29:50 Modern Astronomy
Fall 1999
Lecture 20 ...October 11, 1999
Black Holes II

New visions in astronomy. Today Carinae. A highly massive, evolved star. If we know so much about massive star evolution, we should be able to catch one about to blow up. Carinae may be such a star.
Optical picture of Eta Carinae .
X-ray picture .

Last time I began talking about Black Holes as the most extreme end products of stellar evolution. I first talked about understanding Black Holes from ideas of classical physics; i.e. an object with so much mass crammed into such a small volume that the speed of escape was the speed of light.

The second, and more physically correct expression was via the laws of General Relativity. There are two ingredients of General Relativity. The first is that objects move between two points in spacetime A and B on the shortest path between the two points. The fancy mathematical term for this is a geodesic.
The second principle of GR is that presence of mass warps or bends spacetime, so that the geodesics are no longer straight lines, but instead more complicated, curved paths. The larger the amount of mass in the smaller the region, the stronger the warping or bending of spacetime is. The central statement of General Relativity is the Einstein Field Equation which directly relates the curvature of spacetime to the presence of mass.

This is a difficult concept to grasp, and the ``gimmicks'' that are presented in introductory astronomy groups are really more metaphors for what is really happening. Nonetheless, they do convey something of the physical reality.

Motion along geodesics in curved spaces can look strange. The classical (and best illustration) involves motion along a two dimensional space embedded in a three dimensional one. This is characterized by a sheet of paper or the latex sheet here in the demonstration.
Demonstration of sheet.
Particularly note the overhead view which gives us a truly 2D view of things. If the sheet is is uncurved, or flat, the geodesics are straight lines and things are simple.

If however, the two dimensional surface has curvature, the geodesics are not straight lines in two dimensions. The most familiar illustration of this the paths taken by airplanes on transcontinental flights. Plotted on a two dimensional map, it looks they go way north out of their way. However, you can be sure the airlines are quite interested in flying the minimum distance (with the minimum fuel use) possible. They go on great circle routes which are geodesics on the surface of a sphere.

Let's do a little General Relativistic dynamics here with this experiment. As the ball moves across this sheet, it takes the path of minimum available distance from point A to point B. If there is no curvature, this is a straight path, and everything is simple.

If we now curve the space in which it is moving, it is still following a geodesic. However, as viewed things in a two dimensional space (as it appears on the TV screen) the trajectory appears more complicated and one would interpret its motion as due to a force between the large mass and the small one. I would recommend further reading in the book, beginning on p468, for a description of the curvature of spacetime and Black Holes.

The idea of mass warping spacetime may seem unbelievably odd, but there is direct observational evidence for it. Light does not have mass, so according to Newton's ideas of physics, light should not feel a gravitational force. However, if space is warped like a sheet of plastic, then the light rays will be bent by the warped spacetime. Einstein realized that the positions of stars close to the Sun should be displaced or ``deflected'' from their normal positions.

The obvious problem with carrying out this experiment is that normally one can't see stars in the vicinity of the Sun. This is not the case during a total eclipse of the Sun, so the experiment which verified this prediction was first carried out in 1917.

Now above I said that the greater the mass concentrated in a smaller region, the larger the warping or curvature of spacetime. Einstein's theory of General Relativity predicts that for a sufficiently large concentration of mass in a sufficiently small volume, it is possible for the curvature to become infinite. Infinite curvature can be visualized as an infinitely deep hole, or a rip in the metric.

The conditions for this infinite curvature to occur are the same as for the Schwarzschild radius described above. Thus Black Holes can be viewed as a puncture or abyss in spacetime.

Obviously this all sounds like a pretty extensive yarn, but we have seen in the past couple of lectures that astronomers have found the other weird byproducts of stellar evolution. How would one go about looking for Black Holes?

