project #1

by Arthur Thomason

White dwarfs are characterized by being much less luminous and by having a much smaller radius than a star of equal mass in the main sequence, according to Chandrasekhar. They also have a much higher effective temperature than main series stars of comprable luminocities.
**p** = (m_{o}**v**)/(1-(**v**/c)^{2})^{1/2}

Both medium-mass stars (up to 7 solar masses) and red dwarfs will eventually become white dwarfs once they run out of fuel. However, the vast majority of the billions of white dwarfs in our galaxy are the remains of stars similar to the sun.

According to Schwarzschild, once a star (like the sun) runs out of hydrogen and helium, mass loss reduces the star to a mass less than 1.44 solar masses (Chandrasekhar limit) before C^{12} burning begins. At this point, the star no longer has an internal energy source. Therefore, the weight of the star can no longer be supported by the internal pressure due to fusion and the weight of the matter crushes inward until the gas becomes degenerate. Hence, white dwarfs differ from main sequence stars because thier weight is supported by the refusal of electrons on nucleon to pack themselves together, rather than by energy flowing outward.

White dwarf structures are determined by a pressure/density equation of state, hydrostatic equilibrium, and the composition. However, the matter in white dwarfs becomes so dense that they can no longer be described by by the equation of state that governs main sequence stars. As described by Clayton, to accurately predict the structure of zero-temperature white dwarfs we must adopt a new equation of state. The pressure depends on both the electron pressure and ionic pressure, but because of the small contribution from the ionic pressure, we can neglect it. We can make accurate approximations by assuming relativistic complete degeneracy. The closer the electrons get to one another, the greater the maximum momentum in a completely degenerate electron gas grows. When the density is great enough, the most energetic electrons become relativistic. Thus, we must adopt a relativistic expression for momentum:

or,

**v**=(**p**/m_{o})/[1+(**p**/m_{o}c)^{2}]^{1/2}.

Plugging into the electron pressure equation,

P_{e} = (1/3) _{0}ò^{po}(**pv**_{p})8pp^{2}dp/h^{3},

(where **v**_{p} is the velocity associated with the momentum, **p**) we find:

P_{e} = (8p/3m_{o}h^{2})_{0}ò^{po}(p^{4}dp)/[1+(p/m_{o}c)^{2}]^{1/2}.

To simlify integration, let

sinh J = p/m_{o}c

and

dp = m_{o}cosh J dJ.

Hence,

P_{e}=(8pm_{e}^{4}c^{5}/3h^{3})_{0}ò^{Jo} sinh^{4}J dJ

which yeilds our equation of state:

P_{e} = (8pm_{e}^{4}c^{5}/3h^{3}){[(sinh^{3}J_{o} cosh J_{o})/4]-[3sinh 2J_{o}/16]+3J_{o}/8}.

In fermi momentum:

P_{e}=(8pm_{e}^{4}c^{5}/3h^{3}) [c (2c^{2} - 3) (c^{2} + 1)^{1/2} + 3 sinh^{-1} c].

(Equation 1)

where,

c = [h/m_{e}c](3n_{e}/8p)^{1/3}

or, using:

n_{e} = r/m_{p}m_{e},

c = (3h^{3}r /8pm_{e}^{3}c^{3}m_{e}m_{p})^{1/3} = (r/b)^{1/3}.

(Equation 2)

Therefore,

n_{e} = [8pm_{e}^{3}c^{3}/3h^{3}]c^{3}

and

U = (8pm^{4}c^{5}/3h^{3}){8c^{3}[(c)^{2} + 1]^{1/2} - 1] - f(c)}

where,

f(c) = c (2c^{2} - 3) (c^{2} + 1)^{1/2} + 3 sinh^{-1} c.

According to Mestel the mean energy per electron is then given by:

E_{o} = U/n_{e}.

Thus, the zero temperature pressure can then be written as:

P_{o} = P_{e} = -¶E_{o}/¶(1/n_{e}).

For substitution into the equation of hydrostatic support, it is convenient to express the pressure as:

P_{o} = af(c)

where,

r = bc^{3}.

By taking the gradient of both sides of (Equation 1) and then dividing both sides by r (as demonstated with spherical polytropes), we can obtain an expression for the gravitational potential (f) of a white dwarf. Using the potential and the Poisson equation ([d^{2}/dr^{2}]f = 4pGr), the structure of the system can be determined.

Although white dwarfs have a unique equation of state, certian types of white dwarfs can be aproxomated using a polytropic equation of state. According to Schwarzschild, for low mass white dwarfs with central densities less than 10^{6} g/cm^{3} (c<<1), the degenerate electrons are non-relativistic and we can approximate the equation of state using a polytropic index of n=1.5. So that,

P_{e}=[(3p^{2})^{2/3}h^{2} / 10p^{2}m_{e}](r/m_{p}m_{e})^{5/3}
.

In these low mass (nonrelativistic) white dwarfs, the mass and radius of the structure are related as follows:

M » R^{-3}.

For white dwarfs with large masses and central density greater than 10^{6} g/cm^{3}, r (from equation 2) gets large, as does c. As c>>1, the other terms in the equation become less significant. Therefore,

P_{e} » ( r/b)^{1/3} (2 r/b)^{2/3} ( r/b)^{1/3} = 2( r/b)^{4/3}.

Thus, we can approximate the equation of state for white dwarfs with large masses by using a poytropic index of 3:

P_{e} = [hc (3p^{2})^{1/3} /4] [ r/ m_{e}m_{p}]^{4/3}.

In this case (where the average electron is relativistic), the mass has no radius dependence. Rather, as stated by Mestel, since r^{4/3} is fixed (once m_{e} is determined), the theory of polytropes predicts that, in the extreme relativistic case, only one equilibrium mass is possible:

M_{Ch} = 4 p(2a/pG)^{3/2} (1/b^{2}) (- x^{2}[dJ_{3}/d x])_{1} = [5.80/ m_{e}^{2}] Solar Masses.

If the dwarf has evolved beyond hydrogen burning then we can assume m_{e} » 2. Hence,

M_{Ch} = 1.44 solar masses.

This is called the Chandrasekhar mass limit, which is the upper limit on the mass of a white dwarf. If the mass of the structure exceeds 1.44 solar masses, it can no longer be a white dwarf because the degeneracy will not provide enough pressure to keep the system from collapsing to a radius of zero. In cases where the average density is greater than 10^{6} g/cm^{3} and core temperatures are greater than 10^{7} K, the limit is reduced to 1.2 solar masses or even 1.1 solar masses for Fe^{56}.

**Bibliography**

Chandrasekhar, S., __An Introduction to the Study of Stellar structure__, 1937, 1967

Clayton, D.D., __Principles of Stellar Evolution and Nucleosynthesis,__ 1968

Mestel, L., "The Theory of White Dwarfs," __Stellar Structure,__ University of Cambrage, England

Schwarzschild, M., __Structure and evolution of the Stars,__ 1958