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Lecture 12
Stellar structure equations
Convection
A bubble of gas that is lower density than its
surroundings will rise buoyantly
 From the ideal gas law: if gas is in approximate pressure
equilibrium (i.e. not expanding or contracting) then pockets of
gas that are hotter than their surroundings will also be less
dense.
P
kT
mH
Convection
Convection is a very complex process for which we don’t
yet have a good theoretical model
The first law of thermodynamics
dU  dQ  dW
For an ideal, monatomic gas:
3k
CV 
2mH
CP 5


CV 3
The first law of thermodynamics
dU  dQ  dW
For an adiabatic process (dQ=0):
From the
ideal gas law
5

3
 1
P  V    
P T
 /  1
for ideal, monatomic gas
• In a stellar partial ionization zone, where
some of the heat is being used to ionize the
gas.
• In isothermal gas
Polytropes
A polytrope is a gas that is described by the equation of state:
P  
  5/3
  4/3
 1
For an adiabatic, monatomic ideal gas
For radiative equilibrium, or degenerate matter
For isothermal gas
Convection
Assume that the bubble rises in pressure
equilibrium with the surroundings. What
temperature gradient is required to
support convection?
dT  1  T dP
 1  
dr    P dr
Using the ideal gas law and the
equation for hydrostatic
equilibrium:
dT  1  mH GM r
 1  
dr    k
r2
Convection
Compare the temperature gradient due
to radiation:
with that required for convection:
When will convection dominate?
Simulation of
convection at
solar surface
dT
3 Lr

dr
64 r 2T 3
dT  1  mH GM r
 1  
dr    k
r2
Observations
of granulation
on solar
surface
Static Stellar structure equations
Hydrostatic
equilibrium:
Mass conservation:
dT
3 Lr

dr
64 r 2T 3
Radiation
dP
GM r 

dr
r2
dM r
 4r 2 
dr
or
Equation of state: P 
Energy generation:
Polytrope
kT
mH
dLr
 4r 2 
dr
P  
dT  1  mH GM r
 1  
dr    k
r2
Convection
Break
Derivation of the Lane-Emden equation
1. Start with the equation of hydrostatic equilibrium
dP
GM r 

dr
r2
2. Substitute the equation of mass conservation: dM r  4r 2 
dr
3. Now assume a polytropic equation of state:
4. Make the variable substitution:
1 d  2 d 
1 /  1
x





x 2 dx  dx 
x
P  K 
 
,    

 c 
r
K n  1 n 1
2
 
c
4G
1
 1
The Lane-Emden equation
So we have arrived at a fairly simple differential equation for the
density structure of a star:
1 d  2 d 
n
x





x 2 dx  dx 

x ,  

c
r
n
n
1
 1
K n  1 n 1
2
 
c
4G
This equation has an analytic solution
for n=0, 1 and 5. This corresponds to
=∞, 2 and 1.2
1
n=0,1,2,3,4,5
(left to
right)
x
Stellar structure equations
For the polytropic solution, we can easily find the temperature
gradient, using the ideal gas law and polytropic equation of state.
This is equal to the adiabatic temperature gradient:
dT  1  mH GM r
 1  
dr    k
r2
Finally, to determine the luminosity of the star we use the equation
dLr
 4r 2 
dr
Where the energy generation  depends on density, temperature and
chemical composition.