Jheri tagged me to a challenge making its way around:

Tell me something amazing about your field in a sentence and then spend five or ten minutes on something that isn't well known.

It's probably meant to be a video chat challenge, but I'll write and go with astrophysics. You might want to try this for your own field or maybe a passion.

It takes more energy to melt a snowflake than all of the energy received by all of the radio telescopes on Earth since they were invented in 1932.

Now for something that would take about five minutes to talk about. Something most people don't know about black holes using nothing more than a bit of first semester high school physics and your imagination. (If you don't remember much of that you can skip to the last paragraph.)

Let's say you have a rocket and want to escape the Earth's gravity - really escape rather than go into orbit. There are two energies to consider. The energy of motion of your rocket - its kinetic energy, and the energy from the gravitational attraction between your rocket and the Earth - the gravitational potential energy.

The kinetic energy is just 1/2 mv^{2}, where m is the mass of the rocket and v is its velocity. The gravitational potential energy is -GMm/r where G is the gravitation constant, M is the mass of the Earth, r is the radius of the Earth (the distance from the center of the Earth to the surface where you launch the rocket from), and m is the mass of your rocket.^{1} So the total energy is

E = 1/2 mv^{2} - GMm/r

If you keep increasing the velocity you get to a point where E becomes is no longer negative. When that happens you've escape Earth's gravity. The escape velocity is when the kinetic energy equals the gravitational potential energy. Setting them equal and a bit of algebra we get.

v^{2} = 2GM/r about 11.2 kilometers per second for Earth.

You probably saw this in your physics class and are probably wondering what this Newtonian result has to do with black holes. This is where it gets interesting.

About a century after Newton figured this out an English cleric named John Michell wondered what would happen if you kept increasing the mass M. He knew the speed of light was large, but not infinite. If you were on a body that was massive enough there was a point where the escape velocity would be greater than the speed of light! People didn't know the speed of light was constant or that light could be influenced by a mass, but Michell had discovered idea of a black hole in 1784. Remarkably the result is within a factor of two of the value you get using General Relativity.

Let's play around a bit. c^{2} = 2GM/r is the classical escape velocity for light. Rearrange to get (on the path to something interesting)

M/r = c^{2}/2G

Now consider the density (we're getting close to the fun bit!). The mass of an object is proportional to its density times its volume or M ~ ρ r^{3} (ρ is the density). Now substitute in and solve for density

ρ ~ c^{2}/2Gr^{2}

You're probably used to thinking of a black hole as an incredibly dense object. But density falls as the square of the radius of the black hole. So as black holes get bigger the density you need gets smaller! A black hole with the mass of the Sun has a radius of about 3 kilometers and is extremely dense. The supermassive black hole at the center of M87 that was imaged about a year ago has a diameter roughly four times that of Neptune's orbit - what we usually think of as our Solar System would easily fit inside - but its density is a bit less than that of water!

__________

^{1} The negative sign is from the convention that the Earth has no influence over the rocket at an infinite distance. The negative potential means the rocket is bound to the planet. It's trapped until it has enough energy to escape.

So some black holes would float? Hm.

Posted by: Horace Dediu | 04/28/2020 at 03:20 PM

Yes .. so would Saturn:-)

Posted by: steve | 04/28/2020 at 03:36 PM