The Interstellar Pressure
Astronomi 3: Planetsystemer og stjernedannelse
Joachim Mortensen
February 24, 2009
Astronomi 3: Planetsystemer og stjernedannelse
J. Mortensen
Contents
1 Introduction
2
2 Density and Pressure in the interstellar phases
3
3 Heating and Cooling of the Interstellar Gas
3.1 Heating by Cosmic Rays and interstellar Radiation
3.1.1 Cosmic Rays . . . . . . . . . . . . . . . . .
3.1.2 Interstellar Radiation . . . . . . . . . . . . .
3.1.3 Carbon Ionization . . . . . . . . . . . . . .
3.1.4 Photoelectric Heating . . . . . . . . . . . .
3.1.5 Interstellar X-Rays . . . . . . . . . . . . . .
3.2 Cooling by Atoms, Molecules and Dust . . . . . . .
3.2.1 Fine-structure Splitting . . . . . . . . . . . .
3.2.2 Emission from OI and CII . . . . . . . . . .
3.2.3 Molecules and Dust . . . . . . . . . . . . . .
3.3 A simple model . . . . . . . . . . . . . . . . . . . .
3.3.1 Heating . . . . . . . . . . . . . . . . . . . .
3.3.2 Cooling . . . . . . . . . . . . . . . . . . . .
3.3.3 Connecting the dots . . . . . . . . . . . . .
The Interstellar Pressure
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3
3
3
4
5
5
5
5
6
6
6
6
7
8
9
*1
Astronomi 3: Planetsystemer og stjernedannelse
1
J. Mortensen
Introduction
This purpose of this assignment is to give a physically acceptable derivation of the pressure-model shown
in figure 1[2].
Figure 1: The equilibrium pressure P/kB = nT as a function of the number density n of particles in the
interstellar cloud. The dashed line is the empirical mean pressure value in the interstellar medium of
3 · 103 cm−3 K.
In short, the model pertains to explain how gas of the so-called warm neutral medium (WNM) at point A
and clouds of the so-called cold neutral medium (CNM) at point C can be kept in balance by the unstable
pressure-density state at point B.
Interpreting the model is easy. At point A and C, any contraction of the gas will increase its density
and thereby also its thermal pressure which then acts to expand the gas until it has returned to the
equilibrium pressure. If the gas density decreases, the pressure drops and the gas re-compresses until the
density is high enough to maintain the equilibrium pressure.
Since these processes must occur at the points where the gas is in either of the two phases, then,
in order to have a positive slope at these two points, there must be a mechanism by which the slope
between these two states decreases and becomes negative, giving rise to a point B as showed in the
model. A discontinuoues solution would not be physically realizable, and this model is the simplest
description that meets these requirements.
If at point B the gas density falls the slightest below the mean pressure equilibrium, the pressure rises
and compresses the gas until it reaches the density at point A, where equilibrium can be maintained.
Going the other way, an increase in density at point B will yield a lower pressure assuring a continuous
contraction of the gas until equilibrium is reached again at point C.[1]
The reason that these two phases can coexist is due to the two phases maintaining the same pressure.
Equality of pressure in all the phases of the ISM is based on the theoretical assumption that the time
it takes to return to pressure-equilibrium is considerably smaller than timescales such as the mean time
between supernova shocks, recombination timescales and the cooling timescale.[3]
A simplification in the explanation is necessarily made by the fact that we are only able to detect
and measure the particle (thermal) pressure of the ISM, whereas the magnetic field pressure and the
pressure due to cosmic particles are not negligible. This also goes for the so-called “turbulent” pressure.
For a theoretical model to work, the relationship between theses types of pressure has to be properly
understood.[3]
Since the temperature of the gas is determined by the balance of the heating and cooling of the
gas, the problem lies in determining whether these processes actually purvey a reasonable and coherent
explanation that accounts for a pressure-density relation like the one in this model.[1]
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The object of this assignment therefore lies in illustrating how these processes mainly occur, simplified
as per [1], [2] and [3], and then putting them together to show, in the end, how this relatively simple
relationship is established by the empirical and theoretical groundwork described mainly in [1] and [2].
