Download –1– 4. Energy transport in stars Stars are hotter at the centre, hence

–1–
4.
Energy transport in stars
Stars are hotter at the centre, hence the energy must flow from the centre to the surface. There
are three modes of energy transfer: conduction, radiation and convection. There is no principle
difference between electron conduction and photon radiation. In both cases, energetic particles
collide with less energetic ones resulting in an exchange of energy, and heat is transfered by the
random motion of particles. Convection is a different phenomenon and is not well understood;
only semi-empirical laws exist to describe it. In this section, we will explain the basic ideas of
energy transport in stars, and hence filling in the second piece of the key physics that supplement
the four stellar equations; the others being the equation state and nuclear energy generation rate.
4.1.
order of magnitude estimate
We first illustrate the order of magnitude of heat transfer using the Sun.
Let us estimate the mean free path of a typical photon inside the Sun. Photons cannot escape
directly from the centre to the surface, because they collide very frequently with electrons. The
mean free-path of photons is given by:
1
l=
,
(1)
nσ
where n is the number density of colliding targets (e.g., electrons), and σ is the collision
cross-section.
A lower limit of the cross-section is provided by the Thomson scattering; other processes of
interaction will only enhance the cross-sections. The Thomson cross-section is given by:
1
e2
4π0 me c2
8π 2 8π
σT =
r ≡
3 e
3
!2
= 6.65 × 10−29 m2
(2)
The mean density of the Sun is about 1.4 × 103 kg m−3 , which implies an electron density
ne ≈ ρ̄/ mp ≈ 8.3 × 1029 m−3 . So a ‘typical’ mean free path is then l ≈ 1.8 cm. The mean free path
is shorter at the centre due to the higher electron density and longer close to the surface due to
the lower electron density. It is clear that the mean free path for photons is very short inside the
Sun, in other words stars are nearly opaque to radiation.
Now the temperature gradient of the Sun is about
dT
Tc
1.56 × 107 K
≈
=
= 2.3 × 10−2 K m−1
dr
R
7 × 108 m
(3)
So the temperature difference experienced by a photon in its mean free path is only
δT ≈ l
dT
≈ 4.1 × 10−4 K
dr
(4)
So at the centre of the Sun the fractional difference of temperature in different directions is only
δT
≈ 2.6 × 10−11
T
(5)
Clearly the radiation field is very close to being isotropic and the frequent collisions between
the particles maintain the local thermal equilibrium effectively in stellar interiors. Normally we
–2–
can ignore such small differences in the temperature, however, this small difference is crucial for
radiative transfer. This is because the flux scales as σT 4 , and hence the flux at the centre is about
a factor of (1.6 × 107 /5770)4 ≈ 6 × 1013 higher, so even if a tiny fraction of this flux escapes from
the Sun, it is sufficient to power the flux of the Sun observed at the surface. So we must consider
the non-vanishing net flux surplus of outgoing radiation over the inward-going radiation.
4.2.
Heat transfer due to random motions
A rigorous treatment of radiative transfer in stars is somewhat involved, in these lectures we
will be satisfied with a simplified intuitive picture.
Let us consider the energy flow across across a surface element dA at radius r. We assume all
particles have mean free path of l and move radially with velocity v. The particles pass through
the surface both from below r to the top and from above to below the surface. The particles that
pass through from below the surface will be on average from a radius of r − l, and from the top
from a radius of r + l. The net energy exchanged is roughly
dE ≈ u(r − l) l dA − u(r + l)l dA = l dA [u(r − l) − u(r + l)] ,
(6)
where u(r − l) and u(r + l) are the energy densities at radii r − l and r + l. This energy is
exchanged over a period of dt = l/v, i.e., the time taken for particles to move a distance of l.
Therefore the energy flux through the surface is
f≡
dE
du
= v [u(r − l) − u(r + l)] ≈ −2vl ,
dtdA
dr
(7)
where we have used u(r − l) − u(r + l) ≈ −2l du/dr. Notice we have not specified what particles are
moving across the surface, so the same formalism applies to the energy transfer by both photons
and electrons.
For realistic situations, we have to make a number of modifications to eq. (7):
• In the interior of stars, the particles move in all directions, but on average there are about
1/6 particles move in each direction, so taking into account this fact, we have
1 du
f = − vl
3 dr
(8)
This is in fact in agreement with a rigorous treatment of averaging over all directions of
particles.
