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Introduction:
The stars are huge gas spheres, hundreds of thousands or millions of times more massive than
the Earth. A star such as the Sun can go on shining steadily for thousands of millions of years. This is
shown by studies of the prehistory of the Earth, which indicate that the energy radiated by the Sun
has not changed by much during the last four thousands years. The equilibrium of a star must remain
stable for such period.
The simplest picture of a star is that of an isolated body of gas sufficiently massive that the
only significant forces are self-gravity and internal pressure. In the simplest case, we can assume
spherical symmetry and neglect the influence of rotation, magnetic fields and external gravitational
influences.
Analysis the light of stars, collected from ground-based and orbital telescopes, allows
astronomers to determine a variety of quantities related to their outer layers, such as effective
temperature, luminosity, and composition. However with the exception of the detection of neutrinos
from the Sun and supernova 1987A, no direct way exists to observe the central regions of stars. To
deduce the detailed internal structure of stars requires the generation of computer models that are
consistent with all known physical laws and that ultimately agree with observable surface features.
Despite all of the successes of such calculations, however, numerous questions remain unanswered.
The solution of many of these problems requires a more detailed theoretical understanding of the
physical processes in operation in the interiors of stars.
Mathematically the conditions for the internal equilibrium of a star can be expressed as four
differential equations governing the distribution of mass, gas pressure and energy production and
transport in the stars. These equations will now be derived.
Consider a spherical system of mass M and radius R.
Surface: r=R
The internal structure of this system is described by the
distribution of various quantities, including radius (r), mass
M(r) contained within spherical shell of radius r, luminosity
r
Centre: r=0
X,Y,Z
0,0,Pc,Tc
m,l,P,T
L(r) passing through spherical shell of radius r, temperature
M,L,0,Teff
T(r), pressure P(r), and density (r). Although these are expressed as a function of radius r, running
from r = 0 at the stellar centre to r = R at the stellar surface, we could have equally chosen a different
independent variable, for example mass, M. The problem of stellar structure is to determine the run
of these quantities throughout the star, hence we must derive equations relating these quantities
throughout the star.
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3.1) Mass conservation equation (Mass continuity equation)
For a spherically symmetric star, consider a shell of mass
dM r and thickness dr , located a distance r from the center.
Assuming that the shell is sufficiently thin (i.e. dr  r ), the
volume of the shell is approximately dV  4r 2 dr . If the local
density of the gas is ρ, the shell’s mass (mass = volume x
density) is given by dM r   (4r 2 dr ) . We can write the previous
equation as a differential equation:
dM r
 4r 2 
dr
(3.1)
This is the first fundamental equation of stellar structure, which states how the interior mass of a
star must change with distance from the center.
Problem 1: Calculate the average density of the Sun.
3.2) Hydrostatic equilibrium equation
Stellar evolution is the result of a constant fight
against the strong pull of gravity. Because the gravitational
force is always attractive, an opposing force must exist if a
star is to avoid collapse. This force is provided by pressure.
To calculate how the pressure must vary with depth,
consider a cylinder of mass dm whose base is located a
distance r from the center of a spherical star (Fig 3.3). The
areas of the top and bottom of the cylinder are each A and the cylinder’s height is dr . The radial
forces acting on the cylinder are:
(1) Gravity (inward):
Fg = - G
M r dm
.
r2
(3.2)
(2) Pressure (net force due to difference in pressure between upper and lower faces):
FP  ( FP ,b  FP ,t )
(3.3)
Because the pressure force is always normal to the surface, the force exerted on the top of the
cylinder must necessarily be directed toward the center of the star, while the force on the bottom is
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directed outward. Writing FP , t in terms of FP , b and a correction term dFP that accounts for the change
in force due to a change in r results in
FP ,t   ( FP ,b  dFP )
(3.4)
The net force due to difference in pressure between upper and lower faces is FP  dFP .
Applying Newton’s second law ( F  ma ) to the cylinder:
dm
d 2r
 Fg  FP  Fg  dFP
dt 2
(3.5)
Pressure is defined as the amount of force per unit area exerted on a surface ( P º F / A ). Allowing
for a difference in pressure dP between the top and the bottom of the cylinder due to the different
amount of force exerted on each surface dFP  AdP . Substituting by the previous equation and Eq.
(3.2) into Eq. (3.5) gives
dm
d 2r
GM r dm

 AdP
2
dt
r2
(3.6)
Assuming that the density of the cylinder of gas is ρ, its mass is just dm  Adr , where Adr is the
cylinder’s volume. Using this expression in Eq. (3.6) and dividing both sides by the volume of the
cylinder, yields

