Analysis of the 1995 April 15 partial lunar eclipse -

Appendices A, B and C

Analysis of the 1995 April 15 partial lunar eclipse -

Appendices A, B and C

APPENDIX A

Circumstances and crater location

A1 ECLIPSE CIRCUMSTANCES

The eclipse occurred as shown in figures A-1 and A-2:

APPENDIX B

Adjustments to the measured umbral semi-diameter

B1. REFRACTION

The Moon's semi-diameter increases when the observer is on the surface of the Earth and not at its geocentric centre. To find the topocentric value of the Moon's semi-diameter (Sm') an allowance must be made related to the height of the Moon above the observer's horizon, or for the altitude (a) of the moon during the eclipse.

An approximate correction which is zero when the Moon is on the horizon, and a maximum between 14" to 18" at the zenith (Meeus, 1991: 360-361), is found as follows :

where, at mid-eclipse the Moon's horizontal parallax pi = 1.003167 degrees and using the altitude of the Moon for Melbourne (the nearest site to Bethanga and Lavington, as given by Soulsby, 1994a) the Moon's semi-diameters, geocentric Sm and topocentric Sm' are shown in Table B1.

Table B 1. Semi-diameters of the Moon

1995 April 15 Partial Lunar Eclipse

The increased S'm mean value of 1.37% was used to calibrate lunar images from all observing sites prior to umbral size measurements by Image Analyst Revision 8.0.

B2 UMBRAL EDGE DENSITY CHANGE

The Image Analyst software determines the imaged umbral edge by parameters set for the density of the image in the region of interest (ROI), defined for either the lunar limb (this semi-diameter measurement is used to accurately calibrate each image scale) and for the ROI of the umbra. The density, or grayscale of the ROI can be adjusted to obtain semi-diameter measures at varying values of grayscale by adjustment of the image histogram. The best-fit-circle to the limb was improved by masking the umbra edge, to ensure that only the limb image was used for the software fit, as shown in figure 8 in the text.

The contour traces in the 8-bit grayscale images clearly show that the maximum change in image density near the umbral edge (see Figs. 2 and 3 in the text) is most pronounced at the inflexion point. It was first thought that low values of umbral semi-diameter were due to the software not determining the first gradient defined for the ROI at the maximum inflexion point, particularly when the best-fit-circle was found by threshold level rather than by sign of first gradient.

This was investigated by superimposing the image density traces on the measured images and it was found that the Image Analyst software detected the inflexion contour exactly when the correct value of intensity was chosen from the histogram.

B3 UMBRAL SEMI-DIAMETER MEASURES

Measures of image sequences captured from Brian Sture's and John Bennett's video records using Image Analyst Revision 8.0 are shown in Figure B-1.

B4 TOPOCENTRIC CORRECTION

The measured values have been corrected to give topocentric umbral semi-diameters from the computed correction angle (' ) for each site location as illustrated in Figures B-2 and B-3 and as described below. The topocentric values of F2 are shown in Table B-2 and Figure B-4 while the spherical-projection corrected F2 values are illustrated in figure B-5.

The topocentric correction of the measured images was calculated at the local hour angle (BAA Handbook, 1995) for the site longitude at the time of the captured frame, the topocentric declination (Meeus, 1991) of the Moon (pi'm) and the Sun (pi's) from the observing site position latitude (Ø' ) and elevation, and the angle of incidence of the umbra edge at the Moon was computed as:

The nomenclature used is illustrated in Figure B-2 above. The diagram in Figure B-3 shows the correction of the measured image chord (CH) at the umbra-moon image with other measured values AB, BD taken from the image, to give the topocentric F2. The mathematics used are straight forward, and most can be found in Meeus, they are not included here for the sake of brevity.

Table B 2. Topocentric and spherical projection corrected F2

The topocentric calculations were completed for each series of images using the author's compiled Microsoft™ QuickBasic program Avia Apl

B5 DISCUSSION

It is apparent that the size of the topocentric umbral semi-diameter computed from the image measurements falls below that found by the crater timing technique. One is forced to conclude that despite the topocentric corrections applied, that the technique of image analysis of umbral-moon image measurement is not an appropriate technique for finding change in the true umbral semi-diameter.

However, there is one other correction to be included in the image corrections.

B6 SPHERICAL PROJECTION CORRECTION

If the topocentric semi-diameter values are corrected for spherical curvature as illustrated in Figure B-6, then the increase in F2 is by the amount that arc JD exceeds the measured BD, as follows:

Figure B-5 below (also as Figure 9 in the text), show the resulting values of F2 for the umbra topocentric semi-diameter corrected for spherical projection.

