# Macroscopic shape of rocket exhaust

• Alonso Quixano

#### Alonso Quixano

Greetings to everyone. I would like to ask how the shape of a rocket exhaust plume changes with distance, when the rocket operates in a vacuum. What I'm mainly looking for, is to see how large the diametre of the plume would be at a distance of ~20km from the nozzle. We're assuming an ordinary LH2/LOX fuel mix.

I took a look at Rocket Propulsion Elements, but I didn't manage to find the answer. It mainly concerns itself with how the plume changes depending on the surrounding atmosphere, and when it does mention a shape, it does so in a scale of a few metres. So, either it does not mention what I'm after, or I missed it completely.
(What I did find is this: In a small corner of a figure (p. 646, figure 18-4) there is a note that says "Vacuum limit 0.1-10 m dia.". I'm not sure if that answers my question, however, mainly because I'm worried that the diametre will increase with distance.)

(Edit: I'm posting this in "General Physics" rather than "Homework Questions", mainly because I don't know the equations that govern what I'm asking. Please inform me if that was the wrong forum.)

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You can get a rough estimate if you know the temperature of the exhaust gas once it left the rocket. With an exhaust velocity of ~4km/s, the gas had ~5 seconds time to expand. Typical velocities of the gas follow from the gas mixture and its temperature, that allows to calculate a typical length scale. There is no fixed plume width, of course, it just gets thinner the more far away you are, with this typical length scale calculated before.

Oooooooh! Neat!

Still, I will need one more nudge. I tried searching for the rate of expansion of a gas during a free expansion, but I didn't find any relevant equations. A little more help? (Maybe it has something to do with the mean speed of the molecules in it?)

Typical velocities = thermal velocities. The velocity a molecule has with the energy kT where k is the Boltzmann constant and T is the temperature. T will go down as the gas expands, but if you are just interested in a rough estimate it should be fine to take the temperature of the exhaust after leaving the rocket.