• For radiative transfer by photons, the speed of photons is obviously c. However, for
conduction, the electrons have a distribution of velocities, and so we need to use an average
velocity (v̄) and mean free path (¯l), eq. (8) then becomes
1 du
f = − v̄¯l
3 dr
(9)
We can rewrite the previous equation using the heat capacity:
1 du dT
1
dT
dT
f = − v̄¯l
= − v̄¯lCv
≡λ
3 dT dr
3
dr
dr
(10)
–3–
where Cv is the heat capacity per unit volume. The last expression at the right-hand-side is
called Fick’s law of diffusion, with the coefficient of conductivity given by
1
λ = v̄¯lCv .
3
In astrophysics, it is customary to relate the mean free path to an opacity defined by
l=
1
1
nσ
≡
−→ κ ≡
nσ
κρ
ρ
(11)
(12)
The opacity or interaction cross-sections can be provided by a number of processes, we will
discuss these in more detail. The simplest opacity is that due to electron scattering.
l=
1
1
ne σ T
≡
−→ κe =
ne σ T
κρ
ρ
(13)
For a fully ionized gas, we have ρ = µe ne mp = 2ne mp /(1 + X). Hence
κe =
1 + X σT
= 0.02(1 + X) m2 kg−1
2 mp
(14)
• For photon radiation, in general, the opacity depends on the photon energy/frequency.
Therefore, the opacity we just used should be viewed as a kind of average over frequency.
This average opacity over energy/frequency is called the Rosseland mean opacity. We will
give the proper procedure of averaging below.
4.3.
Radiative transfer
For energy transfer by radiation, obviously, we have
v = c, l =
du
1
, u = aT 4 , Cv =
= 4aT 3 .
κρ
dT
(15)
and hence
dT
1
1 4caT 3 dT
c dPr
4aT 3
f =− c
=−
=−
3 κρ
dr
3κρ dr
κρ dr
The total energy flow rate across of a spherical of radius r is then
Lr = 4πr2 f = −
16πacr2 T 3 dT
3κρ
dr
(16)
(17)
In this section, we shall write the mass and luminosity at radius r as Mr and Lr , respectively. And
we denote M , R and L (without subscripts) as the total mass, stellar radius and total luminosity.
The flux at radius r is denoted as f , and the flux at the stellar surface as F .
The previous equation can be re-arranged to give an equation describing the temperature
gradient of the star
dT
3κρ
=−
Lr
(18)
dr
16πacr2 T 3
We have already mentioned that in general, the opacity (collision cross-section) depends on
the photon frequency, so let us consider the differential energy flux induced by photons between
frequency ν to ν + dν,
1 d[uν dν]
1 1 ∂uν
c 1 ∂uν dT
fν dν = − v̄¯l
=− c
dν = −
dν
3
dr
3 κν ρ ∂r
3ρ κν ∂T dr
(19)
–4–
Integrating over the photon frequency, we obtain the total flux:
Z
f=
∞
fν dν = −
0
c dT
3ρ dr
Z
∞
0
1 ∂uν
c du dT
dν ≡ −
κν ∂T
3κρ dT dr
(20)
where we have defined the Rosseland mean opacity
1
=
κ
∞
Z
0
1 ∂uν . du
dν
κν ∂T
dT
(21)
Once κν is known from atomic physics, then the Rosseland mean opacity can be calculated. Eq.
(18) then remains valid with the understanding that κ is a “mean” opacity properly averaged over
all photon frequencies.
4.3.1.
Simple stellar atmosphere model
We will consider a simple model of stellar envelopes and atmosphere in this subsection. We
assume that the electron conduction is negligible, and hence we only need to consider the heat
transfer by photons. The radiative transfer equation we derived assumes that the radiation field is
almost isotropic. Clearly, at the stellar surface this assumption breaks down since at the stellar
surface one has radiation from “below” but no radiation from “above”. It is somewhat surprising
that that the radiative transfer equation we have derived (eq. 18) applies quite well all the way
up to the stellar surface.
