SUN'S SELENOGRAPHIC COLONGITUDE ============================= John Pazmino NYSkies Astronomy Inc www.nyskies.org firstname.lastname@example.org 2007 December 11
Introduction ---------- In spring 2001 a correspondent, long forgotten, in the Internet newsgroup 'sci.astro.amateur' asked about the Moon's phases. He shot some astrophotos of a certain crater on a given evening. The images were blurred due to faulty mechanics of his apparatus. He wanted to shoot the same crater with the same lighting. But when could he next try again? Many correspondents did advise the chap to simply clock off one cycle of phases, more or less one month later. What at first seems like a simple question with a rather trivial answer, 'wait until one lunation later', actually is one of some complexity. But there's more. A lot more.
Charts and atlases ---------------- Before continuing farther with this discussion and specially if you are a regular observer of the Moon, you better get some good charts and computer programs. There are many excellent charts and atlases for the Moon, such as those by Rukl, S&T, Norton's Star Atlas, London Times, US Air Force, and Elger. For this paper you want a cartographic map of the Moon, not just a set of photographs. So the CLA, Clementine, Orbiter images are of little help for the moment. Be sure the coordinate grid on the map is true lat-lon and NOT a rectangular mesh. Look at the polar areas of the Moon. Do the longitude lines hug the globe and converge at the poles? OK, you got the correct map. If the map in the polar zones still have neat vertical and horizontal lines running all over it, you're out of luck. So Hatfield and Wilkins & Moore, as examples, aren't what you want now. I don't mean you must discard those maps! It just isn't the right one for the purpose at hand.
Computer programs --------------- The computer program must give you certain data about the Moon, more than just its position among the stars or over your horizon. You need from the program the Moon's elongation and the libration. Most planetarium programs fall short by missing out the libration and by giving the phase only by age or illumination. It is a real bonus f the program issues the SSC as one of the parameters for the Moon. I find that this parameter tends to be the schematic one, a simple (90 - elongation) number. I explain this later. You could be lucky and have the true SSC. Read the program's litterature. One program I find very quick and simple is Moon Calculator. It outputs both text and graphics. An other favorite is Astronomical Algorithms. It outputs text tables of lunar (and planetary) data. Both are DOS programs which runs perfectly well under all varieties of Windows. Both are free downloads from astronomy software websites.
Directions on the Moon -------------------- The Moon has TWO distinct schemes of direction for its surface. The traditional one, before the space age, is based on the celestial sphere. North and south are toward the north and south celestial pole, that's no problem. The east and west sides of the Moon correspond to this sense of east and west on the sky. On your lunar map east is the limb or edge near Grimaldi crater. West is near Neper crater. This system is the astronomical system. Maps made since the lunar explorations consider the Moon as a globe in space. On this globe the directions are like those on Earth. East is near Neper and west is near Grimaldi. In other words, a compass rose on the Moon's surface would read the same way as one on the Earth. This is the astronautical method. For sanity's sake you better add the E-W directions on your moon map for the system opposite from the one it was printed with. Now you got both systems on the map for any future purpose. An other glitch to watch out for is an end for end rotation of the Moon. The astronomical charts have south at the top to better match with a telescopic view. The astronautical maps have north at the top like typical Earth maps.
Selenographic latitude -------------------- Like Earth there is a lat-lon system for the Moon. By good fortune the Moon's rotation axis is pretty much perpendicular to us so the poles are near the north and south limb and the lunar equator is straight east-west across the disc. The main deviation from this ideal situation is due to libration, which tilts the lunar globe one way or another. The 'geographic' coordinate scheme for the Moon is called selenographic. Similar schemes exist for the other planets, such as Mars (areographic) and Sun (heliographic). Selenographic latitude on the Moon is just like that on Earth. Positive latitude is toward the north; negative, south. Longitude is also like the Earth but with the 'Greenwich' meridian in the dead center of the disc homed with zero libration..
