Oberth Effect - Detailed Proof

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:

\begin{align} e_k &= \frac{1}{2} V^2 \\ &= \frac{1}{2} (V_\text{esc} + \Delta v )^2 \\ &= \frac{1}{2} V_\text{esc} ^ 2 + \Delta v V_\text{esc} + \frac{1}{2} \Delta v^2 \end{align}

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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