Thermal Equilibrium in Nebulae1
For an ionized nebula under steady conditions, heating and cooling processes that in
isolation would change the thermal energy content of the gas are in balance, such that
the gas is in thermal equilibrium, with a stable temperature.
I. Heating. When a photon ionizes an atom, part of the photon’s original energy hν is
consumed in overcoming the atom’s ionization potential χi and the remainder is converted
into kinetic energy of the liberated electron (what is referred to as a “photoelectron”).
This kinetic energy represents an addition to the thermal energy of the gas and the
photoionization process itself thus acts as a heating process.
The amount of photoelectric heating per unit volume will depend on the number density
of atoms that are being ionized, the rate at which ionizing photons encounter those atoms
(which will depend on the radiation intensity), the ionizing photon energies, and the crosssection for photoionization. For a pure hydrogen nebula, we can write the rate of heating,
or energy Gained per unit volume per unit time, as
0
∞
Γ(H) = n(H )
ν0
4πIν h(ν − ν0 ) σν (H0 ) dν ,
hν
(1)
where n(H0 ) is the number density of neutral hydrogen, Iν is the specific intensity
averaged over all directions, ν0 is the photon frequency corresponding to the threshold
for ionization (i.e., hν0 = χi = 13.6 eV), and σν (H0 ) is the cross-section for hydrogen
ionization as a function of frequency.
Under steady conditions, the hydrogen at each point in the nebula will be in ionization
equilibrium, which means that we can equate the photoionization and recombination rates:
0
∞
n(H )
ν0
1
4πIν σν (H0 ) dν = np ne αB (T ) ,
hν
(2)
Based on Osterbrock & Ferland (2006), Astrophysics of Gaseous Nebulae and Active Galactic Nuclei, 2nd
Ed., Chapter 4.
1
so that
n(H0 ) = ∞
np ne αB (T )
(3)
4πIν 0
hν σν (H ) dν
ν0
and substitution into eq. (1) yields
∞
Γ(H) = np ne αB (T )
ν0
4πIν h(ν − ν0 ) σν (H0 ) dν
hν
∞ 4πIν σν (H0 ) dν
hν
ν0
.
(4)
Note the following:
• The ratio of integrals in eq. (4) is just the average kinetic energy per photoelectron;
since photoionization is balanced by recombination, the rate at which photoelectrons are
liberated per unit volume is given by np ne αB , and the combined product in eq. (4) thus
gives the heating rate per unit volume.
• Hotter stars produce higher energy photons that will lead to an increase in average
kinetic energy per photoelectron, and hence a larger Γ(H), all other things being equal.
When comparing the ionizing spectrum from two different stars, a common description
is that the hotter star produces a “harder” spectrum, i.e. the average photon energy is
larger, with correspondingly greater photoelectric heating.
• The full heating rate Γ will include the contributions of photoelectric heating from
ionization of helium, Γ(He), and heavier elements, in addition to hydrogen. Physically
these contributions will also be described with expressions like eq. (4), modified to include
the appropriate bookkeeping for ionization.
II. Cooling. Cooling occurs when thermal energy in the gas particles is converted to
radiation which escapes from the nebula. The net result is that thermal energy is extracted
from the gas. Several processes are important for cooling.
A. Energy Loss by Recombination. When a free electron is captured to a bound state
in an atom, a photon is emitted that carries away the free electron’s kinetic energy plus
2
the binding energy corresponding to the state it is captured to. The rate at which energy
is Lost from the plasma in this process per unit volume is thus
ΛR (H) ≈ np ne αB (H0 , T ) kT .
(5)
Note the following:
• Equation (5) is an approximation. The average kinetic energy per particle in the gas
is
3
2 kT ,
but the slower particles are preferentially captured, so that the kinetic energy
converted to radiation is somewhat less than the average value; here we represent it by
∼ kT .
• In this treatment of cooling as well as heating (eqs. 2–4), we assume Case B applies and
neglect recombinations to the ground state.
• Equation (5) can be modified to estimate the contributions to recombination cooling by
helium, ΛR (He), and by heavier elements.
• Since the photoionization heating and recombination cooling processes scale with the
density of the ion, for cosmic abundances elements heavier than helium can generally be
neglected.
B. Energy Loss by Free-Free Radiation. In an ionized gas, an electron will radiate
when it undergoes acceleration through Coulomb interaction with an ion. The emergent
radiation is known as bremsstrahlung or free-free emission (reflecting the fact that the
radiating particle is in an unbound state both before and after the emission occurs). The
resulting rate of cooling by this process by ions of charge Z, integrated over all frequencies,
is given by
ΛF F (Z) = εF F
25 πe6 Z 2
= 3/2
3 hme c3
2π kT
me
1/2
g f f n+ ne ,
(6)
where n+ is the ion density and gf f is the mean Gaunt factor for free-free emission; the
Gaunt factor is a dimensionless quantity that parameterizes quantum mechanical effects,
3
and has a value of order unity. An average value appropriate for nebular conditions is
gf f ≈ 1.3. Plugging in values for the constants in eq. (6) yields
ΛF F = (1.42 × 10
−27
−1
erg s
cm
−3
)Z
2
T
K
1/2
gf f
n n +
e
.