2
Density and Pressure in the interstellar phases
The density fraction of a low-density (WNM) cloud to a high-density (CNM) cloud of interstellar gas lies
in the ranges (using the points in the plot as reference to each phase):
nA
∼
nC
(
2 : 105
,
2 : 1015
(1)
with the highest densities predominantly found in areas of star formation. Assuming the above mentioned
pressure-equilibrium for the two phases of the ISM, and assuming that the thermal pressure is only type
of pressure needed to be taken into account, the equation of state (in units of the Boltzman constant) for
the two phases can be written:
P/kB = nT,
(2)
and with the left side being thus equal for both phases, dividing the high-density cloud state by the
low-density state and rearranging gives:
TA
nC
=
1.
nA
TC
(3)
Where the relative densities in the two phases given by (1) ensures the last condition holds.
Since the interstellar gas is largely composed of hydrogen, and the increase in particle density may
only increase/decrease by a factor of two, as atomic hydron either ionizes or recombinates into molecular
hydrogen, the reason for the large relative densities are mainly due to the value of n in the two phases.
And pressure-equilibrium therefore requires the reciprocal relation between the temperatures to change
likewise, as indicated above.
3
Heating and Cooling of the Interstellar Gas
The heating and cooling processes that occur in the ISM involve a series of reactions between cosmic
rays and the ISM and the collisional excitation and subsequent re-radiation respectively. Before deriving
equations that amoount to an explanation of the aforementioned plot in figure 1, I give a brief summary
of the different heating and cooling processes described in chapter 7 in [2].
3.1
Heating by Cosmic Rays and interstellar Radiation
The different reactions that cause heating of the ISM are shown in table 3.1.
3.1.1
Cosmic Rays
The most important source of heating in opaque interstellar clouds are cosmic rays (CR) consisting mainly
of protons accelerated to relativistic velocities by the magnetized shocks of supernova remnants (SNRs)
of both intra- and extragalactic origin. They mainly ionize molecular hydrogen by the reaction shown in
the table. The actual heating is caused by the ejected electron that, if having enough energy, dissociates
more H2 . Other reactions produce ions such as OH the detection of which can be used as an indirect
measure of the CR ionization rate.
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Astronomi 3: Planetsystemer og stjernedannelse
Process
J. Mortensen
Reaction
Cosmic rays on HI
Cosmic rays on H2
Carbon-ionization
Photoelectric ejection
Dust irradiation
Stellar X-rays
p+ + H −−→ H+ + e – + p+
–
+
p+ + H2 −−→ H+
2 +e +p
+
C + hν −−→ C + e –
H + hν −−→ H+ + e –
Table 1: Cloud Heating.
The volumetric rate of heating ΓCR (X) can be calculated from:
ΓCR (X) = ζ(X)nX ∆E(X),
(4)
where X denotes either H2 or HI. ζ(X) is the ionization rate (probability) of a single quantity of X, nX
the volume density of X, and ∆E(X) is thermal energy added to the gas as a result of each ionization
event.
To calculate the heat rates requires knowledge of the interstellar flux of low-energy CR. Using measurements of OH can, as mentioned above, be used to measure these rates indirectly. For H2 and HI, the
heating rates can be found to be[2]:
n HI
eV cm−3 s−1
−3
cm
n
H2
ΓCR (H2 ) = 2 · 10−13
eV cm−3 s−1
cm−3
ΓCR (HI) = 1 · 10−13
3.1.2
(5)
(6)
Interstellar Radiation
The interstellar radiation is composed of different types of radiation which becomes apparent when
looking at figure 3.1.2 from [2]
Figure 2: Mean intensity of interstellar radion as a function of frequency.
Assuming perfect blackbody radiation, the figure shows four maxima for four different components of
the insterstellar radiation field. The blackbody spectrum of the cosmic microwave background radiation
peaks at the lowest frequency. Far infrared peak comes from interstellar dust warmed by starlight. The
third component lies in the visible interval of the spectrum and consists of light from field stars. A more
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J. Mortensen
narrow maximum lies in the ultraviolet interval and is caused by SNRs and stellar winds from massive
stars. No explanation is given for the lacking data, but at the rightmost frequencies, there is a minor X-ray
contribution.
