Stellar Structure: TCD 2006: 2.1 2 equations of stellar structure.

Stellar Structure: TCD 2006: 2.1

2 equations of stellar structure


a stellar interior


assumptions

Isolated body – only forces areself-gravity internal pressure

Spherical symmetry

Neglect: rotation magnetic fields

Consider spherical system of mass M and radius R

Internal structure described by:

r radius

m(r) mass within r

l(r) flux through r

T(r) temperature at r

P(r) pressure at r

[ (r) density at r ]

m,l,P,T

r

M,L,0,Teff

X,Y,Z

0,0,Pc,Tc

Centre: r=0

Surface: r=R


mass continuity

Consider a spherical shell of radius rthickness r (r <<r)density

Its mass (volume x density):

m = 4 r2 r

As r0:

dm/dr = 4 r2 2.1

Also:

m= 4 r2 dr

r

r

24d

dr

r

m


hydrostatic equilibrium

Consider forces at any point. A sphere of radius r acts as a gravitational mass situated at the centre, giving rise to a force:

g = Gm/r2

If a pressure gradient (dP/dr) exists, there will be a nett inward force acting on an element of thickness r and area A:

dP/dr r A m / dP/dr

(element mass is m = r A)

The sum of inward forces is then

m ( g +1/ dP/dr ) = - m d2r/dt2

24d

dr

r

m

r

rz

P+P

P

g

In order to oppose gravity, pressure must increase towards the centre. For hydrostatice equilibrium, forces must balance:

dP/dr = - Gm / r2 2.5


Virial theorem

T h e w h o l e s y s t e m i s i n e q u i l i b r i u m i f 2 . 5 i s s a t i s f i e d a t a l l r ,

w h e n c e i t i s p o s s i b l e t o d e r i v e a s i m p l e r e l a t i o n b e t w e e n a v e r a g e

i n t e r n a l p r e s s u r e a n d t h e g r a v i t a t i o n a l p o t e n t i a l e n e r g y o f t h e

s y s t e m .

M u l t i p l y i n g b o t h s i d e s b y 4 r 3 a n d i n t e g r a t e f r o m r = 0 t o r = R :

RR

drrr

Gmdr

dr

dPr

0

2

0

3 44

I n t e g r a t e l h s b y p a r t s ( d u = d P / d r . d r , d v = 4 r 3 ) ,

a n d s u b s t i t u t e d m = 4 r 2 d r

R MR

dmr

GmPdrrPr

0 0

20

3 434

S i n c e P ( R ) = 0 , t h e f i r s t t e r m i z z e r o . S u b s t i t u t i n g 4 r 2 d r = d v

V M

dmr

GmPdv

0 03


Virial theorem (2)

V M

dmr

GmPdv

0 03

l h s : s i m p l y 3 < P > V , w h e r e < P > i s t h e v o l u m e - a v e r a g e d p r e s s u r e ,

r h s : i s t h e g r a v i t a t i o n a l p o t e n t i a l e n e r g y o f t h e s y s t e m E g r a v .

T h u s t h e a v e r a g e p r e s s u r e n e e d e d t o s u p p o r t a s y s t e m w i t h

g r a v i t a t i o n a l e n e r g y E g r a v a n d v o l u m e V i s g i v e n b y

V

EP grav

3

1 2 . 6

T h i s i s t h e V i r i a l T h e o r e m .

T h e p h y s i c a l m e a n i n g o f p r e s s u r e d e p e n d s o n t h e s y s t e m i t s e l f , b u t

i t c a n b e a p p l i e d t o c l u s t e r s o f g a l a x i e s , c o o l i n g f l o w s , g l o b u l a r

c l u s t e r a s w e l l a s t o i n d i v i d u a l s t a r s .


Virial theorem: non-relativistic gas

In a star, an equation of state relates the gas pressure to the translational kinetic energy of the gas particles. For non-relativistic particles:

P = nkT = kT/V and Ekin = 3/2 kTand hence

P = 2/3 Ekin/V 2.7

Applying the Viral theorem: for a self-gravitating system of volume V and gravitational energy Egrav, the gravitational and kinetic energies are related by

2Ekin + Egrav = 0 2.8

Then the total energy of the system,

Etot = Ekin + Egrav = –Ekin = 1/2 Egrav 2.9

These equations are fundamental.If a system is in h-s equilibrium and tightly bound, the gas is HOT.

If the system evolves slowly, close to h-s equilibrium, changes in Ekin and Egrav

are simply related to changes in Etot.


