Inter-Orbital Kinetic Energy Exchanges: Part I

Electrical power can be transferred between planets using high velocity masses. Kinetic Energy Exchanges are an efficient concept that can output more energy than it consumes and only gets better with distance.

Guest writer Zerraspace (Zach Hajj) works out the details and reveals just how impressive the concept is. 

This blog post was written by Zach Hajj, who provided descriptions, workings and calculations. He is also known as ZerraSpace and is part of the team behind the very promising hard science-fiction game Starfighter Inc (@StarfighterInc). Find his works on DeviantArt and come have a chat with us on the ToughSF Discord.

This concept was based on discussions in the earlier "How to Live on Other Planets" entries for Jupiter and Saturn, detailing the possibility of energy exchange from the outer worlds to inner ones via orbital mass transfer, what might also be called interorbital kinetic energy exchange.

Diagram of a simple Inter-Orbital Kinetic Energy Exchange between Saturn (top) and Earth (bottom).

In summary, a projectile is launched retrograde from a high orbit, such that it can reach a near standstill relative to the sun. At this point it will plummet towards the inner solar system, trading its initial gravitational potential energy for kinetic energy, thus speeding up as it falls farther. With a carefully placed catcher to intercept, this kinetic energy can be harnessed as electricity, and may be many times that required for launch.

The key to this is the relative velocity at each location. Were we to launch such a projectile from Saturn orbit, we could cancel its initial orbital velocity with a 9.7 km/s retrograde boost, enabling freefall. Via energy conservation, we can determine the velocity developed as it falls farther and farther:

Where: 

G is the gravitational constant (6.67408 × 10^-11)

M is the mass of the sun (1.989 × 10^30 kg)

a_0 is our semi-major axis at the fall point (km)

a_1 the axis at the point of study (km)

Thus, falling from Saturnian orbit at 9.6 AU, by the time our projectile has reached 1 AU, it will have accelerated to a full 39.8 km/s. Supposing we have timed this to encounter the Earth on its fall, the planet will be moving orthogonal to it at 29.8 km/s along its orbit, thus the projectile’s relative velocity can be determined via Pythagorean sum to be 49.7 km/s. 

A catcher so located would then intercept it at 5.1 times the initial boost velocity; that is to say, for an initial 47 MJ/kg investment of kinetic energy, we could receive 1.235 GJ/kg, 26 times what we started with. 

This on its own seems quite impressive, but it makes a large assumption – that the projectile velocity on either end is relative to that of the relevant planet. This would be the case if we launched from a space station in a similar orbit to Saturn, and received at another in a similar orbit to the Earth, but not if we were to launch and receive from within the spheres of influence of the planets themselves. 

There are two more factors that must be accounted for there, and as it turns out, we can utilize these to radically improve our figures.

The first thing to understand is that the projectile does not maintain its velocity while passing by a celestial body. Much like it falls towards the sun, it also falls towards the planet and is sped up as it draws closer. This can be more easily understood via energy analysis in the planet's frame of reference – the projectile does not gain or lose energy, and so coming closer to the body, it exchanges its gravitational potential energy for kinetic.  

Again, we can demonstrate through conservation of energy:

Where:

V_inf is the hyperbolic excess velocity

V_e = sqrt(2*G*M/r) is the planetary escape velocity. 

A similar equation can be used to derive the Oberth effect. 

The dependence on escape velocity is very important here, as the escape velocity varies with distance from the body. As we get deeper within the planet’s gravitational well, escape velocity increases, and so too must our object velocity increase. Clearly, the greatest speedup occurs when the escape velocity is significantly higher than the hyperbolic velocity, so for high relative velocity, we want massive bodies and very low orbits.

Around Earth, escape velocity at a 400 km LEO is 10.8 km/s (the often-cited 11.2 km/s is at its surface), so for our originally determined hyperbolic excess velocity of 49.7 km/s, velocity at periapsis becomes 50.9 km/s. This may not seem like a significant increase, but it makes quite a bit more difference at Saturn: assuming we wanted to launch from Titan orbit, local escape velocity is 7.9 km/s, which is more comparable with the intended V_inf of 9.7 km/s, giving a periapsis velocity of 12.5 km/s.

