Static Stellar Structure. 2 Evolutionary Changes are generally slow and can usually be handled in a...

Static Stellar StructureStatic Stellar Structure

2Static Stellar Structure

Static Stellar StructureStatic Stellar Structure

Evolutionary Changes are generally slow and can Evolutionary Changes are generally slow and can usually be handled in a quasistationary mannerusually be handled in a quasistationary manner

We generally assume:We generally assume: Hydrostatic EquilibriumHydrostatic Equilibrium Thermodynamic EquilibriumThermodynamic Equilibrium

The Equation of Hydrodynamic EquilibriumThe Equation of Hydrodynamic Equilibrium

Most of the Life of A Star is Spent in Equilibrium

2

2 2

( )d r Gm r Pdt r r

3Static Stellar Structure

Limits on Hydrostatic EquilibriumLimits on Hydrostatic Equilibrium

If the system is not If the system is not “Moving” - accelerating “Moving” - accelerating in reality - then din reality - then d22r/dtr/dt22 = = 0 and then one recovers 0 and then one recovers the equation of the equation of hydrostatic equilibrium:hydrostatic equilibrium:

If ∂P/∂r ~ 0 then If ∂P/∂r ~ 0 then which is just which is just the freefall condition for the freefall condition for which the time scale is which the time scale is ttff ff (GM/R(GM/R33))-1/2-1/2

2

( )Gm r P dPr r dr

2

2 2

( )d r Gm rdt r

4Static Stellar Structure

Dominant Pressure GradientDominant Pressure Gradient When the pressure gradient dP/dr When the pressure gradient dP/dr

dominates one gets (r/t)dominates one gets (r/t)22 ~ P/ρ ~ P/ρ This implies that the fluid elements must This implies that the fluid elements must

move at the local sonic velocity: cmove at the local sonic velocity: css = ∂P/∂ρ. = ∂P/∂ρ. When hydrostatic equilibrium appliesWhen hydrostatic equilibrium applies

V << cV << css

ttee >> t >> tffff where t where tee is the evolutionary time scale is the evolutionary time scale

5Static Stellar Structure

Hydrostatic EquilibriumHydrostatic Equilibrium Consider a spherical star Consider a spherical star

Shell of radius r, thickness dr and density ρ(r)Shell of radius r, thickness dr and density ρ(r) Gravitional Force: Gravitional Force: ↓↓ (Gm(r)/r (Gm(r)/r22) 4πr) 4πr22ρ(r)drρ(r)dr Pressure Force: Pressure Force: ↑↑ 4r 4r22dP where dP is the pressure dP where dP is the pressure

difference across drdifference across dr Equate the two: Equate the two: 4πr4πr22dPdP == (Gm(r)/r(Gm(r)/r22) 4πr) 4πr22ρ(r)drρ(r)dr rr22dPdP == Gm(r) ρ(r)drGm(r) ρ(r)dr dP/drdP/dr == -ρ(r)(Gm(r)/r-ρ(r)(Gm(r)/r22)) The - sign takes care of the fact that the pressure The - sign takes care of the fact that the pressure

decreases outward.decreases outward.

6Static Stellar Structure

Mass ContinuityMass Continuity

m(r) = mass within a shell =m(r) = mass within a shell = This is a first order differential equation which needs This is a first order differential equation which needs

boundary conditionsboundary conditions We choose PWe choose Pcc = the central pressure. = the central pressure.

