How to harness Jupiter's gravitational energy?

I am currently orbiting Jupiter, the rent is pretty cheap and the view is magnificent.

However, my former electricity provider refuses to connect me to earth's grid for petty reasons like "are you insane?" and "how did you get there anyway?" Still, my electricity requirements are pretty high as I wish to do heavy science stuff and watch cat videos in the winter.

Problem is, the sun is far away often hidden by Jupiter, there is no wind in space and mining does not seem like a good idea around here.

Now, I have a lot of potential energy, as Jupiter is massive. I know that tidal power uses the moon's mass to generate energy, and Jupiter is allegedly bigger than the moon. Thanks to a lot of money, planning and duct tape, I can build pretty much anything I want.

Here are now my actual questions:

• Can I convert Jupiter's gravitational pull to electricity?
• What is the best way to do it?
• Is it a good idea with respect to other potential power sources?
• How much energy can I provide? Enough for a house? A city?

My actual orbit is left as a choice to you, I'd rather it to be stable and with a nice view.

• Comments are not for extended discussion; this conversation has been moved to chat. Nov 7 '16 at 17:55
• Why does everybody want to send you the energy with lasers? Just wondering. May 3 '17 at 0:18
• @Gryphon I guess that's the best way to transfer energy in space.
– PatJ
May 3 '17 at 15:27
• OK, OK, it just seemed like a lot of lasers. May 3 '17 at 19:57
• @Gryphon isn't that a good thing? Lasers are awesome.
– PatJ
May 4 '17 at 14:33

There's plenty of ways to extract energy from Jupiter's gravity. The problem is how do you do so without lowering your house's orbit?

Tidal Flexing? (probably not worth it)

When two objects interact with gravity there is a tidal force. The sides closest to each other feel a stronger tug than the sides furthest away. This causes them to stretch just a little bit. If they're rotating, this causes a cycle of stretching and compressing which extracts energy from gravity. Eventually it will slow the body's rotation until they are tidally locked.

Another way to extract the tidal force is through a highly elliptical orbit. As the bodies get close to each other the tidal force increases and squeezes. As they move away the tidal force decreases and they expand.

This tidal flexing provides a heating effect which keeps many of Jupiter's small moons much warmer than they would otherwise be.

Let's say you built a huge sphere, it can be hollow because mass does not matter, and set it into a highly elliptical orbit and extracted the energy due to tidal flexing. First problem is where did you get all that material? Second is that putting it into a highly elliptical orbit will cost a lot of energy, so there's a big up-front investment. Will you make that up? I doubt you'll get a solid ROI in your lifetime, but someone is welcome to do the math.

Use Jupiter's Magnetic Field? (unsustainable)

Jupiter has an enormous magnetic field, and your house is moving through it. You could wrap some wire around a coil and extract electrical energy from this field! Brilliant!

...except by doing so you're creating an oppositely charged magnet which Jupiter's magnetic field draws in creating a drag on your house's orbit. You're mortgaging your house's orbit for electricity. This is a theme.

You could kick the electro-magnet out into space. Then it will be in its own orbit. It would send the energy back to you with a laser (also a theme) and eventually crash into Jupiter.

You Can't Just "Drop" Things From Orbit

When you're in orbit in a vacuum you can't just "drop" things. If you let go of something it will continue merrily along with you in orbit traveling at thousands of kilometers per hour. It has inertia keeping it going, just like your house.

In order to lower a thing's orbit you have to slow it down. This has to be done either with drag or with energy. There's no atmosphere to drag against in orbit, so you need to provide energy to slow it down.

You don't need to provide all the energy to slow it down, just enough so it scrapes the atmosphere. That will provide drag to slow it down further and eventually fall into Jupiter.

Space Trash Yo-Yo? (unsustainable)

You extract energy from a gravitational field by dropping things into it. The potential energy converts to kinetic energy which can be harnessed like water falling down a waterfall and turning a turbine. But if you raised and lowered your house you'd more energy lifting yourself back out (because friction and entropy) than you gained lowering yourself.

