Energy costs of giving Mars and Venus moons: moving gas giant moons vs. capturing a rogue planet?

Question

As many here likely know, one of the factors that make both Mars and Venus uninhabitable to humans are their lack of a magnetosphere to keep solar winds out.

In the setting I am working on, I wanted Mars and Venus to be terraformed. Among the steps taken in the terraforming process; they would each have a moon put into orbit in order to introduce tidal forces to the planets.

I've thought of three possible solutions to giving either a moon:

• Scientists discover a window in which they can launch a gas giant moon out of orbit on a path that, with possible adjustments made along the way, would result in it being caught in a new orbit without disrupting any other planetary bodies. Earth sends out crews to set up automated systems that will act once the window arrives. It could possibly create two colonization ready worlds in one go depending which moons are commandeered.
• A small rogue planet is spotted on a path through/near our solar system and unmanned vessels are sent to adjust its course to capture it into a stable orbit around Venus or Mars. (Only one of them would be terraformed in this scenario, which is fine.)
• Dwarf planets and larger asteroids are gathered to create an artificial moon which is then placed into orbit around its new home. Assembly can either take place either in orbit, before being moved or on the way to its new destination. As it turns out, there is insufficient material in the asteroid belt to accommodate this method. Which would bring us to ripping moons out of orbit, but multiple smaller ones rather than a single big one.

Assuming that thrusters on the scale to safely move planets are created (even if they're seen as somewhat ludicrous in-world) and that the proposed moons from all three methods are the same proportion in mass to each of the planets they will be sent into orbit of, which of these methods would be the most energy efficient?

I know all three are going to have crazy expensive energy/time demands, but I want to know which one is the least crazy of the three.

Writer's note: The ludicrous scale and increased taxes to fund this massive project are meant to be part of the backstory and one of the factors that drives the outer colonies to declare independence from Earth's interstellar government.

As a side question, would it be possible to use the insertion of said moon into Venus's orbit to also increase its astonishingly slow rotation speed? To kill two birds with one stone. (Just as something to save the time/fuel/money.)

Answer 1

You can compare using delta-v

The delta-v budget of a mission tells you the total change in velocity that you will need to get an object out of one orbit and into another one. This change in velocity is related to fuel consumption by the rocket equation.

$$\Delta v = v_{exhaust}\log\frac{m_0}{m_f}$$

The energy of the delta-v burn can be calculated as the kinetic energy of the fuel exhaust. Exhaust velocity is $v_{exhaust}$, while fuel used is $m_0-m_f$. The energy of a trip is then $$\frac{1}{2}\left(m_0-m_f\right)v_{exhaust}^2.$$

Thus, the energy that is needed to transfer orbits depends in a great deal upon the means of propulsion. Since you didn't specify a propulsion system or its characteristics, then we should calculate energy requirements in terms of delta-v.

Transfer orbit requirements

A Hohmann orbit is an elliptical orbit used to transfer between two other orbiting objects (like planets). Atomic Rockets has pretty complete Hohmann transfer tables.

For Venus and Mars from the four gas giants, the asteroid belt, and deep space, here are the potential delta-v's. The total column is total delta-v; insert is the burn to insert the moon into a transfer orbit, while arrive is the burn to exit the transfer orbit and start orbiting the planet. Note that the deep space object is not orbiting the sun, it starts in a transfer orbit of sorts. All units in km/s.

                  Insert    Arrive       Total
Asteroid-Venus       6.1       6.4        12.5
Jupiter-Venus       18.0       8.1        26.0
Saturn-Venus        11.1       9.1        20.2
Uranus-Venus         6.8       9.8        16.6
Neptune-Venus        7.3      10.0        17.3
Deep Space-Venus              17.8+       17.8+

Asteroid-Mars        2.5       2.4         5.0
Jupiter-Mars        17.8       4.2        22.0
Saturn-Mars         10.9       5.5        16.4
Uranus-Mars          6.7       6.5        13.2
Neptune-Mars         7.2       6.9        17.3
Deep Space-Mars               11.2+       11.2+

Commentary

First off, the orbital transfer delta-v doesn't tell the whole story at all. There are also the orbital characteristics of what you are moving. For example, say you wanted to move Triton, moon of Neptune. Its orbit is both retrograde (backwards) and highly inclined. So this will take you much much more energy than a simple Hohmann transfer. So these are just guidelines at best.

That being said, Mars is easier to get to than Venus, no matter what your starting destination is. Jupiter is harder to move moons out of, it being the most massive gas giant. Uranus, the least massive, is the easiest to move moons out of its orbit.

The deep space calculation should be taken with a grain of salt too. What I listed is the solar escape velocity for an object starting in that planet's orbit. That means the minimum speed to escape the solar system, which is also the minumum speed to capture an object that is not orbiting the sun (which is the case for a rogue planet). However, the rogue planet might be moving considerably faster, so the delta-v requirement will scale up with the rogue planets (unknown) velocity. What I have listed is simply a minimum.

Finally, you discounted the asteroid belt as a source of a moon, but consider that there isn't that much mass just hanging out in the solar system waiting to be captured. There are only 2 Kuiper Belt objects (Pluto and Eris) and seven moons (Luna, Titan, Triton and the four Jovians) that are definitely bigger than the asteroid belt. After the asteroid belt, the next easiest place to move thing from is Uranus. But Uranus's four biggest moons are not that much better than what you can get from the Asteroid Belt. By comparison in $10^{21}$ kgs of mass:

Titania        3.5
Oberon         3.0
Asteroid Belt  3.0
Ariel          1.3
Umbriel        1.2
Ceres          0.9

So as far as getting a moon goes, you're almost better off getting asteroids than Uranus's moons. You get about 1/3 the total mass, and its a lot cheaper energy wise.

Conclusion

The best bet is probably the asteroid belt, even though the moon might be small. The next best bet is a rogue planet passing through, as long as it is going slow and just the right size. The chances of that are basically nil, but thats what stories are for. After that, if you really want a good sized moon, Titan is pretty much the only option. Triton has bad orbital characteristics, Uranus's moons are too small as well, and the Jovian moons are pretty hard to pry out of Jupiter's gravity well.