Download The major properties of the Interstellar Medium (ISM) are described

ASTROPHYSCS IV
Convenor
PH712
Dr S. Serjeant
ECTS Credits 7.5
Teaching Provision: 36 lectures
Prerequisites: PH507, PH607
Aims: To provide in-depth study of selected astrophysics material to allow a student to proceed to entry to a
research degree in the field of astronomy and astrophysics.
Learning Outcomes: To have gained an appreciation of current knowledge in the fields of extragalactic
astrophysics and the interstellar medium.
SYLLABUS:
Interstellar Medium (22 lectures)
The major properties of the Interstellar Medium (ISM) are described. The course will
discuss the characteristics of the gaseous and dust components of the ISM, including their
distributions throughout the Galaxy, physical and chemical properties, and their influence
the star formation process. The excitation of this interstellar material will be examined for
the various physical processes which occur in the ISM, including radiative, collisional and
shock excitation. The way in which the interstellar material can collapse under the effects of
self-gravity to form stars, and their subsequent interaction with the remaining material will
be examined. Finally the end stages of stellar evolution will be studied to understand how
planetary nebulae and supernova remnants interact with the surrounding ISM.
Extragalactic astrophysics (14 lectures)
Review of FRW metric; source counts; cosmological distance ladder; standard candles/rods.
High-z galaxies: fundamental plane; Tully-Fisher; low surface brightness galaxies; luminosity functions and
high-z evolution; the Cosmic Star Formation History
Galaxy clusters: the Butcher-Oemler effect; the morphology-density relation; the SZ effect
AGN and black holes: Beaming and superluminal motion; Unified schemes; Black hole demographics; high-z
galaxy and quasar absorption and emission lines; gravitational lensing [if time].
Assessment Methods: Examination 90%; Coursework 10%
Recommended Texts:
Dyson & Williams, The Physics of the Interstellar
Medium, IOP Publishing ISBN 0 7503 0460 X [QB790.D97 1997];
Extragalactic textbook TBA.
[Note: Minor changes to the syllabus may occur during the year]
Revised January 6, 2006
1
Contents
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
2
Introduction
Galactic Structure and morphological structure of clouds
Dust
Properties of ionised gas, H, Recombination lines
Structure and physics of Ionised Nebulae
Excitation of atoms and molecules
Structure of Molecular Clouds
Star Formation mechanisms
Stability of Molecular Clouds
Chapter 1
Introduction
The interstellar medium is defined as the space between the stars. The average number of particles atoms,
molecules and larger particles, in a cubic metre of interstellar space is only about one million. By comparison
a cubic metre of gas in the lecture theatre contains about 1025 particles. Under the very best vacuum
conditions we can achieve in the laboratory there will still be about 1011 particles - so by any definition the
interstellar medium is a very hard vacuum. In this course we will find that it is by the study of material in this
very rarefied environment that we have learnt most of our modern-day knowledge about star formation, the
abundance of atoms, molecules and elements in space, and modern-day observations are even leading us to
add in improved understanding of star formation and the beginnings of life.
3
The bulge is the elliptical-shaped center part of the Galaxy about 1 - 2 kpc in radius. It had lots of star
formation early on, so now it is made of tens of billions of old, metal-rich! stars.
The disk is the thin pancake-shaped part about 400 pc thick and 15 - 20 kpc in radius with the Sun 8 kpc from
the center, containing over 98% of the dust and gas in the Galaxy and has a few hundred billion stars. Some
stars continue to form so the disk has some young metal-rich stars.
The gas and dust are found in a layer that is thinner than the star layer (the gas/dust layer is the thin dark line at
the midplane of the disk in the picture above and the star layer is the thicker light band).
The stellar halo is a roughly spherical distribution of hundreds of millions of old, metal-poor stars that has
increasing concentration of stars toward the center of the galaxy. It is about 20 - 30 kpc in radius and it may
contain small amount of hot gas, but the disk contains the vast majority. Most of the globular clusters are found
in the halo and, like the halo stars, the number of them increases toward the galactic center. If we were at the
center, we would see approximately the same number of globular clusters in any direction we looked in the sky.
Since the globulars are found bunched up in one part of the sky, i.e., they are swarming around some other
point in the galaxy, we know that we are not at the center. The dark matter halo is denser toward the center. It
extends further out than the stellar halo.
The dark matter halo's presence is indicated from the rotation curve---the balance of the outward acceleration
(from the orbital speed) and the inward pull of gravity so that the mass inside the orbit's distance R from the
galactic center Mencl = Vorbit2 × R / G. The rotation curve is flat even though the light-producing matter's
distribution says it should be falling. The dark matter halo may go out as far as 60 - 80 (or more!) kpc! We don't
know what it is made of (brown dwarfs, black holes, neutrinos with mass?).
The first thing to notice about the interstellar medium is that it is a very in homogeneous place, or as
astronomers say it is very clumpy. An example of this is shown in the figure below, which is of a well-known
nebula.
4
Figure
In the this photograph of the Horsehead nebula we see both a dark region, called a dark cloud, as well as
glowing red emission which is due to the emission of a lying of hydrogen gas. We can identify in the
interstellar medium various kinds of regions which are distinguishable by some combination of that
temperature, density, chemical make up. Temperatures in the interstellar medium range from about 10 degrees
above absolute 0 up to over one million degrees. The densities are much lower, covering a range from
between 10,000 up to more than one million million per cubic cm is all molecules are cubic metre. The
number of atoms all molecules are cubic cm is often known as the number density.
Another feature of this photograph is the horses head. This is part of a dense dark cloud or as it is sometimes
called a molecular cloud. These are often appear as black clouds because the star right emitted by background
stars is scattered and absorbed by small dust particles in the cloud before the light reaches us. The density in
clouds like this is somewhat higher between 106 and 1012 particles m-3, which is relatively dense compare it
with other gas found in the interstellar medium. However that temperature is quite low typically it is around
15 to 100 degrees above absolute zero. Much of the gas in these dark clouds is made not of ionised gas but
instead of molecules. Indeed it is out of the most abundant molecule H2, molecular hydrogen, that stars and
young planetary systems will be formed.
•
The Interstellar Medium, ISM , is the space between the stars in our galaxy - i.e. all regions outside
stellar photospheres. In our Galaxy it is known as the Milky Way.
•
Contains both particles and fields - both of which contain energy
•
Average conditions in the ISM (note astronomers normally use non SI units - such as
cm, gm, pc (=3.08 1018 cm));
gas density ~ 1 H2 cm-3
ρ = no mH ~ 2 x 10-24 gm cm-3
gas ~ 0.025 Mo pc-3
dust ~ 0.002 Mo pc-3
~ 95% of ISM is gas, with ~ 70% bound in H2 molecules and ~ 1% in
dust
cosmic rays ~ 0.5 eV cm-3
magnetic field ~ 1 mG ~ 0.2 eV cm-3 (cf. Earth's = 0.31 Gauss = 3.1 10-5 T)
Starlight ~ 0.5 eV cm-3
Average Temperature 10-15 K
The ISM is concentrated in the disc of our Galaxy in a layer ~ 200 pc thick, having a total mass ~
1011 Mo (~ 1041 kg).
•
•
•
5
Coronal Gas
o Hot (106 K) - highly ionised, bremsstrahlung radiation, soft X-rays
Warm HI
~8000 K - neutral hydrogen, 21cm radio emission, interstellar absorption lines
Cold HI
o ~80 K - neutral hydrogen, 21cm radio emission and absorption. In this cold, neutral phase,
atomic and molecular gas is concentrated in clouds ranging from a few to hundreds of light
years in size with temperatures of up to ~ 100 K (average ~ 15 K) and densities between 1
and 106 cm-3. For example, 22.4 liters of air contains roughly 6 1023 molecutes at room
temperature and pressure. That is 600,000,000,000,000,000,000,000 molecules in about one
tenth the space occupied by your refrigerator! Typical interstellar clouds contain only a few
•
•
•
•
•
•
•
•
6
hundred to ten million molecules in the same volume. For all practical purposes , space really
is a vacuum.
Molecular Clouds
o –cold ~ 10 K - H2, CO & other emission lines H2 UV absorption lines OH Masers
Dust
o
10-40 K graphite or silicates absorption and reddening IR emission dark and reflection
nebulae
HII Regions
o ~8000 K - ionised hydrogen, optical emission lines, radio recombination lines,
bremsstrahlung, ionised region maintained by UV radiation from O/B stars
Planetary Nebulae
o similar to HII regions, ejecta from old stars, ionised by a central white dwarf
Supernovae Remnants
o hot ~ 106 K ionised gas heated by the kinetic energy of the explosion bremsstrahlung (Xrays), synchrotron (radio)
Cosmic Rays
o relativistic electrons & particles, synchrotron, direct detection at earth
Magnetic Fields
o synchrotron (radio), Zeeman splitting of 21cm line, dispersion / Faraday rotation,
polarization of starlight
Study of this material ⇒ • Star formation (optically invisible process)
• Early evolution of young stellar objects - dynamics
• Nucleosynthesis (from returned species to ISM from
mass loss)
• Dynamics of our galaxy
• Structure and evolution of our galaxy - chemical
abundance gradients and variation
Atomic Gas
Molecular Gas
Neutral
Solids
Matter
Liquids
Photon ionised
ISM
Ionised
Collisionally
ionised
Cosmic Rays
Magnetic
Fields
Gravitational
Radiation
Figure 2 Fields and particles in the ISM
7
8
9
10
11
Shocks
IR
Starlight
Star
formation
UV
energy absorbed by the
gas and dust grains
Heats dust
Dust grains collide with the
gas - heating it in the
collision
12
Molecular line radiation
Thermal dust emission
Chapter 2
Molecular Cloud properties
Mass
GMC Complex
GMC
SMC Complex
SMC
Globule
n(H2)
size
Vel dispersion
Temp
-3
-1
Mo
cm
pc
km s
K
_______________________________________________________
106 - 107
200 - 105
100 - 300 10 - 20
5 - 100
104 - 106
200 – 105
20 - 150
5 – 15
5 - 100
102 - 105
200 - 105
10 - 50
2 - 10
5 - 30
3
1 - 10
200 - 104
1 - 10
0.5 - 5
5 - 25
3
4
1 - 500
10 - 10
0.1 - 10
0.2 - 2
5 - 15
Solids
Small particles (dust) with radii ≤ 10-4 cm - seen on photographic plates by their optical obscuration.
The absorb and scatter light of all λ's with effective cross sections generally increasing with decreasing λ ⇒
selectively attenuates light of shorter λ's ⇒ reddens spectra of stars. Measured by the colour excess EB-V
which is ∝ the mass of grains present in the column from the earth to the object, where approximately
NH
≈ 5.4 x 1021 atoms cm−2 magnitude−1
EB−V
which translates to a dust-to-gas ratio by mass ~ 160
Photon ionised gas
•
Hot stars produce significant numbers of photons with λ < 912 angstroms (91.2 nm or E > 13.6 eV)
which can directly ionise H. [Note: At high frequencies it is often expedient to consider electromagnetic
radiation as a stream of photons (rather than a wave, characterised not by ν or λ, but by energy, where;
E = hν =
h
hc
ω=
2π
λ
Here, h = 6.626 10-34 J s is Planck's constant. An often used quantity is the electron volt, eV, where
1eV = 1.602 10-19 J, which corresponds to a frequency 2.418 1014 Hz, and whose equivalent wavelength is
given by;
1.240 10 −6
1240
E(eV) =
=
λ (metres) λ (nm)
].
This results in the emission of line (actually recombination line) and continuum emission. Such regions close
to young stars (particularly O & B stars) are called HII regions. Emission from these regions is an useful
diagnostic of the ISM;
Intensity of continuum radiation
Ratios of atomic line intensities
∫ ne2 dL
⇒ ∫ ne dL
⇒
⇒ hence use ratio of one over the other to solve for ne.
13
Material is very clumpy – e.g. about 1/30 th of the total volume of the Orion region, and temperature
of this gas from line ratios and continuum studies is ~ 7 - 10 thousand K. On average ~ 1% of the ISM mass is
photo ionised, and fills about 10 percent of the volume.
Collisionally ionised
Shocks from for example supernovae explosions heat the gas to ~ 106 K, ⇒ collisional ionisation of
the gas, generation of x-rays and highly ionised states.
Fields
• Magnetic
Pulsars used to give:
< neB// > from Faraday rotation
< ne2 > from dispersion
and find B// ~ 1 x 10-6 G parallel to the galactic plane
Effect - aligns non-spherical grains ⇒ optical
polarisation of starlight.
Mostly protons (np/ne ≈ 100 at 109 eV). Total energy
density 1012 erg cm-3 (~ 0.5 eV cm-3). Isotropic arrival directions up to maximum
• Cosmic Rays
observed energies ~ 1020 eV) . Collisions with interstellar gas ⇒ γ-rays.
General considerations
•
Pressure in the ISM
The pressure in the ISM is straighforward to calculate, using the gas law P = n k T . For a typical dark
cloud in which the particle density n = 1010 m-3 and T = 30 K;
P = n k T = 1010 m-3 x 1.38 x 10-23 J K-1 x 30 K = 4 x 10-12 Pa
(c.f. atmospheric pressure at 300 K at sea level ~ 105 Pa).
Relationship between typical densities and temperatures in the ISM
Phase
Dense clouds
Diffuse clouds
Planetary Nebulae
Diffuse Nebulae
Intercloud gas
High-temp gas
n
m-3
1010
108
109
108
3 x 105
104
log
n I
F
G
Hm J
K
T
K
-3
10
8
9
8
5.5
4
30
70
15000
8000
6000
106
log
TI
F
G
HK J
K
1.5
1.8
4.2
3.9
3.8
6.0
We can see this better graphically – from the last equation, taking logarithms of both sides,
n=
P
kT
log n = − log T + log
14
PI
F
G
K
Hk J
P
Pa
4 x 10-12
1 x 10-13
4 x 10-10
2 x 10-11
2 x 10-14
1 x 10-13
The three phases that represent most of the volume of the ISM: diffuse clouds, intercloud gas and high
temperature gas, all have similar pressures … quite surprising in view of their different densities and
temperatures … this tells us that pressure in the ISM tends to even out … that the ISM is close to pressure
equilibrium in those phases which occupy most of the volume.
•
Many phenomena are hydrodynamic, and involve shocks or processes resulting from
of material moving supersonically. The sound speed, co is given by;
co =
γ
where
the motions
γ kT
γ RT
=
µo m H
µ
γ = ratio of specific heats, and is a property of the gas
In general, PV = a constant, where:
µo
µ
= 1 for neutral H gas
= 1/2 for ionised gas
= weight of one mole of gas
for an monatomic gas γ = 5/3, and γ = 7/5 for a diatomic gas
Galactic Structure
Oort (of Oort cloud fame) and Lindblad in the 1920s independently developed evidence that the galaxy disk is
rotating, from studies of the space velocities of stars. It was even possible to measure the orbital velocity of the
sun, which when combined with Shapley's measurement of the distance from the sun to the center of the galaxy
made possible the determination of the mass inside the orbit of the sun (according to Newton, the effect of the
mass outside the orbit of the sun on the sun's motion will cancel out). Suppose R is the distance of the sun from
the center of the Milky Way, and v is its orbital velocity. Then,
15
Assuming the orbit is a circle, the semi-major axis is a=R=8000 pc, and from Kepler III:
If we are able to measure the orbital velocity v(R) at a series of points at various distances R from the center of
the Milky Way, we can find out how much mass is inside each value R, and thus get the mass distribution in the
galaxy. A plot of v(R) versus R is called a rotation curve. We can re-cast Kepler III so that we don't have to
solve for the period each time. Ignoring the mass of the sun, and using the cgs form, we have:
where M is the mass inside R, and v is the rotation velocity at R. When we analyze the rotation curve of the
Milky Way, we find that the mass out to 40 kpc is 6 x 1011 solar masses -- but the visible part of the Milky Way
is largely done by 16 kpc, within which there are about 2x1011 solar masses. So the Milky Way is surrounded
by an invisible halo of dark matter material whose nature remains unclear.
The shape of the galactic rotation curve is shown below
16
The rotation curve is a combination of Rigid Body rotation v ∝ r close to the centre, and Keplerian
motion v ∝ r -0.5 further out, then flatter at large galacto-centric distance -possibly as result of dark material.
This is the ideal case - in practice the effect of invisible material external to the sun's galacto-centric circle,
causes the curve to be much flatter than Keplerian orbits would predict.
17
This curve is determined experimentally from:
• Observations of the velocities and distances of nearby stars and cepheid variables for
distances of up to a few kpc from the sun - after which absorption and scattering extinguish
light.
• λ21cm, recombination line and CO radial velocity measurements of gas clouds in the ISM for
distance greater than a kpc. It is essential that objects of a known distance such as HII regions
are used in conjunction with these objects, especially where stars of known luminosity are
present.
18
19
20
The velocities are determined using the familiar Doppler Effect. For a source of radiation moving relative to
the observer at velocity v, the shift in frequency, from fo, will be given by the Doppler shift equation:
∆f v
=
f0 c
or in terms of wavelength;
∆λ
λ
where c is the velocity of light.
=−
v
c
For example, for a central line frequency of 1 MHz, the shift in frequency, ∆f for a line emitted by a source
moving at 1 km s-1 is given by:
∆f =
106 x 103
= 3.3 Hz
3 x 108
Lets now see how the velocities of the gas can be used to build up a 3D picture of the Galaxy;
21
R is the star - GC distance
d is the star - sun distance
Θo is the solar circular orbital velocity
Θ is the star's circular orbital velocity
l is the galactic longitude
ωο is the solar galactic angular speed (=Θo / Ro)
ω is the star's galactic angular speed (=Θ / R)
α is angle between l.o.s to star and its orbital velocity vector
First solve for Vr;
V r = θ cos α − θ o sin l
from the sine law;
sin l sin(90 + α ) cos α
=
≅
R
Ro
Ro
also since
ω=
then
θ
R
and
ωo =
θo
Ro
Vr = Ro (ω − ω o ) sin l
Expand (ω - ωo ) as a Taylor expansion;
ω − ωo =
dω
dR
R = Ro
(R − Ro ) + 1 d
2
ω
2 dR
(R − Ro )2
2
+ ......
R = Ro
and in practice only the first term is significant since (R-Ro ) is small.
Now, for the solar neighbourhood with d << R, then (Ro-R ) ~ d cos l, and from the
of Oort's constant;
22
definition
A=−
Ro dω
2 dR
R = Ro
we can write an equation for Vr as;
Vr = -2A (R-Ro ) sin l
⎡
dω
since ω − ω o =
⎢⎣
dR
⎤
R− Ro
(R − Ro )⎥
⎦
and thus
Vr = 2 A d sin l cos l
therefore
Vr = A d sin 2l
Lets look at a numerical example of this taken from the 1991 BSc Examination;
The 21 cm line of neutral hydrogen gas has a rest frequency of 1.420406 x 109 Hz. An interstellar
gas cloud, observed towards l = 15o, has an emission line which is observed to be at 1.420123 x 109 Hz
(corrected for the motion of the sun towards the local standard of rest). Calculate the radial velocity of the
cloud with respect to the observer, and hence estimate its distance.
