Formulas needed for science fiction book

[Firesmith]

I am writing a science fiction book with the following premise:

A starship drive (and weapon) has been discovered that acts as a laser that turns ordinary matter into gamma rays at 100% efficiency; matter goes in one end of a tube and the light comes out the other end, driving the spaceship the opposite direction to conserve momentum. The ships contain a single pilot who has the job of collecting matter (in the form of asteroids or comets) and setting the destination. The ship is attached to the "fuel" and pushes the fuel in front of it (to also act as a shield for radiation when the ship is moving at relativistic speeds through the interstellar vacuum). Even though the pilot is in "stasis" floating in a vat of cloroflorocarbon liquid to provide oxygen and minimize g-forces, the pilot can only withstand a given acceleration long term. I would like the travel times and asteroid/comet sizes to be scientifically realistic, even if the drive itself is not.

Assume that the ship has a mass of Ms (say roughly that of a navy destroyer).
Assume that the ship accelerates (and decelerates) at a constant rate As (some reasonable multiple of g).
Assume that the ship accelerates for the first half of the trip, rotates, and then decelerates (at the same rate) for the second half of the trip.
Assume that the ship takes time Tr (ship time) to rotate at the halfway point. This should be relatively short (e.g., a couple of days)
Assume that the trip distance (in light years) is Dt. I am assuming that the trip distances will probably be less than 50 light-years; less if the resulting asteroid/comet mass needed is unrealistic.
How massive should the asteroid/comet (Mac) be to be able to accelerate and decelerate the ship (and its ever decreasing mass) so that the ship can arrive at the destination with a safety mass of Macr remaining?
How long will it take (ship time) to reach the midpoint (Tsm) and destination (Tsd)?
How long will it take (start/destination time) to reach the midpoint (Tsdm) and destination (Tsdd)?

Also, the drive will be used as a weapon upon reaching the destination star.
Assuming that the remaining asteroid/comet mass is Macr and the ship decelerates (when using the drive as a weapon) at Dw, what is the energy rate being delivered? Also, assuming that the gamma laser drive has a reasonable diameter (e.g., 10 meters say), how big of a spot would that be on the target planet when fired within a stellar planetary system? The beam and power levels need to be great enough to do catastrophic damage from a reasonable distance (to avoid detection) at a reasonable deceleration towards the planet.

There may be a technical problem that needs to be dealt with. When decelerating towards the target star, the gamma rays are speeding ahead of the ship and will provide early warning of the ship's arrival (the ship will be decelerating for half the trip so the gamma rays should beat the ship there by some time). Yet, the ships needs to be considered an ultimate weapon that is almost impossible to stop until it is too late. Thus, the ship can't decelerate directly towards the target until it is close or else the same kind of drive/weapon can be aimed at it. This means that the approach path may need to be slightly off. What would be a reasonable dispersion rate for the gamma laser, and how might this impact the trip time and mass needs? Would the laser beam affect the stray atoms in interstellar space, the target star's stellar wind, or comets in the Ort cloud in a manner easily detectable by the target planet orbiting the target star?

Any help with providing the needed formulas would be greatly appreciated.
Don Firesmith

 

[Nickelodeon]

I can't help with the mathematics but I think the g forces would be way beyond being practical. Deceleration would take years despite the pilots immersion.

Just a couple of comments - I think the sacrificial lump in front sounds a bit cumbersome and not very high tech. Secondly, isolation from the interstellar vacuum is quite handy should you need to exceed light speed (hypothetically) but I'm pretty sure the acceleration /deceleration will prove to be the limiting factor.

 

[Naty1]

Just look up relativity in Wikipedia and copy a few formulas that look cool...maybe change a few super or subscripts...nobody will know what they are...be sure to include an infinity here and there, and integral and a summation sign or three...or use Roger Penrose THE ROAD TO REALITY for a lot of mathematical symbols you've likely never even seen...

There are no technical problems that need to be delt with...that why it's called science FICTION!
GOOD LUCK...

 

[JesseM]

Most of your questions can be answered on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html page. First it gives formulas for onboard time vs. Earth time for constant acceleration--if you want to deal with the case of accelerating half the trip and decelerating the second, I think you can just apply the formulas to the case of traveling half the distance to the destination, and then just double the times (you can double-check if these works by seeing if it gives the same answers as the ones in the second table, like 6.6 years of onboard time to reach Vega 27 ly away in Earth at an acceleration of 1G). In the "how much fuel is needed?" section it also deals with the ratio of fuel mass to payload mass in the case of a perfectly efficient "photon drive" which turns all the rest mass energy of the fuel into kinetic energy. For example, in the case of traveling to Vega 27 ly away, if you want to accelerate at 1G for the first half and decelerate at 1G the second, with the perfectly efficient engine the mass of the initial fuel must be 886 times greater than the mass of the payload that you want to actually reach Vega.

 

[Ich]

If he doesn't brake, that would make an ultimate, undetectable weapon. But nothing new in SF.