Tsiolkovsky’s rocket equation

I was fiddling with a TL-8 fission rocket design for going to the moon and back as cheap as possibly when I noticed something strange; going for more advanced materials would lower the mass of the ship and thus increasing its acceleration but it had no effect on delta-V? The design was for an upcoming article on landings, takeoffs, aerobraking, docking and ramming. It turned out to be just a bug and together with this post you can download the updated design spreadsheets, designs etc at the usual location.

Then I realised that the rules doesn’t dwell much on low tech rocket design, delta-V, mass ratios and such. These are the bread and butter of ‘real’ rocket design, and at the core of this rocket science art is the Tsiolkovsky’s rocket equation. Cool name for a post covering up my spreadsheet blunder and here we are.

Tsiolkovsky, Russian rocket pioneer and visionary did all the theoretical work for rocketry way before anyone really thought of rockets in space. He calculated the velocity needed to go to orbit and that to achieve it one should do it in a multi-stage rocket fueled by liquid Hydrogen and Oxygen, this was in 1903. Even before that, in 1896, he derived his famous rocket equation.

A real rocket accelerates by pushing stuff out the back, the faster it pushes and the heavier the stuff it pushes the higher the acceleration. Now, the tricky part is that as the rocket expends reaction mass it gets lighter which also increases acceleration. A rockets acceleration is at its lowest when it starts and at its highest just before it runs out of reaction mass. All this makes it hard to calculate just how much total velocity change a given rocket will have, twice the fuel will not give you twice the velocity change but more etc. Mr Tsiolkovsky helps us here with this simple formula:

Tsiolovsky rocket equaton

  • dV is the total change in velocity (m/s)
  • Vexh is the exhaust velocity (m/s)
  • M0 is the fully fueled mass of the ship (kg)
  • M1 is the empty mass after all fuel is gone (kg)

ln is the natural logarithm (logaritmus naturale) but you already knew that, right.
A derivation of the rocket equation and more facts about the great Konstantin Tsiolkovsky is available at Wikipedia.

So, whenever you design a ship with a fission or fusion rocket you now know how it gets its endurance value. Pay attention to the mass of components and if you can afford it you should try increasing the Material quality as this will reduce mass and increase acceleration Gs and endurance.

Whenever your friends complain about you fiddling with Intercept just tell them that you’re doing rocket science!


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