# Spacecraft propulsion, Thrust, Delta V

For a spacecraft on an interplanetary mission (e.g. Mars), after deciding the propellant ISP value and Delta V required, how do you calculate the thrust required?

$$F = \dot{m} V_e + (p_e - p_0) A_e$$

According to this equation, thrust depends upon the design of nozzle. So what is designed first? Is thrust determined first before designing the sub systems of rocket/spacecraft or the thrust is calculated after designing the nozzle, thrust chamber, etc.

If the nozzle is to be designed with respect to the value of thrust, then how is the thrust value determined first? Is there a relation between thrust and Delta V?

• I've converted your equation to MathJax, double check and edit further if you like. Can you also add to your post a short description of what each term in the equation stands for? Thanks!
– uhoh
Nov 7 '20 at 17:26

Part 1, finding thrust

The deciding factor for what thrust is required is what acceleration is required

$$F = ma$$

And acceleration requirements are in turn defined by "achieving delta-v in a timely manner". At the root of it, the available time is the deciding factor.

The available time varies a lot depending on the domain:

• Spiralling in slow interplanetary transfers, ion engines have months and years available.
• For impulse burns in orbit, manoeuvrers would need to be performed within a few minutes, otherwise the location has drifted a bit.
• At launch, every second counts as gravity is mercilessly pulling the rocket towards the ground (we want $$a > g_0$$)

These all give very different constraints to optimise for.

And that's the problem here, it's an optimisation problem. The entire business of launching rockets is a complex optimisation problem, with many many variables. And these variables can't nicely be worked out bottom-up. There is feedback.

Example of feedback:

• The thrust required depends on the spacecraft mass.
• The engine mass depends on the require thrust.
• The spacecraft mass depends on the engine mass.

Nevertheless, here are some formulas that may be useful, assuming you have a spacecraft in orbit, and want it to be able to make it to Mars.

The "inverse" rocket equation:

$$\frac{m_{full}}{m_{dry}} = e^{\Delta v/v_e}$$

$$m_{full}$$ is the total mass of the spacecraft before the burn, and $$m_{dry}$$ after the burn (both of these include the engine mass). $$v_e$$ is exhaust velocity as in your formula, calculated as $$v_e = I_{sp} \cdot g_0$$.

The difference in mass is the propellant $$m_{propellant} = m_{full} - m_{dry}$$

Given some timely limit ($$t$$) to the manoeuvrer, you want the engine to run through all that propellant within that time

$$\frac{m_{propellant}}{\dot{m}} < t$$

At that point, you have both $$\dot{m}$$ and $$v_e$$ for your thrust equation. The second term is usually very small, so ignore mostly ignore it.

Part 2, designing engines

Knowing what thrusts are required, you can start designing engines. Or contract out your requirements. Or use existing engines.

But the requirements are coming from the estimate in part 1). And that estimate needed some key engine parameters like Isp and engine mass.

So it really is a loop, where you use estimated engine parameters to calculate required engine parameters. Once you have an engine matching the original requirements, you have to plug those numbers back into the original estimate to get a better estimate, from where you can design an engine that matches better.