The second thing we must now account for is that the relevant velocity is not relative to the planet, but to an object along a point in its orbit. Depending on the point selected, we can utterly change the direction of motion, and thus the magnitude of the relative motion as well.

A catcher station in Earth orbit can add or decrease its orbital velocity to the intercept velocity.

Consider our catcher at 400 km LEO: orbital velocity here is roughly 7.9 km/s, so if our catcher were to intercept the projectile while it is moving in the same direction as its orbit, the orbital velocity would be subtracted for a relative velocity of 50.9 – 7.9 = 43 km/s. On the other hand, if the projectile were to arrive moving opposite the catcher, orbital velocity would be added, yielding 50.9 + 7.9 = 58.8 km/s. At Saturn, the effects are more notable: taking into account the 5.6 km/s orbital velocity at Titan orbit, launching in the direction of the catcher will yield a relative velocity of 6.9 km/s, while launching against it requires 18.1 km/s.

Clearly, to maximize gain, we wish to maximize the relative velocity on the receiving end and minimize it at the launching end. Thus, we should launch when the launcher is moving in the same direction as we intend for our projectile, and we should receive when the catcher is moving opposite its intended arrival. Getting the exact windows would require precise timing, but this doesn’t necessarily have to be factored entirely at the launch site. The projectile could feasibly feature a small propulsion bus, triggered to fire just at the edge of the destination body’s SOI, where only a minor adjustment would be needed to correct the intercept.

Earth's Sphere of Influence extends out to 1.5 million km. 

Thus applied, we have reduced the initial launch investment to 23.8 MJ/kg and increased the received energy to 1.729 GJ/kg, roughly 73 times what was needed to start. We have nearly tripled our relative energy multiplier. 

However, we have neglected another important dimension of the problem: what goes for the planets goes too for their moons.

By launching from the low orbit of a moon such as Titan, additional free energy is available.

Earth is sadly lacking in natural satellites at 400 km LEO, but at Titan, a 1000 km orbit corresponds to an escape velocity of 2.2 km/s (we could go lower but risk significant atmosphere drag from the low scale height). 

Titan around Saturn.

Here the hyperbolic excess speed is the same as the periapsis velocity we expected around Saturn, 6.9 km/s if Titan is moving in the desired direction, so the periapsis velocity at the moon itself can then be determined to be 7.2 km/s. Launching in the direction of station motion, we can shave off its orbital velocity of 1.6 km/s, and hence further reduce launch velocity to 5.6 km/s.

Now initial investment is 15.7 MJ/kg, increasing our energy multiplier to 110. We have more than quadrupled our original estimate.

Such a system makes for potent gains, but comes with a huge catch: exchange can only happen at appropriate intervals, where the inner planet will end up right beneath the projectile as it crosses its semi-major axis. These small windows are separated by the two worlds’ synodic period, the time necessary for them to repeat their relative positions. For far apart worlds, this is much closer to the inner world’s period than the outer, since the outer world barely moves relative to the inner, so for Saturn and Earth, the synodic period is just slightly longer than an Earth year.

Synodic periods of the planets relative to Earth.

This may seem bearable, but the wait quickly becomes prohibitive for farther out pairs; for Jupiter and Saturn, the synodic period is 19.9 years; for Uranus and Neptune, it’s an agonizing 171.4 years! Even for shorter durations, it becomes difficult to utilize this setup as a continuous energy source. In order to do that, we need to find ways to launch and receive outside our original window, and we can do that by adjusting our launch velocity.

As we reduce our initial boost, we maintain an increasing aphelion velocity, and our projectile changes from a straight-line fall into a highly eccentric orbit. With some proper care and adjustment, we can tailor this to intersect our destination’s orbit at any point. The greater the angular offset from our freefall point, the less boost we need, and this applies whether said angle is ahead of or behind the original point – the projectile orbit is symmetric, and so will eventually reach either side. The greatest slowdown will be necessary to access the point directly opposite the freefall point, where our projectile’s periapsis exactly matches our destination’s semi-major axis. In this case, we’ve launched it into a regular Hohmann transfer orbit.