Selenographic longitude ---------------------- Now comes the tricky part. Longitude for the Moon is dimensioned in FOUR ways. One is to run the angle 0 to 360 degrees clear around the Moon from the zero meridian thru astronomical west, the rear face, east, and back. Neper is near 90 degrees longitude, the farside meridian is 180, and Grimaldi is near 270 degrees. On an astronautical chart the numbers run the same way, except the sense is eastward. Hence, for this system both an astronomical and an astronautical map yield the same value for the longitude of a given point. The second method runs the longitude in reverse, from the central meridian, thru Grimaldi, around the far side, thru Neper, and back. This is a nonstandard way but it shows up from time to time. The last two ways send the angle eastward for positive 0 to 180 degrees and westward for negative 0 to 180 degrees. The wrinkle in this dimensioning is the use of east and west longitude. Traditionally these referred to the astronomical east and west but modern maps will often bank them off of the astronautical directions. Result: A crater of 45 degrees west longitude on your old map from the 1950s will now have longitude of 45 degrees east on your new map. Your lunar map will use one of these systems and it doesn't matter which. You may want to pencil in the other two longitude marks next to the ones your map has. I give here sketches to clarify the dimensioning. The line is the lunar equator and north is up. If your chart has the traditional telescopic view with south up, merely turn this page end for end and read upside down.
N E/W (-----+-----+-----+-----+-----+-----) W/E 270 300 330 0 30 60 90 astronomical and astronautical
N E/W (-----+-----+-----+-----+-----+-----) W/E 90 60 30 0 330 300 270 nonstandard
N E (-----+-----+-----+-----+-----+-----) W 90E 60E 30E 0 30W 60W 90W -90 -60 -30 +30 +60 +90 astronomical
N W (-----+-----+-----+-----+-----+-----) E 90W 60W 30W 0 30E 60E 90E -90 -60 -30 +30 +60 +90 astronautical
Libration ------- If you're a regular watcher of the Moon you'll know that the Moon does a little wiggling day by day. It nods and sways a bit in each direction so we see more of one edge and less of the opposite edge. A crater near the edge today may in a few days be far in from it and new craters normally on the far side are brought into view. Conversely some features near the limb tonight may in a week be carried off to the far side and be hidden. This phaenomenon is libration; it was discovered by Cassini and studied by Hevelius in the mid 1600s. The maximum swing in libration is plus and minus 7.5 degree. By libration, combined latitude and longitude, we can eventually see about 59% of the entire lunar globe. Of course, only 50% of the globe at any instant is visible. The other 9% is a band around the limb extending into the farside. Libration is sometimes split into two parts, an 'optical' and a 'physical' part. For our work we take the total of the two altho the physical component is always less than one degree. We worry here only about the libration in longitude, that which wobbles the Moon around its north-south axis to expose or hide the east and west limbs. There is a libration in latitude, too, which turns the north and south edges toward and away from us. This we happily can neglect here. We further miss out the libration component due to our perspective on the Moon as it moves from rise to transit to set. This effect is almost always neglected for it is both at most only one degree and does not shift the terminator on the lunar lat-lon grid. The value of libration is the longitude of the lunar globe in the dead center of the disc. A libration of -2.8 degrees means that on your Moon chart the longitude line for -2.8 degrees, and not the 0 longitude, is straight north and south thru the middle of the disc. An alternative visualization is that libration is the Earth's selenographic coordinates, or the lat-lon of that point where the Earth is in the local lunar zenith. This is easiest to appreciate on a moon map with +/-180 degree longitude markings. A positive libration pulls Neper in from the limb, and pushes Grimaldi out toward the limb. A negative libration pushes Neper toward the limb and pulls Grimaldi in from the limb.
Polar regions ----------- As long as you stay inside (lower latitude) of about 75 degrees north and south latitude, the libration in latitude may be neglected. Because this libration nods the poles to and from you, dire errors can arise when figuring out the visibility of a feature near the poles if you miss out the latitude libration. Unhappily, the behavior of latitude libration is far tougher to appreciate, and we don;t need it here. Stay away from the poles for the rest of this paper.
