Then it occurred to me that departure and destination points on the Lunar surface could be corners of a Lambert space triangle. Of course, the third point would be the moon's center.
So basically it's like trying a to throw a baseball fast enough and at the right trajectory to reach the other side of the Moon. To put it in layman's terms. I suppose a mass driver could be used to effect a propellantless journey from on part of the moon to another in this way.
The second focus of the ellipse would lie in the middle of the chord connecting departure and destination points. Major axis of the ellipse would be r(1+sin(θ/2)). e would be cos(θ/2) / (1+sin(θ/2)). Attached is a pic that might make this clear.Also attached is a spreadsheet giving velocity and angle of suborbital launch from surface. In this spreadsheet I put the angular separation between the pole and an equatorial point - 90 degrees. This suborbital launch would be 1.53 km/s at an angle of 22.5 degrees. Typing into the pink cells user can change angle between departure and destination. The user can also type in radius and mass for other bodies.
Also, I would like to better understand the physics that supports the conclusion that the second focus is the center of the chord connecting the launch and landing sites!
Why is this in Avanced Concepts?
I don't want to offend anyone, but honestly any good student in a first year university physics class should be able to solve this problem.
In the image I found it useful to shade in the Moon. Regarding what this is saying, it might be a way to find the angle and velocity at which to launch from point A so that you land at point B, using minimum energy. But I think it minimizes the launch energy, with the expectation that doing so also minimizes the sum of launch and landing energies, and that both launch and landing are instantaneous impulses.
Allow me to propose an alternative closed-form version of this. This follows directly from the calculations in your spreadsheet. In fact, it is just a reduced form of all the formulas there.optimal launch angle = (Pi - theta) / 4 = (180 - 90) / 4 = 22.5
Quote from: ChrisWilson68 on 04/29/2014 05:53 AMWhy is this in Avanced Concepts?It's about transportation between points A and B on a roughly spherical body like the Moon. It could also be applied to other airless, spherical bodies like Ceres, Mercury, Enceladus, etc.Taking off from one points A to B on the lunar surface assumes a railgun, sling, or perhaps ISRU propellant. Else, why not just go directly from earth to point B?I'd like to see suborbital hops on the moon become routine. But right now it's Advanced Concepts, in my opinion.
And yes, freshmen aerospace gives the vis viva equation.Simple, yes. Obvious? Well, then you are being insulting as it took a few years for this to occur to me.
Quote from: sdsds on 04/29/2014 06:45 AMAlso, I would like to better understand the physics that supports the conclusion that the second focus is the center of the chord connecting the launch and landing sites!Recall that the energy of an elliptical orbit of semi-major axis a is -0.5GM/a. Therefore, minimizing the energy of the transfer orbit is equivalent to minimizing a. Let be T the angle between the major axis and a point on the ellipse as seen from the moon's center, where the other focus is at T=0 (in other words, T is theta/2 in sdsds's diagram, above). Then the radius of a point on the ellipse is r = (1 - e2) a / (1 - e cos T), where e is the eccentricity (usually the denominator is 1 + e cos T, but the other focus is taken to be at T=180o).Now set r to the moon's radius. In fact, it's convenient to measure distances in units of the moon's radius, so r=1. Solve the radius equation for a as a function of e and minimize. The distance from one focus to another is 2ea, which, at minimum a, turns out to be cosT. QED.
Thinking of various possibilities, the shortest possible colored leg seems to be going to the focus on the center of the chord connecting the two points.