The spacecraft's acceleration depends on the total thrust and the spacecraft's mass. For most purposes, we don't care about this. The spacecraft can be theoretically any size. The equation is
A = F / Mc
- A = spacecraft's acceleration (m/s), divide by 9.81 for Gs
- F = spacecraft's thrust (newtons)
- Mc = spacecraft's current mass (kg)
As a short cut, you can calculate acceleration using the Transit Time Nomogram
We don't care about acceleration, that is, with the major exception of landing and take-off. If the Polaris is taking off from Terra, and it does not produce acceleration greater than 1 g, it is not going to move even a millimeter higher. For these calculations, for Mc use the spacecraft's mass with full propellant tanks. As a general rule, you want the spacecraft capable of doing 1.5 g, though 1.3 g will do in theory, and 10.0 g will really reduce the gravitational drag. 1.5 g = 14.72 m/s. The value you pick will be what you will use to calculate Apg in the gravitational drag formula.
Oh, and another thing: keep the acceleration below 30g to avoid injuring the astronauts.
On the Transit Time Nomogram, the minimum liftoff values are labeled on the Acceleration scale for your convenience. In the example above, a 46 metric ton spacecraft with a particle-bed nuclear thermal propulsion system can accelerate at 0.5 g. Glancing at the chart, you can see that the spacecraft has no trouble lifting off from Mercury, Mars, and the various moons; but cannot lift off from Venus or Earth.
This means that the engine's so-called "thrust to weight ratio" has to be higher than 1.0 if the rocket is expected to take off from Terra. (You can get away with less on smaller planets. Maybe.) Sometimes you are lucky and can find this value while researching propulsion systems. Lucky you, I included this data in the engine table above. Bottom line: do not use any engine marked "no" in the T/W>1.0 column if the spacecraft has to be capable of takeoff or landing.
At this website, they suggest that the optimum thrust to weight ratio varies from 1.15 to 1.2.
By rearranging the equation for acceleration, given the ship's thrust we can calculate the maximum mass of the ship with full propellant tanks (the mass of the ship with full tanks is often called Gross Lift Off Weight or GLOW).
GLOW(kg) = Thrust(newtons) / accel(m/s)
GLOW(kg) = Thrust(newtons) / 14.72(m/s)
A single Gas Core engine has a thrust of 3,500,000 newtons. If Polaris has one GC engine, its maximum liftoff mass is 237.8 tons, which is pretty disappointing. ( 3,500,000 / 14.72 = 237,771 kg )
But if it had five GC engines, it would have a liftoff mass of 1188.9 tons. That's more like it.
This also can be calculated with the Transit Time Nomogram
The other major exception is that a ship's acceleration affect maneuverability. This is important if somebody is shooting at you. It is hard to jink when your acceleration is measured in humming-bird powers.
Note that adding more engines only increases the acceleration and thrust (and the rate of propellant consumption). It does nothing to the deltaV or exhaust velocity. It also cuts into the payload mass. Also note that if an engine has a thrust to weight ratio below one, it doesn't matter how many of them you add, it still won't be able to lift-off.
Multiple engines produce other problems that have to be taken into account. If they are too close together, they inflict their waste heat on each other, increasing the heat radiator requirements. If they are too far apart and are of a type that emits nuclear radiation, they increase the number of shadow shields required, which cuts into the payload.
Remember that the ship has to be balanced around the axis of thrust or it will tumble. Cargo will have to be stowed in a balanced manner, and logged in a mass distribution schedule (sometimes called a "Center-of-mass and moment-of-inertia chart).