You need some disposable mass you can toss down the hole. Since you don't want to dispose of your lead figure collection, or your vintage Bricks Of The World set, you can use your trash. Not terribly sustainable, but it'll work for a while. But how do you extract energy from it?

You could put the trash into a special bucket with a turbine in it, wind a wire around the turbine, attach the other end to your house, and drop it like a yo-yo. The wire unwinds as it falls, spins the turbine, and the wire transmits power back to your house. Perfect!

Except it costs more energy than you'll extract.

Even if you let the trash-turbine fall off the end of the wire, the wire is tugging on your house with equal energy as you're extracting and pulling it into a lower orbit. You'll need to expend more energy (not equal because, again, entropy) to keep yourself in orbit.

Space Wind Laser Turbines! (unsustainable or won't work)

Instead, your trash generator... bomb will have a wind turbine attached to it. As it falls through Jupiter's atmosphere the wind generated will turn the turbine and extract energy from its fall. A laser transmits the energy back to your house.

But eventually you'll run out of trash. Can we make this sustainable? Yes!

Cut out the middle man. Instead of using gravitational energy to create wind power, use Jupiter's own copious wind power directly. Drop floating wind turbines into Jupiter's atmosphere and have it shoot the power back to you with lasers. They'll float around in Jupiter's atmosphere until they wear out. With a little engineering they can be made to steer and avoid storms.

How much power can you provide? The potential wind energy of the Earth is about 250 TW. Jupiter's atmosphere is enormous in comparison and more turbulent, so it's limited by how many wind turbine dirigible parts you brought along and how long you can make them last.

...except, as @anaximander pointed out in the comments, it won't work. The atmosphere is a fluid, and the turbine is floating in that fluid. It will move with the wind. This is like putting a water wheel on a boat in a river. The turbine has to be anchored to something and there's nothing to anchor it to.

• Comments are not for extended discussion; this conversation has been moved to chat. Nov 7 '16 at 17:55
• I think you're underestimating the practicality of putting a number of coils of wire around the planet. Then it's one big generator. A series of sails around the outside of the coil (one side white, the other black) could use solar pressure to counter the rotation that would build in the coil as you extract electricity. Jul 14 '18 at 17:39
• @Basic That's 420,000 km of wire for one loop, and that's assuming you can put it right near the surface. Once you put it there it's unstable for the same reason rigid Dyson rings are unstable. If you have to use energy to keep it in orbit to balance the energy you're extracting you could skip the coil and harness that energy directly. There's no need for a full coil, any object in orbit is moving through the field, and I covered that. It's all trading orbital energy for electrical energy. Jul 14 '18 at 17:51
• Relevant to "you can't just drop things from orbit" — If we threw a baseball from the ISS, could we deorbit the ball? Dec 15 '18 at 2:55

You do not

Your question is akin to asking "How do I harness the potential energy of an ideal frictionless hole?".

The short answer is: you do not.

The longer answer is: you find something to push down the hole, converting the "object's" potential energy to kinetic energy, and then harvest that kinetic energy to drive a generator.

The most obvious example of this is hydro power dams. A small-scale example is for instance gravity driven pendulum clocks.

Your problem here is that you are in orbit. And any such "object" you wish to push down the hole is in orbit with you. You also do not have a nice solid anchor to which you could attach your generator.

This is one of the great annoyances of being in space: it is one of the places where you have Spherical Cow conditions. So unlike on Earth where friction against ground and/or air is of great help to you, these do not exist in space. Once you have kinetic energy of an object in space, it is a right nuisance trying to convert it into some other form of energy (which is what happens when you generate exergy, i.e. usable energy).

Contrary to what previous answers said, you can get a lot of energy out of Jupiters enormous gravitation alone. Io already does this: it's undergoing strong tidal flexing, and this heats the interior so that moon is much more geologically active than any of the planets. So, one way to answer your demand would be: build a geothermal plant on Io. If you don't actually want to settle on Io, you can transfer the energy to your orbiting station via laser.
This is of course not easy, but definitely more feasible than harnessing wind power in Jupiter's own atmosphere (where, even if you could manage to sustain the floating turbines in the upper atmosphere for long enough to be useful, it would be crazy difficult to point the laser steady enough in the right direction).

