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Applications of Physics to Astronomical Systems
Lecture 2
3) Balancing Gravity with Kinetic Energy – the Virial Theorem
The previous discussion neglected the kinetic energy of the particles that make up the bodies. So
long as a body is hot the random motions will provide a pressure which can support it against the
force of gravity. This is what keeps the sun and most other stars at their present diameters. In those
cases the pressure, P, is generated by the motion of the individual particles in the gas, according to
P = n kB T where n is the number of particles per unit volume and T is the temperature. (Inside
stars, the gas is highly ionised so n is the total number of free electrons and ions.) Radiation
pressure also makes a contribution in very hot stars.
In conglomerations of stars, such as globular clusters and galaxies, the random motions of the stars
themselves can, in a loose sense, be thought of as playing the same role as that of the atoms in a
gas. If the stars are moving around fast enough, a cluster will not be subject to gravitational
collapse. Even in clusters of galaxies it is the random motions of the galaxies that stop them all
falling into the centre of mass. Note also that more orderly motions can also balance the force of
gravity. In the case of the solar system, the orbital motion of the planets gets them from falling into
the sun, and in a spiral galaxy like the Milky Way, the rotation (i.e. angular momentum) plays a big
role in determining its size.
Note however, that unlike the pressure exerted by quantum effects, a balance based on kinetic
energy cannot last indefinitely. A star is radiating strongly and therefore loosing energy. If there
were no input of energy available, the sun would cool in about 30 million years. In fact it will last
much longer than that (it is about half way through its 10 billion year life) because it gets energy
from nuclear reactions in its core, but when this source runs it will contract to become a white
dwarf. It is the same for all other stars and in fact the more massive ones have much shorter lives
which end in spectacular collapses. As we have already seen collisions between stars themselves
are rare so there is very little dissipation of kinetic energy in clusters of stars and isolated galaxies.
These will last (as clusters of “cold” stellar remnants) for a long time, but they too will decay
eventually.
Now let us look at this balance in a little more detail. Consider the case of orbital motion. The
gravitational potential energy of an object of mass m in a circular orbit, radius r, about a much
larger body, mass M, is
EG = – G M m / r and the kinetic energy is Ek = ½ m v 2 = G M m / 2r.
We see that Ek = – ½ EG and that the total energy, Ek + EG = – Ek = ½ EG. It turns out that
this is a general result, called the “virial” theorem, for any system where an equilibrium has been
reached between kinetic motions and gravity. (Note that it does not apply to orbits in general and
certainly not to the radial collapse of e.g. a gas cloud. For collapse from rest we must of course
have Ek = – EG .) A somewhat unexpected result is that, if we take energy out of such a system
(e.g. by allowing it to radiate), the total energy must become more negative and it becomes more
compact, but the kinetic energy must increase, so that it gets hotter. Stars do not get colder as they
loose energy but hotter!
Because it relates mass to velocity and size, both of which are observable, we can use the virial
theorem to estimate the mass of things like clusters of galaxies and molecular cloud complexes,
although we need to take case that the assumption that equilibrium has been established is valid.
4) Redshift and the Expansion of the Universe
Finally we look at the application of these ideas on the largest scales of all. If we consider only the
forces which we know about from laboratory physics it seems clear that gravity must dominate on
very large scales and therefore that the universe should be in a state of collapse. We can try to
argue that if it is uniform and infinite then there is no preferred direction for any object to move in
so nothing happens. It is clear however that, at best, any such situation would be unstable – any
local perturbation producing higher density would collapse locally. (This is indeed the picture we
have of how galaxies were formed.) This has however clearly not happened on very large scales.
Einstein realised that this was a problem when he applied his theory of General Relativity to
cosmology. He assumed that the universe was static and found that no such solution was possible,
even with his more sophisticated theory of gravity. As an escape from it he added an extra term to
his field equations, called the cosmological constant , which essentially amounted to postulating
an additional repulsive force which would balance out gravity on very large scales.
What was wrong of course was the assumption that the universe is static. Hubble showed that the
distant “nebulae”, as external galaxies were then known, are in fact receding from us at a speed
which is approximately proportional to their distance. More accurately, what he showed was that
the apparent brightness of similar objects (large spiral galaxies in this case) declined as the square
of the redshift. The measured quantities in astronomical parlance were: the magnitude m where
m = – 2.5 log10(F / F0 ) where F is the flux (i.e. W/m-2) and F0 is the flux of a standard star
and redshift z = ( observed –  lab ) /  lab .
When redshift is interpreted as a Doppler shift we find that this corresponds to a recessional
velocity V = c z. (Note that this is a non-relativistic definition. Because of the way z has been
defined, values of V > c would occur if this formula is used on the objects we nowadays observe.
In fact it is better to think of the quantity (1 + z ) =  observed /  lab as being a measure of the size of
the universe now compared to its size when the light was emitted.)
Hubble’s law showed that the universe was in a state of uniform expansion and led to the idea of the
Big Bang. This follows from the simple argument that if the distance between objects is increasing
by some fixed fraction in a given time, there must have been a moment in the past when the objects
were all on top of each other. To determine how long ago this was we need to convert the units
above into real distances and velocities and measure the slope of the best-fitting line, which is
known as Hubble’s constant H0. This proved hard to do, but the value is now quite well determined
at 65 +/ – 7 km/s per Mpc. The inverse of this is of course a time, called the Hubble time.
Here are some of the plots from Hubble’s papers. The data on which the original discovery of
expansion was based in 1929 (left), was marginal at best, but by 1931 (right) the correlation was
convincing. Notice however that the calibration of the distance scale was out by nearly a factor of
10 relative to the modern determinations.