Hohmann transfer velocity from Saturn to Earth is 5.4 km/s, requiring a Titan launch velocity of merely 3 km/s, 4.5 MJ/kg. This will reach a relative velocity of 10.3 km/s to the Earth, and relative to our LEO station, up to 22.8 km/s, 260 MJ/kg, for an energy multiplier of 25.7. This is far less than we’d have received from freefall, but it does represent the lower limit for potential gains. For the rest of the year, figures will be between this and freefall.  

We could also apply this same principle another way. Consider that for freefall, we’ve pushed our projectile to a near standstill. With a little extra boost, we could then regain an aphelion velocity, but moving retrograde around the sun, effectively doing the same trick in reverse. Now greater speedup leads to less eccentric higher-reaching orbits, until at maximum boost, the projectile would reach the inner world at perihelion, but moving in the opposite direction – a retrograde Hohmann transfer orbit.

For a Hohmann transfer from Saturn to Earth, solar velocity at aphelion is 9.7-5.4 = 4.3 km/s, so to reach this velocity moving retrograde to Saturn, we must then boost by 9.7+4.3 = 14 km/s. Optimized launch velocity from Titan then becomes 10.3 km/s, and velocity at perihelion will be 40.1 km/s. 

Here is where the advantage of this technique becomes apparent: because we are moving in the opposite direction to the Earth, we add its full velocity to our relative motion, rather than the Pythagorean sum, yielding an incredible 69.9 km/s at planetary periapsis, 78.6 km/s relative to our LEO station. Our launch investment has drastically increased to 53 MJ/kg, but so too has the received energy at the Earth, now up to 3.089 GJ/kg. The energy multiplier has dropped to 58, but energy per launched mass is far greater, almost double what we had from freefall. 

Launching from a low velocity to high velocity planets allows the difference in orbital velocities to be exploited.

Again, we have bound potential energy gains depending on the time of year, but in the other direction. Now freefall is our lower limit, and full retrograde is our maximum.

With this, the energy transfer system can be made to run regardless of the relative position of the planets involved.

However, thus far only one side has really benefited. Titan has to expend energy to send the projectile, whereas Earth gets both the free energy and the material of the projectile to boot. Earth could share back some of its dividends via laser and cargo shipments, but there’s an easier way to send up what was given, and that is by using the system itself.

Thus far, we have only focused on a projectile falling from the outer solar system towards the inner. By playing this game in reverse, launching projectiles outwards as well as inwards, we can ensure those farther reaches receive their due energy as well.

Railgun launcher from The Expanse. It could be used for peaceful purposes.

Launching a projectile along the launcher’s orbit, at 400 km LEO a boost of 7.3 km/s for 26.7 MJ/kg is sufficient to reach out all the way to Saturn. Conversely, we can receive this same projectile at Titan moving opposite its orbit and a local catcher, at a relative velocity of 15.2 km/s for 115.5 MJ/kg. This is more than twice the 53 MJ/kg needed for a retrograde Hohmann launch, and four times the 23.8 MJ/kg for a simple freefall. Earth’s investment is a drop in the bucket next to the potential 1.729 to 3.089 GJ/kg it can receive.

Thus, with the projectile passed back and forth, both Titan and Earth can make gains, and moreover, the same mass can be recirculated and continually supply power.

Of course, this is the minimum required energy for transmission, and the minimum Titan can expect to receive. Can we boost further to reach the outer world at any time of the year? 

In this case, it’s not as useful a solution. Boosting up to even the local solar escape velocity will access less than half of the outer world’s orbit, and even infinite boosts along our planet’s direction of motion cannot access points behind it. To reach those, we need to boost radially to our orbit… and for points trailing, we must ultimately boost retrograde.