Near the stellar surface, the luminosity and radius can be taken as the total luminosity L and
the stellar radius R. The radiative transfer equation becomes
dT
3κρLr
3κρ
L
3κρ
L
=−
=−
=−
F, F =
dr
16πacT 3 r2
4acT 3 4πR2
4acT 3
4πR2
(22)
We can rewrite the previous equation as
3F
dT 4
=
,
dτ
ac
(23)
where we have defined a dimensionless optical depth
dτ ≡ −κρdr = −
dr
, τ = 0 at r = R
l
(24)
The solution is then
3F
3F
τ −→ ur = ur,0 +
τ
(25)
ac
c
Now as we have discussed stars radiate approximately like a blackbody, and we have introduced
the concept of effective temperature by
T 4 = T04 +
4
L = 4πR2 σ Teff
.
(26)
This blackbody radiates energy per unit area as the real stars does. Now, for a perfect blackbody
with Teff , the energy density just above the surface will be
1
4
ur = a Teff
,
2
(27)
–5–
which is one half of the isotropic blackbody value. The factor of 1/2 arises because the radiation
field is only coming from the hemisphere facing the star, but nothing from the “upper” hemisphere.
We adopt an approximation that at the stellar surface, i.e., at τ = 0, the radiation energy
4 /2, in analogy with the blackbody case. Combining with eq. (25), we find that
density is just a Teff
aT04 =
4
1
a Teff
−→ T0 = 1/4 Teff ,
2
2
(28)
at the stellar surface. The temperature distribution close to the stellar surface is
T4 =
1 4
3F
4
Teff +
τ = Teff
2
ac
1 3
+ τ
2 4
(29)
4 = ac T 4 /4. Therefore at
where we have used the fact that at the stellar surface F = σ Teff
eff
τ = 2/3, we have T = Teff . The optical depth τ = 2/3 corresponds to the so-called photosphere,
i.e., the radius at which the temperature is equal to the effective temperature.
The diffusion approximation to the radiative transfer near the stellar surface, and the
approximation that T0 = Teff /21/4 is known as the Eddington approximation. And this provides
one boundary condition at the stellar surface.
Let us further find the pressure at the photosphere. The radius of the photosphere is very
close to the stellar radius, R. So from dτ = −κρdr, we can define a mean opacity
∞
Z
κρdr ≡ κ̄
τ=
Z
∞
ρdr.
(30)
R
R
From the hydrostatic equilibrium, we have dP/dr = −GMr ρ/r2 , hence we can integrate the
equation from R to ∞, and obtain
Z
∞
P (r = R) =
R
GMr ρ
GM
dr ≈ 2
2
r
R
Z
∞
ρdr,
(31)
R
where in the last step we have the fact that close to the stellar surface, Mr = M, r = R. Combining
with the previous equation and using τ = 2/3 for the photosphere, we find that
P (r = R) =
GM 1 2
.
R2 κ̄ 3
(32)
Equation (32) and T0 = Teff /21/4 provide two approximate but improved boundary conditions for
the four stellar equations, as derived in §2.
4.3.2.
Eddington Limit
From the general formalism, for photon diffusion, we have1
1 1 dur
1 dPr
dPr
κρ
κρ Lr
f =− c
= −c
, −→
=− f =−
,
3 κρ dr
κρ dr
dr
c
c 4πr2
1
The derivations given in the lectures are different and somewhat more intuitive.
(33)
–6–
where we have used ur = 3Pr in the first step (Pr is the radiation pressure). Now from hydrostatic
equilibrium, we have
dP
GMr
=− 2 ρ
(34)
dr
r
Dividing the two previous equations, we find that
κLr
dPr
=
dP
4πcGMr
(35)
At the stellar surface, Mr ≈ M and Lr ≈ L, so
dPr
κL
=
dP
4πcGM
(36)
When the stellar luminosity is very high then the density in the stellar atmosphere is very low,
in this case, the opacity is dominated by electron scattering, κ = κe , which only depends on the
hydrogen mass fraction (cf. eq. 14). The right-hand-side of the previous equation is therefore a
constant. The differential equation can be readily solved to yield
Pr − Pr,0 = (P − P0 )
κe L
.
4πcGM
(37)
where P0 , Pr,0 are the total and radiation pressures at the surface, respectively. Even slightly
below the surface, the pressure becomes much higher than that at the stellar surface. We can
ignore P0 , Pr,0 to obtain
κe L
Pr ≈ P
.
