Download Lecture 5/10 The interstellar medium and star formation Ulf

Lecture 5/10
The interstellar medium and star formation
Ulf Torkelsson
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The interstellar medium
The interstellar medium is the gas and dust that we can find inbetween the stars. Through its
absorption and scattering of light from distant light sources it causes interstellar extinction. This
causes stars to appear to be fainter than they really are. For this reason we have to add an extra
term aλ to the expression for the apparent magnitude
mλ = Mλ + 5 lg d − 5 + aλ .
(1)
The interstellar medium has an optical depth τλ , so that we can also write the intensity of a star
as
Iλ
= e−τλ .
(2)
Iλ0
Thus the change in the apparent magnitude due to the interstellar extinction is
mλ − mλ0 = −2.5 lg
Iλ
= −2.5 lg e−τλ = 2.5τλ lg e = 1.086τλ ,
Iλ0
(3)
so we have
aλ = 1.086τλ .
(4)
On the other hand we can relate the optical depth to the number of scattering dust particles
Z s
Z s
τλ =
n (s) σλ ds = σλ
n (s) ds = σλ Nd ,
(5)
0
0
where Nd is the column density. The composition of the dust particles is not known in detail, but
it is likely that they contain graphite and silicates.
The dominating component in the interstellar medium is hydrogen, which may occur as atomic,
molecular or ionised hydrogen. Atomic hydrogen in the interstellar medium is in its ground state,
and thus could be observed through its absorption lines in ultraviolet, but it is far more common
that it is observed at radio waves. There are two different states in the n = 1 level of hydrogen.
In the one state the spin of the proton and electron are aligned and in the other state they are
antialigned. The energy of the state in which they are aligned is slightly more energetic, and the
electron may then switch to the other state by emitting a photon of frequency 1420 MHz or wave
length 21.1 cm. This transition is forbidden, that is it has a very low probability, and a hydrogen
atom can stay in the more energetic state for millions of years before it changes, but because
hydrogen is so abundant it is still a strong line in radio astronomy. Atomic hydrogen can be found
in diffuse HI clouds, where the temperature is 30 to 80 K, and the number density of hydrogen is
108 - 8 × 108 m−3 . Such clouds have masses of 1 to 100 M .
Thicker clouds will instead have molecular hydrogen. Molecular hydrogen forms on the surfaces
of the dust particles in the denser clouds, and the high gas density is also shielding the molecules
from the ultraviolet radiation from the stars, which would otherwise dissociate the molecules.
Unfortunately molecular hydrogen does not have any spectral lines in optical or radio. Instead
one investigates the molecular gas using other molecules that have suitable spectral lines in radio,
for instance CO, which has a spectral line at 2.6 mm. There is a wide range of molecular clouds
ranging from the translucent molecular clouds that are similar to the diffuse HI clouds to the
giant molecular clouds. The translucent molecular clouds have temperatures of 15 - 50 K, and
number densities of 5 × 108 to 5 × 109 m−3 . Their masses range from 3 to 100 M . The giant
molecular clouds on the other hand have temperatures of 20 K, and number densities between 108
and 3 × 108 m−3 , but their masses are on the order of 106 M . Inside these clouds their are dense,
opaque cores with temperatures of 100 - 200 K, and number densities of 1013 - 1015 m−3 . These
cores have radii of 0.05 to 1 pc and masses of 10 to 1000 M .
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Gravitational collapse and the formation of protostars
In order for a gas cloud to be in equilibrium it must fulfill the virial theorem that on average
2K + U = 0.
(6)
If 2K > |U | then the cloud will expand, and on the other hand if 2K < |U | then the cloud will
contract under its own gravity. For a spherical cloud of constant density the gravitational potential
energy is
3 GMc2
,
(7)
U =−
5 Rc
and the kinetic energy of the particles in the cloud is
K=
3
N kT,
2
(8)
N=
Mc
.
µmH
(9)
where the number of particles is
The condition for that the cloud should collapse is then
3Mc kT
3 GMc2
<
.
µmH
5 Rc
(10)
We can express the radius using the original density of the cloud
Rc =
3Mc
4πρ0
1/3
4πρ0
3
1/3
.
(11)
GMc5/3 ,
(12)
We then get
3
3Mc kT
<
µmH
5
and we find that
Mc >
5kT
GµmH
3/2 4πρ0
3
1/2
= MJ ,
(13)
which is the expression for the Jeans mass. The Jeans mass for a diffuse hydrogen cloud is
1500 M , but it is only 17M for one of the cores in a giant molecular cloud. Thus we expect that
gravitational collapse and star formation starts in the giant molecular clouds.
We can construct a simple model of this collapse. Assume that the collapse is essentially free
fall, that is the only important force is gravity. Then we can write the equation of motion for a
spherical shell at the initial radius r0 in the cloud as
d2 r
Mr
= −G 2 .
dt2
r
(14)
We now notice that since the entire shell is collapsing it must be that the mass Mr inside this shell
is constant and is Mr = 4πr03 ρ0 /3. We introduce this into our equation of motion and multiply
the equation with dr/dt
dr d2 r
4π
1 dr
= − Gρ0 r03 2 .
(15)
dt dt2
3
r dt
We integrate this equation over time and get
1
2
dr
dt
2
=
4π
1
Gρ0 r03 + C1 .
3
r
2
(16)
C1 is an integration constant, which we can fix by requiring that the collapse velocity is 0 when
r = r0 , so that
4π
C1 = − Gρ0 r02 .
(17)
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Thus we can write
1/2
dr
8π
2 r0
=−
Gρ0 r0
−1
.
