Math need for interstellar voyage

[RGClark]

Hi there
I'm working on a short story placed on an interstellar ship. But I need you guys to help me with the math and authentic technology involved.
The story is taking place in the 22th century. I'm talking of an interstellar colony ship carrying 10'000 colonists in cryo-sleep and a crew of 12. I'm imagine a taildragger ship with a length of about 2 km. It's mass could be around 10'000 tons (e.g. at least 500kg per colonist of lifesupport equiptment + 5000 tons of crew habitat, reactor, engines, fuel, structure, etc). Plausible?
The ship should accelerate continuosly until reaching max speed (between 0.6-0.8c), then turn around and deccelerate the other half of the journey. The entire traveling time should fit within of one to two decades. The destination is our closest star with a distance of 4.3 ly.
What kind of technology could be feasable by then to achive this criterias?
And how would the timing of such a trip look like? How long would it take to leave our solar system? Is it even possible to "spiral-in" directly on a planet in the destination system or use swing-bys at departure and arrival?
To be honest, inspiration came a bit from James Cameron's "ISV Venture Star" from Avatar. But in contrary to said ship, I imagine a ship that constatly thrust on lower levels until it reaches max speed, then turns around and constantly thrusts to brake down for arrival. I thought about ion-drives and a low-g acceleration with the help of a solar sail for acceleration and braking. But they are far to weak, to get a realistic setup for the framework I want to use. Any other ideas, e.g. VASIMR?
Thanks for you support!
Richy

Nice article describing the different possibilities:

How Interstellar Space Travel Works (Infographic).
by Karl Tate, SPACE.com Infographics Artist | July 01, 2013 08:12am ET
http://www.space.com/17619-how-interstellar-travel-works-infographic.html

In regards to what is technically possible now it would probably be the Project Orion method of pulsed atomic bomb explosions:

Project Orion (nuclear propulsion)
4. Interstellar missions.
http://en.wikipedia.org/wiki/Project_Orion_(nuclear_propulsion)#Interstellar_missions

Carl Sagan once observed it would be an excellent way to get rid of all the nuclear weapons we now have.


Bob Clark

 

[wingnut]

Ship A is happy with always having a velocity less than, say, 0.2c in the galaxy frame. It has an engine that chucks light out.

Ship B, like ship A, is content with traveling "slow". Its engine however chucks out mass at relativistic speeds.

Ship C ain't got time for that and flies "fast" (over 0.6c or whatever in the galaxy frame), and uses an engine that chucks out light.

Ship D is in a hurry too, but its engine chucks out matter.

It may be obvious but I've never seen this explained anywhere:

What would the trajectory of the ships A to D look like if they were visualized with videnie for example?

I suppose interstellar travel trajectories for slow ships would also look like a Hohmann transfer orbit while the fast ships would be on a one-tangent transfer orbit?

 

[Thorsten]

This is fascinating, but Thorsten, what equation would you use to compute deltaV based on exhaust velocity, mass ratio, and whatever other parameters needed?

Always start from differential momentum conservation

dp_ship = - dp_propellant

Dependent on what regime you're in, you insert an expression for the momentum of the propellant.

In the non-relativistic limit, that is

dp = dm v

In the relativistic limit, that is

dp = dm gamma v

and for light that is

dp = dE

Then you find the relevant limit - total reaction mass stored, or total energy available in the case of light, and integrate that equation, which gives you the total momentum gain of the ship. Inserting the ship's residual mass, you get the total velocity gain from that. In the non-relativistic case that gives you the rocket equation, if the ship gets relativistic it gives you the rocket equation in rapidity, and with light as propellant that gives you something like a generalized rocket equation based on energy reserves rather than mass reserves.

 

[RGClark]

Ah, but the propellant gets relativistic before the rocket does, so if you're using a near-c matter stream, then you get the relativistic mass corrections on the propellant, so a given propellant mass can by you an arbitrary amount of momentum, the relation is no longer
Delta p= Delta m v
but
Delta p = gamma Delta m_0 v
with
gamma = 1/sqrt(v^2/c^2)
so for a near-c matter stream gamma can be arbitrarily high and the momentum gain from a kg of matter may be hundred times that of the non-relativistic limit.
Provided you have the energy to accelerate your matter stream to near-c. Doesn't work with light as propellant, because that has no rest mass, so the relativistic momentum is different.
...

That's interesting. A big part of the loss of efficiency for the Project Orion propulsion method is that the explosion products are allowed to stream out in all directions. If we could constrain the products as with a nozzle so that they only steam out the back we would have a more efficient propulsive method.
There is a concept of a magnetic nozzle being investigated for plasma propulsion, such as VASIMR. The question is how much power would you need to run such nozzles to contain the power of a nuclear explosion? And how huge would be the strength of the magnetic fields needed? Would it take so much power to run that you might as well let the explosions proceed unconstrained?

Bob Clark

 

[Richy]

Thanks so far guys! That's a lot to work with! Glad to be part of th OF!

As it seems, I wont be able to avoid the use of "magic". But as we all know, magic is only technology, we don't yet understand.