These values are now reasonably close to that found by crater timing analyses and illustrate a general increase in the umbral semi-diameter throughout this partial eclipse. Ignoring one value for the Ellenbrook images lower than the general trend, at 12.35 hours, a trend curve can been drawn to show change in umbral size.

B7 CONCLUSION

If correction for spherical projection is applied to the topocentric values of F2, then reasonable semi-diameter values are obtained. This correction has been included in the author's Microsoft™ QuickBasic compiled program Avias Apl.

B8 REFERENCES

Meeus, J., 1991. Astronomical Algorithms, pp 263. Willmann-Bell Inc. 429 pp.

Taylor, G.E., 1995. The Handbook of the British Astronomical Association, 96 pp.

APPENDIX C

Improved primary contact times

C1 ALGORITHM

An algorithm has been prepared based on the statistical umbral oblateness theory (Soulsby, 1980) and a linear relationship (Soulsby, 1986) of umbral semi-diameter and the cosine of twice the position angle (cos2phi) of its contact with the Moon (and as shown in Figure 10).

The algorithm finds the predicted primary contacts using an oblateness equal to the Earth's geoid (1/298.257) and then the time of true contacts, using the observed oblateness (1/80), or for other predictions, the mean umbral oblateness (1/102). The circumstances of the eclipse, illustrated in Figure A-1, influences the difference between the improved predictions and the geoid time of each primary contact.

Table C 1. Predicted ILEE primary contact times

With Geoid 1/Fe = 298.257, observed umbral oblateness 1/Fo = 80 for the eclipse of 1995 April 15 and an overall mean observed oblateness of 1/Fo = 1/102 for the 1996 eclipses.

C2 IMPROVED PRIMARY CONTACT TIMES

The ILEEP algorithm has the following steps:

1. Find tan phi; = y/x at each primary contact, from the Besselian coordinates (x, y) of the Moon's centre (Soulsby, 1994), and the position angle of the primary contact (phi) measured from the umbral equator towards the pole.

2. Find the re (or ro) value from re = re pole * F/(F-1) for each contact from the Earth's geoid value (1/Fe = 1/298.257), and the values from the observed umbral oblateness of the 1995 April 15 eclipse (1/Fo = 1/80, see Figure 10), and from the overall mean observed umbral oblateness value (1/Fo = 1/102, Soulsby, 1990) for the 1996 eclipse predictions.

3. Find cos 2phi and the true value of the umbral semi-diameter from the linear best fit graph (ro versus cos 2phi in Soulsby, 1986 and in Figure 10) by similar triangles.

4. Find the values of (re phi) for each primary contact position angle, and deduct the relevant Re pole value from the (re phi) value to find the difference in umbral radius (delta Re) at the position of each primary contact.

5. Convert these radii to distance vectors (delta Re vector) along the direction of the Moon's motion, at a path angle found from the Besselian coordinates of first and fourth contact.

6. Find the time difference (delta t) by dividing each delta Re vector by the relevant value of the motion of the Moon (in degrees/min), and convert the difference to seconds in time.

7. For primary contacts preceding mid-eclipse time, add each delta t to the (Geoid) predicted primary contact time, as oblateness gives a smaller umbral semi-diameter and hence later contact, and deduct (delta t) from the primary contacts after mid-eclipse, as these contacts occur earlier.(Corrected on 17 June 1996).

8. Should the value of observed umbral oblateness differ greatly from the established mean of 1/Fo = 102, used for general prediction, then substitute this value at step 2 (as in the case of the 1995 April 15 eclipse).

9. The described algorithm has been included in the author's revised and compiled Microsoft™ QuickBasic computer program, ILEEP Apl.

C3 REFERENCES

Soulsby, B. W., 1980. Umbra oblateness estimates, Proc. 9 th Natn. Aust. Conv. amat. Astrer., Geelong, April 4 - 7: pp 149-159.

Soulsby, B. W., 1986. Oblateness of the Earth's outer atmosphere, Aust. J. Astr. (1), 4: 157-165.

Soulsby, B. W., 1990. Improved lunar eclipse ephemerides, J. Brit. astr. Ass., (100), 6: 293-305.

Soulsby, B. W., 1994, 1995. The lunar eclipse observer, Calwell Lunar Observatory, (1),4 & 6.