Applications of Physics to Astronomical Systems

Lecture 1

1)The Scale of the Universe

Here are the sizes and masses of some of the identifiable objects in our universe.

"Radius" (m) Mass (kg) Luminosity(W)

Nucleus 5 x 10-16 1.6 x 10-27
Atom 5 x 10-11 1.6 x 10-27
Dust Grain 10-07 4 x 10-18
Bacteria 3 x 10-06 10-13
Human 3 x 10-01 10+02 (10+02)
Comet Nucleus 3 x 10+03 10+14
Earth6.4 x 10+06 6 x 10+24
Jupiter 7 x 10+07 2 x 10+27
Brown Dwarf 2 x 10+07 10+29
Sun 7 x 10+08 2 x 10+30 3.8 x 10+26
Solar System 5 x 10+12 2 x 10+30
White Dwarf 6 x 10+06 <2.8 x 10+30 10+23
Neutron Star 10+04 <6 x 10+30
Red Giant Star 4 x 10+10 10+31 10+30
Interstellar Cloud 3 x 10+16 10+33
Globular Cluster 10+17 10+36
Active Galactic Nucleus <10+15 10+39 10+39
Large Galaxy 5 x 10+20 10+42 10+37
Cluster of Galaxies 5 x 10+22 10+45 10+39
Super Cluster 10+24 10+47
Observable Universe1.5 x 10+26 ~10+53
Radius of Earth’s Orbit1.5 x 10+11m is 1 Astronomical Unit (AU),
1 light year is ~ 10+16 mand 1 parsec (pc) = 3600x180/AU is 3x10+16 m.
For a Black Hole, the Schwarzschild radius = 2GM/c2 is ~ 3 x Mass / Msun km.

We can now measure distances inside the solar system very accurately by timing radar signals bounced off the surfaces of planets and the round-trip times for radio signals sent to spacecraft. This means that the AU is very well defined. We measure the distances to nearby stars by measuring the parallax – the change in apparent position (relative to much more distant stars) as seen from opposite side of the earth’s orbit. This technique is the origin of the unit of distance called the “parsec”, which is the distance at which 1 AU subtends an angle of 1 arc second.

Obviously in most cases these numbers only represent typical values. The sizes and masses of interstellar clouds and galaxies, in particular, range over several orders of magnitude.

Particular points to notice are:

1)The bulk of the matter, in our neighbourhood at least, is in the form of stars and the larger objects are basically conglomerations of stars.

2)There is a huge ratio between the size of a star and the typical space between the stars (typically ~107 which is ~1021 in terms of volume). If stars form from roughly uniform clouds of gas, this is the amount of compression that has to take place. It also means that collisions between stars will hardly ever happen except in very dense clusters of stars.

3)By contrast, the separations between galaxies are only ~100 times their typical sizes. This means that collisions are relatively common. These scramble up the galaxies and often lead to them merging.

4)A plot of Mass against radius shows many interesting features:

2)The Support of “Cold” Matter against Gravity

It is worth asking which of these features arise directly from the laws of physics and which are the result of more complicated history. The forces of nature to consider are: strong nuclear, weak nuclear, electro-magnetic and gravity. On large scales we would expect gravity to be the dominant factor because the nuclear forces are short-range and positive and negative electrical changes balance out. Gravity is however a purely attractive force and it can only pull things together. By itself, therefore, it will not produce objects of particular sizes. Something has to balance gravity.

We see that, for very small objects up to large planets, mass scales as radius cubed. These objects are made of “normal” solid matter with density of a few grams per cc. It is clear that this characteristic density is set by the typical size of atoms, which is in turn related to the Bohr radius, a0 = ( 40 / e2 ) . ( ħ2 / me ) = 5.10-11m. This shows that atomic sizes are controlled by the interaction of the electromagnetic force and quantum mechanics. Specifically the electrostatic attraction between the electrons and the nucleus is balanced by the uncertainty principle. To obtain this result by “order of magnitude” arguments, just put a0pe = ħ, where pe is the momentum of the electron and set the kinetic energy of the electron equal to the electrostatic energy ( e2 / 40a0).

Note that at a deeper level something else is stopping the electrons and protons combining to form neutrons. If that were to happen, the electromagnetic forces would no longer be involved and much higher densities would result. We will return to this.

Objects up to perhaps 100m in size will be held together by chemical forces (the residual electro-magnetic effects due to the distribution and movement of charges in the atoms). Beyond that gravity becomes the main binding force. Note that a spherical configuration will have the lowest energy and that the more massive an object is, the more it will tend to form to a precise sphere.

