I've been recently performing some computations regarding the Oberth effect and I'd like to share the results.
I've always thought the term "Oberth effect" was a bit overused around here and somewhat misunderstood. It is not the hard and fast rule that everyone seems to thinks it is. For example, the place where I usually hear the Oberth effect evoked is when the conversation is about interplanetary ejection burns and/or orbit insertion burns. It is usually stated that performing the burn closer to the planet is always better. However, for low orbits very close to a planet, the usual thinking isn't always correct.
A transfer trajectory to another planet requires a certain hyperbolic excess velocity, denoted V∞. This is the velocity left over after a spacecraft escapes a planet's gravity. The V∞ required to reach a specific destination during a specific launch window is the same regardless of the orbital altitude from which the spacecraft is ejected. The equation for V∞ is,
V∞2 = Vbo2 - Vesc2
where Vbo is the burnout velocity and Vesc is the escape velocity.
Vbo is the initial orbital velocity plus the ejection Δv, i.e. Vbo = Vorb + Δv. Therefore, by substituting and rearranging the equation we get,
Δv = (V∞2 + Vesc2)0.5 - Vorb
Of course, Vorb and Vesc are both functions of the orbital altitude. Therefore, for any given orbital altitude and V∞, we can compute the Δv required to eject our spacecraft.
Let's take the example of a spacecraft on a transfer trajectory to Duna where V∞ = 900 m/s. Computing Δv and plotting it on a graph, we get
Note that it initially takes less Δv the higher the orbit (significantly less in this case). It is not until we pass an altitude of 8145 km that the Δv begins to increase (i.e. the Oberth effect).
The shape of the curve varies dramatically with different values of V∞. For instance, lets' say we are going to Jool during a launch window where V∞ = 2800 m/s. The graph now looks like this,
Again, the Δv initially decreases with increasing altitude, though the decrease is much less than the previous example. Also note that the Δv starts to increase just past an altitude of 303 km, which is much lower than the Duna case.
What these results don't take into account is that, although it may require less Δv to eject from a higher orbit, it takes more Δv to reach the higher orbit in the first place. When we add together both the launch Δv and the ejection Δv, the lower orbit results in less total Δv.
When planning a mission, you can always use Alex Moon's Launch Window Planner and adjust the orbit altitudes to see how the ejection and insertion Δv change. However, note that when arriving at a planet, the orbit that you want to enter into will be the one that produces the lowest overall mission Δv. You don't want to enter into an orbit just because it has a low insertion Δv if its going to make subsequent maneuvers more costly.