Still, I think electrodynamic tethers are a better solution, as they harness the energy directly in your own orbit. Schwern is right though that in this case, the braking effect is significant. Basically, the electrical energy you harvest comes just out of your own kinetic energy, which really is something you should not be wasteful with (crashing into Jupiter is not a pleasant experience). You can still harvest a lot of energy before that happens – 260 kWh per kilogram you bring into orbit with you, which corresponds to about 37 times the energy density of coal – but that's it then.

It is possible to get more if – like with the Io geoelectric plant – you tap on the much vaster kinetic energy of one of Jupiter's moons. To do this, you need to place your orbit near one of the stable Lagrange points, best in Ganymede's L5 point. The electrodynamic braking will then be compensated by Ganymede's gravitational tug, some way ahead in your orbit.

• Electrodynamic tethers and using a moon sounds like the most feasible plan I've seen so far. Essentially you're leveraging the massive energy of the moon rather than using your own. Nov 4 '16 at 11:32

As others have mentioned, you are not going to be able to convert gravitational energy to electric while you are still in orbit. But, thanks to all of that planning and money you mentioned you can still set up something to allow you to get the energy you need while maintaining your current lifestyle.

Dyson Wept

What we are going to do here is take the concept of a Dyson Swarm, turn it sideways, and then squint our eyes at it until it does what we want. And isn't that what Science is all about?

A Dyson swarm is a series of solar panels that orbit a star, collecting energy and then transmitting it to some other location. Unlike a Dyson sphere, which is a solid shell, the swarm is made up of many small panels with spacing between them. So how does this help you around Jupiter?

The Belt of Jupiter

Instead of placing your solar energy collectors around the Sun, you can instead place them around Jupiter itself. Figure out what your orbital path will be ahead of time, and then seed the entirety of Jupiter with a ring of those solar collectors that will be near (but not intersecting!) that path. This way, no matter where Jupiter is in relation to the sun there are always some collectors aligned to absorb energy from it. Any collector your ship is in range of would transmit its power to you, and ones that were not in range would include a battery pack to store it in. As you travel along the orbit you would come into range of different collectors, triggering their batteries to start transmitting to you while the previous collectors refill their charge.

Don't Reinvent the Wheel

Of course, if you are going to go through the efforts of making something that looks like a Dyson swarm around Jupiter, you could just as easily do the same thing but around the Sun as they were originally designed. The wireless energy transmission would be harder to calibrate, but on the plus side you would gain much more energy using the same number of collectors, or conversely the same energy using far fewer.

• This plan as a problem: the efficiency of solar collectors at Jupiter is very poor. Jupiter is 5AU out, compared to the Earth's 1AU. Solar intensity falls with the square of the distance so at 5 times further out solar collectors at Jupiter will be 25x less efficient. At Earth's orbit solar energy is 1360 W/m<sup>2</sup>. In Jupiter's it's 54 W/m<sup>2</sup>. A square meter of good solar panels in Earth's orbit could run 3 or 4 high end gaming desktops. At Jupiter it would barely power a laptop. Nov 3 '16 at 0:06
• Also putting solar panels "all around Jupiter" requires a lot more solar panels than "all around earth". I'm not sure it fits my budget.
– PatJ
Nov 3 '16 at 15:15
• @PatJ You hadn't specified an upper limit to the budget, which would of course limit how many panels you can use. But I do want to point out that you don't need them "all around Jupiter" on every axis, just a single ring of them encircling the area that would get the most sunlight. And per Schwern's concerns, that is why I suggest just using a typical configuration around the sun and then beaming that energy back to the Jupiter station. Nov 3 '16 at 16:35
• Does Jupiter emit enough radiation of the proper type to activate the photovoltaics? In other words, would it be possible to rebrand solar panels as Jovian panels? Oct 5 '18 at 15:05

Since the question is asking for gravitational energy, then you could consider the answers here to be the "backup" plan (although given the difficulty of harnessing gravitational energy, this is actually plan "A").