(18)
dt
3
r
Note that we choose the negative root since we are looking for a collapse solution. We re-write this
equation using the substitutions θ = r/r0 and K = (8πGρ0 /3)1/2 , which gives us the equation
dθ
= −K
dt
1
−1
θ
1/2
.
(19)
We further make the substitution θ = cos2 ξ, which gives us
and
d
dξ
cos2 ξ = −2 cos ξ sin ξ ,
dt
dt
(20)
1
1 − cos2 ξ
sin2 ξ
−
1
=
=
.
cos2 ξ
cos2 ξ
cos2 ξ
(21)
We can now write our equation as
cos2 ξ
1 + cos 2ξ dξ
K
dξ
=
= ,
dt
2
dt
2
(22)
which we can easily integrate
1
K
1
ξ + sin 2ξ = t + C2 .
(23)
2
4
2
We can evaluate the new integration constant by saying that r = r0 when t = 0, that is θ = 1 or
ξ = 0 at t = 0. This is fulfilled if C2 = 0. We now have the equation
ξ+
1
sin 2ξ = Kt.
2
(24)
We can now determine the free-fall time for the cloud to collapse by calculating when r = 0, that
is θ = 0 or ξ = π/2. Therefore we have that
π
= tff K,
2
(25)
which gives
π
tff =
=
2K
r
3π
.
32Gρ0
(26)
One can make an important observation here. The free-fall time is independent of r0 , that is all
the spherical shells reach the centre at the same time, which is called homologous collapse.
In practice we will not get this simple spherical collapse. One reason for this is that the Jeans
mass that we calculated before is much larger than the typical mass for a star. Apparently the
cloud must fragment into smaller pieces while it is collapsing.
For simplicity we will follow a cloud that is collapsing into a single star of the same mass as
the Sun. At first the cloud is collapsing in free fall, but one can expect that the density is higher
in the central parts of the cloud, so that the central part is collapsing more rapidly. When the
density increases to 10−10 kg m−3 it becomes optically thick and the cloud starts to heat up. The
collapse in the central part slows down and the central part within 5 AU is almost in hydrostatic
equilibrium.
Outside this protostellar core matter is still in free-fall, but when it meets the core it develops a
shock wave in which most of its kinetic energy is converted to heat. The dust will evaporate when
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the temperature reaches 103 K. The opacity will then drop and the surface with optical depth
τ = 2/3, at which the radiation is emitted, will move inwards. Since the emitting surface decreases
the temperature must increase in order for the luminosity to remain high. At 2000 K the molecular
hydrogen dissociates, which costs energy and the core starts to collapse again. Equilibrium will be
re-established when the core has collapsed to a radius that is 30% higher than the current radius
of the Sun, and we have got a pre-main sequence star.
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Pre-main sequence stars
The temperature of the star increases until the H− ions provide a strong opacity, which makes
the star convective. In some cases the convection zone will reach all the way into the centre of
the star. During the continuing collapse the star will then follow the Hayashi track downwards
through the HR-diagram. Already during this collapse phase the first nuclear reaction will start,
deuterium burning, but the main energy source of the star will be its potential energy. Later on the
increasing central temperature decreases the opacity in the centre and the core becomes radiative.
When this happens the star will start to increase in luminosity again. Almost at the same time
the first two steps in the proton-proton-chain will start to convert 1 H to 3 He, and the first steps
in the CNO-cycle will convert 12 C to 14 N. Because of the steep temperature-dependence of the
CNO-cycle there will be a strong temperature gradient in the core, and it will become convective.
When all 12 C in the core has been consumed the core will make the last adjustments to become a
radiative core deriving its energy from the proton-proton-chain. The star then settles down onto
the zero-age main sequence.
In case the star is lighter than 0.5M then the central core will not become hot enough to burn
12
C. Furthermore in these light stars the opacity is always large enough that the core will remain
convective. Heavy stars on the other hand are hotter and start the nuclear reactions early and as
main-sequence stars they use the CNO-cycle to burn hydrogen.
Hot stars emit copious amounts of ultraviolet light as main-sequence stars. This ultraviolet
radiation ionises the surrounding hydrogen gas and we get HII regions. In these regions there is
a balance between the ionisation of neutral hydrogen atoms by the ultraviolet photons and the
recombination of protons and electrons. During the recombination events the electrons cascade
down through the energy levels and among other spectral lines they generate the Balmer lines that
can be seen in ordinary light. In general these HII regions appear red because of the Balmer-α
line. In large star formation regions the radiation from these massive stars may be strong enough
to stop the formation of the lighter stars before they have fully collapsed.
Low-mass pre-main sequence stars are often observed as highly variable stars with strong emission lines, so called T Tauri-stars. One makes a distinction between classical T Tauri-stars and
weak-lined T Tauri-stars, which have much fainter emission lines. Detailed observations have shown
that the classical T Tauri-stars are surrounded by extensive discs of gas and dust. These discs
exist because the collapsing cloud is rotating from the beginning, and while the cloud contracts
it spins up and becomes more and more flattened so that eventually we get a pre-main sequence
star surrounded by a disc. It is probably in these discs that planets are formed. The weak-lined T
Tauri-stars appear to have lost most of their discs, which agrees with that these are more evolved
and are close to becoming main-sequence stars. However thin dust discs have been observed around
some young main-sequence stars such as Vega and β Pictoris.
Another fascinating aspect of these T Tauri-stars is that they may have very extended jets
that shoot out from the star or the innermost part of its disc. These jets can extended over vast
distances, and when it is colliding with the surrounding gas it forms shock waves that are observed
as Herbig-Haro objects.
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