The aim of this report is to give a clear summary of the main points surrounding the often confusing topic of black holes.

Philip Jenkins

Physics Coursework - Research and Report

BLACK HOLES

The aim of this report is to give a clear summary of the main points surrounding the often confusing topic of black holes. Many people are fascinated by these phenomena, but how many could say for sure what a black hole is?

So to start, I feel that a proper definition of a black hole is needed.

A black hole is an object from which nothing can escape, or to put more accurately, a black hole is an object small enough that its escape speed exceeds the speed of light.

“Escape speed?” You may ask, but more on this later. The first part of the report will concentrate on the history of the black hole theory and how it has developed over time.

History Lesson

The name “black hole” was created in 1967, but the idea is not so modern as the beginnings of the theory can be traced back to the eighteenth century. The theory of the black hole can be linked to the theory of light, which culminated in Einstein’s General Theory of relativity in the early twentieth century. This theory married space and time, and could only be achieved thanks to the relative weakness of gravity as a force. As you approach a black hole, gravity becomes much stronger, so it dominates over any other force distorting the common rules of space and time, causing weird and wonderful phenomena which one cannot imagine or relate with.

Einstein’s theory can be adapted to black holes (see later), but before his era, the notion of black holes had been toyed with.

It was a clergyman called John Michell who initially proposed the idea of black holes way back in 1783. He calculated that an object could have a large mass (equal to the Sun or even greater) and could also be very small - thus creating an immense gravitational pull. He also stated that there could be a large number of these objects and they would remain invisible due to their relatively minute (7 km) size.

Thirteen years later, the famous scientist Pierre Laplace also found these facts, independently of Michell. Due to his stature however, he was credited for the initial discovery, until Michell was later found to have discovered it first.

After this promising initial theory, the idea of black holes became unpopular for many years, only re-emerging with Einstein and his contemporaries in the twentieth century.

The Physics Behind Black Holes

Black holes have some interesting physics behind them, mostly based on energy.

There are 2 fundamental equations which we can start with:

This is the simple equation for kinetic energy - one half mass multiplied by velocity squared.

This equation is the gravitational potential energy of an object of mass m at a distance R from a large object (or “parent body”) mass M (e.g. a planet or a star).G is the gravitational constant 6.67 x 10-11 Nkg-2m2.

Black holes have an immensely strong gravitational attraction, but what could escape it? How could we work out what speed is needed to escape?

As stated before nothing can escape a black hole - so there is no attainable speed which an object can reach for it to escape from the potential well of a black hole. This is useful for defining an object that might be a black hole, as the earlier equations can be used to determine whether or not the object is a black hole if the speed needed to escape it is greater than or equal to the speed of light.

If an object is to escape from any parent body, then its initial kinetic energy must be greater than the maximum gravitational potential energy which the parent body can provide. To calculate the escape speed needed for any parent body, the following method can be used.

Using the two equations on the previous page and the assumption that the loss in kinetic energy is equal to the gain in gravitational potential energy the following statement can be written:

multiply by 2:

cancel m and square root:

So this will give the speed necessary to escape a parent body of mass M and radius R. If the value of v is greater than the speed of light, this would signify that the object is a black hole.

The radius which an object of mass M needs to be compressed to in order to become a black hole can also be calculated.

Take the final equation:

For a black hole:

where c is the speed of light.

This can be rearranged thus:

So the radius ...

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multiply by 2:

cancel m and square root:

For a black hole:

where c is the speed of light.

This can be rearranged thus:

So the radius of a black hole is very small indeed, as c2 is on the bottom of the equation this meaning that the number produced for R is always going to be a minute fraction of the mass of the object.

Their diminutive size means that black holes are impossible to view directly. Instead other methods have been developed for their detection.

Detecting Black Holes

Black holes are invisible, so you may think that it is impossible to detect their presence if you can’t see them. This is not the case however, as by measuring the effects a black hole’s gravity has on its surrounding space (and any objects therein) is fairly simple, and there are several characteristics which are thought to be unique to black holes.

