# 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,
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
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.
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
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
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Again, that’s Brilliant.org/minutephysics which lets Brilliant know you came from here.
Good luck problem solving!