Eclipse geometry and the Saros

Talking points

• USNO glossary: syzygy is a configuration where three or more bodies are approximately in a straight line, and also the times of new and full Moon. A node is either point where the plane of an orbit crosses a reference plane.
• NASA eclipse site: the Moon's orbit is inclined at a mean 5.145 degrees to Earth's. At the ascending node the Moon crosses the ecliptic from south to north, at the descending node from north to south.
• NASA: greatest eclipse at 03:23:22 TD, umbral magnitude 1.844. The Saros 130 catalogue gives gamma 0.0124 (NASA's key: the distance from the axis of Earth's shadow to the Moon's centre, in Earth equatorial radii), so the Moon passes almost through the middle of the shadow, and 1 hour 41 minutes 53 seconds of totality, the longest of its series. Lunar Saros 130 eclipses happen at the Moon's ascending node.
• The viewer works out the umbral magnitude from its own geometry and gets 1.834. The clock runs at five minutes a second.

Talking points

• NASA: 12 June 2029, partial, magnitude 0.458, Saros 118, seen from the Arctic, Scandinavia, Alaska, northern Asia and northern Canada. The Saros 118 catalogue gives gamma 1.2943: NASA's key defines gamma as the distance of the shadow cone's axis from Earth's centre in equatorial radii, so here the axis passes above the North Pole. The viewer's geometry gives 1.2964.
• NASA: if new Moon takes place within about 17 degrees of a node, a solar eclipse will be visible from somewhere on Earth. The actual value ranges from 15.39 to 18.59 degrees with the eccentricity of the Moon's and Earth's orbits.
• An estimate of that limit: seen from Earth's centre, some observer on Earth sees the discs touch if the Moon's ecliptic latitude β is less than the Moon's horizontal parallax minus the Sun's plus both semi-diameters. With NASA's mean distances and radii (Moon 384,400 km and 1,737.4 km, Sun 149.598 million km and 695,700 km, Earth's equatorial radius 6,378.1 km) those are 0.951, 0.002, 0.259 and 0.266 degrees, so β < 1.474 degrees. At a distance u from the node the Moon's latitude is given by sin β = sin i · sin u, so u < arcsin(sin 1.474° / sin 5.145°) = 16.7 degrees.
• The estimate treats the Moon as fixed at conjunction; NASA's figure comes from the full calculation.

Talking points

• USNO: full Moon on 26 June 2029 at 03:22 UT and new Moon on 11 July 2029 at 15:51 UT. The viewer finds the new Moon two minutes later.
• The check passes within 3 degrees of new Moon, about six hours either side.
• The view is side-on to the ecliptic at true scale, so the Moon's height above or below the ecliptic shows the tilt of its orbit.

Talking points

• NASA: the Sun moves along the ecliptic at 0.99 degrees a day and takes 34.5 days to cross the 34 degree wide eclipse zone round each node. The synodic month is 29.53 days, so every eclipse season has one solar eclipse and possibly two. The mid-points of eclipse seasons are 173.3 days apart.
• NASA: 11 July 2029, partial, magnitude 0.230, Saros 156. The Saros 156 catalogue gives gamma −1.4191: the axis passes below the South Pole. That series began with a partial eclipse on 1 July 2011, so this is its second eclipse.
• NASA (Periodicity of Solar Eclipses, table 5): two solar eclipses one lunation apart have Saros numbers that differ by 38, and 118 + 38 = 156.
• Both solar eclipses of this season are at the Moon's descending node (Saros 118 and 156 catalogues), the lunar eclipse between them at the ascending node (lunar Saros 130): at full Moon the Moon is on the far side of Earth from the Sun, at the opposite node. NASA's lunar catalogue marks it "pp": the solar eclipses before and after it are both partial.
• Intervals from NASA's TD times: 13.97 days from the June solar eclipse to the lunar one, 15.51 days from that to the July solar eclipse, 29.48 days between the two solar eclipses.

