Work and World

# The Black Hole Tipping Point

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
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