Mass-Volume Relation Chandrasekhar realized, that there has to be a maximum mass for a white dwarf Equating the central pressure estimate with the electron degeneracy pressure yields 2 3 π G ρ 2 R 2 = ( 3 π 2 ) ⅔ ħ 2 Z ρ 5 m A m H 5/3 Subrahmanyan Chandrasekhar (1910 1995) Assuming constant density yields an estimate for the radius of the white dwarf R ( 18 π ) ⅔ ħ 2 Z 10 G m M ⅓ A m H 5/3
Mass-Volume Relation R ( 18 π ) ⅔ ħ 2 Z 10 G m M ⅓ A m H The estimate is small by about a factor of two One consequence of the relation is 5/3 Subrahmanyan Chandrasekhar (1910 1995) M R 3 = constant The more massive the white dwarf, the smaller it s size However, there is a limit to the amount of matter that can be supported by electron degeneracy pressure
Chandrasekhar Limit In the extreme relativistic limit, the speed of light has to be used for the electron velocity This leads to a corrected degeneracy pressure P = ( 3 π 2 ) ⅓ Z ρ ħ c 4 A m H 4/3 Subrahmanyan Chandrasekhar (1910 1995) We can obtain an estimate for the maximum white dwarf mass be equating the above equation with the initial pressure estimate and using Z / A = 0.5 The radius cancels and we obtain M Ch 3 ( 2 π ) ½ ħ c 3/2 Z 8 G A m H 2 = 0.44 M
Chandrasekhar Limit The precise derivation gives for the actual mass limit M Ch = 1.44 M No white dwarf has been discovered with a mass exceeding the Chandrasekhar limit Subrahmanyan Chandrasekhar (1910 1995) Mass-radius relation with Sirius B
The Cooling of White Dwarfs Most stars end their lives as white dwarfs With no fusion heat source inside and no gravitational contraction, the white dwarfs cool slowly down Thermal energy reservoir slowly depletes
Energy Transport In most star s interiors, photons are the most effective form of energy transport In white dwarfs, degenerate electrons can travel for large distances without scattering off nuclei Scattering very likely would lead to loss of energy Lower energy states are already occupied by other electrons Electrons conduction is very effective means of energy transport leading to almost isothermal interior Only non-degenerate surface layers show temperature gradient
Energy Transport The pressure can be expressed as a function of temperature P = 4 16 π a c G M k 17 3 L κ 0 µ m H ½ 17/4 T Using the ideal gas law leads to an expression for the density ρ = 4 16 π a c G M µ m H 17 3 L κ 0 k ½ 13/4 T Equating with the expression for the degeneracy leads to L = 4 D 3 16 π a c G m H 7/2 µ M T 17 3 κ c 0 k = C T c 7/2
Energy Transport The luminosity is proportional to the core temperature to the power of 3.5 From Stefan-Boltzmann law we know that the luminosity varies with the effective temperature to the power of 4 Outer layers cool slower than the isothermal interior of the white dwarf
Cooling Timescale The kinetic energy is primarily carried by the nuclei, as the degenerate electrons cannot give up much energy to the already occupied lower energy states Assuming all mass of the white dwarf contained in the nuclei U = M 3 A m H 2 k T c Assuming a carbon nucleus and a core temperature of 28 million K results in thermal energy available of 6 10 40 J Estimating the cooling time scale leads to ~170 million years U τ 0 = = L 3 M k 2 A m H C T c 5/2
Cooling Timescale The 170 million years is an underestimate As the core temperature decreases, the cooling time scale increases Because the depletion of the internal thermal energy provides the luminosity, we get d U - = L dt d M 3 - k T = C T dt A m c c H 2 7/2 Integration with T 0 temperature at t = 0 5/2 5 A m H C T 0 T c ( t ) = T 0 1 + t 3 M k - ⅖
Cooling Timescale 5 t T c ( t ) = T 0 1 + 2 τ 0 - ⅖ Inserting this result into the equation for the luminosity yields L = C T c 7/2 2/7 5/7-7/5 5 A m H C T 0 = L 0 1 + t 3 M k 5 t = L 0 1 + 2 τ 0-7/5 with L 0 = C T 0 7/2
White Dwarf Crystallization Theoretical cooling curve for a 0.6 M white dwarf model The white dwarf crystallizes from the inside out as it cools down Crystallization occurs when the nuclei start to settle into a crystal lattice structure maintained by their electrostatic repulsion The nuclei s latent heat of 1 kt per nucleus is released during this phase change, leading to slight warming The final white dwarf is a crystallized carbonoxygen sphere of the size of the Earth