Gravitational energy is not something you can directly harvest. On Earth, we generally tap into gravity by taking the potential energy from something like a mill pond or a reservoir at a high point, and convert it into kinetic energy by having the water run down a raceway to a lower point, turning a turbine or waterwheel somewhere between point "A" and point "B".

The closest analogy I can think of would be to allow an electrodynamic tether to pass through Jupiter's magnetosphere. The wire will generate a current as it spirals down through the magnetosphere and energy can be beamed from the decaying orbit to a suitable receiver. This paper provides calculations and gives you some idea of the magnitude of the task you are setting yourself, for example, a tether for a small spacecraft needing an average of 180W while in a 5 day orbit around Jupiter would need a tether 4.75km long by 1mm in diameter. The natural analogue is the Moon Io passes through the Jovian magnetosphere and generates an electric "arc" running between the moon and Jupiter that carries 2 trillion watts of electrical energy.

The other takeaway from this paper is that useful power is generated inside the orbit of Europa, which is also inside the intense radiation fields of Jupiter itself, so you would either be inside a heavily armoured space habitat, or in orbit beyond Callisto and thus outside of the radiation fields.

So very long tethers could be built and sent on elliptical orbits around Jupiter, beaming power back to you. As the energy is beamed from the tether, the orbit will naturally decay (you don't get something for nothing). You could beam energy back to the tether, and then it acts as the armature of an electric motor, being "pumped" against the magnetic field of Jupiter and raising the orbit. Given efficiency losses, the amount of energy being pumped back into the tether to raise its orbit will be more than you harvested as it was extracting energy from the magnetosphere, so part of your infrastructure is going to be building and launching new tethers on a periodic basis.

Not quite. But you can use it's moons, thanks to the slingshot effect. What you do is pass behind them, given their direction of orbit. The gravity of your own spacecraft will pull the moon back, slowing it down just a tiny bit, while the moon's gravity will speed you up. You basically transfer momentum from the moon to yourself.

With a good aim, you can then fly through Jupiter's magnetic field at high speed and extend a tether like the one AndyD273 mentioned, converting the kinetic energy you have into useful electrical energy. This will slow you down, but if you plan it correctly, you will have enough energy to pass another moon.

The downside is that you have to pass over the poles of Jupiter, and Jupiter's gravity will pull you out the orbital paths of the moons. This doesn't make it impossible, but you could spend an awful lot of time making weird orbits and not much time leeching momentum from moons or turning it into power. Solar cells, even this far out, may provide more power.

• This answer has promise, but is not hard science. Perhaps you could describe in a more rigorous manner the orbits you would follow and the methods you would use to generate electricity? Nov 3 '16 at 18:32
• "The downside is that you have to pass over the poles of Jupiter, ..." I don't get this. Why would you have to pass over the poles, and why would it be a downside even if you did? Nov 4 '16 at 5:00

Energy can't be created or destroyed so... (for simplicity, I'm assuming your home is Callisto)

You total potential energy from orbit is $$\mathrm{Mass \times Gravity \times Height} \\ = 10.8×10^{22} \times \mathrm{3.86×10^{17} N \times 1,880,000km \\ = 725,842,965,000,000,000,000,000\times10^{22} Joules}$$

(Use this to calculate gravity easily)

A Watt is 1 Joule/Second, so assuming you live 100 years, divide Joules by Seconds (3,153,600,000 seconds is close enough)

So about $724,842,965,000,000\times10^{22}$ Watts if you crash yourself into Jupiter (but you'd die long before you reached the surface)

And then for scale, a TV uses 80-400 watts. The average household uses an average of 660,000,000 watts (converted from 10,932 kilowatthours).

roughly $100,000(\times 10^{22})$ Watts left brings you close enough to the limit to know you'd die long before harvesting enough power from gravity alone. (From Jupiter's atmosphere)

Ok, Math was off, so this may be possible if some magic device existed, but next point invalidates this plan anyways.

