Solar Time and the Equation of Time #

Time in everyday life is ultimately based on the Sun. But the Sun is an imperfect clock: its apparent motion across the sky is not uniform. This article explains why, defines the two kinds of solar time, derives the equation of time that connects them, and shows why this matters for chart calculation and for understanding the relationship between clock time and astronomical time.

Apparent Solar Time #

Apparent solar time is the time directly read from the Sun’s position. The apparent solar day is the interval between two successive transits of the Sun across the observer’s meridian — two consecutive solar noons. Apparent solar noon is the moment when the Sun crosses the upper meridian and reaches its highest altitude for the day.

A sundial measures apparent solar time. The shadow rotates at a rate determined by the Sun’s actual motion, and the hour marks on the sundial correspond to 1/24 of the apparent solar day.

The problem is that apparent solar days are not all the same length. Some solar days are up to 30 seconds longer than 24 hours; others are up to 20 seconds shorter. Over weeks and months, these differences accumulate to shifts of many minutes. A sundial and a mechanical clock, synchronized at one moment, will gradually drift apart, reconciling only at certain times of year.

Why the Apparent Solar Day Varies #

Two independent effects cause the non-uniformity of apparent solar time.

The Eccentricity Effect #

The Earth’s orbit around the Sun is an ellipse, not a circle. By Kepler’s second law, the Earth moves faster near perihelion (early January) and slower near aphelion (early July). This means the Sun’s apparent eastward motion along the ecliptic — its daily shift against the stars — varies through the year. Near perihelion, the Sun moves about 1.019° per day; near aphelion, about 0.953° per day.

A faster-moving Sun takes longer to “lap” the meridian (the Earth must rotate slightly more than 360° to bring the Sun back to the meridian), so apparent solar days near perihelion are longer than average. Near aphelion, they are shorter.

The eccentricity contribution to the equation of time has a period of one year (one anomalistic year, to be precise) and an amplitude of about 7.7 minutes.

The Obliquity Effect #

Even if the Earth’s orbit were perfectly circular, apparent solar days would still vary because of the tilt of the ecliptic relative to the equator. The Sun moves along the ecliptic, but time is measured by the Sun’s motion projected onto the celestial equator (because the equator, not the ecliptic, is the plane of the Earth’s rotation and therefore of timekeeping).

Near the equinoxes, the ecliptic crosses the equator at an angle of $\varepsilon \approx 23.4°$. The Sun’s motion along the ecliptic has a substantial north-south component that does not contribute to its east-west (right ascension) progress. The Sun’s right ascension advances more slowly than its ecliptic longitude.

Near the solstices, the ecliptic is parallel to the equator (both are at the same declination extreme). The Sun’s motion is entirely east-west, and 1° of ecliptic longitude translates to more than 1° of right ascension (because the ecliptic is compressed onto the equator at higher declinations: $\Delta\alpha \approx \Delta\lambda / \cos\varepsilon$ when $\beta = 0$). The Sun’s right ascension advances faster than its ecliptic longitude.

The obliquity contribution has a period of half a year and an amplitude of about 9.9 minutes.

Mean Solar Time #

To obtain a uniform time scale, astronomers invented the mean Sun: a fictitious body that moves along the celestial equator at a constant rate, completing one full circuit in one tropical year. The mean solar day is the interval between successive transits of the mean Sun. It is exactly 24 hours of mean solar time, by definition.

Mean solar time is the hour angle of the mean Sun plus 12 hours (the “+12” shifts the day boundary from noon to midnight). Civil clocks keep mean solar time — or rather, they keep a standardized version of it (UTC), which is uniform to high precision.

The mean Sun is a mathematical construction. Its right ascension increases at a constant rate:

$$ \alpha_{\text{mean}} = \alpha_0 + \frac{360°}{T_{\text{year}}} \times t $$

where $T_{\text{year}}$ is the length of the tropical year in mean solar days, and $t$ is time elapsed from some epoch. The tropical year is approximately 365.2422 mean solar days, so the mean Sun advances about $0.98565°$ per day, or $3.94 \text{ min}$ of RA per day.

The Equation of Time #

The equation of time (EoT) is the difference between apparent solar time and mean solar time:

$$ \text{EoT} = \text{apparent solar time} - \text{mean solar time} $$

Equivalently, it is the difference in right ascension between the true Sun and the mean Sun, converted to time:

$$ \text{EoT} = \alpha_{\text{true Sun}} - \alpha_{\text{mean Sun}} \quad \text{(converted to minutes of time)} $$

When the EoT is positive, the true Sun is ahead of the mean Sun (apparent noon occurs before mean noon; the sundial is “fast”). When negative, the true Sun is behind (the sundial is “slow”).

Approximate Formula #

An approximate equation of time, accurate to about 1 minute, can be computed from the Sun’s mean anomaly $M$ and ecliptic longitude $\lambda$. Let $D$ be the day number of the year (1 = January 1), then:

$$ B = \frac{360°}{365.25}(D - 81) $$

$$ \text{EoT} \approx 9.87 \sin 2B - 7.53 \cos B - 1.5 \sin B \quad \text{(minutes)} $$

The $9.87\sin 2B$ term is the obliquity contribution (period ≈ 182.6 days). The $-7.53\cos B - 1.5\sin B$ terms represent the eccentricity contribution (period ≈ 365.25 days).

