English is currently available.
Aerospace Engineering DISCUSSION
I am trying to understand launch vehicle sizing with the rocket equation, Δv = Isp × g0 × ln(m0 / mf). Low Earth orbit needs an orbital speed of about 7.8 km/s, and I have read that the launcher has to deliver more than that.
For an engine with a specific impulse of 350 s, what mass ratio does that imply, how much of the rocket must be propellant, and how exactly does staging help if the total Δv is the same?
The launcher has to supply the orbital speed plus the losses on the way up: gravity loss while it is thrusting against its own weight, aerodynamic drag, and steering losses. A typical total to low Earth orbit is somewhere around 9 to 9.5 km/s; take 9.4 km/s.
Exhaust velocity is Isp × g0 = 350 × 9.81 = 3433 m/s. The required mass ratio is m0 / mf = exp(9400 / 3433) = exp(2.74) = about 15.5. So the final mass is 6.5 percent of the lift-off mass, and 93.5 percent has to be propellant.
That 6.5 percent has to contain everything that is not propellant: tanks, engines, structure, avionics and the payload. Suppose the dry structure is 8 percent of the stage's own mass (structure plus propellant), which is already a light design. Then the structure alone exceeds the 6.5 percent allowed, and the payload comes out negative. With that engine and that structure, a single stage cannot reach orbit at all, and with better numbers the payload is still a very small fraction.
The exponential is the problem: every extra 1 km/s multiplies the mass ratio by exp(1000 / 3433), about 1.34.
Staging helps because empty tanks and engines are thrown away instead of being accelerated to orbital speed. Split the 9.4 km/s into two stages of 4.7 km/s each. Each stage needs a mass ratio of exp(4700 / 3433) = 3.93, so its final mass is 25.4 percent of its initial mass. With the same 8 percent structural fraction, each stage can carry a payload of (0.254 - 0.08) / 0.92 = 19 percent of its own lift-off mass.
The first stage's payload is the whole second stage, so the payload to orbit is 0.19 × 0.19, about 3.6 percent of lift-off mass, where the single stage had less than nothing.
The other lever is specific impulse. Hydrogen and oxygen engines reach roughly 450 s in vacuum; at that value the single-stage mass ratio for 9.4 km/s is exp(9400 / 4415) = 8.4, leaving about 12 percent for dry mass and payload. That looks better, but liquid hydrogen has a very low density, so the tanks are large and the structural fraction is harder to keep low.
Also note that Isp is lower at sea level than in vacuum for the same engine, because of back-pressure on the nozzle, so first-stage and upper-stage figures should not be mixed.
A detail about the equation itself: it assumes no external forces, so it gives the ideal velocity change from burning the propellant. Gravity and drag losses are handled by adding them to the Δv requirement, as done above, and they depend on the trajectory and the thrust-to-weight ratio. A vehicle that lifts off with a low thrust-to-weight ratio spends longer fighting gravity and needs more Δv.
Earth's rotation gives a small credit in the other direction: up to about 0.46 km/s for an eastward launch from the equator, less at higher latitude.
