Stellar evolution is ultimately a battle against gravity. All fuel runs out eventually, but gravity never does! The battle ends in a stalemate for stars with initial main sequence mass below about 10x the mass of the Sun - these end up as a white dwarfs or neutrons star depending on the initial mass. But for stars which start out more massive than this then there are no known alternatives other than a complete triumph of gravity, where nothing can hold up ultimate collapse into a black hole.
But theory is all well and good - whats the evidence that these exotic objects actually exist ? Can you really squeeze the mass of the sun into something the size of the Earth (a white dwarf!) - the Sun's diameter is about 100x bigger than that of the Earth. A teaspoonfull of a white dwarf would weight about 10 tons!! Can this really happen ? And neutron stars are even more extreme - mass of the Sun into something the size of Newcastle (10km diameter!). A teaspoonfull of this would be as heavy as the average mountain! Can they REALLY be held up by this odd property of electrons and neutrons that they don't like being squashed in a box - wouldn't it be more plausible to say that they all collapse into black holes ? Alternatively you might view the concept of black holes with deep distrust - is there REALLY no way to halt the collapse of a core of more than 3x the mass of the Sun ?
White dwarfs and neutron stars cool, so will emitt blackbody radiation. We've already seen how the shape of the blackbody radiation depends only on temperature - the wavelength at which the intensity peaks gives you a direct measure of the temperature of the material. But another key property is that for a given temperature then the luminosity of a blackbody is directly proportional to the surface area emitting it. White dwarfs much smaller than a Sun-like star, so a white dwarf with the same temperature as the sun should then be much less luminous. Plotting Hertzsprung-Russel diagrams (temperature versus luminosity) show that there are such dim objects (click on the picture and the colour-magnitude diagram to get the full images) exactly as predicted by white dwarf models. Some of these are even in binary systems - so we can get a mass via the doppler shift! This was first done for Sirius B (binary companion of Sirius A, the brightest star in the sky). The doppler shifts, together with the binary separation give the mass of Sirius B to be about the same as that of the Sun, yet the radius derived from its temperature and luminosity is 100x less than that of the Sun.
But for a neutron star, the siz is much much smaller even than that of a white dwarf. Their surface area is TINY, so any radiation associated with their cooling from the supernovae explosion will also be very small. And of course, there is no surface to cool for a black hole. So this is not a good way to look for neutron stars and black holes. As in the quote from Red Dwarf: 'The thing about a black hole, its main distinguishing feature, is its black. And the thing about space, your basic space colour, is its black! So how are you supposed to see them ?'
One key property of the formation of neutron stars is that they should be accompanied by a dramatic burst of neutrinos as all the electrons interact with protons proton + electron -> neutron + neutrino. And neutrinos HAVE been detected from the supernova explosion of SN1987A! But all this tells you is that neutrons formed from electrons and protons, not that a neutron star is formed - the core could have collapsed further into a black hole.
The observational discovery of neutron stars was a consequence of some entirely different physics. They were first discovered in 1967 by a graduate student, Jocelyn Bell, at Cambridge in the early days of radio astronomy. She saw these regular pulses every 1.3s from an object which moved with the stars (rising 4 minutes earlier each night). Such regular pulses imply a very stable mechanims - orbital timescale, or oscillation timescale or rotation timescale (or alien lifeforms ???). She's written an article on the history of the discovery of pulsars. Soon she discovered 3 more - one of which had a period of 0.25s. What can these be ?
Well, its easiest to say what it can't be! It can't be from a white dwarf - the timescale to get anything orbiting around a white dwarf is longer than 0.25s, and a white dwarf would break up if it was made to rotate as fast as 4x per second (period of 0.25s) and its oscillation period is longer than 1 second also. So its not a white dwarf - it must be smaller and/or denser..... Only thing smaller and/or denser than a white dwarf is a neutron star - so suppose it was one of these. Something in orbit around a neutron star would produce short dips rather than short pulses, and neutron stars are so dense that their oscillations are much much shorter than a second. But how about rotation ?
Stars rotate and when a rotating star collapses then its rotation gets faster and faster as its radius gets smaller and smaller. I'm sure the rotating lecturer was unforgetable - but for those of you who missed it, think of a spinning skater: they start spinning with their arms out, then as they bring their arms towards their body they spin faster. Its called conservation of angular momentum. Stars also have a magnetic field, and in a similar way the magnetic field gets stronger as the radius of the star gets smaller. So neutron stars can end up with very strong magentic fields, spinning very fast. And in these extreme magnetic fields, charged particles can be accelerated. Accelerating particles radiate, so this produces two light (or radio or X-ray) beams pointing out from the magnetic poles. And these can sweep across our line of sight due to the neutron star rotation, giving rise to regular pulses of light - see this nice animation of a pulsar. And there are some (most famously the crab pulsar) which are assocated with supernovae remnants). There are even a few pulsars in binary systems so we can weigh them accurately! and sure enough, they have masses of about 1-2x the mass of the Sun, yet must have tiny radii in order to have such fast rotation periods.
But what about black holes ? We saw last lecture that these were defined to be where the escape velocity was bigger than the speed of light. This is the definition of then event horizon around a black hole - its the closest we can go into the immensely strong gravitational field and still send signals back out to the Universe. The collapsed core itself is much smaller - it may be truely point-like (a signularity) - we don't know! So you can get much further in. You just can't ever get out!
So how can we ever find out anything about black holes ? Well, we've said before that most stars are in binary systems, rather than being single stars like our Sun. If the black hole has a binary companion star which is VERY CLOSE to the compact object, then the outer layers of the star can be more strongly attracted by the gravity of the compact object than they are to the center of the star itself. So they just fall off, forming an accretion disk, spiralling in towards the event horizon. As the material falls it can heat up due to the gravitational energy released. For a white dwarf then the disk gets so hot that it emitts in the blue/ultraviolet part of the spectrum, while for a neutron star or a black hole the gravitational energy released by the material falling in is SO large that the disk gets X-ray hot, forming an X-ray binary system.
But how can we tell if a strong X-ray source is a neutron star or a black hole? The best way is to measure its mass by looking at the orbit of the companion star. If its more than 3x the mass of the sun then it cannot be a neutron star, and the only thing we know of that it can be is a black hole.