Atoms are so “stiff” that they keep their same basic size even under the enormous pressures that occur inside large planets. To find the pressure, Ps, at which they start to get squashed, just set the compressive force equal to the pressure times the area ~ a02 and put this equal to the magnitude of the electrostatic force, e2 / 40a02. We then need to compare that with the pressure at the centre of a planet. Dimensional argument gives us that this must be PG = k G M 2 / R 4 where M is the mass of the planet and R is its radius. To obtain the dimensionless constant k we have to integrate the pressure down from the surface to the centre. For a sphere of uniform density we find k = 3/8.

We now equate these and make use of the fact that the planet is made up of N atoms each of mass about equal to the mass of the proton, mp. (As far as we know massive planets are mostly made of hydrogen.) Neglecting the dimensionless constants we have:

G M 2 / R 4 ~ e2 / 40a04 and N = M / mp ~ R3 / a03 .

HenceG N2 mp2 ~ N4/3e2 / 40 i.e. N2/3 ~ e2 / 40 G mp2 .

This is the expression for one of the famous “large dimensionless numbers” of nature – the ratio of the electro-magnetic force to the gravitational force between two protons. (Since both are inverse-square forces the ratio does not depend on the distance between the protons.) Its value is 1.24.10+36.

Hence N is ~ 1.4.1054 corresponding to a mass of ~ 2.3.1027 kg, which is just a little larger than that of Jupiter. In objects which are larger than this the atoms will be squashed and as a result the density rises. These are known as “degenerate” objects because the pressure is essentially provided by the fact that the electrons, being Fermions, refuse to be forced into the same states. The smaller the volume they are forced to occupy the higher their energy becomes. It turns out that in this regime the radius of an object scales with its mass to the minus one-third power.

According to this simple picture, degenerate objects form a continuous sequence from massive planets, to brown dwarfs, to white dwarfs. The latter can be as massive as our sun but only about the size of the earth. The mean density in a white dwarf is about 2.109 kg m-3, i.e. a million times “normal” solid matter. Here is this part of the mass/size relationship on an expanded scale.

Red Giant

Neutron Star

White Dwarf Sun

Brown Dwarf

Jupiter

There is a strict limit, however to the mass that can be supported by electron degeneracy pressure, called the Chandrasekhar limit. This arises because at very high densities the energy of the electrons becomes large enough for their velocities to become relativistic. This means that the relationship between energy and momentum changes from E ~ pe2 / 2me to E ~ pe / c. The result is that, if more mass is added to a white dwarf where the electrons are already relativistic, the degeneracy pressure does not increase fast enough to counter the increase in gravitational pressure. This leads to collapse. (We actually observe this to occur, in most spectacular fashion – it is seen as a supernova of type I.) This limit is reached when the objects contain ~ ( hc / G mp ) 3/2 atoms, which corresponds to about 1.4 times the mass of the sun.

Complete collapse may however not occur because the densities eventually get high enough to make it energetically advantageous for the electrons and protons to combine to form neutrons (releasing neutrinos). The neutrons are of course Fermions too and so they also exert a degeneracy pressure. Because these are ~2000 times more massive than the electrons they can do this up to much higher energies than the electrons. A body consisting almost entirely of neutrons with stellar mass can therefore be stable, with density of order 1018 kg m-3. We observe these through the radio pulses they give off as they rotate, hence the name “pulsars”. When these are in binary systems the masses can be estimated from their orbits and they are found to be very close to the Chandrasekhar limit. – i.e. consistent with them having been formed by collapse of a degenerate star.

In this case of degenerate neutrons there is an upper limit of ~3 times the mass of the Sun at which the neutrons would become relativistic. According to these simple ideas then, anything more massive than this should collapse and form a black hole. A black hole is an object where the gravity is so intense that it overwhelms all the other forces. The only characteristic size is the Schwarzschild radius, which is about 3km for one solar mass.

There therefore needs to be some other source of support acting against gravity to explain the existence of the more massive objects in the list above.

Scale Models of Big Things

Try dividing by 1 million (1km = 1mm):

Historic Cambridge is now about this big
and the earth is 13m is diameter, but the solar system
is still ~ 10,000km across

Try another million, i.e. 1012

Now the earth is a spec, 13 microns in diameter, the sun is 1.4 mm in diameter, the earth’s orbit is 150mm in radius and the solar system, ~ 10m diameter, nearly gets in the lecture theatre. But the nearest star is still 40km away.

Another, so now dividing by 1018

Stars are now brilliant points, ~ 30 mm apart, the disk of our galaxy is in our locality about 100pc thick, so fits between the floor and ceiling, and the galaxy itself covers central Cambridge. Now it is the Andromeda galaxy that is ~20km away.

Finally, when we get to a factor of 1024:

We have galaxies an inch or so apart and the largest known structures ~ 100 Mpc across (now ~3m) can be imagined in this room. Cosmological effects would come into play if we ventured outside it and the “horizon” would be at ~150m.