Co-Orbital Rendezvous

[Enchanter468]

I'm sorry if this is not the place to discuss this, but I wasn't really sure where else to go (I was doing some Google searching on orbital mechanics and this forum popped up).

I've written a certain amount of fanfiction (yeah, yeah, I know), most of which takes place in the Gundam franchise, and as such involves space combat. As I'm a science major, I like to try and apply as much real science to the stories as I can, but as I'm a biology major, I'm not so great when it comes to certain aspects of physics. I've been able to fill in a lot of the gaps courtesy of Nyrath's Atomic Rockets website and other online sources, but I've hit one or two questions that I just can't find an answer for, which brings me here.

In one of my stories, a hostile military fleet departs their base on the Moon and heads for the space colonies at Lagrange point 5. I want to know how long it would take them to get there.

Given that the Lagrange points are on the Moon's orbital path around the Earth, I'd thought the fleet could simply accelerate and then fly along the Mon's orbit until they reached L5, but it turns out that if they did that it would boost them into a higher orbit. I looked up how a co-orbital rendezvous works and learned about how it's done by NASA (dropping into a phasing orbit and exiting it when your target loops back around and catches up to you). The problem is that using that method would take over three weeks (about 23 days).

It occurred to me, however, that the phasing orbit method may just be the rendezvous method that uses the least DeltaV, and that there might be other ways that are faster but have a higher DeltaV cost (sort of like how a brachistochrone orbit is faster than a Hohmann transfer orbit, but uses much more DeltaV).

Does anyone know of this hypothetical faster method?

Again, I'm sorry if I've brought this to the wrong place; I know this is more for people who play the KSP game. I just figured it couldn't hurt to ask.

[Rakaydos]

As you speed up, cut a chord inside the Moon's orbit. If you have the delta V, you can "shortcut" the orbit and use the natural "fling outward" effect to bring you back out to the lagrange point. (where you have to slow down again)

[Scotius]

Even "short cut" will take significant amount of time. And since there is no stealth in space, defenders would have plenty of time to prepare for attack. Generally, i think war in space is very similiar to a chess match between two expert players. They can see the board and all pieces, they know exactly what are the rules and what can be done with them. If anyone would win, i presume it would be more intelligent of the two - able to plan farther ahead and formulate a cunning strategy.

[Nibb31]

A phasing orbit can be above or below the rendez-vous orbit. You phase lower to catch up with your target (you go faster) and you phase higher to get caught up by it. So to reach EML5 from the Moon, you would just boost higher and wait for the EML5 to catch up with you, and then do a classic orbital transfer back down to lunar orbit for the RV. You could probably optimize it with a single parabolic burn that takes you outward and brings you back down where the EML5 station will be.

Generally speaking, EML5 is a poor place to put a colony space station. It's both hard to reach from Earth and hard to reach from the Moon. You would be better off locating your colony at EML1 or 2.

[Whirligig Girl]

If you have enough Delta-V, you could just use a brachistrone transfer. Burn towards your target, then turn around to slow down half way through.

[Enchanter468]

Even "short cut" will take significant amount of time. And since there is no stealth in space, defenders would have plenty of time to prepare for attack.

Very true, and the exact reason I asked about the shortcut. I figure the more time the defender has to prepare for the attack, the worse it's going to be for the attacker.

Generally speaking, EML5 is a poor place to put a colony space station. It's both hard to reach from Earth and hard to reach from the Moon. You would be better off locating your colony at EML1 or 2.

That part actually wasn't my decision; the colonies are at L5 in the show.

[Armchair Rocket Scientist]

If you have enough Delta-V, you could just use a brachistrone transfer. Burn towards your target, then turn around to slow down half way through.

This. For example, approximating the trajectory as a 400,000 km long straight line trip, with an acceleration of 1 G, the trip would take about 3.5 hours... but require 125 km/s of dV.