PH507 The Hydrostatic Equilibrium Equations for Stars

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PH507 The Hydrostatic Equilibrium Equations
for Stars
Dr J. Miao
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We will cover the following topics

• Equation of Mass distribution
• Equation of hydrostatic support
• - Virial Theorem
• -Centre pressure and mean temperature of a
  star
• -Different time scales for stars
• -How do we know we are right?
• 3. Equation of Energy generation
• 4. Equation of Energy transportation

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The Equations for stars
How does a star exist?
Internal pressure gradient
Force of gravitation

1. Stars are spherical and symmetric about their
   centers
2. Stars are in hydrostatic equilibrium

Two fundamental assumptions
Four equations of stellar structure
1. Equation of mass distribution
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2. Equation of hydrostatic support
The gravitational mass m(r) situated at the
centre gives rise to an inward gravitational
acceleration equal to
Newtons second law
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2.1 What can we know from the equation of
hydrostatic support
i) What will happen if there is no pressure
gradient to oppose the gravity?
Each spherical shell of matter converges on the
centre ? free fall of the star
The mass of the thin shell is ?M with an initial
radius of r0
The mass included in the sphere of radius r0 is
m0,
When the thin shell collapses to the distance r
?KE ?GE
It follows that the free fall time to the centre
of the sphere is given by
(See Appendix)
For the sun, tff 2000s
In fact, collapse under gravity is never
completely unopposed. During the process,
released gravitational energy is usually
dissipated into random thermal motion of the
constituents, thereby creating a pressure which
opposes further collapse ?The internal pressure
will rise and slow down the rate of collapse. The
cloud will then approach hydrostatic equilibrium
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ii) What will happen if a star is in hydrostatic
equilibrium state?
an element of matter at a distance r from the
centre will be in hydrostatic equilibrium if the
pressure gradient at r is (d2r/dt2 0 in eq.1.3)
The whole system is in equilibrium if this
equation is valid at all radii.
Eq. (1.4) implies that the pressure gradient
must be negative, or in other words, pressure
decreases from the inner central region to the
outer region
The three quantities m, r, ? are not
independent
2.2 From Hydrostatic equilibrium equation to
Virial Theorem
If m is chosen as the independent space variable
rather than r,
Divide 4??r2 into two sides of (1.4)
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Integrating the left-hand side of above by parts,
the equation can be written
Which is gravitational potential energy of the
star
Using the symbol
Note Vc0, and dm ?dV
If the star were surrounded by a vacuum, its
surface pressure would be zero
This is the general, global form of the Virial
Theorem and will be used very often later on. It
relates the gravitational energy of a star to its
thermal energy.
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2.3 What can Virial theorem tell us for classical
idea gas system ?
the equation of state of a classical gas is
known as
the internal energy per unit mass is
the system is stable and bound at all points
This is also the Virial theorem in another form?
Because the total energy (binding energy) E ?
U ? - ? / 2 ? / 2 , ? and E are always
negative, if a star is in stable.
Virial theorem tells that in a contracting gas
system (protostar ),
the energy for radiation is provided by half of
the decreased gravitational potential energy ?E
?? / 2.
When there is no energy from contraction, the
radiation of a star is provided by thermonuclear
reactions.
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2.4 Estimate the minimum pressure at the centre
of a star
Integrating eq.(1.5) from the centre to the
surface of the star
On the right-hand side we may replace r by the
stellar radius to obtain a lower limit for the
central pressure i.e, 1/r gt 1/R
The pressure at the centre of the Sun exceeds 450
million atmospheres
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2.5 Estimate the minimum mean temperature of a
star
We can use this result to estimate the average
internal temperature of a star
In the gravitational potential energy expression,
r is less than R everywhere
? between two stars of the same mass, the denser
one is also hotter.
For the Sun, Eq. (1.9) gives us T gt 4 ?106 K
if the gas is assumed to be atomic hydrogen
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2.6 Estimate the importance of the radiation
pressure
the corresponding expression for radiation
pressure is
with T 4 ?106 K and
( ?1.4 ?103 kgm-3, a 7.55?10-16 Jm-3K-4),
