Applications of Physics to Astronomical Systems
Lecture 8
13)
Star Formation – the Current Picture
Our discussion so far has left out two key elements – angular momentum and magnetic fields, both
of which turn out to be important. Even quite dense, cold interstellar clouds remain weakly ionised
because of the passage of cosmic rays (charged high energy particles produced in supernova
remnants) and X-rays (from e.g. white dwarf stars). This means that they are very good conductors
and any magnetic fields that are generated tend to be very long lasting. Because the intervals
between collisions are long, charged particles tend to spiral round the magnetic fields and not move
across them, so on large scales the field is trapped in the gas. The differential rotation and general
turbulence in the interstellar medium tends to wind up the fields until there is roughly equal energy
density in the magnetic fields and in the kinetic energy in the random motions of the clumps of gas.
Observations of the Zeeman splitting of the 21cm line and of molecular spectral lines confirm that
magnetic fields have roughly the strength required for this to be so: from ~ 2.10 –10 T in the diffuse
clouds to ~ a few times 10 –7 T in dense ones.
If a cloud containing a trapped magnetic field contracts, then we expect the flux through a particular
surface to remain fixed so the field, B, scales with radius, R, as R –2 and so the energy density in the
magnetic field, which is also a measure of the pressure it can exert, scales as R –4 and the total
magnetic energy in the cloud scales as R –1. This is the same dependence on radius as the
gravitational energy, G M 2 / R (unlike the thermal pressure). This means that for a given initial
magnetic field strength, clouds below a certain size cannot collapse, even if they are unstable
according to the Jean’s criterion. It appears therefore that the initial timescale for the shrinkage to
form dense condensations is set by the rate at which the neutral gas can slide past the charged
particles. This is called ambipolar diffusion. This timescale can be 10’s of millions of years, so the
presence of the magnetic fields explains the long lifetimes of the interstellar clouds.
When sufficient gas has drained through the field to become gravitationally unstable, more rapid
collapse can begin. There will, however, be some angular momentum present and if this were
conserved the gas could still not condense to form an object the size of a star. What happens
instead is that the gas forms a thin rapidly rotating disk and the material drains inwards at a speed
set by the rate at which angular momentum can be carried away. The magnetic field almost
certainly plays a role here in coupling the disk to the more distant envelope. The winding up of the
magnetic field produces a torque which slows down the disk and removes angular momentum.
The most dramatic observational result of this process is the formation of stellar jets. These are
highly collimated and are observed to move at velocities of ~200 km/sec. Then can extend up to a
parsec from the star and push bow-shocks of molecular gas in front of them. They carry a large
amount of energy and momentum into the cloud. It appears that essentially all protostars have these
jets and that the formation of such a jet along the axis of rotation is an inevitable result of accretion
in the presence of angular momentum and magnetic fields. (They have strong similarities in form
to the jets seen from galactic nuclei and smaller black holes, although the scales and energies are
quite different.)
14)
Stellar Structure and Evolution
Once the infalling gas has heated up sufficiently to form a stable object – a protostar in hydrostatic
equilibrium – the evolution proceeds on the Kelvin-Helmholtz timescale where the energy lost
through radiation is replaced by gravitational potential energy gained through a slow shrinkage.
This is about 20 million years in the case of a one solar mass object. As we have seen, this
shrinkage means that the temperature rises but it turns out that the luminosity stays the same. On a
plot of temperature versus luminosity, the Hertzsprung-Russell (HR) diagram, the star therefore
moves horizontally during this “pre-main-sequence” evolution.
When the temperature at the core of the star reaches about 10 million K the nuclear reactions that
convert hydrogen into helium turn on. These are extremely temperature sensitive (the rate of
energy production increases as a very high power of temperature) and one might expect that a
massive explosion would result. This is prevented by the stabilising effect of the relation between
radius and temperature which, from the virial theorem, is T  1/R. Any increase in the rate of
energy production increases the pressure and this increases the radius, lowering the temperature and
restoring the energy production to the rate which just matches the outward leakage of energy. The
result is that the stars luminosity and temperature remains almost constant at a position on the
“main sequence” in the HR diagram during the whole period of helium burning in the core.
Models can be constructed using the following 4 physical relationships:
1) Hydrostatic equilibrium
2) Mass continuity
3) Energy generation
4) Energy transport
Models can be constructed which demonstrate the expected relationship between the surface
temperature and the luminosity and how these depend on mass. It turns out that the luminosity
scales as M 3 for more massive stars and M ~4 for lighter ones, which means that the main-sequence
lifetimes are proportional to M –2 or M –3 respectively. (The amount of hydrogen available for
burning is proportional to M.
Once the hydrogen runs out in the core the structure changes with the nuclear reactions taking place
in a thin shell and the outer envelop becoming very extended. The star becomes a red-giant. The
addition of more and more helium onto the core compresses it, raising its temperature further, until
eventually (in high mass stars) the helium burning process ignites creating carbon. A series of
further cycles like this occur with heavier elements being formed in each but these provide less and
less energy and so happen quicker and quicker. Finally when elements heavier than iron are formed
the nuclear reactions start to absorb energy instead of release it and a rapid collapse occurs
producing a supernova (type II). This overall process is thought to have produced almost all the
heavy elements in the universe today.