Calculating a Black Hole’s Mass and Radius

The first and probably most obvious measurable property of a black hole is its mass. Clearly, black holes must be extremely massive and also dense, since each consists of most of the matter of a giant star compressed into an unimaginably tiny space.

Kip Thorne of the California Institute of Technology is one of the leading experts on black holes and their strange properties.

Yet how can scientists on Earth measure the mass of a black hole or other body lying trillions of miles away? That depends on whether any stars or other large bodies happen to lie near the black hole. If it is floating through space alone, far from any such objects, scientists will have no way to measure its mass.

In contrast, if the black hole and a star are orbiting each other (actually, each orbiting a common center of gravity), scientists can use the formula for universal gravitation to compute their masses. First, using sensitive instruments and mathematics, they measure the distance between the two objects. Then they compute their orbital velocity (the speed at which they move in orbit). Finally, they plug these figures into an equation that determines the mass. As Thomas Arny says, this method “can be used to find the mass of any-body around which another object orbits. Thus, gravity becomes a tool for determining the mass of astronomical bodies.” In the case of black holes, scientists often express their masses in multiples of the Sun’s mass. A black hole is said to contain 8, 12, 20, or some other number of solar masses.

The mass of a black hole, which is measurable, directly affects the nature of other properties of the hole that are not measurable. One of these is the size of its Schwarzschild radius. The easiest way to understand this fundamental property of a black hole is to visualize the hole moving through space. From time to time, it encounters gas, dust, asteroids, and other forms of matter, which are naturally attracted by its huge gravitational pull. When the matter gets close enough, it is torn apart and reduced to atoms; then it is sucked into the black hole, where the debris spirals into the bottomless gravity well, never to be seen again.

The crucial part of this scenario of annihilation is that, on its way into the black hole, the matter passes what might be called “the point of no return,” which scientists call the event horizon. As long as the matter manages to stay outside the horizon, it has a chance of escaping. Once it crosses the horizon, however, it will disappear into the black hole’s gravity well. The distance from the center of a black hole, called the singularity, to the event horizon is the Schwarzschild radius, named after German astronomer Karl Schwarzschild, who discovered it in 1915.

Mathematical equations show conclusively that this radius will vary according to a black hole’s mass. The more massive the hole, the longer the Schwarzschild radius, and conversely, the less massive the hole, the shorter the radius. A black hole of one solar mass will have a Schwarzschild radius of 1.86 miles; and a hole of ten solar masses will have a radius of about 20 miles. In the latter case, therefore, the point of no return for any matter approaching the black hole lies 20 miles from the singularity, or center.

Tragedy Cuts Short a Brilliant Career

German astronomer Karl Schwarzschild, who was born in 1873, made major contributions to knowledge about superdense objects and their effects on space and time. He became director of the Astrophysical Observatory in Potsdam in 1909. In 1915, while serving his country in World War I, he heard about Einstein’s work on the theory of general relativity. Schwarzschild contacted Einstein and kept him informed about his own efforts to describe the geometry of spacetime around a superdense object occupying a single point, or singularity. Among Schwarzschild’s mathematical discoveries was that the singularity would be separated from the event horizon by a certain distance, which scientists later named the Schwarzschild radius in his honor. Tragically, he contracted a skin disease while in the military and grew gravely ill. Einstein presented his colleague’s groundbreaking ideas to the scientific community only months before Schwarzschild died in May 1916 at the age of forty-two.