I'm doing a few calculations to understand how to go from the Earth to the Moon. I used an elliptical (Hohmann Transfer) orbit to go from LEO to a circular Moon's orbit performing two $\Delta v$ (resulting in $\Delta v$ ~ 3.94 km/s). I don't know exactly how to perform the landing process and calculate the required $\Delta v$. I few papers in the literature show that it is between 1.5 km/s - 2 km/s. I think I should use an hyperbolic orbit but I'm not sure how. Any clues for a simple calculation? also, does it depend on the altitude above the Moon's surface?

Thank you

  • $\begingroup$ i.imgur.com/AAGJvD1.png $\endgroup$ – Organic Marble Apr 2 '20 at 17:43
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    $\begingroup$ @OrganicMarble That's probably the theoretical value for an ascent to 100km; descent needs a little additional safety margin. $\endgroup$ – Russell Borogove Apr 2 '20 at 17:46
  • $\begingroup$ @RussellBorogove You need a safety margin going up, also. So long as your measurement of your location and your landing point is accurate you shouldn't need any more margin going down. (Now, if your measurements are wrong....I've killed too many Kerbals with MechJeb's land anywhere mode.) $\endgroup$ – Loren Pechtel Apr 3 '20 at 4:00
  • $\begingroup$ @LorenPechtel It’s not symmetrical in practice. For ascent, no rocks to hit in LLO, and you know your surface point is safe. If your endpoint is off by a km or two, you can fix it up at leisure with RCS. $\endgroup$ – Russell Borogove Apr 3 '20 at 4:06
  • $\begingroup$ @RussellBorogove Yeah, if there's the slightest question as to your landing you need reserve fuel. Just look at Apollo 11. I'm talking about a situation where it's routine, you know exactly where you're going. $\endgroup$ – Loren Pechtel Apr 3 '20 at 4:53

The absolute minimum delta-v for getting from a circular orbit at 110km altitude to the the surface of a perfectly spherical moon would be around 1736 m/s -- a 25 m/s periapsis-lowering-to-surface burn, coast, and instantaneous 1711 m/s terminal burn -- this is effectively a Hohmann to zero altitude combined with an additional burn to go from orbital velocity at zero altitude to surface velocity, which is obviously impractical and unsafe.

The Apollo landers budgeted around 2125 m/s for their landing, or about 22% more than the theoretical minimum.

If you are using a more modern, computer controlled, automatic landing at a prepared pad with a radio beacon on it, you could probably cut that quite a bit. 2050 m/s is probably a good rule of thumb figure.

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    $\begingroup$ Thanks Russell, how did you compute those numbers? $\endgroup$ – Jorafb Apr 2 '20 at 18:03
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    $\begingroup$ Hohmann transfer from 110km to 0, plus orbital speed of Hohmann at zero that has to be cancelled out, plus surface rotation speed of the moon -- it assumes that you're coming in retrograde as the Apollos did; save 10m/s if you're orbiting prograde. $\endgroup$ – Russell Borogove Apr 2 '20 at 18:15
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    $\begingroup$ @RussellBorogove While at JPL I came up with a very similar maneuver (not practical, but a good initial reference estimate for ∆V required) for landing on moons of Jupiter or Saturn. I had the final burn occur at a very low altitude (1-10 m) and then just drop to the surface. I called it a "Stop'n'Drop"! $\endgroup$ – Tom Spilker Apr 2 '20 at 19:31
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    $\begingroup$ Thanks. I think that my calculations just considered the transfer from LEO to the Moon's orbit around the Earth, but it does not include the transfer into a moon's lower orbit. $\endgroup$ – Jorafb Apr 2 '20 at 20:07
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    $\begingroup$ @Jorafb 3940 m/s is about right for LEO 250km to LLO 100km according to the chart Organic Marble commented with above. $\endgroup$ – Russell Borogove Apr 2 '20 at 20:34

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