225 May 2020
Shape of Big Rockets
Thomas Hebbeker
RWTH Aachen University, Physics Institute III A
Abstract
Recent publications discuss the size of chemical rockets for long-distance travel, tobe launched from a planet. Here I point out that such a rocket cannot be tall and slimbut will be short and fat. a r X i v : . [ phy s i c s . pop - ph ] M a y Motivation
Recently, M. Hippke calculated the size a chemical rocket must have to leave the gravitationalfield of a planet [1, 2]. He found that the rocket mass, given to a large extent by the fuel, becomesenormous for a massive (exo)planet. For simplicity only single stage rockets are considered.The author illustrates in Fig. 1 of reference [1] how big such a rocket would be in comparisonto the Saturn V and other rockets that have been used on earth in the past, by scaling up theirlinear dimensions, without changing the shape.However, such a rocket has not only to be able to exceed the escape velocity of the planet andof the stellar system it lives in, it must also be able to overcome locally the gravitational field atthe surface of the planet. This requires a huge boost during launch, which implies a wide rocketengine and thus a large diameter of the rocket.
We assume a planet with mass M P and radius R p . The single stage chemical rocket’s mass is m R which is equal to the total fuel mass m F at the start, thus we neglect the mass of the emptyrocket and the payload. We denote by v F the exhaust velocity of the burnt fuel relative to therocket, which we assume to be constant. The mass of the expelled burnt fuel per time, the massflow rate ˙ m F = ˙ m R should not change either, till the fuel is used up. Atmospheric friction andother disturbances are not taken into account.In the launch position the rocket generates a lift off force F R = v F · ˙ m R . (1)This force F R must exceed the gravitational force F G between planet and rocket to make a liftoff possible: F R > F G (2) F G is the rocket’s weight at the surface of the planet, F G = g P · m R , (3)where g P = G N · M P R p (4)is the surface gravitational acceleration and G N denotes the gravitational constant. The condi-tion (2) translates into ˙ m R > G N M P v F R p · m R (5)In fact, F R (cid:29) F G is desirable, else lots of fuel is burnt just to balance the gravitational pullduring launch, but in the following estimate we use the minimum requirement (2).Let’s assume the rocket is of cylindrical shape with cross sectional area A and height H , so thatthe volume is V = A · H = m R /ρ R with the density of the unburnt fuel ρ R = ρ F . Ideally theburnt fuel can be expelled over the full area A , so the engine nozzle which produces the exhaustjet covers the bottom of the rocket completely: ˙ m R = v F · A · ρ E (6) here ρ E (cid:28) ρ F is the mass density of the exhaust gas. Putting equations (5) and (6) togetheryields A > G N M P v F R p ρ E · m R = g P v F ρ E · m R (7)Thus the minimal cross section A min grows proportional to the rocket mass m R , if all othermodel parameters are kept fixed. This implies that the corresponding height H = m R / ( A ρ F ) is constant, and the ratio H/A decreases, the rocket becomes fat.
Finally let’s look at a simple numerical example, inspired by the first stage of the Saturn Vrocket as used for the first manned moon flights: g P = 10 m / s = g Earth v F = 3 · m / s ρ F = 10 kg / m ρ E = 0 . / m A = π · (5 m) = 80 m H = 40 m m R = 2 · kg This gives for the smallest area A min fulfilling (7) A min = g Earth v F ρ E · m R = 20 m (8)Not surprisingly, this is well below A = 80 m . The main reason for A min /A < is that theSaturn V is designed such that F R is a factor of about three bigger than F G , in addition the totalnozzle area is smaller than the geometrical cross section A .For a given ratio of A min /A a big rocket with m R = 4 · kg , as discussed in [1, 2], wouldhave a cross section 200 times larger than the Saturn V, translating into a radius of
70 m insteadof . The height would remain unchanged, thus the diameter would exceed the height: weget a big fat rocket.
References [1] M. Hippke, Super-earths in need for extremely big rockets, arXiv:1803.11384v2[physics.pop-ph].[2] M. Hippke, Spaceflight from super-earths is difficult, arXiv:1804.04727v2 [physics.pop-ph].[1] M. Hippke, Super-earths in need for extremely big rockets, arXiv:1803.11384v2[physics.pop-ph].[2] M. Hippke, Spaceflight from super-earths is difficult, arXiv:1804.04727v2 [physics.pop-ph].