Fusion candle - getting Earth to Jupiter

Question

In A World Out of Time, per Schlock Mercenary, Earth is moved to a new home by lighting a fusion candle. The fusion candle idea is discussed in the Schlock quote:

Building a gas-giant colony ship is not as difficult as it looks.

1. Build a fusion candle. It's called a "candle" because you're going to burn it at both ends. The center section houses a set of intakes that slurp up gas giant atmosphere and funnel it to the fusion reactors at each end.
2. Shove one end deep down inside the gas giant, and light it up. It keeps the candle aloft, hovering on a pillar of flame.
3. Light up the other end, which now spits thrusting fire to the sky. Steer with small lateral thrusters that move the candle from one place to another on the gas giant. Steer very carefully, and signal your turns well in advance. This is a big vehicle.
4. Balance your thrusting ends with exactness. You don't want to crash your candle into the core of the giant, or send it careening off into a burningly elliptical orbit.
5. When the giant leaves your system, it will take its moons with it. This is gravity working for you. Put your colonists on the moons.
6. For safety's sake, the moons should orbit perpendicular to the direction of travel. Otherwise your candle burns them up.
7. They should also rotate in the same plane, with one pole always illuminated by your candle (think "portable sunlight")
8. The other pole absorbing the impact of whatever interstellar debris you should hit (think "don't build houses on this side")

But the logistics are mind-boggling. It seems things would get tricky piloting Jupiter to catch earth (but not impossible) - the maneuvers for getting into orbit seem basic, but the candle needs to be rotated pretty fast to not burn the earth (going out of the ecliptic, I'd imagine).

So my question revolves on the possibility of getting Earth to Jupiter instead. Rockets, explosions, throwing the moon like a baseball towards the sun - whatever it takes to get over there. Given the rocket equation's thirst for mass, how much of an Earth is left over by the time we get into orbit around Jupiter? Is it habitable? Main Question: Is this idea plausible or will we be forced to pilot Jupiter to pickup Earth like a hobo jumping on a moving train?

I'll assume a Hohmann transfer orbit, with methane-oxygen combustion to start, but who am I kidding? This is saving the Earth. If you want to Rich Purnell a couple of Venus Flybys to save the planet, go for it. If you want to light up a fusion candle on earth that slurps up the ocean, go for it. If you want to light all the nuclear bombs in the world on one side of a really big metal plate, go for it. Throw the moon into the sun and be propelled backwards? Go for it. The Earth will be grateful you did - I'm just looking for plausible solutions. Of course, the other logistics of having Earth orbit Jupiter (Tidal forces, radiation) - is bad, but that's another question.

Answer 1

Asteroid flybys

Description

1. Use a convenient asteroid of small mass, pick one already close (in $\Delta V$ terms) to one of Jupiter's Lagrange point. Asteroid Hektor is a good candidate. It's already in Jupiter's L4 point and masses about $1 \cdot 10^{19} kg$
2. Put a solar power stations in several spots on the body.
3. Plant mass driver/railgun/coilgun on its surface so that the recoil of at least one of them points through the body's center of mass & connect them to your solar panel farms.
4. Even distribute the other railgun/coilguns around the body to provide for maneuvering.

5. Begin launching materials in such a way that you begin moving your asteroid (and you can also launch those materials to where they are needed - so you get a twofer).

6. Now steer that body through the nearest Lagrange point so you can begin using the Interplanetary Transportation Network. (see references).
   

The Interplanetary Transport Network (ITN)1 is a collection of gravitationally determined pathways through the Solar System that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little or no energy. These points have the peculiar property of allowing objects to orbit around them, despite lacking an object to orbit. While they use little energy, the transport can take a very long time.

