Astronomy & Celestial Mechanics
the sky, computed
Concept

Sundial — where it appears

An instrument reading apparent solar time from the direction of a shadow. It disagrees with a clock by up to sixteen minutes through the year, and reconciling the two is what the equation of time is for.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

What an eccentricity does to the shape, and what it does to the offset. The fractional flattening 1 − b/a and the fractional focal offset c/a, against eccentricity. The offset is first order in e and the flattening is second, so at Earth's e = 0.0167 the outline is 0.014% from a circle while the Sun sits 1.67% of the semi-major axis off the centre — a factor of 120.

An orbit can look exactly like a circle and still not be one

Earth's orbit departs from a circle by fourteen parts in a hundred thousand. The Sun's offset from its centre is a hundred and twenty times larger, and everything interesting is in the offset.

orbits · The ellipse The equation of time, and its two causes. The difference between a sundial and a clock over the year, in minutes, computed from Kepler's equation and the tilt. The eccentricity term has one cycle a year and the obliquity term has two; their sum runs from −14.2 to 16.4 minutes.

The Sun is a bad clock, by up to sixteen minutes

Solar noon and twelve o'clock are not the same instant, and the discrepancy runs through a fixed annual cycle. It has two causes, one from the shape of the orbit and one from the tilt of the axis.

sky · Equation of time An hour that was a twelfth of the daylight. The length of a daylight hour when the time from sunrise to sunset is divided into twelve, through a year, at Alexandria (31.2°N), Rome (41.9°N), London (51.5°N), Stockholm (59.3°N). At Alexandria the hour runs from 51 minutes at the winter solstice to 71 at the summer solstice; at Rome the hour runs from 46 minutes at the winter solstice to 76 at the summer solstice; at London the hour runs from 39 minutes at the winter solstice to 83 at the summer solstice; at Stockholm the hour runs from 30 minutes at the winter solstice to 93 at the summer solstice. The dashed line is sixty minutes, the length it has at both equinoxes everywhere. This is the hour of the ancient Mediterranean world and of medieval Europe until mechanical clocks: an hour defined by the Sun, which a sundial with suitably drawn lines reads exactly, and in which the equation of time does not exist, because nothing is being compared with a uniform clock. Sunrise, sunset and noon are each defined by the Sun, and the clock that would disagree with them had not been built.

An hour that stretched with the season

For most of recorded history an hour was a twelfth of the daylight — seventy-six minutes at a Roman midsummer and forty-six in midwinter — and a sundial read it exactly. In that system there was no equation of time, because nothing uniform was being compared with the Sun. The sixteen-minute correction became real only when the hour was made equal, and measurable only when clocks could keep time more steadily than the Sun by more than it.

sky · Equation of time

Named alongside it

The objects these essays reach for when they reach for this one.

Equation of timeAnalemmaDeclinationEccentricityEquinoxMean solar timeObliquitySolsticeApparent solar timeDay lengthFocusInsolation

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