1. Mass
Due to its huge mass, a black hole causes strange things to occur around it. If a black hole is near another star, then the star’s matter is pulled towards the black hole. The black hole is said to accrete matter into itself - which forms an Accretion Disc around the circumference of the black hole

Picture courtesy NASA

By observing what occurs to objects around a black hole (factors such as rotation speed) the mass of the black hole can be estimated using Keplar’s Third Law. The picture below shows the core of a galaxy called NGC 4261. The brown disc at the centre is the same size as our solar system, but the body which it is orbiting around weighs 1.2 billion times as much as the Sun. For this mass to be compressed into such a relatively small space the most feasible explanation is a black hole.

Photo courtesy NASA/Space Telescope Science Institute
Credit: L. Ferrarese (Johns Hopkins University) and NASA

2. Discharge of X-Rays and Jets

When matter has been drawn into a black hole’s accretion disc, the gravitational force accelerates the material to immense speed, causing it to heat up to millions of degrees Kelvin. This superheated material emits strong x-ray radiation and can be detected using a high power x-ray telescope. The most powerful x-ray telescope is the Chandra X-ray Observatory, in orbit above the earth.
Another form of discharge from black holes is a jet, a high speed blast of material which can be detected with either radio telescopes or regular high power telescopes. The picture on the next page shows an image of a jet from galaxy M87. The top left and bottom images are from radio telescopes and the top right image is from the Hubble Space Telescope.

Photo courtesy NASA/Space Telescope Science Institute
Credit: NRAO, NSF, Associate Universities, Inc., NASA, and John Biretta (STScI/Johns Hopkins University)

3. Gravity Lens

It was Einstein’s General Theory of Relativity which predicted that gravity could bend space. This theory has since been proven correct, and large distortions in space have been attributed to black holes. Below is a picture from the Hubble Space Telescope. Notice that the rightmost image seems to show 2 stars very close to each other. In fact there is only one star, but there is an object in between the star and the telescope. This object has bent the space (and therefore the light) as a lens would - the explanation is a black hole. Invisible but very powerful.

Photo courtesy NASA/Space Telescope Science Institute
The Structure of a Black Hole

There are two proposed types of black hole:

Schwarzschild

Kerr

Both the Schwarzschild and Kerr black holes have two features in common, the singularity and the event horizon. The singularity is what remains of the core of the collapsed star. It sits right in the centre of the black hole, in a huge gravitational potential well. Below is an artist’s impression of a black hole - notice the huge dip in the fabric of space time at the singularity.

Photo courtesy NASA

The event horizon is the periphery of the black hole, and is the point of no return for any object which travels past it on its way into the black hole. Nothing can escape a black hole once it has crossed the event horizon.

The Kerr black hole is the most common, and the main difference between this and the Schwarzschild hole is that it rotates. A normal star rotates, so when it collapses its momentum is conserved, causing the core to rotate. This, coupled with the massive gravity causes the space around the event horizon to be pulled in a circular motion around the black hole. This area of spinning space is called the Ergosphere. It is usually egg-shaped and it is what causes an accretion disc to form if material is sucked into it. The edge of the ergosphere, where it meets ordinary still space, is called the Static Limit.
With current physics knowledge, it is not known what exactly happens to an object after it has crossed the event horizon. The gravity from the black hole is such that time itself will stop for any observers outside the hole trying to look in - so observing an object falling into a black hole could not be achieved. It is extremely doubtful that anyone could survive near a black hole due to the immense forces involved.

Cygnus X-1 - A Black Hole Candidate

The most likely candidate for a black hole is called Cygnus X-1. It was found in the early 1960’s as a strong source of x-rays in the Cygnus (swan) constellation. The suspected black hole is part of what is known as a binary system, where planets orbit two stars which are very close to each other. One of the stars in this particular system has collapsed into Cygnus X-1. The other star, snappily titled HDE 226868, is being pulled into the black hole forming an accretion disc.

It was observed that the star increased in brightness every 5.6 days. This was attributed to the fact that the star appears egg shaped when the black hole and the star are viewed side-by-side. The increased surface area of the star in this position means that more light is produced than when the black hole is in front or behind the star, when the star will appear circular.

However, not only a black hole could produce this effect. There could be a very small star next to HDE 226868, or a neutron star (left behind when a smaller star dies). There was however, evidence to suggest that this was not the case.

A small star was a reasonable explanation for the change in brightness of HDE226868, as the huge distance of the Earth from the Cygnus system would mean that such a small object would not be distinguishable. Further examination of the evidence though proved this idea unlikely. The object was emitting powerful x-rays - something which small stars are not commonly known to do.

