Synodic and sidereal periods

Talking points

• NASA Moon Fact Sheet: revolution period 27.3217 days, synodic period 29.53 days. The fact sheet notes call the first the sidereal period, the time to orbit once relative to the fixed background stars.
• USNO glossary: the sidereal month is the mean period of revolution with respect to the background stars, approximately 27.322 days; the synodic month is the period between successive new moons as seen from the geocentre, approximately 29.531 days. A lunation is the time between two consecutive new moons.
• NASA eclipse site: the mean synodic month is 29.53059 days, nearly 2.21 days longer than the sidereal month (27.32166 days). After one revolution with respect to the stars, the Moon must go a little further to catch up to the same position relative to the Sun and Earth.
• USNO puts this new Moon at 20:24 UTC on 7 January 2027; the viewer's ephemeris finds it two minutes later. The view is at true scale, from above the ecliptic.

Talking points

• Derivation: the angle between the two directions grows at ω₁ − ω₂ = 360/P₁ − 360/P₂ degrees a day. It returns to the same value after a gain of 360 degrees, so S = 360/(ω₁ − ω₂), that is 1/S = 1/P₁ − 1/P₂, with P₁ the shorter period. In general 1/S = |1/P₁ − 1/P₂|.
• For the Moon the slower turn is the Sun's direction as seen from Earth, which goes round once a sidereal year (NASA: 365.256 days). The Moon gains 360/27.3217 − 360/365.256 = 13.1763 − 0.9856 = 12.1907 degrees a day on the Sun.
• USNO's glossary defines a planet's synodic period the same way: the mean interval between successive conjunctions of a pair of planets, as observed from the Sun.
• In one sidereal month Earth goes 27.3217 × 0.9856 = 26.93 degrees round the Sun; at 12.19 degrees a day relative to the Sun, the Moon needs 2.21 more days to line up again.
• The clock runs at six hours a second.

Talking points

• The check passes within 1.5 degrees of new Moon. Near this new Moon the Moon gains about 11 degrees a day on the Sun, so that is about three hours either side, and every pause inside it gives a lunation between 29.68 and 29.95 days.
• USNO: new Moon on 7 January 2027 at 20:24 UTC and on 6 February 2027 at 15:56 UTC. The viewer finds the second two minutes later.

Talking points

• NASA eclipse site: the length of a lunation varies from its mean by up to seven hours. When new Moon occurs near perigee the lunation is shortest, and near apogee it is longest: the Moon has further to go in its orbit to reach new Moon, and moves slower there.
• NASA: Earth's elliptical orbit also matters, giving shorter lunations near aphelion and longer ones near perihelion. USNO: Earth's perihelion was on 3 January 2027 at 02:33 UTC.
• JPL Horizons, the Moon's geocentric distance: apogee on 7 January 2027 at 08:07 UT (406,607 km), half a day before the new Moon, and on 3 February 2027 at 13:30 UT (406,189 km); 402,711 km at the new Moon of 6 February.
• Over the five thousand years of its catalogue NASA finds lunations from 29.26574 to 29.84089 days. When the Moon's perigee and Earth's perihelion point the same way, they run from 29.273 to 29.820 days.

Talking points

• USNO glossary: opposition is when two bodies have apparent ecliptic longitudes that differ by 180 degrees as seen from a third body, usually tabulated as seen from Earth.
• JPL Horizons: Mars reaches opposition on 19 February 2027 at 15:50 UT, 101.4 million km from Earth. The viewer's calendar (src/sky/events.js) has it at 15:44; it uses JPL's approximate elements, with a nominal error of 40 arcseconds for Mars and 20 for the Earth-Moon barycentre.
• NASA Mars Fact Sheet: sidereal orbit period 686.980 days, synodic period 779.94 days.
• This view is at true scale, from above the ecliptic. Both planets go round anticlockwise from here, Earth faster.

Talking points

• The check reads the date through the Sun's longitude of date: it passes within 4 degrees of 330.8 degrees, the Sun's longitude at this opposition, which is about four days either side (the Sun moves 1.01 degrees a day in February). There is no check on Mars itself, so the question asks for the interval.
• JPL Horizons: the previous opposition was on 16 January 2025 at 02:38 UT and this one is on 19 February 2027 at 15:50 UT, 764.55 days apart. The viewer's calendar has 02:37 and 15:44.

Talking points

• JPL Horizons, Mars oppositions (UT): 13 October 2020 23:25, 8 December 2022 05:41, 16 January 2025 02:38, 19 February 2027 15:50, 25 March 2029 07:49, 4 May 2031 12:03, 28 June 2033 01:29, 15 September 2035 19:38. The gaps: 785.26, 769.87, 764.55, 764.67, 770.18, 785.56 and 809.76 days.
• At opposition Mars lies in the direction of Earth's own heliocentric longitude, the Sun's longitude plus 180 degrees: 21.1, 76.1, 116.2, 150.8, 185.0, 223.8, 276.7 and 352.8 degrees. (These are longitudes of date.) So between oppositions Mars goes once round plus 55.0, 40.1, 34.6, 34.2, 38.9, 52.9 and 76.1 degrees, and the gap grows with that extra arc.
• Mars's longitude of perihelion is 336.04 degrees in NASA's J2000 mean elements, so aphelion is at 156.04 degrees (precession since 2000 moves both by under half a degree). The shortest gaps lie round aphelion, where Mars is slowest, and the longest, into 2035, runs through perihelion. Horizons has Mars at aphelion, 249.2 million km out, around 5 March 2027.
• The clock runs at a week a second.

Talking points

• NASA Venus Fact Sheet: sidereal orbit period 224.701 days, synodic period 583.92 days. NASA Mercury Fact Sheet: 87.969 and 115.88 days, and the formula gives 115.88 too.
• Five synodic periods of Venus, 2,919.6 days, fall 2.4 days short of eight sidereal years (8 × 365.256 = 2,922.0 days), so the same Earth, Venus and Sun arrangement comes back at nearly the same time of year every eight years.
• JPL's Basics of Space Flight puts Venus launch opportunities about every 19 months and minimum-energy opportunities to Mars about every 25 months. In months of 30.44 days, 583.92 days is 19.2 months and 779.94 days is 25.6.