Let us derive another form of the hydrostatic Let us derive another form of the hydrostatic equation using the mass continuity equation. equation using the mass continuity equation. Express the mass continuity equation as a differential: Express the mass continuity equation as a differential:

dm/dr = 4πrdm/dr = 4πr22ρ(r). ρ(r). Now divide the hydrostatic equation by the mass-Now divide the hydrostatic equation by the mass-

continuity equation to get: dP/dm = Gm/4πrcontinuity equation to get: dP/dm = Gm/4πr44(m)(m)

2

0( ) 4 ( )

rm r r r dr

7Static Stellar Structure

The Hydrostatic Equation in Mass The Hydrostatic Equation in Mass CoordinatesCoordinates

dP/dm = Gm/4πrdP/dm = Gm/4πr44(m)(m) The independent variable is mThe independent variable is m r is treated as a function of mr is treated as a function of m The limits on m are:The limits on m are: 0 at r = 00 at r = 0 M at r = R (this is the boundary condition on the M at r = R (this is the boundary condition on the

mass equation itself).mass equation itself). Why?Why?

Radius can be difficult to defineRadius can be difficult to define Mass is fixed.Mass is fixed.

8Static Stellar Structure

The Central PressureThe Central Pressure Consider the quantity: P + Gm(r)Consider the quantity: P + Gm(r)22/8πr/8πr44

Take the derivative with respect to r:Take the derivative with respect to r:

But the first two terms are equal and opposite so the But the first two terms are equal and opposite so the derivative is -Gmderivative is -Gm22/2r/2r55..

Since the derivative is negative Since the derivative is negative it must decrease outwardsit must decrease outwards.. At the center mAt the center m22/r/r44 → 0 and P = P → 0 and P = Pcc. At r = R P = 0 therefore . At r = R P = 0 therefore

PPcc > GM > GM22/8πR/8πR44

2 2

4 4 5

( ) ( ) ( )8 4 2

d Gm r dP Gm r dm Gm rPdr r dr r dr r

9Static Stellar Structure

The Virial TheoremThe Virial Theorem

4

3

3 2

30 0 0

2

4 ( )

4

(4 ) 4 3

3(4 ) |

4

M MM

dP mGdm r mdP mGrdm r

d dr mGr P r Pdm dm r

P mGr P dm dmr

r rdr dm

Remember:

10Static Stellar Structure

The Virial TheoremThe Virial Theorem The term is 0: r(0) = 0 and P(M) = 0The term is 0: r(0) = 0 and P(M) = 0 Remember that we are considering P, ρ, and r as Remember that we are considering P, ρ, and r as

variables of mvariables of m For a non-relativistic gas: 3P/ = 2 * Thermal energy For a non-relativistic gas: 3P/ = 2 * Thermal energy

per unit mass.per unit mass.

30(4 ) |Mr P

0

3 2M Pdm U for the entire star

GM dm the gravitional binding energyr

11Static Stellar Structure

The Virial TheoremThe Virial Theorem -2U = -2U = ΩΩ 2U + 2U + ΩΩ = 0 Virial Theorem = 0 Virial Theorem Note that E = U + Note that E = U + ΩΩ or that E+U = 0 or that E+U = 0 This is only true if “quasistatic.” If This is only true if “quasistatic.” If

hydrodynamic then there is a modification hydrodynamic then there is a modification of the Virial Theorem that will work.of the Virial Theorem that will work.

12Static Stellar Structure

The Importance of the Virial The Importance of the Virial TheoremTheorem

Let us collapse a star due to pressure imbalance:Let us collapse a star due to pressure imbalance: This will release - ∆This will release - ∆ΩΩ

If hydrostatic equilibrium is to be maintained If hydrostatic equilibrium is to be maintained the thermal energy must change by:the thermal energy must change by: ∆∆U = -1/2 ∆U = -1/2 ∆ΩΩ

This leaves 1/2 ∆This leaves 1/2 ∆ΩΩ to be “lost” from star to be “lost” from star Normally it is radiatedNormally it is radiated

13Static Stellar Structure

What Happens?What Happens? Star gets hotterStar gets hotter Energy is radiated into spaceEnergy is radiated into space System becomes more tightly bound: E System becomes more tightly bound: E

decreasesdecreases Note that the contraction leads to H burning Note that the contraction leads to H burning

(as long as the mass is greater than the critical (as long as the mass is greater than the critical mass).mass).

