## Escape velocities

Written on 25 September 2019, 10:38pm

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Back in the secondary school I worked on a physics project about the escape velocities. Really interesting stuff – it was the first time when school intersected space – but looking back 20 years I can’t help but notice that I was missing the big picture. Here’s why.

The escape velocity is the minimum speed need for an object in order to escape from the gravitational influence of a celestial body. For the Moon, it’s 2.38km/s at its surface. For Jupiter, it’s 60km/s. For a black hole, it’s infinity.

There are 3 important things to remember when talking about the escape velocity:

1. it is independent of the mass of the object. It doesn’t matter if you’re shooting a small bullet or a big spaceship. However, the escape velocity depends on the mass of the celestial body
2. it assumes that the object travels in vacuum, not in an atmosphere. So no friction
3. it assumes that the object is no longer subject to thrust after reaching the escape velocity

With these 3 things in mind, a good example to illustrate the escape velocity would be an imaginary cannon launching a projectile on the Moon. Launch it with an initial speed inferior to 2.38km/s and it will come back; anything above that will cause the projectile to leave the gravitation influence of the Moon (well, until it reaches the gravitational sphere of another massive body).
On the other hand, using the Earth as the celestial body is a terrible example. Not only Earth has an atmosphere (until an altitude of about 100km) which introduces friction and makes the object mass and shape important, but in a subliminal way, everybody would imagine that the object reaching the escape velocity is a rocket. But at the Earth surface, the escape velocity is 11.2km/s, and no material known to man would resist the combined effect of the brutal acceleration and aerodynamic forces or heating.

So how did Apollo missions did reach the Earth’s escape velocity? Well, in short, they didn’t.

### Going up

The Saturn V rocket had more ‘stages’, which thrusted the spacecraft away from the Earth:

• first stage S-IC from lift-off to about 68km altitude. Through the atmosphere, more than 2 minutes thrust, until a maximum speed of 2.7km/s.
• second stage S-II from 68km to 175km altitude. Partially through the atmosphere (up to 100km – the Karman line), then vacuum, 6 minutes of thrust, until a max speed of about 7km/s.
• third stage S-IVB from 175km to 191km altitude. Through vacuum, more than 2 minutes thrust, until the parking orbit, when the orbital velocity is about 7.8km/s
• translunar injection (TLI) – J-2 engine of the S-IVB stage – from 191km to 334km altitude. Through vacuum, about 6 minutes thrust until reaching a velocity of 10.4km/s.

At that altitude of 334km, when the TLI is over, the Earth escape velocity is only 10.9km/s.
Apollo did not have to reach this escape velocity because it was not trying to escape the gravitational field of the Earth. The goal was only to get close enough to the Moon so that the spacecraft uses its gravitation. After 323.000km (about 84% of the Earth-Moon distance, also called the first Lagrange point), the spacecraft entered the gravitational sphere of the Moon. The Apollo missions were put on a ‘free return’ path: a trajectory where the moon’s gravity would automatically pull them round and back towards Earth if the lunar orbit insertion (LOI) burn failed.

### Coming down

A somehow lesser known fact is that the escape velocity also apply for an object approaching a celestial body. An object with no initial speed, approaching a celestial body from far, far away, will hit the body with the escape velocity. Coming back to the Moon example, an object approaching from far away, without any initial speed, will hit the surface of the Moon with 2.38km/s.

Of course, if the body has an atmosphere, this will slow down the object, so either it will disintegrate in the atmosphere or it will hit the body with a speed smaller than the escape velocity. Because of the atmosphere, the mass and more importantly, the surface of the object will become relevant: if it survives the atmospheric re-entry, it’s possible that the terminal velocity comes into play.

Apollo command modules were approaching Earth from a big enough distance (323.000km), so just before they hit the atmosphere they were falling down at nearly escape velocity. The fastest was Apollo 10 with a re-entry speed of 11.0685 km/s. “From Mars, the best trajectories have re-entry velocities similar to the Moon. 11.5 – 12.5 km/s”

### Summary

• when talking about escape velocities, the Moon is a better example than the Earth
• we can talk about escape velocity for spacecraft leaving Earth only at high altitudes (above the atmosphere layer) and only after the burn (thrust) stopped
• the Apollo missions got really close to the Earth escape velocity during the TLI, but it did not reach it because it didn’t need to – they also used Moon’s gravity
• we also talk about escape velocity when an object approaches a body from far away. Apollo got really close to the escape velocity on re-entry, right before the atmosphere slowed it down