# What is the theoretical maximum speed of a rocket-powered aircraft in the atmosphere?

I read that the scram jet has a theoretical maximum speed of Mach 24. What is the theoretical maximum speed of a rocket-powered aircraft designed to operate in the atmosphere?

• Why do you think such a value exists? Given a big enough engine, there may well be no upper limit at all. Feb 8, 2018 at 18:54
• @abelenky: Not entirely true. The speed of light is an absolute maximum, of course. I'd also guess that there's a maximum acceleration you can get from a rocket exhaust, so you'd be limited by how long that acceleration takes you to get out of the atmosphere. And of course you have practical considerations like how much fuel you can carry... Feb 8, 2018 at 19:19
• @jamesqf, the assumption is clearly that you accelerate in a way to stay in the atmosphere. And the thing with practical considerations is that they don't give you a theoretical limit, just a practical one. Feb 8, 2018 at 20:15
• @abelenky size of the engine alone doesn't buy you very much: a bigger engine will consume more fuel, which is heavy, requiring a bigger engine... it's the curse of the rocket equation. That still doesn't imply a theoretical hard limit though, just that it gets exponentially expensive to attain higher speeds. Feb 8, 2018 at 22:19
• This would depend on the altitude, (or should I say, the air density). At an arbitrarily high enough altitude, where the air is thin enough to be at whatever arbitrary density you choose, just about any speed is attainable. Or are you asking what is the maximum attainable indicated/calibrated airspeed? I'd guess that would be based on the dynamic pressure that would cause the best possible insulator (think space Shuttle Heat Shield tiles), to melt or break apart from heat stress. Feb 8, 2018 at 22:25

The upper limit is given by the L/D of the rocket-powered vehicle and its thrust.

In order to follow the curvature of the Earth so it will stay within the atmosphere, the vehicle must create enough downforce by flying inverted so it can bend its flight path enough to not escape to outer space.

A practical altitude for flying this way would be dozens of kilometers up in the higher atmosphere, so drag and heat loads stay manageable. For the L/D I use the approximation given by Dietrich Küchemann (replace with a more recent one if handy): $$\left(\frac{L}{D}\right)_{max}=\frac{4\cdot(Ma+3)}{Ma}$$ which would result in an L/D of 4 for Mach growing very large. Add to this the effect of gravity and your maximum downforce is the weight of the vehicle plus four times its rocket thrust. This force now has to balance the centrifugal force which results from flying at constant altitude: $$m\cdot g + 4\cdot T = \frac{m\cdot v^2}{R_{Earth}+h}$$ where $$R_{Earth}$$ is the earth radius of approximately 6367 km and $$h$$ the flight altitude above ground. Now solve for flight speed v: $$v=\sqrt{\left(g + 4\cdot\frac{T}{m}\right)\cdot\left(R_{Earth}+h\right)}$$ By flying in dense air close to the ground the lift force will be immense, but so will be the required thrust. Since flight altitude is small compared to the radius of the Earth, the equation can be simplified with a small loss in accuracy: $$v=\sqrt{\left(g + 4\cdot\frac{T}{m}\right)\cdot6.400.000 m}$$ A more precise result would now add the effect of Earth's rotation, so flying eastwards along the equator will reduce flight speed by 464 m/s and add the same amount to the speed calculated above when flying westwards.

• Using a modern Raptor engine with a 200 "marketing" TWR we get a bit over 220km/s. I'm not sure Kuechemann accounted for drag due to nuclear fusion :) Apr 25, 2021 at 8:44
• @Sanchises: Add a realistic amount of fuel and structure and the speed will be more realistic, too. However, it will always be a multiple of escape velocity so your unit of km/s is appropriate. But you are right, Küchemann's is only a crude approximation. Apr 25, 2021 at 9:39
• Could your proposed limit be exceeded simply by pointing the nose down to tilt the thrust vector downward, using thrust rather than aerodynamic lift to provide the required downforce? I suppose not, as long as the achieved L/D is better than 1:1-- Apr 25, 2021 at 12:15
• @quietflyer:Yes, but the efficiency of that is L/D lower than just flying in more dense air, so this only helps when L/D drops below 1, as you rightly point out. Apr 25, 2021 at 12:50

In theory the max speed within the atmosphere is the speed that makes the air resistance equal to the strenght of the material the rocket is build from.

• Is this common knowledge? Do you have a source to reference?
– bogl
Sep 23, 2018 at 10:12

Assuming we're talking about an aircraft, that must complete a trip between point A and B, without destroying itself or its environment. If we also assume that we have the capacity of creating somehow the propulsion necessary to achieve any speed, then the limits would be related to:

1. Escape Velocity, we don't want the aircraft to leave Earth, thus, there's at least one upper bound equal to 40280 km/h.

2. Aerodynamic heating, fuselage's material must not melt nor break from higher temperature and air pressure.

Lockheed SR-71 Blackbird, which has been labelled as one of the fastest aircrafts ever, was made of titanium alloy, titanium alloy has a melting point around 1700 °C, even if you're using tungsten you're limited to its melting point of 3414 °C.

1. At high speeds, a shock wave forms, which deflects the air from the stagnation point and insulates the flight body from the atmosphere. This can affect the lifting ability of a flight surface to counteract its drag and subsequent free fall. This depends on Reynolds number.

I'm not an expert so I do not have the means to determine a number, but a good first order estimation can be made taking this two factors into account

• Re insulates the flight body from the atmosphere -- then how do supersonic aircraft fly? Are you perhaps referring to plasma and not shock waves? Either way, a reference would be nice. Welcome to the site.
– user14897
Apr 24, 2021 at 15:44
• @ymb1 thanks for the welcome, best source I could found online is history.nasa.gov/SP-60/ch-5.html Shock waves are actually an issue for both supersonic and hypersonic flight Apr 24, 2021 at 19:20
• The fastest aircraft ever was the X-15, which was powered by a rocket engine, btw. Apr 25, 2021 at 9:35
• @PeterKämpf I'll edit my answer Apr 26, 2021 at 5:31