The velocities quickly become staggering and require massive setups. Because we can launch in the same direction as the launcher and receive moving opposite to it, we can always make sure that a retrograde launch is cheaper than a retrograde catch and thus net some energy, but it will still drain most of what we’d expect to gain from the partner. Whether this would be deemed an acceptable tradeoff depends on one’s priority. Do we want regular shortly-interspersed transfers, or maximum energy per transfer?

Mercury and the Sun.

In some cases, this is not much of an issue. Mercury has such a short synodic period with essentially any other world (on the order of five months with Venus and three/four months with anywhere else) that it would not be much trouble to simply wait for the optimal launch window and use inexpensive Hohmann transfers for launch while receiving the return retrograde. For other worlds, a compromise is possible: if we consider the reverse freefall as a practical upper limit, we make sure that launch is always cheaper than any reception regardless of the time of year.

Applied to our ongoing example, for our LEO to Titan launch, we can reduce the maximum (reverse freefall) velocity from the received 58.8 km/s, 1.729 GJ/kg to 43 km/s, 924 MJ/kg; in turn, we can expect 58.8 km/s to 78.6 km/s, 1.729 GJ/kg to 3.089 GJ/kg returns. Thus, we ensure that we will always make some gain, and still allow mutual launch for roughly three-quarters of the year.

Can we improve this even further?

We are essentially adding on the orbital velocity when receiving and subtracting it when launching, so to maximize the difference, it pays to increase the orbital velocity, and for that we must move deeper.

Mimas is the closest of Saturn's largest moons, and therefore orbits the fastest.

For Saturn, we’ll go as far as Mimas (there are other moons closer in, but they are blocked behind the rings). This deep in Saturn’s gravity well, orbital velocity is an impressive 14.3 km/s, and escape velocity is an even more substantial 20.2 km/s. Mimas itself has a low mass and miniscule gravity – at 100 km orbit, orbital velocity is a mere 92 m/s and escape velocity is barely 130 m/s – so we could feasibly ignore its gravitational contributions to the projectile. 

Mimas is tiny. 

As before, we want the catcher and moon to be moving in the opposite direction of the projectile when intercepting, which leads to a relative velocity of 35.3 km/s, 623 MJ/kg. For a full retrograde return, we can do with a velocity of 10.2 km/s, 52 MJ/kg is sufficient. Thus, where Titan can make 4 times what it must send, Mimas can make a more impressive 12.

How far could one go with this? 

For the highest possible relative velocities, we should drop from the highest possible location to the lowest possible location, from Neptune to Mercury. 

The lonely blue planet.

We have already discussed the advantages of Mercury, but as for Neptune, it is so high up, its orbital velocity is miniscule (a mere 5.4 km/s) making for cheap returns, and little extra investment is needed to boost from freefall to retrograde (the aphelion velocity for our projectile in this case will be a mere 0.9 km/s).

Neptune is very far away...

The initial projectile can be launched from Mercury at 17.0 km/s, 144.5 MJ/kg; intercepting at Triton yields relative velocity of 13.1 km/s, 85.8 MJ/kg. Triton can then launch it back at 3.7 km/s, 6.8 MJ/kg, so it will arrive at Mercury at a whopping 118 km/s, 6.962 GJ/kg. Relative to its own launch, Mercury has multiplied the energy 48-fold, while Triton effectively multiplies launch energy 12.5-fold, and from Triton launch to Mercury reception, we have multiplied the energy a thousand-fold.

The launcher has to take into account Triton's inclination.

Similar systems can be worked out for other worlds, with figures worked out below. Note these assume low orbit launch (400 km unless specified), and the moons of the gas giants used are Callisto, Titan (1000 km), Miranda (100 km) and Triton:

Find full file here: Google Document.

Energies involved depend on the direction of transfer. For outward transfers, the pattern is simple: energy required at the inner world always increases and energy received at the outer world always decreases for further spaced partners. 

For inward transfers, it’s more complicated. Hohmann trajectories are also more expensive with distance, while freefall velocity is invariant of the inner world, and retrograde transfers end up being less expensive since we’re aiming for a decreasing aphelion velocity. Generally speaking, all inward transfers become less expensive as we move farther out, owing to the lower solar system velocities involved, which might be enough reason to justify colonizing the Neptune system – not only do we get more energy, it’s easier to send. 