The approach which has been used is to look for very massive, compact objects. By compact I mean the following. We know of stars which are 10, 20, 40 solar masses. But these objects are extremely luminous and very large in size. They are larger in radius than our Sun. Massive objects which are normal stars are easy to pick out. Black Holes are expected to be extreme end products of stellar evolution and therefore very small in size. As mentioned Monday, a Black Hole of one solar mass would only be a couple of miles in diameter. Furthermore, these Black Holes wold not be expected to radiate starlight.

We can determine the masses of objects in binary star systems. Throughout this semester we have used the circular orbit equation to calculate the mass of objects,

Therefore if we measure the orbital speed and orbital radius of a small (low mass) object orbiting a large mass object, we can determine the mass of the massive object.

It turns out the general problem in which the two objects have comparable masses is not much more difficult, although I won't go through the algebra steps here. The net result is that one can often determine the masses of both objects in a binary star system.

Given this, the idea is to look for binary star systems in which one of the stars is a product of advanced stellar evolution. The charming term which is used in the astronomical literature for such objects is ``degenerate objects''. From measurements of the orbit, one solves for the mass of both objects. At this point you can apply some criteria to determine whether one or both objects is a Black Hole.

If the mass of the degenerate object is less than or equal to 1.4 , we would conclude the degenerate object is a White Dwarf, since WD's can exist up to this mass.

What if it is more than 1.4 . As I have mentioned in passing, astrophysicists are not exactly sure what the maximum mass a neutron star can have. The theoretical estimates range from a low value of about 1.5 to an extreme high value of about 2.7 . If one finds a binary system with a degenerate object, and the degenerate object has a mass less than or equal to 2.7 , the degenerate object can be identified as one of the ``garden variety'' objects we have talked about so far, White Dwarfs or Neutron Stars.

If, however, one finds a binary system with a degenerate object more massive than 2.7 , the only conclusion (based on current astrophysical knowledge) is that the degenerate object is a Black Hole.

This is a fairly straightforward recipe for finding something, and astronomers have known about it for some time. An intensive search for Black Holes began about 30 years ago. Already in the 1970's there was a very good candidate, which is called Cygnus X-1. Its name means that it is in the constellation of Cygnus, and it is called X-1 because it is the brightest emitter of x-rays in that part of the sky. This means that there must be lots of extremely hot material in this star system (Why?) so that unusual things are happening.

The stellar component of Cygnus X-1 is a spectral class O supergiant. It is in a binary system with an orbital period of 5.6 days, and the O supergiant is really being swung around in its orbit. From analysis of its orbit, we conclude that the mass of the companion is in the range of 10 - 15 . If this were a normal star, we should see evidence of a B or O star light, but we don't. There is also evidence that the shape of the O star not spherical, but teardrop-shaped, as if it were subject to an incredibly strong gravitational field.

The Picture of Cygnus X-1 which arises is this.
Picture of Cygnus X-1.
Here we see the O supergiant, with a stream of matter being pulled off by the extremely strong gravitational field of the object at right. This matter flows into a pancake-like disk of hot matter called an accretion disk. It is this which produces the x-ray emission. Finally, the matter flows into the degenerate companion.

The mass of this compact companion is well above the threshold for a Black Hole. For many years, Cygnus X-1 was the only object of its kind, although astronomers were suspicious of many other objects. Nowadays however, astronomers are certain of a total of ten systems.

Transparency with Black Hole candidates
This table lists a number of characteristics, such as the distance of the system, the orbital period, and the type of star which is the companion. The most important column is the mass of the degenerate object. As you can see, in all cases, it is well above the 2.7 maximum neutron star mass.

So, barring the unknown existence of a new type of collapsed star, these objects are Black Holes, and the most extreme end products of stellar evolution do indeed exist.

A final question is a simple one. Why do black holes shine? By the original definition, we would expect them to emit no light and be invisible. Yet they are some of the most conspicuous objects in the sky. The reason for this is the emission of the accretion disk, or hot pancake of matter that is orbiting and slowly spiraling into the black hole.

Galactic-scale black holes.

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