Uranus surface gravity, escape velocity and orbit calculator
Everything about orbiting a body follows from one number, its gravitational parameter GM, together with its radius.
Escape velocity
21 375.20m/s
- Surface gravity
- 9.0076 m/s2
- Circular orbit speed at that altitude
- 15 055.30
- Orbital period
- 2:57:48
- Orbital period in minutes
- 177.80
- Stationary orbit altitude Where an orbit matches the rotation period. Meaningless for a retrograde or very slow rotator.
- 57 324
- Mean density
- 1270 kg/m3
- Orbital kinetic energy of the payload
- 113 331 000 000
- Energy to lift the payload out of the well
- 228 450 000 000
Show the working
- Orbit altitude above the surface = 200
- Payload mass = 1000
- gm = 5.793939e15 = 5 793 940 000 000 000
- radius = 2.5362e7 = 25 362 000
- Surface gravity = 5 793 940 000 000 000 / 25 362 000^2 = 9.007 55
- Escape velocity = sqrt(2 * 5 793 940 000 000 000 / 25 362 000) = 21 375.2
- orbit_radius = 25 362 000 + 200 * 1000 = 25 562 000
- Circular orbit speed at that altitude = sqrt(5 793 940 000 000 000 / 25 562 000) = 15 055.3
- Orbital period = 15 055.3 > 0 ? 2 * pi * 25 562 000 / 15 055.3 : 0 = 10 668.1
- Orbital period in minutes = 10 668.1 / 60 = 177.801
- Orbital kinetic energy of the payload = 1000 * 5 793 940 000 000 000 / (2 * 25 562 000) = 113 331 000 000
- Energy to lift the payload out of the well = 1000 * 5 793 940 000 000 000 / 25 362 000 = 228 450 000 000
- geo_radius = (5 793 940 000 000 000 * (-17.24 * 3600 / (2*pi))^2)^(1/3) = 82 685 900
- Stationary orbit altitude = (82 685 900 - 25 362 000) / 1000 = 57 323.9
- Mean density = 5 793 940 000 000 000 / 6.6743e-11 / (4/3 * pi * 25 362 000^3) = 1270.37
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- g = GM / r^2
- escape velocity = sqrt(2 GM / r)
- circular orbit speed = sqrt(GM / r)
- period = 2 pi r / v
- escape velocity is always sqrt(2) times the circular orbit speed at the same radius
Escape velocity assumes no atmosphere and no further thrust. It is the speed at which kinetic energy equals the depth of the gravity well.
Mean density is derived by dividing GM by G, so it inherits the uncertainty in G — about one part in twenty thousand.
Where the numbers come from
- GM and mean radius from planetary ephemeris and spacecraft tracking solutions.