(38)
4πcGM
The radiation pressure must be smaller than the total pressure,
Pr
κe L
4πcGM
=
<1→L<
≡ Ledd
P
4πcGM
κe
(39)
The Eddington luminosity is
Ledd =
4πcGM
2.5 × 1031 J s−1 M
6.5 × 104 L M
=
=
κe
1+X
M
1+X
M
(40)
This is a very powerful statement: any spherically symmetric star in hydrostatic equilibrium must
radiate below the Eddington limit, irrespective of its energy source. Physically one can show that
a star radiating at the Eddington luminosity, its gravity is precisely balanced by the pressure
exerted by the outgoing radiation. A higher luminosity will inevitably cause mass loss from stars,
thus violating the assumption of hydrostatic equilibrium.
4.4.
Electron conduction
The same formalism applies to energy transfer by electron conduction. Eq. (10) applied to
the electrons gives
1
λe = v̄e ¯le Cv,e
(41)
3
To evaluate this, we need to estimate the average electron velocity, mean free path, and heat
capacity in turn. To see the efficiency of electron conduction relative to the radiation, we compare
these quantities with those for photons.
–7–
The mean thermal energy is 3kT /2 for each electron, so the average velocity of electrons
q
ve2
ve2 3kT 1/2
=
c
me c2
3kT
v̄e
=
−→
≈
me
c
(42)
The thermal energy per unit volume for the electrons is 3ne kT /2, so the heat capacity per unit
volume is
du
3
3Pe
Cv,e =
= ne k =
.
(43)
dT
2
2T
For photons, Cv = 4aT 3 = 12Pr /T , and hence
Pe
Cv,e
=
Cv
8Pr
(44)
To find the mean free path of electrons, let us estimate the collision cross-section between electrons
and ions of charge Ze. Suppose an electron moves along the x direction with velocity v when
unperturbed. So the momentum of the particle is p = px = mv. Now we put in an ion with a
minimum impact parameter b. Clearly most deflections occur at when |x| <
∼ b, and hence the
duration of the interaction is ∼ 2b/v. Let us consider the y-component of the momentum gained
by the electron by assuming that the electron trajectory is unchanged
py = Fy δt ≈
1 Ze2 2b
1 2Ze2
×
=
2
4π0 b
v
4π0 bv
(45)
The deflection angle of the trajectory is θ ∼ py /px . If py px , then the deflection angle θ 1, so
the electron trajectory is virtually unchanged, i.e., there has been hardly any interaction between
the electron and the ion. However, when py ≈ px , the deflection angle θ ∼ 1, the trajectory is
significantly perturbed, i.e., the electron and ion have interacted strongly. This happens when
b∼
1 Ze2
2 Ze2
∼
4π0 mv 2
4π0 kT
(46)
At this impact parameter, it is clear that the thermal energy kT is comparable to the Columb
energy, 1/(4π0 ) × Ze2 /(kT ). It is therefore reasonable to identify the collision cross-section as
2
σe ≈ πb ≈ π
1 2Ze2
4π0 3kT
!2
(47)
Now if we assume the photon interaction cross-section is the Thomson cross-section (eq. 2) due to
electron scattering then
¯le
1
1
σT
6
kT 2
=
/
≈
= 2
(48)
l
ni σ e ne σ T
σe
Z
me c2
where ni is the ion number density and is approximately equal to the electron number density ne .
Combining eqs. (42, 44, 48), we finally obtain the ratio of coefficient of conductivity by electrons
and that by photons:
√
λe
3 3 1 Pe
kT 5/2
≈
(49)
λ
4 Z 2 Pr me c2
For the centre of the Sun, one can easily show that
Pe ≈ 103 Pr , kTc ≈ 10−3 me c2
(50)
–8–
and hence
λe
≈ 4 × 10−5
(51)
λ
At the centre of the Sun, the heat transfer due to photons is much more efficient than that by
electrons. This happens because although the electron heat capacity is larger than that of photons,
the electrons have much smaller mean free path and thermal velocity, as a result, the electrons are
less efficient energy carriers.