The radial velocity of the cloud is given by;
v ν0 − ν 1.420406 x 109 − 1.420123 x 109
−4
−1
=
= 1.99 x 10 ≡ 59.7 km s
=
9
ν
c
1.420123 x 10
Since Vr = A d sin 2l , and given A = 15 km s-1 kpc-1, solve for d;
d=
Distance Ambiguities
23
vr
59.7
=
= 7.96 kpc
A sin 2l 15 sin (2 x 15o )
The angular velocity increases inwards. In this schems, the greatest velocity along a line of sight is at the cloud
4 shown above, where the line of sight is tangential to the orbit at that point. This means that the radial
velocity of clouds in a fixed direction grows with distance up to a maximum value at cloud (4 in this case),
where
V r,max = RK (ω − ω o )
where RK = Ro sinl
for the point K (where cloud 4 is). Then the distance, PK of the cloud from the Sun is
r = Ro cosl
And the resultant spectrum will look like;
24
The distance of the cloud Pk from the sun is r = R0 cos l . It is clear that there as inherent distance ambiguities
which need to be considered.
Line maps
25
Position Velocity plots
26
27
Rotation and Mass Distribution
We saw previously how to measure the maximum velocity at some galactic radius, Rk, and galactic longitude,
l. Thus we can determine the angular velocity of the gas at different distances from the galactic centre. In this
case, the rotation curve o = o(R),
28
29
Local Standard of Rest
To set a local reference frame, velocity measurements of the Sun are made by taking the average velocity of
material in the local solar neihbourhood, obtained by averaging the space velocities of the closest hundredd or
so stars. The LSR is then the dynamical rest frame of local material, which moves around the sun in a circular
orbit, at a velocity defined on the Galactic rotation curve, appropriate to the Galactocentric distance of the
Sun. The Sun's orbit is not perfectly circular, so with respect to the LSR, the solar motion is 19.5 km s-1
towards the constellation of Hercules (l = 56 degrees, b = 23 degrees).
Density Wave Theory
The stars and gas and dust clouds in the disk congregate in a spiral pattern. There are four parts to the spiral
pattern in our Galaxy called spiral arms.
There are many stars that are also in-between the spiral arms, but they tend to be the dimmer stars (G, K, Mtype stars). Long-lived stars will move in and out of the spiral arms as they orbit the galaxy. Star formation
occurs in the spiral arms because the gas clouds are compressed in the arms to form stars. The very luminous,
short-lived O and B-stars and H II regions around them enhance the spiral outline. They outline the spiral
pattern the same way christmas lights around the edges of a house will outline the borders of the house at night.
The O and B-type stars live for only a few million years, not long enough to move outside of a spiral arm. That
is why they are found exclusively in the spiral arms.
30
Differential rotation provides an easy way to produce a spiral pattern in the disk. Differential rotation is the
difference in the angular speeds of different parts of the galactic disk so stars closer to the center complete a
greater fraction of their orbit in a given time. But differential rotation is too efficient in making the spiral arms.
After only 500 million years, the arms should be so wound up that the structure disappears. Also, the spiral
pattern should occupy only a small part of the disk. The observations of other galaxies contradicts this: spiral
galaxies rarely have more than two turns. Galaxies are billions of years old so the spiral pattern must be a longlasting feature. What maintains the spiral pattern?
31
Density wave theory explains the persistence of spiral arm structure against destruction by differential galactic
rotation.
This painting illustrates the fact that the disk of our Galaxy rotates differentially-- stars close to the center
take less time to orbit the Galactic center than those at greater distances. If spiral arms were somehow tied to
the material of the Galactic disk, this differential rotation would cause the spiral pattern to wind up and
disappear in a few hundred million years. Spiral arms would be too short-lived to be consistent with the
numbers of spirals actually observed.
How then do the Galaxy's spiral arms retain their structure over long periods of time in spite of differential
rotation? A leading explanation for the existence of spiral arms holds that they are spiral density waves-coiled waves of gas compression that move through the Galactic disk, squeezing clouds of interstellar gas and
triggering the process of star formation as they go.
This explanation of spiral structure avoids the problem of differential rotation because the wave pattern is not
tied to any particular piece of the Galactic disk. The spirals we see are merely patterns moving through the
disk, not great masses of matter being transported from place to place. The density wave moves through the
collection of stars and gas comprising the disk just as a sound wave moves through air or an ocean wave
passes through water, compressing different parts of the disk at different times. Even though the rotation rate
of the disk material varies with its distance from the Galactic center, the wave itself remains intact, defining
the Galaxy's spiral arms.
32
In fact, over much of the inner part of the Galactic disk (within about 15 kpc of the center), the spiral wave
pattern is predicted to rotate more slowly than the stars and gas. Thus, Galactic material actually catches up
with the wave, is temporarily slowed down and compressed as it passes through, then continues on its way.
(For a more down-to-earth example of an analogous process, see Interlude 23-2 below.) As shown in Figure
23.17, the slowly moving spiral density wave is outrun by the faster rotation of the disk. As gas enters the arm
from behind, it is compressed and forms stars. Dust lanes mark the regions of highest-density gas. The most
prominent stars -the bright O and B blue giants -live for only a short time, so OB associations, young star
clusters, and emission nebulae are found only within the arms, near their birthsites. Their brightness
emphasizes the spiral structure. Further downstream, ahead of the spiral arms, we see mostly older stars and
star clusters. These have had enough time since their formation to outdistance the wave and pull away from it.
Over millions of years, their random individual motions distort and eventually destroy their original spiral
configuration, and they become part of the general disk population.
Density-wave theory holds that the spiral arms seen in our own and many other galaxies are actually waves of
gas compression and star formation moving through the material of the galactic disk. Gas (red arrows) enters
the arm (white arrows) from behind, is compressed, and forms stars. The spiral pattern we see in this painting
at right is delineated by dust lanes, regions of high gas density, and newly formed O and B stars. The inset
photograph at left shows the spiral galaxy NGC 1566, which displays many of the features described in the
text. Note, incidentally, that although the figure shows a "two-armed" spiral, astronomers are not completely
certain how many arms make up the spiral structure in our own Galaxy (see Figures 23.10 and 23.12). The
theory makes no strong predictions on this point.
33
Assume that the material in the galaxy coexists with a density wave pattern which rigidly propagates around
the galaxy. This will perturb the local gravitational potential, effectively propagating a longitudinal
compression wave through the gas. This model predicts that the rotation curve should be perturbed
and that star formation will be preferentially triggered along the inner edges of spiral arms, in the
region of minimum gravitational potential. Both effects are observed to occur in our galaxy. Observations give
the best value for this pattern speed to be ~ 13.5 km s-1 kpc-1; i.e. in our neighbourhood the density wave
moves around the Galaxy at ~ half the velocity of the stars. In this case, the density wave is continually
overtaken by matter in the local ISM. This matter is compressed in the region of the potential well of the
density wave. In the cloud, dark clouds are produced by the compression, as well as an increased density in the
synchrotron radiation and a shock wave (fully drawn curve in the Figure below). In the course of ~ 107 years,
young bright stars and HII regions are formed in this 'shocked' gas. The major portion of the potential well is
filled with HI.
34
35
In a galaxy the spiral region of greater gravity concentrates the stars and gas. The spiral regions rotate about as
half as fast as the stars move. Stars behind the region of greater gravity are pulled forward into the region and
speed up. Stars leaving the region of greater gravity are pulled backward and slow down. Gas entering spiral
wave is compressed. On the downstream side of wave, there should be lots of H II regions (star formation
regions). This is seen in some galaxies with prominent two-armed spiral patterns. But there are some
unanswered questions. What forms the spiral wave in the first place? What maintains the wave?
Self-propogating Star Formation
Another popular theory uses the shock waves from supernova explosions to shape the spiral pattern. When a
supernova shock wave reaches a gas cloud, it compresses the cloud to stimulate the formation of stars.
36
Some them will be massive enough to produce their own supernova explosions to keep the cycle going.
Coupled with the differential rotation of the disk, the shock waves will keep the spiral arms visible.
Computer simulations of galaxy disks with a series of supernova explosions do produce spiral arms but they are
ragged and not as symmetrical and full as seen in so-called ``grand-design'' spirals that have two arms. There
are spiral galaxies with numerous, ragged spiral arms in their disks (called ``flocculent'' spirals), so perhaps the
self-propogating star formation mechanism is responsible for the flocculent spirals.
Transient Spirals
In this scenario the spiral arms come and go. This behavior is seen in computer simulations of galactic disks. It
is possible that all three theories may be correct. Some galaxies (particularly the grand-design spirals) use the
density wave mechanism and others use the self-propogating star formation or transient spiral mechanisms. Our
galaxy may be an example of a spiral that uses more than one. The spiral density waves could establish the
overall pattern in the disk and the supernova explosions could modify the design somewhat.
37
38
(a) Self-propagating star formation. In this view of the formation of spiral arms, the shock waves produced by
the formation and later evolution of a group of stars provides the trigger for new rounds of star formation. We
have used supernova explosions to illustrate the point here, but the formation of emission nebulae and
planetary nebulae are also important. (b) This process may well be responsible for the "partial" spiral arms
seen in some galaxies, such as NGC 300, shown here in true color. The distinct blue appearance derives from
the vast numbers of young stars that pepper its ill-defined spiral arms.
An important question (one that unfortunately is not answered by either of the theories just described) is:
Where do these spirals come from? What was responsible for generating the density wave in the first place, or
for creating the line of newborn stars whose evolution drives the advancing spiral arm? Scientists speculate
that (1) instabilities in the gas near the Galactic bulge, (2) the gravitational effects of our satellite galaxies (the
Magellanic Clouds), or (3) the possible asymmetry within the bulge itself may have had a big enough
influence on the disk to get the process going. The first possibility is supported by growing evidence that many
other spiral galaxies seem to have experienced gravitational interactions with neighboring systems in the
39
relatively recent past. However, many astronomers still regard the other two possibilities as equally likely. For
example, they point to isolated spirals, whose structure clearly cannot be the result of an external interaction.
The truth of the matter is that we still don't know for sure how galaxies--including our own--acquire such
beautiful spiral arms.
40
Chapter 3
Interstellar Dust
About 1 % of the mass of the ISM is in the form of solid dust grains, whose characteristic dimensions
-5
are 10 to 10-6 cm. These grains are found throughout the interstellar gas, in HI and HII regions, in
molecular clouds and in Planetary Nebulae.
The presence of dust particles in the ISM is revealed by many types of observations. One of the most
important is the reduction in apparent brightness of starlight - known as interstellar extinction.
41
Starlight passing through a dusty region of space is both dimmed and reddened, but spectral lines are still
recognizable in the light that reaches Earth.
This arises from scattering and absorption of the starlight by the grains (in roughly equal proportions).
The scattered light reappears as a general diffuse 'glow' in the ISM, while the absorbed light is re-radiated at
infrared and submillimetre wavelengths. The degree of extinction is measured by a number of stellar
magnitudes of extinction, or absorption, Av .
42
One way to estimate this was developed by Wolf, usinf a measure of star counts that has become known as the
Wolf diagram technique
The Horsehead Nebula in Orion is a striking example of a dark dust cloud, silhouetted against the bright
background of an emission nebula.
43
The density of interstellar matter is extremely low. It averages roughly 106 atoms per cubic meter—just 1 atom
per cubic centimeter--but densities as great as 109 atoms/m3 (1000 atoms/cm3) and as small as 104 atoms/m3
have been found. Matter of this low density is far more tenuous than the best vacuum - about 1010
molecules/m3--that we can make in laboratories here on Earth. Interstellar matter is about a trillion trillion
times less dense than water. Interstellar dust is even rarer than interstellar gas. On average, there are only
about 10-6 dust particles per cubic meter--that is, 1000 per cubic kilometer. Interstellar space is populated with
gas so thin that harvesting all the matter in an interstellar region the size of Earth would yield barely enough
matter to make a pair of dice.
44
Grains made of
Ices
Silicates (particularly Mg and Al silicates)
SiC
Graphite
Lodestone / Iron
Volatile Mantles
At present, astronomically derived information on interstellar grains comes primarily from observations of
extinction, scattering, polarization, and infrared emission. The UV-IR extinction curve requires several IS dust
components: small (less than 0.01 micron-size) grains to explain the far-UV extinction; graphitic carbon to
produce the 0.22 micron bump; and somewhat larger particles of size 0.1 micron, giving rise to the visual
extinction. A spectral feature near 10 microns is evidence for small amorphous silicate grains. Finally, a series
of IR emission bands is ascribed to polycyclic aromatic hydrocarbon (PAH) molecules [Hudgins et al. 1994] or
hydrogenated amorphous carbon grains. In addition to astronomically derived data, new information has come
from laboratory studies of interstellar SiC, graphite and diamond that have been identified in meteorites as trace
constituents. These samples have been identified by their peculiar isotopic compositions. With the exception of
a few alumina grains most of the laboratory IS grains appear to have formed around carbon rich stars. The
phases identified are very robust materials which aids their survival in the interstellar medium (ISM) the solar
nebula, and the extreme chemical processing in the laboratory that is used to isolate them from the bulk of
meteoritic minerals. Besides direct information on the chemical, mineralogical and isotopic composition of a
selected set of IS grains, these samples provide proof that at least some wonderfully crystalline IS grains grow
to sizes of at least 20 microns in circumstellar outflows, and that they survive residence in the interstellar
medium for appreciable amounts of time with their mineralogical and isotopic compositions intact.
Interstellar dust forms by condensation in circumstellar regions around evolved stars, including red giants,
carbon stars, AGB stars, novae and supernovae. The process gives rise to silicate grains when there is more
oxygen than carbon in the star, and carbonaceous grains when the carbon content exceeds that of oxygen.
Pristine grains will retain the isotopic signatures of their formation environment and such signatures have been
detected as rare components of primitive meteorites. Interstellar dust accumulates volatiles in molecular clouds.
Grains are sputtered in intercloud regions, they experience shocks, and they undergo cycles of destruction and
re-formation in the interstellar medium. When the grains are subsequently exposed to UV and cosmic rays in
the interstellar medium (ISM), processing may convert the icy mantles to refractory organic material.
Dust is the major form incorporating heavy elements in the Galaxy that are not inside stars. Due to its high area
to mass ratio, dust plays important roles in interstellar processes. One of its most important properties is its light
absorbtion that permits the formation of cold dense clouds, where molecular species can both form and be
shielded from the otherwise destructive effects of ultraviolet radiation. The cooling effect of dust in some
clouds assists in their collapse to form new generations of stars and planetary systems. Interstellar grains are the
major repository of condensible elements in the interstellar medium and dust influences nearly all types of
astronomical observations including obscuration of visible light from most of the stars in our Galaxy.
A typical grain cross-section in a dense cloud might look like:
45
Interstellar Extinction Curve
Dust grains scatter ( and to a lesser extent) absorb radiation.
46
Optical Image
47
IRAS image
Radio image
48
X-ray image
If the intensity of light near a source at a wavelength λ is Iλ (0);
Iλ = Iλ (0) e − τ λ
and of course there is an equivalent expression for Iν .
49
The Figure below shows a calculation of the amount of power which is absorbed by absorbing slabs
of various opacities - note the exponential dependence;
100
50
0
0
2
4
6
Optical depth τ
τ
%
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
100
60.7
36.8
22.3
13.5
8.2
5
3
1.8
1.1
0.7
Since the particle size is ~ 100 nm (just smaller than the wavelength of optical or NIR radiation ~ 500
or 2000 nm respectively), the dependence of scattering with wavelength is ~ proportional to wavelength;
Eλ ∝
1
λ
∝τλ
where Eλ is the fractional proportion of energy lost from the starlight at wavelength λ .
A curve of the interstellar extinction curve is shown below;
50
When stars form deep inside dense molecular clouds, the surrounding gas and dust become
part of the infalling envelope feeding the central object. In the earliest stages, the nascent
protostars are extinguished by hundreds to thousands of magnitudes, so that only the
circumstellar gas and dust can give a glimpse of what is happening inside. A study of their
evolution is therefore key to understanding solar origins. Part of this gas and dust ends up in
the rotating discs surrounding the young stars, and forms the basic material from which icy
planetesimals, and ultimately planets, are formed. Therefore modern studies seek quantitative
information on the chemical building blocks available during planet formation.
Due to the high dust obscuration, star birth is best studied at long wavelengths. Most young
stellar objects (YSO’s) have been found through IRAS and groundbased infrared surveys,
and have the peak of their spectral energy distribution at farinfrared wavelengths. The spectra
of the coldest protostellar objects (ages of ~104 yr since collapse began) peak around 100 µm
and such sources are best studied with submillimeter telescopes. Once the dense envelopes
start to dissipate due to the effects of outflows, the objects become detectable at infrared
wavelengths, around ages of ~105 yr. Both regions have their advantages for studying
circumstellar material. In the submillimeter, thermal continuum emission from cold (Tdust .
10 .50 K) dust is seen, as well as spectral lines from a plethora of gas-phase molecules.
Owing to the heterodyne technique, the spectral resolving power of these data is intrinsically
extremely high, R=λ/.λ>106 or .V < 0.1 km/s, so that the detailed kinematics of the region can
be studied. Until the advent of large millimeter interferometers such as ALMA, however, the
spatial resolution of these data remains poor.
51
Mid-infrared spectroscopy has the advantage that the composition of both the gas and the
dust can be studied. Solidstate material has characteristic broad vibrational transitions in the
infrared, but no strong bands at millimeter wavelengths. In this case, the features are often
seen in absorption against the hot (Tdust >300 K) dust in the immediate surroundings of the
young star (Fig. 1). At R >2000, the gas-phase lines – which are intrinsically much narrower
– also become visible, albeit only for the most abundant molecules.
The earliest mid-infrared spectra of YSO’s were obtained in the 1970’s and 1980’s, mostly
with the Kuiper Airborne Observatory and UKIRT (see van Dishoeck & Tielens 2001 for a
historical review). At the low spectral resolution of those data, only solid-state bands were
detected, but the spectra of some massive protostars already revealed a surprising wealth of
features. These included not only the anticipated bands of the silicate grain cores at 9.7 and
18 µm due to the Si-O stretching and bending modes, but also other broad features. Thanks to
detailed interaction with laboratory astrophysicists, these could soon be ascribed to ice
mantles, in particular H2O ice and CO ice.
The big step forward came in 1995 with the launch of the Infrared Space Observatory (ISO).
The Short Wavelength Spectrometer (SWS) on ISO provided the first opportunity to obtain
mid-infrared spectra over the entire 2.5–20 µm range unhindered by the Earth’s atmosphere.
