→ nootherimportantproperties → time-independent → Newtoniangravity → barotropicEOS p = p ( ρ ) Simpleself-gravitatingbodies: R OTATING P OLYTROPES

R OTATING P OLYTROPES

Simple self-gravitating bodies:

→ barotropic EOS p = p(ρ)

→ Newtonian gravity

→ time-independent

→ no other important properties

S IMPLE BUT NOT TRIVIAL :

1. Polytropic stars:

p = Kρ ^γ

2. Cold white dwarfs – degenerate electron gas EOS

3. Isothermal interstellar gas clouds:

2

E ULER AND CONTINUITY EQUATIONS

∂v

∂t + (v ∇)v = − 1

ρ ∇p − ∇Φ g

∂ρ

∂t + div(ρv) = 0

P URE R OTATION

We assume motion in our star in a form of simple

rotation:

v = r Ω(r, z)e _φ

in cylindrical coords:

S ELF - GRAVITATING , ROTATING GAS IN FULL

MECHANICAL EQUILIBRIUM

r Ω(r, z) ² e _r = 1

ρ ∇p + ∇Φ g

∂ρ

∂t + Ω(r, z) ∂ρ

∂t = 0

C ONTINUITY E QUATION – S OLUTION

ρ(r, z, φ; t) = F (r, z, φ − Ω t)

F – arbitrary function

∂ρ = 0 ↔ axial symmetry

I NTEGRABILITY C ONDITION

∇ × 

r Ω(r, z) ² e _r



= ∇ ×  1

ρ ∇p + ∇Φ g



2 r Ω ∂Ω

∂z e _φ = ∇  1

ρ



× ∇p

But p = p(ρ): ∇ 

1

ρ

 × ∇p = − _ρ ¹ 2

∂p

∂ρ ∇p × ∇p ≡ 0 so:

∂ Ω(r, z)

∂z = 0 ↔ Ω = Ω(r)

C ENTRIFUGAL P OTENTIAL

Φ _c (r) = −

r

Z

0

Ω(˜ r) ² r d˜ ˜ r

∇Φ c (r) = −r Ω(r) ² e _r

E NTHALPY

h(ρ) =

Z 1

ρ dp

∇h(ρ) = 1

ρ ∇p

Integration constant is defined to be such that:

h(ρ = 0) = 0

Euler equation becomes sum of gradients:

∇ h(ρ) + Φ _c + Φ _g  = 0

with solution:

h(ρ) + Φ _c + Φ _g = C = const

“R OTATING STAR ” E QUATION

h(ρ) + Φ _c + Φ _g = C = const

∆Φ _g = 4πG ρ

Φ _g (r) =

Z ρ(r ⁰ )

|r − r ⁰ | d ³ r ⁰

C ANONICAL FORM OF INTEGRAL EQUATION

Hammerstein, A. 1930 Acta Mathematica, 54, 117-176

h(ρ) + R(ρ) + Φ c = C

f = R [F (f)]

S OLUTION M ETHOD

f ₁ = R[F (f 0 )],

f ₂ = R[F (f 1 )],

· · ·

f _n = R[F (f n −1 )]

· · ·

Iteration succesfully applied numerically:

Self-consistent field method (Ostriker, J.P., Mark, J.W.-K. 1968 ApJ, 151, 1075)

HSCF (Hachisu, I. 1986 ApJS, 61, 479)

Z ERO - ORDER APROXIMATION

C − Φ c − h(ρ 1 ) = R(ρ 0 )

Using non-rotating ρ ₀ :

h(ρ ₀ ) + R(ρ 0 ) = C ₀

We can eliminate integral operator R:

F IRST - ORDER APROXIMATION

h(ρ ₁ ) = h(ρ ₀ ) − Φ c + C − C 0

or simpler, using enthalpy h(ρ ₀ ) ≡ h 0 , h(ρ ₁ ) ≡ h 1 :

h ₁ = h ₀ − Φ c + C − C 0

h ₁ = h ₀ − Φ c + ∆C

C ⁽³⁾

C ⁽²⁾

C ⁽¹⁾

C ⁽⁰⁾

h 0

h 0 −Φ c

−Φ c

h(r, z=0)

V ALUE OF ∆C

By substitution of our formula into basic equation we

get:

∆C = Φ _c (r)

This holds only if ∆C = 0, Φ _c ≡ 0. Instead, we can use

mean value:

∆C = −b Φ _c = −  4

3 πR ₀ ³

 ₋₁ Z

V ₀

Φ _c d ³ r

V ALUE OF ∆C

We can avoid negative enthalpy with:

∆C = −Φ c (R ₀ )

E XAMPLE : P OLYTROPIC EOS

Enthalpy is:

h(ρ) = Kγ

γ − 1 ρ ^{γ −1}

Zero-order – n-th Lane-Emden function w _n :

ρ ₀ = ρ _c (w _n ) ⁿ

Approximate formula:

ρ ₁ =



ρ ^1/n _c w _n − 1

nKγ (Φ _c + ∆C)

 _n

E XAMPLE :

ELEMENTARY FUNCTIONS

For n = 1, Ω(r) = Ω ₀ /(1 + r ² /A ² ) and

∆C = Φ _c (R ₀ = π) we get:

ρ ₁ (r, z) = sin √

r ² + z ²

√ r ² + z ² + 1

2

Ω ² ₀ A ² r ²

1 + ^{r 2}

A ²

− 1

2

Ω ² ₀ A ² π ²

1 + ^{π 2}

A ²