Outline

● Properties of white dwarfs

● White dwarf cooling

● Minimum stellar mass (brown dwarfs)

Equations of Stellar Structure

● Which are valid, relevant for white dwarfs?

dP(r) −GM (r)ρ(r) dT (r) −3 L(r) κ(r)ρ(r) = = 2 3 dr r2 dr 16 π r ac T (r) dM (r) dL(r) =4 π r2ρ(r) =4 πr2ρ(r)ϵ(r) dr dr

Scaling Relations

● Assume P(r) ∝ rβ, same for M(r), ρ(r), then dP/dr ∝ P/r, etc...

● Find scaling relations from equations of stellar structure dM (r) = 4 π r2ρ(r) → M ∝r3 ρ dr dP(r) GM (r)ρ(r) G M ρ G M 2 = − → P∝ ∝ dr r2 r r 4

● Equation of state for degenerate electron gas 2/3 2 5/3 5/3 3 h −5/3 Z 5/3 5/3 M P = ( π ) m p ρ = bρ ∝b 5 20m ( A ) r

● Equate pressures to find that radius decreases with mass 5/3 −1/3 b − / Z M r ∝ M 1 3 r=2.3×109 cm ( ) G A ( M Sun ) Limiting mass of White Dwarf

● As mass increases, radius decreases towards zero. At what point does this break down?

Pressure

● In calculating pressure, assumed v = p/m.

● This breaks down as v→c. Instead, v ≈ c.

4 1 ∞ 1 ∞ 8 π c p P≠ ∫ n( p) p v dp instead P= ∫ n( p) p v dp= f 3 0 3 0 3h3 4

● Then equation of state is

1/3 4/3 3 hc Z / P = ρ4 3 ( π ) 4/3 ( ) 8 4 m p A

● Note different exponent on density.

● Why no dependence on mass of electron?

Scaling Relations

● Scaling relations from equations of stellar structure G M 2 P∝ r 4

● Equation of state for degenerate electron gas is P ∝ ρ(4+ε)/3 with ε = 1 for non-relativistic gas, ε = 0 for ultra-relativistic gas.

● Equate pressures to find dependence of radius on mass r ∝M(ϵ−2)/3ϵ as ϵ→0, r →M−∞=0

● When electrons are ultra-relativistic, pressure increases too slowly with density to support the star against collapse as mass increases. Maximum mass, the “Chandrasekhar mass”,

3/2 3/2 hc − hc M≈ m 2= m ( G ) p ( G m2 ) p p

● Accurately calculated value is 1.4 solar masses. White Dwarf Cooling

● Thermal energy of star while still supported by thermal pressure is

2 G M 3 M GMm p Eth≈ = N kT≈ kT →kT≈ 2r 2 m p r

● For M and r of typical white dwarf, find kT ~ 10 keV, T ~ 108 K.

● White dwarf radiates as blackbody with it surface temperature.

● Electrons are good conductors in most of star, leading to a near uniform temperature, but a thin layer of normal matter on the surface insulates the star and slows cooling. Ignore this layer to estimate cooling time, which will then be an upper limit.

−dEth 2 4 3 M k dT 3 M k −4 =L → 4 π r σ T = → dt= 2 T dT dt 8 mp dt 32π σ m p r

● Solution T ~ t-1/3. Cooling time is on order of 109 years. Brown Dwarfs

● As a protostar collapses, fusion turns when the core temperature gets high enough. Hydrogen fusion requires T > 107 K.

● We have assumed that radius will keep contracting, until a sufficiently high temperature is reached. But, what is degeneracy pressure stops contraction first?

● Use same “white dwarf” radius calculated before. Now Z/A = 1 since gas is hydrogen. Z 5 /3 M −1/ 3 GMm r=2.3×109 cm ( ) kT≈ p =107 K A ( M Sun ) r

● Equate r and solve for M = 0.07 MSun.

● Nuclear fusion in stars of this mass or smaller never turns on. They are “brown dwarfs”.

Homework

● For next class: – Problem 4-2