John Pazmino 
 NYSkies Astronomy Inc 
 2007 December 11
    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 
    You could be lucky and have the true SSC. Read the program's 
    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 
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. 
  E/W (-----+-----+-----+-----+-----+-----) W/E 
      270   300   330   0     30    60    90 
      astronomical and astronautical  
  E/W (-----+-----+-----+-----+-----+-----) W/E 
      90    60    30    0     330   300   270 
    E (-----+-----+-----+-----+-----+-----) W 
      90E   60E   30E   0     30W   60W   90W 
      -90   -60   -30         +30   +60   +90 
    W (-----+-----+-----+-----+-----+-----) E 
      90W   60W   30W   0     30E   60E   90E 
      -90   -60   -30         +30   +60   +90  
    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. 
    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 
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. 
    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. 
    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). 
    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 
  E/W (-----+-----+-----+-----+-----+-----) W/E 
      90    60    30    0     330   300   270 
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. 
    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 
                = (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 
    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. 
    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.