Solar Eclipse Predictions and the Mean Lunar Radius
Mean Lunar Radius 'k'
A fundamental parameter used in the prediction of solar eclipses is the Moon's mean radius 'k', expressed in units of Earth's equatorial radius. The actual radius of the Moon varies as a function of position angle and libration due to the irregularity of the lunar limb profile. From 1968 through 1980, the United States Naval Observatory Nautical Almanac Office (USNO) used two separate values for k in their eclipse predictions. The larger value (k=0.2724880), representing a mean over lunar topographic features, was used for all penumbral (i.e., exterior) contacts and for annular eclipses. A smaller value (k=0.272281), representing a mean minimum radius, was reserved exclusively for umbral (i.e., interior) contact calculations of total eclipses (Explanatory Supplement, 1974). Unfortunately, the use of two different values of k for central eclipses introduces a discontinuity in the case of hybrid or annular-total eclipses.
In August 1982, the International Astronomical Union (IAU) General Assembly adopted a value of k=0.2725076 for the mean lunar radius. This value is currently used by USNO for all solar eclipse predictions (Fiala and Lukac, 1983) and is believed to be the best mean radius, averaging mountain peaks and low valleys along the Moon's rugged limb. The adoption of one single value for k eliminates the discontinuity in the case of annular-total (i.e., hybrid) eclipses and ends confusion arising from the use of two different values. However, the use of even the best 'mean' value for the Moon's radius introduces a problem in predicting the character and duration of central eclipses, particularly total eclipses. A total eclipse can be defined as an eclipse in which the Sun's disk is completely occulted by the Moon. This cannot occur so long as any photospheric rays are visible through deep valleys along the Moon's limb (Meeus, Grosjean and Vanderleen, 1966). But the use of the IAU's mean k guarantees that some annular or annular-total eclipses will be misidentified as total. A case in point is the eclipse of 1986 Oct 03. The Astronomical Almanac for 1986 identified this event as a total eclipse of 3 seconds duration when in it was in fact a beaded annular eclipse. Clearly, a smaller value of k is needed since it is more representative of the deepest lunar valley floors, hence the minimum solid disk radius, and ensures that an eclipse is truly total.
Of primary interest to most observers are the times when central eclipse begins and ends (second and third contacts, respectively) and the duration of the central phase. When the IAU's mean value for k is used to calculate these times, they must be corrected to accommodate low valleys (total) or high mountains (annular) along the Moon's limb. The calculation of these corrections is not trivial, but is essential, especially if one plans to observe near the path limits (Herald, 1983). For observers near the central line of a total eclipse, the limb corrections can be closely approximated by using a smaller value of k, which accounts for valleys along the profile.
Solar eclipse predictions appearing on EclipseWise.com use the IAU's accepted value of k (k=0.2725076) for all penumbral (exterior) contacts. However, the predictions here depart from IAU convention by adopting the smaller value for k (k=0.272281) for all central (interior) contacts. This is done in order to avoid eclipse type misidentification and to predict central durations that are closer to the actual durations observed at total eclipses.
Using this smaller value of k for central eclipses is consistent with predictions published in Thousand Year Canon of Solar Eclipses and Five Millennium Canon of Solar Eclipses. Consequently, the smaller k produces shorter central durations and narrower paths for total eclipses when compared with calculations using the IAU value for k. Similarly, the smaller k predicts longer central durations and wider paths for annular eclipses.
Duration of Total Eclipse with Different Values of k
A recent example of eclipse predictions for the total solar eclipse of 2017 August 21 using different values of k can be seen in the following table. They show the predicted duration of totality on the central line in Shawnee National Forest, Illinois (89°, 15' 24" W & 37° 38' 12" N). This is near the position where the total eclipse lasts the longest along the entire eclipse path.
- 2 minutes 44.3 seconds from USNO (using IAU value of k [k=0.2725076])
- 2 minutes 40.3 seconds from EclipseWise.com - 2017 Total Solar Eclipse (using smaller value of k [k=0.272281])
- 2 minutes 41.5 seconds from Eclipse Bulletin: Total Solar Eclipse of 2017 (using actual lunar limb profile)
- 2 minutes 41.4 seconds from 2017 Interactive Eclipse Map (X. Jubier) (using actual lunar limb profile)
There is to right or wrong answer in the above calculations. They simply use different assumptions for calculating the duration of totality at a given point and they have different levels of accuracy. That being said, 3) and 4) are the most precise predictions. They are in near agreement in spite of the fact that they were calculated totally independent of each other with completely different software.
The information presented on this web page is based on material originally published in Five Millennium Canon of Solar Eclipses: -1999 to +3000 and Thousand Year Canon of Solar Eclipses.
Permission is freely granted to reproduce this data when accompanied by an acknowledgment:
"Eclipse Predictions by Fred Espenak (EclipseWise.com)"
Astronomical Almanac for 1986, Washington: US Government Printing Office; London: HM Stationery Office (1985).
Espenak, F., and Meeus, J., Five Millennium Canon of Solar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE), NASA Tech. Pub. 2006-214141, NASA Goddard Space Flight Center, Greenbelt, Maryland (2006).
Espenak, F., Thousand Year Canon of Solar Eclipses, Astropixels Publishing, Portal, Arizona (2014).
Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, Her Majesty's Nautical Almanac Office, London, 1974.
Herald, D., "Correcting predictions of solar eclipse contact time for the effects of lunar limb irregularities," J. Brit. Ast. Assoc. , 93, 6, 1983.
Meeus, J., Grosjean, C.C., and Vanderleen, W., Canon of Solar Eclipses, Pergamon Press, Oxford, United Kingdom (1966).