The different InterOrbital Kinetic Energy Exchange trajectories possible.

Interestingly, this choice makes little difference to the inner world, as far enough out, relative velocity and energy change only a little even with greatly increasing distance. This is because projectile velocity is bracketed between the planet’s own orbital velocity and local solar escape velocity, and with this, we could set bounds upon the expected energy.

For two-way transfers, what’s most important is the location of both bodies in the solar system. For the terrestrial inner worlds, solar velocity dominates all exchanges, and so there are orders of magnitude difference between the cheapest sendoff/returns (Hohmann) and the most expensive (retrograde).

While significant energy is gained, much must be committed, so the overall multiplier for the whole exchange is low, and the outer world will usually earn a small fraction of what was initially given. For the gas giant outer worlds, the high orbital velocities and low solar velocities allow for small differences between different sendoffs. Launch can be made much cheaper than reception, so the two-way energy multiplier is much higher, and energy can be multiplied going both inwards and outwards.

But the gas giants need not limit themselves to interplanetary transfers. The same factors we’ve applied to interplanetary transfers can also be applied to interlunar transfers, between moons of the same planet. High orbital velocities and deep gravity wells are needed to make this work, which is exactly what the gas giants provide. Here the deepest moons almost hug the planet and can reach near-planetary velocities, while there remain far out moons for high gravitational potential.

Jupiter's spaghetti-orbits of its multiple moons. 

For maximum benefit, we try between the closest and farthest possible moon, ignoring oddly inclined outer satellites. Around Jupiter, this gives us Metis and Callisto. 

Callisto would make for an ideal colony location.

Launching from Metis at 11.6 km/s, 67.3 MJ/kg, we can receive the projectile at Callisto at 7.3 km/s, 26.6 MJ/kg. Launching it back at 9.8 km/s, 48 MJ/kg, it will be received at Metis going 74.6 km/s, 2.783 GJ/kg, 41.3 times what it cost to send.

Saturn has a huge collection of moons to choose from.

Around Saturn, we exchange between Mimas and Titan, again ignoring deeper moons within the rings (Hyperion might make a better choice than Titan in terms of distance, but Titan’s significant gravity gives it a greater advantage for changing velocities). Mimas sends out at 4.5 km/s, 10.1 MJ/kg, to be received at Titan at 5.1 km/s, 13 MJ/kg; launching back at 7.1 km/s, 25.2 MJ/kg, it is received at Mimas going 33.3 km/s, 554.4 MJ/kg, 55 times what it cost to send.

Similar systems could be set up around Uranus and Neptune, but would suffer due to the lower velocities and smaller spacing between moons.

These could be the trajectories of projectiles between a Saturnian moon and a low orbit station.

In both these cases, the farther moon suffers a net loss, and must be resupplied by the inner moon. Though we could make Callisto and Titan break even by launching into freefall rather than retrograde Hohmann, this would significantly reduce the energy available to the other world. 

In this case, the shorter distances make energy transfer via other means more viable, so Metis/Mimas could have an easier time resupplying. Clearly, this describes an optimal window; launch at other times is possible, but in this case the synodic period due to the inner moon is short enough (a day or less) that there is really no point in doing so.

What does this mean for the solar system at large? 

For one, gas giants can be relatively self-sufficient in terms of energy. Once enough energy is stockpiled for the initial launch and a redistribution system is set up to make up for the losses at the outer moon, this original stockpile can multiplied again and again to sustain local colonies.

A kite-drawn wind turbine in Saturn's high velocity winds. 

Alternatively, if the outer moon has its own alternative means of producing energy, this orbital transfer system could be used to share it. One could easily envision this opening up the Saturn system – wind turbines on Titan powering launchers to all the inner moons, which receive many times what Titan had to put in.

Such a system requires trust on both ends. 

Mimas cannot produce energy if Titan is not willing to send mass back to it, but it also won’t have any reason (or indeed, the means) to beam back energy in turn. 