So with a small star discounted, the scientists went on to disprove the theory of a neutron star. Neutron stars rotate very fast, and emit light at regular intervals (they are also known as pulsars for this very reason). Below is a graph showing the emissions of the neutron star Hercules X-1.

Note how the wave period is almost constant, and the amplitude is also fairly regular - this is a very typical example of a neutron star. When this graph is compared to the one from Cygnus X-1, it becomes obvious that it is not a neutron star.

The other factor which swung Cygnus X-1 in favour of the black hole theorists was its mass. It was estimated that Cygnus X-1 had a mass of around 9 solar masses, that is to say that it is 9 times more massive than the Sun. Neither a neutron star or a white dwarf (the other 2 types of dead star) could weigh so much. A white dwarf cannot weigh more than 1.4 solar masses1 and a neutron star cannot weigh more than 3 or 4 solar masses.
So scientists are almost completely convinced that Cygnus X-1 is a black hole, and the proof of this would be the culmination of years of theory and data collection.

1Subrahmanyan Chandrasekhar discovered this limit - it is now known as the Chandrasekhar limit.

2J.R. Oppenheimer and G.M. Volkoff determined the upper mass of a neutron star. It is called the Oppenheimer-Volkoff mass.

Going Further

A black hole of 9 solar masses is fairly typical, however some cosmologists think that there are hugely massive black holes, with masses that exceed a billion times our own Sun. Some think that these holes are at the centre of spiral galaxies (such as our own milky way) and the spiral effect is like a hue accretion disc surrounding the hole.

There are theories (mainly speculative) that link black holes with the science fiction world of wormholes. Einstein stated that wormholes could exist, but never proved it. A wormhole is said to be a way of travelling from one area of space to another very quickly, and speculation suggests that black holes may be linked together. This would certainly be useful for interstellar travel in the far future if proven correct, but the majority of the science world seems sceptical. However if a probe was sent into a black hole and was found to have emerged in a different section of space, this would definitely have implications on the future of space travel.

The fascination with black holes was mainly fuelled by Hollywood in the 1950’s and 60’s, when science fiction was in its heyday. Films and magazines depicted black holes as cosmic vacuum cleaners, objects of destruction which eventually sucked up everything. This however is not the case, for example if the Sun somehow turned into a black hole, the Earth would continue to orbit it as before, apart from the lack of light
This general ignorance and fear amongst the public seemed to make black holes among the most popular areas of physics, but if more knew the truth, then maybe black holes would not have the same mysterious effect.

Conclusion

There is overwhelming evidence, in my opinion at least, that black holes exist. What lies inside a black hole is not for me to speculate. No matter what occurs inside a black hole it is sure to be fascinating whenever it is discovered. After researching this topic, I have found a great deal of information which was new and surprising to me and it has consolidated my interest in the topic.

"The universe is not only stranger than we imagine, but stranger than we can imagine."

J.B.S Haldane

Bibliography

“A brief history of black holes” Richard Buckley (2000) From “Physics Review” pp 18-20

A useful source, containing the mathematics behind black holes and a concise history of their discovery.

“The identification of Cygnus X-1 as a black hole” Steve Degennero (1996)
http://www.owlnet.rice.edu/~spac250/steve

Fantastic case study of Cygnus X-1, however needed some careful reading due to the fact that this included university level physics.

“The look of a black hole”
http://www.leyada.jlm.k12.il/proj/black/look.htm

A smaller web site, with no obvious author, so reliability not the greatest.

“How Black Holes Work” Craig C. Freudenlich, Ph.D. (2001)
http://www.howstuffworks.com/black-hole.htm

The best source on black holes which I found. Many of the pictures in this report are from this site. Also included many useful links for black hole resources.

“Imagine the Universe - Black Holes”
http://imagine.gsfc.nasa.gov
Imagine the Universe is a service of the High Energy Astrophysics Science Archive Research Center (HEASARC), Dr. Nicholas White (Director), within the Laboratory for High Energy Astrophysics at NASA's Goddard Space Flight Center.

The Imagine Team
Project Leader: Dr. Jim Lochner
Curator:Meredith Bene Ihnat
Responsible NASA Official:Eunice Eng

A good Advanced Level site, with another case study on the black hole Cygnus X-1. NASA backing makes this one of the most reliable sources.

The aim of this report is to give a clear summary of the main points surrounding the often confusing topic of black holes.

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