High quality data were obtained for about a dozen YSO’s, revealing several new features and
allowing a much more reliable identification of other species (Fig. 2). Several important
ingredients of ices, such as CO2, CH4 and CH3OH, were firmly established with abundances
ranging from < 1% up to 30% of that of H2O ice. However, the ISO-SWS only had the
sensitivity to observe sources forming massive O or B stars with luminosities >104 L_.
The advent of 8–10 m class telescopes equipped with infrared spectrometers with largeformat arrays has opened up the possibility to study low-mass protostars with luminosities
comparable to that of our Sun. Although hampered by the atmosphere, the sensitivity of these
facilities is such that a large sample of objects can be surveyed in a relatively short time. The
main goals to understand this are to: (i) obtain an inventory of the major and minor ice
components in a large set of low- and intermediate-mass YSO’s (< 103 L_) and some
circumstellar discs, and study evolutionary and environmental effects by comparison with
high-mass sources and comets; (ii) use gaseous and solid-state features to probe the physical
conditions and thermal history of the protostellar environment; and (iii) constrain the basic
ice structure through comparison with experimental data available from astrochemists.
52
53
54
We note that the extinction is a function of λ , being strongest at short wavelengths. (the extincted
stars are said to be reddened ). Note particularly that the amount of extinction depends strongly on
wavelength. This means that if an astronomer wishes to 'see' deeply into a cloud of absorbing material (such as
a molecular cloud), observations at infrared wavelengths are able to penetrate more deeply than optical
observations. Practically there are limitations in that the Earth's atmosphere itsel is highly absorptive at many
infrared wavelengths.
55
Reddening helps explain the bluish appearance of reflection nebulae - nebulae rich in gas grains but
with no internal source to disperse them, and which lie close to bright OB stars - so their radiation (the stars')
is scattered (analogy to Earth's atmosphere).
•
From the IR to the near UV the normalised extinction is almost linear in frequency
which implies
•
•
1
λ
Charged particles give a scattering that is independent of wavelength, and atoms and molecules give a
λ-4 dependence
need grains of size comparable with visible light, i.e. ~0.1µm
Now, if rd » λlight
56
τ∝
⇒
dust grains absorb all photons
⇒
not seen ∴ rd ∉ λlight
rd « λlight
⇒
⇒
Rayleigh scattering and extinction
not seen ∴ rd ⊇// λlight
rd ≅ λlight
⇒
sizes ~ 2 10-5 cm.
∝
1
λ
4
A useful approach is to look at colour excess. We usually define a colour excess, E , as the actual
colour index minus the unreddened index. Thus in the UBV photometric system we have;
EU-B = (U-B ) - (U - B )o
and
EB-V = (B-V ) - (B-V )o
where the subscript o represents unreddened values. The reddening ratio is then given by;
EU −B
EB − V
We can make a simple estimate of the number density of grains, and the mass density of interstellar
dust. Starlight in our part of the galaxy travels a distance of about 1 kpc during which it suffers about 1
magnitude of extinction i.e. it encounters an optical depth of absorbing material ~ unity. The opacity, τ, is
given in terms of the extiction factor in the visual, V, part of the spectrum by;
τ ν ≅ Qe ,V π a 2 Nd L
so if the grain radius a = 0.3 µm, and Qe,V ~ 1, then;
1 = 1 π a Nd L
2
and thus the mean dust particle density is;
π (0.3 10
−6
1
) (3.08 10
2
16
3
x 10
)
= 1.15 10 −7 m−3
and given that tha average density ρ ~ 3000 kg m-3, then the mass density in
medium is
volume x dust density x number of particles ;
57
the interstellar
58
Thus the mass of dust is only ~ 1 % of the total mass of material in the interstellar medium - in fact
there is only ~ 1 dust particle per 1011 hydrogen atoms. However, in the optical region the extinction due to
dust particles is still dominant owing to their enormous cross-section (π a2 ~ 3 10-13 m2 ) compared to atomic
or molecular scattering cross-sections.
Thermal balance of dust grains
The properties of grains are studied using Mie Thoey, this is quite complex and beyond the scope of
this course. However, we can say more simply that the extinction factor of grains can be calculated for various
material or constituent materials for simple geometric forms - spheres, long needles etc. This is composed of
an absorption part and a scattering part;
Qe = Qabs + Qsca
However this alone does not get us a good approximation to the interstellar absorption curve, in
particular since the particles may have a range of different radii, and that they are partially oriented by the
galactic magnetic field (see later), and that they have non-spherical shapes.
The interstellar dust is subjected to the radiation field of stars in the Galaxy, and is 'heated' to an
equilibrium temperature Tdust , which corresponds to anequilibrium between irradiation and re-readiation. For
a (spherical) dust grain of radius a, which is strick by radiation flux fν(r) at a distance r from a star;
∞
∞
∫π a
0
2
Qabs,ν fν (r) dν = ∫ 4 π a2 Qabs,ν Bν (Tdust ) dν
0
where the re-radiation is in the form of Balck-body radiation for temperature Tdust . In practice, most
of the absorption occurs of UV radiation, ad the re-radiation is predominantly at IR wavelengths;
Qabs,UV fUV (r) = 4 Qabs, IR σ Tdust
4
where σ is the Stefan-Boltzmann constant we have met before.
Depending on the size and composition of interstellar dust grains, we find that dust in the vicinity of
hot stars may be up to ~ 100 K, and the IR spectrum will peak at wavelengths ≤ 30 µm.
Surprisingly the IRAS satellite discovered the radiation flux from the nearbu A0V star Vega (α Lyr)
to be ~ 10 times greater than theoretical models suggested, even though they did reproduce the optival and UV
spectra well. This IR excess can be attributed to a circumstellar dust disc of size ~ 100 AU and a temperature
~ 85 K. The dust particles were found to be larger (10 µm) than those common in the ISM. The total mass of
dust grains in this sytem was ~ the mass of our solar system - this is then an example of a proto-planetary
system. About 4 other similar systems have been discovered to date.
Dusr grains evaporate at temperatures ≥ 1500 - 1800 K, and may form in the gas in the cooler outer
atmospheres of Red Giant stars.
Polarisation
A further effect of dust particles is to introduce linear polarisation into starlight. This has a
wavelength dependence which peaks in the optical region, and has typical values of ~ 1 percent per magnitude
of extinction. Although polarisation can easily be introduced in reflected light, for the conditions in the ISM,
the only way for it to occur is for the particles to be non-uniform in shape, and co-aligned - giving the effect
rather like the ruled glass plate that makes up a pair of polarised glasses. The most likely shapes are elongated
needles, which contain some sort of ferromagnetic material, allowing the grains to be aligned by the ambient
ISM magnetic field ( ~ 10-9 to 10-10 Tesla (cf. Earth's = 0.31 Gauss = 3.1 10-5 T)).
59
(a) Unpolarized light waves have randomly oriented electric fields. When the light passes through a Polaroid
filter, only waves with their electric fields oriented in a specific direction are transmitted, and the resulting
light is polarized. (b) Aligned dust particles in interstellar space polarize radiation in a similar manner.
Observations of the degree of polarization allow astronomers to infer the size, shape, and orientation of the
particles.
In the figure below, the orientation of polarisation across the Galactic plane is illustrated.
60
The intensity of a star will vary in magnitude as a polarising filter is rotated in front of it. The
polarisation is a function of wavelength. If the max and min intensity are given by Iλ.max and I λ.min , then the
polarisation, P is given by;
P=
61
Iλ . max − I λ. min
Iλ . max + I λ. min
Typically λmax is in the range 450 - 800 nm (in the optical), with an average of ~ 550 nm (yellow).
In early 1995, new detectors at the UK's JCMT will allow the forst long wavelength infrared (submm)
observations to be made to study the polarisation of material in molecular clouds - but rather than relying on
scattered light, now being emission revealing intrinsic properties of the re-radiated thermal infrared emission
of grains.
Processes which could lead to grain alignment
The Davis-Greenstein Effect (paramagnetic relaxation)
62
This occurs for a spinning aspherical grain whose spinning motion is slowed (remember the grain
may contain Fe and Lodestone), so that the axis of the greatest moment on inertia becomes parallel to the
angular momentum axis. The inertia axis is ⊥ to the long axis of a grain and ∴ paramagnetic relaxation aligns
the elongated grains with their long axes ⊥ to the magnetic field.
INSERT FIGURE with GAS STREAMING (ie alignment
by gas streaming past)
Grains may be charged, either positively or negatively, as electrons or ions stick to the grain, building
up a net charge on its surface.
63
The dust particles also reveal their presence by their own direct emissions. The grains may
be heated by the ambient UV field to several hundred K, and re-emit radiation at near and mid IR
wavelengths.
IR emission / absorption features in the near IR from graphite, silicates, unidentified bands etc.
Grain rotation can come from
a) gas collisions
b) momentum of incident photons
Rotation of Interstellar Grains
For rotating grains
1 2 1
Iω ≈ kT
2
2
where
I is the moment of inertia
ω is the angular velocity
since
2 2
mr
5
I=
where
r = the grain radius
m = 4/3 πr3ρ
and
ρ = the grain density, then
2
ω =
So,
if
r
=
10-7
m
and
5 3kT
2 4π r 5 ρ
ρ
=
103
kg
m-3
and
T
=
80K,
5 3
1
1
ω2 = ⎧⎨
1.38 10−23 x 80⎫⎬
−35
3
10
⎩ 2 4π 10
⎭
and thus
ω = 2.57 x 105 radians s-1.
The importance of grains is that they provide a surface for the hydrogen atoms to accrete onto, where
they are retained sufficiently long for H2 to form before the hydrogen atoms are ejected. The grains are
important in forming a localised site where three or more body reactions can occur, permitting the various
chemical reactions to progress. The probability of gas-phase three (or even two) body reactions is very low,
and therefore is an ideal site for atoms and molecules to stick to, and then by migrating around the grain, to
interact with others. The grains also act as a shield against dissociating UV radiation, allowing molecules
remain un-dissociated.
64
Growth of grains
The manner of formation of grains remains unclear. We suspect that particles of dust may condense
in the outer regions of stars such as cool supergiants, RCrB stars, novae etc. This however still leads to too
low a production rates, and it seems likely that they may also form inside dense molecular cloud cores. Lets
look at this latter scenario for grain formation in a cloud, for this we will assume that some sort of 'seed grain'
is present in a gas cloud, and that material from the cloud can accrete onto the grain surface, putting down
successive layers and allowing its radius to increase - this process takes a long time - maybe ~ 109 years ?;
Consider the growth of a grain with radius r at time t. Interstellar atoms and molecules impinge on
the grain at velocity v . Then for accreting atoms and molecules;
4πr2
2
dr πr n m v α
=
dt
ρ
where
r is the grain radius
n is the number density of accreting atoms and
molecules
m is the average mass of accreting atoms and
molecules
α is the sticking coefficient (= unity)
v is the average velocity of the incident atoms
and mols
thus;
dr n m v α
=
dt
4ρ
and
∫
r
0
dr =
∫
t
0
nmvα
dt
4ρ
solving this;
r(t) =
nmvαt
+c
4ρ
where in the limit as t -> 0, c = r(0), the initial radius of the grain at t = 0, so that;
r(t) = r(0) +
nmvα t
4ρ
assuming that the seed grain starts out very small and can therefore be neglected, and the sticking
coefficient, α = unity;
t=
4 ρ r(t)
nmv
To get to a size 10-5 cm requires ~ 3 109 years or more. Therefore it is unlikely that grains can grow
in the diffuse ISM - it is more likely that they grow inside denser regions (in molecular clouds), and in stellar
atmospheres (due to condensation of ejected material)
65
Lets do a simple calculation to illustrate the time taken for a grain to grow by this accretion process
to a size 10-7 m, if ρ = 2500 kg m-3 , and the gas number density = 2 103 cm-3 ;
First estimate the velocity of the ambient (potentially accretable) molecules;
v=
3 x 1.38 x 10−23 x 80
3kT
=
m
3 x 10−26
= 332 metres per second
then with our previous equation;
t=
10−7 x 4 x 2500
3
−26
2 x 10 x 3 x 10
x 332 x 1
= 5.02 x 1016 seconds = 1.6 x 109 years
This time is comparable to the age of the Galaxy, and much greater than the lifetime of a typical
molecular cloud. It is therefore unlikely that the grains could be formed in regions such as this, unless the gas
densities are much higher (such as in a dense molecular cloud core), or in the cooling atmospheres of cool
stars.
The likely destruction mechanisms for grains are a) direct evaporation due to the internal temperature
of the grain, and b) sputtering which is the ejection of atoms or molecules from the grain by the impact of
other atoms or ions moving at thermal velocities. Sufficient energy to evaporate the ices may need a relative
velocity of ~ 1 km s-1 , and > 10 km s-1 to evaporate the iron based core. Other possible destruction
mechanisms are c) UV photos - providing heating, and cosmic rays - which carry enough energy to evaporate
grains completely.
66
Chapter 4
Ionised gas
Atomic Energy Level Structure
In the following we will need to know something about the energy levels present in a simple atom Hydrogen. The first thing is that transition between two energy levels with energies Eu and El result in the
emission (or absorption) of a light quantum of energy hν, where;
hν =
h
= Eu − El
2π
The full specification of an atomic energy level involves four quantum numbers: n Principle
Quantum Number), l (Angular Momentum Quantum Number), s (Spin Quantum Number) , and m (Magnetic
Quantum Number). The energy level described by the 'principle' quantum number alone is thus a composite of
several different levels that have almost the same energy. In the language of quantum mechanics the level is
degenerate. For example the n = 2 level of hydrogen is made up of the following sub-levels;
j = 3 /2
m = -3 /2, -1 /2 , 1/ 2, 3 /2
j = 1/ 2
m = -1 /2, 1 /2
j = 1/ 2
m = -1/2 , 1/ 2
l=1
n=2
l=0
this gives a total of eight sub-levels. As a general approximation, the number of sub-levels (the
degeneracy) of principle quantum number n is 2n2. This is also called the statistical weight of level n, and is
commonly denoted by the term gn. In effect, the statistical weight of an energy level n is the number of
distinct quantum states it contains.
Permitted and forbidden transitions
•
•
67
Angular momentum quantum numbers
• In quantum mechanics each energy level is split into groups of states with different angular
momentum states l = 0, 1, 2, … n-1
• These can have different directions which are also quantised by the z-component m= -l, -l+1, …,
0, … , l
• Electron states can also have intrinsic spin S = ± 1/2
• Thus each states has 2(2l + 1) states
Transitions (∆E) can occur by either
• emission or absorption of photons
• collisions
•
•
•
•
•
•
•
Photons carry off momentum
“Permitted” transitions ∆l = ± 1, ∆m =0, ± 1
• common, 108 s-1
“Forbidden” transitions
• –inconsistent with γ angular momentum transfer
• –have a very low probability
• –usually written e.g. [OII] 372.6 nm
• –time-scales ~10-10 s-1
“Strictly Forbidden” transitions
–never occur
–e.g. Hydrogen n = 2, l=0 → n=1, l=0
Two photon transitions
• these can allow non-permitted transitions
• produce continuum photons ∆E = ∆E1 + ∆E2
• –low probability
• 10 s-1
•
High Density regions
–Permitted transitions common
–Collision excitations and de-excitations common
• Low Density regions
–2 photon transitions can occur
–Forbidden transitions from states without permitted transitions can also occur
Excitation of Interstellar Matter
Since it isn't possible to take a thermometer out to measure material in the ISM, are there other ways in which
temperature can be measured ? We normally have to use observations of spectral lines to enable us to estimate
temperatures.
•
Measurements of either their intensities or widths can tell how the energy is distributed across
various energy levels
You probably remember that atoms and molecules do not all move at the same speed. Their velocities are
governed by the Maxwell Boltzmann distribution:
f(ν) = probability of finding a particle with energy E = 1/2 mv2 in the velocity range ν to (ν + dν) at
temperature T [where f(ν) dv = 1]. f(v) is actually given by the equation;
3
⎛ m ⎞2
⎟
f (v) dv = 4π ⎜
⎜ 2π kT ⎟
⎠
⎝
⎛ mv 2
exp⎜⎜ −
⎝ 2kT
⎞ 2
⎟⎟ v dv
⎠
or can be approximated by
⎛ mv 2
f (v) dv =∝ exp⎜⎜ −
⎝ 2kT
and the most probable speed of a particle is;
v=
and the average kinetic energy per particle is;
68
2 kT
m
⎞ 2
⎟⎟ v dv
⎠
KE =
3kT
mI
F
v =
G
J
H2 K 2
2
where m is the particle's mass and T is the absolute kinetic temperature of the gas.
The rms speed is defined as;
vrms = v
1
2 2
1
2
F3kT I
=G J
Hm K
which tells us that the mean speed of paticles increases with temperature, and decreases with the
atomic or molecular mass. From the Maxwell-Boltzmann distribution we see that:
So, one way of measuring temperatures in the ISM is to determine the way that molecular speeds are distributed
over the various possible states, and to assume that this is governed by a Maxwell Boltzmann distribution. Such
a temperature measured in this way is known as the kinetic temperature.
69
The fact that the lines are broadened means that there will be a net doppler shift of lines towards us, or away
from us:
Since the atoms and molecules will be moving relative to each other, they will frequently collide, transferring
energy amongst themselves. Sometimes this just results in a change in velocity, but other times it will lead to
one (or both) of the atoms or moleculaes being excited into and electronic, vibrational or rotational state above
the ground state. If the gas is in equilibrium, then the number of atoms in a particular state will be given by the
Boltzmann equation,
Nb gb −
=
e
Na ga
( Eb − E a )
kT
where Ea and Eb are known as the excitation energies and are fixed by the chemical species in the atom, and the
statistical weight, ga, gb, quantify how easily an electron can reach the specific energy levels.
The temperature determined in this way is known as the excitation temperature. An example of the excitation
temperature differing from the kinetic temperature might be in a gas which is illuminated by a strong radiation
field – this situation is sometimes encountered in the gas around hot, young stars. In such regions the
moleculaes are not really in equilibrium, and we can't really say that the atoms or moleculaes have a true
'temperature', since this can only be strictly applied in equilibrium. However these various temperature
70
estimates do tell us about conditions in the ISM, and are a convenient way of expressing line intensities, but we
have to be careful about specifying how the temperatures were determined …
It is difficult to see where a spectral line begins and ends, since the velocities will be spread over a very large
range of values. A more convenient way is to express the width of a spectral line by measuring the width at half
height.
So, this is the width at half the height of the line. This increases with the square root of the temperature. If the
atoms or molecules obey Maxwell Boltzmann statistics then for a kinetic temperature of Tk, it can be shown
that:
bW gmc
2
TK =
2
1/ 2
555
. k f 02
where k is Boltzmann's constant and f0 is the frequency of the centre of the line.
