Transcript Document

2 equations of stellar structure
Stellar Structure: TCD 2006: 2.1
a stellar interior
Stellar Structure: TCD 2006: 2.2
assumptions
Isolated body – only forces are
Surface: r=R
self-gravity
internal pressure
Spherical symmetry
Neglect:
rotation
X,Y,Z
magnetic fields
r
Consider spherical system of mass M
and radius R
Internal structure described by:
r
radius
m,l,P,T
m(r)
mass within r
Centre: r=0
l(r)
flux through r
0,0,Pc,Tc
T(r)
temperature at r
P(r)
pressure at r
[ (r)
density at r ]
Stellar Structure: TCD 2006: 2.3
M,L,0,Teff
mass continuity
Consider a spherical shell of radius r
thickness r (r <<r)
density 
Its mass (volume x density):
m = 4  r2  r
As r0:
dm/dr = 4  r2 
r
r
dm
 4r 2 
dr
2.1
Also:
m= 4   r2  dr
Stellar Structure: TCD 2006: 2.4

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
g
If a pressure gradient (dP/dr) exists,
there will be a nett inward force acting
dm on
 4r 
r
an element of thickness r and areadA:
dP/dr r A  m /  dP/dr
(element mass is m =  r A)

zP
2
The sum of inward forces is then
m ( g +1/  dP/dr ) = - m d2r/dt2
P+P
r
r
In order to oppose gravity, pressure
must increase towards the centre.
For hydrostatice equilibrium, forces
must balance:
dP/dr = - Gm  / r2
Stellar Structure: TCD 2006: 2.5
2.5
Virial theorem
The whole system is in equilibrium if 2.5 is satisfied at all r,
whence it is possible to derive a simple relation between average
internal pressure and the gravitational potential energy of the
system.
Multiplying both sides by 4r3 and integrate from r=0 to r=R:

R
0
R Gm
dP
4r
dr   
4r 2 dr
0
dr
r
3
Integrate lhs by parts (du=dP/dr.dr, dv=4r3),
and substitute dm = 4r2dr
4r P 
3
R
0
R
 3 4r 2 Pdr   
0
M
0
Gm
dm
r
Since P(R)=0, the first term iz zero. Substituting 4r2dr=dv
V
 3 Pdv   
0
M
0
Gm
dm
r
Stellar Structure: TCD 2006: 2.6
Virial theorem (2)
V
 3 Pdv   
0
M
0
Gm
dm
r
lhs: simply 3<P>V, where <P> is the volume-averaged pressure,
rhs: is the gravitational potential energy of the system Egrav.
Thus the average pressure needed to support a system with
gravitational energy Egrav and volume V is given by
1 E grav
P 
3 V
2.6
This is the Virial Theorem.
The physical meaning of pressure depends on the system itself, but
it can be applied to clusters of galaxies, cooling flows, globular
cluster as well as to individual stars.
Stellar Structure: TCD 2006: 2.7
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 nonrelativistic particles:
P = nkT = kT/V and Ekin = 3/2 kT
and 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.
Stellar Structure: TCD 2006: 2.8
Virial theorem: ultra-relativistic gas
For ultra-relativistic particles:
Ekin = 3 kT
and 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).
Stellar Structure: TCD 2006: 2.9
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
element
If  is the energy produced per
unit mass, then
l+l = l +  m  dl/dm = 
Since dm = 4r2  dr,
dl/dr = 4r2  
2.12
We will consider the nature of
energy sources, , later.
Stellar Structure: TCD 2006: 2.10
l+l
r
l
r
m
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 4r2 to obtain a
luminosity l, whence
L = - (4r2c / 3) d(aT4)/dr
 dT/dr = 3/4acT3 l/4r2
Stellar Structure: TCD 2006: 2.11
2.16a
radiative energy transport (2)
Writing the temperature gradient as
d ln T
:
d ln P
dT  GmT


2
dr
r P
2.16
where, in radiative equilibrum
d ln T
3
lP
  rad 
d ln P
16ac GmT 4
2.17
These equations are obtained by combining 2.16a with h-s equilibrium and
taking logs.
Stellar Structure: TCD 2006: 2.12
convective energy transport
An element of gas is at some radius r.
Consider its upward displacement by a
*
distance r, allowing it to expand
*
P2 ,2
P2,2
adiabatically until the pressure within is
equal to the pressure outside. Release
r
the element. If it continues to move
upwards, the layer in question is
*
*
P1 ,1
P1,1
convectively unstable.
Let the pressures and densities be denoted by P*, *, and P,  respectively.
Initially P*1=P1 and *1=1. After the perturbation, P*2=P2 and
*2=*1(P*2/P*1)1/, where PV=c and =5/3 for a highly-ionized gas. For
radiative equilibrium, we require *2>2 so that the net force
(bouyancy+gravitation) is downwards and the element will return to its
starting position.
Stellar Structure: TCD 2006: 2.13
convective energy transport (2)
Eliminating asterisks and writing
*
P1=P2+dP, we obtain
P d 1

 dP 
*
P2 ,2
2.18
P2,2
r
for radiative equilibrium.
*
*
P1 ,1
P1,1
This condition is related to the
temperature gradient assuming some equation of state (e.g. P=kT/m) so
that 2.18 becomes
d ln T
 1
    ad 
d ln P

Stellar Structure: TCD 2006: 2.14
2.19
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/4r2 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.
Stellar Structure: TCD 2006: 2.15
energy transport
From 2.17 we have in radiative equilibrium
 rad 
d ln T
3
lP

d ln P 16ac GmT 4
2.20
and from 2.19 we have in adiabatic convective equilibrium
ad 
 1

2.21
In formulating the stellar structure problem we often require a single
expression for the temperature gradient and write
d ln T
   1    rad  ad
d ln P
2.22
where  is represents a convective efficiency such that
1.=0: radiative equilibrium
2.=1: adiabatic convection
3.0<<1: non-adiabiatic convection -  must be determined from convection
theory
Stellar Structure: TCD 2006: 2.16
equations of stellar structure
We have derived four time-independent equations of stellar structure.
These form a set of coupled first order ode’s in one independent variable, r,
and four dependent variables, m,l,P,T, which describe the structure of the
star
dm
 4r 2  .
dr
dP
Gm
 2
dr
r
2.1
2.5
dl
 4r 2  .
dr
2.12
dT  GmT


2
dr
r P
2.16
Stellar Structure: TCD 2006: 2.17
ode’s: Lagrangian form
Note that any variable could be used as the independent variable.
In an Eulerian frame, r, the spatial coordinate is the independent variable.
However, in dealing with most problems in stellar structure and evolution it
is more appropriate to work in a Lagrangian frame, with mass as the
independent variable.
dr
1

dm 4r 2 
2.23
dl

dm
2.24
dP
Gm

dm
4r 4
2.25
dT
Gm T


4
dm
4r P
2.26
Stellar Structure: TCD 2006: 2.18
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=4R2Teff4.
Thus we have four first order odes (2.24-2.26) and four bcs (2.27-2.30).
Stellar Structure: TCD 2006: 2.19
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)
Stellar Structure: TCD 2006: 2.20
2.35
2 equations of stellar structure -- review
Assumptions: spherical symmetry, …
Mass continuity
Hydrostatic equilibrium
Conservation of energy
Radiative energy transport
Convective energy transport
The Virial theorem
Eulerian and Lagrangian forms
Boundary Conditions
Constitutive equations
Stellar Structure: TCD 2006: 2.21