Virial theorem: ultra-relativistic gas

For ultra-relativistic particles:

Ekin = 3 kTand hence

P = 1/3 Ekin/V 2.10

Applying the Viral theorem:

Ekin + Egrav = 0 2.11

Thus h-s equilibrium is only possible if Etot = 0.As the u-r limit is approached, ie the gas temperature

increases, the binding energy decreases and the system is easily disrupted. Occurs in supermassive stars (photons provide pressure) or in massive white dwarfs (rel. electrons provide pressure).


conservation of energy

Consider a spherical volume element dv=4 r2 dr Conservation of energy demands that energy out must equal energy in + energy produced or lost within the elementIf is the energy produced per unit mass, then

l+l = l + m dl/dm =

Since dm = 4r2 dr,

dl/dr = 4r2 2.12

We will consider the nature of energy sources, , later.

r

mr

l+l

l


radiative energy transport

A temperature difference between the centre and surface of a star implies there must be a temperature gradient, and hence a flux of energy. If transported by radiation, then this flux obeys Flick’s law of diffusion:

F = -D d(aT4)/dr

where aT4 is the radiation energy density and D is a diffusion coefficient. We state (for now) that D is related to the “opacity” (actually: D = c/)

The flux must be multiplied by 4r2 to obtain a luminosity l, whence

L = - (4r2c / 3) d(aT4)/dr

dT/dr = 3/4acT3 l/4r2 2.16a


radiative energy transport (2)

W r i t i n g t h e t e m p e r a t u r e g r a d i e n t a s Pd

Td

ln

ln:

Pr

TGm

dr

dT2

2 . 1 6

w h e r e , i n r a d i a t i v e e q u i l i b r u m

416

3

ln

ln

GmT

lP

acPd

Tdra d

2 . 1 7

T h e s e e q u a t i o n s a r e o b t a i n e d b y c o m b i n i n g 2 . 1 6 a w i t h h - s e q u i l i b r i u m a n d

t a k i n g l o g s .


convective energy transport

A n e le m e n t o f g a s is a t s o m e ra d iu s r .

C o n s id e r its u p w a rd d is p la c e m e n t b y a

d is ta n c e r , a llo w in g it to e x p a n d

a d ia b a t ic a lly u n t il th e p re s s u re w ith in is

e q u a l to th e p re s s u re o u ts id e . R e le a s e

th e e le m e n t . I f i t c o n t in u e s to m o v e

u p w a rd s , th e la y e r in q u e s t io n is

c o n v e c t iv e ly u n s ta b le .

L e t th e p re s s u re s a n d d e n s it ie s b e d e n o te d b y P * , * , a n d P , re s p e c t iv e ly .

In it ia l ly P *1 = P 1 a n d *

1 = 1 . A f te r th e p e r tu rb a t io n , P *

2 = P 2 a n d

*2 = *

1 (P *2 /P *

1 ) 1 /, w h e re P V= c a n d = 5 /3 fo r a h ig h ly - io n iz e d g a s . F o r

ra d ia t iv e e q u il ib r iu m , w e re q u ire *2 >

2 s o th a t th e n e t fo rc e

(b o u y a n c y + g ra v ita t io n ) is d o w n w a rd s a n d th e e le m e n t w il l re tu rn to its

s ta r t in g p o s it io n .

P 1 , 1 P 1* , 1

*

P 2* , 2

* P 2 , 2

r


convective energy transport (2)

E lim in a t in g a s te r is k s a n d w r it in g

P 1 = P 2 + d P , w e o b ta in

1

dP

dP 2 .1 8

fo r ra d ia t iv e e q u il ib r iu m .

T h is c o n d it io n is re la te d to th e

te m p e ra tu re g ra d ie n t a s s u m in g s o m e e q u a t io n o f s ta te (e .g . P = k T / m ) s o

th a t 2 .1 8 b e c o m e s

1

ln

ln adPd

Td 2 .1 9

P 1 , 1 P 1* , 1

*

P 2* , 2

* P 2 , 2

r


convective energy transport (3)

There are two main circumstances under which 2.19 will fail.

1.In the centre of main-sequence stars, the radiation flux l/4r2 can become

very large, whilst remains small. Thus the temperature gradient

dlnT/dlnP required for radiative equilibrium becomes large, and the

material becomes convectively unstable. This gives rise to convective

cores in massive stars.