14Static Stellar Structure

An Atmospheric Use of PressureAn Atmospheric Use of Pressure We use a different form of the equation of We use a different form of the equation of

hydrostatic equilibrium in an atmosphere.hydrostatic equilibrium in an atmosphere. The atmosphere’s thickness is small compared The atmosphere’s thickness is small compared

to the radius of the star (or the mass of the to the radius of the star (or the mass of the atmosphere is small compared to the mass of atmosphere is small compared to the mass of the star)the star) For the Sun the photosphere depth is measured in For the Sun the photosphere depth is measured in

the 100's of km whereas the solar radius is 700,00 the 100's of km whereas the solar radius is 700,00 km. The photosphere mass is about 1% of the Sun.km. The photosphere mass is about 1% of the Sun.

15Static Stellar Structure

Atmospheric PressureAtmospheric Pressure Geometry: Plane ParallelGeometry: Plane Parallel dP/dr = -Gm(r)ρ/rdP/dr = -Gm(r)ρ/r22 (Hydrostatic Equation) (Hydrostatic Equation)

R ≈ r and m(R) = M. We use h measured with respect to R ≈ r and m(R) = M. We use h measured with respect to some arbitrary 0 level.some arbitrary 0 level.

dP/dh = - gρ where g = acceleration of gravity. For the dP/dh = - gρ where g = acceleration of gravity. For the Sun log(g) = 4.4 (units are cgs)Sun log(g) = 4.4 (units are cgs)

Assume a constant T in the atmosphere. P = nkT and Assume a constant T in the atmosphere. P = nkT and we use n = ρ/μmwe use n = ρ/μmHH (μ is the mean molecular weight) so (μ is the mean molecular weight) so

P = ρkT/μmP = ρkT/μmHH

16Static Stellar Structure

Atmospheric Pressure ContinuedAtmospheric Pressure Continued PP== ρkT/μmρkT/μmHH dPdP== -g ρ dh-g ρ dh dPdP== -g P(μm-g P(μmHH/kT) dh/kT) dh Or Or dP/PdP/P== -g(μm-g(μmHH/kT) dh/kT) dh

Where: PWhere: P00 = P and ρ = P and ρ00 = ρ at h = 0. = ρ at h = 0.

0

0

:H

H

gm hkT

gm hkT

Integrate P P e

e

17Static Stellar Structure

Scale HeightsScale Heights

H (the scale height) is defined as kT / μmH (the scale height) is defined as kT / μmHHgg It defines the scale length by which P decreases It defines the scale length by which P decreases

by a factor of e.by a factor of e. In non-isothermal atmospheres scale In non-isothermal atmospheres scale

heights are still of importance:heights are still of importance: H ≡ - (dP/dh)H ≡ - (dP/dh)-1-1 P = -(dh/d(lnP)) P = -(dh/d(lnP))

18Static Stellar Structure

Simple ModelsSimple Models

The linear model: ρ = ρThe linear model: ρ = ρcc(1-r/R)(1-r/R) Polytropic Model: P = KρPolytropic Model: P = Kργγ

K and γ independent of rK and γ independent of r γ not necessarily cγ not necessarily cpp/c/cvv

To do this right we need data on energy generation and energy transfer but:

19Static Stellar Structure

The Linear ModelThe Linear Model

Defining EquationDefining Equation Put in the Hydrostatic EquationPut in the Hydrostatic Equation Now we need to deal with m(r)Now we need to deal with m(r)

2

(1 )

( ) (1 )

c

c

rR

dP Gm r rdr r R

20Static Stellar Structure

An Expression for m(r)An Expression for m(r)

Equation of Continuity

Integrate from 0 to r

2

32

3 4

3

3 4

3 4

4 (1 )

44

4( )3

( )3

4 3( )

c

cc

c c

c

dm rrdr R

rrR

r rm rR

RTotal Mass M m R

r rSo m r MR R

21Static Stellar Structure

Linear ModelLinear Model• We want a

particular (M,R)

