Equation of State Michael Palmer

Equations of State Common ones we’ve heard of Ideal gas Barotropic Adiabatic Fully degenerate Focus on fully degenerate EOS Specifically on corrections made to this EOS

White Dwarfs Physics of WD’s provide a lot of opportunity to improve EOS of a fully degenerate gas Fully degenerate core Partially degenerate towards surface Crystallization (starting in the interior) Physics of crystallization

Crystallization Interior of WD, ions can not be treated as ideal Must consider Coulomb interactions Electrons not effect at screening ions  favourable for ions to rearrange themselves When 3kT/2 is of the order of -Ze 3kT/2 => Thermal energy -Ze => Coulomb energy per ion

Crystallization Coulomb coupling ratio  = (Ze) 2 / (r i kT) (Kippenhahn and Weigert, pg 134) –r i is the mean separation between ions »Defined as (3/(4  n i )) 1/3 »n i is the number density of ions –k is the Boltzmann constant and T is temperature Gives insight into the strength of the Coulomb interactions If  >> 1, kinetic energy’s role not significant and ions will try and settle into a lower energy state Crystallization at  ≈ 175 for one component plasma Intermediate values results in phase transition from a gas to a liquid

Crystallization T m ≈ (Z 2 e 2 /  c k)(4  /3  0 m u ) 1/3 critical temperature obtained from  and density relation (Kippenhahn and Weigert, pg 134) Phase from liquid to solid can not be gradual Symmetry properties First order phase transition => lose latent heat Phase transition found to be first order (Winget et al. 2009) PUT IN FIGURE!!!! Latent heat ~ kT per ion, slows cooling Observed as bump => PUT IN FIGURE!!!!

Crystallization Lattice ions oscillate Coulomb energy -E C = 2Z/(A 1/3 )  6 1/3 keV »As T-> 0 ions not at rest Oscillate about points of equilibrium Frequency  E 2 ~ Z 2 e 2 n 0 /m 0 ZE zp = 3  E h (bar) /2 and E zp = (0.6/A)  6 1/2 keV E zp << E C so does not contribute as much to E = E 0 + E C + E zp ~ E 0 + E C Find EC influences the pressure by lowering it as compared to an ideal Fermi gas –From P  -dE / d(1/n)

Crystallization Specific heat C v effect When  << 1 the ions in the interior of the white dwarf behave as an ideal gas, C v = 3k/2 As ions form lattice, energy goes into lattice oscillations, results in additional degrees of freedom which raise C v to a maximum of 3k  = 2c v MT / 5L

Crystallization Electron polarization, Coulomb crystal f ie = -f ∞ (x r )  [1 + A(x r )(Q(  )/  ) 8 ] »f ie is the correction to free energy »f ∞ (x r ) = b 1 √(1 + b 2 /x 2 ) »A(x r ) = (b 3 + a 3 x 2 ) / (1 + b 4 x 2 ) »Q(  ) classically defined as ≈ q  »  is T P /T »xr = p F /m e c »b1,b2,b3,b4,a3 are all constants »q = Form of Q(  ) is redefined as Q(  ) = [ln(1 + e (qh) 2 )] 1/2 [ln(e - (e - 2)e -(qh) 2 )] -1/2

PC EOS & MESA Low temperature high density region Carefully handles mixtures of carbon and oxygen Accounts for all corrections to EOS laid out earlier and more Ex => Inverse Beta Decay Can handle both classical and quantum Coulomb crystals, Coulomb liquid interactions (weak or strong coupling),… Default for  > 80