Precise Computation #

For precise work, the equation of time is computed from the Sun’s true geocentric right ascension $\alpha_\odot$ and the mean Sun’s right ascension $\alpha_{\text{mean}}$:

$$ \text{EoT} = \alpha_\odot - \alpha_{\text{mean}} $$

The Sun’s true right ascension is obtained from its ecliptic longitude (computed from orbital mechanics) via the ecliptic-to-equatorial transformation:

$$ \tan\alpha_\odot = \cos\varepsilon \tan\lambda_\odot $$

where $\lambda_\odot$ is the Sun’s true ecliptic longitude and $\varepsilon$ is the obliquity. Since the Sun’s ecliptic latitude is zero, the transformation simplifies to this single equation.

The mean Sun’s right ascension is a linear function of time, defined to increase uniformly. The difference $\alpha_\odot - \alpha_{\text{mean}}$ gives the EoT in angular units; multiply by $4 \text{ min/degree}$ (or $24 \times 60 / 360$) to get minutes of time.

Care must be taken with the wraparound: both $\alpha_\odot$ and $\alpha_{\text{mean}}$ are modular quantities (0° to 360°), and their difference must be normalized to $(-180°, +180°]$ before conversion.

Annual Pattern #

The equation of time varies between approximately $-14.2$ minutes (around February 12) and $+16.4$ minutes (around November 3). It passes through zero four times per year: around April 15, June 13, September 1, and December 25 (dates vary slightly year to year).

The extreme values correspond to the constructive interference of the two sinusoidal components:

The February minimum is where both the eccentricity component (Sun past perihelion, moving fast) and the obliquity component (Sun near the equinox, RA advancing slowly) combine to make the true Sun lag the mean Sun.
The November maximum is where both components combine to make the true Sun lead.

The Analemma #

The analemma is the figure-eight pattern traced by the Sun’s position in the sky at the same civil clock time on successive days throughout the year. If you photograph the Sun from the same location at the same clock time every few days for a year, the images form an elongated figure-eight.

The north-south extent of the analemma reflects the Sun’s changing declination (the $\pm 23.4°$ swing between solstices). The east-west extent reflects the equation of time: the Sun is east or west of its mean position by the EoT, which reaches up to about $\pm 4°$ in azimuth at the horizon.

The analemma is asymmetric (the lobes are unequal) because the eccentricity and obliquity contributions have different periods and amplitudes. The northern lobe (June–December) is larger than the southern lobe (December–June).

Solar Time and Chart Calculation #

The equation of time connects civil clock time to the Sun’s actual position, but for chart calculation, the primary bridge is different: civil time → UTC → UT1 → sidereal time. The equation of time does not appear explicitly in this pipeline. It is implicitly embedded in the Sun’s computed ecliptic longitude: when the chart software computes the Sun’s longitude from the ephemeris, it already reflects the Sun’s non-uniform motion.

Where the equation of time matters in practice:

Local Mean Time (LMT): before the adoption of standard time zones, civil time was local mean time — mean solar time at the observer’s geographic longitude. For births before the local adoption of time zones (often before the late 19th century), the recorded time is LMT, and converting it to UTC requires only the geographic longitude, not the equation of time. However, if the recorded time is local apparent time (sundial time, which occurred in some regions before LMT was adopted), then the equation of time is needed to convert it to LMT and thence to UTC. See Time Zones and Historical Time.
Sundial conversions: if a historical birth time is recorded from a sundial (apparent solar time), convert to mean solar time by subtracting the EoT for that date, then to UT by subtracting the longitude correction.
Sun–MC relationship: the MC equals the local sidereal time (expressed as right ascension). The Sun’s right ascension equals the sidereal time at apparent solar noon (when the Sun is on the meridian). The equation of time is the offset between apparent noon and mean noon — so it also measures the offset between the Sun’s right ascension and the mean sidereal time at mean noon.

Worked Example #

Date: February 12. Location: longitude 10°E. Recorded sundial time: 14:30 apparent solar time.

Step 1 — Look up the equation of time for February 12: EoT ≈ $-14.2$ minutes.

Step 2 — Convert to local mean time:

$$ \text{LMT} = \text{Apparent solar time} - \text{EoT} = 14\text{h},30\text{m} - (-14.2\text{m}) = 14\text{h},44.2\text{m} $$

(Subtracting a negative EoT adds the correction: the sundial is “slow” by 14.2 minutes, so mean time is ahead.)

Step 3 — Convert LMT to UT:

$$ \text{UT} = \text{LMT} - \frac{\text{longitude}}{15°/\text{h}} = 14\text{h},44.2\text{m} - \frac{10°}{15°/\text{h}} = 14\text{h},44.2\text{m} - 0\text{h},40\text{m} = 14\text{h},04.2\text{m} $$

The longitude correction is $10°/15 = 0.667\text{ h} = 40 \text{ min}$, eastward, so subtract from LMT.

This UT can then enter the standard chart calculation pipeline.

References #

Meeus, J. (1998). Astronomical Algorithms, 2nd ed. Willmann-Bell. Chapter 28.
Explanatory Supplement to the Astronomical Almanac, 3rd ed. (2013). University Science Books. Chapter 6.
Hughes, D. W. et al. (1989). “The equation of time.” Quarterly Journal of the Royal Astronomical Society, 30, 51–73.
Dershowitz, N., & Reingold, E. M. (2008). Calendrical Calculations, 3rd ed. Cambridge University Press. Chapter 14.