A gas of hydrogen atoms emits radiation of frequency 3 1015 Hz. The width at half height is 1.61 1010 Hz, what
is the kinetic temperature of the gas ?
. x 10 hHz
c161
T =
10 2
k
2
x 1674
.
x 10−27 kg x 9 x 1016 m2 s −2
c
5.55 x 1.38 x 10-23 J K -1 x 3 x 1015
. gx 1674
.
x 9 x 10
b161
=
2
5.55 x 1.38 x 107 x 9
hHz
2
2
9
K = 57 K
Often in astronomy, we use units of speed, and so the linewidths are given in m s-1.
A line in the rotational transition of HC3N in a nearby dense cloud emitted at a frequency of 9.1 x 109 Hz has a
width at half height of 0.2 km s-1. What is the kinetic temperature ?
A line width of 0.2 km s-1 corresponds to a width in frequency units of:
2 x 102 m s-1 x 9.1 x 109 Hz
= 6.06 x 103 Hz
3 x 108 m s-1
Now the linewidth will be f0 (v/c), where v is the width in m s-1. Then for the HC3N line:
71
Tk =
51 x 1674
.
x 10−27 kg x 4 x 104 m2 s −2
= 45 K
5.55 x 1.38 x 10-23 J K -1
In the ISM, optically thin lines are often determined by the kinetic temperature, unless there are anomalous
velocities in the gas (i.e. that it is turbulent, or has a systematic velocity structure, such as rotation, collapse,
shocks etc).
Excitation Temperature
Suppose we have an amount of energy and a number of atmos and molecules, then the energy can be distributed
among that atoms or molecules by raising them to disrete higher levels
There are several ways of doing this – at equilibrium, for a large number of atoms or molecules (so we can do
meaningful statistics), the distribution of population over the energy levels is given by the Boltzmann
distribution, which tells us that at a temperature T, the number of molecules nl, in a level whose energy ie εi
above the ground state is given by:
ni =
F
I
G
H JK
ε
gi
n0 exp − i
g0
kT
where no is the number of molecules in the ground state, go is the number of states with the ground
state energy, and gi is the number of states with energy εi, and k is the Boltzmann constant.
For example, if we wanted to know the population of the n = 2 level of hydrogen at room temperature (T = 300
K), and εi = 1.634 x 10-18 J, which is the energy of the n = 2 level above the n = 1 level.
72
The value of gi is equal to the number of states with energy equal to that of the n = 2 level. You may remember
from atomic physics that the 2p state of a hydrogen atom has the quantum numbers n = 2 and l = 1. But we must
also add the magnetic quantum number (m), which has (2l + 1 values), -1, 0, +1. There are this three states
corresponding to 2p. To this, we have to ad one for the 2s state, which has the same energy as the 2p states.
Finally we have to allow for the electron spin. This doubles the number of states for both n = 1 and n = 2. Thus
gi = (3 + 1) x 2 = 8. The population of the n = 2 level compared to the n = 1 level is therefore:
F
G
H
IJ
K
8
16.3 x 10-19 J
≅ 4.09 x 10-171 n0
x n0 x exp −
2
138
. x 10-23 J K -1 x 300 K
so the fraction of hydrogen atoms in the n = 2 level at room temperature is thus negligable.
In the figure below, we see the Boltzmann distribution for a sample of 1000 hypothetical molecules with ten
equally spaced levels (spaced by εi) – the graph is a series of lines, since the atoms or molecules can only
occupy certain levels.
How can this be used to measure temperature in the ISM ?
This simplification comes about since the dominant components of the ISM are H, He, e- and their
ions; collisions between these are ELASTIC for energies < 10 eV (~ 105 K). Thus many transfers of KE
occur between particles before an inelastic collision occurs - resulting in this maxwellian distribution of
velocities. In this circumstance the particles are characterised by the kinetic temperature Tk .
Thermal ionisation becomes important where kT is comparable to the binding energy. In H (EB ~
13.6 eV ≅ 2.18 10-18 J), this occurs for T ~ 105 K. By contrast excitation requires only ~ 104 K. At lower
73
temperatures ionisation or excitation will not be completely efficient, and the gas will contain a mixture of
states.
Designate the energy of an atomic level n by En and the number of atoms in this level by Nn. Using
Boltzmann's law, and denoting the number of particles in the state n by gn , we have;
Nn = A e
−
En
kT
gn
The total number of atoms in all levels is then;
∞
∞
N = ∑ Nn = A∑ gn e
n=1
−
En
kT
≡ A Z (T )
n=1
where Z(T) is called the Partition Function, and A is a constant. Using these last two equations to
eliminate A,
Nn =
E
− n
N
gn e kT
Z(T )
or
Z(T ) = ∑ gn e
−
En
kT
n
The relative populations of two levels are then given by;
Nn gn −
=
e
Nm gm
( E n − Em )
kT
(where the Partition Function cancels in this case ).
Lets now call the levels with the subscript u for upper and l for lower;
If we observe the spectral lines of a particular atom or molecule, then we can obtain the column density of the
molecule in each of the levels. Each column density is proportional to the no of atoms or molecules in a
particular level (as the cloud path is the same). Let the two levels be u and l. If the two levels are n and m with
energies εi above the ground state, then;
nu = n0
F
IJ
G
H K
gu
ε
exp − u
g0
kT
and
nl = n0
and combining these:
F
I
J
G
H K
gl
ε
exp − l
g0
kT
ε I
F
− J
G
HkT K= g expF
g
∆ε I
=
exp
− J
G
H
ε
g
kT K
F
I g
− J
G
HkT K
u
nu
nl
u
l
u
l
l
where ∆ε = (εu – εl). Since the column densities are proportional to the numbers of molecules;
74
F
IJ
G
H K
N u gu
∆ε
=
exp −
N l gl
kT
where Nu and Nl are the column densities of the two levels.
The two lowest rotational levels of CN are separated in frequency by 1.134 x 1011 Hz, and the ration of the
column densities for these levels in a particular direction is N(J = 1) / N(J = 0) = 0.32. The energy separation
∆ε = ∆f x h, where h is Planck's constant. We will see later that there are 2J + 1 rotational states for eac energy,
so that gu = 3 and gl = 1. We can now determine the excitation temperature of CN, by rearranging the previous
equation;
Tex =
∆ε
k x 2.30 log
F
g N I
G
Hg N JK
u
l
l
u
so, for CN,
Tex =
1.13 x 1011 Hz x 6.63 x 10-34 J s
= 2.4 K
3
138
. x 10-23 J K -1 x 2.30 log
0.32
F
I
G
H J
K
It is now generally accepted that the populations of these two levels are due to the molecule interacting with the
radiation left over from the big bang at the formation of the universe.
Temperatures of diffuse nebulae are determined from populations of energy levels calculated from populations
of energy levels calculated from the emission lines due to species such as the hydrogen 'Balmer' lines, and
'forbidden' lines of O+, O++ and N+. In ionised gas the temperatures are typically in the range 8,000 – 10,000
K.
The standard thermometer in molecular clouds in the CO molecule, because it is the most abundant molecule
after H2.
Types of Radiation
Blackbody radiation
Many astronomical objects radiate as blackbodies - we studied this in the PAS course. The intensity
of radiation emitted at frequency ν , into frequency range ∆ν by a body of temperature T is given by;
⎛ 2hν 3 ⎞ ⎡ 1 ⎤
I(ν )∆ν = ⎜ 2 ⎢ hν
⎥
⎝ c ⎠ kT
⎣ e − 1⎦
or in units of wavelength where
νλ = c and hence dν = −
c
λ2
dλ
then
⎛ 2hc 2 ⎡ 1 ⎤
I( λ )∆λ = ⎜ 5 ⎞ ⎢ hc
⎥
⎝ λ ⎠ λkT
⎣ e − 1⎦
commonly Bν (T ) = Iν (T ) and Bλ (T ) = Iλ (T )
75
at short wavelengths, where hν >> kT , this equation takes the form of Wien's Law;
hν
2hν 3 − kT
Bν (T ) = 2 e
c
At long wavelengths, where hν << kT , we make the approximation;
hν ⎞
hν
exp⎝⎛
−1 ≅
kT ⎠
kT
this equation takes the form of the Rayleigh-Jeans Law;
Bν (T ) =
2ν 2 kT
c2
The area under the Blackbody curve is obtained by integrating Bν over all frequencies, the total
energy flux, πB , of the blackbody is [note: this can be related to the Luminosity of and object - an important
value to determine];
76
πB = σ T 4
where σ is the Stefan constant
2π 5 k 4
σ=
= 5.669 10 −8 Watts m−2 K −4
2 3
15 c h
The brightness of a blackbody increases as the fourth power of its temperature, the total energy
output per unit time of the object (its luminosity) is then obtained with the relationship;
L = 4π R2 σ T 4
since the surface area of a sphere of radius R is 4π R2 .
The peak of the blackbody curve can be found with the Wein displacement Law;
νm ≈
λm ≈
3kT
≈ 6 1010 T Hz
h
or
0.3
where λ is in cm
T
for example the continuum spectrum from our Sun peaks at ~ 500 nm; therefore the surface
temperature is ~ 5800 K. By contrast, a protostar, whose surface temperature may be ~ 100 K, will emit most
of its radiation at wavelengths ~ 1 mm (~ 300,000 MHz = 300 GHz).
The majority of astronomical objects have spectra which are ~ like blackbodies: they have been
particularly useful in understanding stars and forming stars (protostars). We will also see later how basic
atomic constants are related to this function.
77
For a blackbody whose radius is r and whose distance is D , the flux density Sν incident at the earth
is;
Sν (T ) = Ω s Bν (T ) ≈
π r2
D
2
Bν (T )
where Ωs is the solid angle subtended by the source and Bν(T) is the brightness of the source at
frequency ν and temperature T . Note that radio astronomers measure flux density in flux units (denoted
Janskys - or Jy) where 1 Jy = 10-26 watt m-2 Hz-1 .
Another common way this is expressed, which takes account of the opacity of the dust (it is usually
optically thin at mm wavelengths) is;
Sν = 2 kTd
Definition of temperature
78
ν 2Ω S
c 2τ d
Temperature is rarely uniquely defined in astronomy - and we have to specify which form of
temperature we refer to. The various forms are;
Temperature
Brightness
Colour
Basic Law / Equation
Planck Curve
Planck Curve
Effective
Stefan-Boltzmann Law
Excitation
Boltzmann Law
Ionisation
Saha Equation
Kinetic
Thermal Doppler broadening
Observations
Flux at one wavelength
Flux at two or more
wavelengths
Luminosity (power) and
radius
Relative equivalent line
widths of spectral lines of
the same element
Relative equivalent line
widths of spectral lines of
adjacent stages of ionisation
Widths and profiles of
spectral lines
Temperature can also be assigned to a radiation field (for example starlight) and to matter. In the
latter case it will be based on the velocity distribution ( a kinetic temperature), the degree of ionisation and the
degree of ionisation.
Bremssrahlung (Free-free) emission
Bremssrahlung is a German word which literally means 'braking'. This refers to the fact that
electrons encounter a coulombic force as they pass close to bare nuclei (protons) which deflects their forward
motion, slowing (or 'braking') their motion.
An example of a source showing Bremsstrahlung emission is the Orion Nebula;
79
This is described by the Blackbody equation to consist of optically thick and thin material. Before we
apply the Blacbody Equation, we note that the optical depth for Bremsstrahlun emission is given by;
κ ν ∝ ν −2.1Te−1.35
80
The position of the turnover frequency (see the above Figure) marks the point at which the
opacity, τν = 1.
Bν (T) ≈
2ν2 kT
c2
thus
Iν ∝ ν
2
now, if on the other hand τ << 1 then
Iν ~ Bν (T) τν
and therefore;
2
Iν ∝
2kTν
c
so that
or if
−2.1
ν
2
−1.35
e
T
Iν ∝ ν−0.1
τ >> 1
Iν ~ Bν (T)
Iν ∝
2kTν2
2
∴ Iν ∝ ν
2
c
At low frequencies, the spectral index ~ 2, and the emission here is optically thick - or opaque. At
higher frequencies it is ~ -0.1, and the emission is optically thin. [note: since
TB =
c2
2 Iν
2k ν
and
Iν [W m
−2
−1
−1
Hz sr ] = 3.08 10
−28
(ν [MHz ] T [K ])
2
B
that
Iν ∝ ν
−0.1
−2.1
≡ TB
]
.
Determining the Electron Temperature
The electron temperature can be estimated by three methods
a)
the absolute thermal radio continuum intensity at low frequencies where τ >> 1
(since Iν = Bν (Te) ~ ν2 Te ).
b)
from the observed intensity ratios of two collisionally excited optical emission lines (such as
OII 372.9/372.6 nm), for which the extinction will be the same for both.
c)
81
intensities of radio recombination lines.
d)
Take the ratio of the recombination line relative to that of nearby continuum emission, and;
L
c
TL = τ ν Te and Tc = τν Te
Then assume Ne = NP + N(He+) and;
−1.15
2.1
TL ∆νL
T e
4 ν
≈ 2.3 x 10
TC
N(He+)
1+
NP
Recombination line emission
In an ionised gas and electron and ion may pass close enough so that the electron and ion recombine,
with a suitable release of energy, and with the electron in an energy level having a large principle quantum
number. Most of these electrons immediately jump to the ground state, releasing energy in emission lines
known as resonance lines. In a few cases the electrons cascade downwards from level to level, releasing their
energy as recombination lines. Typically the number in these highly excited states is ~ 10-5 of the number in
the ground state. These lines are astronomically important since the ratio of the energy in a line to that of the
underlying continuum emission gives the ratio of bound to free electrons, and hence a measure of gas
temperature, and also the line width can be broadened by pressure effects, giving a measure of the gas density.
In addition, Doppler shifts measured from the line centres give information on the velocity fields in the gas.
82
The line frequencies can be simply calculated. Levels having large principle quantum numbers differ
little in energy, and the line emission between adjacent levels appears in the radio wavelength part of the
spectrum. The frequency of a recombination line transition can be calculated with the well known Rydberg
formula;
⎡1
1
⎤
ν = R c Z2 ⎢ 2 −
(n + ∆n)2 ⎥⎦
⎣n
where R is the Rydberg constant for a hydrogen atom (10.96776 µm-1), c is the speed of light, Z the
effective electronic charge e , and n is the lower principle quantum number, and ∆n is the change in n . For
recombinations to singly ionised atoms, Z = 1; to doubly ionised atoms, Z = 2, etc.
FIGURE 8-8 The Bohr model for hydrogen. The orbitals are the stable energy levels of the electron,
from n = 1 (ground state) and up. Shown are the sequences of electronic transitions for the Balmer
(absorption and emission), Lyman, and Paschen series.
Transitions occur over different shell jumps. Depending on the start and finish energy levels, the
emission lines occur in what are known as Series, as shown above. The Lyman and Balmer lines are at UV and
optical frequencies respectively, and the Paschen, Bracket and Pfund lines normally seen at infrared
wavelengths.
The first hydrogen lines discovered were the Balmer Series, and they are designated Hα (∆n = 1), Hβ
(∆n = 2), Hγ (∆n = 3), etc. For the example of the Hα line (a jump from n = 3 ⇒ 2);
1
λH α
⎛1
1 ⎞⎟
−1
= 10.96776 ⎜ 2 −
= 1.52330 µm
2
⎝2
(2 + 1) ⎠
λ Hα = 656.3 nm
Similarly, the Lyman-α line has the wavelength 121.6 nm (in the ultraviolet), and the H60α line at λ
~ 1 cm - in the radio wavelength region. There is clearly a lower limit to the frequency of a recombination line
83
transition, when the interval between depopulation processes is less than the orbital period of a bound electron
- in practice this is for n values of ~ 740 which occurs at ~ 16.3 MHz.
Figure 8-9 Hydrogen energy-level diagram. Transitions for the first three hydrogen series are shown in
emission. On the left are shown the general types of transitions for absorption, emission, and ionisation - note
on the right the energies (in eV) of corresponding photons.
To completely ionise the H atom requires 2.18 10-18 J = 13.6 eV , photons with suitable wavelenghts
(<91.6 nm) occur in the ultraviolet region of the spectrum. Therefore the most likely place to find ionised H is
close to UV sources - for example hot young stars (Tsurface ~ 25,000+ K).
HII Regions
84
The Strömgren Radius
Consider a thin shell of gas of thickness dR at radius R from a young star.
The number of particles in the shell = 4π R2 n dR, where n is the particle density.
If the star emits dN ionising photons per second, then the radius of the ionised region grows by dR,
where
dN
dR
= 4π R2 n
dt
dt
assuming each photon ionises a hydrogen atom. However, if the ions and electrons recombine at a
rate a, a new photon will be needed to ionise the reformed hydrogen atom, and then the above equation must
be modified to allow for this;
3
dN
dR 4πR
2
= 4πR n
−
ni ne α
dt
dt
3
A final equilibrium is reached when
dR
=0
dt
i.e. when the emission rate of ionising photons exactly balances the rate of recombination, thus the
first term on the right hand side of the modified equation above = 0, and then:
85
dN 4π R3
ni ne α
=
3
dt
and
dN
= S*
dt
the stellar ionising photon emission rate, then solve for R
1
3
⎛ 3 S* ⎞⎟
R = ⎜⎝
= Rs
4π nineα ⎠
where Rs is the Strömgren radius.
Since ni = ne = no, the gas density in the neutral cloud, hence
1
⎛ 3 S* ⎞ 3
⎟
Rs = ⎜
⎝ 4π n2o α ⎠
During the process of ionisation, the number of particles inside the ionised region will
double, creating extra internal pressure. This causes the HII region to expand beyond that predicted by the
Strömgren equation.
So, in equilibrium
2 ni kTi = nokTo
where the subscripts i and o refer to the ionised and neutral gas respectively.
Setting the pressure P = 2 ni kTi, and making the simplifying assumption that this corresponds to the
pressure difference between the inside and outside of the bubble, then since acceleration, a, is force per unit
mass,
2
pressure x surface area 2ni kT x 4π R
⎛ 8π ni kT ⎞ 2
a=
≈
≈⎝
R
M ⎠
M
mass
where M corresponds to the mass of material being accelerated. In practice there would be additional
pressure from material outside the expanding shell, but for this simple treatment, its effects are ignored.
Evolution of the ionisation front
86
If the number of Lyman photons per unit area falling on the ionisation front per second is ILy , then
with the geometry shown below;
Neutral gas
with density n /m^3
Ionised gas
*
I
Ly
At time t, the
distance moved
by the IF is R
then for the front to move a small amount from R to (R+dR) in time dt;
ILy dt = n dR
∴
ILy
dR
=
dt
n
This is then the velocity at which the IF propagates. Note that so far recombination has been
neglected. Taking this into account;
87
2
S* = 4πR ILy
∴ ILy =
S*
2
4πR
4πR3 neni α
+
3
−
Rneni α
3
so, now we can rewrite for the IF velocity
S*
dR
Rnα
=
−
dt
3
4πR2 n
In practice, a good approximation to an HII region's growth is given by (proof not required );
3
R3 = Rs { 1 − exp ( −nαt ) }
Thus, although Rs is finally approached, the development of the HII region is initially rapid, on a
characteristic timescale ~ 105 years, and with the radius growing with a characteristic velocity ~ the sound
speed (~14 km s-1).
