Ever wondered how high lunar mountains are or how deep craters are. Well, determination of lunar serface heights is very straightforward. All that is needed is a good fotograph showing a lunar feature, measurmment of its shadow it casts and a lunar ephemeris.

The determination of heights is based on the shadow a features casts. Therefor the exact location of the feature must be known (from a lunar atlas), the observer's location, the length of the shadow (from the fotograph) and the position of the Sun with regard to the feature (calculated from the ephemeris). The calculation is very simple.

cos X= sin bs sin bp + cos bs cos be sin(colong - Le)

X is the distance from the apparent centre of the lunar disk to the sub-polar point.

S is the lenth of the shadow corrected for the angle of the Sun at the time of measurement.

sin A = sin bs sin bp - cos bs cos bp sin(colong - Lp)

A is the altide of the Sun as seen from the top of the feature.

H is the height of the feature in units of a decimal fraction of the apparent lunar radius. By multiplying this value by 1738, the height of the observed feature above its surroundings is given in meters.

To get some idea of the error or uncertainty on H, the absolute error is calculated. This is based on the relative error made on the measurements of the shadows. Notice that a small shadow results in a reltively large error. A few examples of the determination of lunar surface heights are given below.

The picture above was taken on 08/04/2003 on 1915 UT. It shows the area around Abulfeda, Almanon and Geber. Measurments of the depth of these craters were made as of some hills. Geber has a diameter of 45 km and is 3.510 km deep (Rükl, 1996). This compares very well with the calculated values. Almonon has a width of 49 km and a depth of 2.480 km according to Rükl. These values compare very well with the calculated values. The calculated value for Abulfeda is slighter larger than the value given by Rükl (3.110 km). Notice that a crater rim does not have to be smooth. The rim of for instance Abulfeda has higher and lower parts. This is nicely visible on the irregular shadow the rim casts on Abulfeda's floor.

The picture above shows Plato and Mons Pico (10/04/2003, 1920 UT). The rim stands approximately 2 kilometres above its interior but this can be as high as 3.3 kilometres for the highest peaks. Plato itself has a diameter of 101 kilometres wich should result in a crater depth of 4.5 kilometres (according to the relations of Baldwin, see note below). This difference is obviously due to the infill of lava which is thus approximately 2 kilometres (4.5 - ~2.5).

Mons Pico is over 2 kilometres high, also three heights are determind in the hummocky Montes Alpes surrounding Plato.

The picture below shows Ptolemaus, Alphonsus, Klein and part of Albategnius (07/06/2003, 2100 UT). Ptolomaeus is approximately 2.4 kilometres deep which corresponds well with the measerments. Notice that this depth is the sum of difference heights. The rim of the crater is very complex of different stages. The shadows which are cast on Ptolemaeus floor are due to walls with heights up to 2 kilometres high. On these, higher peaks are found up to 0.4 kilometres. The net result is that Ptolomaeus is about 2.4 kilometres deep. The same story can be told of Aristarchus. All craters in the picture are filled by material (ejecta and/or lava). As was the case for Plato, the craters thus were even much deeper just after their formation. An estimation of the thickness of this infill is made in the table below with ther relations of Baldwin.

The height of the central peaks of Alphonsus and Albategnius could also be calculated as well as a crude estimation of the height of the small mountain ranges situated in Alphonsus.

In a last example the height the famous Rupes Recta is determined (20/04/2002, 2030 UT). It is actually a giant fault of 120 kilometres length. It is situated central in an old crater which rims can still be seen north and south of Thebit and in the west as mare ridges. Subsidence of the western part of this old crater resulted in the now observed impressive fault. The picture is of very low quality because it is part of a larger picture. It was zoomed in to measure the shadow lengths cast by the rupes. Different heights can be found in literature ranging from 200 to 400 metres. My measurements are 300 metres for the central part of the fault. Towards the north and the south this diminishes until the fault disappears in the north or is blocked by mountains in the south. Two of the highest peaks on the rim of Thebit were also calculated. Their values correspond well with the overall depth of 3.3 kilometres found in literature. This is also the depth expected for a crater with diameter of 54 kilometres. Thus no major lava or ejecta deposits are to be found within this crater.

Note: Baldwin's relations

The figure on the next page illustrates these relations graphically together with some lunar examples. Depth of the crater increases with diameter but a larger crater is relatively less deep than a small crater. Lunar examples show that many of the small craters follow this relation. The larger craters, however, are less deep than expected because of volcanism and/or deposits of ejecta. The strength of the target material is also of importance explaining why some craters are deeper than expected. Height of the rim also increases with diameter. Some of the lunar craters have rim heights which are double as high as expected.