### Looking for loft with all the comforts of home

I ran across this post over at the systemic blog about ballooning in the upper atmosphere of Venus and Jupiter. Presumably, there are sweet spots in their atmospheres where the temperature and pressure are close enough to STP at Earth's surface that a human could go ballooning in these locales with little more than breathing apparatus and attire suitable for surviving the climate (an acid-resistant full-body bathing suit for Venus, or a heavy parka for Jupiter).

In the case of Jupiter, however, the post leaves out one important detail: gravity. On Venus, the gravity is comparable to that on Earth. However, if one were to go ballooning among the cloud tops of Jupiter, one would likely spend a lot of time pinned to the floor of the gondola by about 2.5 g's (2.5 times the Earth's gravity).

The article reminded me of a little exercise in orbital mechanics that I had once worked out. I was looking for places in the solar system where one could experience about 9.8 m/s

^{2}of gravitational acceleration and a 24 hour day. In addition to the Earth's surface, I thought that a space station orbiting a gas giant could possibly provide the proper environment. Simply place the center of gravity of the station in a 24 hour orbit around the planet, and extend a pair of tethers up and down from there such that a full 1G would be felt at either end. At the lower end, the force of the planet's gravity exceeds the centripetal force due to the orbit by exactly 1G, and at the upper end the reverse is true.

Let R denote radial distance from the center of the planet, and H be the height above the commonly accepted "surface" of each body. The numbers given below are for each planet around which this theoretical orbiting station could exist.

For Jupiter:

R[lower] = 110385 km

R[orbit] = 288208 km

R[upper] = 1861246 km

H[lower] = 39015 km

For Saturn:

R[lower] = 61187 km

R[orbit] = 192841 km

R[upper] = 1856417 km

H[lower] = 787 km

For Uranus:

R[lower] = 24196 km

R[orbit] = 103222 km

R[upper] = 1854656 km

H[lower] = 666 km

For Neptune:

R[lower] = 26324 km

R[orbit] = 109229 km

R[upper] = 1854715 km

H[lower] = 4024 km

If this same process is applied to the Earth, we get the recipe for a space elevator.

For Earth:

R[lower] = 6363 km

R[orbit] = 42233 km

R[upper] = 1854357 km

H[lower] = 0 km

Curiously, we see that no matter what planet we are orbiting, the upper end of the tether is always at approximately the same radial distance. As I look back at my equations now, I see that there is a weak dependence on M, however the coefficient is exceedingly small. The first two terms of the series expansion are:

R[upper] = 1854336 + M*3.6695e-16 + ...

where R is in km and M is in kg. I'm sure this is probably a well established conclusion in orbital mechanics or general relativity, but it's the first time I've encountered it.

Labels: orbital mechanics