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Section 4
Structure equations
[review of 2112 material]
For reference, we recall some basic equations of stellar (and atmospheric) structure:
4.1
Hydrostatic (pressure) equilibrium
Consider a volume element of density ρ, thickness dr, area dA. The gravitational force on the
element (its weight) is the gravity times the mass:
g × ρ dA dr.
In hydrostatic equilibrium (HSE), the downward force of gravity is balanced by the upwards
pressure:1
dP dA = −gρ dA dr;
i.e.,
dP
= −ρ g
dr
1
(4.1)
Pressure is force per unit area, so the force is pressure times area
dA
r + dr
P
ρ
r
P+dP
gdm
Figure 4.1: Hydrostatic equilibrium.
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(the minus sign indicating that the two forces act in opposite directions). In spherical symmetry
(e.g., for a star) the local gravity is Gm(r)/r2 , where m(r) is the total mass contained within
radius r (measured from the centre), so in this case HSE can be expressed in the form
dP (r)
Gm(r)
= −ρ(r)
dr
r2
(4.2)
which is useful in discussing stellar atmospheres.
Using eqtn. (3.6), we can express eqtn. (4.1) in terms of optical depth:
g
dP (τ )
=
dτν
κν
4.1.1
(4.3)
Equations of state
The principal sources of pressure throughout a ‘normal’ (non-degenerate) star are gas pressure,
and radiation pressure.2 We will take the corresponding equations of state to be, in general,
PG = nkT ;
(4.4)
= (ρkT )/(µm(H))
1
PR = aT 4
3
(4.5)
for number density n at temperature T , density ρ. Here µ is the mean molecular weight (§ 4.4),
k is Boltzmann’s constant, and m(H) is the hydrogen mass; a is the radiation constant,
a = 4σ/c, with σ the Stefan-Boltzmann constant.
4.2
Mass Continuity
The quantities m(r) and ρ(r) appearing in eqtn. (4.1) are not independent, but are related
through the equation of mass continuity. For static configurations, the mass in a spherical thin
spherical shell of thickness dr at radius r is
dm(r) = 4πr2 ρ(r) dr;
that is,
dm(r)
= 4πr2 ρ(r).
dr
(4.6)
2
Electron degeneracy pressure is important in white dwarfs, and neutron degeneracy pressure in neutron stars.
Magnetic pressure, B 4 /(4π), and turbulent pressure, ρv 2 /2, can also be significant under some circumstances.
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In a dynamic medium, the mass passing through some unit area per unit time is just the mass
density times the component of the velocity normal to the area. In particular, for a spherically
symmetric flow, such as may apply in a stellar wind with mass-loss rate Ṁ , we have
Ṁ = 4πr2 ρ(r)v(r)
(4.7)
where ρ(r), v(r) are the density and (radial) flow velocity at radius r.
4.3
Energy continuity
Our final equation of continuity is energy continuity; by inspection,
dL
= 4πr2 ρ(r)ǫ(r)
dr
(4.8)
where
• r is radial distance measured from the centre of the star
• P (r) is the total pressure at radius r
• ρ(r) is the density at radius r
• g(r) is the gravitational acceleration at radius r
• m(r) is the mass contained with radius r
• L(r) is the total energy transported through a spherical surface at radius r
• ǫ(r) is the energy generation rate per unit mass at radius r
The stellar radius is R, the stellar mass is M ≡ m(R), and the emergent luminosity L ≡ L(R)
(dominated by radiation at the stellar surface).
4.4
Mean Molecular Weight
The ‘mean molecular weight’, µ, is3 simply the average mass of particles in a gas, expressed in
units of the hydrogen mass, m(H). That is, the mean particle mass is µm(H); since the number
3
Elsewhere we’ve used µ to mean cos θ. Unfortunately, both uses of µ are completely standard; but fortunately,
the context rarely permits any ambiguity about which ‘µ’ is meant. And why ‘molecular’ weight for a potentially
molecule-free gas? I don’t know, though I suspect it may be because this nomenclature originated in other contexts
(such as hydrostatic equilibrium in the Earth’s atmosphere).
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density of particles n is just the mass density ρ divided by the mean mass we have
n=
ρ
µm(H)
and
P = nkT =
ρ
kT
µm(H)
(4.9)
For an un-ionized gas
X
Y
X+ +
(fi /Ai )
4
µ=
i
!−1
where fi , Ai are the mass fraction and atomic weight of element i, which has atomic number
zi . 4
For a fully ionized gas of mass density ρ the number densities are:
Element:
No. of nuclei
No. of electrons
H
He
Metals
Xρ
m(H)
Xρ
m(H)
Yρ
4m(H)
2Y ρ
4m(H)
Zρ
Am(H)
(A/2)Zρ
Am(H)
where A is the average atomic weight of metals (∼16 for solar abundances), and for the final
entry we make the approximation that Ai ≃ 2zi .
The total number density is the sum of the numbers of nuclei and electrons,
n≃
ρ
ρ
(2X + 3Y /4 + Z/2) ≡
m(H)
µm(H)
(4.10)
(where we have set Z/2 + Z/A ≃ Z/2, since A ≫ 2). We see that
(4.11)
µ ≃ (2X + 3Y /4 + Z/2)−1
= 2 (1 + 3X + Y /2)−1
(making use of the fact that X + Y + Z ≡ 1).
We can drop the approximations to obtain a more general (but less commonly used) definition,
µ−1 =
X zi + 1
i
Ai
fi
(where, of course, the summation now includes hydrogen and helium).
By way of illustration, for a fully ionized pure-hydrogen gas (X = 1, Y = Z = 0), µ = 1/2;
for a fully ionized pure-helium gas (Y = 1, X = Z = 0), µ = 4/3;
for a fully ionized gas of solar abundances (X = 0.71, Y = 0.27, Z = 0.02), µ = 0.612.
4
Of course, we could include hydrogen and helium implicitly in this definition, but it’s normal practice to
identify them separately.
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