In the early 1700s, Philippe de La Hire made a curious observation about Earth's umbra. The predicted radius of the shadow needed to be enlarged by about 1/41 in order to fit timings made during a lunar eclipse (La Hire 1707). Additional observations over the next two centuries revealed that the shadow enlargement was somewhat variable from one eclipse to the next. According to Chauvenet (1891):
Chauvenet adopted a value of 1/50, which has become the standard enlargement factor for lunar eclipse predictions published by many national institutes worldwide. The enlargement enters into the definitions of the penumbral and umbral shadow radii as follows.
penumbral radius: Rp = 1.02 * (0.998340 * Pm + Ss + Ps) (1-3)
umbral radius: Ru = 1.02 * (0.998340 * Pm - Ss + Ps) (1-4)
where: Pm = Equatorial horizontal parallax of the Moon,
Ss = Geocentric semi-diameter of the Sun, and
Ps = Equatorial horizontal parallax of the Sun.
The factor 1.02 is the enlargement of the shadows by 1/50. Earth's true figure approximates that of an oblate ellipsoid with a flattening of about 1/300. Furthermore, the axial tilt of the planet towards or away from the Sun throughout the year means the shape of the penumbral and umbral shadows vary although the effect is small. It is sufficient to use a mean radius of Earth at latitude 45° to approximate the departure from perfectly circular shadows. The Astronomical Almanac[1] uses a factor of 0.998340 to scale the Moon's equatorial horizontal parallax to account for this (i.e., 0.998340 ≅ 1 - 0.5 * 1/300).
In an analysis of 57 eclipses covering a period of 150 years, Link (1969) found a mean shadow enlargement of 2.3%. Furthermore, timings of crater entrances and exits through the umbra during four lunar eclipses from 1972 to 1982 (Table 1-3) closely support the Chauvenet value of 2%. From a physical point of view, there is no abrupt boundary between the umbra and penumbra. The shadow density actually varies continuously as a function of radial distance from the central axis out to the extreme edge of the penumbra. However, the density variation is most rapid near the theoretical edge of the umbra. Kuhl's (1928) contrast theory demonstrates that the edge of the umbra is perceived at the point of inflexion in the shadow density. This point appears to be equivalent to a layer in Earth's atmosphere at an altitude of about 120 to 150 km. The net enlargement of Earth's radius of 1.9% to 2.4% corresponds to an umbral shadow enlargement of 1.5% to 1.9%, in reasonably good agreement with the conventional value.
| Table 1-3. Umbral Shadow Enlargement from Craters Timings | |||
| Lunar Eclipse Date |
Crater Entrances % Enlargement |
Crater Exits % Enlargement |
Sky & Telescope Reference |
| 1972 Jan 30 | 1.69 [420] | 1.68 [295] | Oct 1972, p.264 |
| 1975 May 24 | 1.79 [332] | 1.61 [232] | Oct 1975, p.219 |
| 1982 Jul 05 | 2.02 [538] | 2.24 [159] | Dec 1982, p.618 |
| 1982 Dec 30 | 1.74 [298] | 1.74 [ 90] | Apr 1983, p.387 |
Some authorities dispute Chauvenet's shadow enlargement convention. Danjon (1951) notes that the only reasonable way of accounting for a layer of opaque air surrounding Earth is to increase the planet's radius by the altitude of the layer. This can be accomplished by proportionally increasing the parallax of the Moon. The radii of the umbral and penumbral shadows are then subject to the same absolute correction and not the same relative correction employed in the traditional Chauvenet 1/50 convention. Danjon estimates the thickness of the occulting layer to be 75 km and this results in an enlargement of Earth's radius and the Moon's parallax of about 1/85. Since 1951, the French almanac Connaissance des Temps has adopted Danjon's method for the enlargement Earth's shadows in their eclipse predictions as shown below.
penumbral radius: Rp = 1.01 * Pm + Ss + Ps (1-5)
umbral radius: Ru = 1.01 * Pm - Ss + Ps (1-6)
where: Pm = Equatorial horizontal parallax of the Moon,
Ss = Geocentric semi-diameter of the Sun,
Ps = Equatorial horizontal parallax of the Sun, and 1.01 ≅ 1 + 1/85 - 1/594.
The factor 1.01 combines the 1/85 shadow enlargement term with a 1/594 term[2] to correct for Earth's oblateness at a latitude of 45°.
Danjon's method correctly models the geometric relationship between an enlargement of Earth's radius and the corresponding increase in the size of its shadows. Meeus and Mucke (1979) and Espenak (2006) both use Danjon's method. However, the resulting umbral and penumbral eclipse magnitudes are smaller by approximately 0.006 and 0.026, respectively, as compared to predictions using the traditional Chauvenet convention of 1/50.
For instance, the umbral magnitude of the partial lunar eclipse of 2008 Aug 16 was 0.813 according to the Astronomical Almanac (2008) using Chauvenet's method, but only 0.806 according to Meeus and Mucke (1979) using Danjon's method.
