For an elliptic orbit with specific angular momentum h given by h 2 = μ μ p = μ μ a ( 1 − − e 2 ) we use the general form of the specific orbital energy equation, ε ε = v 2 2 − − μ μ r with the relation that the relative velocity at periapsis is v p 2 = h 2 r p 2 = h 2 a 2 ( 1 − − e ) 2 = μ μ a ( 1 − − e 2 ) a 2 ( 1 − − e ) 2 = μ μ ( 1 − − e 2 ) a ( 1 − − e ) 2 Thus our specific orbital energy equation becomes ε ε = μ μ a [ 1 − − e 2 2 ( 1 − − e ) 2 − − 1 1 − − e ] = μ μ a [ ( 1 − − e ) ( 1 + e ) 2 ( 1 − − e ) 2 − − 1 1 − − e ] = μ μ a [ 1 + e 2 ( 1 − − e ) − − 2 2 ( 1 − − e ) ] = μ μ a [ e − − 1 2 ( 1 − − e ) ] and finally with the last simplification we obtain: ε ε = − − μ μ 2 a