole.

It is thought to be the end state in the stellar evolution of

over 95% of the stars present in the Milky Way Galaxy.

The closest white dwarf to the Solar System is Sirius B, 8.6

light years away.


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White Dwarf: Origin

A star spends most of its main sequence lifetime

in a state of dynamic equilibrium, where the

inward crushing force of gravitation is balanced

by the thermal pressure exerted by the usage of

nuclear fuel in energy releasing fusion reactions

within the star.

When the nuclear fuel is exhausted, the inner

layers of the star naturally contracts, causing thecore to collapse inward. The outer layers are

ejected due to the thermal instabilities caused on

the surface of the red giant and is ejected finally,

as a planetary nebula, leaving behind the hot,

dense and degenerately stable white dwarf star.

This process allows the parent star to be up to 4

M in mass but the white dwarf by itself can only

be a maximum of ~1.4 M in size due to

Chandrasekhar's limit.

An artists

impression of thesequence of stages in

the evolution of a Sun-

like star. The white

dwarf is the last stage.

IMAGE COURTESY:Google


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Stars of masses less than 4 solar

masses (M ) are thought to end

their evolution in the White Dwarf

stage.

There is an equilibrium state in

white dwarfs created by the

inward gravitational collapse and

the outward electron degeneracypressure.

In 1931, Chandrasekhar correctly

predicted that the electrons

exhibit relativistic behaviour,unable to support the crushing

force beyond a limit, ~ 1.4 M

which is called Chandrasekhar's

Limit.

Diagrammatic

representation of electron

degeneracy pressure governed

by Paulis exclusion principle.

IMAGE COURTESY: Michael

Richmond


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White Dwarfs: Characteristics White dwarfs have small diameters, comparable to planets than

stars themselves. They can vary from anything as small as ~1000

miles to ~19,000 miles which is comparable to the size of theEarth.

Although they are governed by the Chandrasekhars limit, they

mostly vary in size from 0.5 M0.7 M .

White dwarfs are under-luminous due to their nature, and range

in magnitude of brightness from 9.0 16, 16 being lower. Thefaintest of white dwarfs can be 100,000 times fainter than the Sun

which has a magnitude of -26.7.

They range in temperature from 5,000 Kelvin to 70,000 Kelvin

depending upon their size and age, and the oldest are said to be

the remains of one of the first stars born in the Universe.

Their densities are very high owing to the nature of compressive

forces that formed them. Sirius B, one of the most extensively

studied white dwarfs has a density of 125,000 gm/cm3. The

densest white dwarf may be 10,000 times more than this! Only

quantum and statistical mechanics can explain such high densities.


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Finding Chandrasekhars Limit

Our main objective is to arrive at the Chandrasekhars

limit using suitable computational techniques to generate

the required values critical to the limit and the equations

that precede it.

A suitable C program using simple algorithm is written to

extract the generated solutions of the LaneEmdenequation, which is a second-order differential equation

relating the radius of the star with the density.

These extracted solutions are suitably converted into a

graph using pre-existing software, from which we canobtain the desired values of the equation, which we will

use to calculate the limit.


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The LaneEmden EquationTo find the equations, we start with the equation of continuity

(1)

And, the hydrostatic equilibrium in the star means, that we canequate the pressure gradient with the gravitational force

times the density of the star,

(2)

Putting M(r) from (2) in (1), we obtain,

(3)

This is the density-radius relation. The next step is the pressure

density relation, for which we must bring in the polytropic

relation relating pressure and density,

(4)

Putting (4) in (3) and re-arranging,


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This results in the following equation after differentiation,

This is a second order differential equation that can be solved

by putting , the central density at r=0 and at

r=R.

We now put the equation in dimensionless form by making

suitable assumptions,

And

Where n is known as the polytropic index, we get the

following equations,

(i)

(ii)

Since


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We substitute (ii) in (i) to get,

Finally we make complete the non-dimensionalisation in theabove equation to arrive at the LaneEmden Equation.

Putting

and (5) (a) and 5 (b)

We get the LaneEmden Equation:

(6)

Using suitable computational techniques, a program is written

to obtain values for different values.


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Solving the LaneEmden equation

using C Programming We now have the LaneEmden equation which we have derived

convincingly using the method of non-dimensionalisation. The

challenge is now to correctly translate this equation into a program

that will produce the solutions of the equation for various values of

the polytropic index n.

The trick lies in simplifying the C program so as to not tangle itself in a

knot of confusing loops and nested loops!

I have converted the second order differential equation to two first

order differential equations by making suitable variable substitutions.

Here, we establish a new set of boundary conditions. We can say that

the central density = 1 when = 0, and furthermore d/d= 0 when

= 0. Here, we need to find the values of() for every = 0 for

different polytropic indexes n. The values of the x intercepts (from the

graph of the solutions) are labeled as 1 for different values of n. The

algorithm that I use in the C program is by assuming a new variable to

render the LaneEmden equation in first order differential form.