• ρc = 3M/πR3 and take Pc = P(ρc)

• Substitute m(r) into the hydrostatic equation:

3 42

2

2 32

2

2 3 42

2

4 13

4 73 3

2 73 9 4

.

c

c

c c

c

dP G r r rdr r R R

r r rGR R

r r rSo P P GR R

where P is the central pressure

22Static Stellar Structure

Central PressureCentral Pressure

2 2

2 2 2 2

2 6

2

4

0

5( ) 036

5 5 936 36

54

cc

cc

c

At the surface r R and P

G RP R P

G R GR MPR

GMPR

23Static Stellar Structure

The Pressure and Temperature The Pressure and Temperature StructureStructure

The pressure at any radius is then:The pressure at any radius is then:

Ignoring Radiation Pressure:Ignoring Radiation Pressure:

2 3 42 2

2 3 4

5 24 28 9( ) 136 5 5 5c

r r rP r G RR R R

2 32

2 3

5 19 9136 5 5

H

Hc

m PTk

G m r r rRk R R R

24Static Stellar Structure

PolytropesPolytropes P = KρP = Kργγ which can also be written as which can also be written as P = KρP = Kρ(n+1)/n(n+1)/n

N ≡ Polytropic IndexN ≡ Polytropic Index Consider Adiabatic Convective EquilibriumConsider Adiabatic Convective Equilibrium

Completely convectiveCompletely convective Comes to equilibriumComes to equilibrium No radiation pressureNo radiation pressure Then P = KρThen P = Kργγ where γ = 5/3 (for an ideal where γ = 5/3 (for an ideal

monoatomic gas) ==> n = 1.5monoatomic gas) ==> n = 1.5

25Static Stellar Structure

Gas and Radiation PressureGas and Radiation Pressure

PPgg = (N = (N00k/μ)ρT = βPk/μ)ρT = βP PPrr = 1/3 a T = 1/3 a T44 = (1- β)P = (1- β)P PP== PP PPgg/β/β== PPrr/(1-β)/(1-β) 1/β(N1/β(N00k/μ)ρT k/μ)ρT ==1/(1-β) 1/3 a T1/(1-β) 1/3 a T44

Solve for T: Solve for T: TT33== (3(1- β)/aβ) (N(3(1- β)/aβ) (N00k/μ)ρk/μ)ρ TT== ((3(1- β)/aβ) (N((3(1- β)/aβ) (N00k/μ))k/μ))1/31/3 ρ ρ1/31/3

26Static Stellar Structure

The Pressure EquationThe Pressure Equation

True for each point in the starTrue for each point in the star If β ≠ f(R), i.e. is a constant thenIf β ≠ f(R), i.e. is a constant then P = KρP = Kρ4/34/3

n = 3 polytrope or γ = 4/3n = 3 polytrope or γ = 4/3

0

14 3 4

0 33 1

N k TP

N ka

27Static Stellar Structure

The Eddington Standard Model n = 3The Eddington Standard Model n = 3 For n = 3 ρ For n = 3 ρ T T33

The general result is: ρ The general result is: ρ T Tnn Let us proceed to the Lane-Emden EquationLet us proceed to the Lane-Emden Equation ρ ≡ λφρ ≡ λφnn

λ is a scaling parameter identified with ρλ is a scaling parameter identified with ρcc - the central - the central densitydensity

φφnn is normalized to 1 at the center. is normalized to 1 at the center. Eqn 0: P = KρEqn 0: P = Kρ(n+1)/n(n+1)/n = Kλ = Kλ(n+1)/n(n+1)/nφφn+1n+1 Eqn 1: Hydrostatic Eqn: dP/dr = -Gρm(r)/rEqn 1: Hydrostatic Eqn: dP/dr = -Gρm(r)/r22

Eqn 2: Mass Continuity: dm/dr = 4πrEqn 2: Mass Continuity: dm/dr = 4πr22ρρ

28Static Stellar Structure

Working OnwardWorking Onward2

2

2

1

12

2

1 ( ) : ( )

