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White Dwarfs (Degenerate Dwarfs) WHITE DWARFS (DEGENERATE DWARFS) White dwarfs are stars supported by pressure of degenerate electron gas, i.e. in their interiors thermal energy kT is much smaller then Fermi energy EF . We shall derive the equations of struc- ture of white dwarfs, sometimes called degenerate dwarfs, in the limiting case when their thermal pressure may be neglected, but the degenerate electron gas may be either non-relativistic, somewhat relativistic, or ultra-relativistic. We shall introduce variable x defined as a dimensionless electron momentum: x ≡ p/mc, xF ≡ pF /mc. (wd.1) Following the derivations in the chapter ”Equation of state” we may write density and pressure as a function of dimensionless Fermi momentum xF 3 2 ρ = Aµ x , µ = , (wd.2a) e F e 1+ X xF x4dx P = B 1 2 , (wd.2b) (1 + x2) / Z0 where X is hydrogen abundance by mass fraction, µe is the mean number of nucleons per electron, and 8π mc 3 A ≡ H =0.981 × 106 [g cm −3], (wd.3a) 3 h 8π mc 3 B ≡ mc2 =4.80 × 1023 [erg cm −3]. (wd.3b) 3 h The equation of hydrostatic equilibrium may be written as dP dP dx x4 dx = = B 1 2 = (wd.4) dr dx dr (1 + x2) / dr GM GM 3 − r ρ = − r Aµ x , r2 r2 e where we wrote x instead of xF , for simplicity. The equation of mass conservation may be written as dM r =4πr2ρ =4πr2Aµ x3. (wd.5) dr e We shall introduce dimensionless variables x1, x2, and x3, defined with: 2 r ≡ αrx1, Mr ≡ αmx2, 1+ x ≡ x3. (wd.6) Combining equations (wd.4) , (wd.5) , and (wd.6) we obtain equations in dimensionless variables dx3 2GA αm x2 1/2 = − 2 x3 , (wd.7) dx1 B α x r 1 3 dx2 αr 2 1.5 = 4πAµe x1 (x3 − 1) . (wd.8) dx1 α m wd — 1 The two scaling parameters αr and αm may be adjusted so as to make the constants in the square brackets in the equations (wd.7) and (wd.8) equal to one. This is accomplished by setting 1/2 B 1 8 −1 −1 α = =5.455 × 10 µ (cm)=0.00784 R⊙µ , (wd.9) r 8πG Aµ e e e 1.5 1 B 1 33 −2 −2 α = =2.00 × 10 µ (g)=1.005 M⊙µ . (wd.10) m 1/2 G 2 e e (2π) (2Aµe) With these scaling parameters the two equations (wd.7) and (wd.8) may by written as dx3 x2 1/2 = − 2 x3 , (wd.11) dx1 x1 dx2 2 1.5 = x1 (x3 − 1) . (wd.12) dx1 The dimensionless equations (wd.11) and wd.12) have to be supplemented with the boundary conditions. These are: x3 = x3,c, x2 =0, at x1 =0, (inner boundary conditions), (wd.13) x3 =1, x2 = x2,s, at x1 = x1,s, (outer boundary condition), (wd.14) where x3,c is the central value of the variables x3, and x2,s and x1,s are the surface values of dimensionless mass and radius, respectively. The total mass and radius of a white dwarf are given as M = αmx2,s, R = αrx1,s. (wd.15) As we have two inner boundary conditions (wd.13) , and there are two ordinary differential equations (wd.11) and (wd.12) , we may treat (wd.13) as the initial conditions for the integrations, with x3,c being a free parameter. For a given value of x3,c we may calculate central density using equations (wd.2a) and (wd.6) , integrate numerically equations (wd.11) and (wd.12) , calculate x1,s and x2,s corresponding to x3 = 1, i.e. ρ = 0 at the white dwarf surface, and finally calculate the total mass and radius with equations (wd.15) . In this way we may obtain the mass - radius relation for white dwarfs. The mass - radius relation for white dwarfs may be estimated using the usual algebraic ap- proximation to the differential equations of stellar structure and an analytical approximation to the equation of state for degenerate electron gas. The equations of stellar structure may be approximated with: M M ≈ R2ρ, i.e. ρ ≈ , (wd.16) R R3 P GM GM 2 GM 2 ≈ ρ ≈ , i.e. P ≈ . (wd.17) R R2 R5 R4 The equation of state may be approximated as 1 2 −2 −2 − / 5/3 4/3 P ≈ K1ρ + K2ρ . (wd.18) Equations (wd.16) , (wd.17) , and (wd.18) may be combined to obtain 8 10 8 −2 ≈ R ≈ R R P 2 4 2 10/3 + 2 8/3 . (wd.19) G M K1 M K2 M This may be rearranged to have wd — 2 2 4 3 1/2 K1 G M / R ≈ 1 − . (wd.20) GM 1/3 K2 2 The last equation should have the correct asymptotic form, but there may be dimensionless coeffi- cients of the order unity that our approximate analysis cannot provide. However, we may recover the coefficients noticing that in the two limiting cases, ρ ≪ 106 g cm −3, and ρ ≫ 106 g cm −3, the equation of state (wd.18) is very well approximated with a polytrope with index n =1.5 and n = 3, respectively. In these two limiting cases we have exact mass - radius relations: K1 R = for n =1.5, (wd.21) 0.4242 GM 1/3 1.5 K2 34 −2 −2 M = =1.142 × 10 µ (g)=5.74 M⊙µ , for n =3, (wd.22) 0.3639 G e e Combining equations (wd.21) and (wd.22) we may write (wd.20) as 1 2 4/3 / K1 M R ≈ 1 − = (wd.23) 0.4242 GM 1/3 M " Ch # 1 2 5 3 −1 3 4 3 / 2 / M / M / 0.0126 R⊙ 1 − , µ M⊙ M e " Ch # with the Chandrasekhar’s mass 2 2 M =1.435 M⊙ . (wd.24) Ch µ e The analytical formula (wd.23) approximates the exact numerical mass - radius relation for white dwarfs with an error smaller than 15% for masses near Chandrasekhar limit, and a much better accuracy at lower masses. The following table gives a comparison between the numerical and analytical values of white dwarf radii for µe = 2. The first column gives the logarithm of central density, the second white dwarf mass in units of M⊙, the third and fourth give numerical and analytical white dwarf radii, respectively, in units of R⊙, and the fifth column gives the fractional error of the analytical radii. log ρc M/M⊙ R/R⊙ error numerical analytical 4 .04811 .03448 .03446 .0008 5 .14600 .02339 .02335 .0015 6 .39366 .01566 .01558 .0048 7 .80146 .01013 .00997 .0158 8 1.16176 .00619 .00593 .0411 9 1.34619 .00353 .00325 .0803 10 1.41096 .00188 .00165 .1230 wd — 3.
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