The situation is very different for degenerate stars such as white dwarfs. In such cases, the
conduction by electrons is often much more efficient than the radiative transfer for the following
two reasons. First, the mean free path of electrons is much larger, because the quantum cells of
phase space are filled up such that collisions in which electron momentums are changed become
more difficult. The collision cross-section is reduced, and hence the mean free path of electrons is
much longer. Second, the energy of electrons is raised, because the energy levels are filled all the
way up to the Fermi energy EF kT . Hence, the electrons move with velocities much larger than
the thermal velocity. Due to the increased mean free path and velocity, the energy transfer by
electron conduction becomes much more efficient in degenerate stars.
In practice, the heat transport is dominated by radiation when a star is not degenerate and
dominated by electron conduction for a degenerate star. The transition between these two energy
transfer mechanisms is very rapid.
We mention in passing that the heat transfer by ions is much smaller than that by electrons,
since they move with a mean velocity that is a factor of ( mp /me )1/2 slower (see eq. 42). So we
can ignore the effect of ions on heat transfer.
4.5.
Opacity
The calculations of opacities is one of the most difficult problems in stellar astrophysics.
Extensive opacities can be found in tabular forms 2 . Here we will be satisfied a rough sketch of
various opacities occurring in stars. The opacity usually depends on the mass density, temperature
and chemical composition. One often parameterises the opacity by power-laws:
κ = κ0 ρn T −s .
(52)
Throughout this subsection, X, Y, Z are the usual definitions used in metallicities, and the
opacities are in units of m2 kg−1 .
4.5.1.
Electron scattering
The simplest form of opacity is due to electron Thomson scattering (see eq. 2). The associated
opacity is given by
1 + X σT
κe =
= 0.02(1 + X) m2 kg−1
(53)
2 mp
2
e.g., see http://heasarc.gsfc.nasa.gov/topbase/home.html
–9–
Note that the electron opacity depends only on the hydrogen mass fraction X for a fully ionized
gas. The above formula, however, only applies when the gas is fully ionized and when the electrons
are non-relativistic and non-degenerate. In the relativistic and degenerate cases the opacity is
reduced. A more general formula taken into these corrections is given by
ρ3
κe = 0.02(1 + X) 1 + 2.7 × 10
T2
11
−1 "
1+
T
4.5 × 108
0.86 #−1
m2 kg−1
(54)
where we have defined ρ3 = ρ/(103 kg m−3 ).
4.5.2.
“Kramers” opacity
Photons can be absorbed by an electron aided by a nearby ion. Depending on whether the
electron is initially free and the resulting electron is free, the process is called free-free, bound-free,
and bound-bound absorption. In the free-free absorption, the electron is free before and after the
absorption. In the bound-free absorption, the electron is initially bound to an ion, but the photon
energy is sufficient to free the electron from the ion binding. In the bound-bound absorption, the
electron makes a transition from a lower bound state to a higher bound state.
The opacity due to the bound-bound, bound-free and free-free transitions can be approximated
by the so-called “Kramers” formula:
κK ≈ 4 × 1024 (1 + X)(Z + 0.001)ρ3 T −3.5 , ρ3 = ρ/(103 kg m−3 ).
(55)
This source of opacity is important when hydrogen and helium and other heavier elements are
4
partially ionized, for T >
∼ 2 × 10 K.
4.5.3.
Opacity due to H − ion
The H − opacity is very important for cool stars such as the Sun. H − is formed by attaching
one extra electron to the hydrogen atom. The ionisation potential is 0.75eV.
The presence of H − atoms depends both on the temperature and the availability of electrons.
If the temperature is too high, then the H − ions are completely ionized. If the temperature is too
low, then there will be no free electrons available; the free electrons are provided by both ionized
hydrogen and by heavier metals that have lower ionisation potentials. In practice, the opacity due
to (H − ) is important for 4 × 103 ≤ T ≤ 8 × 103 K. An approximate formula for the H − opacity is
given by
7.7
κH − = 1.1 × 10−26 Z 0.5 ρ0.5
(56)
3 T
Notice that the H − opacity is very temperature dependent.
4.5.4.
Other sources of opacity
At temperatures between 1.5 × 103 ≤ T ≤ 3 × 103 K, the water molecules and CO molecules
dominate the opacity, approximated by
κm = 0.01Z
(57)
– 10 –
At very high density and low temperature, the electrons become degenerate. As we have discussed
the electron conduction becomes very efficient. This electron conductivity can also be expressed
in a form of “opacity”:
"
−8
κc = 2.6 × 10
T2
hZ i 2 1 +
ρ3
∗
ρ3
2 × 106
2/3 #
(58)
where hZ ∗ i is the average electric charge per ion.