Of course, the small magnitude difference between the two methods is difficult to observe because the edge of the umbral shadow is poorly defined. The choice of shadow enlargement method has the largest effect in certain limiting cases where a small change in magnitude can shift an eclipse from one type to another. For example, an eclipse that is barely total according to Chauvenet's method will be a large magnitude partial eclipse if calculated using Danjon's method. Table 1-4 shows five such instances where a shallow total eclipse calculated with Chauvenet's method becomes a deep partial eclipse with Danjon's.
| Table 1-4. Total (Chauvenet) vs. Partial (Danjon) Lunar Eclipses: 1501-3000 | |||
| Calendar Date | Umbral Magnitude (Chauvenet) |
Umbral Magnitude (Danjon) |
Magnitude Difference |
| 1540 Sep 16 | 1.0007 | 0.9947 | 0.0060 |
| 1856 Oct 13 | 1.0017 | 0.9960 | 0.0057 |
| 2196 Jul 10 | 1.0007 | 0.9960 | 0.0047 |
| 2413 Nov 08 | 1.0042 | 0.9993 | 0.0049 |
| 2669 Feb 08 | 1.0016 | 0.9951 | 0.0065 |
Similarly, small umbral magnitude partial eclipses using Chauvenet's method must be reclassified as penumbral eclipses of large penumbral magnitude when calculated with Danjon's method. A recent example was the eclipse of 1988 Mar 03, which was partial with an umbral magnitude of 0.0028 according to Chauvenet's method, but was penumbral with an umbral magnitude -0.0017[3] by Danjon's method. A similar case will occur on 2042 Sep 29. For a list of all such cases from 1501 through 3000, see Table 1-5.
| Table 1-5. Partial (Chauvenet) vs. Penumbral (Danjon) Lunar Eclipses: 1501-3000 | |||
| Calendar Date | Umbral Magnitude (Chauvenet) |
Umbral Magnitude (Danjon) |
Magnitude Difference |
| 1513 Sep 15 | 0.0036 | -0.0003 | 0.0039 |
| 1900 Jun 13 | 0.0012 | -0.0040 | 0.0052 |
| 1988 Mar 03 | 0.0028 | -0.0017 | 0.0045 |
| 2042 Sep 29 | 0.0027 | -0.0031 | 0.0058 |
| 2429 Dec 11 | 0.0020 | -0.0033 | 0.0053 |
| 2581 Oct 13 | 0.0017 | -0.0054 | 0.0071 |
| 2678 Aug 24 | 0.0007 | -0.0036 | 0.0043 |
| 2733 Aug 17 | 0.0037 | -0.0040 | 0.0077 |
| 2669 Feb 08 | 1.0016 | 0.9951 | 0.0065 |
Finally, in some cases, the shadow enlargement convention can make the difference between a shallow penumbral eclipse (Chauvenet) or no eclipse at all (Danjon). Table 1-6 lists nine small magnitude penumbral eclipses over a 500-year interval as determined using Chauvenet's method (Liu and Fiala, 1992). When the eclipse predictions are repeated using Danjon's method, no lunar eclipses are found on these dates.
| Table 1-6. Penumbral Lunar Eclipses (Chauvenet): 1801-2300 | |
| Calendar Date | Penumbral Magnitude (Chauvenet) |
| 1864 Apr 22 | 0.0237 |
| 1872 Jun 21 | 0.0008 |
| 1882 Oct 26 | 0.0059 |
| 1951 Feb 21 | 0.0068 |
| 2016 Aug 18 | 0.0165 |
| 2042 Oct 28 | 0.0077 |
| 2194 Mar 07 | 0.0085 |
| 2219 Apr 30 | 0.0008 |
| 2288 Feb 18 | 0.0204 |
Practically speaking, the faint and indistinct edge of the penumbral shadow makes the penumbral eclipse contacts (P1 and P4) completely unobservable. So the small magnitude differences discussed here are only of academic interest. Still, it is important to note which shadow enlargement convention is used because it is critical in comparing predictions from different sources. In the Five Millennium Catalog of Lunar Eclipses: -1999 to +3000 (NASA TP-2009-214173), Earth's penumbral and umbral shadow sizes have been calculated by using Danjon's enlargement method. Other sources using Danjon's method include Meeus and Mucke (1979), Espenak (2006) and Connaissance des Temps. Several sources using Chauvenet's method are Espenak (1989), Liu and Fiala (1992), and Astronomical Almanac.
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[1] The Astronomical Almanac is published annually by the Almanac Office of the U.S. Naval Observatory.
[2] Connaissance des Temps uses a value of 1/297 for Earth's flattening. At latitude 45°: 1/594 = 0.5 * 1/297.
[3] A negative umbral magnitude means that the Moon lies completely outside the umbral shadow and is, therefore, a penumbral eclipse.
Astronomical Almanac for 2008, Almanac Office, U.S. Naval Observatory, U.S. Government Printing Office, Washington; London: HM Stationery Office (2006).
Chauvenet, W.A., Manual of Spherical and Practical Astronomy, Vol. 1, edition of 1891, (Dover reprint, New York, 1960).
Danjon, A., "Les eclipses de Lune par la penombre en 1951," L'Astronomie, 65, 51Ð53 (1951).
Espenak, F., "Eclipses During 2007," Observer's Handbook - 2007, Royal Astronomical Society of Canada (2006).
Espenak, F, and Meeus, J. "Five Millennium Canon of Lunar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE)," NASA Tech. Pub. 2008-214172, NASA Goddard Space Flight Center, Greenbelt, Maryland (2009).
Espenak, F, and Meeus, J. "Five Millennium Catalog of Lunar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE)," NASA Tech. Pub. 2008-214173, NASA Goddard Space Flight Center, Greenbelt, Maryland (2009).
Kuhl, A., 1928, "Ueber den Einfluss des Grenzkontrastes auf Prazisionsmessungen," Physikalische Zeitschrift, 29, 1-34.
La Hire, P., Tabulae Astronomicae (Paris 1707).
Link, F., Eclipse Phenomena in Astronomy, Springer-Verlag, New York (1969).
Liu, B.-L., and A.D. Fiala, Canon of Lunar Eclipses 1500 B.C. - A.D. 3000, Willmann-Bell, Richmond, Virginia, p. 215 (1992).
Meeus, J., and H. Mucke, Canon of Lunar Eclipse: -2002 to +2526, Astronomisches Buro, Vienna, Austria (1979).
