Download THE INTERSTELLAR MEDIUM (ISM) Summary Notes: Part 2

THE INTERSTELLAR MEDIUM (ISM)
Summary Notes: Part 2
Dr. Mark WESTMOQUETTE
The main reference source for this section of the course chapter 5 in the Dyson and Williams (The Physics
of the Interstellar Medium) book. Beware: this book is very technical!!
Note: Figure numbers refer either to the chapter and figure number (N-n) in Freedman and Kauffman,
or to the Dyson and Williams textbook.
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PHOTOIONIZED NEBULAE: H ii REGIONS
H ii regions (like the Orion nebula) are regions of ISM gas in which an OB star (or OB
stars) are embedded. For simplicity we shall start by assuming an idealized nebula which
is spherical and at its centre is a single O or B star. The radiation from the central OB
star PHOTOIONISES the nebula gas, and it is this process and its inverse (RECOMBINATION) which provides the physical basis for much of the nebula’s physical properties.
2.1
The photoionisation process
Photons from the central OB star, which has a high effective temperature (&20 000 K at
B0 and up to ∼50 000 K for an O5 star) can have a sufficiently high energy such that when
they are absorbed by atoms in the gas they cause the removal of a bound electron from
the atom. This is termed IONISATION.
We will consider Hydrogen since it is the most abundant element in the gas: our basic
picture of the atomic structure of H is that it consists of a proton and a single electron,
which can occupy any bound energy state, denoted by a ‘quantum number’ n = 1,2,3,4, etc.
The n = 1 state is known as the GROUND STATE and has zero energy (E = 0; see Fig
5-25). As we go to very large n-values the energy levels become very close together and
there comes a point where the electron is no longer in a bound state but can occupy a
range of possible CONTINUUM states where it has a certain free kinetic energy.
The removal of a bound electron from an atom to form an ion+electron is called IONIZATION. If the energy supplied to ionise the atom comes from an incoming photon, the
process is called PHOTOIONISATION.
Using Hydrogen as an example (Fig 5-25) we see that the energy required to take the
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Table 1: Hydrogen recombination lines (wavelengths in nm)
α
β
γ
δ
Lyman
Balmer
Paschen
n0 → n = 1 n0 → n = 2 n0 → n = 3
122
656.3
1875
103
486.1
1282
97
434.0
1094
95
410.2
94
electron from the ground state (n = 1, E = 0) to the continuum is 13.6 eV (electron
volts). This value is known as the ionisation energy for Hydrogen, IH . Thus to ionise H
from the ground state we need photons with energy >13.6 eV, or in wavelength terms
photons with wavelengths LESS than 912 Å∗ . Only hot (OB) stars have radiation fields
with significant numbers of photons at these short (UV) wavelengths. Thus only OB stars
can photoionise the hydrogen gas in the surrounding nebulae.
NB. the low density of the gas in H ii regions means that the EXCITATION of the H-atoms
is very low, so that nearly ALL the H-atoms will have their electrons in the ground state
(n = 1), so photoionisation need ONLY be considered from the n = 1 state.
The strong (Coulomb) attraction between the negatively charged electron (e− ) and the
positively charged proton (p+ ) produces the inverse process of RECOMBINATION, so we
have a REVERSIBLE reaction:
H + hν p+ + e−
(1)
In recombination the energy of the photon which is produced depends on two things: (a)
the Kinetic Energy (KE) of the recombining electron (in its continuum state) and (b) the
binding energy of the bound-level into which the recombination occurs.
If the recombination occur directly to some upper level n0 , the electron will try and go to
the ground state by jumping down to lower n-levels, and each n0 → n00 jump produces a
photon at a specific energy or wavelength which corresponds to some particular spectral
line in Hydrogen. For transitions from n0 → n = 2 give rise to the Balmer series of lines
seen at optical wavelengths (Hα is the 3→2 transition and occurs at 6563 Å, Hβ is the 4→2
transition and occurs at 4861 Å). Transitions from n0 → n = 3 give the Paschen series in
the near IR, whilst those from n0 → n = 1 give the Lyman series in the far-UV (see Table
2.1). This is the main process by which the Hydrogen emission lines are produced in H ii
region spectra – and this produces the RECOMBINATION EMISSION LINE spectrum
of the nebula.
Equilibrium occurs when there is a balance between the forward and backward rates of
the above equation – this is called the CONDITION OF IONIZATION BALANCE.