4.5.5.
Total opacity
The total opacity is a proper sum of all the opacities that we have discussed. In general, the
opacity depends on the density, temperature and chemical composition. For rigorous calculations
of stellar models, one needs to use tabulated forms of opacity, as we mentioned at the beginning
of this section.
4.5.6.
Pulsation instability
Throughout this course, we have assumed stars are in hydrostatic equilibrium, in other words,
the stars neither expand nor contract. However, in reality, many stars are observed to be variable.
In fact, these stars play fundamental roles in astrophysics. In this subsection, we shall discuss the
so-called κ-mechanism in pulsating variable stars. There are many kinds of variable stars, and
here we limit ourselves to the so-called Cepheid variables.
Cepheid variables (one example is shown in Fig. 1) change their brightness periodically. Their
period is generally in the range of 1-50 days. They are very luminous with typical luminosities
between 300 − 4 × 104 L , so they can be seen to large distances.
Fig. 1.— The light curve for a very bright Cepheid variable, Delta Cephei. The period is about 5
days and the amplitude of variation is about a factor of 2.
The pulsation is driven by zones in the envelope in which He+ is ionizing to He++ . For
a partially ionized Helium envelope, its opacity, like the H − opacity, increases as temperature
– 11 –
increases. (This is contrary to the trend seen in most normal materials, e.g. Kramers opacity.)
When the material is compressed, its temperature increases, its opacity increases, i.e. it becomes
more opaque and traps radiation flowing from below. This trapping causes more heating and
raises the internal energy of the gas. This creates a reservoir of thermal energy. Eventually the
gas reaches maximum compression and rebounds. As the high-pressure gas expands, it drives the
material to a higher velocity than the case when no extra photons are trapped. As the material
expands, it cools, and it becomes more transparent to photons, as a result, the photons escape
more freely. In fact, the photons cool too freely: at the maximum expansion, the temperature and
density of the gas are so low that the pressure is insufficient to balance the gravity, and therefore
the envelope falls back. This initiates a compression phase, and the cycle continues as long as the
star is in the so-called “instability” strip.
Cepheid variables are very important in astrophysics/cosmology, because it turns out that the
period has almost a one-to-one correspondence with the luminosity: the most luminous Cepheids
have longest periods. Hence from the observed period, we can infer its intrinsic luminosity.
Combined with the observed flux, we can infer the distance to a Cepheid variable3 . Because
Cepheids are very luminous, they can be seen in external galaxies. In fact, they are the the first
steps to determine the extragalactic distance scales in cosmology.
4.6.
Convection
So far we have only discussed how energy generated in the stellar interior can be transported
by radiative diffusion and electron conduction; both are due to random collisions between particles.
There is yet another important mode of energy transport, this is called convection. Convection
occurs when the temperature gradient is very steep.
Suppose there is a hot parcel of gas rising in the atmosphere, it will continue to rise if its
ambient gas is cooler and denser, due to the buoyancy. Similarly, a falling parcel of gas will
continue to fall if its ambient gas is hotter and less dense. In other words, this system is unstable.
If we perturb a parcel of gas radially, instead of returning to its original position, the parcel will
move further and further away from the original position. Energy is transferred in this process,
because a rising hot bubble will eventually merge with its cooler environment, and, similarly, a cold
descending bubble will descend and merge with its hot environment in the lower atmosphere. In
convection, one has simultaneously rising hot bubbles and dropping cold bubbles. The Sun has a
convective surface layer, with a width of about 0.3R . This is manifested as the so-called granules
(see Fig. 2). In convection, turbulence usually results. Because of this and other complications,
there is no complete (astro-)physical theory for convection.
4.6.1.
Convective instability
It is, however, relatively simple to derive the critical temperature gradient above which
convection occurs. Let us imagine we move a unit mass of gas parcel of gas with pressure P , and
3
There are some small but important complications, e.g. metallicity effects.
