Launching a Generation Ship

Question

The human race has colonized the solar system with permanent settlements on all of the inner planets and outposts beyond. Vast space stations pepper the voids between orbits and massive mining/refining facilities stand vigil along the frontier. Interplanetary travel is routine and 25 Billion prosperous humans live within Sol’s gravity well. But not a single human lives beyond our sun’s reach.

FTL is still an elusive dream. We have all the energy we could ever use and have complete mastery over biology and conventional physics, but no miracle solution has been found for getting to other stars. We are still vulnerable and can become extinct from anything that can take out our civilization’s only sun.

This vulnerability troubles many, so after much debate, we decide to solve the problem. With nothing more than slower-than-light travel, humanity will reach for the stars.

We have chosen a large Kuiper belt object as the basis for both our ship and its launcher. The idea is to divide the asteroid into two pieces, one three times as large as the other. The smaller portion will be hollowed out to become the ship’s hull and the larger, the counterweight, will be sacrificed during the launch to help the ship obtain solar escape velocity.

Today’s question is about the launching process. Once the ship is ready and has been firmly reattached to the counterweight, the pair will be pushed out of orbit and dropped on a very deep slope into our sun’s gravity well. Picking up speed during the inbound journey, the pair will miss the sun by a barely survivable margin and then race past on its way out of our solar system. At the optimal moment, the ship and counterweight will detach from each other and the ship will push itself forward using rockets, nuclear explosives and a rail gun-type launcher which is secured to the counterweight. All of that force will push back against the counterweight, stealing its inertia and springing the ship out of our system with enough speed to escape Sol’s gravity well completely.

The counterweight will then fall back sun-ward to either be salvaged before or slagged when it plummets into the sun.

So the question… What percentage of light-speed would a launch technique like this imbue upon the ship portion of the asteroid, after it left our gravity well? Assuming the original asteroid was similar to MakeMake: 5x10^21 kg with a starting point for its sun-dive of 50 AU. Also assume that the rockets, nukes and rail gun successfully transfer 90% of the inertia from the counterweight to the ship.

Also, am I missing anything? Are there other ways to add speed to such a massive ship during its launch?

Answer 1

Physics time!

You split the rock into two parts, one is 3 times larger in mass than the other. You state "Also assume that the rockets, nukes and rail gun successfully transfer 90% of the inertia from the counterweight to the ship." This doesn't make much sense, because inertia can't be transferred, but I'm presuming you meant to say momentum. So here's what we can say:

• We have a large rock, with momentum $p$ and mass $m$, which undergoes an event splitting it in two.
• After the breakup, the smaller rock, with mass $0.25m$ has a momentum of $0.9p$
• How fast is the smaller rock traveling after the breakup, with respect to the original velocity.

Because the smaller rock has a momentum of $0.9p$ and mass $0.25m$, we can write the momentum equation $0.9p=0.25m\cdot v^\prime$, where $v^\prime$ is the velocity of the small rock after the breakup. Rearranging slightly, that gives us $p=\frac{0.25}{0.9}m\cdot v^\prime$. Since the original rock had a momentum equation $p=m\cdot v$, we can combine those equations to:

$$m\cdot v = \frac{0.25}{0.9}m\cdot v^\prime$$ $$v^\prime = \frac{0.9}{0.25}v = 3.6v $$

So this says our railgun trick increases our velocity to 3.6x that of the velocity of the rock before we fire them.

Now let's take a moment and realize just how bad news this is. In the span of a nuke/railgun event, we need to provide an acceleration to reach 3.6x faster than we got to by accelerating from 50AU out from the sun. This says the vast majority of the speed we achieved was from the railgun/nuke firing. In fact we'd probably have done just fine by using those railguns without the solar plunge. Needless to say, the accelerations here are going to be beyond mind blowing. Maybe should I say less "mind boggling" and more "splattering bloody bits of brain on the bulkhead behind you." Even genetically engineered humans in liquid suspension aren't going to enjoy this. In fact, I'm not certain if there is a material in existence that will survive!

That being said, the slingshot around the sun is a tricky one. I had to do some research on gravity assist for this question, but it looks like gravity assists around the sun are not all that effective. Here's the deal, as best as I understand it when explained by others. Gravity assist starts to make sense once you think about a particular coordinate system you're interested in. For example, for most interplanetary jumps, it makes sense to have the coordinate system fixed on the center of gravity of the solar system, which is rather close to the center of mass of the sun. If you have an object moving in this frame, such as Neptune, you can plan your orbit to rob some of its momentum. In any coordinate system, the trajectory will look the same, but in the one we cared about at the time, that trajectory happens to look like we stole momentum.

In the case of the sun, it doesn't have very much velocity with respect to the center of mass of the solar system, so you can't really use it effectively for a gravity assist (except perhaps by taking a dip into the solar core...). Some mentioned that there might be some value in a a gravity assist in a galactic coordinate system, where the sun has a velocity of 220km/s, but if we do that, we find our asteroid is also moving at roughly that same velocity. An interstellar traveler might use our sun as a slingshot, but you can't use it on your home turf! Frustrating!

There is a related gravity assist called the Oberth maneuver, where you do a burn while close to a large mass. The effect appears to be a multiplication factor on the specific energy of the burn by $\sqrt{1+\frac{2V_{esc}}{\Delta V}}$ This effect becomes smaller as your attempted delta-v becomes larger. $V_{esc}$, the escape velocity, is 617.5km/s for the sun, which is rather fast, but piddly when put in "fraction of the speed of light" terms: $0.00205c$. The faster you get with respect to that, the less useful Oberth maneuvers are.

So we are left with the situation we had before: a giant railgun/nuke, an asteroid, and a few space engineers who had too much to drink the night before and thought it'd be a great prank to throw a spitball at Alpha Centauri!

Just for fun, let's say a nuke takes about a second to go off. They look roughly mushroom shaped 0.25s after firing, so a second seems reasonable for fun mathematics. Let's say you want to reach 0.01c, or 2,997,925 m/s. Trivially we can see that is an average acceleration of 2,997,925 m/s^2, or 305598 gees. That's twice the acceleration of a 9mm bullet in the barrel of a gun. It's three times the acceleration of an ultracentrifuge, a device designed to separate out samples by molecular mass.

It's 30 times more than the acceleration of a Mantis Shrimp punch.

Ow.