During the process of ionisation, the number of particles inside the ionised region will double,
creating extra internal pressure. So, in equilibrium
2 ni kTi = nokTo
where the subscripts i and o refer to the ionised and neutral gas
respectively.
Setting the pressure P = 2 ni kTi, and making the simplifying assumption that this corresponds to the
pressure difference between the inside and outside of the bubble, this then leads to an acceleration of the gas,
and expansion. Consequently the boundary between the ionised and neutral material propagates into the
surrounding medium. The radius of the region is approximately;
2
⎛ 2Ti ⎞ 3
⎟ Rs
R = ⎜
⎝ Tn ⎠
where Ti and Tn are the temperatures of the ionised and surrounding neutral gas (proof not required here).
Eventually a stationary point would be reached, but in practice other factors, such as external shocks,
interaction with other objects, inhomogeneities in the surrounding ISM etc., will result in the HII region
merging into the diffuse ISM.
88
The time evolution of the size of an HII region is roughly given by;
⎧
(− n H α t)⎫
Ri3 = RS3 ⎨1 − e
⎬
⎭
⎩
where α is the recombination coefficient, nH is the hydrogen density and t is time. The
graph below shows the way the radius grows with time - it clearly approaches some equilibrium state
asymptotically;
Ionisation fraction
H is the most abundant element and it is useful to compare an HII region to a nebula of pure
hydrogen – made of HI and HII:
HI + photon ⇔ HII + e
If the nebula is in ionisation balance, the forward and reverse rates will be equal. So, concentrating on HII,
Recombination (loss) rate = photoionisation (production) rate
89
Nr = Ni
Where Nr is the number of recombinations per unit volume per second and Nr is the number of ion pairs (HII
+ e) produced by ionisation per unit volume per second.
We can estimate the degree of ionisation by writing the expression for Nr and equating it with an expression
for Ni .
The expression for Ni depends on three factors:
n(H+) is the number density of HII
n(e) is the number density of electrons
α is the recombination coefficient, which is a measure of the rate of recombination of electrons and HII
For a nebula of pure hydrogen, we can set n(e) = n(H+)
The rate of recombination Nr is then:
Nr = α n(H+) n(e) = α n(H+)2
The expression for Ni contains three factors, n(H) is the number density of hydrogen atoms, φH is the flux of
ionising photons from the star that is responsible for ionising the gas and σ i is the cross-section for ionisation
– in other words the probability that an atom of H will absorb an ionising photon.
The rate of ionisation is:
Ni = n(H) φH σ I
Note that the expression for Nr contains n(H+) and for Ni contains n(H) - so that the degree of ionisation,
which is traditionally denoted by the symbol x , is:
x=
n( H + )
n( H + )
=
n
n( H ) + n( H + )
Assuming that there is ionisation balance, we have
α n(H+)2 = n(H) φH σ I
Now divide by n2 and substitute for n(H) from n = n(H+) + n(H)
n − n( H + )
n( H + )2
α
=
φ Hσ i
n2
n2
Now substituting the value of x for (n(H+) / n) from earlier, gives
x2
σφ
= i H
1− x
αn
To find the value for x we take typical values for a diffuse nebula and substitute them into this equation:
φH = 1015 s-1 m-2
σ I = 7 x 10-22 m2
α = 2 x 10-19 m3 s-1 at T = 10,000 K
n = 108 m-3
90
7 × 10 −22 ( m 2 ) × 1015 (s-1 m -2 )
x2
=
= 3.5 × 10 4
1 − x 2 × 10 -19 (m 3 s-1 ) × 10 8 (m -3 )
Solve as a quadratic equation x = 0.99997. Therefore only 0.003% of the hydrogen is neutral – in other words
the gas is almost totally ionised.
Photodissociation regions (PDR's)
Massive stars have a strong effect on their environment. At a distance of a few pc from a massive Ostar (> 106 Lo), the flux of photons is ~ 1000 times larger than the average radiation field of the solar
neighbourhood. Massive stars also have supersonic winds with velocities up to several thousand km s-1
carrying between 0.1 and 1% of the stars luminosity in the form of kinetic energy. Massive stars affect their
environment very soon after the stellar core has formed. At the end of its lifetime the star explodes as a
supernova, ejecting ~ 1051 erg of energy into its surroundings. The radiation, shocks and winds from nearby
OB stars affect the physical conditions, structure and chemistry of dense molecular clouds. The most important
global application of the effects of star formation is to understand the effects of massive star formation as a
whole in starburst galaxies. For example the average flux in the central 700 pc of the IR luminous galaxy M82
is about the same as that 5pc from an O-star.
Far UV radiation (912-2000 Å) from stars such as this dissociate most molecules and ionises a
number of abundant atoms such as C, Si, S and Fe. When neutral gas clouds are exposed to far-UV radiation,
there are zones where most of the gas is in atomic or partially ionised forms. Examples include the interfaces
between HII region and surrounding neutral molecular clouds, neutral shells around planetary nebulae and the
diffuse interstellar medium in our galaxy. The size of the photodissociation region is determined by the UV
penetration depth. The most efficient absorbers are dust and H2 , and the thickness of a PDR thus extends into
a cloud until τ ~ a few, which for dust corresponds typically to a hydrogen column density ~ 1022 cm-2.
PDR's may extend more deeply into clouds if the gas is very clumpy. The stratification structure of a PDR near
a molecular cloud is shown below;
91
PDR
Molecular Cloud
HII/HI
HI/H
2
CII/CI
UV
OI/O
2
Flux
1
10
10
21
10
100
22
Photoelectric heating and UV pumping of H2
mechanisms in PDR's
e
10
23
Av (magnitudes)
n(H) cm
-2
are probably the most important gas heating
-
+
+
+
+ +
DISSOCIATION
Collisional
deexcitation
Photoelectric
Heating
UV pumping of H2
Reflection nebula often appear blue, and stars seen through intervening clouds are often reddish due to the
process of light scattering and extinction. Small droplets and dust grains will scatter light, but nonuniformly.
The blue components of a beam are preferentially scattered while the red hardly scatter at all. The degree of
scattering was determined by Physicists in the last century to be prportional to the frequency of the light to the
fourth power. When a beam of light passes through a cloud of interstellar matter, much of the blue is scattered
out. The beam that emerges appears reddened because it has lost it's blue component.
92
Reflection nebula often appear blue, and stars seen through intervening clouds are often reddish due to the
process of light scattering and extinction. Small droplets and dust grains will scatter light, but nonuniformly.
The blue components of a beam are preferentially scattered while the red hardly scatter at all. The degree of
scattering was determined by Physicists in the last century to be prportional to the frequency of the light to the
fourth power. When a beam of light passes through a cloud of interstellar matter, much of the blue is scattered
out. The beam that emerges appears reddened because it has lost it's blue component.
93
Chapter 6
HI emission
Most of the hydrogen gas in the interstellar medium is in cold atomic form or molecular form. In 1944 Hendrik
van de Hulst predicted that the cold atomic hydrogen (H I) gas should emit a particular wavelength of radio
energy from a slight energy change in the hydrogen atoms. The wavelength is 21.1 centimeters (frequency =
1420.4 MHz) so this radiation is called 21 cm radiation. The atomic hydrogen gas has temperatures between
100 K to about 3000 K.
It corresponds to a transition within the 1 s2 S1/2 ground state of HI, between its two hyperfine levels with
total spin 1 (nuclear and electronic spins aligned) and spin 0 (spins anti-parallel). The small energy difference
(~ 6 10-6 eV) gives a line in the radio frequency range at;
λo - 21.1 cm or
νo= 1420.4 MHz
Most of the hydrogen in space (far from hot O & B-type stars) is in the ground state. The electron moving
around the proton can have a spin in the same direction as the proton's spin (i.e., parallel) or be in the direct
opposite direction as the proton's spin (i.e., anti-parallel). The energy state of an electron spinning anti-parallel
is slightly lower than the energy state of a parallel-spin electron. Remember that the atom always wants to be in
the lowest energy state possible, so the electron will eventually flip to the anti-parallel spin direction if it was
somehow knocked to the parallel spin direction. The energy difference is very small so a hydrogen atom can
wait on average a few million years before it undergoes this transition. (If atoms could tell us what's
happening:)
94
Even though this is a RARE transition, the large amount of hydrogen gas means that enough hydrogen atoms
are emitting the 21 cm radiation at any one given time to be easily detected with our radio telescopes. Our
galaxy, the Milky Way, has about 3 billion solar masses of H I gas with about 70% of that further out in the
galaxy from the Sun. Most of the H I gas is in disk component of our galaxy and is located within 220 pc from
the midplane of the disk. What's very nice is that 21 cm radiation is not blocked by dust! The 21 cm radiation
provides the best way to map the structure of the Galaxy.
95
The energy difference between the two states is ~ 9.42 10-18 erg. Using the relationship between
photon energy and wavelength;
λ=
hc
E
where (using cgs units in this case) λ is in cm, h is Planck's comstant (6.626 10-27 erg-sec) and c is
the velocity of light = 3 1010 cm s-1, and E is the energy difference (ergs). Substituting 9.42 10-18 for E, we
find;
96
−27
λ=
10
6.626 10 x 3 10
≅ 21.1 cm
9.42 10 −18
The intensity of the 21 cm emission line depends on the density of the neutral atomic hydrogen along
the line of sight. This transition is 'forbidden' having an extremely small transition probability (Einstein
coefficient), A = 2.87 10-15 s-1. Now the average lifetime in the upper level is A-1 = 1.1 107 years. Under ISM
conditions, collisions are more frequent, ~ every 400 years, and so the real rate of transitions is much shorter
than the 'forbidden' rate would imply.
The mean particle density of HI in the ISM ~ 4 105 m-3, or 0.4 cm-3, between ~ 4 and 14 kpc,
decreasing on going outwards. Sky syrveys show the spiral arms to be composed of many smaller cloud
fragments - characteristic sizes for an HI cloud are a diameter ~ 5 pc, mean density ~ 2 107 m-3, temperature ~
80 K and mass ~ 30 Mο. The total mass of HI in the ISM is ~ 2.5 109 Mo, distributed in the form of a flat disc
along the galactic plane, with a half-width of ~ 240 pc.
Molecular line emission
Different types of atoms can combine in the coldest regions of space (around 10 K) to make molecules.
The cold molecules are detected in the radio band. Most of the molecules are hydrogen molecules (H2) and
carbon monoxide (CO). Actually, molecular hydrogen does not emit radio energy but it is found with carbon
monoxide, so the radio emission of CO is used to trace the H2. Other molecules include such familiar ones as
H2O (water), OH (hydroxide), NH3 (ammonia), SiO (silicon monoxide), CO2 (carbon dioxide) and over a
hundred other molecules. Many of the molecules have carbon in them and are called organic molecules. The
organic molecules and the water and ammonia molecules are used in biochemical reactions to create the
building blocks of life: amino acids and nucleotides. The presence of these molecules in the interstellar medium
shows us that some of the ingredients for life exist throughout the Galaxy. Some carbonaceous meteorites
reaching the Earth have amino acids in them---apparently, amino acids can be created in conditions too harsh
for normal biological processes.
Quote from a paper in Astrophysical Journal, 191, 229, 1975;
Ethyl alcohol has been of interest to mankind since the dawn of civilisation.
During early October 1974, we detected a truly astronomical source of ethyl alcohol
located in the direction of the Galactic Centre in our Galaxy. Preliminary estimates
indicate that the alcoholic content of this cloud (SgrB2) yield approximately 10 28 fifths at
200 proof. This exceeds the total volume of all man's fermentation efforts since the
beginning of recorded history.
The first interstellar molecules were in fact discovered in absorption against starlight;
We have already seen that the spectrum of a star forming region such as the Orion Nebula is very rich
in the emission lines of many molecular species. To date there are some hundred plus, ranging from simple
97
lines such as CO (detected 1968), through to the first amino acid Glycine (detected 1994) [Hoyle and
Wickramasingh have long speculated that life started by DNA and amino acids hitting the Earth on comets and now we are indeed starting to detect DNA constituents in space ].
In the diffuse HI clouds in the ISM, where hydrogen occurs mainly in atomic form, there are only a few
simple molecules such as CH, CH+ and CN, which have long been known from their optical absorption
lines; the hydroxl radical OH, and the molecules H2, HD and CO which are known from their UV absorption
lines. Following the discovery in 1968/9 of the radiofrequency lines of H2O and NH3 and H2CO, searches
have led to the discovery of ~ 100 interstellar molecular line species.
Molecular hydrogen H2 does not produce radio emission. It produces absorption lines in the ultraviolet.
However, the gas and dust become so thick in a molecular cloud that the ultraviolet extinction is too large to
accurately measure all of the H2 in the interior of the cloud. Fortunately, we see evidence of a correlation
between the amount of CO and H2 so we use the easily detected CO radio emission lines (at 2.6 and 1.3 mm)
to infer the amount of H2. The CO emission is caused by H2 molecules colliding with the CO molecules. An
increase in the density H2 gas results in more collisions with the CO molecules and an increase in the CO
emission.
Another nice feature of the CO radio emission is that its wavelength is small enough (about 100 times
smaller than 21-cm radiation) that only medium-sized radio telescopes are needed to map the distribution of
the molecular clouds. Large radio telescopes can be used to probe the structure of individual molecular
clouds. There is some controversy about how the molecules are clumped together in the clouds. Is one gas
cloud actually made of many smaller gas clouds? There is some evidence that indicates that 90% of the H2 is
locked up in 5000 Giant Molecular Clouds with masses greater than 105 solar masses and diameters greater
than 20 pc. The largest ones, with diameters greater than 50 parsecs, have more than a million solar masses
and make up 50% of the total molecular mass. Other studies indicate that the giants are actually made of
smaller clouds grouped together into a larger complex.
98
Introduction: Since the "discovery" of the first dense molecular
clouds 30 years ago, it had been suspected that over 99% of the
molecular matter in these clouds was bound up in the "invisible" H2
molecule. Similarly, H3+, also invisible to radio observations, was
expected to be the cornerstone of the chemistry that forms most
other molecules in these clouds. In the intervening years,
astronomers had resorted to using CO as a tracer of H2, and
molecular ions like HCO+ and N2H+ to estimate the relevance of
H3+. In recent years, the advances in infrared and submillimeter
photometry and spectroscopy has highlighted the importance of
these molecular clouds to the process of star formation, for which
we have only a prescriptive understanding.
Where is the molecular hydrogen? Although H2 is thought to
encompass well over 99% of the molecular material in an starforming cloud, it is very unfortunately nearly invisible in its cold,
quiescent form. Why is this? Symmetry; H2 is a homonuclear
diatomic molecule. Although this makes it a simpler system to study
from a quantum mechanical point of view, its symmetry also
naturally places the center of charge and the center of mass at the
same exact location. The electric dipole moment of the molecule is
therefore identically zero, and so there are no permitted dipole
transitions available by which the H2 molecule can radiate. It does
not therefore couple to a radiation field and therefore has does not
glow. At high gas temperatures, or if subjected to an ultraviolet
radiation field, it can radiate weakly via forbidden electric
quadrupole transitions and can be observed directly in the infrared.
However, only the tiny gas fraction at 2000 degrees Kelvin can
radiate in such infrared spectral lines; the vast majority of the gas is
much cooler (10-50 Kelvin) and remains invisible.
However, even if H2 cannot glow by
its own light, it is in principle possible
to detect H2 in absorption via those
same intrinsically-weak infrared
quadrupole lines or strong electric
dipole lines in the vacuum ultraviolet.
Ultraviolet detection of H2 was
performed by the Copernicus satellite
in the mid-1970's toward very tenuous
diffuse interstellar clouds, and
attempted in the 1980's via infrared
line absorption towards dense
molecular clouds, but high-resolution
infrared spectrometers were not
sensitive enough to detect the very
weak H2 lines. With substantial recent
improvements in the sensitivity of
99
infrared detectors and high-resolution
cooled array grating spectrometers,
we can now detect cold H2 in dense
clouds for the first time!
Like H2, the pivotal H3+ ion is similarly invisible. H3+ is expected to be responsible for
initiating the complex ion-neutral reactions that lead to the formation of most other
molecules. Detection of this ion is critical for verification of our basic understanding of
interstellar chemistry. Furthermore, H3+ results from nearly every cosmic-ray photoionization
of H2, so measurement of H3+ can be used as a direct probe of the cosmic ray ionization rate
in dense clouds, which is a critical and fundamental quantity for which no direct measurement
exists! Via the same absorption line techniques as used for H2, we are also able to directly
search for H3+ and can probe directly the physics of molecular clouds, addressing very basic,
fundamental questions about their structure and physical processes that have not been
previously and directly studyable.
So how do we do it? Young "protostars" are heavily embedded in the shroud of dusty
molecular material from which they form, so they are not seen in visible light. However,
longer-wavelength infrared light can penetrate these dusty cocoons, and these young stellar
objects are readily detected as luminous infrared sources. We use these sources as simple
"candles", as background light sources that can be used to probe the circumstellar and
interstellar material in absorption along the line of sight from us to the star. One such source
is the embedded massive star, unimaginatively called "IRS 2" (InfraRed Source #2) in NGC
2024, the Flame Nebula immediately adjacent to the famous Horsehead Nebula and the left
(east) most belt star of Orion. The following is a multi-wavelength overview of the region.
Looking towards the bright infrared source IRS 2 with NOAO's new Phoenix spectrometer (a
cryogenic high-resolution echelle spectrometer for use from 1-5 microns) at the Kitt Peak 2.1meter, we have made the first simultaneous detections of cold H2 and H3+!
100
Furthermore, we can also easily observe CO along the same lines of sight in absorption,
allowing us to compare the structure, excitation, and abundances of the infrared
measurements of CO to more conventional emission-line (sub)millimeter wave studies of
CO! Notice that many CO lines can be observed simultaneously, versus having to observe
one line at a time at radio wavelengths, each time with a different receiver and often a
different telescope!
101
Thus high-resolution infrared spectroscopy has an important role in interstellar
studies: crucial non-polar molecules like H2 and H3+ can be observed, and the
abundances and excitation conditions can be cleanly referred to the same
milliarcsecond absorbing column of gas. This allows much more accurate
measurements of the environments of star formation than can be achieved by radio
techniques.
