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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