ISV VentureStar (Avatar): acceleration problem

[TRodrigues]

Greetings,

I've been perusing the great site http://www.projectrho.com/rocket/index.php" , a fictional spaceship used in the movie Avatar.

In there, the ship is described to be accelerated by a bank of lasers and a light-sail at a constant 1.5 g acceleration for 0.46 year until they reach a cruising speed of 70% of the speed of light. The specs of the VentureStar don't include a mass, so I wanted to run those numbers a bit to see if I could get some sort of constraint on it.

However, I never even got that far. The http://hermes.physics.adelaide.edu.au/~dkoks/Faq/Relativity/SR/rocket.html" tells me that a rocket undergoing a constant acceleration a will, after (coordinate) time t reach a velocity (measured from Earth) of v = a t / sqrt(1 + (a t / c)²). However, plugging in the values for acceleration and coordinate time into that equation (1.55 ly/y² and 0.46 y, respectively) yields a velocity of only 0.58 c, not 0.7 c as it was given in the source. According to my calculations, the acceleration would need to be a bit over 2.09 g (2.15 ly/y²) to reach the projected speed in the allotted time. So I ask: did I make a mistake in my calculation, does the above formula not apply for this case, or are the numbers given really wrong?

 

[JesseM]

Apparently they either didn't realize the difference between proper acceleration and coordinate acceleration, or they specifically meant the ship had a coordinate acceleration of 1.5g. For coordinate acceleration A, the velocity after coordinate time t is just v = At, so since 1g=1.03227 lyr/yr^2, that gives v=(1.5*1.03227)*0.46 = 0.71c (close, although if they wanted 70% light speed it should have been 0.452 years).

Given that the ship is being pushed by a laser at rest in the Earth frame, which should (ignoring the spreading of the beam due to decollimation) deliver a constant amount of energy per unit time to the ship in the Earth frame, perhaps it's not implausible that the acceleration would be constant in the Earth frame rather than the ship having constant proper acceleration.

edit: well, even a perfectly collimated beam wouldn't deliver a constant amount of energy per unit time since as the ship's velocity increased less photons would hit it per unit time...

 

[TRodrigues]

I see, so the numbers check out (to a significant figure) with coordinate acceleration of 1.5 g. Ok, that's probably what was meant, then. As for the laser, I definitely noticed this: As the ship accelerates, the lasers redshift from the ship's point of view and thus the energy hitting the sails falls by a factor roughly proportional to the gamma parameter, even ignoring decollimation.

So, to calculate the actual acceleration I would have to take the laser's power, scale it by the redshift factor, and apply the proper acceleration formula (well, one for non-constant acceleration, at any rate), correct? I don't know how that would go in Earth's frame of reference, though. I guess integrating the power to get how the kinetic energy evolves and then calculate how the speed evolves as the mass increases relativistically, too?

I might be able to estimate the rest mass of the ship, after all...