Terminator -------- The Moon is a solid ball of rock lighted by the Sun. As it circulates around the Earth we see more or less of the lighted hemisphere, thus presenting to us in the sky the various shapes associated with the Moon's phases. They run from no moon in the sky thru thin crescent, fat crescent, half moon, gibbous, full, gibbous, and the others in reverse order. We usually do not see the dark or night side against the dark sky, so to a casual viewer the Moon does seem to grow and shrink over the days. When the Moon is a thin crescent within a couple days of new Moon, the night side is sometimes visible as a gray zone completing the round disc of the Moon. This glow is light reflected from the Earth and is the earthshine phaenomenon. The frontier on the lunar surface between day and night is the terminator. It is the line along which the Sun is rising or setting, exactly like the terminator line on Earth. The sunrise terminator is the half we see when the Moon is in our evening sky between new to full via first quarter. The sunset terminator is on the front face when the Moon is between full and new via last quarter. The patterns of light and dark that show up the relief of the lunar surface is entirely due to the local altitude of the Sun. With it rising or setting near the terminator, the shadows are longest, the valleys are filled with shadow, the tops of mountains are hit by sunlight. Hence, the most impressive part of the Moon at first glance is the region straddling the terminator. Places on the nightside of the Moon are essentially invisible, except for the effect of earthshine. Places on the dayside far from the terminator tend to be flat with short or no shadows; they are sometimes overlooked. Hence, the place of the terminator on the lunar globe is of major importance for the home astronomer; that's where the action is. The terminator on the front side in our evening is sometimes called the 'evening' terminator. This is a most misleading term. It is the sunrise or morning terminator from the eye of a person on the Moon.
Speed of terminator ----------------- You've seen how within a night the daylight on the Moon intrudes into a crater; this is for viewing in the evening with a sunrise terminator. The motion of the terminator is surprisingly rapid considering that it takes a month to circle the lunar globe. Leaving out the details, the terminator creeps on the lunar ground 0.51 degrees of longitude or 15.3 kilometers per hour, on the average. It would, therefore completely cross a smaller crater in an hour or it covers 100 kilometers on the lunar ground in 6-1/2 hours. An other useful equivalence is that one degree of angle on the Moon is quite 30 kilometers. With the smaller diameter of the Moon, and greater curvature of its horizon, it is in fact possible to stand in the middle of a larger crater and not see the walls. They would be over the local horizon! The steep curvature is demonstration by the gradation of sunlight on a mare or in a large flat crater when the terminator lies across it.
Phase --- The phase of the Moon is specified in one of several ways and you must be conversant with all of them. The oldest and most traditional method is to state the 'age' of the Moon. This is the time elapsed since the last new Moon. A Moon of age 3.24 days is one which is now 3.24 days after new Moon. Age may be given in days and decimal or in days and hours. The oldest the Moon can be is 29.5 days, the time to complete one round of phases from new thru full to new again. An other way is to state the elongation of the Moon from the Sun, or the difference between the ecliptic longitude of the Moon and Sun. This is measured downrange along the ecliptic and ignores the slight displacement of the Moon north or south of the ecliptic. The elongation doesn't increase in a uniform way due to the varying speed of the Moon in her orbit. The elongation may be expressed from 0 thru 90, 180, 270, to 360 degrees. Or it may run from 0, thru +90, +/-180, -90, back to 0. On the average the elongation increases 12.191 degrees per day. A third, less common, means of specifying the phase is to state the percent or fraction of the Moon illuminated by the Sun. This ranges from 0% or 0.0 at new Moon, to 50% or 0.50 at first quarter, to 100% or 1.00 at full Moon. Beyond full Moon the percent or fraction is negative: last quarter is -50% or -0.50. Again from the irregular motion of the Moon this parameter does not flow at a constant rate. The last method, and one of considerable confusion, is the Sun's selenographic colongitude. It has its own section below.
