# Space and Time

## Part III: Orbits

### By Mark Lawrence

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Tidal Force: 2M/r3

Orbital Period: 2πr sqrt( r / M )

Orbital Period, proper time: 2πr sqrt( (r - 3M) / M )

Orbital Velocity: sqrt( M / r )

Escape Velocity: sqrt( 2M / r )

## Orbits and Escape

here are some interesting things about black holes with respect to orbits and escape. Obviously this is correct on a large scale - the whole point of black holes is that anything that goes in never gets out. Maybe. Well, more on that later. But it's also true that things that get near can get in real trouble.

Because of the space program, we all have an idea of what an orbit is, an intuition. This is new, if you asked people 100 years ago about orbiting the Earth, I think an average person wouldn't have a clue what you were talking about. To orbit a planet, you need to build up some speed. Close up orbits require more speed than orbits that are more distant. Also, there's another idea, which is to escape entirely. The space shuttle has plenty of power available to orbig the Earth, but no where near enough to even get to the moon, much less to escape the Earth's gravity entirely and make it to Mars or further. Things near black holes are even more complicated.

The basic rules are pretty simple. To orbit a planet of mass M at a radius from the center of the planet of r, you need a speed of the square root of M/r. That's all there is to orbital velocity. If you're sitting on a planet of mass M, and you're sitting a distance r from the center of the planet, to escape the planet entirely you need a speed of square root of 2M/r. That's it for escape velocity.

Let's work out some real numbers. The Space Shuttle and the Space Station both orbit at about 400 km. We have to add the Earth's radius onto that, another 6400 km. Orbital velocity is square root( M/r ) = square root( 1.1e32 / ( (6400+400) / 4.04e-38 ) )  = 2.56e-5. This is a velocity, so in our natural units it comes out as a fraction of the speed of light. To get more normal units we multiply by the speed of light. This is 2.56e-5 * 3e8 = 7670 meters / second = 17,000 miles per hour.

Communications relay satellites and TV broadcast satellites orbit in what is called Geosynchronous orbit. This means they orbit high enough so that their orbit the Earth once each day. Since the Earth turns once each day, these satellites appear to be in the same spot in the sky all the time. Let's work out how high they have to orbit. At an orbital radius of r, the satellite travels 2πr per orbit. The orbital velocity is square root ( M/r ). So the orbital period is the distance divided by the velocity, P = 2πr  / square root ( M/r ). This has to be 24 hours. 24 hours is 24 * 60 * 60 seconds / 1.346e-43 seconds / planck = 6.42e47 plancks. Now, (P/2π)2 M = r3.  1.04e94 * 1.1e32 = r3.  cube root( 1.15e126 ) = 1.05e42 plancks = 1.05e42*4.037e-35 = 4.23e7 meters = 26,300 miles. So, communications and tv satellites are all orbiting at 26,300 miles above the center of the Earth, about 22,300 miles above the Earth's surface.

The escape velocity is the speed at which you would have to throw a rock so that it would leave the planet and never come back. This is square root( 2M / r ), 1.414 * the orbital velocity for a given radius. At the edge of a black hole, r = 2M, so the escape velocity is infinite just as we expect. At the edge of a black hole, the orbital velocity is square root( M / r ) = square root( 1/2 ) = .7. So it seems that to orbit a black hole right at the edge, you have to be going 70% of the speed of light. This result is not exactly correct, however: this is what things look like to an observer far away from the black hole. We have to ask what things look like to a guy in a space ship attempting the orbit. This guy has his clocks running slower than we think, so he thinks his speed is higher than we do. The clock correction gives us a new formula for the orbital velocity, square root( M / ( r - 3M ) )

## Our Solar System

Below are plots of the inner and outer solar systems, seen first from the top, then edge on. In these plots, planet orbits are shown as rings, asteroids are shown as yellow dots, and comets are shown as white wedges. In these plots you can easily see the LaGrange points in Jupiter's orbit - these are gravitationally stable points at 60° before and after each planet. Asteroids tend to collect near these points, as you can see. These plots were created by Dr. Paul Chodas and are located on the NASA/JPL web page at http://ssd.jpl.nasa.gov/orbit_diagrams.html. The inner planets and asteroids, seen from the top. The inner planets and asteroids, seen from the side. The outer planets and asteroids, seen from the top. The outer planets and asteroids, seen from the side.

## Mercury

Semimajor axis a
Perihelion a - c
Aphelion a + c
Mean orbital velocity
Max. orbital velocity
Min. orbital velocity
Orbit eccentricity
57.91m km
46.00m km
69.82m km
47.87 km/s
58.98 km/s
38.86 km/s
0.2056 b2 = a2 (1 - e2)
c = ae