The equation of time and the analemma

Talking points

• USNO glossary: "equation of time: the difference apparent solar time minus mean solar time." Apparent solar time is "based on the diurnal motion of the true Sun"; mean solar time on "the fictitious mean Sun, with uniform motion along the celestial equator". Universal Time is "loosely, mean solar time on the Greenwich meridian (previously referred to as Greenwich Mean Time)".
• Hughes, Yallop and Hohenkerk (1989) write it as E = GHA(apparent Sun) − GHA(mean Sun), or equally the right ascension of the mean Sun minus that of the apparent Sun, and note that older textbooks reverse the sign.
• NOAA: solar noon in minutes of UTC is 720 − 4 × longitude − E. At Greenwich on 3 November 2027, E = +16.49 minutes, so noon is at 11:43:30. JPL Horizons has the Sun's local hour angle at Greenwich passing zero at 11:43:33.5 UTC, and the US Naval Observatory lists the upper transit at 11:44 (to the minute).
• The viewer computes no equation of time for the globe. It turns the Earth by the IERS Earth rotation angle and points it at a Sun placed by JPL's elements, and on this date that puts the Sun over Greenwich at 11:43:28 UTC. Greenwich is marked at the Airy transit circle, latitude 51° 28′ 38″ N, longitude 0 by definition.
• This scene opens paused at 11:43:30 UTC, with the Sun-overhead marker and Greenwich on the same meridian to within a few seconds of time. The view is from the Sun's side at true scale.

Talking points

• USNO glossary: the rate of the true Sun's diurnal motion "undergoes seasonal variation caused by the obliquity of the ecliptic and by the eccentricity of the Earth's orbit".
• Derivation. The mean Sun has right ascension L, the Sun's mean longitude. The true Sun's ecliptic longitude is λ = L + 2e sin M + (5/4)e² sin 2M + …, M the mean anomaly. Its right ascension follows from tan α = cos ε tan λ, which expands as α = λ − tan²(ε/2) sin 2λ + ½ tan⁴(ε/2) sin 4λ − … (Hughes et al., eq. 5). So E = L − α ≈ −2e sin M + tan²(ε/2) sin 2L, Hughes et al.'s eq. 7, which they find within 18 seconds of the full expression at the present epoch.
• Hughes et al.: the ellipticity term has a period of a year, an amplitude under 8 minutes, and crosses zero at perihelion and aphelion; the obliquity term has an amplitude of about 9.9 minutes and crosses zero at the equinoxes and solstices, and is only quasi-sinusoidal because the seasons differ in length.
• The viewer's equationOfTimeMinutes (src/orbits/earthframe.js) is NOAA's calcEquationOfTime term for term: y sin 2L₀ − 2e sin M + 4ey sin M cos 2L₀ − ½y² sin 4L₀ − (5/4)e² sin 2M, with y = tan²(ε/2), turned into minutes at 4 minutes a degree. NOAA says its calculations are based on Meeus's Astronomical Algorithms.
• Constants: e = 0.0167 and obliquity 23.44° (NASA Earth Fact Sheet; the formula uses e = 0.016708634 and ε = 23.4393° at J2000). One radian of hour angle is 180/π × 4 = 229.18 minutes of time. Earth's spin axis is drawn in this view.

Talking points

• USNO: perihelion on 3 January 2027 at 02:33 UTC and aphelion on 5 July at 05:06 UTC. The scene opens at that perihelion.
• Hughes et al.: the ellipticity component crosses zero at perihelion and at aphelion, and its amplitude is under 8 minutes.
• The first-order term is −2e sin M, so its amplitude is 2e radians of hour angle.

Talking points

• From the viewer's equationOfTimeMinutes: minimum −14.22 minutes on 11 February, maximum +3.64 on 14 May, minimum −6.57 on 26 July, maximum +16.49 on 3 November; zero on 15 April, 13 June, 1 September and 25 December.
• JPL Horizons puts the Sun on the Greenwich meridian (local apparent hour angle zero) at 12:14:12 UTC on 11 February, 11:56:22 on 14 May, 12:06:33 on 26 July and 11:43:34 on 3 November, and within 6 seconds of 12:00 on each zero date. The US Naval Observatory's transits, to the minute, agree: 12:14, 11:56, 12:07, 11:44 and 12:00. The viewer's globe is within 15 seconds of Horizons on all eight dates (12:14:07, 11:56:13, 12:06:24, 11:43:28).
• Why the two big turns differ: at 12:00 UTC on 3 November the two first-order terms are +6.7 (eccentricity) and +9.8 (obliquity) minutes; on 11 February they are −4.7 and −9.6. Perihelion (3 January 2027, 02:33 UTC) comes 12.2 days after the December solstice (21 December 2026, 20:50 UTC), so the yearly term is not in step with the half-yearly one.
• This scene opens at 12:14 UTC on 11 February 2027, the Sun over Greenwich 14 minutes after mean noon.

Talking points

• Solar noon is 12:00 UTC minus the equation of time at Greenwich: 12:00 − 16.5 minutes = 11:43:30. JPL Horizons gives 11:43:33.5, the viewer's globe 11:43:28 and the US Naval Observatory 11:44.
• The check passes within 3 minutes of 11:43:30 with the clock paused, so mean noon (12:00) and the wrong sign (12:16:30) both fail. The clock under the task shows UTC to the minute, and the readout gives local mean time at Greenwich, which there is UTC.

Talking points

• 12:00 UTC + 14.2 minutes = 12:14:13 from the formula; JPL Horizons gives 12:14:12, the viewer's globe 12:14:07 and the US Naval Observatory 12:14.
• The check passes within 3 minutes of 12:14:13 with the clock paused.

Talking points

• At a fixed UT the subsolar point sits at latitude δ and longitude −E/4 degrees (E in minutes; earthframe.js gives the Sun's longitude as −15° × (UT − 12 h) − E/4). Over a year it draws the analemma on the globe; on the sky the same figure is drawn by the Sun at a fixed clock time.
• Shape: δ runs once up and down in a year (from −23.44° to +23.44°, NASA's obliquity), while the obliquity term of E runs twice. A curve whose x has twice the frequency of its y is a figure of eight with equal loops; the yearly eccentricity term added to x makes one loop larger than the other.
• Size in 2027: from −14.22 to +16.49 minutes is 30.7 minutes of time, 7.7° of hour angle east to west, against 46.9° north to south.
• In this scene, 12:00 UTC on 3 November 2027, the Sun-overhead marker is west of Greenwich by E/4.