7. You will arrange for the small asteroid to begin a very long series of alternate flybys between Jupiter and a large asteroid.
8. On one end of the orbit, the asteroid flyby will transfer some momentum between the small asteroid and the large one.
9. On the other end of the orbit, the asteroid flyby will transfer some momentum between the small asteroid and Jupiter.
10. Each of these flybys will mimic a purely elastic momentum transfer ($m_{asteroid} \cdot \Delta v_{asteroid} = m_J \cdot \Delta v_J$ collision, but the net result is that you are transferring momentum between the target and Jupiter by way of your small asteroid.

The goal is to get the large asteroid to its nearest (in terms of $\Delta V$) Lagrange point so it enters the Interplanetary Transportation Network too. You will repeat the above steps only the Earth will be the target of movement.

Rather than moving the Earth into the Interplanetary Transportation Network, you will gradually increase the orbital radius of the Earth until you get it to the desired location.

Warning, although we could start doing this sort of Solar System Engineering now, it will likely take thousands of years to complete the operation.

Calculations

Basically we're performing momentum transfer interactions (modeled as collisions). We get one complete round trip momentum transfer exchange for every round trip between Jupiter, the target, and back to Jupiter. Because the orbit phase (where the object is in its orbit around the sun) will constantly change and that for most exchanges, the asteroid will have to complete a whole orbit of the Sun before it gets another chance to interact.

Assumptions:

• Approximate the time it takes to complete a single momentum exchange as the time it takes to complete an elliptical orbit with perihelion at Earth's orbit and aphelion at Jupiter's (5.2 years).
• Asteroid we use to transfer momentum is Vesta
• $M_{Vesta} = 2.6 \cdot 10^{20}$
• $M_{Earth} = 5.9 \cdot 10^{24}$
• $M_{Jupiter} = 1.9 \cdot 10^{27}$
• Max $\Delta V$ from Jupiter encounter is $35 \frac{km}{sec}$ but realistically we'll be lucky to get $10 \frac{km}{s}$
• Max $\Delta V$ from Earth encounter is about $10 \frac{km}{sec}$

So the momentum transfer from each encounter will be about the same and it equals: $$ m_{Jupiter} \cdot \Delta V_{Jupiter} = m_{Earth} \cdot \Delta V_{Earth} = m_{Vesta} \cdot \Delta V_{Vesta} $$

So each interaction will give the following $\Delta V$:

• $\Delta V_{Vesta} = 10 \frac{km}{sec}$
• $\Delta V_{Earth} = 4.4 \cdot 10^{-4} \frac{km}{sec}$
• $\Delta V_{Jupiter} = 1.37 \cdot 10^{-6} \frac{km}{sec}$

$\Delta V$ required to raise Earth to Jupiter's orbit $= 3.4 \frac{km}{sec}$.

Divide the $\Delta V$ requirement by the amount of $\Delta V$ provided by a single interaction: $$N_{interactions} = \frac{\Delta V_{required}}{\Delta V_{Interaction}} = \frac{3.4}{4.4 \cdot 10^{-4}} = 7,624 interactions$$

Vesta flybys of Earth. At an average of 5.2 years per flyby, this will take you $$7,624 \times 5.2 = 39,648 years$$

But you can significantly accelerate the process by using more than one asteroid to transfer momentum.

Related discussion on how to move planets using asteroid flybys.

Answer 2

As usual for me, I am substituting ancient wisdom for hard science...

How do you eat an elephant? One bite at a time.

Start by building a planetoid in Jupiter's orbit using materials from the asteroid belt and the existing moons. That should give you a reasonable core for your "captured earth".

Now start shipping your favorite parts of Earth to Jupiter, in whatever quantities your technology allows. As each new part arrives, add it to your planetoid, building it up like a layer cake. Keep this up until there is nothing left on Earth worth saving.

Whether your extra-planetary exports include all the humans is up to you, but you will want to take at least one mating pair of each major animal species. More would probably be better.

The final step is to edit the history books, so that in a few generations, everyone thinks that your now-earth-sized planetoid IS the original Earth.

Problem solved! Earth now orbits Jupiter.