File:Speed of moon in 104-103 BC.png

English: Comparison of speed of moon with values given in a Babylonian tablet, 104-103 BC

There is a cuneiform tablet discussed in the first part of Die Babylonische Mondrechnung by Franz Xaver Kugler (1900). It is like a spreadsheet, with 11 columns and 39 rows. The rows correspond to new moons from March 23, 104 BC, to April 18, 101 BC. The columns give various kinds of information about the moon, with columns based on earlier columns, till the final column which gives the date of the new moon, that is, on what day of the ending lunar month the new moon falls.

Column "F" gives the speed of the moon (at the time of the new moon) in degrees, arc minutes, and arc seconds per day, but it's a calculated value. The method used was to let the speed increase linearly by 36 arc minutes per day each month during the seven months when the new moon occurs closer and closer to perigee. If it would pass the value of 15°16'5'' while going up by 36', then after getting to that value it goes down by the amount that is left of the 36'. It then goes down by 36' every new moon until it "bounces off" the lower limit, which is 11°5'5'' in the same way as it does from the upper limit. This gives a zigzag function instead of a sine wave sort of function, which would be more realistic.

In this graph, the first 15 values of the tablet are displayed as blue squares, and the turning points at the upper and lower limits as red triangles. The yellow curve is the actual speed of the moon, calculated using the formula for longitude in T. C. van Flandern & K. F. Pulkkinen, Low-Precision Formulae for Planetary Positions (1979), taking the analytic derivative. Only the "great inequality", the "evection" and the "variation" were included (see Mondbahn), since the other terms do not make a visible difference.

The graph shows one "full moon cycle" of approximately 14 lunar months. The speed of the moon goes up and down every anomalistic month, and one can see that the Babylonian values are fairly close to the true value at the time of the new moons.

It so happens that the range between the upper and lower limits is 251 arc minutes, and together with the fact that the change per synodic month was taken as 36 arc minutes this means that 251 synodic months will bring the situation back to the starting point. This occurs after 18 cycles like the one shown here, and there will be 18 more anomalistic months than synodic months, making 269. This equivalence is quite accurate and is mentioned in the Almagest, but it is not known for sure whether the Babylonians realized the equivalence.

The red triangles correspond to the moments when the sun is lined up with the perigee or apogee of the moon's orbit. The full moon near the first and third red triangle will be "supermoons" because the moon will then be near perigee, and the new moon near the second red triangle will also be a supermoon near perigee. There were solar eclipses on July 30, 104 BC, and January 22, 103.