2.In ionisation zones, a) the adiabatic exponent becomes close to unity,

and b) the opacity may become very large. Hence radiative equilibrium

may be violated for small values of the temperature gradient. This gives

rise to convective envelopes in cool stars.


energy transport

F r o m 2 . 1 7 w e h a v e i n r a d i a t i v e e q u i l i b r i u m

4rad 16

3

lnd

lnd

GmT

lP

acP

T

2 . 2 0

a n d f r o m 2 . 1 9 w e h a v e i n a d i a b a t i c c o n v e c t i v e e q u i l i b r i u m

1

ad

2 . 2 1

I n f o r m u l a t i n g t h e s t e l l a r s t r u c t u r e p r o b l e m w e o f t e n r e q u i r e a s i n g l e

e x p r e s s i o n f o r t h e t e m p e r a t u r e g r a d i e n t a n d w r i t e

adrad1lnd

lnd

P

T 2 . 2 2

w h e r e i s r e p r e s e n t s a c o n v e c t i v e e f f i c i e n c y s u c h t h a t

1 . = 0 : r a d i a t i v e e q u i l i b r i u m 2 . = 1 : a d i a b a t i c c o n v e c t i o n 3 . 0 < < 1 : n o n - a d i a b i a t i c c o n v e c t i o n - m u s t b e d e t e r m i n e d f r o m c o n v e c t i o n

t h e o r y


equations of stellar structure

W e h a v e d e r i v e d f o u r t i m e - i n d e p e n d e n t e q u a t i o n s o f s t e l l a r s t r u c t u r e .

T h e s e f o r m a s e t o f c o u p l e d f i r s t o r d e r o d e ’ s i n o n e i n d e p e n d e n t v a r i a b l e , r ,

a n d f o u r d e p e n d e n t v a r i a b l e s , m , l , P , T , w h i c h d e s c r i b e t h e s t r u c t u r e o f t h e

s t a r

24 rdr

dm . 2 . 1

2r

Gm

dr

dP 2 . 5

24 rdr

dl . 2 . 1 2

Pr

TGm

dr

dT2

2 . 1 6


ode’s: Lagrangian form

N o t e t h a t a n y v a r i a b l e c o u l d b e u s e d a s t h e i n d e p e n d e n t v a r i a b l e .

I n a n E u l e r i a n f r a m e , r , t h e s p a t i a l c o o r d i n a t e i s t h e i n d e p e n d e n t v a r i a b l e .

H o w e v e r , i n d e a l i n g w i t h m o s t p r o b l e m s i n s t e l l a r s t r u c t u r e a n d e v o l u t i o n i t

i s m o r e a p p r o p r i a t e t o w o r k i n a L a g r a n g i a n f r a m e , w i t h m a s s a s t h e

i n d e p e n d e n t v a r i a b l e .

24

1

rdm

dr 2 . 2 3

dm

dl 2 . 2 4

44 r

Gm

dm

dP

2 . 2 5

P

T

r

Gm

dm

dT44 2 . 2 6


boundary conditions

To solve a set of odes, boundary conditions are required.

For this 1d description of a star, the boundaries are at the centre and the

surface.

In the centre, the enclosed mass and luminosity are defined

r(m=0) = 0 2.27

l(m=0) = 0 2.28

At the surface the temperature and pressure can be defined, to first

approximation, by

T(m=M) = Teff 2.29

Pgas(m=M) = 0 2.30

Teff is related to the stellar luminosity and radius by L=4R2Teff4.

Thus we have four first order odes (2.24-2.26) and four bcs (2.27-2.30).


constitutive relations

In addition, , , refer to energy generation, density and energy transport,

the last depending on and , the convective efficiency and opacity. These

quantities describe the physics of the stellar material and may be

expressed in terms of the state variables (P and T) and of the composition

of the stellar material (X,Y,Z or Xi). These constitutive relations are required

to close the system of ode’s:

= (P,T,Xi) (equation of state) 2.31

= (,T,Xi) (nuclear energy generation rate) 2.32

= (,T,Xi) (opacity) 2.33

= (,T,Xi) (convective efficiency) 2.34

= (,T,,,Xi) (energy transport) 2.35


2 equations of stellar structure -- review

Assumptions: spherical symmetry, …

Mass continuityHydrostatic equilibriumConservation of energyRadiative energy transportConvective energy transport

The Virial theorem

Eulerian and Lagrangian formsBoundary ConditionsConstitutive equations

Stellar Structure: TCD 2006: 2.1 2 equations of stellar structure.

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