Where orbital transfer is the primary source of power, it could force partnership based on mutual dependence: however each side feels about the other, its needs the system for sustenance, contingent on the others’ cooperation, and so they will put aside their differences at least insofar as it enables them to continue working together.

This breaks down if one side has an alternative energy source, in which case it gains a measure of power over the other partner. 

Consider Titan, powering the Saturn system through its wind turbines. Since it can make do without the beamed returns, yet is critical to circulating mass, it could use this to force its influence on the other moons, threatening to withhold all launches if they won’t subject themselves to its demands.

Such trust becomes even more critical when it’s done between planets, where shipments must be planned years in advance. The same energy politics could be extended to the solar system at large, though in this case, the threat is slightly different: rather than stopping the mass stream, which would not be felt for years, one world could radio propellant busses near their destination to move off course, thus immediately denying the catcher. 

The passive move is to order the projectiles to miss. The aggressive one... is to hit. (by JimHatama)

Mercury would seem to be best situated to take advantage of this, since it is the best partner in terms of frequency and energy availability per transfer, but also has plentiful solar power it can turn to should the launches cease.

Of course, this assumes that each side controls transfer in only one direction. It would certainly be in one side’s interests to control stations on both ends of the exchange, and thus avoid having to make any concessions to the other party.

This does not necessitate hostile acquisition – such a situation could arise historically from one colonizing power providing stations to develop its new outposts, and another could even offer these services to a more developed world selling the energy in exchange for local goods. 

A Kinetic Energy Exchange can serve as a transport system that supplants freighters. 

Other forms of dominance would require significant disparity between the strength of the dominator and the dominated, as one side must be capable of maintaining forces far from home to reliably defend assets right above the surface of their opposition. Justifying such presence would likely require significant energy to be at stake, and it may well be that some worlds’ advantages position them to be taken advantage of: Mercury and Neptune could well benefit, but they could also be at risk from those vying for the unique capabilities they’d offer.  

Having said that, the orbital transfer system could be a means of independence as well as suppression, because there is one thing that it almost unique about it.

It is an effective low-investment means of providing energy in the outer solar system.

This is critical; although many other means have been discussed on this blog, they require either large structures or rare resources. 

Wind turbines on a massive scale could be built within the atmospheres of the gas giants themselves, harnessing their powerful convective currents, but these would require delving straight into their deep gravity wells and creating structures in mid-air capable of surviving the harsh local conditions. 

Huge installations would be needed to extract significant energy from Enceladus's heat.

Large geo (luna?)-thermal plants could be built on Europa and Enceladus, making use of the temperature gradient between their frozen surface and subsurface ocean, but these would require digging tens to hundreds of kilometers through their crusts. 

Electrodynamic tethers can be used in reverse to produce electricity.

Electrodynamic tethers could be launched at the gas giants themselves, obtaining power as they de-orbit within each planet’s magnetic field, but these would require regular consumption of metal, a resource that is in short supply in the largely icy outer solar system.    

This far out from the sun, it’s most likely early colonies would rely on nuclear power, and to get that, they would need regular shipments of fissionable material from the inner solar system. At least for a while, they would be dependent on them for all their energy needs.

The orbital transfer system provides an alternative, both via interactions between moons and the planets themselves. It is not necessarily a good means of sending energy from the inner solar system to the outer, given how little energy is received relative to the cost of launch outside of optimal windows (as little as 2% for a retrograde Mercury-Neptune transfer), but it is great for sending energy from one gas giant to another, seeing as the high orbital velocities mean that is increased both ways.

If the inner worlds threaten to withhold shipments a little too often, Jupiter and Saturn could quickly set up networks within their own moon systems, having to hold on only long enough to get the launchers and receivers set up before they could power on and manage on their own. 

In the longer term, these two gas giants could set up exchanges between one another, forming the basis for further development. The turbines on Titan could be used to set up metal mines on Callisto, providing both with energy and metal, and in one fell swoop, they would cover two of the greatest limitations to their worlds’ self-dependence.

Suffice to say, the interorbital kinetic energy exchange allows for many different paths of development, making for a more interesting image of the solar system.