CAN A HUMAN JUMP OFF AN ASTEROID?

Before we can answer this, please be sure to read: Jump on Mars

Now then...

Q: Can a human jump off an asteroid?

A: Yes, if it is under (about) 12km in diameter.

To get an upper bound on human launch velocity, suppose local gravity g_x is zero.

v_x² ≤ 2 · h_e · g_e + 2 · L · ( g_e - g_x )
      = 2 · 1.16m · 9.807m/s² + 2 · ½m · ( 9.807m/s² - 0.00m/s²)
      = 32.56m²/s²

v_x ≤ 5.71m/s = 12.77mph

This means the fastest any human can launch by jumping, even in zero gravity, is 5.71m/s.

Asteroids range widely in density. A solid ball of salt water has density 800kg/m³. A solid ball of iron and nickel has density 8000kg/m³. A typical asteroid is near the lower end of this range.

A uniform object of radius r and density ρ has mass:

M = ρ × 4/3 π r³

A uniform object of mass M and radius r has escape velocity:

v² = 2 G M / r

Therefore:

v² = 8 π ρ G r² / 3

v = r × (8 π ρ G / 3)^1/2

r = v / (8 π ρ G / 3)^1/2

Plugging in density values here yields an upper limit. In principle, a human could jump off a solid ball of ice up to 17km in diameter, or a solid ball of metal up to 5.4km in diameter.

These are extreme cases of density. If we consider a much more typical loose aggregate asteroid with a density of 1600kg/m³ the maximum diameter for "self launch" is 12km.

Other typical asteroids are the captured moons of Mars. Phobos has a diameter of 23km (14mi) and an escape velocity of 18m/s (41mph). Self launch is impossible. However, Deimos is oblong, at 15km x 12km x 11km. Standing at the end of its longest axis, the escape velocity drops to 5.5m/s. So in principle, it is possible that a human could self launch straight off of Deimos, and right into a orbit around Mars.

Q: Can a human jump from Deimos to Phobos?

A: No way. Deimos itself orbits at nearly 5000kph. Your jump may exit you from the Deimos gravity well, but your orbital velocity is almost exactly the same. You won't pass through the 7700kph orbit of Phobos.