We have
Therefore it certainly appears that radiation
pressure is unimportant at an average point in
the Sun!
This is not true of all stars, however. We shall
see later that radiation pressure is of
importance in some stars, and some stars are much
denser than the Sun and hence correction to the
idea gas are very important.
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2.7 How accurate is the Hydrostatic Assumption?
If a mass element starts from rest with this
acceleration, its inward displacement s after a
time t
if we allow the element to fall all the way to
the centre of the star, we can replace s in the
above equation by r and then substitute
The time t is that it would take a star to
collapse if the forces are out of balance by a
factor ?
But fossil and geological records indicate that
the properties of the Sun have not changed
significantly for at least 109 years(3?1016s)
If t gt 31016s ? ? lt (tff / t)2 lt 10-27
most stars are like the sun and so we may
conclude that the equation of hydrostatic
equilibrium must be true to a very high degree of
accuracy !
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2. 8 How valid is the spherical symmetry
assumption?
Departure from spherical symmetry may be caused
by rotation of the star.
?
2 ? 10-5
? of the Sun is about 2.5 ?10 -6
Departures from spherical symmetry due to
rotation can be neglected.
This statement is true for the vast majority of
stars. There are some stars which rotate much
more rapidly than the Sun. For these stars, the
rotation-distorted shape of the star must be
accounted for in the equations of stellar
structure. r ?(r, ?, ?)
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3. Energy generation in stars
3.1 Gravitational potential energy
It is a likely source of the stellar energy and
has the form
The total energy of the system
Assuming a constant density distribution
the gravitational potential energy
the total mechanical energy of the star is
What can this tell us?
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Assuming that the Sun were originally much larger
than it is today
how much energy would have been liberated in its
gravitational collapse?
If its original radius was Ri, where Ri gtgt R?,
then the energy radiated away during collapse
would be
Further assuming Lsun is a constant throughout
its lifetime, then it would emit energy at that
rate for approximately
Is the Kelvin-Helmholz time scale
We have already noted that fossil and geological
records indicate that the properties of the Sun
have not changed significantly for at least 109
years (3 1016 s)
But the Sun has actually lost energy L 3
1016 1.2 1043 J
Gravitational potential energy alone cannot
account for the Suns luminosity throughout its
lifetime !
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3.2 Nuclear reaction
the total energy equivalent of the mass of the
Sun, .
If all this energy could be converted to
radiation, the Sun could continue shining at its
present rate for as long as
is called nuclear timescale
The Sun just have consumed its mass
Hence, for most stars at most stages in their
evolution, the following inequalities are true
td ltlt tth ltlt tn. (1.9)
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3.3 How do we include the energy source?
Define luminosity L (r) as the energy flow across
any sphere of radius r. The change in L across
the shell dr is provided by the energy generated
in the shell
where ? (r) is the density ? (r) is the energy
production rate per unit mass
This is usually called the energy-generation
equation
The energy generation rate depends on the
physical conditions of the material at the given
radius.
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4. How is energy generated transported from
center to outside?
4. 1 Convection.
Energy transport by conduction (and radiation )
occurs whenever a temperature gradient is
maintained in any body
But convection is the mass motion of elements of
gas, only occurs when.
Consider a convective element of stellar material
a distance r from the centre of the star
define ?P, ?? as the change in pressure and
density of the element
?P, ?? , as the change in pressure and density of
the surroundings
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If the blob is less dense than its surroundings
at r?r then it will keep on rising and the gas
is said to be convectively unstable.
The condition for this instability is therefore
Whether or not this condition is satisfied
depends on two things
a) the rate at which the element expands (and
hence decreases in density) due to the decreasing
pressure exerted on it
b) the rate at which the density of the
surroundings decreases with height.
We can make two assumptions about the motion of
the element

1. The element rises adiabatically, i.e. it moves
   fast enough to ensure that there is no exchange
   of heat with its surroundings