– 12 –
Fig. 2.— A high-resolution photograph of the solar surface taken by the Skylab space station. The
visible surface is sprinkled by regions of bright and dark gas known as granules. This granulation
of the solar surface is a direct reflection of solar convection. Each bright granule measures about
1000 km across-comparable in size to a continent on Earth – and has a lifetime of between 5 and
10 minutes. Together, several million granules constitute the top layer of the convection zone,
immediately below the photosphere.
density ρ adiabatically from r to r + dr, with 0 < dr r. The equilibrium pressure and density at
r + dr is P + dP and ρ + dρ.
Let us suppose when the bubble arrives at r + dr, its pressure is P + δP , density ρ + δρ,
and temperature is T + δT . If δρ > dρ, the bubble is denser then the environment, the parcel
will return to its original position. However, if δρ < dρ then the bubble is less dense than
the environment, then it will experience an upper-ward buoyancy. It will continue to rise, and
therefore the gas is convectively unstable.
In general, there will be some energy exchange between the bubble and the environment.
Deep inside stars, the gas can be assumed to rise adiabatically. The gas then satisfies the adiabatic
condition:
P V γ = P ρ−γ = constant,
(59)
where γ is the adiabatic index and we have used the fact that for a unit mass, V = 1/ρ. Taking
derivative of the above equation, we have
δP
δρ
1ρ
=γ
−→ δρ =
δP
P
ρ
γP
(60)
Now the bubble will, in general, adjust to the pressure of the ambient medium by expansion.
This expansion occurs with the velocity of sound, which is usually much larger than any other
motions of the bubble. Therefore we can assume the bubble is always in pressure equilibrium, i.e.,
δP = dP . Which gives
1ρ
δρ =
dP
(61)
γP
Convective instability sets in if
δρ =
1ρ
dP
dρ
dP < dρ −→
<γ
γP
P
ρ
(62)
– 13 –
If the gas is ideal, then we have
P =
ρ
kT ∝ ρT .
µ mp
(63)
Taking derivative of ln P , we have
dP
dT
dρ
=
+
P
T
ρ
(64)
dT
γ − 1 dP
<
T
γ P
(65)
Combining eqs. (62) and (64), we have
Dividing by dr and taking the absolute value of the above equation (note that dT /dr < 0 so we
have to reverse the inequality), we find that convections occur when
dT γ−1
dr > γ
T dP P dr (66)
This instability criterion is called the Schwarzschild criterion, and is due to K. Schwarzschild, the
astronomer who also discovered the Schwarzschild black hole solution of Einstein’s field equation.
Notice that
• convection occurs when the temperature gradient is steeper than the Schwarzschild critical
value on the right-hand-side.
• the Schwarzschild criterion is a local one, as it only uses the local temperature and pressure
gradients.
4.6.2.
When do convections occur?
To see more clearly when do convections occur, let us recast the Schwarzschild criterion in
terms of the logarithmic slope of the temperature vs. pressure. From eq. (65), we obtain (note
dP < 0 and so we have to reverse the inequality)
∇T ≡
d ln T
γ−1
>
≡ ∇ad
d ln P
γ
(67)
We show below that (γ − 1)/γ is the the logarithmic slope corresponding to an adiabatic process.
The Schwarzschild criterion then states that convections set in when the actual temperature slope
is higher than the adiabatic slope.
To derive the adiabatic slope, we use the familiar relation
P V γ ∝ P ρ−γ = const.
(68)
for an adiabatic process. The equation of state for an ideal gas is P = ρkT /(µ mp ), which gives
ρ ∝ P/T . This, combined with eq. (68), then gives
P 1−γ T −γ = const. −→ ∇ad ≡
∂ ln T
∂ ln P
=
S
γ−1
γ
(69)
– 14 –
Suppose we know ∇T by some means, then what is the luminosity that can be transported by
radiation? We already know part of the answer from eq. (17), where the energy transported Lr
should be more properly written as Lrad ,
Lrad = −
16πacr2 T 3 dT
,
3κρ
dr
(70)
where the subscript “rad” indicates that the energy is transported by radiation. Now we want to
recast the temperature gradient dT /dr with ∇T . This can be done as follows:
dT
dT dP
=
=
dr
dP dr
T d ln T
P d ln P
dP
=
dr
T
∇T
P
dP
dr
(71)
Now from hydrostatic equilibrium, dP/dr = −GMr ρ/r2 , therefore we have
Lrad
16πacr2 T 3
=−
3κρ
T
∇T
P
GMr ρ
− 2
r
=
16πacGT 4 Mr
∇T
3
Pκ
(72)
Now if the convection occurs, part of the energy is transported by convection on top of that by
radiation. This means that the total luminosity Lr = Lrad + Lconv will be larger than Lrad ,
Lr > Lrad =
16πacGT 4 Mr
∇T
3
Pκ
(73)
In other words, the convection occurs when
3
P κ Lr
> ∇T
16πacG T 4 Mr
(74)
Astronomers commonly define the left-hand-side of the previous inequality as the so-called
“radiative” temperature gradient
∇rad =
3
P κ Lr
16πacG T 4 Mr
(75)
This is the local logarithmic slope of temperature vs. pressure which would be required if all
the given luminosity were to be carried by radiation. The previous equation and eq. (67) can be
combined to give the final condition when the convection occurs when
∇rad > ∇T > ∇ad
4.6.3.
