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White Dwarf Stars

White dwarf stars are stars supported by "degeneracy pressure". Degeneracy pressure is a pressure that matter can exert even in the absence of any heat. It stems from the Pauli exclusion principle of quantum mechanics, which says that not more than one electron may occupy the same quantum state.

To calculate the structure of a white dwarf star, we need to know an equation of state, which relates the pressure of a gas to its denstiy. To calculate this pressure, we use another fundamental principle of quantum mechanics is the uncertainty principle, which says that a particle confined to a finite space, Dx, must have a minimum momentum, Dp, given by:

DpDx > h, (1)

where h = 6.626 x 10^-27 ergs s is called Planck's constant.

Let's suppose that we have a gas of electrons at zero temperature and density n_e. Then, a cube with volume V= Dx³ will contain N_e = n_eDx³ electrons. If we pick the value of Dx so that N_e = 2, we find Dx = (2/n_e)^1/3. (We are allowed to put two electrons in the same box because the electron has two possible spin states and each of these states is a different quantum state.) Now, each of the electrons in this box must have minimum momentum p_e = h/Dx, or minimum energy

E_e = p_e²/2m_e = (h/Dx)²/2m_e. (2)

Now, the pressure of a non-relativistic gas is related to its energy density by P = (2/3)(E/V). [This relationship should be familiar to you in the case of a hot gas, where P = nkT and E/V = (3/2)nkT, but it is more general than that.] Thus, from equation (1) we find

P_e = (2/3)(2E_e)/Dx³ = (2/3)(h²/m_e) Dx^-5 = (2/3)(h²/m_e)(n_e/2)^5/3. (3)

Finally, we need to relate the electron number density, ne, to the mass density, r. White dwarf stars are not made of hydrogen, they are made of carbon and oxygen, both of which have twice as many nuclear particles (neutrons and protons) as electrons. These elements have two hydrogen masses for every electron, so n_e = r/(2m_H). Plugging this value into equation (3), we find the desired equation of state:

P_e = (2/3)(h²/m_e)(r/4m_H)^5/3. (4)

[Actually, the value of pressure given by equation (4) is a bit low because our estimate of momentum from the uncertainty principle was not exact. The exact value of pressure is greater by a factor 2.16.]

Now, just as we did with main sequence stars, we can estimate the radius of the star roughly by equating this pressure with the gravitational pressure:

P_G = GM_*²/(4pR_*⁴). (5)

Once again, we assume a constant density star, so that r = 3M_*/(4pR_*³). Putting this expression into equation (4) and equating the result to P_G (eq. 5), we find:

R_* = 0.04(h²/Gm_em_H²)(M_*/m_H)^-1/3 = 10⁴ km (M_*/M_Sun)^-1/3. (6)

In fact, the estimate of equation (6) is about 20% less than the result from the exact theory, better than we had a right to expect given the rough approximations that we have made.

Note that equation (6) says that R_* decreases as M_* increases (see the mass-radius diagram for white dwarf stars).

The Chandrasekhar Limit: To derive equation (6), we assumed that the electrons were moving non-relativistically, so that their energy was related to their momentum by E_e = p_e²/2m_e (eq. 2). But when the electrons are moving close to the speed of light, we must replace equation (2) by the relativistic equation E_e = m_ec² = p_ec = hc/Dx. Thus, we find

m_ec = h[3M_*/(8pm_HR_*³)]^1/3. (7)

Putting the expression for R* from equation (6) into equation (7), we find an expression for the value, M₁, of the star's mass for which the electrons are moving at the speed of light. Plugging in the constants, we find M₁ = 0.67 M_Sun.

When the electrons are moving at nearly the speed of light, the pressure due to the electrons increases with density as P_e ~ r^4/3, which is a slower rate of increase than that given by equation (4). In fact, this more slowly increasing pressure with increasing density can't keep up with the increasing pressure due to gravity (eq. 5), as the mass of the star increases and its radius decreases. Gravity wins out and the star collapses.

We estimated that this collapse would happen when M_* = 0.67 M_Sun, but this was a only rough estimate. A detailed calculation shows that this collapse happens when M_* = M_C, where the value of M_C is called the Chandrasekhar limit and has the value M_C = 1.4 M_Sun. No white dwarf star can exist with M_* > M_C.