Day-by-day -------- It is a normal temptation to collect pictures of the Moon at each say of age for a complete lunation. It is actually nearby impossible to do so. There are two geometrical reasons, apart from weather, twilight, mundane obligations. First, the exact moment of new Moon can be at any clock hour. If you see that a new Moon occurred on, June 30 in a certain year, you would suppose that on July 1 you would see a one-day old Moon. Each evening thereafter you will see a Moon one day older. This virtually is never true. If new Moon happened at local sunset on June 30, then one day later at sunset the Moon will in deed be one day old. But if new Moon happens at any other hour, the very first Moon you see on the day after new is NOT 'one-day old'. It's as much as 12 hours early or late from geometric one-day old. You could be seeing a Moon of perhaps 1.3 days of age. And each successive evening you see a 2.3, 3.3 &c day old Moon, assuming you go out at the same hour each evening. The second reason is that you can not over a whole lunation observe at the same hour each night. For the very young Moons you have to watch at or right after sunset; the Moon sets too soon to see her later in the evening. After a few days you observe the Moon in the darker hours of the evening, throwing off your equal intervals of age. After full Moon, the Moon is not in the early evening sky at all and you must wait for a late hour. Eventually, you find yourself outside in the dawn hours to see the waning crescent a couple days before the next new Moon. The result is that while you'll ultimately get a nice sequence of Moon pictures, they will not be so nicely spaced in age as you hoped. By examining pictures taken on all assorted dates for, say, 'four day old Moon', you'll find some whose phase seems better suited for a 3- day Moon and some more appropriate for a Moon of 5 days. Such is life.
Sun's selenographic colongitude ----------------------------- This mouthful of a phrase is actually a bad mistake. It may relate to a longitude point on the Moon and is indeed so calculated by the major ephemerides like those of USNO or JPL. But it is also commonly cited as a schematic angle, not based on an actual longitude on the Moon. The problem is that usually you don't know which method is used in a particular instance. The Sun's selenographic colongitude, SSC, is supposed to specify the longitude on the lunar equator where the sunrise terminator falls at a given moment. At some point on the Moon the Sun is at his noon position and this point has a certain longitude. The sunrise terminator is 90 degrees away to the east (astronautical west) and the sunset terminator is 90 degrees away to the west (east).
Uh-oh! ---- The next parts are very very important; read carefully. The sunrise terminator is in the center of the disc at the instant of first quarter and its angle is then zero. The angle increases with age or elongation thereafter to 90 degrees at full, 180 degrees at last quarter, and 270 degrees at new. Recall that the elongation of the Moon from the Sun is 0 for new; 90, first quarter; 180, full; and 270, last quarter. See that SSC LAGS elongation by 90 degrees. Got that? SSC = (elong) - (90d). This number just derived is NOT the actual longitude on the Moon where the terminator sits! If you look at the sketches above for the various longitude dimensionings, you'll see that the terminator marches from right to left. The 'longitude' of the terminator stated by SSC ends up being a 0-360 angle running OPPOSITE to the 0-360 angle of longitude itself! Or you may think of it being read off of am OVERLAY of a new longitude scale. I show this below so you can match it with the sketches above. Having just said all that, you may find that your chart in fact uses the nonstandard revered scale of longitude, the second scheme explained earlier.
N E/W (-----+-----+-----+-----+-----+-----) W/E 90 60 30 0 330 300 270 colongitude
Longitude or not? --------------- If we calculate SSC from the elongation, by subtracting 90 degrees from it, we got a number that at first looks like we can mark it on a moon map and see where the terminator sits. Not quite. What we got is a sort of schematic angle telling where the terminator is on the visible disc. By this simple method the SSC (which it's still called even this it is NOT the true SSC) is DEFINED to be zero at the moment of first quarter, 90 degrees at full Moon and so on. This number ends up being just an other way to specify the phase. What's missing is the effect of libration. It is easy to miss out on the libration factor because libration is not a standard feature of planetarium programs. Elongation is, or can be figured out from other information given by the program. To get things right you have to include the libration of longitude. This must be subtracted from the raw SSC. The complete formula for the SSC is
(elongation) - (90 degrees) - (libration)
with the caveat that we use the reverse longitude scale of the last sketch above.
Confusion! -------- All the above seems so convoluted, I know. I did say that the answer to the question of when the lighting on a given crater repeats is not simple. But we are now ready to attack that question. I long forget the exact situation from 'sci.astro.amateur' but let's get some numbers in this problem. At a starviewing I attended in Cadman Plaza, Brooklyn, on 31 May 2001 at about 20h EST (9 PM daylight time) I saw the morning terminator sitting across Mons la Hire. I didn't recognize this mountain in Mare Imbrium at the session; I looked it up when I got home. When will it again be so situated?
