Copyright 2012 Robert Clark
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:
SITE ACCESSIBILITY AND CHARACTERISTIC VELOCITY
REQUIREMENTS FOR DIRECT-DESCENT LUNAR LANDINGS.
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.