How much fuel will my astroship need?

[DaveC426913]

I'm working on a story. It's retro-fantasy, Jules Vernian or Wellsian "First Men in the Moon" style, perhaps a space opera. (more Vernian, since he leans a little more toward sceince than Wells did) In a nutshell: space travel in 1855.


Some explorer/engineer stumbles upon an outcrop of handwavium, which, when properly stimulated, produces handwavium rays. The rays temporarily suppress the charge on protons and electrons, causing the target material to fly apart. Stick that in a chamber, drop any inert material into the hopper and boom, you've got a rocket engine.

What I want to check is thus:

As long as I set a sufficiently high specific impulse (I_sp), I can reduce the mass of the bulk inert matter to an arbitrarily small amount.
I don't know if it's this simple but: I imagine a ten-fold increase in SI might require a ten-fold decrease in fuel mass.

chemical propellants: I_sp in the mid 100's
fission rocket (theoretical): I_sp in the 1000's
fusion rocket (theoretical): I_sp in the 10s/100s of thousands
handwavium rocket: I_sp sufficient to enable reduction of inert fuel mass significantly


I now have a spaceship that isn't 99% fuel, and it can go where it wants without undo concern for fuel limits.

This allows my protagonists (who are astronomers and engineers but not orbital mechanics specialists) to naively blast off from Earth and head straight out of Earth-environs without knowing about orbital mechanics or escape velocities. They point and go.

Say the rocket can achieve a max acceleration of 1.5g (it'll need >1g thrust just to lift off from Earth).
Say their destination is the asteroid belt. That's just 3-5 days at full accel and decel.
Say their little ship is on the order of 100 tons.

How much fuel (ballpark) might such a trip need? I am only concerned with plausibility - the fuel/payload ratio (to zero decimal places). Can I have a ship that is roomy for a few people without a vast majority of it being fuel?


(Note: it will run out of fuel by the time it reaches its destination - perhaps because they didn't plan a better trajectory - so I don't want an arbitrarily high efficiency).

 

[anuttarasammyak]

Though it is not your case FYI the case of photon rockets
$$\frac{v}{c}=\frac{1-r^2}{1+r^2}$$
where v is final velocity achieved and
$$r=\frac{initial\ total\ mass\ of\ rocket\ - \ consumed\ as\ fuel\ mass}{initial\ total\ mass\ of\ rocket}=\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}$$
I think it gives a theortetical limit for your case.

[EDIT] After full accell of 1.5 g in four days speed is
$$v=1.5*9.8*60*60*24*4 \approx 5*10^6 \ m/s \approx 0.017 \ c$$
$$ r=0.98$$
2% consumption. Almost 8 % consumption of initial rocket mass for go-return.

 

[DaveC426913]

Nah. They throw mass into the hopper and it is exhausted as plasma.

What im trying to achieve is - not acceleration - but a better fuel/mass ratio than chemical rockets (so they can get to the asteroid belt even though theyre not aerospace engineers - which won't even exist for another century) in a small craft.


But not so efficent that they don't run out of fuel.

 

[anuttarasammyak]

Nah. They throw mass into the hopper and it is exhausted as plasma.

What im trying to achieve is - not acceleration - but a better fuel/mass ratio than chemical rockets (so they can get to the asteroid belt even though theyre not aerospace engineers - which won't even exist for another century) in a small craft.I

Ideally all exhasted plasma has momentum of same direction backward. Exhasuted plasma would have the least mass and the maximum momentum by a proposed reaction. Say
$$p >> mc$$
$$E=\sqrt{p^2 c^2 + m^2c^4} \approx pc$$
Superrelativistic. It seems close to photon rocket to me.

 

[Filip Larsen]

don't know if it's this simple but: I imagine a ten-fold increase in SI might require a ten-fold decrease in fuel mass.

Yes, as mentioned in your other thread, for small fuel mass ratio $f = m_f/m_0$ a first order approximation for the the rocket equation is $\Delta v = -v_e \log(1-f) \approx v_e f$, so after ejecting factor $f$ of the total mass as propellant you would get factor $f$ of the exhaust velocity. This is the same as saying that for small $f$ the accumulation of the rockets speed change is approximately that as can be determined from conservation of momentum after a single impulsive (instantaneous large speed change) maneuver.

 

[Filip Larsen]

Say the rocket can achieve a max acceleration of 1.5g (it'll need >1g thrust just to lift off from Earth).
Say their destination is the asteroid belt. That's just 3-5 days at full accel and decel.
Say their little ship is on the order of 100 tons.

How much fuel (ballpark) might such a trip need? I am only concerned with plausibility - the fuel/payload ratio (to zero decimal places). Can I have a ship that is roomy for a few people without a vast majority of it being fuel?

If the rocket spends the fuel mass ratio $f = m_f/m_0$ on thrusting with local acceleration $a$ for time $t$ then we have $\Delta v = a t = -v_e \log(1-f)$, which means $$m_f = (1 - e^{-\frac{at}{v_e}}) m_0 \approx \frac{at}{v_e} m_0$$ with the approximation valid for small $f$. Inserting your numbers (a = 1.5G, t = 4 days) and using $v_e = c$ to get the theoretical maximum efficient rocket engine possible (thus giving a lower limit on the spend fuel mass) both expressions give $m_f \approx 1.7\%\, m_0$.

For a less than maximum efficient rocket, if we let $v_e = \lambda c$, then for a fuel mass ratio of 0.5 (i.e. half the initial mass is fuel), we get that $\lambda = \frac{\Delta v}{c \log 2}$ which with your numbers give $\lambda \approx 2.5\%$ meaning you need an exhaust speed at $2.5\% c$ or better to use half or less of your rocket mas for fuel for the trip.

 

[sbrothy]

I hate to repeat myself but this site is of immense help for questions of this matter:

Atomic Rockets

 

[Klystron]

Consider using fresh water as fuel. The crew need water anyway. Verne might have them swim in the giant water tanks. H.G. would have them paddle about in a small boat or sub. This could also lead to the asteroid belt disaster emergency when an eccentric crew member refuses to wear a hair net on their 19th Century long locks while swimming and accidentally fouls the fuel injectors. Or the water/fuel slowly freezes rupturing the tank. Cue science lecture on water states.

In your scenario where they run low on fuel they discover a method to capture an ice asteroid and extract pure water using the core drive. So, basically a steam rocket, HMS (or USS) Teakettle.

 

[DaveC426913]

... you need an exhaust speed at $2.5\% c$ or better to use half or less of your rocket mas for fuel for the trip.

Wow.

 

[DaveC426913]

Consider using fresh water as fuel.

Yes. Probly.

The crew need water anyway. Verne might have them swim in the giant water tanks. H.G. would have them paddle about in a small boat or sub. This could also lead to the asteroid belt disaster emergency when an eccentric crew member refuses to wear a hair net on their 19th Century long locks while swimming and accidentally fouls the fuel injectors. Or the water/fuel slowly freezes rupturing the tank. Cue science lecture on water states.

In your scenario where they run low on fuel they discover a method to capture an ice asteroid and extract pure water using the core drive. So, basically a steam rocket, HMS (or USS) Teakettle.

The conversion works on any matter. Anything that will fit in the hopper/tank. Including, in a pinch, parts of the spaceship itself. Good thing they have vacuum suits...

But that's not incompatible with your good suggestions.