3.1.3
Carbon Ionization
The ultraviolet radiation is a source of thermal energy. The radiation ionizes C in HI clouds shown in
the third reaction. Again it is the ejected electrons which provide the actual heating. Assuming that the
carbon is mostly in neutral form, the heating rate ΓCI is shown to be:
ΓCI = 4 · 10−11
3.1.4
n H
eV cm−3 s−1
cm−3
(7)
Photoelectric Heating
Another source of heating electrons is the liberation of electrons off the surface of dust grains by the
photoelectric effect. The process causes the grains themselves to heat, although not at the same rate as
the electrons. The photoelectric heating rate ΓPE is:
ΓPE = 3 · 10−11
n H
eV cm−3 s−1
cm−3
(8)
Irradiation of the grains happens when the electrons do not leave the grains but instead raises the thermal
temperature of the grains. The heating rate Γd of the internal electrons from optical light is:
Γd = 2 · 10−9
n H
eV cm−3 s−1 ,
cm−3
(9)
but the cloud density has to be high before the internal energy of the dust grains can be transferred to
the gas by collisions.
3.1.5
Interstellar X-Rays
The last type of heating is one that only occurs locally in molecular clouds where stars are formed. The
different molecules in the gas are heated directly by excitation from the X-rays created in the newborn
stars. The heating rate ΓX is found to be (for r < rX ):
ΓX = 2 · 10−13
−8/3
n 13 LX
r
H
eV cm−3 s−1 ,
cm−3
1030 erg s−1
0.1 pc
(10)
where LX is the star luminosity in the X-ray band, and rX is the characteristic distance, within which
most of the energy is deposited, and which is defined as:
rX = 2
3.2
−1
nH
pc.
103 cm−3
(11)
Cooling by Atoms, Molecules and Dust
Clouds in the ISM cool themselves by emission from the atoms, molecules and dust grains in the gas. The
different reactions are shown in table 3.2.
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Process
Reaction
O collisional excitation
C+ fine structure excitation
CO rotations excitation
Dust thermal emission
Gas-grain collisions
O + H −−→ 0 + H + hν
C+ + H −−→ C+ + H + hν
CO + H2 −−→ CO + H2 + hν
Table 2: Cloud Cooling.
3.2.1 Fine-structure Splitting
The electrons in hydrogen and helium are described quantum-mechanically by one-particle wave functions without angular momentum (l = 0). It means that these electrons do not display any fine-structure
splitting. Atomic oxygen does, since its ground state have four p-electrons (l = 1).
The individual angular momenta can be written as components of a vector giving the total orbital
quantum number L = 1, and similarly the spin-vector is S = 1.
The ground state of atomic oxygen is written 3 P , where P denotes L = 1 and the exponent is determined by 2S + 1. This 3 P state is a multiplet of three slightly different energies denoted by the possible
sums of J ≡ L + S, with possible values 0, 1 and 2. 3 Pj=2 is then the ground state of OI, and 3 Pj=0 the
level with the highest energy.
3.2.2
Emission from OI and CII
In OI, the energy difference between 3 P1 and 3 P2 is 2.0 · 10−2 eV corresponding to a temperature of 230
K. Photons of this temperature have a wavelength of 63 µm, well inside the infrared. Since this radiation
can escape the clouds it plays a significant role in the cooling of the system. Similarly, ionized CII creates
ultraviolet photons, from (2 P3/2 to 2 P1/2 ), which also escapes the clouds. CII is both easily ionized and
abundant, so the cooling rate is even higher for this process. The cooling rate ΛX for these atoms is found
to be:
2
230 K
nH
exp −
eV cm−3 s−1
ΛOI = 2 · 10
103 cm−3
Tg
2
92 K
nH
−9
exp −
eV cm−3 s−1
ΛOII = 3 · 10
103 cm−3
Tg
−10
3.2.3
(12)
(13)
Molecules and Dust
Compared to atoms the molecules in the gas have closely spaced rotational energy levels, which make
them highly apt to receive energy through collisions. CO, among the different molecules present in the
cloud, is the most dominant cooling agent. The transition causing the cooling is J = 1 −−→ 0, because it
is optically thick.
