Oberth Effect in Solar Sails: Where does the extra energy comes from?

Suppose, the reflectors of Solar Sails are opened for same amount of time at both periapsis and apoapsis. How does Oberth effect work in this case? Will Delta-V be same and hence change in Kinetic energy higher at periapsis because of the velocity squared function?

Or, to provide same delta-v, we need to open the reflectors for longer period at periapsis? In that case, I think the "extra" solar energy will be responsible for extra change in Kinetic energy.

In case of conventional rockets with exhaust mass being ejected, I believe the higher change in Kinetic energy comes from higher change in Kinetic energy of the exhaust mass. (Correct me, if I am wrong).

When two spacecrafts moving at different speeds are accelerated by solar sails, with everything else being equal, the $$\Delta V$$ is the same for them (or, rather, almost the same for non-relativistic speeds; there may be a relative difference on the order of magnitude of $$v/c$$). So, indeed, a faster moving spacecraft gets more kinetic energy out of it. Where does it come from?

Only a tiny part of incoming light's energy is converted to spacecraft's kinetic energy: a photon with momentum $$p$$ has energy $$pc$$, but if you impart a small momentum $$\Delta p$$ to a spacecraft moving with velocity $$v$$, its kinetic energy will change by approximately $$v\cdot \Delta p<. Almost all energy either goes away with reflected light or is converted into heat. So for a faster spacecraft, the reflected light will have slightly less energy, and the spacecraft will be heated slightly less.

• So can we say, ultimately, the extra energy is coming from the sun? Commented Mar 15, 2021 at 12:21
• @RajArjit Yes. . Commented Mar 15, 2021 at 12:59
• "Heated less" <-- for simple analysis, assume 100% reflectivity, which simplifies the $\Delta p$ the spacecraft receives per photon. Then it's easier (I think) to see that two craft of equal mass receive equal $\Delta p$ and thus different $\Delta KE$ Commented Mar 15, 2021 at 13:28
• @CarlWitthoft: And to receive equal Δp, Sun will need to provide higher energy to the one at periapsis, right? Is it possible for that to happen if reflectors are opened for exact time at both periapsis and apoapsis? Commented Mar 15, 2021 at 18:00
• @PM2Ring: Not necessarily it is less at apoapsis. There can be a case where the rocket is revolving around another planet in polar orbit such that the apoapsis and periapsis are at same distance from the sun. Commented Mar 16, 2021 at 5:57

The Oberth effect exists because imparting a fixed change in momentum increases kinetic energy more when traveling at a faster speed than at a slower speed. This is due to the fact that kinetic energy is proportional to the square of velocity. Let's look at an example. Assume your spacecraft has a mass of 1 kg, and you have a rocket engine that can provide a delta V of 1 m/s. If you do the maneuver at 5 m/s, your kinetic energy goes from 5^2=25 J to 6^2=36 J. If you do the maneuver at 10 m/s, your kinetic energy goes from 10^2=100 J to 11^2=121 J. With the same change of 1 m/s, you gain 11 J in one case and 21 J in another case.

For a solar sail, the "propellant" is actually the light from the sun itself. The photons impart a small change in momentum on the spacecraft, as explained in Litho's answer. This phenomenon is referred to as solar radiation pressure.

Regarding the effect of opening solar sails at periapsis and apoapsis, this depends on many factors. First, as you mentioned, the speed at periapsis and apoapsis are different, and therefore you'll spend less time near periapsis than you will near apoapsis. The eccentricity of your orbit will determine the difference in speeds. So depending on how eccentric your orbit is will determine the tradeoff on additional time needed at periapsis versus how much more efficient the periapsis maneuver is. The orientation of your orbit is also important, because if you are in Earth's shadow during periapsis or apoapsis then of course you will not receive any delta V at that point. Similarly, if your line of apsides is in line with the sun, then you won't be able to get much delta V in velocity direction at either apse.