The Black Hole Tipping Point

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Black holes are very heavy astronomical objects (which gives them all sorts of cool behaviors and properties), but to make a black hole it takes more than just a lot of mass. It takes a lot of density, that is, a lot of mass crammed into a sufficiently small space. Precisely how much mass, or how small it needs to be crammed, will vary. Black hole formation is complicated, but there are essentially two possible paths: start with a fixed amount of matter and compress it smaller and smaller until it reaches the tipping point where it’s dense enough to become a black hole (this is how supernovas turn the core of supergiant stars into black holes), or keep adding matter to an existing object until it reaches the tipping point where it’s so big it becomes a black hole (for example, if two neutron stars merge they can form a black hole).

You can do a very rough calculation of these tipping points yourself knowing just two things: the equation for what’s called the Schwarzschild radius of a black hole, and the equation for the mass of a spherical object. The Schwarzschild radius is the distance from the center of a black hole below which nothing, not even light, can escape ; you may have heard it called the “event horizon” and how big it is depends only on the black hole’s mass; The G and c squared here are constants that help convert from kilograms to meters, so the equation can also be written in SI units as 1.49*10^-27 times mass, but the important thing is that the heavier the black hole, the bigger the Schwarzschild radius. Schwarzschild, by the way, means “black shield” in German, which is bizarrely appropriate for the physicist after whom black hole event horizons are named! Now let’s blindly use this equation to start calculating Schwarzschild radii for other objects: the Schwarzschild radius of the sun is about , the Schwarzschild radius of the Earth is about 1 cm , and the Schwarzschild radius of a cat is about 0.01 yoctometers. What do these mean? Well, nothing, since the sun, the earth, and the cat aren’t black holes. Yet. In principle, any object that gets squeezed down to around the size of its Schwarzschild radius will become a black hole. It’s hard to imagine squeezing the whole earth until it literally becomes this big; but when supergiant stars die, their supernovae explosions are so powerful they can compress the star’s already-dense cores past their Schwarzschild tipping points to become black holes. But suppose you don’t have access to supernova-strength compression; you can instead make a black hole by adding more mass to your object.

The equation you want is here: it describes how the mass of a spherical object is equal to the density of the material in question times the volume it takes up. Or, rearranged a little bit, it says that the radius of that sphere is proportional to the cube root of its mass. Now, the Schwarzschild radius of an object is proportional to its mass directly, no cube roots involved, so as an object’s mass increases, its Schwarzschild radius will increase much faster than its actual radius. Double the mass, double the Schwarzschild radius, but only 1.26 times the actual radius. Now, remember, the Schwarzschild radius starts off really really small and doesn’t really mean anything until the entire object can fit inside the Schwarzschild radius; but it’s mathematically guaranteed that straight lines eventually catch up to cube roots, so we just need to keep adding matter to the earth – eventually it will fit inside its own Schwarzschild radius and collapse into a black hole! For the Earth, which has the density of rock , this tipping point occurs at a size of around 140 million kilometers – basically the distance to the sun.

Though to be honest rock definitely isn’t strong enough to sustain the pressure necessary and we’d probably collapse into a neutron star long before getting that big. As for neutron stars themselves, the tipping point numbers tell us that they will become black holes if they get bigger than about 6 times the mass of the sun, and about 20km in size ! This is a simplified result from a simplified equation –I mean, neutron stars aren’t constant density, for one–, but it’s pretty darn close to both astronomical observations, and much more sophisticated theoretical predictions for the maximum possible mass (and size) of neutron stars. Only off by a factor of two or three. So to recap: if you want to turn your cat into a black hole, you have two options: either compress it down to a trillionth the size of an atomic nucleus, or cover it in a pile of other cats that reaches beyond the sun. You may have noticed I just said “beyond the sun”, not “almost to the sun” as was the case with the earth. That’s because cats aren’t as dense as rock, so they’ll have a different black hole tipping point – I challenge you to figure it out using the Schwarzschild radius and mass of a sphere equations and leave the answer in the comments.

And after that, you could head over to this video’s sponsor, Brilliant.org, for more interactive quizzes and mini courses on physics and math. In fact, they even have an introductory quiz specifically on black holes and gravity which guides you through deriving the Schwarzschild radius formula and other cool stuff like that, with just the right balance between hand-holding and creative problem-solving – I’ll link to it in the video description . And the first 314 people to go to either that link or Brilliant.org/minutephysics will get 20% off a premium subscription to Brilliant. Again, that’s Brilliant.org/minutephysics which lets Brilliant know you came from here. Good luck problem solving!

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