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Logic and C Program

In equation (6), we take,

6 (a)

Then the inverse would be,

6 (b)

Putting 6 (a) in 6, we get,

6 (a) an (b) gives us our two first order diff. equations:

6 (c) and 6 (d)


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Structure of the C Program

Initiating C

Library

Declaration of variables

and boundary conditions

Initiate loop; the main loop code that is a syntax expression

of the equations 6 (c) and 6 (d):

theta = theta + dxi * (phi/(xi * xi))

phi = phi dxi * (xi * xi * pow(theta, n))

File print command (fprintf)

to obtain output in a file

Exit loop, exit

program


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We are only concerned with the limits > -0.3 and < 10.0 because, we can obtain the

necessary intercepts of(,) for = 0, at the surface of the star, within that range. With

the intercept values it is easy to find the mass M and the radius R of the star in termsof the central density and then to find the Chandrasekhar's limit.


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From the graph we can obtain the particular values

needed to illustrate the solutions. We can see that only

values up to n = 5 are taken and not higher than that. This

is because, for n > 5, the solutions produce an asymptotic

curve that doesn't intercept the x-axis.

The solutions we are particular about are for the

relativistic and non-relativistic cases (as proposed by Sir

Eddington called Eddington's Approximation).

(7)

(8)

I will illustrate the purpose of finding theXiterms on the

right hand side above.

With these values, we can establish a relationship

between mass M and radius R, in terms of the central

density of the starc.

i l C l l i d i i h


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Final Calculations and arriving at the

limit

(9)

R l ti i ti d N l ti i ti


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Relativistic and Non-relativistic

approximations From equation 5 (a), we can approximate the non-

relativistic case governed by (7) and infer,

And using (9) and 5 (b), we get the following relations:

And hence we deduce,


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This gives us an important result - the mass of the star is

inversely proportional to the volume of the star, hence

adding mass, will shrink the volume of the star.

Hence, as I said before in the sub-section on electrondegeneracy, the more the mass, the lesser the space

available for electrons to move about and because of the

Heisenberg Uncertainty Principle, decrease in space

decreases the uncertainty in positionx, but increases the

uncertainty in momentump. Eventually, the particles

will approach the speed of light and relativistic conditions

come into force.

Hence it is only appropriate to consider the ultra-

relativistic approximation, where from (8),

And from (9),


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This gives a good result, implying that the radius is more

sensitive to fluctuations in the mass of the polytropic

object and at a critical mass, the radius becomes zero,

unless there are internal forces (degeneracy pressure) thathelp balance it out. Hence we use the value of1

2 |(1)|

from (8) in (9), we get,

And finally we get, from 5 (a), calculated after using n=3,

we get,

(10)

The value ofKis determined by the Fermi-Dirac equation

and calculation of energy density, and subsequently

pressure assumed to be equal to a momentum flux, and

the rest mass density.


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Consider,

Putting g =2, and using the condition:f(E) (1, EEF) and

f(E) (0, E>EF), we get,

Where,pF, is the Fermi momentum. IfmB is the mean

Baryon mass and Ye, the mean number of electrons perbaryon, and ne the number density, we get an expression

for, density, which is equal to


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Now that we have got separate equations for pressure

and density, we can relate them using P = K, the

polytropic equation to get the value ofKfor ultra-

relativistic case as,

(11)

Ye = for stars which have matter (constituents) up to

Helium, and using the known values in Kin (11) and finally

that in (10):


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The polytropic equation of state, the non-relativistic and

ultra-relativistic approximations ofKwere all used by

Chandrasekhar in his path-breaking discovery.

There were successive corrections added for electrostaticeffects and inverse beta decay and the limit was fine-

tuned further.


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Conclusion The calculation for Chandrasekhar's limit was perhaps the

most sensational discovery in the field of astrophysics in

the 1930's. It was not accepted by many scientists at thatpoint of time because it was thought to be over-

sensational to the point that it would confirm the

existence of black holes, about which very little was

known. I have, to a small degree, simplified the calculationof the values used to find the Chandrasekhar's limit, using

a C program and obtained the limit by introducing

Eddington's approximation which was included in

Chandrasekhar's original paper published in 1930.


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Future Work

As part of future work, a probable direction would be to

continue to study on how the limit is affected by certain

approximations and error corrections involving

electrostatic and decay effects and whether the same canbe applied to heavier and denser bodies such as neutron

stars (stabilised by neutron degeneracy pressure) and

black holes (defined by the Schwarzschild's radius).


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Watch the stars, and from them learn. Albert Einstein

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Stellar Evolution- White Dwarfs