∗
Photon energy is described using the formula: E = h ν, where h is the Planck constant and ν is the
frequency of the light (photon). This can be related to the wavelength using c = ν λ, where λ is the
wavelength, thus E = h c / λ. In this way we can relate energy in eV to wavelength.
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2.1.1
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Thermalisation of gas in a pure Hydrogen nebula
As eqn. 1 shows, the nebula gas contains three types of particles: neutral atoms (H),
protons (p+ ) and electrons (e− ).
Photoionisation feeds energy continuously into the nebula via the kinetic energy (KE) of
the ejected (photo)electrons. The energy of the electrons is determined by the energy
distribution of the ionising photons from the ionising OB star. This is determined by its
effective temperature, Teff .
Collisions between the particles take place which THERMALIZES the particle velocity distributions and transfers energy from one type of particle species to another, and this occurs
on such short timescales as to be regarded as instantaneous compared to the timescale for
successive photionisation/recombination.
After thermalization, all the particles in the photoionised nebula have a ‘Maxwellian Distribution’ of velocity (and hence KE), and thus we can assign a single KINETIC temperature,
TK , of the gas particles which is the same for all three types of particle.
The speeds at which the particles interact and share energy follows the order: (i) hot,
photoelectrons share their energy with other electrons, (ii) electrons transfer energy to
protons via collisions (needs a longer time since the proton mass is > electron mass), (iii)
protons share their energy by collisions with neutral atoms.
2.1.2
Ionisation balance in (pure H) H ii regions
We need to be able to calculate the forward and background rates to get the Hydrogen
ionisation ratio.
The PHOTOIONISATION rate is given by:
NPI = a nHI J
[m−3 s−1 ]
(2)
– nHI is the number density of neutral H (H i)
– J is the number of ionising photons crossing unit area per unit time at some point in
the nebula,
– a is the ‘photoionisation cross-section’ for H (from level n = 1). To a first approximation
we can assume a = 6.8 × 10−22 m2 .
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The CROSS-SECTION of an interaction can be thought of as the ‘effective area’ that the
interacting particle sees. The term ‘cross-section’ is just a way to express the PROBABILITY (likelihood) of interaction. See Fig. 1.
Figure 1: Interaction cross-section. Geometric vs. effective area.
The RECOMBINATION rate into level n of an H-atom is:
NR = ne np αn = (ne )2 αn (Te )
[m−3 s−1 ]
(3)
– ne is the electron density
– np is the proton density†
– αn = RECOMBINATION COEFFICIENT of level n, and is the RATE of ion–electron
recombination per unit volume (units cm3 s−1 ). It varies as a function of Te (the electron
temperature of the gas). Note: here the interaction cross-section is already incorporated
in αn .
To get the TOTAL rate of recombination we need to sum eqn. 3 over all levels of H
(n = 0→∞)
Photoionisation is assumed to ONLY occur from level n = 1 (due to the low excitation
of the gas) BUT recombination can occur to any n-value level. Thus any recombination
directly to the n = 1 level is IMMEDIATELY followed by a photoionisation.
For this reason, astronomers make it easy for themselves when calculating equation 3,
and ONLY take recombinations to level n = 2 and above. This is known as CASE
B RECOMBINATION. Under the assumption of case B, αn becomes αB , and a good
−3/4
approximation is: αB = 2 × 10−16 Te
Calculations often use the DEGREE of IONIZATION, X, of H which is given by the
simple relation:
np = ne = X n
where n is the number density of hydrogen nuclei (i.e. the sum of the proton and neutral
atom number densities). Thus nHI = n − Xn, and note that X lies between 0 and 1.
†
ne np = (ne )2 because in equilibrium, the number of electrons equals the number of protons
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Figure 2: Ionisation structure of two homogeneous H + He model H ii regions.
The condition of IONIZATION EQUILIBRIUM says that eqn. 2 = eqn. 3, i.e. the total
photoionisation rate = total recombination rate.
This allows us to calculate X for any given value of J and for a specified gas density (and
hence ne ) at some distance r from the star. If the star is emitting S? ionising photons per
second (i.e. below 912 Å), then
J = S? /(4πr2 )
[m−2 s−1 ]
If we take a typical O-star energy distribution for an O6V star (like those of the Trapezium
stars in Orion), we find S? ∼ 1049 photons per sec, and adopting for example a gas density
of n = 108 m−3 and r = 1 parsec, gives J ∼ 8.4 × 1014 m−2 s−1 . This gives X = 0.999999,
i.e. the H gas is almost completely ionised.