As well as being a cause of heating, the dust also provides a means for cooling of the gas. Collisions
with the gas atoms and/or molecules lead to lattice-vibrations in the dust grains. These vibrations decay
through emission of optical and ultraviolet photons.
3.3
A simple model
The following model of the heating and cooling processes that occur in the ISM is easy to follow and
explicitly conveys a derivation of the plot in figure 1.
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3.3.1
J. Mortensen
Heating
Heating of interstellar gas is mainly conveyed by collisional ionization by low energy CR and interstellar
X-rays. ::Where they come from::. Cosmic radiation is most commonly in the form of protons (p+ ), and
the interaction between themand atomic hydrogen H is:
p+ + H −−→ p+ + H+ + e−
(14)
For non-relativistic protons, the model of heating can be obtained as follows: take H to be the rate of
heating [J s−1 ], I as the rate of ionization [s−1 ] and K as the kinetic energy of each ejected electron [J].
Then:
H = IK.
(15)
If σ(E) is the collision cross section of the CR proton [m2 ], and nH is paricle density of cloud of hydrogen
[cm−3 ], and N (E) the flux of CR protons [W/m2 ], the collision rate is given by:
collision rate = nH σ(E)v(E)
= nH σ(E)
X
=
nH σ(E)N (E).
(16)
(17)
(18)
E
This can be assumed to be equal to the ionization rate I. Defining E(E) as the energy lost per CR proton
per unit mass of traversed interstellar gas, the kinetic energy per ejected electron is:
K = me E(E) − x,
(19)
where x is defined as the potential ionization energy. The heating H of the gas by (non-relativistic) CR
protons is therefore given by:
H=
X
nH σ(E)N (E)(me E(E) − x)
(20)
E
= αnH ,
(21)
X
(22)
where:
α≡
σ(E)N (E)(me E(E) − x)
E
It is clear, then, that the heating of interstellar gas is proportional to nH . The argument for utilizing this
model as a fair approximation lies in the fact that the uncertainties in the observable values that make
up the constant α are considerably larger than the estimates and assumptions made by the model. The
relation is shown in figure 3.3.1.
However, problems with heating by CR protons may arise, firstly, from the fact that it assumes a large
flux of non-relativistic (low-energi) protons, an amount which requires extrapolating from the observed
presence of high-energy CR protons.
Secondly, there is a problem concerned with the upper limit CR proton ionization rate of 10−15 s−1 ,
which is set by observed limits of spallation products from collisions.
Thirdly, the same ionization rate is also set by observed abundance of H2 , which puts a limit on the
rate of H2 dissociation, something that is not a problem for X-rays, since they can not penetrate denser
clouds of H2 .
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J. Mortensen
Figure 3: Heating and cooling rates as a function of gas density.
Also, if CR protons are a product of supernovae, it means that only local heating can occur, since
the the protons can not diffuse quickly enough to produce uniform heating throughout the interstellar
gas.[1]
As X-rays are also a significant source of heating of the gas by processes similar to the ones involving CR
protons, this type of heating distinguish itself from that of CR protons, if heating is mostly by X-rays. That
is, since helium has an ionization cross section σHe = 20σH , the electron density rises significantly by
ionization of helium, and photons produced by recombination hereof are in turn able to ionize hydrogen.
If the degree of ionization ne /nH ≤ 0.1, the CR protons lose much of their energy due to collisions
with the electrons.
3.3.2
Cooling
Cooling of the ISM happens when thermal electrons collide with atoms or ions in the gas, subsequently
causing heat to be radiated away. At high temperatures (T ≤ 104 − 108 K), excitation of hydrogen and
helium are the most common reactions. At colder temperatures, these reactions cease, and ions such as
C+ , Si+ and Fe+ that have lower excitations energies, are the ones that drive the cooling – though at a
factor of 10−4 lower than cooling by hydrogen and helium.