Initial steps ----------- From your map you see that Mons le Hire has a longitude of quite 25.4 degrees west, 25.4 degrees east, -25.4 degrees, or 334.6 degrees, all depending on the longitude scale on the map. Because we are now going to compare this longitude with the SSC, we better convert it to the wrong-way numbering scheme of SSC. Mons le Hire is at the 25.4 degree mark on the SSC scale. We crank up Astronomical Algorithms and look at the Moon for 2001 May 31, 20h EST, to find where the terminator was. We find elongation is 117.7 degree and libration is +3.8 degree. (The libration consists of two parts; take the total.) The terminator sat at
SSC = (elongation) - (90) - (libration) = (117.7) - (90) - (+3.9) = 23.9 degree.
If you use Moon Calculator, the colongitude it gives IS in fact the true SSC with no further doctoring to do on it. The libration is ALREADY factored in. This is a degree ahead of Mons la Hire, but still pretty much right on it, as I saw on that evening. We must match this location of the terminator to see the same lighting on Mons la Hire.
Our first cut ----------- As a first guess we examine the Sun's selenographic colongitude one lunation later, more or less. We try 2001 June 29 20h. We pick off the elongation as 114.2 degrees and the libration of longitude as +6.0 degrees. The actual longitude of the terminator is
SSC[20h EST] = (114.2) - (90) - (+6.0) = 18.2 degrees
This falls short of Mons la Hire by 5.7 degrees. Libration varies over a timescale of a month so that within one night it'll remain about the same. We can shift the terminator to Mons la Hire merely by waiting; it'll edge along at 0.51 degree per hour. We have to wait
delta(hour) = delta(lon)/(0.51 deg.hr) = (5.7)/(0.51) = 11.2 hours
On the clock this is (20) + (+11.2) = 31.2 -> 7.2 = 7h12m on the following morning, the 30th. Checking with Astronomical Algorithms, we have elongation of 120.0 degree and libration of +5.8 degree. From these, the terminator is at
SSC[20h EST] = (120.0) - (90) - (+5.8) = 24.2
This is close! Can we accept this as is, or should we correct for the slight overrun of the terminator past the mountain? Not quite. Where is the Moon in mid morning of June 30th? It's under the horizon! The program tells us that moonset was at 01h09m on the 30th and the next moonrise is at 14h49m on that same day. So this date is no good and we must keep looking. We could almost have ruled out a date one lunation later by the synodic or phase period of the Moon. It's 29.5 days. If we see the Moon in our sky at a certain phase, the very next occurrence of that same phase will be one half day off. Most likely the Moon will be under the horizon.
Our second cut ------------- We try the second lunation, July 29th, also at 20h EST. We find the elongation is 121.7 degrees and libration is +6.8 degree. The actual longitude of the terminator at this moment is
SSC[20h EST] = (121.7) - (90) - (+6.8) = 24.9
This is also close! We're (24.9) - (23.9) = 1.0 degree too far at 20h EST. We must back up
delat(hour) = (1.0)/(0.51) = 2.0
to put the terminator on Mons la Hire. The new time is (20) + (-2.0) = 18.0 -> 18h00m. Wait! Is the Moon in the sky at that hour? Yes. Moonset is at 00h50m on the next morning, the 30th, giving us almost 7 hours of viewing. We can watch Mons la Hire emerge into full daylight before moonset. Thus, we now know when to try for an other go at Mons la Hire.
Conclusion -------- Was that so difficult? Yes and no. You did need the charts and computer program. You did need the smarts to check for the place of the Moon in the sky. You had to mind the various ways longitude on the map could be dimensioned. You have to watch if a quoted SSC is the raw value (an expression of phase) or the true one (the longitude of the sunrise terminator). And, because libration of latitude was neglected, you must keep away from the polar regions. Keep within =/- 75 degrees latitude. On the other hand, you did no horrible math or graphing. which we oldtimers had to do in our young years in astronomy! All you did was a few keypresses on a computer and calculette.