This can be done for a range of r-values (distances from the central star) to give X as a
function of r – see Fig. 2.
2.2
Sizes of photoionised H ii regions: STRÖMGREN SPHERES
A particular star cannot ionise an indefinite volume of ISM gas. The volume it can ionise
depends on the volume at which the total recombination rate is exactly equal to the rate
at which the star emits ionising photons (S).
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If this region has a radius RS , the condition of ionisation balance says that:
S=
4π 3 2
R n αB
3 S
For a given value of S and n (the nebula gas density) we can solve the equation easily to
find RS , which is called the STRÖMGREN RADIUS for the ionised gas around an OB
star.
Typical values are: S = 1049 photons per sec, n = 108 m−3 , which gives RS = 3 parsecs.
The radius of a Strömgren Sphere (the region of fully ionised H gas around an OB star)
will INCREASE if the gas density is LOWER, and will also INCREASE the HOTTER
is the central OB star – since it will emit more ionising photons. For low densities (say
n = 106 ) and for an O6V star we can get RS ∼ 100 parsecs.
2.3
Heating and cooling processes in photoionised nebulae
Photoionisation feeds energy into the nebula gas since it creates hot, fast photoelectrons.
Without some ENERGY LOSS mechanisms the gas temperature would rise indefinitely.
Observational analysis of H ii region spectra suggest electron temperatures of about 10 000
K (= 104 K), significantly lower that the effective temperatures of the ionising OB stars.
If we have a pure H nebula, energy is released in H recombination. The KE of the
recombining electron is converted to photons, which may escape from the nebula and cool
it. However, by comparing the heating and cooling rates from this process we find that
higher nebula temperatures are expected.
On average each recombination removes an energy (3/2) k Te from the gas, and the energy
loss, or COOLING RATE, is:
L=
3
k Te NR
2
[Joules m−3 s−1 ]
The energy input, the HEATING RATE, is given by:
G = NI Q
[Joules m−3 s−1 ]
where Q is the energy injected into the gas in each photoionisation. For star with an
approximate Blackbody spectrum, one can show that Q ∼ (3/2) k Teff . In equilibrium,
L = G, and NR = NI , and we expect to find Te ∼ Teff , i.e. a nebula gas electron temperature
comparable to the star’s effective temperature. For a typical OB star Teff ∼ 30 000–60 000
K, which is much higher that values of Te ∼ 10 000 K for nebulae derived from their
spectra.
Inclusion of other possible H-cooling process like Balmer-line emission, and Hydrogen freefree emission (which occurs mainly at radio frequencies) still gives too high temperatures,
This implies that we have neglected some important cooling process . . .
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Forbidden line cooling of H ii regions (and PNe)
Real nebulae contain other elements than just Hydrogen. We must, therefore, relax our
assumption of a pure H nebula. A special emission line formation process in “metals”
can give emission line radiation which can escape from the gas and cool it. This is called
FORBIDDEN LINE emission.
Bound electrons in atoms can be excited/de-excited in two ways: by photons (resulting in
absorption/emission) or by collisions (non-radiative process, meaning there is no way to
‘see’ it) between atoms or atoms and electrons.
The Balmer (n → 2) and Lyman (n → 1) transitions in Hydrogen are excited/de-excited by
the absorption/emission of photons, and give rise to what are called ALLOWED transitions
in quantum mechanics. The downward (de-excitation) transitions of electrons within the
atom occur spontaneously with a very high probability (typical transition probabilities are
∼109 per sec).
However, some energy levels in metals (and their ions) are split into ‘sub-levels’ or ‘fine
structure states’. Downward transitions amongst these low-lying energy levels can have
a very low probability of occurring spontaneously (typically ∼10–100 per sec – or 1 per
0.01–0.1 secs). Because of this they are often also called METASTABLE states. Photons
emitted in this way result in what are called FORBIDDEN LINES.
The formation process of these Forbidden Lines provides the cooling of the nebula gas in
the following way:
(a) In the ISM, the atom density is too low for atom–atom collisions to occur. However
atom–electron collisions do occur. Bound electrons in the lower levels of e.g. O+ and O++
are excited to higher levels by COLLISIONS with free electrons in the gas REMOVING
some of the Kinetic Energy of the colliding electrons. These free electrons come from the
photoionisation of H.