Sadly though, you will probably be expending power to MAINTAIN your orbit. And since messing with your potential energy will decay your orbit... This really is a bad idea. The above equations assume perfect conversion. You will likely only ever harness a tiny fraction of that. You want cheap, reliable power! But the only power sources left now are the sun and what you bring from Earth... And you aren't getting nearly enough sunlight unless you build some massive solar array system.

Your only 'reasonable' (everything is unreasonable at that distance) is to use a Radioisotope thermoelectric generator (output based on your model/design, though expect about 2-5 Watts/kg). (At least from a modern tech standpoint, this is what the smart people at NASA do. Just try not to make the NASA Office of Planetary Protection upset.)

• Potential energy is acceleration due to gravity times mass times height. Where did you get mass for your figure of 3.8e17 newtons. Also, I wish there was a 'Use 10 times too many significant digits in a hard-science question' badge for this site. Nov 2 '16 at 16:45
• I got the Mass of Jupiter and Calypso from Wikipedia (and their distance apart) and used the calculator I linked to get g. Callisto is 10.8×10^22kg so that would actually put my answer about 20 orders of magnitude too small. Nov 2 '16 at 16:53
• Gravitational potential energy is $\frac{-GMm}{r}$. You can harvest half of the difference by changing $r$ if you stay in a circular orbit. No need to calculate local values of $g$. Energy harvested by lowering Callisto to cloud tops is around $8.7\times 10^{31}\text{J}$. But that's a huge mass. Nov 3 '16 at 6:38
• "660,000,000 watts (converted from 10,932 kilowatthours)." - Erm, you are out by a few orders of magnitude. 11000kWh ~= 40GJ, which assuming you draw energy at a constant rate over the course of a year, that corresponds to a power draw of 40GJ/(60*60*24*365) ~= 1250W. But either way, 80x10^48 J from the planet divided by 40GJ per year is roughly 2x10^39 years, so your point is still valid. Nov 5 '16 at 11:15

Use a MHD Generator. Problem solved.
You directly tap into the magnetospheric plasma's kinetic energy by using the Lorentz force, without affecting your orbital energy OR angular momentum (the difference between both seems to elude everyone here). This one is free recharge of batteries forever.

The problem with "tidal flexing" is none. First of all, there are two types of tidal internal heating on moons. One dissipates eccentricity (and thus angular momentum, not energy). That's tidal flexing.
The other, weaker one, but also active in orbits with zero eccentricity (like you know, every moon in the solar system, just look at some numbers) that works through actual tidal forces, is so strongly dependent on size and mass of both objects, that your house wouldn't have a problem with it.

• Sounds promising! Where do I find the plasma?
– PatJ
Nov 4 '16 at 9:58
• The whole magnetosphere is filled with it. Plasma is essentially gas that is ionized to a varying degree. Magnetospheric plasma is usually fully ionized (meaning every atom has at least one charge) and consists of electrons and all kinds of elements of varying charge numbers. So there are lots of charges flying around. But Magnetospheric physics is more complicated. You want to avoid regions with heavy non-thermal highly-energetic particles flying around, but also float in a region of sufficient current density to charge your MHD Gen. Nov 4 '16 at 10:35
• If I'm in that sphere, would there not be friction slowing me down and eventually crashing me into the planet?
– PatJ
Nov 4 '16 at 10:53
• @PatJ: You would have some limited friction. But Electromagnetic forces are really strong. Look up the extension of magnetospheres. They're huge. You can easily pick a region that has sufficient charge density and low enough friction to make your plan feasible. Nov 4 '16 at 23:30

Direct gravitation energy is a lost cause in freefall. If you're close enough to get gravity, then you're no longer in orbit and will die soon.

There is a work around though, which does use gravity, indirectly at least.

By using an Electrodynamic Tether you can tap into all the kinetic energy you have and harvest all the electricity you want.

Electrodynamic tethers (EDTs) are long conducting wires, such as one deployed from a tether satellite, which can operate on electromagnetic principles as generators, by converting their kinetic energy to electrical energy, or as motors, converting electrical energy to kinetic energy. Electric potential is generated across a conductive tether by its motion through a planet's magnetic field.

