Notes on Relativity and Cosmology - Physics Department, UCSB
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Notes on Relativity and Cosmology - Physics Department, UCSB

Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB

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27.09.2014 Views

264 CHAPTER 9. BLACK HOLES way that these black holes were found was that astronomers found certain stars which seemed to be emitting a lot of high energy x-rays. This is an unusual thing for a star to do, but it is not so odd for a black hole (as we will discuss shortly). On closer inspection of the star, it was found that the star appeared to “wobble” back and forth. This is just what the star would seem to do if it was in fact orbiting close to a small massive dark object that could not be seen directly. This is why they are called binary systems, since there seem to be two objects in the system. These massive dark objects have masses between 5 and 10 solar masses. Actually, there are also cases where the dark companion has a mass of less then 2 solar masses, but those are known to be neutron stars (see below). We had a discussion in the previous section about how our knowledge of normal matter led to the conclusion that Sagittarius A ∗ is a black hole. Well, we also have a pretty good idea of how star-like objects work in the solar mass range. In actual stars, what happens is that the objects become dense enough that nuclear fusion occurs. This generates large amounts of heat that increases the pressure in the matter (remember the ideal gas law?) far above what it would be otherwise. It is this pressure that keeps the object from collapsing to higher density. Thus, the reason that a star has a relatively low density (the average density of the sun is a few times that of water) is that it is very hot! This of course is also the reason that stars shine. Now, the dark companions in the binary systems do not shine. It follows that they are not hot. As a result, they must be much smaller and much more dense. Our understanding of physics tells us that massive cold objects will collapse under their own weight. In particular, a cold object greater than 1.4 times the mass of the sun will not be a star at all. It will be so dense that the electrons will be crushed into the atomic nuclei, with the result that they will be absorbed into the protons and electron + proton will turn into a neutron. Thus the object ceases to be normal matter (with electrons, protons, and neutrons) at all, but becomes just a big bunch of neutrons. This number of 1.4 solar masses is called the Chandrasekhar limit after the physicist who discovered it. In practice, when we look at the vast numbers of stars in the universe, we have never found a cold star of more than 1.4 solar masses though we have found some that are close. So, any cold object of more than 1.4 solar masses must be at least as strange as a big bunch of neutrons. Well, neutrons can be packed very tightly without resistance, so that in fact such ‘neutron stars’ naturally have the density of an atomic nucleus. What this means is that one can think of a neutron star as being essentially one incredibly massive atomic nucleus (but will all neutrons and no protons). The density of an atomic nucleus is a huge 10 18 kg/m 3 . (This is 10 15 times that of normal matter.) Let us ask: suppose we had a round ball of nuclear matter at this density. How massive would this ball need to be for the associated Schwarzschild radius to be larger than the ball itself? The answer is about 4 times the mass of the sun. So, working with a very simple model in which the density is constant (and always equal to the density of normal nuclei, which are

9.5. BLACK HOLE ASTROPHYSICS AND OBSERVATIONS 265 under significantly less pressure) inside the object, we find that any cold object with a mass greater than four solar masses will be a black hole! It turns out that any model where the density increases with depth and pressure yields an even stronger bound. As a result, modern calculations predict that any cold object with a mass of greater than 2.1 solar masses will be a black hole. As one can see from the observational data in the handout, the dark companions in the binary systems all have masses significantly greater than 2 solar masses. By the way, it is reassuring to note that every neutron star that has been found has been in the range between 1.4 and 2.1 solar masses. 9.5.3 A few words on Accretion and Energy Even with the above arguments, one might ask what direct measurements could be made of the size of the dark companions. Can we show directly that their size is comparable to the Schwarzschild radius? To do so one needs to use the energy being released from matter falling into a black hole. This leads us to a brief discussion of what are called accretion disks. The idea is shown on one of the pictures that I have handed out. In general, matter tends to flow into black holes. This addition of matter to an object is called “accretion.” Black holes (and neutron stars) are very small, so that a piece of matter from far away that becomes caught in the gravitational field is not likely to be directed straight at the black hole or neutron star, but instead is likely to go into some kind of orbit around it. The matter piles up in such orbits and then, due to various interactions between the bits of matter, some bits slowly loose angular momentum and move closer and closer to the center. Eventually, they either fall through the horizon of the black hole or hit the surface of the neutron star. In cases where the compact object is in a binary system, the matter flowing in comes mostly from the shining star. This process makes the accreting matter into a disk, as shown in the picture 8 below. This is why astronomers often talk about ‘accretion disks’ around black holes and neutron stars. 8 This picture was created by Jillian Bornak, a past PHY312 student.

9.5. BLACK HOLE ASTROPHYSICS AND OBSERVATIONS 265<br />

under significantly less pressure) inside the object, we find that any cold object<br />

with a mass greater than four solar masses will be a black hole! It turns out<br />

that any model where the density increases with depth <strong>and</strong> pressure yields an<br />

even str<strong>on</strong>ger bound. As a result, modern calculati<strong>on</strong>s predict that any cold<br />

object with a mass of greater than 2.1 solar masses will be a black hole.<br />

As <strong>on</strong>e can see from the observati<strong>on</strong>al data in the h<strong>and</strong>out, the dark compani<strong>on</strong>s<br />

in the binary systems all have masses significantly greater than 2 solar masses.<br />

By the way, it is reassuring to note that every neutr<strong>on</strong> star that has been found<br />

has been in the range between 1.4 <strong>and</strong> 2.1 solar masses.<br />

9.5.3 A few words <strong>on</strong> Accreti<strong>on</strong> <strong>and</strong> Energy<br />

Even with the above arguments, <strong>on</strong>e might ask what direct measurements could<br />

be made of the size of the dark compani<strong>on</strong>s. Can we show directly that their<br />

size is comparable to the Schwarzschild radius? To do so <strong>on</strong>e needs to use the<br />

energy being released from matter falling into a black hole. This leads us to a<br />

brief discussi<strong>on</strong> of what are called accreti<strong>on</strong> disks.<br />

The idea is shown <strong>on</strong> <strong>on</strong>e of the pictures that I have h<strong>and</strong>ed out. In general,<br />

matter tends to flow into black holes. This additi<strong>on</strong> of matter to an object is<br />

called “accreti<strong>on</strong>.” Black holes (<strong>and</strong> neutr<strong>on</strong> stars) are very small, so that a<br />

piece of matter from far away that becomes caught in the gravitati<strong>on</strong>al field is<br />

not likely to be directed straight at the black hole or neutr<strong>on</strong> star, but instead<br />

is likely to go into some kind of orbit around it. The matter piles up in such<br />

orbits <strong>and</strong> then, due to various interacti<strong>on</strong>s between the bits of matter, some<br />

bits slowly loose angular momentum <strong>and</strong> move closer <strong>and</strong> closer to the center.<br />

Eventually, they either fall through the horiz<strong>on</strong> of the black hole or hit the<br />

surface of the neutr<strong>on</strong> star.<br />

In cases where the compact object is in a binary system, the matter flowing in<br />

comes mostly from the shining star. This process makes the accreting matter<br />

into a disk, as shown in the picture 8 below. This is why astr<strong>on</strong>omers often talk<br />

about ‘accreti<strong>on</strong> disks’ around black holes <strong>and</strong> neutr<strong>on</strong> stars.<br />

8 This picture was created by Jillian Bornak, a past PHY312 student.

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