Relativistic Stars: White Dwarfs

Relativistic Stars: White Dwarfs

Late Stages of Stellar Evolution The star enters the Asymptotic Giant Branch with an active helium shell burning and an almost dormant hydrogen shell Again the stars size and luminosity increase, leading to a deepening of the convective zone (dredge-up) The hydrogen shell reignites, possibly several times, leading to pulsation of the star Eventually the helium burning stops and the core collapses Outer layers will be blown off and a planetary nebula forms in a Nova

Planetary Nebulae The glow of planetary nebula shells is caused by ultra-violet light emitted by the dwarf star Abell 78 NGC 39 Eskimo

White Dwarf Sirius B In 1844, Friedrich Wilhelm Bessel concluded from observations of Sirius that it was a binary star system Sirius is the brightest star in the sky, mostly because of its proximity of 8.6 ly In 186, Alvan Graham Clark discovered Sirius B with the new 18.5 Lick refractor Sirius A is ~1,000 times brighter than Sirius B Friedrich Wilhelm Bessel (1784 1846) White dwarf Sirius B at the five o clock position

White Dwarf Sirius B Some parameters for Sirius B M = 1 M " L = 0.03 L T e = 7,000 K R = 0.008 R ρ = 3 10 9 kg m 3 g = 4.6 10 6 m s

Spectral Classes of White Dwarfs White dwarfs have their own spectral class D DA display on pressure-broadened hydrogen absorption lines (about two thirds of white dwarfs) DB display only helium absorption lines (8% of white dwarfs) DC display no lines (14% of white dwarfs) DQ display some carbon absorption lines DZ display some metal lines

Cores of White Dwarfs We derived the period-density relation using P ( r ) = π G ρ ( R r ) 3 At the central core of the white dwarf we have P ( 0 ) π G ρ R 3 3.8 10 N m Sirius B

Cores of White Dwarfs Estimate the core temperature using d T dr = 3 4 a c κ ρ T 3 L r 4 π r Radiation T T c R 0 = 3 4 a c κ ρ T c 3 L 4 π R T c 3 4 a c κ ρ L 4 π R ¼ T c T 7.6 10 7 K Sirius B

Cores of White Dwarfs Sirius is a class DA white dwarf with hydrogen lines in the spectrum Given the core temperature the energy production rate due to hydrogen fusion in the core should lead to much higher luminosity than observed Therefore, hydrogen is present only at the dwarf s surface Most white dwarfs consist of ionized carbon and oxygen cores which cannot undergo fusion at the given temperatures What acts against the gravitational force inside a white dwarf?

Electron Degeneracy Pressure Answer: Pauli exclusion principle at work Any particle or wave in an ensemble occupies a quantum state identified by a set of quantum numbers In case of fermions with spin ½, each fermion has to have a different set of quantum numbers As an ensemble of fermions cools down, more and more of the lower energy states are being filled At zero temperature, all of the lower energy states are occupied and the fermion gas is completely degenerate The maximum energy of an electron in a completely degenerate gas is called Fermi energy ε F

Electron Degeneracy Pressure Consider electrons as standing waves in a box of sides L The momentum is given by p x ( y, z ) = h N x ( y, z ) L N x ( y, z ) integer quantum numbers Which yields for the total kinetic energy ε = h N 8 m L N = N x + N y + N z

Electron Degeneracy Pressure Total number of electrons in gas corresponds to total number of unique quantum numbers times two (for two spins states) N e = 1 4 π N 3 for positive N 8 3 x ( y, z ) Solving for N = 3 N e π ⅓ yields ε F = ħ m ( 3 π n e ) ⅔ with n e = N e / L 3

Electron Degeneracy Pressure Write for the number density of electrons # electrons # nucleons Z n e = = nucleon volume A ρ m H And obtain for the Fermi energy ε F = ħ 3 π m Z ρ A m H ⅔ This gives for the condition for a degenerate electron gas 3 k T < ħ 3 π m Z ρ A m H ⅔

Electron Degeneracy Pressure This gives for the condition for a degenerate electron gas With the definition T ħ < 3 π Z ⅔ ρ ⅔ = D 3 m k m H A D = 161 K m kg - ⅔

Electron Degeneracy Pressure The calculation of the degeneracy pressure makes use of Pauli exclusion principle Heisenberg s uncertainty principle Δx Δp x ħ Without derivation, we obtain for the pressure due to a completely degenerate, non-relativistic electron gas P = ( 3 π ) ⅔ ħ Z ρ 5 m A m H 5/3 Using Z / A = 0.5 for a carbon-oxygen white dwarf gives P 1.9 10 N m Sirius B