Talking points

• NASA: the lunar nodes rotate westward along the ecliptic at 0.05295 degrees a day; one complete rotation takes 18.6 years (6,793.48 days) with respect to the fixed stars. USNO glossary: the eclipse year, between successive passages of the Sun through the same node, is approximately 346.62 days.
• Derivation: against the stars the Sun moves east at 360/365.25636 degrees a day (JPL's sidereal year) and the node west at 360/6,793.48, so they meet again after 1/(1/365.25636 + 1/6,793.48) = 346.62 days. Half of that, 173.31, is NASA's 173.3 days between the mid-points of eclipse seasons.
• The same reasoning gives the draconic month, node to node: 1/(1/27.32166 + 1/6,793.48) = 27.2122 days, NASA's 27.21222.
• NASA: 6 February 2027 is annular (Saros 131, magnitude 0.928, 7 minutes 51 seconds at greatest eclipse) and 2 August 2027 total (Saros 136). They are 176.75 days apart, close to six lunations (177.18 days), and NASA's table 5 says six lunations on adds 5 to the Saros number: 131 + 5 = 136.
• NASA's Saros 131 catalogue: the annular eclipse of 26 January 2009 (07:59:45 TD, lunation 112) is one Saros before this one (16:00:48 TD, lunation 335), 6,585.33 days and 223 lunations earlier.

Talking points

• NASA (Periodicity of Solar Eclipses): synodic month 29.530589 days, anomalistic month (perigee to perigee) 27.554550 days, draconic month (node to node) 27.212221 days, their values for 2000.
• NASA: 242 draconic months are 6,585.3575 days and 239 anomalistic months 6,585.5375 days. 223 × 29.530589 = 6,585.3213 days; NASA prints 6,585.3223 next to a day count of 6,585 d 07 h 43 m, which matches 6,585.3213. That is about 18 years 11 days 8 hours.
• NASA: two eclipses one Saros apart happen at the same node, with the Moon at nearly the same distance from Earth and at the same time of year. 19 eclipse years, 6,585.78 days, are about 11 hours longer than the Saros, so the Sun is back near the same node too.
• The scene opens at the greatest eclipse of 2 August 2027, side-on to the ecliptic at true scale. Saros 136 eclipses are at the descending node (NASA's catalogue).

Talking points

• NASA: 22 July 2009, total, greatest eclipse 02:36:25 TD, Saros 136, magnitude 1.080, 6 minutes 39 seconds at greatest eclipse; totality over India, Nepal, China and the central Pacific.
• NASA's path table: greatest eclipse at 02:35:18.1 UT, 24° 13.3' N, 144° 07.1' E, with ΔT = 65.9 s, and the longest totality, 6 minutes 39.5 seconds, at 02:29:16 UT. The scene opens at greatest eclipse.
• NASA's Saros 136 catalogue numbers this eclipse's lunation 118 and the 2027 eclipse's 341 (lunations counted from 6 January 2000): 223 apart. Gamma grows from 0.0698 to 0.1421 between them, as the Moon moves north with each eclipse of the series.
• The viewer's calendar runs from 2026, so this eclipse cannot open in eclipse mode. The shadow here comes from the same geometry, whose axis meets the ground at 24.18° N, 143.89° E at this moment, about 0.2 degrees from NASA's point, with gamma 0.0701.

Talking points

• NASA's path table: greatest eclipse at 10:06:37.7 UT, 25° 30.3' N, 33° 11.0' E, ΔT = 71.7 s, with 6 minutes 22.6 seconds of totality there. The viewer's calendar opens it at 10:06:40 UTC, NASA's TD time less 69 seconds.
• NASA's Saros 136 catalogue, greatest-eclipse longitudes: 105.2° W in 1991, 144.1° E in 2009, 33.2° E in 2027, 78.5° W in 2045, each about 111 degrees farther west. NASA: the extra third of a day means Earth turns an extra 120 degrees or so each cycle, and the series returns to about the same region every three Saros (the Exeligmos, about 54 years 34 days).
• NASA: even-numbered solar Saros series are at the descending node and their gamma increases: 0.0698, 0.1421 and 0.2116 for 2009, 2027 and 2045.
• The lunar eclipses repeat too. The total lunar eclipse of 31 December 2028, in the viewer's calendar, is one Saros after that of 21 December 2010: both are lunar Saros 125, with umbral magnitudes 1.246 and 1.256, 6,585.36 days apart. Five 29 Februarys fall between them, so the calendar gap is 18 years 10 days.
• The clock runs at a minute a second.