Excitation potentials of these fine structure levels are ∼1 eV, which is approximately equal
to the energy of the electrons (kT ≈ 1 eV at Te = 10 000 K). Note: all levels of H and He
have much higher excitation potentials, meaning they cannot be excited by collisions with
electrons of the same temperature.
(b) Fine structure states are metastable, so the electron sits there for a relatively long time
(0.01–0.1 secs). Remember, collisions with atoms are negligible. This gives the electron the
chance to RADIATIVELY de-excite by itself (spontaneously), thus emitting a forbiddenline photon. The energy is now transferred to the photon.
Note: if the gas density is too high, the jump to the lower level can be induced by another
collision before spontaneous de-excitation has time to occur. This is called collisional deexcitation – no photon is emitted but kinetic energy is returned to the gas. For most
H ii regions the rate of radiative (spontaneous) de-excitation is greater that collisional
de-excitation.
Every forbidden transition has a ”critical density”, above which the collisional de-excitation
(by electron–atom collision) rate exceeds the radiative (spontaneous) de-excitation rate.
In these circumstances we say the forbidden line becomes “quenched”.
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Figure 3: Energy level diagram for left: O ii (O+ ) and right: O iii (O++ ).
Despite the low abundance of metals (compared to H), forbidden transitions make a significant contribution because their excitation potentials (1–few eV) are so low. Only
electrons with thermal (energy ∼ kT ) energies are needed, rather than ionising photons
(E > 13.6 eV).
(c) because the photon emitted in the Forbidden Line transition has only a very weak
probability of being absorbed the emitted photon easily escapes from the nebula gas,
carrying away photon energy and this cools the gas.
By this process we have REMOVED kinetic energy from the gas and transformed this to
photon energy which ESCAPES from the gas easily.
When this process is included in calculations of the heating and cooling rates it turns out
that irrespective of the Teff of the central OB star, we end up with electron temperatures
of between 7000–10 000 K, as observed.
The term ‘forbidden’ is really a misnomer! It was coined before we understood all the rules
governing electron transitions. A more intuitive name for forbidden lines is “collisionallyexcited lines” (CELs), since they are emitted following collisions of free electrons with ions
in the nebular plasma.
Note: the ionisation potential energy of neutral O is about the same as for H (13.5 eV),
whilst that of O+ is about 35.1 eV. Thus only the hottest O stars can produce significant
O++ in their H ii regions.
Not all CELs result from forbidden transitions – it just happens that most optically observable CELs are forbidden. Moving only slightly into the UV non-forbidden CELs can
start to be seen (e.g. C iv λλ1548, 1551, or Si iv λλ1394, 1403).
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For OXYGEN, the low energy states of singly ionised Oxygen (O+ ) and doubly ionised
Oxygen (O++ ) are illustrated in Fig. 3. “Forbidden Transitions” amongst these levels
include the lines at 3726, 3729 Å for O+ and 5007, 4959, 4363 Å for O++ .
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Figure 4: Energy level diagram for N ii (N+ ) and O iii (O++ ), showing the most commonly
used temperature diagnostic ratios. The arrow indicates increasing energy.
2.4
How to derive temperatures, densities and the chemical composition of emission-line nebulae (H ii regions, PNe)
It turns out that the observed relative intensities of forbidden lines in species like O+ , O++
and other forbidden lines in elements like Sulphur, Argon, Chlorine provide good ways of
measuring the electron temperatures and electron densities in H ii regions.
The most widely used nebular ‘thermometer’ is the intensity ratio of the [O iii] lines
λ4363/λ5007, or equivalently λ4363/λ4959 [or even (λ4959+λ5007)/λ4363] – see Figs 4
and 5. Intuitively, the temperature dependence can be seen from the fact that collisions
with more energetic (hotter!) free photoelectrons are needed to push the bound electrons
of the O++ ion in the upper state from where 4363 Å line is emitted than to populate
the lower energy levels from where 4959 or 5007 Å are emitted. So the line strength ratio
λ4363/λ5007 immediately tells us how hot the electron plasma in a nebula is. In a cool
nebula the λ4363/λ5007 ratio will be lower than in a hotter nebula.
Similarly, the most commonly used density measure is the intensity ratio of the [S ii] lines
λ6717/λ6731 or the [O ii] lines λ3726/λ3729. Fig. 6 shows the energy level diagrams for
these ions, where the transitions used are highlighted. Fig. 7 plots the density dependence
of these line ratios for a given nebula temperature, Te . It is immediately obvious that the
usefulness of these indicators only spans a finite range in densities.
