Physics Question - Ion Propulsion

[Arrow]

I need some help with this problem, mainly because I don't know the physics equations.

Lets say I want to accelerate a ship with ion engine with exhaust exiting at .95c (roughly 2,850,000 km/s; something that should easily achievable with a particle accelerator). Lets assume the fuel is Xenon (atomic number 54 and an atomic mass of roughly 131.3). The ship I want to accelerate has a mass of 102,300 metric tons (roughly 5 times the mass of an Iowa class battleship, with rounding errors from converting from Standard to Metric). I also want the rate of acceleration to be 960 m/s^2 (100 G) and I want to accelerate to a speed of .3c (900,000 km/s). The time for this process is about 10 minutes. The question is how much propellent (Xenon) mass do I need?

[Admiral Valdemar]

To get to ".3c" that is 900,000km/s would be quite a task...

[Arrow]

To get to ".3c" that is 900,000km/s would be quite a task...

But it is doable. A trip from the sun to pluto, by my math, would take a little over 18 hours. Also, the acceleration to that speed from a dead stop would take 15 minutes (oops, I said 10 in my first post). The acceleration would have to ramp up to 100 G, since you don't want to turn the crew of this theoritical space ship into piles of goo. And I believe modern particle accelerators can get particles up to speeds of .9c. The main problem with the feasability of getting a ship up to that speed would possibly be the amount of Xenon needed to move, which is why I asked the question.

[Sir Sirius]

.3c = 900,000 km/s
100 G*15 minutes = .3c

You'd better check your math...

[Enlightenment-alternate]

The ship I want to accelerate has a mass of 102,300 metric tons (roughly 5 times the mass of an Iowa class battleship, with rounding errors from converting from Standard to Metric). I also want the rate of acceleration to be 960 m/s^2 (100 G) and I want to accelerate to a speed of .3c (900,000 km/s). The time for this process is about 10 minutes. The question is how much propellent (Xenon) mass do I need?

Your acceleration numbers don't add up.

Given 960m/s^2 acceleration:
v = at
600s * 960m/s^2 = 576000m/s

To reach .3c in 10 minutes:
v/t = a
0.3c / 600s = 149896m/s^2 = 1529g

Running the numbers to reach .3c using an engine with an exhaust velocity of .95c and a non-fuel mass of 102,300t, I get a wet mass of 140,198t tons, or 37898t of fuel.

Getting a high enough mass flow at .95c to manage an acceleration of 1529g will be a bit of a challenge, however. Going with an antimatter drive would be a much better bet than using a particle accelerator.

[Enlightenment-alternate]

The acceleration would have to ramp up to 100 G, since you don't want to turn the crew of this theoritical space ship into piles of goo.

Ramping up won't help. Humans will black out under prolonged exposure to 8g; going much higher will be fatal.

[Admiral Valdemar]

.3c = 900,000 km/s
100 G*15 minutes = .3c

You'd better check your math...

c is 300,000km/s, 0.3 of that would be a lot less and 900,000km/s is FTL so good luck at getting that.

Plus, accelerate at any decent rate to get to a fraction of c and you're paste on the rear wall.

[Arrow]

Well so much for that idea. Looks like I dropped a decimal somewhere.

[ClaysGhost]

I need some help with this problem, mainly because I don't know the physics equations.

Lets say I want to accelerate a ship with ion engine with exhaust exiting at .95c (roughly 2,850,000 km/s; something that should easily achievable with a particle accelerator). Lets assume the fuel is Xenon (atomic number 54 and an atomic mass of roughly 131.3). The ship I want to accelerate has a mass of 102,300 metric tons (roughly 5 times the mass of an Iowa class battleship, with rounding errors from converting from Standard to Metric). I also want the rate of acceleration to be 960 m/s^2 (100 G) and I want to accelerate to a speed of .3c (900,000 km/s). The time for this process is about 10 minutes. The question is how much propellent (Xenon) mass do I need?

Note: 0.95c = 0.95 * 300,000 km/s = 285,000 km/s.
0.3c = 0.3 * 300,000 km/s = 90,000 km/s.

Relativistic rocket equation:

delta_v = c * tanh[(v_ex / c) * ln (m_i/m)]

m_i = m_f + m = initial ship's mass (reaction mass + structure)
m_f = reaction mass (desired quantity)
m = mass after thrusting (structure)
delta_v = change in velocity due to ejection of (m_i - m) kg of reaction mass
v_ex = exhaust velocity

m_f/m = (exp[(c / v_ex) * atanh[delta_v / c]] - 1)

m_f = (exp[(1 / 0.95) * atanh[0.3]] - 1) = 0.3852 * m = 39,400 tonnes of reaction mass.

Acceleration constraints:

a = v_ex * m_dot / (m(t) * gamma^3)
gamma = (1 - (v(t)/c)^2)^-0.5

a = acceleration
m_dot = rate of decrease of ship's mass m(t) - rate of ejection of reaction mass, rate at which reaction mass is used up, in other words
v(t) = velocity of ship at time t

m_dot = m(t) * a * gamma^3 / v_ex = 1.023e8 * 100 * 9.81 * (1 - 0.3^2)^-1.5 / 2.85e8
m_dot = 405.6 kg/sec

m_dot will not be constant since m(t) and gamma will change (assuming you desire fixed acceleration, a), so the following is estimated (using values appropriate to when the velocity of the craft is approaching your requirement, i.e. 0.3c):

Burn time ~= mass of "fuel" / rate at which "fuel" is used up, so
t = m_f / m_dot = 39.4e6 / 405.6 ~= 97,000 seconds, or roughly 1.1 days.

I am not sure, but your estimate of 10 minutes appears to be derived by mistaking the distance (s = 0.5 * a * t^2) equation for the velocity (v = u + a * t) equation of motion under constant acceleration. If you use v = u + at, you should find a burn time of just over a day, comparable with that extracted above.

If you are ejecting ~400kg of Xenon per second from your rocket at 0.95c, you will need to supply engine power P:

P = gamma(v_ex) * m_dot * c^2 = 3.203 * 405.6 * (3e8)^2 = 1.17e20 Watts, or 117 Exawatts if you're into prefixes. With that energy requirement, the mass ratio calculations are seriously in error, since you will need over 100,000 tonnes of fuel to generate power, and that's assuming an antimatter/matter reactor with perfect efficiency. As you can see, you actually would need three times more fuel mass (to power your drive) than reaction mass!

[Arrow]

Hehe! Yeah, I made a big mistake in my math (which explains why I'm a programmer...).

[Enlightenment-alternate]

In case anyone's wondering, the reason my fuel mass total differs from the value cauclated by ClaysGhost is that he took relativity into account any my mass ratio calculator applet uses Newtonian mechanics.