Lecture 9: Hydrostatic Equilibrium
We now develop the equations of stellar structure. The geometry is spherical
and stars are assumed to be functions only of the radial variable, r, which runs
from the center of the star (r=0) to its surface (r=R). There are only four
fundamental equations of stellar structure and two relate to the stability of the
star imposed by the condition of Hydrostatic Equilibrium. Balancing the pressure
gradient force with the gravitational force, yeilds:
2
G(4πr
ρdr)Mr
2
− [P (r + dr) − P (r)] 4πr =
r2
Rearranging and taking the limit as dr --> 0 yields:
dP
−GMr ρ
=
dr
r2
This is the first fundamental equation of stellar structure. The definition of Mr
(mass interior to r) is the second fundamental equation of stellar structure:
dMr
= 4πr2 ρ
dr
These equations by themselves cannot be solved since there are three
unknowns, P, rho and r and only two equations. However, they can be integrated
to give a limit on the central pressure of a star. Dividing the two boxed equations
yields:
dP
−GMr
=
dMr
4πr4
which can be integrared from center (where P = P(central)) to surface, (where
P=0, to a good approximation):
! surf ace
! surf ace
−G
Mr
dP = −Pc =
dMr
4
4π center
r
center
Integrating by parts leads to
GM 2
Pc ≥
8πR4
We can, furthermore, estimate the central temperature of the Sun if we assume
the ideal gas law applies and the Sun is made of Hydrogen.
ρ
P =
kT
µmH
and taking rho to be the mean density of the Sun, we find T > 10 million K
at the center or the Sun. (The actual temp is a bit higher ... around 17 million.)
It is necessary to look a bit closer at the concept of pressure, since it plays a key
role in stellar structure. The book goes into a good deal of detail on this which is
quite good, so be sure to read that. Here we just summarize a few highlights.
Pressure is force per unit area or the rate of transfer of momentum across an
area. As the book shows in detail, it can be written quite generally (for any type
of particle, including relativistic ones and photons) that:
! ∞
1
P =
n(p)pvdp
3 0
!
where p is the magnitude of the momentum vector: p = p2x + p2y + p2z
For non-relativistic particles in stars, n(p) is given by the Maxwell-Boltzmann
relationship, so it is straightforward to calculate the pressure as:
ρ
Pgas = nkT =
kT
µmH
For photons, we have p = E/c and since E = h nu, the relevant distribution
function is the Planck law. See the book for the details, but the integration yields
what we expected:
1 4
Prad = aT
3
Pressures add in a simple way such that the total pressure is just due to the sum
of the partial pressures contributed by each component. An important partial
pressure is the pressure contributed by the electrons. Sometimes it is written
separately from the part of the gas pressure contributed by the atoms. In
general, one can write:
!
1 4
Ptotal = Pgas + Prad = Pa + Pe + Prad = (
na + ne )kT + aT
3
all atoms
Note that in most cases of interest in stellar structure, the radiation pressure is
much less than the gas pressure and can often be neglected. The exception is the
atmospheres of very hot stars, where the gas density is so low and the
temperature so high that radiation pressure becomes important. As you can
imagine, this pressure is sometimes so high that it starts a wind blowing from the
surface of the star. Winds are a common feature of massive stars. Also, at some
point, the radiation pressure may make it impossible to form a more luminous
star than a certain limit. This is known as the Eddington limit and we’ll talk about
it more later. It sets a limit on the luminosity that an object of any given mass can
have.
Polytropes:
If an additional relationship between pressure and density can be found
(or assumed) then the equations above can be solved for a unique structure. The
simplest useful assumed relationship is a “power law”, which can be written as:
P = Kργ
where K and gamma are constants. It is useful to look at the solutions that arise
from this assumption, that are called “polytropes” for several reasons. First, some
stars actually obey this relationship, including white dwarfs and pre-main
sequence stars. Also, historically, polytropes were the first structures studied that
came close to being accurate representations of real stars. Homer Lane, an
American physicist, made contributions to the problem and the equation of
polytropic structure known as the Lane-Emden equation memorializes that.
The idea of polytropes arose historically from considerations of fully
convective gases. In this picture, the gas is “turned over” repeatedly by
convection; hence the term polytropic. Convection is assumed to be rapid enough
that the rising and falling parcels of gas don't have enough time to exchange
appreciable amounts of heat with their surroundings, so their changes are
adiabatic. This condition leads to the power-law relation between the pressure and
the density given above, as we shall see next time. For adiabatic expansion or
compression, the exponent γ turns out to be the ratio of specific heats, Cp/Cv.
However, in general we are not dealing with free convection and adiabatic
processes, and the exponent γ has no physical significance. It's just a “fudge
factor” — an adjustable parameter, whose numerical value is chosen to
approximate actual conditions. Historically, the problem has been put in the
context of the “polytropic index”, n, defined as:
1
n=
γ−1
We combine the equations of hydrostatic equilibrium and mass conservation to
get:
! 2
"
dMr
1 d r dP
=−
dr
G dr ρ dr
! 2
"
1 d r dP
Which is:
= −4πGρ
2
r dr ρ dr
! 2
"
k d r d γ
Or:
(ρ ) = −4πGρ
2
r dr ρ dr
Historically, this equation was rewritten in terms of dimensionless variables:
ρ = ρc θn and r = αξ
Substituting we see that the constant alpha has the value:
1
!
(n
−1) " 12
(n + 1)Kρc
α = const =
4πG
The final equation, then has the form known as the Lane-Emden equation:
!
"
1 d 2 dθ
n
ξ
=
−θ
ξ 2 dξ
dξ
It has analytic solutions for only three values of n, namely n= 0, 1 and 5. The
forms are given in the book and graphed below (the book uses D for theta):
Increasing polytropic index corresponds to increasing central concentration of
the “star”. Two cases are important for real stars:
n=1.5, corresponding to gamma=5/3 is a good representation of a premain sequence star and of low mass white dwarfs for reasons we will see later
n=3, corresponding to gamma = 4/3 is a decent representation of the Sun
and also an excellent representation of massive white dwarfs.