The strengths of certain forbidden lines of heavy ions in an H ii region or a Planetary
Nebula, combined with knowledge of the electron temperature and density in the nebula,
allow us to determine the ABUNDANCE of these ions (and collectively of their respective
element) relative to Hydrogen. The abundance of O++ , for example, can be calculated
using the following formula:
ne I(λ4959)
N (O2+ )
∼
+
N (H )
f (Te ) I(Hβ)
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Figure 5: [O iii](λ4959+λ5007)/λ4363 intensity ratio as a function of temperature. Reproduced from Osterbrock (1989).
Figure 6: Energy level diagram for O ii (O+ ) and S ii (S+ ), showing the most commonly
used density diagnostic ratios.
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Figure 7: Variation of [O ii] (solid line) and [S ii] (dashed line) intensity ratios as a function
of ne at Te = 10000 K. Reproduced from Osterbrock (1989).
where ne is the electron density; f (Te ) is the fraction of O2+ ions able to emit at 4959 Å
(this has a strong dependence on nebular temperature – see Fig. 8) and I(λ4959)/I(Hβ)
is the flux of the [O ii] 4959 Å line relative to Hβ.
Now we can measure the strength of the forbidden lines from all the ionic stages of an
element (e.g. O, O+ , O2+ ) and add up all the abundances to find the TOTAL abundance
relative to Hydrogen.
The technique is as follows: we first observe the nebula using a spectrograph coupled
to a telescope. We measure the line strengths (“fluxes”) relative to the strength of a
representative Hydrogen recombination line. The one typically used is Hβ 4861 Å (since
it’s strong and easily accessible in the optical part of the spectrum).
We use the observed ratios of the strengths of certain forbidden lines to determine the
temperature and density of the nebular plasma.
We then use expressions such as that shown above to find out the abundance of ions relative
to H+ , and add up the abundances of all the ionic stages to find the total abundance of a
heavy element (e.g. C, N, O, Ne, S, Ar, Cl, etc.) relative to Hydrogen in the nebula. Only
with large telescopes can the weaker lines of the above elements be observed from nebulae
in our Galaxy and beyond, allowing us to study the composition of distant galaxies. This
information is vital for a proper understanding of the chemical evolution of the universe
and the recycling of matter within galaxies (from stars to the ISM and vice-versa).
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Figure 8: Fraction of O++ ions able to emit the 4959 Å line as a function of nebula
temperature.
2.5
Additional information
UV forbidden lines (e.g. C iii] 1908 Å; [O iii] 1663 Å) are sampled from space (e.g. from the
IUE or FUSE satellites). For IR lines (e.g. [O iii] 52, 88 µm; [Ne iii] 57 µm) we need a high
altitude plane flying above most of water vapour in the atmosphere, since water absorbs
in the IR (e.g. the Kuiper Airborne Observatory – KAO) or a satellite (e.g. Spitzer ). For
optical lines (e.g. [O iii] 4363, 4959, 5007 Å, [O ii] 3727, 3729 Å, [N ii] 6548, 6584 Å, [Ar iii]
7005 Å, [Ar iv] 4740 Å, [Ne iii] 3968 Å, etc.), we can use ground based telescopes.
HELIUM, which makes up about 10% (by number) of the gas in nebulae, does not emit
forbidden-lines, but recombination lines just like Hydrogen does. The most prominent of
these, and most accessible, are in the optical part of the spectrum: He i 4471 Å, 5876 Å,
6678 Å, and He ii 4686 Å. Doubly ionised Helium however can only be found in Planetary
Nebulae, as the cooler stars of H ii regions do not emit photons energetic enough to ionise
He+ . Hence the He ii 4686 Å line (produced when He++ recombines with an electron to
produce He+ ) is NOT observed in H ii regions.
The abundance of Helium in nebulae is derived from the observed strengths of the aforementioned Helium recombination lines relative to Hβ. Sometimes He i recombination lines
in the radio part of the spectrum are also used. Because of the way that Helium and
Hydrogen recombination lines are created, the strengths of these lines do not depend very
sensitively on the electron temperature in a nebula. Thus, even without a very accurate
measurement of the nebular temperature, a rather accurate Helium abundance (relative
to H) can be derived.
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Ionized metal atoms can also emit recombination lines (allowed/permitted transitions),
but are usually very weak (due to their low abundances). Like H recombination lines,
they exhibit little or no dependence on nebular temperature or density so are not very
useful.