Introduction
When images of the eclipsed Moon are captured and then measured, the projection of the conical shadow of the Earth's atmosphere onto the curved surface of the Moon causes fore-shortening of the curvature of the umbral imaged edge, particularly near the edges of the Moon, that is at first, second, third and fourth contacts.
This Appendix describes a method for estimating this fore-shortening, the calculation of the expected umbral curvature, or its geometry as seen from a topocentric location on the Earth and a comparison is made with this estimated value with the measurements made of the imaged umbra at five minute intervals throughout the lunar eclipse.
The comparison determines if there is any substantial change in the umbral geometry throughout the eclipse, if there is then the change is due to variation in the condition of the Earth's upper atmosphere during the time of the eclipse.
Schematic
The following figure shows two views of the Moon, the upper section is a side elevation with the umbra (centre O and semi-diameter F2) and Moon (centre M and semi-diameter Sc) with the lower section of the figure with the Moon in plan, or at 90 degrees to the upper view showing the leading umbral edge at an angle ^ (delta) to a line vertical to the projection plane passing through the Moon's centre.
In the upper view, where the umbra meets the limb of the Moon at C, a half chord is drawn from C to H on the line OM with point B the point at which the umbral edge meets OM. Point B is projected to the Moon in plan below and intersects the upper limb of the Moon at the lower point G and at the projection plane passing through M at the lower point B. The the leading edge of the umbra is at an angle ^ between the vertical projection line of B between the two views, and intersects the Moon's limb at point G and the projection plane through the Moon's centre at point A.
Geometry
Using the geometry described in the figure, most values are known from the lunar eclipse circumstances and the computed Besselian elements of the eclipse. The centre of the Moon at M is at coordinates (x,y) referenced to the centre of the umbra O at (0,0) the coordinate origin, values of the semi-diameter of the Moon Sc are known and values of F2 are also found from the circumstances of the lunar eclipse.
Then, from the upper part of the figure, the values of OM, CH, BH and of the angle CMB (theta) which is 0 at first contact, 90 degrees at second contact, can be found from the following equations, as well as the values of BG and AB shown in the lower part of the figure:
In the lower figure the inclination of the umbral edge, angle ^ (delta) is found from the difference between the Sun's semi-diameter and its horizontal parallax plus the difference between the topocentric declination of the Moon and of the Sun at a given time during the eclipse, as follows:
At any time during the eclipse we can estimated the value of the fore-shortened umbral semi-diameter Fi from the equations below, and compare this with the measured umbral semi-diameter;
Theoretical Results
Values of the Chord CH at five minute intervals and the fore-shortened umbral semi-diameter Fi have been estimated by the above formulae using a value of ^ (delta) for the geographic coordinates of the Observatory in an MS Excel 98 spreadsheet, the results are shown below:
Measured Results
The value of the inclination of the umbral edge, angle ^ (delta) varies with time at each observer's location. Delta values have been computed for the geographic location of the Calwell Lunar Observatory at longitude 149d 06m 00.9s E, latitude 35d 26m 31.4s S and an elevation of 630 metres using my original QuickBasic program Via.BAS.
Measures of the imaged umbra from all three CCD cameras at the observatory have been compared with the estimated Fi values over the whole eclipse, the comparison is shown below, firstly for the Sony CCD 511E images, then for the Sony CCD 450E images and lastly for the Mintron images:
When these values are collated for each phase of the eclipse the results from the three CCD cameras can be compared and it is immediately evident that some spread in the data occurs, however a trend can be seen which is discussed in the analysis.