I will not paraphrase the whole derivation as done above for the heating rate, but only present the
main result from [1], where the cooling rate H 0 is given by:
1
H 0 ∝ bT − 2 ne n,
(23)
where b is the so-called guillotine factor with b(T ) = 0 for 3kB T /2 < the excitation energy of the specific
atom or ion. n and ne are the volume density of atoms/ions and electrons respectively. T is the temperasture of the interstellar gas. The actual constant of proportionality is given in the source paper, but is not
really relevant in this assignment.
Looking again at a system largely composed of hydrogen, some helium and metals such as mentioned
above, the electron density ne depends on the ionization of hydrogen. Written ne ≡ βnH , the degree of
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ionization β is generally about 10−1 . At temperatures & 104 K, where hydrogen is ionized by thermal
electrons, β ≈ 1, i.e. ne = nH .
The cooling rate the reads:
1
H 0 ∝ bβT − 2 n2H ,
(24)
where:
H
1
∝ bH T − 2 n2H .
0
(25)
T &104 K
For Lower temperatures, the guillotine factor for hydrogen becomes zero, and cooling is now only done
by the ions in the gas. With their much lower rate of cooling, the equation for this process becomes:
H
1
∝ 10−4 · bi βT − 2 n2H .
0
(26)
T <104 K
Both functions are shown in figure 3.3.1. It is clear that cooling at either temperatures have a much
steeper slope compared to the heating rate. This means that there are two densities at which the rate of
heating and cooling are the same (marked in the plot as P and Q. At these two points the temperature
change is zero, beacuse the two processes that occur at these densities cancel each other out.
3.3.3
Connecting the dots
How do the two equilibrium points P and Q in figure 3.3.1 translate into something like figure 1? If we
have a gas of density nH and the temperature is lower than the ionization temperature for hydrogen,
then, as nH decreases, the cooling rate (26) drops faster than the heating rate (20). When the value of
nH is below the point of equilibrium in P, heating dominates, and the temperature of the system goes up.
1
As this happens the cooling rate drops even faster since it is also proportional to T − 2 .
The temperature dependance on nH is shown in figure 3.3.3, where points A, B and C represent the
same stable and unstable situations shown in figure 1.
Figure 4: The temperature dependance on nH .
The temperature change is steep, and as it reaches the ionization temperature for hydrogen, this in turn
causes a sudden change in β = ne /nH which goes up as shown by the arrow in figure 3.3.3. The change
in density causes the cooling rate to rise again, until equilibrium at P is reached, or, perhaps more likely,
it is a gradual process, where the system oscillates still closer about the equilibrium point, until finally
stabilized.
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If, instead, the density is increased at P, the cooling rate rises faster than that of heating, and the
temperature drops (figure 3.3.3). When it is too low for hydrogen to be ionized, the cooling due to (25)
stops, since bH (T ) = 0, and cooling is now only purveyed by the ions in the gas, at a rate 10−4 times
lower (26). When this happens, the cooling rate is below that of heating, but, as the density increases, a
new equilibrium is reached at Q in a manner analogous to the mechanisms in the previous situation.
Figure 5: The electron density ne as a function of the density nH of hydrogen.
What remains now is to look at the total gas pressure P/kB = (ne (nH )+nH )T of the system as a function
of the density nH of hydrogen atoms in the gas. This relation is shown in figure 3.3.3:
Figure 6: The total gas pressure P/kB = (ne (nH ) + nH )T of the system as a function of the density nH
of hydrogen atoms in the gas.
This plot evidently accounts for the simple requirements that were stated in the introduction.
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References
[1] Vincent C. Reddish. Stellar formation / by V. C. Reddish. Pergamon Press, Oxford; New York, 1st ed.
edition, 1978.
[2] Steven W. Stahler and Francesco Palla. The Formation of Stars. Wiley-VCH Verlag GmbH & Co. KGaA,
Weinheim, 2004.
[3] Gerrit L. Verschuur, Kenneth I. Kellermann, and National Radio Astronomy Observatory (U.S.). Galactic and extra-galactic radio astronomy, by the staff of the National Radio Astronomy Observatory. Edited
by Gerrit L. Verschuur and Kenneth I. Kellermann, with the assistance of Virginia van Brunt. SpringerVerlag, New York, 1974.
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