How Big is a Black Hole?

by Nicholas Mee on January 10, 2022

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A sufficiently massive star may collapse under its own gravity at the end of its life until it becomes so dense that nothing can escape its gravitational attraction. It has become a black hole.

This is the ultimate collapsed body. The black hole does not have a solid surface. It is simply a spherical region of space from which nothing can escape – not even a ray of light – hence the name. The sphere that defines the boundary of the black hole is known as the event horizon of the black hole.

An artist’s conception of the black hole known as Cygnus X-1. Credit: NASA.

The existence of black holes was predicted by Einstein’s theory of gravity – general relativity – which was published in November 1915, during the First World War. Einstein’s theory accounts for gravity as spacetime curvature. According to the theory massive objects curve the spacetime in their vicinity and then other objects, including light beams, travel along the shortest paths through this curved spacetime. Solutions of Einstein’s equation describe the shape of spacetime surrounding a massive body. The equations were solved in the very important case of a spherically symmetric mass by the German theorist Karl Schwarzschild shortly after Einstein’s theory was published. What makes his achievement all the more remarkable is that Schwarzschild was under fire in the trenches on the Eastern Front and was suffering from a serious skin disease called pemphigus. Tragically, Schwarzschild died just a few months after his ground breaking work was published.

Schwarzschild’s solutions describe the shape of spacetime inside and outside a spherical mass which is very useful as many objects such as stars and planets are spherical to a good approximation. Schwarzschild’s exterior solution tells us that for a sufficiently large collapsed mass there is a radius at which space becomes so warped that all roads lead inwards – so even light that is radiated outwards is directed inwards towards the centre. It was several decades before the implications were fully understood – but this is what we know as a black hole. The radius of no return is known as the Schwarzschild radius, and it is given by this formula:

Rs = 2GM/c2

where R subscript s is the Schwarzschild radius, G is Newton’s gravitational constant, c is the speed of light in a vacuum, and M is the mass of the collapsed object. This formula allows us to work out the size of a black hole. So what does it tell us? If the Earth collapsed under gravity to form a black hole, the radius of its event horizon – its Schwarzschild radius – would be about nine millimetres. So we could easily hold it between our forefinger and thumb. Not that this would be advisable, of course. We would rapidly end up swallowed by the black hole.

The Schwarzschild radius of Jupiter is about three metres, so if Jupiter collapsed to form a black hole, it would be a lot bigger than an Earth-mass black hole, but it would still fit within a house.

The mass of the Sun is just over one thousand times the mass of Jupiter, so the Schwarzschild radius of the Sun is about three kilometres.

There is no known way to compress a mass the size of the Earth, Jupiter or even the Sun into a black hole. But it is useful to remember that the Schwarzschild radius of the Sun is three kilometres. The mass of a black hole is usually quoted in solar masses, and as the Schwarzschild radius is proportional to the mass, this enables us to estimate the size of the black hole as a multiple of the Sun’s Schwarzschild radius.

The first good candidate for a real black hole. An artist’s impression of this black hole is shown above. It is known as Cygnus X-1. It is an intense source of X-rays in the constellation Cygnus – hence the name. The X-rays are emitted from matter that streams away from a blue giant companion star to form an extremely high temperature accretion disc before eventually spiralling into the black hole, as depicted in this image. Recent estimates suggest that the mass of the Cygnus X-1 black hole is about twenty solar masses. This means that its Schwarzschild radius is about sixty kilometres, so it would easily fit in Central England. 

Real world black holes, such as Cygnus X-1, spin at a frantic pace and this reduces the size of their event horizons. The reason for this is that a spinning black hole is like a spacetime tornado. As it spins, the black hole drags space around with it. This means that light travelling around the black hole in the same direction that it is spinning can approach closer to the centre of the black hole and still just about make its get away. So the event horizon of the black hole is closer to its centre than would be the case for a non-rotating black hole of the same mass. Spinning black holes are described by the Kerr solution to Einstein’s Equation. It was discovered by the New Zealander Roy Kerr in 1963 and it describes the shape of spacetime around real world rotating black holes. The Kerr solution tells us that the radius of the event horizon of a black hole depends on the rate at which it spins, but it is always between a half and one times the Schwarzschild radius.

We now know of much bigger black holes than Cygnus X-1. We have even detected the spacetime ripples produced by the cataclysmic collisions and mergers of pairs of black holes. The first such signal was detected in September 2015 by LIGO in the United States and gravitational wave observatories are now routinely detecting these gravitational wave signals. So far the biggest black hole merger that has been detected was between black holes of 66 and 85 solar masses to produce a 142 solar mass black hole that would be about seven times the size of Cygnus X-1.

The LIGO gravitational wave observatory at Livingston, Louisiana. Credit: LIGO.

But this is a mere baby. We now know that there are supermassive black holes at the centres of every galaxy – even our own Milky Way galaxy. Reinhard Genzel and Andrea Ghez and their teams have monitored the stars at the very centre of the galaxy and have produced conclusive evidence that they are orbiting a black hole with a staggering mass of 4.3 million solar masses. The Schwarzschild radius of this monster is 14 million kilometres – so this black hole is huge. By comparison the radius of the Sun is 700,000 kilometres, so the black hole at the centre of our galaxy has about twenty times the radius of the Sun.

The orbits of the stars at the centre of the galaxy as mapped out by Andrea Ghez and her team. Credit: Keck/UCLA Galactic Center Group.

But, even this monster pales by comparison with other supermassive black holes that we know are out there. The Event Horizon Telescope recently published an image of the accretion disc around the supermassive black hole at the centre of the giant elliptical galaxy M87.

The mass of this black hole is estimated to be about 6.5 billion solar masses, which gives it a Schwarzschild radius of almost 20 billion kilometres. Neptune is the outermost planet in the solar system – it orbits the Sun at a distance of 4.5 billion kilometres. So the orbits of all the planets around the Sun would easily fit within the M87 supermassive black hole.

An artist’s impression of a supermassive black hole.
Credit: ESO/L. Calçada.

And this black hole isn’t even the biggest black hole that we know of.


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