1( ) 2 : 4

( 1)

1 ( 1) 4

nnn

nn nn

n

r dPSolve for m r m rG dr

d r dPPut m r in Gr dr dr

dP dK ndr dr

d r dK n Gr dr dr

29Static Stellar Structure

SimplifySimplify12

2

12

2

12

2

11 2

22

1 ( 1) 4

1( 1) 4

1 1( 1)4

( 1)4

1:

nn nn

n

nn

nnn

nn

d r dK n Gr dr dr

d dK n r Gr dr dr

d dK n rG r dr dr

K nDefine aG

r so ad dra

d dSod d

n

30Static Stellar Structure

The Lane-Emden EquationThe Lane-Emden Equation

Boundary Conditions:Boundary Conditions: Center Center ξξξ = 0; φ = 1 (the function); dφ/d ξ = 0ξ = 0; φ = 1 (the function); dφ/d ξ = 0

Solutions for n = 0, 1, 5 exist and are of the form:Solutions for n = 0, 1, 5 exist and are of the form: φ(ξ)φ(ξ) == CC00 + C + C22 ξ ξ22 + C + C44 ξ ξ44 + ... + ... == 1 - (1/6) ξ1 - (1/6) ξ22 + (n/120) ξ + (n/120) ξ44 - ... for ξ =1 (n>0) - ... for ξ =1 (n>0) For n < 5 the solution decreases monotonically and φ For n < 5 the solution decreases monotonically and φ →→ 0 at 0 at

some value ξsome value ξ11 which represents the value of the boundary level. which represents the value of the boundary level.

22

1 nd dd d

31Static Stellar Structure

General PropertiesGeneral Properties For each n with a specified K there exists a family of For each n with a specified K there exists a family of

solutions which is specified by the central density.solutions which is specified by the central density. For the standard model:For the standard model:

We want Radius, m(r), or m(ξ)We want Radius, m(r), or m(ξ) Central Pressure, DensityCentral Pressure, Density Central TemperatureCentral Temperature

14 3

04

3 1N kKa

32Static Stellar Structure

Radius and MassRadius and Mass1

1122

1

2

0

3 2

0

22

3 2

0

( 1)4

( ) 4

4

1

( ) 4

nn

a

n

n

R a

n KG

M r dr

a d

d dd d

d dM a dd d

1

3 2

0

3 2

1

332 22

( ) 4

4

( )

( 1)44

nn

dM a dd

dad

Now substitute for a and evaluate at boundary

n K dMG d

33Static Stellar Structure

The Mean to Central DensityThe Mean to Central Density

1

1

1

332 22

33(1 )3 2 32

1

1

1

( 1)44

44 ( 1)

3 3 4

3

3

c

nn

nn

c

n K dG dM

R n KG

dd

dd

34Static Stellar Structure

The Central PressureThe Central Pressure At the center ξ = 0 and φ = 1 so PAt the center ξ = 0 and φ = 1 so Pcc = Kλ = Kλn+1/nn+1/n

This is because P = KρThis is because P = Kρn+1/nn+1/n = Kλ = Kλ(n+1)/n(n+1)/nφφn+1n+1 Now take the radius equation:Now take the radius equation:

1122

1

1112 22

1

1 2

21

( 1)4

( 1)4

41)

nn

nn

nn

n KRG

n KG

GRSo Kn

35Static Stellar Structure

Central PressureCentral Pressure

1

1

1

1 12 2

22

212

3 2 2 6 2

22 2

12 2 6

22

4

4 1( 1) 3 /

4 163 9

4 1 9( 1) 3 / 16

14 ( 1) /

n nn n

c c

c

P K K

GRn d d

But M R so M R

GR MPn d d R

GMR n d d

36Static Stellar Structure

Central TemperatureCentral Temperature

0

0

g c c c c

c cc

c

N kP T P

which meansPT

N k