Notes on the Combined Parallax - Semi-Diameter Corrections for the Moon

The last two correction steps in the correction scheme for Sextant measurements are the corrections for Parallax and for the Semi-Diameter of the observed body. For stars these two steps can be omitted. For the planets Mars and Venus the Parallax effect should be corrected, whereas the Semi-Diameter correction can be ignored. For the Sun, the Parallax effect is small but the Semi-Diameter correction is large and finally, for the Moon both the Parallax and Semi-Diameter correction are necessary.

Because of the large Parallax effect of the Moon the Parallax Correction Tables have a section, which is used for the Moon only. In order to simplify the Sextant Correction Scheme, the combined correction for Parallax and Semi-Diameter effects can be compiled. Such combined Moon Correction Tables will have to be compiled for upper-limb observations and lower-limb observations separately. This results in more pages in the tables for the convenience of a simpler correction scheme.

The picture on the left, shows the situation for an observation of the lower limb of the Moon. The line-of-sight after correcting the Sextant Altitude for Dip and Refraction is shown in red. This gives the observer (Obs) the topographic Altitude of the lower limb. This Altitude must be further corrected to obtain the geocentric Altitude as would be measured at the center of the Earth. The difference between these two altitudes is δ = Hgeo - Htopo. The combined Parallax and Semi-Diameter correction δ can be calculated from the outer triangle (in which the semi-diameter correction is performed by shifting the line-of-sight through the center of the Moon) by applying the Law of Sines: sin(δ) = sin(90°+Htopo) (rE + rM/cos(Htopo)) dEM which can be reformulated as:

δ = asin( (cos(Htopo)/dEM) * (rE + rM/cos(Htopo)) ) = asin( rE/dEM * cos(Htopo) + rM/dEM) The values rM, rE and dEM are the radius of the Moon, the radius of the Earth and the distance Earth-Moon (center-to-center) respectively.

A similar picture can be drawn for the observation of the Moon's upper limb with the line-of-sight after correcting the Sextant Altitude for Dip and Refraction shown in red. Again the required correction is the difference between the two altitudes: δ = Hgeo - Htopo. The combined Parallax and Semi-Diameter correction δ can be calculated from the inner triangle (in which the semi-diameter correction is performed again by shifting the line-of-sight through the center of the Moon) by applying the Law of Sines: sin(δ) = sin(90°+Htopo) (rE - rM/cos(Htopo)) dEM which can be reformulated as:

δ = asin( (cos(Htopo)/dEM) * (rE - rM/cos(Htopo)) ) = asin( rE/dEM * cos(Htopo) - rM/dEM)