David.Smith@cmu-cs-ius.arpa (04/07/84)
Problem: How close to horizontal must you fire one of those high-powered rifle bullets for it to stay in orbit? Let's treat the moon as a perfectly smooth sphere of 2160 miles diameter, and assume that our marksman fires from a height of 6 feet. The horizon is 0.083 degrees below the "horizontal," and is located 1.57 miles away. If one fires below the horizon, the bullet will surely hit the moon, whatever its velocity. If one fires above that with infinite velocity, it will miss the moon. 0.083 degrees corresponds to an elevation error of 5.2 inches in 100 yards. But the fastest the bullet can be fired into a return orbit is just below parabolic velocity. To prevent a bullet with parabolic speed from dropping more than six feet, the maximum depression angle is 0.0588 degrees, which corresponds to an aiming error of 3.7 inches in 100 yards. At this angle, the bullet will impact 2.2 miles away. There had better not be a six-foot hill within 4.4 miles of the firing point. By symmetry, this maximum depression angle is also the maximum elevation angle. What if the moon is littered with three-foot hills? Then you had better not fire at an elevation or depression of more than 0.04 degrees, corresponding to an error of 2.6 inches in 100 yards. If the bullet has less than parabolic velocity, these margins dwindle, until they reach zero at circular velocity. (If the bullet goes around at a constant six feet, it will run into highlands sometime.) Rougher terrain also cuts into the margins. Then there is the moon's rotation, which over a month's time is bound to put something in the way of that six foot periapsis. As I recall, Apollo spacecraft circled the moon in about two hours. That puts circular velocity at about 3400 mph, and parabolic velocity at 4800 mph. Mighty fast rifle bullets. David Smith
raymond@zps.UUCP (Doug Raymond) (04/11/84)
Maybe the lunar infantrymen were JUMPING before pulling their triggers, thereby obtaining enough altitude at the point of injection...
karn@allegra.UUCP (Phil Karn) (04/11/84)
There are also numerous perturbations to a lunar satellite due to the moon's triaxial ellipticity. These perturbations are much larger in relation to the two-body forces than they are for earth satellites, and many have the effect of changing perilune altitudes. The Apollo flights had to be very careful of these effects when in lunar orbits with perilunes of 10 km or so. Another interesting item is that the Apollo 11 landing overshot its intended target by some number of kilometers (5-7 comes to mind) because the precise nature of the moon's gravitational field was still unknown, even with all the experience from the Lunar Orbiter series. A primary goal of Apollo 12 was to achieve demonstratable accuracy in landing (based on Apollo 11 tracking results) by landing next to a defunct Surveyor. All this makes a story about rifle bullets fired from the surface of the moon going into lunar orbit pretty far fetched. Phil