−3
cm
cm−3
(7)
Note the following:
• For a typical case of ionized hydrogen and singly ionized helium, both have Z = 1 so
that n+ = np + nHe II . Heavier elements can again be generally neglected due to their
small abundance.
• Mass of the radiating particle appears in the denominator of eq. (6), which explains why
electrons dominate free-free emission rather than protons or other more massive particles.
C. Energy Loss by Collisionally Excited Line Radiation. Cooling in H ii regions is
often dominated by spontaneous line emission from heavy element ions that have been
collisionally excited. In this case thermal energy of the colliding particle, usually an
electron, is transferred to the internal energy of the excited ion, and subsequently removed
from the nebula by the emitted photon.
For an atom with two energy levels, the cross-section for collisional excitation by an electron
moving with velocity v is
πh̄2 Ω(1, 2)
1
me v 2 > χ12 ;
for
σ12 (v) = 2 2
me v
g1
2
(8)
here Ω(1, 2), known as the collision strength, is a dimensionless quantity that parameterizes
the quantum mechanical effects and is a function of v; g1 is the statistical weight of the
lower level; and χ12 is the excitation potential, i.e. the energy required for excitation from
the lower level to the upper level. From LTE considerations, it is possible to derive a very
similar expression for collisional deexcitation, in which case a transition occurs from level
4
2 to level 1, and the excitation energy χ12 is added to the kinetic energy of the colliding
electron:
πh̄2 Ω(1, 2)
.
σ21 (v) = 2 2
me v
g2
(9)
Note that for a given pair of levels the same collision strength applies for both the excitation
and deexcitation processes.
The total collisional deexcitation rate per unit volume will depend on the density of target
particles (ions in level 2), the density of colliding particles (electrons), the speed at which
the colliding particles are moving around, and the cross-section for deexcitation, with an
integration over velocities since not all particles are moving at the same speed. The total
rate of collisional deexcitations per unit volume per unit time is thus
n2 ne q21 = n2 ne
∞
vσ21 f (v) dv ,
0
(10)
where q21 is the deexcitation coefficient and f (v) is the distribution function for particle
speeds (i.e, the Maxwellian distribution). Carrying out the integration in eq. (10) yields
n2 ne q21 = n2 ne
2π
kT
1/2
h̄2
3/2
me
Υ(1, 2)
,
g2
(11)
where Υ(1, 2) is the velocity-averaged collision strength, so that
q21 = (8.629 × 10
−6
3 −1
cm s
)
T
K
−0.5
Υ(1, 2)
.
g2
(12)
The Boltzmann equation enables us to relate the coefficients for excitation q12 and
deexcitation q21 , with the result that
q12 =
g2
q21 exp(−χ12 /kT )
g1
and
q12 = (8.629 × 10
−6
3 −1
cm s
)
T
K
5
−0.5
Υ(1, 2)
exp(−χ12 /kT ) .
g1
(13)
(14)
Some Υ values for transitions of interest are tabulated by Osterbrock & Ferland in their
Tables 3.6 and 3.7. Note that these tables use a compressed representation for instances
where no j quantum number is indicated. These cases correspond to transitions between
a term consisting of a single level (i.e. single j quantum number) and a term with several
levels distinguished by different j quantum numbers. In this case, the tabulated values
represent the sum of collision strengths for the different possible transitions between the
two terms. The Υ value for a single transition can be obtained from the relation
Υ(SLJ, S L J ) =
(2J + 1)
Υ(SL, S L ) ,
(2S + 1)(2L + 1)
(15)
where the primed quantum numbers correspond to the level of interest in the multi-level
term and the unprimed quantum numbers correspond to the level in the single-level term.
The summed collision strength, Υ(SL, S L ), corresponds to the value listed in the tables.
Note the following:
• The energy level structure for an atomic system is a function of the ionization state, and
not simply the atomic number. Ions that are “isoelectronic” (such as N ii, O iii, Ne v)
have similar level structures shifted in energy by the differing nuclear charge. Ions with
the same number of outer shell electrons (such as O ii and S ii, which fall in the same
column of the periodic table) will also have similar level structures.
• For common ions in nebulae, transitions between the low-lying excited states and
the ground state violate selection rules for electric dipole transitions, but radiative
deexcitations can still occur via magnetic dipole or electric quadrupole transitions.
• The quantum mechanical probabilities for such “forbidden” transitions are small, with
the result that the Einstein A values are small and the deexcitation process tends to be
slow; Einstein A values for transitions of interest are tabulated in Osterbrock & Ferland
Tables 3.12 and 3.13.
6
• Forbidden transitions are indicated in notation by including brackets around the ion; for
example [O iii] λ5007, where 5007 indicates the wavelength in Å. Transitions that violate
only the ∆S selection rule are referred to as intercombination, or semiforbidden transitions,
indicated in notation by a single bracket on the ion: e.g. C iii] λ1909.
