Notes on the Combined Parallax  SemiDiameter Corrections for the Moon
The last two correction steps in the correction scheme for Sextant measurements
are the corrections for Parallax and for the SemiDiameter 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 SemiDiameter correction can be ignored.
For the Sun, the Parallax effect is small but the SemiDiameter correction is large and
finally, for the Moon both the Parallax and SemiDiameter 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 SemiDiameter effects can be compiled.
Such combined Moon Correction Tables will have to be compiled for
upperlimb observations and lowerlimb 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 lineofsight 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 = H_{geo}  H_{topo}.
The combined Parallax and SemiDiameter correction δ can be calculated from
the outer triangle (in which the semidiameter correction is performed by shifting
the lineofsight through the center of the Moon) by applying the Law of Sines: sin(δ) = sin(90°+H_{topo})
(rE + rM/cos(H_{topo})) dEM
which can be reformulated as: 
δ = asin( (cos(H_{topo})/dEM) * (rE + rM/cos(H_{topo})) )
= asin( rE/dEM * cos(H_{topo}) + rM/dEM)
The values rM, rE and dEM are the radius of the Moon, the radius of the Earth and the
distance EarthMoon (centertocenter) respectively.


A similar picture can be drawn for the
observation of the Moon's upper limb with the lineofsight after correcting the Sextant
Altitude for Dip and Refraction shown in red.
Again the required correction is the difference between the two altitudes: δ = H_{geo}  H_{topo}.
The combined Parallax and SemiDiameter correction δ can be calculated from
the inner triangle (in which the semidiameter correction is performed again by shifting
the lineofsight through the center of the Moon) by applying the Law of Sines: sin(δ) = sin(90°+H_{topo})
(rE  rM/cos(H_{topo})) dEM
which can be reformulated as: 
δ = asin( (cos(H_{topo})/dEM) * (rE  rM/cos(H_{topo})) )
= asin( rE/dEM * cos(H_{topo})  rM/dEM)

Notice that the values for the combined correction can be positive and negative for both
upper and lower limb observations. So attention should be paid to copy the correct sign
for this combined correction from the applicable tables.