And Jupiter has a massive magnetic field, which means lots and lots of energy!

This isn't going to solve the bandwidth problems for streaming youtube cat videos though.

EDIT: There is no such thing as a free lunch.
Interacting with Jupiters gravity in any way will cause the house to lose momentum, and eventually deorbit.

You could postpone this by passing the inertia debt off on something else, like a tidally locked ice chunk from Jupiter's rings or a moon.
You could attach the ED tether to it, and use either microwaves or lasers to send the energy to your house, then when they eventually lose enough momentum to deorbit you collect the tether and place it on a new rock.

Gravity isn't the best option

Jupiter is a giant ball of mostly hydrogen, and so is for all practical purposes an unlimited supply of fuel.
At the cost of having to import oxygen you could burn the hydrogen to get energy, and get drinking water as a byproduct.
You could also separate out the hydrogen 3 to use in a fusion reactor for energy, and sell the resulting helium to Earth to pay the high bandwidth bills.

• This will lower the house's orbit. Nov 4 '16 at 18:44
• @Schwern True, but you might be able to strike a balance between drag and power to where a minimal amount of fuel could be expended to boost the orbit once in a while. And you are also really close to a really large stash of hydrogen which could potentially be harvested and used as fuel to increase momentum. Of course if you can do that you could just use the hydrogen to make energy... Nov 4 '16 at 19:45
• Yeah, some calculations on how much power you can generate for how long while staying in a safe orbit would be interesting. Nov 4 '16 at 21:07
• Why not just directly convert the fuel to electricity? Nov 5 '16 at 8:43
• @TimB Because the question was about using gravity to do it. And you can't use the gravity directly when you are in freefall. If you drop something into Jupiter to try to generate power then you will be losing mass, and there is only so much mass you can lose. Burning Jupiter's hydrogen, setting up a small atomic generator, or a fusion generator using Jupiter's hydrogen... Lots of better options than using gravity. There is no such thing as a free lunch/launch. Nov 7 '16 at 14:18

As others pointed out, to extract "gravitational energy" you need to let something fall down, just as in an hydroelectric dam energy is harvested by letting water fall. As the question of what to let fall arises, I suggest to use a couple of Jovian moons.

The solution is quite simple, although there are a few practical problems I let to you to solve - I'm sure by the time you get to Jupiter you will have solved them.

First of all you need a pair of long light and strong tethers - a few millions of kilometres should be enough.

Then you need to attach an end of each tether to a different Jovian moon. Any couple of moons could work.

Attach the other end of the tethers to a sheave with an spring, similar of the device used in extensible cords (example). Keep in mind that the sheave and the spring need to be capable to allocate most of the length of the tethers, but I'm sure that if you have managed to get the tether that won't be a practical concern.

Now you just need to attach an electrical generator to the sheave and you will get plenty of energy that you will be getting from the mechanical energy of a couple of Jupiter moons. Since you will be braking and accelerating them, they are likely to end falling to Jupiter or getting in the same orbit, but don't worry: there is plenty of moons around to keep powering your house.

You might be concerned by the risk of the tethers getting entangled in Jupiter, but we shouldn't worry about that. If it happens - and if Jupiter surface isn't slippery enough to the tether to free itself - the planet will start pulling the moons and you will get even more energy from Jupiter's rotation. You just need to make sure that the tethers are strong enough and firmly attached to the moons.

You can get energy from tidal forces. I think the 'easiest' way would be to put a long rod on either side of your station, and have an equal mass on each rod that can slide up or down the rod. The masses are connected to a mechanism that means they both move inwards or outwards together, for example you could have a crank shaft in your station a little like the piston mechanism in an engine. You could make this 'engine' reciprocate by rotating the station, but I suspect that ends up taking energy from the rotation of your station. Instead, I think it would be best to have an elliptical orbit like Io. Attach a clock spring to your crankshaft, and keep the axis of your rods pointing towards Jupiter (tidal locking does this for free). Then as your altitude changes, so does the differential force on the two masses, so the force on the spring changes, so the crankshaft turns a little, so you can extract a bit of energy. A pitifully tiny amount of energy you say? Just use bigger masses and longer rods!