Under steady conditions the number of ions in a given energy state will obey an excitation
equilibrium, such that the rate of transitions into a given state will equal the rate of
transitions out of it. For our two-level atom under nebular conditions, the upper state
is populated by collisions and depopulated by collisions as well as spontaneous radiative
transitions. In mathematical terms, this means that
n1 ne q12 = n2 ne q21 + n2 A21 ,
(16)
from which it follows that the density of ions in the upper state is
ne q12
1
.
n2 = n1
e q21
A21
1 + nA
21
(17)
The emitted photons remove energy from the nebula, and the cooling rate is thus
ΛC = n2 A21 hν21
so that
ΛC = n1 ne q12 hν21
1
1+
(18)
ne q21
A21
.
(19)
Under interstellar conditions, particle densities are often very low. In the limit as ne → 0,
eq. (19) reduces to
ΛC = n1 ne q12 hν21 .
(20)
In this low density limit, every collisional excitation is followed by emission of a photon.
In the limit of high density, with ne → ∞, eq. (19) becomes
ΛC →
g2
n1 ne q12 hν21
= n1
exp(−χ12 /kT ) A21 hν21 .
ne q21 /A21
g1
7
(21)
Comparison of eq. (21) with eq. (18) implies that
n2 = n1
g2
exp(−χ12 /kT ) ,
g1
(21)
which is just the Boltzmann equation; in the high density limit a collisional equilibrium
applies, with collisions dominating both excitations and deexcitations.
In a multi-level atom, we can calculate the total line cooling by summing expressions like
eq. (18) for each transition of interest, after first solving for the number densities of the
excited states. The level populations will in general be set by an excitation equilibrium
involving both radiative and collisional transitions, which for level i implies
j=i
nj ne qji +
nj Aji =
j>i
ni ne qij +
ni Aij .
(22)
j<i
j=i
The excited state densities sum to the total density of the ion:
nj = n .
(23)
j
Combining the equations (22) and (23) allows solution of the level populations; in practice
only a limited number of states (often just the bottom 5) are considered in carrying out
the solution. The collisionally excited radiative cooling rate for a given ion is then given
by
ΛC =
i
ni
Aij hνij .
(24)
j<i
In the low-density limit, ne → 0, this becomes a sum of terms like eq. (20). However,
if the density is increased, collisions become important and a more complete solution is
required. It is customary to define the critical density for a given level as the density
where collisional removal from the state becomes equal to the radiative de-excitation rate
(which may involve more than one transition):
j<i
ncrit (i) = Aij
j=i qij
8
.
(25)
See Table 3.15 from Osterbrock & Ferland for ncrit values for some levels of interest.
Thermal Equilibrium
In a stable nebula, heating matches cooling, which means that
Γ = Λ = ΛR + ΛF F + ΛC .
(26)
Here ΛC represents the sum of collisionally excited line emission from the different ions
present in the nebula. It is worth noting how each of the terms in eq. (26) depends on the
gas temperature.
• Γ(H), given by eq. (4), depends on T through the recombination coefficient αB (T );
recall that (approximately) αB ∝ T −0.7 , which means that Γ is a decreasing function of
temperature.
• From eq. (5), ΛR ∝ αB kT ∝ T +0.3 , an increasing function of temperature.
• From eq. (6), ΛF F ∝ T 1/2 , an increasing function of temperature.
• Since collisional excitations increase with increasing temperature, ΛC is an increasing
function of temperature. (See eqs. 20 and 21 for behavior in the low and high density
limits, with q12 given by eq. 14.)
• Since each of the relevant cooling processes is an increasing function of T , their sum Λ
is likewise an increasing function of temperature.
Since Γ(T ) is a monotonically decreasing function and Λ(T ) is a monotonically increasing
function, eq.
(26) is satisfied at a single value of T , which will be the equilibrium
temperature for the nebula. See Figures 27.1–27.3 from Draine for illustrative cases (in
his notation Γpe = photoelectric heating, Λrr = recombination cooling, Λce = cooling by
collisionally excited emission).
Note the following:
9
• Given the important and often dominant role of collisionally excited cooling by ions of
heavy elements, it is not surprising that the equilibrium temperature will be sensitive to
the nebular abundances. As the abundance of heavy elements, or “metallicity”, increases,
the larger density of coolants leads to an increase of Λ(T ) and a decrease in equilbrium
temperature.
• A counter-intuitive result is that increasing the nebular abundance can result in a decrease
in the strength of optical line emission from the nebula, as the cooling becomes increasingly
dominated by infrared fine structure lines that have a weak sensitivity to temperature. Fine
structure lines result from transitions within the ground-state term of an ion that differ in
J quantum number.
• Increasing the nebular density will decrease the efficiency of radiative deexcitations
relative to collisional deexcitation, with the result that Λ(T ) gets smaller and a hotter
equilibrium T will result. The emissivity of a specific line is impacted when the nebular
density n exceeds the transition’s critical density ncrit (see eq. 25).
10