Celestial Mechanics and the Lunar Probe

our solar system is a little over 400 years old not the real solar system of course but the solar system as we know it it was handed To Us by Nikolai cernus and it made him very unpopular since he upset the ancient and very human idea that the Earth was the center of the universe in his solar system all of the planets swung in circles about the sun just as the moon swung around the earth and gradually over these 400 years we've accepted his solar system as our solar system we've also accepted reluctantly our Earth's smallness given a sun this size the Earth is this size and this far away 93 million miles not really too far as such things are measured but far enough to make us feel pretty unnecessary but we do have company the Moon is 1/4 our diameter and averages a scant 240,000 m away scant that is until we think about trying to get there the moon's orbit is slightly elliptical and the Earth the Earth is a bit closer to one side than the other a great convenience since it causes the moon each month to approach about 10,000 mil closer to us than its average distance while the moon is going around the earth the earth is moving in a nearly circular orbit around the sun one trip each year throughout this time of course the Earth tipped at an angle of 23 1/ 12° is spinning about itself producing the well-known 24-hour day so we have the Earth spinning on its inclined axis the moon revolving around the earth and both making an annual circuit around the Sun all of this takes place within several larger motions of which we can ignore all but one for now using the Earth's orbit as a straight line going through the center of the sun and the center of the earth we let this line act as a reference as though we're floating the Earth in a tank then let's swing the moon and its orbit around the Earth and make the plane of the moon's orbit a thin disc now while the Moon is continuing counterclockwise the disc is moving clockwise or backwards around the earth keeping the high side always the same distance above the water in reality to complete just one cycle of this continuous motion require slightly more than 18 and 1/2 years to further explain what happens during this backing up motion of the moon's orbit let's look at it from above we find that this is such a slow and stely motion that each time the package makes its yearly trip around the Sun the pl of the moon's orbit backs up only a little less than 20° this line formed where the moon's orbit and Earth's orbit meet is called the line of nodes these then are the Motions with which we're principally concerned the cautious backing of the line of nodes the spinning of the Earth on its axis the path of the Moon circling the earth and there annual swing about the sun now what does all this have to do with the lunar probe everything everything because all of these swings and gyration affect the distance between the Earth and the moon and this distance more than anything else is the Ultimate Enemy we have to overcome thus our Delight that the Earth is not at the center of the moon's orbit since this means means that for nearly half the lunar month the trip is shorter than it is during the other half the difference between the least distance and the greatest distance being almost 10% since our lunar probe launching site is in the northern hemisphere the half of the month in which the moon's orbit lies north of the equator is better for us than the other half again because the moon is closer during this time actually with our present systems there are about 4 days centered about the time the moon's track is farthest north each month which give us our best shot at the moon and naturally when the moon is at the limit of its Northerly passage for the month the center of this 4-day period gives us the best moment the backing of the line of nodes or rather the backing of the pl of the moon's orbit also affects the distance to be traveled now as the plane of the moon's orbit slowly backs around the earth once every 18 and 1/2 years and we look at the Moon the same time every month from the same point on the earth year after year we find that the moon Bobs up and down like a cork float as waves pass under it with 18 and 1/2 years between waves Midway in this this 18 and 1/2 year cycle the moon reaches the low point in the swing the trough in the wave it is then at its most southernly passage slightly over 9 years later as the plane of its orbit keeps backing the Moon is once again at the top of the wave the upper limit of its Northerly passage obviously then since our launching site is in the northern hemisphere at this latitude the nearer our latitude the backing of the moon's orbit allows the moon to pass the better since it puts the Moon closer to passing overhead with the distance to it less and less thus we also have a best year once in every 18 and 1 12 distance saved between 4 and 5,000 Mi last best year 1950 but even the best combination of bests lessens only a little the incredible difficulty of launching a successful lunar probe what follows is one of the simplest of many possible trajectories and sets of conditions each of the three stages charged with boosting the rocket to a speed of 25,000 mph and guiding it along a preset path drops away in turn until finally the fourth stage plunges silently towards the rendevu point with the moon a point a little more than two days and slightly less than a quar of A Million Miles Away the first day the fourth stage slows the second day our rocket is in the moon's path but 1,000 miles ahead of it the retro rocket fires triggered by a signal sent from Earth the probe slows from more than 5200 mph to slightly over 2900 in less than 10 seconds and the moon gathers it in and sweeps away through space with it the probe circling the moon a th mil above its surface at a speed of over 2900 mph makes a complete trip in about 4 hours and a half and so the lunar probe held firmly in its orbit by the moon's gravitational field sends information about its new parent back to its old 240,000 M away simple hardly but then nothing that's really worth doing ever is [Music]