Stars form in the molecular clouds. If the molecular cloud is cold and dense enough it can collapse under its
own gravity. Smaller fragments can form and produce stars (see the stellar evolution document for further
details). The Milky Way has about 2.5 billion solar masses of molecular gas with about 70% of it in a ring at 4 8 kiloparsecs distance from the center. (1 kiloparsec = 1000 parsecs and the Sun is about 8 kiloparsecs from the
galactic center.) Not much molecular gas is located at 1.5 - 3 kiloparsecs distance from center but about 15% of
the total molecular gas mass is located close to galactic center within 1.5 kiloparsecs from the center. Most of
the molecular clouds are clumped in the spiral arms of the disk and stay within 120 pc of the disk midplane.
102
The hydrogen molecule, H2, is by far the most important interstellar molecule - but unfortunaly it has
no spectral lines in either the IR or radiofrequency wavebands, which would otherwise make its distribution
easy to study. The galactic distribution of H2 can however be determined indirectly by observations of the
second most abundant molecule, CO; its abumdane relative to the H2 molecule is [CO]/[H2] ~ 10-4.
Essentially the more abundant hydrogen molecules collide frequently with the CO molecules, and transfer
their information (temperature, velocity etc) to the CO molecules. The CO molecules radiate at a wavelength
of 2.6 mm (115.271 GHz (1 GHz - 1000 MHz - therefore the CO radiates at 115271 MHz)).
The energy states in molecules are essentially determined by the electrons, corresponding to their
quantum numbers. The simplest case are diatomic molecules - such as CO or CS. The important transitions at
radio frequencies are rotational transitions due to the roational quantum number (J). For cool molecular
clouds, transitions between the lowest rotational levels are important.
Diatomic molecules are the snmallest molecules that can exist, just two atoms held together by a chemical
bond. Bonding is caused when two atoms actually share a common electron. The diatomic molecule can rotate
or tumble, and the bond can vibrate, or periodically change length with a very high frequency. The tumbling
speeds of a diatomic molecule are quantized just like the energy levels of an atom, and so are the allowed
frequencies of vibration. A tumbling diatomic molecule can change its spin rate by emitting or absorbing
microwaves. It can change its vibrational speed by emitting or absorbing ultraviolet and visible light, and more
complicated molecules can change their vibrational modes by absorbing or emitting infrared. This is why
water vapor in our atmosphere helps to trap heat radiation.
103
Molecules will be formed that reflect the relative abundances of materials available in space for buliding such
molecules, which are listed below.
Cosmic Abundances in Interstellar Medium
Element
Number of atoms relative to 1 EE 6 H atoms
H
1,000,000
He
100,000
C
450
N
91
O
740
Ne
140
S
19
Fe
33
Ca
2.2
Tracer of Evolution
The appearance of the spectrum of a molecular cloud can be used to trace its age, or evolutionary state:
104
A spectral survey for three sources in a high-mass star-forming region (W3), in one of the small windows of the
submillimeter wavelength region observable from the ground .
The source in the top panel is undergoing infall from its parent molecular cloud and, at the same time, has a
massive outflow. This is reflected in the spectrum through the prominence of sulfur-bearing molecules and a
relatively small number of lines.
The star in the centre of the source in the middle panel has heated its environment. Complex molecules like
alcohols have evaporated from the icy mantles of dust grains. Rotational lines from these molecules dominate
the spectrum. The elevated temperature also causes many other molecular lines to be visible.
In the bottom panel, the star has broken free from its parent cloud and radiate from the back. The resulting
spectrum shows few lines of mainly very simple molecules.
Frequencies of Molecular Line emission (example of a rigid rotor)
First, derive the expression for the frequency of a diatomic molecule;
105
m1
r1
r2
X
m2
centre of
gravity
The kinetic energy of a diatomic molecule is given by
KErot =
where
1 2
Iω
2
I= moment of inertia = m1r12 + m2r22 = µ ro2 and
µ=
m1m2
m1+ m2
[Note this result if for a diatomic molecule … formally the definition of moment of inertia is:
I = ∑ mi ri 2
thus as we form longer linear molecules, the atoms at the end are further from the centre of mass, and there are
more terms to add – because of the larger number of atoms. The MOI thus increases, B decreases, and the
rotational spectrum is at lower frequencies, with the lines more closely spaced]
[Note - the appearnce of this term demonstrates that molecular spectra allow observation of different isotopes;
for example there is a shift of a factor of 1.046 between the lines of 12C16O and 13C16O, corresponding to
the reduced mass of 6.86 and 7.17. We will see later that because of the different abundances of the
isotopomers, we can observe the same molecule in two transitions for which everything is the same (such as
Tex) except the opacity - this will be important in determining column densities]
The angular momentum, L , of the molecule about its centre of mass is given by;
L = Iω
Now the rotational states of molecules are quantised so that the angular momentum can only have
discrete values given by;
KErot =
af =L
1 2
Iω
Iω =
2
2I
2
2
2I
If there are no forces acting on the molecules, then the moment of intertia is:
I = µr 2
(µ is the reduced mass given earlier, and r is the interatomic distance)
The quantisation of L for a rotating system is given by
106
a f
L2 = J J + 1 h 2
(where J = 0, 1, 2 … etc is the rotational quantum number)
Then the KE becomes:
KErot =
a f
J J + 1 h2
2I
This produces a ‘ladder’ of freqencies that radiation is emitted at, with the spacing increasing with J.
The transition energy between an upper level J and a lower level (J - 1) is given by:
a f a f
J J + 1 h2
J − 1 J h2
−
2I
2I
∆E = EJ − EaJ −1f =
cJ + J hh − cJ − J hh
2
=
=
2
2
2I
2
2I
2 J h2 J h2
=
2I
I
Now, since I = µr2, then:
h
1
∆E = J F I
H2π Kµr
2
2
=
Jh
2
4π 2 µr 2
and since ∆E = hν
ν=
Jh
4π 2 µr 2
So, for CS, the reduced mass can be written in terms of the proton mass as:
m=
12 m p × 16 m p
c12 m
p
+ 16 m p
h≅ 7 m
p
given that the interatomic separation is about 1 angstrom:
c
ICS = µr 2 ≈ 7 × 1.7 × 10 −27 × 1 × 10 −10
h≈ 10
2
−46
kg m 2
and the energy difference for a transition from the first excited state to the ground state is:
c10 h~ 10
∆KE ~
−34 2
10 −46
−22
Joules ~ 10 −3 eV
Lets now look at an example; a small part of the millimetre wavelength part of the spectrum towards a
molecular cloud is shown below.
107
30
Antenna Temperature (K)
10
20
0
114.4
114.5
9
Frequency GHz (1 GHz = 10 Hz)
114.6
It is believed that the line is due to the ground state rotational emission of a common diatomic
interstellar molecule. It is believed that carbon must be one of the atomic constituents of the molecule; by
assuming the bond length of carbon bearing diatomic molecules are typically 1.131 x 10-10 m, determine the
atomic weight and identity of the other atomic species, and hence the identity of the molecule.
Now, solve for the line - since its the ground state, J= 1;
µ=
1 x 6.627 x 10−34
2
11
4π x 1.1451 x 10
−10 2
x (1.131 x 10
)
= 1.146 x 10−26
or in amu;
µ=
1.146 x 10
1.67 x 10
−26
−27
= 6.862
Now, assuming C is one of the atoms, therefore we can solve for the atomic mass of the other species;
µ=
m1 m2
m1 + m2
∴ 6.862 =
12 x m2
12 + m2
12 m2 = 82.344 + 6.862 m2 ∴ m2 = 16
Thus, the atomic weight of the other atom is 16, therefore the identity of the atom is O, and the
molecule is CO.
Some molecular lines of importance to astronomers are:
OH
H20
CO (J= 1→0)
CO (J= 3→2)
108
1612,1665, 1667,1720 MHz (λ ~ 20 cm)
22235.1 MHz ⇒ 22.235 GHz (λ~ 1.35 cm)
115.271 GHz
345.796 GHz
HCN
HCO+
88.631 GHz
89.189 GHz
(J= 1→0)
(J= 1→0)
(X-ogen )
Unfortunately, H2, the most abundant molecule,, has no dipole moment, and to date has only been observed by
difficult to observe lines in the ultraviolet part of the spectrum (invisible from the Earth's surface because of
the absorption in the Earth's atmosphere), and in the near-infrared. These lines arise when J changes by 2.
The lines arise from a small quadrupole interaction
Vibrational Spectra
In addition to rotational transitions, it is also possible to have vibrational transitions, the classical analog of
this is two balls joind by a spring. Vibrational transitions are often seen at infrared wavelengths, but often
occur in parts of the spectrum that are inaccesible from the ground due to atmospheric absorption. To first
order, the vibrational frequeny comes from Newton's Laws of Motion:
ν=
1
2π
k
µ
where k is a constant that depends on the stiffness of the 'spring', and µ is the reduced mass we discused
earlier.
Excitation of ISM material
An atom may be excited to a higher energy level in two ways; radiatively or collisionally - i.e.
radiation may shine on an atom increasing its energy, or the atom may collide with another particle (including
electrons) and gain energy. An atom generally remains in an excited state for a short time (~ 10-8 secs), before
re-emitting a photon.
Radiative excitation occurs when a photon is absorbed by the atom; the photon's energy must
correspond exactly to the difference between two energy levels of an atom. This produces absorption lines
superimposed on a background continuous spectrum. Collisional excitation takes place when a free particle
(an electron or another atom) collides with another atom giving up part of its KE (to the atom). The atom is
collisionally excited to a higher state. Such an excited atom returns to its ground state by producing an
emission line spectrum. A radiative de-excitation is also termed a spontaneous transition.
Heating in the ISM
Temperature is an important characteristic of the ISM … but we must be careful about what we mean when we
say the word temperature. We have discussed kinetic temperature which depends on the distribution of atomic
or molecular speeds, and excitation temperature, which depends on the distribution of the electrons between
the various energy levels of the atom or molecule. As an example, you will remember, for example, that we
said radiation could change the populations of the levels, without affecting the kinetic temperature.
In the ISM, the kinetic temperature varies between a few K and a few million K. It is important in determining
molecular formation, which is important for cooling of the ISM – or maintaining the thermodynamic balance.
So, how is the ISM heated and cooled ?
a) Heating
The energy sources available are electromagnetic radiation, cosmic rays, shocks, gravity, X-rays and chemical
heating. The most important form of electromagnetic radiation is ultraviolet radiation from young stars. For
example HII regions often lie close to their associated OB star clusters.
Starlight may cause ionisation and dissociation – leading to heating. On process is photoionisation:
A + photon
109
Æ (A+)* + e
Where A is and atom or molecule, and A+ is the ion. The ion may well be left in an excited state A+* , if the
ejected electron is removed from an inner orbital. If A represents an atom, it can only be excited electronically,
however, if A is a molecule, the photon's energy is used to ionise the molecular ion may be left excited
electronically, vibrationally or rotationally … so the photon's energy not only inises the molecule, but also
excites it … with the rest of the energy being taken away by the ejected photoelectron. Thus, in this situation:
hν = I0 + ∆E + meν2
where meν2 is the KE of the photoelectron, I0 is the ionisation energy and ∆E is the increase in internal
energy of the atom or molecule.
The KE directly contributes to the kinetic temp of the ISM, and becomes 'shared' amongst other particles –
often as a result of collisions. So the energy created is spread out into the general ISM.
A further form of energy input is possible as a result of photoinisation … if the photoelectron is from an inner
orbital, the ion stays in an excited state, returning to the ground state by an electronic transition (as an electron
junps from a higher level to occupy the level of the ejected photoelectron). This results in the emission of a
photon, thus A+* radiates, eventually producing A+ in its electronic ground state:
A+* Æ A+ + photon
UV emission from young stars in normally limited to distances of a few pc from the stars – before absorption
in the ISM results on the UK photons being absorbed by neutral hydrogen.
De-excitation of ISM material
Atoms are always interacting with the electromagnetic field on short time scales ~ 10-8 sec. Because
a photon is re-emitted we call the process radiative de-excitation. Another form of de-excitation is collisional
de-excitation, where the colliding particle gains KE in the exchange. In most astrophysical situations the two
modes of de-excitation compete with each other and need to be considered.
Most spontaneous downward transitions occur on short time scales, certain transitions- because of
quantum mechanical rules - occur more slowly: these are called Forbidden transitions. These usually involve
jumps from excited metastable states to the ground state of an atom. Under most conditions, collisions deexcite the metastable state before it loses energy by a radiative process. Hence forbidden lines are normally
produced when the gas densities are low, so that the chances of collision are small during the time interval
between excitation and radiative de-excitation.
Ionisation
With sufficient energy to liberate an electron from a neutral atom, the atom is said to be ionised;
X + energy → X + + e−
where X is the atom. To denote neutral hydrogen (with no electrons removed) we use HI where the I
refers to the neutral state, and for singly ionised hydrogen; HII - you may sometimes also see this referred to
as H+. A similar system applies to other atoms; O++ = OIII = doubly ionised oxygen (two electrons
removed).
The energy required to ionise an atom depends on the ionisation state of the atom, and the particular
electron to be liberated.
Excitation equilibrium
The width of a spectral line depends directly on the number of atoms in the energy state from which
the transition occurs - so we want to know the fraction of all atoms of a given element that are excited to some
110
specific energy state. The two processes we have looked at - collisional and radiative excitation / de-excitation
both depend on temperature, since the mean kinetic energy of the gas particle goes as;
mv 2 3kT
=
2
2
where m is the particle's mass, v its speed, k is Boltzmann's constant and T is the gas temperature in
degrees Kelvin. The number of photons of a given energy increases rapidly with increasing temperature.
For simplicity, consider the situation where thermal equilibrium prevails and the average number of
atoms in a given state remains unchanged with time (a steady-state situation). Each excitation is, on average,
balanced by a de-excitation. From Statistical Mechanics, the number densities in the various states will then be
related by Boltzmann's Equation;
⎛ {E − Eu }⎞
nu gu
⎟
=
exp ⎜ l
⎝
nl gl
kT ⎠
where n is the number density in the level, g is the multiplicity of the level (also referred to as the
statistical weight function or degeneracy) and E is the energy of the level, the subscripts u and l refer to the
upper and lower levels respectively. Note: the bracketed term in the above equation is always negative, so that
the ratio of upper to lower number densities increases as the temperature increases, i.e.
nu
g
⇒ u as T ⇒ ∞
nl
gl
Kinetic temperature
This follows if the velocity distribution of all the particles is Maxwellian, with a probable speed
2kT
m ,
and for a system in LTE
⎛ εi ⎞
ni = no exp⎝ − ⎠
kT
the kinetic temperature governs the level populations in the relationship . The excitation temperature,
Tex is the kinetic temperature that describes the actual ratio of populations of two levels in terms of the
temperature that would give that ratio in thermodynamic equilibrium.
Thus;
nu gu bu
hν ⎞ gu
⎛ hν ⎞ ∴ 1 = 1 − k ln⎛⎜ bu ⎞⎟
=
exp−⎛⎝
exp−
=
⎝ kTex ⎠
Tex Tkin hν ⎝ bl ⎠
nl
gl bl
kTkin ⎠ gl
The spectrum is characterised by blackbody emission with T ~ 90K, and ~ 1000 K, superimposed
with rotational molecular lines in emission longward of the peak, atomic fine-structure and recombination
lines shortward of the peak. The intensities of the optically thin lines give column densities, and of the
optically thick lines, kinetic temperatures. The frequencies give velocities, and composition, and
recombination lines give electron densities, continuum emission (usually assumed to be optically thin) gives
dust temperature and masses.
The relationship between the emission, jν, and absorption κν, coefficients is given by;
111
jν = Bν(T) κ ν
where Bν is the Planck or Blackbody function.
This is also known as Kirchoff's Law. This is an important relationship that means that in a
state of thermodynamic equilibrium, the ratio of the emission coefficient, jν, to the absorption coefficient, κν,
is related by the universal blackbody equation.
where Bν is the Planck function.
At radio wavelengths, where kT >> hν, the approximation
Bν(T) ∼
can be used, so that
2ν2 kT
c2
2
jν
2ν kT
=
2
κν
c
Emission and absortion of radiation
In thermodynamic equilibrium (i.e. blackbody radiation from a cavity at temperature T with the
emission and absortion of the cavity or element the same), lets consider how radiation propagates through an
absorbing medium as a way of seeing how we can measure temperature.
The emission of radiation from an element of volume dV per unit time into the solid anle dω and in
the frequency intervl ν ... ν+dν will be the energy;
jν dν dV dω
the emission coefficient jν, depends on the frequency ν as well as the state and nature of the emitting
material (chemical composition, temperature and pressure). The overall energy output of an isotropically
emitting volume element dV in unit time is;
∞
dV 4π
∫ jνdν
0
This can be compared with the energy loss by absorption which a beam of intensity Iν experiences in
passing through a layer of matter of thickness ds, and is given by;
dIν
= −κ ν Iν
ds
The coefficient κν, is also termed the absorption coefficient. Combining the changes in intensity
along a path of length ds, we have both emission and absorption;
dIν
= −κ ν Iν + jν
ds
using the optical thickness;
dτ ν = κ ν ds
112
then
dIν
= − Iν + Sν
dτ ν
where Sν is known as the source function, and is the ratio of the emission and absorption coefficients;
Sν =
jν
κν
The source function
What is the source function? Consider the radiative transfer in a black body. By definition, this is a volume of
gas in perfect thermal equilibrium, in which nothing will change with time. A beam of light passing through
such a gas volume will therefore not change either. Inder these circumstances:
b
g
dI λ
= −κ λ + κ λ Sλ = κ λ Sλ − I λ = 0
ds
or:
I λ = Sλ
Since the blackbody, Iλ equals the Planck function, Bλ , then for a blackbody in thermal equilibrium,
Sλ = I λ = Bλ
i.e. in complete thermodynamic equilibrium, the source function equals the Planck function, or, in other
words, the emissivity
ε λ = κ λ Bλ
which is the well known fprm of Kirchoff’s Law, and
Bλ =
2hc 2
λ
1
5
e
hν
λkT
−1
Emission from a line
Now, for a particular species, the energy emitted by downward transitions from level u to l per unit
volume per second per Hz, is related by
4 π jν = nu hν Aul Φn
where
nu = no density in the upper state
Aul = Einstein A for spontaneous down transitions
Φν = probability of a photon being emitted between ν and (ν+dν)
now combining a), b) and c)
κν =
nu Φν ⎛ hν ⎞
A c2
2 ⎝ kT ⎠ ul
8π ν
Optical depth along a path length s, is defined;
113
τ=
∫
∞
0
and by definition
s
τν dν = ∞ dν κ ν ds
∫
∫
0
∫
∞
0
0
Φν = 1
then we can write
τ=
Aul c2 h ν nu
8 π v2 k T
=
Aul c2 h Nu
8 π vk T
where Nu = column density in the upper state = ∫ nu ds.
