Escape Velocity Comparator
v_{\text{esc}} = \sqrt{\frac{2GM}{R}} — how fast you must launch to slip gravity's leash and drift into the endless void
the Schwarzschild radius and black holes 🖖
If a body is compressed so densely that its radius becomes smaller than its Schwarzschild radius (Rs = 2GM/c²), its escape velocity exceeds the speed of light. At this point, even photons cannot escape, forming an event horizon — a black hole. For the Sun, this critical radius is about 3 kilometers; for Earth, it is a mere 9 millimeters.
why the rocket's own mass doesn't matter 🖖
Escape velocity is the launch speed that lets an object coast away forever instead of falling back. The surprise: it depends only on the world you are leaving — its mass M and radius R, through v = √(2GM/R) — never on what you throw. A marble and a battleship both need 11.2 km/s to leave Earth. That is why a single number can sum up each body.
escape is only √2 faster than orbiting 🖖
The speed to circle a body just above its surface is v = √(GM/R), and escape velocity is exactly √2 times that. So breaking free of a stable orbit never takes double the speed — only about a 41% boost. That is the yellow path in this tool's simulation, fired at 0.71× (1/√2) of escape speed: it settles into a circle while the green escape shot leaves for good.
Example problems
- Moon - Moon — 2.38 km/s; so low that meteorite ejecta escape directly to space
- Mars - Mars — 5.03 km/s; too low to retain lighter atmospheric gases over geological time
- Venus - Venus — 10.36 km/s; just high enough to slow atmospheric loss significantly
- Earth - Earth — 11.2 km/s; gas molecules at room temperature are far slower, so the atmosphere stays bound
- Saturn - Saturn — 35.5 km/s; ring particles orbit too slowly to escape yet too fast to fall in
- Jupiter - Jupiter — 59.5 km/s; strong enough to retain hydrogen and helium unlike Earth
- Sun - Sun — 617.5 km/s (0.2% of c); compress to 3 km radius and light cannot escape
- Example 8 - white-dwarf sample case
- Example 9 - neutron-star sample case