PV ? constant
2. The element rises with a speed lt the speed
of sound..
This means that, during the motion, sound waves
have plenty of time to smooth out the pressure
differences between the element and its
surroundings and hence ?P ?P at all times
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By using P/? ? constant
By the second assumption ?P ?P
For an ideal gas in which radiation pressure is
negligible, we have
P kT? / m,
? log P log? log T constant.
This can be differentiated to give relationship
of the changes between P, T and ? of the
surrounding
Substitute (1.12) and (1.13) into (1.11)
The critical temperature gradient for convection
is given by
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Note that the temperature and the pressure
gradients are both negative in this equation, we
can use modulus sigh to express their magnitudes
Eq.(1.15) can also be written as
Convection will occur if temperature gradient
exceeds a certain multiple of the pressure
gradient.
The criterion for convection derived above can be
satisfied in two ways
a) The ratio of specific heats, ?, is close to
unity
In the cool outer layers of a star, the gas is
only partially ionized, much of the heat used to
raise the temperature of the gas goes into
ionization and hence cv and cp are nearly same?
?1. A star can have an outer
convective layer
b) the temperature gradient is very steep
a large amount of energy is released in a small
volume at the centre of a star, it may require a
large temperature gradient to carry the energy
away ? A star can have convective core.
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4.2 Conduction and radiation
Conduction and radiation are similar processes
because they both involve the transfer of energy
by direct interaction,
the flux of energy flow
Which of the two - conduction and radiation - is
the more dominant in stars to transport energy?
particles
photons
Energy

Number density
nparti
nphoton
gt
mean free path
?parti 10 -10 m
?photon 10-2m
ltlt
Photons can walk more easily from a point where
the temperature is high to one where it is
significantly lower before colliding and
transferring energy, resulting in a higher
transport of energy.
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Conduction is therefore negligible in nearly all
main sequence stars and radiation is the dominant
energy transport mechanism over conduction in
most stars.
4.3 Equation of Radiative transport
If we assume for the moment that the conditions
for the occurrence of convection is not
satisfied
we can write down the fourth equation of stellar
structure,
The energy carried by radiation in the flux Frad,
can be expressed in terms of the dT/dr and a
coefficient of radiative conductivity, ?rad,
where the minus sign indicates that heat flows
down the temperature gradient.
The radiative conductivity measures the readiness
of heat to flow
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Astronomers generally prefer to work with an
inverse of the conductivity, known as the
opacity, which measures the resistance of the
material to the flow of heat. Detailed arguments
(see Appendix 2 of Taylors book stellar
evolutions) show that the opacity
where a is the radiation density constant and c
is the speed of light
Combining the above equations we obtain
Recalling that flux and luminosity are related by
the equation of radiative transport
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It is the temperature gradient that would arise
in a star if all the energy were transported by
radiation
It should be noted that the above equation also
holds if a significant fraction of energy
transport is due to conduction, but in this case,
Lr? Lr Lcond.
Then (1.22) can be written as
Clearly, the flow of energy by radiation/conductio
n can only be determined if an expression for ?
is available.
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4. 4 Radiation of Neutrinos
In massive stars late in their lives, the amount
of energy that must be transported is sometimes
larger than either radiation of photons or
convection can account for .
In these cases, significant amounts of energy may
be transported from the center to space by the
radiation of neutrinos.
This is the dominant method of cooling of stars
in advanced burning stages, which also plays a
central role in events like supernovae associated
with the death of massive stars.
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Summary
1. Based on two fundamental assumptions
we derived the four equations of stellar
structure
There are four primary variables M(r ), P(r), L(r
), T(r ) in these equations, all as a function of
radius
We also have three auxiliary equationsP
equation of state, PP(?,T, Xi)? opacity
?(?,T,Xi)? nuclear fusion rate, ?(?,T,Xi).
These are three key pieces of physics and we will
discuss them in detail
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2. From the most important hydrostatic
equilibrium equation
-- Drive the global form of Viral theorem.
With the gravitational potential energy of a stat
If the density of the system is a constant,
Drive another form of Viral theorem
Which tells us a star in hydrostatic
equilibrium is stable and bound at all points
- ?E ?U - ?? / 2. .only half of the
released potential energy can be used as
radiation during the collapse process inside a
star!
-- estimate the minimum center pressure in a star

-- estimate the minimum mean temperature of a
star
3. Criteria for convection
4. Three important time scale
td ltlt tth ltlt tn. (1.9)
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5. Show that the radiation pressure is not
important in Sun-like stars
6. Radiation is more efficient way to transport
energy from place to place than conduction
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Appendix
This may be simplified by introducing the
parameter
to give