This doesn't actually make you spiral in (in fact if you're outside a geostationary orbit it will make you spiral out) due to tidal acceleration. You're actually robbing energy from Jupiter's rotation.

https://physics.stackexchange.com/questions/142435/when-a-planet-is-heated-through-gravitational-pull-where-is-the-energy-taken-fr

https://en.wikipedia.org/wiki/Tidal_acceleration

To harness gravitational energy you have to use a slingshot to transfer momentum. Specifically transfer momentum from something orbiting the thing you're orbiting. (You can't transfer momentum from a star while orbiting that star.) Jupiter has moons right?

Gravity assist and a form of aerobraking give you a method to speed up and slow down. https://en.wikipedia.org/wiki/Gravity_assist

"Realistic portrayals of encounters in space require the consideration of three dimensions. The same principles apply, only adding the planet's velocity to that of the spacecraft requires vector addition, as shown below." Check out the diagram.

https://en.wikipedia.org/wiki/Aerobraking

"During the termination phase of the mission, a "windmill experiment" was performed: Atmospheric molecular pressure exerts a torque via the then windmill-sail-like oriented solar cell wings"

Instead of converting the energy (pressure) to heat, you use a turbine and generate electricity.

I would use multiple drones that travel between Io, Jupiter's upper atmosphere, and Europa.

How much energy? You would calculate this by subtracting the periapsis (closest approach) speed of the Io-Jupiter orbit from the periapsis speed of the Europa-Jupiter orbit.
Io takes 1.77 days to orbit and Europa takes 3.55 days so you should get a charge at least once a week.

https://en.wikipedia.org/wiki/Elliptic_orbit

velocity = sqrt( Jupiter's grav param x ( 2/dist from Jupiter - 1/length of orbit )

https://en.wikipedia.org/wiki/Standard_gravitational_parameter

Jupiter's grav param = 1.27 x 10^17 m^3/s^-2

https://en.wikipedia.org/wiki/Jupiter

https://en.wikipedia.org/wiki/Galilean_moons

Io's orbit radius = 421,700 km

Europa's orbit radius = 670,900 km

Speed before aerobraking = 58,153 m/s

Speed after aerobraking = 57,405 m/s

Let's say your home or the drone has the mass of the Eiffel tower.

https://en.wikipedia.org/wiki/Eiffel_Tower

Ship's mass = 7,300,000 kg (The tower's original weight)

kinetic energy = .5 x mass x velocity^2

The change in kinetic energy = 315,500,000,000 Kilojoule

1 joule = 0.0002777 watt hours

So 87,600,000 Kilowatt hours

If you could collect 20% of this you'd get 17.5 Gigawatts.
That's 910 a year if you can get a charge once a week.

New York uses 60,000 Gigawatts a year. http://engineering.mit.edu/ask/how-many-wind-turbines-would-it-take-power-all-new-york-city

So you'll need 66 drones with the mass of the Eiffel tower or your home will need to be 480,000,000 kg to power a city. Europa's mass is 4.8×10^22 so go ahead and power 10 cities.

If you didn't use drones this would be extremely exciting.

I'm picturing a surf board shaped ship with multiple rows of paddle wheels running across the top and bottom

You can get energy from a gravitational field, but not in keeping your requirement for a stable orbit. If your orbit slowly spirals outwards your kinetic energy reduces.

This could be done using Jupiter's magnetic field, turning the Jupiter/spacecraft system into a giant dynamo.

However energy supply is never limitless: gradually your spacecraft will float further and further away from Jupiter, eventually leaving its gravitational influence, and entering the gravitational influence of another planet.

Also, while this was happening the power generated would also drift downwards to almost zero.

I'm not a physics guy, but I know you can't simply generate energy from gravitational FORCE continuously. But you may get by allowing something to fall(at expense of it's potential energy), but that isn't continuous. You'll have to find some other natural method to take it back (like evaporation -> hydroelectric power from dam).