Saturday, September 15, 2012

Delta-V for "direct descent" to the lunar surface?

Copyright 2012 Robert Clark

 I was trying to get a lower roundtrip delta-V for lunar missions by flying directly to the lunar surface rather than going first into lunar orbit then descending, the "direct descent" mode. Here's a list of delta-V's of the Earth/Moon system:

Delta-V budget.
Earth–Moon space.

If you add up the delta-V's from LEO to LLO, 4,040 m/s, then to the lunar surface, 1,870 m/s, then back to LEO, 2,740 m/s, you get 8,650 m/s, with aerobraking on the return.
I wanted to reduce the 4,040 m/s + 1,870 m/s = 5,910 m/s for the trip to the Moon. The idea was to do a trans lunar injection at 3,150 m/s towards the Moon then cancel out the speed the vehicle picks up by the Moons gravity. This would be the escape velocity for the Moon at 2,400 m/s. Then the total would be 5,550 m/s. This is a saving of 360 m/s. This brings the roundtrip delta-V down to 8,290 m/s.
I had a question though if the relative velocity of the Moon around the Earth might add to this amount. But the book The Rocket Company, a fictional account of the private development of a reusable launch vehicle written by actual rocket engineers, gives the same amount for the "direct descent" delta-V to the Moon 18,200 feet/sec, 5,550 m/s:

The Rocket Company.

Another approach would be to find the Hohmann transfer burn to take it from LEO to the distance of the Moon's orbit but don't add on the burn to circularize the orbit. Then add on the value of the Moon's escape velocity. I'm looking at that now.

Here's another clue. This NASA report from 1970 gives the delta-V for direct descent but it gives it dependent on the specific orbital energy, called the vis viva energy, of the craft when it begins the descent burn:


The problem is I couldn't connect the specific orbital energy it was citing to a delta-V you would apply at LEO to get to that point. Any suggestions on how to accomplish that are appreciated.

  Bob Clark


qraal said...

Not sure I get your difficulty Bob. Standard orbital equation will give you that.

Robert Clark said...

Do you have a reference?

Bob Clark

qraal said...

Hi Bob
Wikipedia's entry on the vis viva equation is good. Any astrodynamics text will have it. Any number of lecture notes online...

Bruce Shapiro gives a derivation here...

My favourite set of lecture notes on Astrodynamics are from the EMA 550 Course at U Wisconsin-Madison, though the set I have are from 2004. Not sure they're accessible anymore. MIT's Open Course ware Astrodynamics materials are good too.

The trick with delta-vee is that you need to factor in gravity-losses while under thrust. There are useful means of estimating such, but full computation requires numerical orbital analysis. I threw together a toy mathematical model of a Saturn V lift-off years ago which comes pretty close, but I think there's free-ware programs out there which can work it out for you.

Robert Clark said...

Found this after a web search:

Lunar Base Studies in the 1990s.
1993: Early Lunar Access (ELA).
by Marcus Lindroos
To save fuel, the LEV makes a direct landing rather than enter an intermediate lunar parking orbit as Apollo did. The vehicle retains sufficient propellant to perform a later ascent burn to return the crew to Earth. For unmanned cargo missions, the LEV carries a heavier payload and uses up all its fuel for landing.[/quote]

I still need to find out how much they were able to save with their trajectory.

Bob Clark

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