(76)
Convection in upper main sequence stars (M >
∼ 1.5M )
In this case, we have convective cores and radiative envelopes.
In massive stars, energy is generated by a reaction cycle involving C, N and O (CNO) cycle.
The energy generated per second per kg, , is
∝ ρT 16
(77)
It has a very high sensitivity on the temperature. As a result, the energy generation is concentrated
in a small central region where the temperature is the highest. This means a very high value
of Lr /Mr , and hence ∇rad is very high. Therefore the core is convectively unstable. Physically,
with a large in the central region, the flux required for thermal balance is so large that the
energy transfer by photon diffusion is insufficient, and some of the energy must be carried away
by convection. Deep inside stars, convections are so efficient that the true temperature gradient is
very close to (but slightly above) the adiabatic value.
– 15 –
4.6.4.
Convection in lower main sequence stars (M <
∼ 1.5M )
In this case, we have radiative cores and convective radiative envelopes.
In the lower main sequence, nuclear energy is generated via the so-called pp-chain, which has
less temperature sensitivity,
∝ ρT 4
(78)
The energy production in the core is spread over a much larger area, so that the flux and ∇rad are
much smaller in the central region. Hence the core regions are convectively stable and radiative.
However, the outer envelopes of stars with low Teff are generally convective. There are two
reasons for this,
• In cool atmospheres, H − ions form, and this increases the opacity κ by orders of magnitude,
as a result, ∇rad becomes very large, and the Schwarzschild criterion is easier to satisfy.
• The adiabatic temperature gradient ∇ad is smaller in the cool atmospheres where hydrogen
and helium are partially ionized. In such a region, the heat capacity is quite large. The
reason is that the input energy is partially used to ionize the zone instead of raising its
temperature appreciably. From thermal dynamics, the adiabatic index is related to the heat
capacity by
T ∂P 2 ∂P −1
γ =1−
(79)
Cv ∂T V ∂V T
and hence a large heat capacity Cv means an adiabatic index (γ) that is close to 1,
∇ad = (γ − 1)/γ is therefore small. The condition ∇T > ∇ad is much easier to satisfy.
The Sun has a convective envelope that reaches down to about 30% of the solar radius.
For more massive stars, the atmospheres are too hot for the hydrogen to be neutral, and this
suppresses the large H − opacity and also increases the value of the adiabatic index γ, both effects
stabilize the envelope against convective instability.
Near the surface of a star convection is not very efficient in carrying heat, and there is no
good theory to calculate its efficiency. For most practical purposes astronomers use the so called
“mixing-length theory”, which parameterises our ignorance about convection with some free
parameter; this is essentially a semi-empirical prescription to derive ∇T by interpolating between
∇rad and ∇ad . We do not discuss this theory any further.
Summary
We have discussed the three energy transport mechanisms in stars. When the temperature
gradient is smaller than the Schwarzschild critical value, energy is transported by the photon
diffusion in non-degenerate stars while the electron conduction dominates in degenerate stars.
Convection occurs in the centres of upper-main sequence stars where the nuclear energy is
generated in a small region and a steep temperature gradient exists in this central region.
Convection is also important in the stellar atmospheres of lower main sequence stars, particularly
in the ionisation zone, as a result of the large opacity and small values of ∇ad . There is currently
– 16 –
only semi-empirical “mixing-length theory” to describe convection, a subject beyond the scope of
the course.