For an optically thin line, Tline - Texτ. So, replacing τ in the last Eqn, where T in the hν/kT is Tex,
then;
Aul c2 hν
Nu
Tline =
8πν2 k
++++++++++++++++++++++++++++++
Optically thick emission lines ⇒
Optically thin lines
Linewidths
Line Doppler shifts
Line frequencies ⇒
Tex ->> Tkin ->> thermometer of the gas
temperature
⇒
column density of the material
⇒
estimate of amount of turbulent motions in the gas
⇒
kinematics of the gas
tracer of the chemistry
Excitation temperature Tex of levels j and (k > j) is defined by;
Nk gk
hν ⎞
=
exp ⎛ −
⎝
Nj
gj
kTex ⎠
where
n is the frequency
N are the level populations
g are the degeneracies
Radiation temperature TR is the observed intensity of a line in excess of the cosmic microwave
(CMB);
⎛ c2 ⎞
⎟⎟ (I − I )
TR = ⎜⎜
⎝ 2 k ν2 ⎠ ν 2.7
where
114
Iν is the specific radiation intensity
background
I2.7 is the intensity of the CMB
For an homogeneous cloud of optical depth τ, and constant Tex, assume a source function Sν;
Sν =
3
2hν
2
c
1
hν
⎧ exp⎛
⎞ − 1⎫
⎬
⎨
⎝
kT
⎭
⎩
ex ⎠
and
I2.7 =
2 h ν3
2
c
1
hν
⎫
⎞
⎧ exp⎛
⎨⎩
⎝ k 2.7 ⎠ − 1⎬⎭
and a radiative transfer equation;
Iν = I2.7 exp (-τ) + Sν (1 - exp (-τ))
∆TR =
hν ⎧
1
1
⎫ (1 − exp{−τ})
−
⎪
⎪
k ⎨ ⎛
hν
hν
⎞
⎞
⎤⎥ − 1 ⎬
⎤
⎡
⎛ exp ⎡⎢
⎪ ⎝ exp ⎢ kT ⎥ − 1⎠
⎪
⎠
⎝
⎣ 2.7 k ⎦
⎩
⎭
⎣ ex ⎦
now since
hν
ν
⎞ 1
≈ 5 ⎝⎛
kT
100 GHz ⎠ T
it will therefore be <<1 for realistic cloud temperatures of ~ 15K, and so we are in the Rayleigh-Jeans
regime. Here for example, v ~ 100 GHz, so that
hν ⎧
1
⎫ ≅ 0.8
⎪
⎪
k ⎨⎧
hν ⎤
⎫
⎬
⎡
−
1
⎬⎭ ⎪
⎪ ⎨ exp ⎢ k 2.7 ⎥
⎩⎩
⎣
⎦
⎭
which is small compared to the first term, and can therefore often be effectively neglected.
Thus;
∆TR = (Tex − 2.7) (1 − exp{−τ})
by assuming that the optical depths are proportional to the abundances, that the excitation
temperatures are equal and that τ (CO) >> 1 and τ (13CO) < 1, then for CO,
12
∆TR = (Tex − 2.7) (1 − exp{−τ[ CO]})
12
∴ Tex = ∆TR ( CO) + 2.7
and for 13CO
115
13
13
∆TR ( CO) = (Tex − 2.7) (1 − exp {−τ( CO)})
note - in practice the CMB temperature may be << Tex
if we use Tex from the 12CO data, and solve for τ (13CO), then;
13
⎛
∆TR ( CO) ⎞
τ ( CO) = − ln ⎜ 1 −
⎟
12
⎝
∆TR ( CO) ⎠
13
Relationship of the Einstein A and B coefficients
116
Lets consider the equilibrium conditions which is set up for an atom, where Aul is the Einstein
coefficient for a spontaneous transition between the two states, Bul is the Einstein stimulated emission
coefficient for a transition induced by radiation of energy density Uνdν in the frequency interval ν to ν + dν .
Level u
A ul
Bul Uν
Blu U ν
Level l
The equation relating the upward and downward transitions can be written;
N l Blu U ν = N u Aul + N u Bul U ν
Uν has the value for an emitting source;
hν
8π hν
4π
kT
(
e
− 1) =
Bν (T )
3
c
c
3
Uν =
117
rearrange to give Uν in the form
Uν
Blu Nu ⎛ Aul
⎞
⎜
=
+ U ν⎟
⎝
⎠
Bul Nl Bul
⎛ Blu Nl ⎞⎟ Aul
=
thus Uν ⎜
+ Uν
⎝ Bul N u ⎠ Bul
and therefore
Aul
1
Bul ⎧ ⎛ Blu Nl ⎞
⎫
⎨ ⎜⎝ B N ⎟⎠ − 1⎬
⎩
ul u
⎭
Uν =
from the Boltzmann equation in LTE
Nl
gl
=
N u gu
1
hν ⎞
exp − ⎛⎝
kT ⎠
so substitute for Nl / Nu into the previous equation from the Boltzmann equation given above
∴ Uν =
Aul
1
Bul ⎧ Blu gl
1
⎫
− 1⎪
⎪B g
⎨ ul u exp − ⎛ hν ⎞
⎬
⎪
⎪
⎝ kT ⎠
⎩
⎭
now since guBul = glBlu in LTE, Blu can be eliminated, and substituting for Uν,
Aul ⎡
1
8πhν3
=
⎢
Bul ⎢ ⎧
hν ⎞
hν ⎞
− 1⎫⎬
− 1 ⎫⎬
c3 ⎧⎨ exp ⎛⎝
exp − ⎛⎝
⎨
⎠
⎠
⎭
⎭
kT
kT
⎩
⎣⎩
3
∴
8πhν
3
c
=
⎤
⎥
⎥
⎦
Aul
Bul
and hence
Aul =
8πhν3
c3
Bul
Heating of gas in the Interstellar Medium
A dense cloud or cloud core in a region of massive star formation is exposed to UV ligt from the
newly formed OB star which ionises surrounding gas and forms an HII region. The ejected electrons are able
to inject energy (by collisions) into the cloud, and to heat it. We have already studied this in the section on HII
regions.
118
Another important way of heating cloud material are a set of heating processes which include the
conversion of mechanical supersonic energy sources (stellar winds and outflows, supernaova explosions) into
thermal energy via shocks. Like radiative heating, shocks shoud predominantly heat the edges of clouds.
Energetic particles (cosmic rays from ~ 1 ⇒ 100 MeV) and hard X-rays (> 250 eV) penetrate much
deeper into clouds and significantly contribute to the heating there by collisional impact ionisation. A cosmic
ray (usually a proton) hitting a neautral hydrogen atom, for instance, produces a proton and a fast electron;
H + c.r.(1 ⇒100 MeV ) → H + + e − (Ekin ~ 35eV ) + c.r.
In the case of molecular instead of atomic hydrogen, the molecular ion H2+ is produced. We will see
later that this is a key ion in the interstellar chemistry that dominates the chemical structure of the Interstellar
Medium. Heating by cosmic rays can plausibly account for the temperatures of the bulk of cold non-star
forming molecular clouds with T ~ 10 K.
X-rays photoionise and heat the gas in much the same way as cosmic rays. In the solar
neighbourhood X-ray ionisation is much smaller than cosmic ray ionisation and plays a minor role. Howver Xray heating may be more important near OB and T-Tauri stars and associations, where it may significantly
contribute to cloud ionisation and heating.
If magnetic fields are present and move (slip) relative to the bulk of material in the cloud (e.g. during
gravitational collapse), the resulting frictional (ambipolar diffusion) heating is important.
119
Magnetism can hinder the contraction of a gas cloud, especially in directions perpendicular to the magnetic
field (solid lines). Frames (a), (b), and (c) trace the evolution of a slowly contracting interstellar cloud having
some magnetism
Cooling of gas in the Interstellar Medium
Gas cooling in PDR's is primarily by fine structure lines of OI (63µm), CII (158µm) and SiII (35µm).
The absolute and relative line strengths of these lines provide the best diagnostic which can distinguish
between excitation by photo ionisation as opposed to collisional excitation by shocks;
Starting from the equation of statistical equilibrium,
nl {ne rlu + U ν Blu} = nu {ne rul + Uν Bul + Aul }
where nl, nu, and ne represent respectively the populations of the lower and upper levels
and the electron density, and assuming additionally that terms containing Uν are negligible (cf. collisions) and
~ 0.
thus we can rewrite the equation given previously for the balance of excitation and de-excitation;
nl ne rlu = nu ne rul + nu Aul
and therefore by manipulation
nu
ne rlu
ne rlu ⎧
1
⎫
=
=
nl
(ne rul + Aul )
Aul ⎪⎨
ne rul ⎪⎬
⎪ 1+ A ⎪
⎩
ul ⎭
Now the cooling rate will be
Λc = nu Aul hνul
and substituting in for nu, we find since
nu =
ne nl rlu ⎧
1
⎫
Aul ⎪⎨
ne rul ⎪⎬
⎪ 1+ A ⎪
⎩
ul ⎭
that
1
⎫
Λc = ne nl rlu hνul ⎧⎪
⎪
n
r
e
ul
⎨ 1+
⎬
⎪
Aul ⎪⎭
⎩
The shape of the cooling function for interstellar gas is shown below. It is clear that ionised gas (T >
104 K) is the major coolant.
120
121
Chapter 8
Stars are fundamental objects in astronomy. Their formation or birth is a complex process spanning;
•
•
•
12 orders of magnitude in mass (1011 ⇒ 10-1 Mo)
12 orders of magnitude in linear size (1023 ⇒ 1011 cm)
7 orders of magnitude in temperature (10 ⇒ 108 K)
and involves many diverse physical phenomena.
The first issue confronting a discussion of star formation (SF) is the origin of stellar masses. To form
these requires either;
•
some process if hierarchical fragmentation which preferentially leads to the formation of
objects with masses ~ 10-1 to 100 Mo.
As an interstellar cloud collapses, gravitational instabilities cause it to fragment into smaller pieces. The
pieces themselves continue to collapse and fragment, eventually to form many tens or hundreds of separate
stars.
___ or ___
•
star formation is essentially an accretion process, with stars being built up by the
gravitational collapse of molecular clouds, and that this accumulation is eventually halted by some
process, such as a stellar wind, which is itself triggered by the onset of thermonuclear fusion.
Lets review the observations and various lines of evidence.
•
Bimodal Star Formation
Bimodal star formation is the idea that the birth of high and low mass stars may
involve different mechanisms. As we will see later, this has a firm observational basis.
However, although the idea of a temporal sequence of different mass stars remains
controversial, the spatial distinction between regions of high and low mass star formation is
certain. Stars only form in dense molecular clloud cores; it is the properties and environment
of these dense cores that preferentially leads to high or low mass star formation.
•
Initial Mass Function IMF
The astronomer Salpeter was the first to empirically derive the initial mass function
from the observed stellar luminosity function in the solar neighbourhood. He found the the
number of stars born with masses between m and m + dm was;
122
f (m) dm ∝ m
−2.35
for stars in the mass range 0.4 - 10 Mo.
This would suggest SF is contagious - ultimately spreading throughout molecular
clouds as gravitational collapse is propagated by compression, for example through
supernova shocks or stellar winds. The problem is that this predicts the collapse of clouds to
form protostars begins "outside - in", whereas current evidence suggests the converse.
•
Binary star and bound cluster formation, and hierarchical fragmentation
- The majority of stars form in binary or multiple systems
- Requires the fragmentation of a rapidly rotating and collapsing clouds core.
However observational objections to this are that large-scale free-fall motions are not
common in molecular clouds.
•
Efficiency of SF
In order to form a bound stellar cluster (such as the Pleiades) the star formation
efficieny must be ≥ 50%. In practice only ~ 10% of stars are born in bound clusters. For the
rest of the Galaxy, the star formation efficiency is ~ 2%, producing ~ 3 - 5 Mo yr-1. [This
should be compared to the rate of return of gas to the ISM via mass loss from evolved stars
of ~ 1 Mo yr-1.
Properties of Molecular Clouds
•
Sizes
Determined from CO observations
Dwarf molecular clouds - little SF, may survive several galactic rotation periods, lifetimes ~
108 years.
GMC's with hot cores - assembled and dispersed as they cross spiral arms on timescales ~
107 years.
•
Mechanical balance
Most clouds have masses » Jeans Mass, a critical mass above which a cloud must fragment.
This poses a problem; if all clouds seen in the Galaxy (~ 5000) are collapsing on a typical free-fall
timescale
1
⎛ 3π ⎞ 2
⎟ ≤ 10 6 years
t ff ≈ ⎜
⎝ 32 G ρ ⎠
then the star formation rate would be
9
SFRtheoretical =
Mclouds 4 10
≅
≅ 1000 Mo yr −1
t ff
4 106
whereas in fact the observed rate is ~ 3 Mo yr-1.
∴
either
a)
Mclouds is overestimated
Unlikely - not observationally supported
123
b)
Clouds are supported against free-fall collapse
c)
SF ceases after only a small amount of mass is converted
to stars
Unlikely since SFE > 50% in Ophiucus, and >
10% in many others.
d)
We live in strange times - anti-Copernicun.
Therefore b) is the most likely
•
Mechanisms which can stabilise clouds against collapse
Simple star formation
In these panels a simplified model of low-mass star-formation is presented. Attention will be
paid to the role of water and the capabilities of HIFI. Pictures courtesy of M. Hogerheijde,
based on Shu et al. (1987).
In the phase that molecular cloud cores condense out of the more tenuous gas, it
becomes impossible to look through the cloud in visible light. The longer
wavelengths accessible to HIFI will, however, still penetrate through the whole
cloud and provide crucial information on its chemical composition and physical
conditions. On the surfaces of dust grains water ice mantles are forming.
124
When infall proceeds, the densities and temperatures rise, allowing more
chemical reactions to proceed, including ones leading to water. The icy grain
mantles will slowly evaporate enriching the molecular gas, giving rise to a
complex chemistry. HIFI will be able to measure the majority of the species in
the gas, and the dynamics of the collapse.
Infall proceeds at a higher rate and the bipolar outflow reaches observable
scales. This shock-chemistry will be traced by observing specific molecules
produced in this environment. Water is certainly one of them. The high spectral
resolution of HIFI will be used to trace the dynamics of the process.
The star has formed and sends out copious X-ray radiation. The specifics of the
particular chemistry resulting from interaction with these hard photons are
directly detectable by HIFI.
125
The molecular disk, now tenuous, is still within reach of HIFIs observing
capabilities. The onset of planetary system formation may be observable
A planetary system may be left-over from the star-formation process. Planets
will not be detected by HIFI outside our own Solar System, but Solar System
objects will be observed with HIFI in unprecedented spectral detail.
Star Formation mechanisms
There are many examples we see where the effects of a first generation of star formation can lead to
second generations being formed (and more). Such a process has been called Sequential Star Formation, and
descrobes the way that once star formation has started, its effects infectiously lead to conversion of a large
126
fraction of the mass of a molecular cloud. This process is by no means always the case, but some aspects of it
are present in almost all high mass star formation regions.
127
A false color image of the teardrop shaped HST 10 star-disk system and immediate neighbors, a silhouetted
disk (top left) and a second star-disk system (bottom right). At the center of HST 10 lies a dark nearly edge on
disk with a diameter approximately the same as Pluto's orbit. Surrounding the system is diffuse hot gas which
has been evaporated from the disk surface. We are witnessing the destruction of a circumstellar disk which if
otherwise left alone would be a strong candidate for producing planets.
The center of the Trapezium cluster showing the four massive energetic stars and a number of evaporating
proto-planetary disks.
128
A gallery of star-disk systems in Orion's Trapezium. The first four objects are being evaporated by the central
massive stars, while the last two disks are visible in silhouette against the background nebula.
Chapter 9 Cloud Collapse
129
Artist's conception of the changes in an interstellar cloud during the early evolutionary stages outlined
previously. Shown are a stage-1 interstellar cloud; a stage-2 fragment; a smaller, hotter stage-3 fragment;
and a stage-4/stage-5 protostar. (Not drawn to scale.) The duration of each stage, in years, is also indicated.
The Jeans Criterion
Assume we start with an isolated sphere of gas
Free fall collapse time
Consider a test particle of mass m at the edge of a cloud of mass M, radius R, initial density ρo.
If the particle falls towards the centre, it moves inward along an elliptical orbit whose semi major axis, a = R / 2, and
eccentricity = 1 (i.e. in a straight line towards the centre). Then,
4πR 3 ρ o 4π (2a ) ρ o 32πa 3 ρ o
=
=
3
3
3
3
M =
130
From Kepler's third Law;
4π
P
=
a3 GM
2
2
4
and since M = π R3
3
⎛ 3π ⎞
⎟
∴ P=⎜
⎝ 8Gρ o ⎠
0.5
now, defining the free-fall time , tff = P/2, so;
⎛ 3π ⎞
⎟
t ff = ⎜
⎝ 32Gρ o ⎠
0.5
For the sun this is about 0.44 hours (26.6 minutes), and for a typical molecular cloud ~ 105 - 106 years.
Now we can re-write this another way since
M=
then substituting for ρ ,
4
π R3 ρ
3
3π 4π R 3
π
R3
t ff =
=
≅
32 x 3 GM
8 GM
R3
GM
An alternative way of looking at this is that assuming their is no internal pressure in a collapsing
cloud, a small mass at the surface will fall unhindered along a path r(t) that satisfies the differential equation;
GM
d 2r
2 =−
2
dt
r
where r = R at t = 0. We can solve this differential equation directly, or with more insight say that
during the fall of the particle over a distance R in a time t the average value of;
R
d 2r
≈− 2
2
t
dt
therefore we expect
-
MG
R
2 ≈ −
R2
t
and hence
t=
R3
GM
with the mass in Mo and radius in Ro, this can be expressed;
131
⎛ M ⎞
3
t ff = 1.59 10 ⎜
⎟
⎝ Mo ⎠
−
1
2
3
⎛ R ⎞2
⎜ ⎟
⎝ Ro ⎠
seconds
For the sun this is about 0.44 hours (26.6 minutes), and for a typical molecular cloud ~ 105 - 106
years.
Collapse and fragmentation of Molecular Clouds
We have seen that the process of star formation involves the collapse of a large interstellar cloud into
a small stellar sized one. What are the physical conditions under which collapse can occur ?
Consider the equilibrium of a single isolated, spherically symmetric cloud ( a good example in
practice may by a BOK globule) under the influence of three forces; internal pressure, self-gravity and surface
pressure of an external medium. Consider a spherical shell in a cloud of thickness dr at radius r;
r
R
r+dr
The mass of the shell is;
dM(r) = 4π r ρ (r ) dr
2
where ρ(r) is the density at radius r . [ρ(r) may vary as a non-linear
function - perhaps as r-2 ].