GM r  dP
d 2r
=2
dt
r2
dr
(3.7)
If we assume further that the star is static, then the acceleration must be zero. In this case Eq. (3.7)
reduces to
dP
GM r 

dr
r2
(3.8)
dP
  g ,
dr
where g  GM r / r 2 is the local acceleration of gravity at radius r. Equation (3.8) is the condition of
hydrostatic equilibrium, and represents the second fundamental equation of stellar structure.
Equation (3.8) indicates that:
1- In order for a star to be static, a pressure gradient dP / dr must exist to counteract the force of
gravity. It is not the pressure that supports a star, but the change in pressure with radius.
2- Pressure always decrease with increasing radius (outward), the pressure is necessarily larger in the
interior than it is near the surface.
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3- Pressure gradient vanishes at r = 0.
4- Condition at surface of a star: P = 0 (to a good first approximation)
If we use enclosed mass as the dependent variable, we can combine these two equations into
one:
dP dP dr
Gm
1

x
 2 x
dm dr dm
r
4r 2 
dP
Gm

dm
4r 4
(3.9)
This is an alternate form of hydrostatic equilibrium equation
Problem 2: Estimate the central pressure of the Sun using the equation of hydrostatic equilibrium, by
considering the whole Sun as one shell.
Problem 3: Calculate the pressure at half the solar radius ( R / 2 ).
Problem 4: Obtain a very crude estimate of the pressure at the center of the Sun, assume
that M = 1 M , r = 1 R
and  =  = 1.41g cm-3 is the average solar density. Assume the surface
pressure is exactly zero (compare your result with problems 2&3).
Problem 5: Show that the hydrostatic equilibrium, Eq. (3.8), can also be written in terms of the
optical depth τ, as
dP g
= . This form of the equation is often useful in building model stellar
d 
atmosphere. Use the relation between the opacity  and the optical depth τ ( d  = - dr ).
Problem 6: Use hydrostatic equilibrium to compare the centetral pressure of the Sun and (a) B0V
star, (b) G0III star, and (c) G0 Iab star (use appendix E).
3.3) Energy generation equation (conservation of energy)
To determine the luminosity of a star, it is now necessary to consider all of the energy
generated by stellar material. The contribution to the total luminosity due to an infinitesimal mass dm
is simply dL = ε dm, where ε is the total energy released per gram per second by all nuclear reactions
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and by gravity, or ε = εnuclear + εgravity. For a spherically symmetric star, the mass of a thin shell of
thickness dr is just dm = 4πr2ρdr. Substituting and dividing by the shell thickness, we have
dLr
 4r 2  ,
dr
(3.10)
where Lr is the interior luminosity due to all of the energy generated within the star interior to the
radius r. Equation (3.10) is the third fundamental stellar structure equation, it expresses the
conservation of energy which require that energy produced in the star has to be carried to the surface
and radiated away. The rate at which energy is produced (Eq. (3.10)) depends on the distance to the
center. Essentially all of the energy radiated by the star is produced in the hot and dense core. In the
outer layers the energy production is negligible and Lr is almost constant.
3.4) Energy transport equation
We have already related the fundamental quantities P, M, and L to independent variable r
through differential equations that describe hydrostatic equilibrium, mass conservations, and energy
generation, respectively (see Eqs. 3.1, 3.8, and 3.10). However we have not yet found a differential
equation relating the basic parameter of temperature, T, to r. Moreover, we have not explicitly
developed equations that describe the processes by which energy is carried from the deep interior to
the surface of the star. Three different energy transport mechanisms operate in stellar interiors:
1) Radiation allows the energy produced by nuclear reactions and gravitation to be carried to the
surface via photons.
2) Convection can be a very efficient transport mechanism in many regions of a star, with buoyant,
hot mass elements carrying excess energy outward while cool elements fall inward.
3) Conduction transports heat via collisions between particles. Although conduction can play an
important role in some stellar environments (compact stars, white dwarfs and neutron stars, where
the mean free path of photons is extremely short), it is generally insignificant in interiors of normal
stars, since the electrons carrying the energy can only travel a short distance without colliding with
other particles.
First consider radiation transport. The gradient in the radiation pressure produces the slight
net movement of photons toward the surface that carries the radiative flux, this process is described
by
dPrad


Frad , where Frad is the outward radiative flux,  is the average Rosseland mean
dr
c
opacity. The radiation pressure gradient may also be expressed as
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dPrad 4 3 dT
 aT
. Equating the
dr
3
dr
two expressions, we have
dT
3 

Frad . Finally, if we use the expression for the radiative
dr
4 ac T 3
flux, written in terms of radiative luminosity of the star at radius r, Frad 
Lr
, the temperature
4r 2
gradient for radiative transport becomes
dT
3   Lr

dr
4 ac T 3 4r 2
(3.11)
This is the fourth stellar structure equation. It gives the temperature change as a function of the
radius. It depends on how the energy is transported.
------------------------------------------------------------------------------------------------------------------------Problem 7: Verify that the basic stellar structure equations are satisfied by 1 M๏ STATSTAR model
found in Appendix I. This may be done by selecting two adjacent zones and numerically computing
the derivates on the left-hand sides of the equations, for example
dP
dr
Pi +1 - Pi
,
ri +1 - ri
and compare your results with results obtained from the right-hand sides using average values of
quantities for the two zones [e.g. M r = (M i + M i +1 ) / 2 ]. Carry out your calculations for the two
shells at r = 1.27x1010 cm and r = 1.34x1010 cm and then compare your results for the right and left
hand sides of each equation by determining relative errors. Note that the model in Appendix I
assumes complete ionization everywhere and has the uniform composition X=0.7, Y=0.292,
Z=0.008. Your results on the right and left hand sides will not agree exactly because STATSTAR
uses a Runge-kutta numerical algorithm that carries out intermediate steps not shown in Appendix I.
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