Detailed Proof
If an impulsive burn of Δv is performed at periapsis in a parabolic orbit then the velocity at periapsis before the burn is equal to the escape velocity (Vesc), and the specific kinetic energy after the burn is:
where
When the vehicle leaves the gravity field, the loss of specific kinetic energy is:
so it retains the energy:
which is larger than the energy from a burn outside the gravitational field by:
It can then be easily shown that the impulse is multiplied by a factor of:
Substituting 50 km/s escape velocity and 5 km/s burn we get a multiplier of 4.6.
Similar effects happen in closed and hyperbolic orbits.
Read more about this topic: Oberth Effect
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