In equilibrium, the inward gravitational force on the shell due to the mass M(r) interior to it must be
balanced by a pressure gradient. For equilibrium;
F=
GMm
4π r 2 dP(r) = −
r2
GM(r ) dM(r)
r2
now setting;
4
3
V(r) = π r
3
132
the volume interior to r , then;
3 V(r) dP(r) = −
G M(r) dM(r )
r
(A)
Integrating this from the centre of the cloud to the edge, where it has radius R , and pressure equal to
that of the external ISM, PEXT ;
3∫
PEXT
PCEN
V (r)dP(r) = − ∫
MCLOUDS
G M(r ) dM
0
r
=
PCEN
MCLOUDS =
where
central pressure of the clouds
Total cloud mass
So first integrate the L.H.S. of the above equation by parts;
Edge
PEXT
3 ∫ V (r)dP(r) = 3V(r)P(r)
PCEN
Vcloud
− 3
∫
P dV
0
Centre
Vcloud
= 3 Vcloud PEXT − 3
∫ P dV
0
where Vcloud = 4/3 πR3 = the total cloud volume.
Now to evaluate the remaining integral from the above, remember that the Internal Energy, εi , per
unit volume of monatomic gas is given by;
εi =
3
P
2
where P is the pressure. We can then write;
Vcloud
∫
0
where
PdV =
2
3
Vcloud
∫ε
i
dV =
0
2
3
τ
τ is the thermal energy content of the cloud
P will be PEXT
Now the RHS of (A) is just the gravitational potential, Ω , of the cloud, and hence (A) can be rewritten;
3 Vcloud PEXT = 2τ + Ω
(B)
which expresses the fact that the system is in hydrostatic equilibrium under the three forces
mentioned earlier. This is a form of the well known Virial Equation.
Although P and ρ may be functions of r , we can simplify by assuming the cloud has an uniform
density ρo . Lets now see if self-gravity is important;
133
Consider an uniform cloud of average pressure Pav ;
∴ 2 τ = 3 Pav Vcloud
(C)
and from (A) the gravitational potential
Mcloud
Ω=−
∫
0
G M(r ) dM(r)
r
Now if, as we said, ρo is constant with r
4
M(r) = π ρo r 3
3
and
dM(r) = 4π ρ o r 2 dr
then we can re-write the equation for Ω ;
R
Ω = −G
∫
0
2
(4π ) ρ2 r5 dr
o
3
r
⎛ 4π ρo ⎞
1 5
= −G ⎜
R
⎟ x3x
⎝ 3 ⎠
5
2
=−
2
1
3G ⎛ 4
⎜ π R 3 ρ ⎞⎟
⎝
⎠
R
5 3
3 GM 2
∴ Ω=−
5 R
Also, remembering from (B) and (C);
1 GM 2
Ω
=−
2τ
5 R Pav Vcloud
Now using typical values of Pav ,
Ω
≈ 0.03
2τ
∴τ > Ω
and therefore the effects of self gravity are not the dominant perturbing force to affect the
stability of an uniform density cloud.
So, how do we get from typical ISM densities to those of a star ? Lets look at a simplified example of
the collapse of an isolated gas sphere (the simplest example possible). Ignoring the effects of surface pressure
on the gas sphere, then (B) becomes
134
2
τ+Ω=0
in equilibrium.
Starting from the given equation for the gravitational potential energy, Ω , of a uniform spherical
molecular cloud having mass M, and radius R,
Ω=−
3 G M2
5R
and also given
τ = 32 PV
τ + Ω < 0,
then from the virial Eqn, for collapse to occur, 2
∴
3 G M2
> 3 PV
5R
or
GM2
> PV
5R
Now, writing an equation for pressure;
P=
ρkT
µmH
and
3M
4πR3
=ρ
then
2
GM
3M kT 4πR
>
3
5R
4πR µmH 3
3
and hence
GM
kT
>
5R
µmH
Since the sound speed csound is given by;
csound =
γkT
µmH
where γ = ratio of specific heats which will be assumed to equal 1, then for collapse to occur,
GM
> c2sound
5R
If we then assume the time taken, tsound, for a sound wave to cross a distance R;
135
R
tsound =
csound
then for collapse,
GM
> C2s ound or
5R
4
1
G πR3 ρ
> c2s ound
3
5R
∴ ⎛⎝
2
⎞ > 15
csound ⎠
4πGρ
R
and hence
1
15 ⎞ 2
tsound > ⎛⎝
4π G ρ⎠
tsound ~ 2 tfree-fall.
Are there any limits to a cloud's mass or radius for collapse to occur ?
Consider a gas moleculae close to the surface of a spherical cloud. On account of its thermal
motion it has a kinetic energy:
mv 2
KE =
2
where m is the mass of the molecule and v is its speed. Because of gravitational
attraction, it will also have a gravitational potential energy:
PE = −
GMm
R
The magnitude of this is equivalent to the amount of enegry that would have to be supplied
to remove the molecule from the cloud.
In order for the molecule to escape from the cloud of gas, it’s KE must bw adequate to raise
its potential energy from –GMm/R to that appropriate to an effectively infinite separation,
i.e. a potential energy of zero. The molecule will however be bound to the cloud if this is not
so – i.e. if:
GMm mv 2
>
2
R
or
2GM
> v2
R
We now note an old relationship between kinetic temperature, T and the individual energies
of the molecules:
136
1 2 3
mv = kT
2
2
A bar should be on top of the first term – representing the fact that it is an average.
Thus:
v2 =
3kT
m
Substitution for v2 gives:
2GM 3kT
>
R
m
2GM
>R
3kT
i.e.
or rearranging:
3
⎛ 2GMm ⎞
3
⎜
⎟ >R
3
kT
⎝
⎠
Now, remember the equation for uniform density:
4
M = πR 3 ρ
3
or
3M
> R3
4πρ
So, substituting for M,
3
3M
⎛ 2Gm ⎞
3
⎜
⎟ M >
4πρ
⎝ 3kT ⎠
thus
3 ⎛ 3kT ⎞
M >
⎜
⎟
4πρ ⎝ 2Gm ⎠
3
2
81 ⎛ kT ⎞
M >
⎜
⎟
32πρ ⎝ Gm ⎠
3
2
M >
137
3
9
1
4(2πρ ) 2
⎛ kT ⎞ 2
⎜
⎟
⎝ Gm ⎠
This says that a cloud of density p and temperature T will be gravitationally bound if its
mass exceeds the quantity on the right hand side. However, such a cloud will not be stable.
It’s temperature is > than that of the 3 K background, and hence it will radiate more energy
than it receives, cooling down. As the T reduces, so will the internal pressure, and hence
gravity dominates and the cloud continues to collapse. Thus the condition that a cloud is
bound, is also that it will collapse under it’s own gravity. On the other hand, if the mass is
less than this limit, the thermal motion of the gas will overcome the gravitational attraction
and the cloud will disperse. This critical mass is called the Jeans Mass.
MJ =
3
9
1
4(2πρ ) 2
⎛ kT ⎞ 2
⎜
⎟
⎝ Gm ⎠
Similarly we can derive an equivalent radius:
Remember the equation we used above for mass,
4
M J = πR J3 ρ
3
or
3M J
= R J3
4πρ
substituting into the Jeans Mass equation:
R J3 =
3
9
4πρ 4(2πρ ) 12
3
3
33
⎛ kT ⎞ 2
⎛ kT ⎞ 2
⎜
⎟ =
⎟
3 ⎜
⎝ Gm ⎠
2 3 (2πρ ) 2 ⎝ Gm ⎠
or taking the cube root of both sides
1
1
3 ⎛ kT ⎞ 2 ⎛ 9kT ⎞ 2
⎟⎟
⎟ = ⎜⎜
R J = ⎜⎜
2 ⎝ 2πρGm ⎟⎠
⎝ 8πρGm ⎠
Thus for a given mass, this is the critical radius of an uniform density sphere that governs
whether the gas will collapse or disperse.
138
Ambipolar Diffusion
The earliest stages of star formation are characterized by a slow gravitational contraction towards a 1/r2 density
distribution, against the opposing forces of magnetic fields and turbulence. Magnetic field support of molecular
clouds is possible through the friction between neutral and charged particles: the neutral particles will follow a
straight trajectory to the centre of gravity, whilst the ionised particles spiral around the magnetic field lines
towards the poles. The magnetic field will eventually weaken because it will diffuse away from the clump with
the charged particles through the poles (ambipolar diffusion).
When enough of the field has diffused away, the molecular clump becomes gravitationally unstable and it will
collapse inside-out resulting in a central objewct called a protostar. The infalling material carries angular
momentum to the protostar, that will speed up and reach a break-up velocity soon. The solution to this long
standing problem in star formatiuon theory is the formation of a rotating disc around the protistar. Most of the
material will first fall onto the disc, and then slowly accrete onto the protostar. As matter diffuses towards the
protostar, angular momentum will be transported outwards by frictional forces (viscous disc). The luminosity is
generated from thermalised potential energy by gravitational collapse
3 GM 2 dR*
L=− ×
×
7
dt
R2
and the photons are reprocessed by the circumstellar material.
139
140
(a) When a protostellar wind encounters the disk of nebular gas surrounding the protostar, it tends to form a
bipolar jet, preferentially leaving the system along the line of least resistance, which is perpendicular to
the disk. (b) As the disk is blown away by the wind, the jets fan out, eventually (c) merging into a
spherical wind.
141
(a) This false-colored radio image shows two jets emanating from the young star system HH81-82
(whose position is marked with a cross at center). This is the largest stellar jet known, with a length
of about 10,000 A.U. (The colors are coded in order of decreasing radio intensity, red, blue, green.)
(b) An idealized artist's conception of a young star system, showing two jets flowing perpendicular to
the disk of gas and dust rotating around the star. (See also the chapter-opening photos of more
stellar jets.)
A model that has been developed on the basis of observations is the following:
142
In this figure the model for star formation may be represented by four conceptually distinct stages of
development that are the culmination of the last several decades of theoretical and observational efforts.
Within molecular clouds, there are cores of material which are denser than the surrounding cloud. Dense
molecular condensations form within these molecular cores as the loss of magnetic support through ambipolar
diffusion allows gas and dust to contract gravitationally. Eventually these condensations become sufficiently
centrally concentrated to undergo dynamical collapse; the inner regions form an evolving protostar and
surrounding disk while the outer regions form an extended infalling envelope of material (the arrows). At
some point, a bi-polar wind breaks out along the rotational poles of the system, while material continues
flowing inward along the equatorial regions. The visual extinction toward the central protostar is typically tens
to thousands of magnitudes at this point, effectively obscuring it from scrutiny at optical wavelengths. Over
time, the angle occupied by the wind broadens, removing surrounding material and halting the inward flow of
material. At this point, the system becomes detectable at near-infrared and even optical wavelengths as a star
plus a disk, commonly recognized as a T Tauri system.
143
144
145
146
Many astronomers regard the passage of a shock wave through interstellar matter as the triggering mechanism
needed to initiate star formation in a galaxy. Calculations show that when a shock wave encounters an
interstellar cloud, it races around the thinner exterior of the cloud more rapidly than it can penetrate its thicker
interior. Thus, shock waves do not blast a cloud from only one direction. They effectively squeeze it from
many directions, as shown below.
Shock waves tend to wrap around interstellar clouds, compressing them to greater densities and thus possibly
triggering star formation.
147
An artist's conception of a cloud fragment undergoing compression on the southerly edge of M20, as shock
waves from the nebula penetrate the surrounding interstellar cloud.
Atomic bomb tests have experimentally demonstrated this squeezing--shock waves created in the blast
surround buildings, causing them to be blown together (imploded) rather than apart (exploded). After shock
waves cause the initial compression of an interstellar cloud, natural gravitational instabilities may divide it into
the fragments that eventually form stars. Figure 19.18 suggests how this mechanism might be at work near
M20.
148
(a) Star birth and (b) shock waves lead to (c) more star births and more shock waves in a continuous cycle of
star formation in many areas of our Galaxy. Like a chain reaction, old stars trigger the formation of new stars
ever deeper into an interstellar cloud.
Disks and jets pervade the universe on many scales. Here, they seem to be a natural result of a
rotating cloud of gas contracting to form a star. Matter falling onto the embryonic star creates a pair
of high-speed jets of gas perpendicular to the star's flattened disk, carrying away heat and angular
momentum that might otherwise prevent the birth of the star. This image shows a small region near
the Orion Nebula known as HH1/HH2, whose twin jets have blasted outward for several trillion km
(nearly half a light-year) before colliding with interstellar matter. (HH stands for Herbig-Haro, after
the investigators who first cataloged such objects.) The next three photos show stellar jets ejected
from three different very young stars. Reproduced here to scale, these images collectively depict the
propagation of a jet through space.
149
(Inset B) This image of HH30, spanning approximately 250 billion km, or about 0.01pc, shows a thin
jet (in red) emanating from a circumstellar disk (at left in grey) encircling a nascent star.
(Inset C) One of HH34's jets is longer, reaching some 600 billion km, yet remains narrow, with a
beaded structure.
(Inset D) HH47 is more than a trillion km in length, or nearly 0.1pc. This photo shows one of its jets
plowing through interstellar space, creating bow shocks in the process.
150
Evidence for other planets
151
152
Chapter 10
Molecular Chemistry
The observation that the interstellar medium contains vast clouds of gas and thus in which are found a variety of
molecules has shown in the astronomer the rich chemistry which exists in the interstellar medium. At the time of
writing almost 100 molecules are unknown to exist in the interstellar medium. Most of these were discovered after
1970 when technological advances were made which allow astronomers to make observations at millimetre
wavelengths. It is in that this part of the millimetre waved spectrum that the rotational transitions of many
molecules can be found, and so that discovery awaited the possibility to observe spectra in this part of the
spectrum.
Diffuse clouds
The first observations of molecules in the interstellar medium were may in diffuse clouds. These are clouds are
optically thin because they have a very low-density. Among the molecules which were observed in diffuse clouds
were the ionic species CH+, C+ and other ions such as Ca+. The H+ ion is also seen, particularly in HII regions, i.e.
diffuse clouds which are irradiated by the radiation of hot young stars which are capable of ionising the gas. We
have already seen that ultraviolet radiation with energy is greater than 13.6 eV is absorbed by hydrogen. Since
hydrogen is a very efficient absorb absorber of ultraviolet radiation, it is often only the outer parts of interstellar
clouds which are heavenly ionise.
Dense molecular clouds
Chemically, these are the most interesting part of the interstellar medium, because in dark clouds and diverse star
forming regions molecules more complex than H2 can exist, and reactions between them can lead to the synthesis
of many more complex molecules, including organic molecules such as the basic amino acids.
Chemically the dense molecular clouds consists of gas, which is mostly molecular hydrogen or H2, plus atomic
helium, plus interstellar dust grains. The effect of the H2 and thus renders the clouds opaque to visible light and
radiation at shorter wavelengths. However radiation at millimetre wavelengths is able to escape from the cloud,
and the detection of its radiation at the earth allow is the astronomer to probe inside what are otherwise quite
opaque clouds.
Emission lines at 1660 MHz to OH were the first interstellar molecular radio frequency lines which were
identified. The first molecules which was detected by radio astronomy was the emission line at 24 GHz of
ammonia. This was followed shortly afterwards by the detection of water vapour emission at 22 GHz. These
detections in 1968 were followed at the year after by the discovery of the ubiquitous molecular tracer carbon
monoxide, CO, which has proven to the one of our most useful probes of the interstellar medium. Many of the
important detections of new molecules in the interstellar medium were made using a telescope in America at the
National Radio Astronomy Observatory (NRAO) at Kitt Peak in Arizona.
By far the most common constituent of molecular clouds is the hydrogen molecule H2. Unfortunately this
molecule does not unit or that's all radio radiation it only emits short wavelength ultraviolet radiation so it cannot
be easily observe. Theoreticians had expected H2 to be very common in interstellar space but proof of its existence
was very hard to obtain. In most cases astronomers observe a different molecule, such as carbon monoxide, and
then infer the presence of H2 because we know that H2 is necessary to collide waved the CO molecules in order for
them to be excited. Only when spacecraft were able to measure the ultraviolet spectra of a few stars which were
located near the edges of some dense clouds was the presence of molecular hydrogen finally confirmed.
In the absence of us being able to use H2 as a probe of the interstellar medium, interest focused on the use of
carbon monoxide as a probe of the molecular interstellar medium. Some other molecules such as hydrogen
cyanide, ammonia, methyl alcohol were also used because they were able to sample material which had a
somewhat higher density than carbon monoxide, even though they were much less numerous than the carbon
monoxide molecules sometimes by as much as 8 or 9 orders of magnitude. These molecules were however very
important because they served as tracers of the structure and physical properties of molecular clouds.
153
To develop an understanding of molecular chemistry, we need to start the story with the formation of
the simplest molecule, H2 .
Formation of H2 in molecular clouds
There is common agreement that the process of H2 formation occurs on the surfaces of grains, where
the grain forms part of a 3-body reaction, effectively acting as a catalyst;
If we assume the catalytic process for;
HI + HI → H2
is completely efficient for all atoms arriving at the grain surface, then the formation rate of
H2 is;
dn( H2 ) 1
2
= n(HI ) π a ng uH
dt
2
for grains of radius a , number density ng and velocity uH . Putting appropriate values;
dn(H2 )
≅ 2 10 −17 n(HI ) nH
dt
cm−3 s−1
where n(HI) = number density of neutral H , and nH is the total number of hydrogen
nuclei, given by;
nH = n(HII ) + n(HI ) + 2 n(H2 )
In fact, in the more diffuse (Av ~ small) H2 is readily dissociated by UV light;
H2 + hν ⇒ H + H + hν
This means that in the diffuse ISM , most of the H is neutral, whereas in the more shielded material
(for example in molecular clouds) the H is bound in the form of H2 molecules.
H2 formation on grains
The site of formation of H2 is on grain surfaces;
154
H
van der Waals
attractive force
H 'sticks' to the
grain surface
The evaporation rate in this case must be less than the arrival rate of other H atoms. If the arriving
atom is not chemically bound, then these weak forces create a potential well of depth q , where q /k ~ 300 3000 K. Therefore;
⎛
q ⎞
n(HI ) σ uH > ψ exp ⎜ −
⎟
⎝ k Tgrain ⎠
⇓ ............................. ⇓
arrival rate
where
evaporation rate
σ is the grain cross sectional area
uH is the velocity of the arriving H
ψ is a constant
q is the potential well depth
Tgrain is the grain's temperature
For grains with Tgrain < 12 K and q /k ~ 300, then this criterion is valid and a second H atom will
arrive before evaporation of its potential partner - so cool grains will be the best catalysts to promote the
formation of H2 .
of
io
n
en
tf
Ef
fic
i
20
o
mon
or
m
at
nd
om
tio
ora
40
Ev
ap
Grain temperature K
ina
H
2
tes
60
lay
,
tion
rma
er fo
is
gas
d
te
ep l e
om
d fr
IS
nto
Mo
ns
grai
Arrival rate
We have already seen in the section on dust how the radius of a grain increases with time (by
accreting material onto the grain surface).
155