Friday, September 25, 2020

[usfsxftv] gravitational potential in the solar system

gravitational potential is gravitational potential energy per unit mass.  its value at a distance x from a point mass M is -G*M/x.  we calculate its value at various distances from the sun.  the product G*M is the standard gravitational parameter, known to high precision.

at the surface of the sun (radius = 6.96e8 m): -190678.79 MJ/kg (mega joule per kilogram)

at the distance equal to the length of the semimajor axis of the orbit of Mercury, 0.387098 au: -2291.74 MJ/kg

Earth, 1 au: -887.12787 MJ/kg. example calculation using the "units" program:

You have: -(gauss_k^2 au^3 day^-2) / 1 au
You want: J / kg
  * -8.8712787e+08

Mars, 1.523679 au: -582.2275 MJ/kg

Jupiter, 5.2044 au: -170.46 MJ/kg

Neptune, 30.07 au: -29.50 MJ/kg

G is very small, solarmass is very big, and distances in astronomy are always very big, so what scale of number results from -GM/x ?

it turns out the gravitational potential has the same units as specific energy, energy per unit mass.  flipping the sign, gravitational potential in the outer solar system happens to be similar to the specific energy of common substances.  we calculate their corresponding distances from the sun:

negative gasoline: -46 MJ/kg = 19 au (Uranus)

negative glucose: -16 MJ/kg = 55 au (slightly beyond the Kuiper belt, which ends at 50 au)

these are the specific energies of combustion with oxygen, and it assumes that the oxygen is free (not included as part of the mass).

perhaps we imagine a gasoline-powered launch platform co-orbiting with Uranus, launching canisters of gasoline at escape velocity out of the solar system.

negative TNT: -4.184e3 MJ/short ton  = -4.612 MJ/kg = 192 au.  (i have yet to find an authoritative source stating whether the "ton" in TNT equivalent explosive yield is a U.S. ton or metric ton.  atomic bombs were first invented in the U.S., with their yields experimentally compared against quantities of TNT measured in the U.S., so i'm guessing U.S. ton.  the units program uses U.S. ton.)  the explosive decomposition of TNT does not require an oxidizer; however, the reaction produces elemental carbon, so if an oxidizer is available, it can produce more energy.  (how much more?)

if we replace the sun with an equivalent point mass and ignore general relativity:

negative plutonium-239 fission (a.k.a. fusion): -83610000 MJ/kg = 1.6e6 m

negative hydrogen fusion (a.k.a. fission): -639780320 MJ/kg = 2.1e5 m

negative matter-antimatter annihilation (a.k.a. pair production): -89875517874 MJ/kg = 1.48e3 m.  This is exactly half the Schwarzschild radius of a 1 solar mass black hole.

energy density values are from Wikipedia.

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