# What calculations should I do to go from LEO to the moon with an ionic motor?

my question is, what calculations should I do to be able to go from LEO to the lunar orbit? If this were done, two of these thrusters would be used https://www.cubesatshop.com/product/ifm-nano-thruster/, I know that once being in the Earth's orbit it is not all thrust, given that in Some parts would be inefficient and we would only spend Xenon, gather a lot of information from the SMART-1 mission and they commented that only in a third of the orbit they activated the motors, then only orbited until returning and redoing that until reaching the lunar orbit.

I read these two previously published articles carefully: Could a cubesat be self propelled to the moon from LEO? and Could a cubesat propel itself to Mars? and the truth is that they have helped me a lot but there are still things that I don't understand and maybe someone can explain to me. Thanks in advance!

You have to calculate the deltaV that these thrusters can provide you.
You can do this by using the Rocket Equation.
You need the ISP value of those thrusters and the starting and final mass of your cubesat (i.e. mass when you are deployed in LEO and mass after you have expended all your Xenon). That will give you the maximum available deltaV.
This needs to be higher than the deltaV you need to reach your destination.

You can use this as a rough guide to many places in the Solar System:
The Solar System: a subway map
From LEO to low Moon orbit it is about 3940 m/s deltaV (3260+680 in the map).

Using a ionic drive, so with very low thrust, you may need long burn times and this may increase the deltaV required (since you cannot do some efficient manuevers such as Hohmann transfer that require a more "instantaneous" burn). However, 3940 m/s deltaV still gives you a first rough estimate. Indeed this question says SMART-1 demonstrated a total 3900 m/s deltaV.

A more precise calculation will have to account for the type of transfer/trajectory you will use from LEO to low Moon orbit. Unfortunately, I do not know the calculations involved.
You may have a look at this other question for calculating a ionic drive transfer by a spiral trajectory as a starting point.

As you may have noticed, deltaV does not depend on thrust. Since the engines are identical and have the same ISP, having two or more of them gives no benefit in terms of deltaV and it is actually detrimental since your final mass will be higher. So, use two or more engines only if you need them for something else (e.g. redundancy).

• One more question, if I put 4 engines instead of 1, the delta-v would be higher, and I would need less xenon in the end, right? Commented Feb 19, 2020 at 20:15
• @ValentinoZaffrani No, precisely the opposite, actually. You would have higher dry mass, and your delta-V (for a spacecraft with the same amount of xenon propellant) would be even lower. Plus you would need 4 times as much electric power, which means 4 times as much weight in solar panels. Commented Feb 20, 2020 at 7:49