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Equation of Time Calculator

Calculate the Equation of Time (EoT) - the difference between apparent solar time (sundial time) and mean solar time (clock time). This varies throughout the year due to Earth's orbital eccentricity and axial tilt.

Equation of Time

Equation of Time

-6m 6s

-6.103 minutes

Interpretation

Sundials run 6.1 minutes SLOW (sun behind mean time)

Contributing Factors

Eccentricity Effect

-1.639 min

Elliptical orbit

Obliquity Effect

7.738 min

Axial tilt (23.5 deg)

Solar Declination

21.1773 deg

Day of Year

199

Annual Extremes

Maximum Fast (Sundial ahead)

+16.38 min

Around day 305 (early November)

Maximum Slow (Sundial behind)

-14.26 min

Around day 45 (mid February)

When EoT = 0 (Sundial matches clock)

~April 15
~June 13
~September 1
~December 25

About the Equation of Time

The Equation of Time is the difference between apparent solar time and mean solar time. Two factors cause this variation: (1) Earth's elliptical orbit causes it to move faster at perihelion (January) and slower at aphelion (July), and (2) the 23.5 degree tilt of Earth's axis means the sun's apparent motion along the ecliptic projects onto the celestial equator at varying rates. The combination produces a maximum variation of about +/- 16 minutes throughout the year.

What Is the Equation of Time?

The Equation of Time (EoT) is the difference in minutes between apparent solar time β€” the time shown by a sundial β€” and mean solar time β€” the uniform time kept by clocks. Throughout the year, a sundial can run up to 16 minutes ahead or 14 minutes behind a clock, following a characteristic figure-eight pattern known as the analemma.

This discrepancy arises from two independent astronomical effects: the elliptical shape of Earth's orbit and the tilt of Earth's rotational axis relative to its orbital plane. Neither effect alone causes the full variation; their sum produces the Equation of Time's distinctive double-humped annual curve.

The concept has been known since antiquity, but was mathematically formalized in the 17th century. Before mechanical clocks, sundials were the primary timekeeping instrument, and the Equation of Time was essential knowledge for navigators, astronomers, and anyone needing to synchronize with civil time. Today it remains important in solar energy engineering, satellite communications, and precision astronomy.

Equation of Time Formula (NOAA Method)

This calculator uses the NOAA (National Oceanic and Atmospheric Administration) algorithm, which expresses the Equation of Time as a Fourier series based on the fractional year angle gamma.

Equation of Time (NOAA)

EoT = 229.18 Γ— (0.000075 + 0.001868Β·cos(Ξ³) βˆ’ 0.032077Β·sin(Ξ³) βˆ’ 0.014615Β·cos(2Ξ³) βˆ’ 0.040849Β·sin(2Ξ³))

Where:

  • EoT= Equation of Time in minutes (positive = sundial fast, negative = sundial slow)
  • Ξ³= Fractional year angle in radians: (2Ο€/365) Γ— (dayOfYear βˆ’ 1)
  • 229.18= Scaling factor derived from Earth's mean solar day length in minutes
  • cos/sin terms= Fourier coefficients capturing orbital eccentricity and axial tilt effects

The Two Physical Causes

The Equation of Time is the algebraic sum of two independent effects:

Cause Amplitude Explanation
Orbital EccentricityΒ±7.7 minutesEarth moves faster near perihelion (January) and slower near aphelion (July), so the sun appears to move at variable speed along the ecliptic
Axial ObliquityΒ±9.9 minutesEarth's 23.5Β° tilt means the sun's motion along the ecliptic projects onto the equator at varying rates; near the solstices the east-west component is smaller

The combined effect reaches its maximum positive value (sundial ~16 min fast) in early November and its maximum negative value (sundial ~14 min slow) in mid-February. There are four dates per year when the Equation of Time equals zero and a sundial perfectly matches a clock: approximately April 15, June 13, September 1, and December 25.

How to Use This Calculator

  1. Set Today or Enter a Date: Click "Set Today" to load the current date, or manually enter a Year, Month, and Day.
  2. Read the Equation of Time: The main result shows the EoT in minutes and seconds, with sign β€” positive means the sundial is ahead of the clock, negative means it is behind.
  3. Read the Interpretation: The plain-language box describes how to correct a sundial reading to obtain mean (clock) time.
  4. Review Contributing Factors: The eccentricity and obliquity components show how much each cause contributes on that specific date. Solar declination indicates how far north or south of the equator the sun is.
  5. Annual Extremes: The extremes panel shows the peak fast and slow values and the approximate day of year they occur.

Real-World Applications

Solar energy engineers use the Equation of Time to predict the exact moment of solar noon β€” the time when a solar panel receives peak irradiance. Without this correction, a fixed-time tracking algorithm could miss solar noon by up to 16 minutes, significantly reducing energy yield. Concentrated solar power plants and high-efficiency PV trackers require accurate EoT calculations for optimal positioning.

Navigators before GPS relied on the Equation of Time to convert observations of the sun's altitude (using a sextant) into accurate longitude measurements. The Nautical Almanac has included Equation of Time tables for over 250 years.

Architects and urban planners use the Equation of Time β€” combined with latitude and longitude β€” to predict exactly where shadows fall throughout the day and year, informing decisions about window placement, shading devices, and daylighting strategies in buildings.

Worked Examples

Equation of Time on November 3

Problem:

What is the Equation of Time on approximately day 307 (November 3)?

Solution Steps:

  1. 1Ξ³ = (2Ο€/365) Γ— (307 βˆ’ 1) = (2Ο€/365) Γ— 306 β‰ˆ 5.267 radians
  2. 2cos(Ξ³) β‰ˆ 0.496, sin(Ξ³) β‰ˆ βˆ’0.868, cos(2Ξ³) β‰ˆ βˆ’0.508, sin(2Ξ³) β‰ˆ βˆ’0.862
  3. 3EoT = 229.18 Γ— (0.000075 + 0.001868Γ—0.496 βˆ’ 0.032077Γ—(βˆ’0.868) βˆ’ 0.014615Γ—(βˆ’0.508) βˆ’ 0.040849Γ—(βˆ’0.862))
  4. 4EoT = 229.18 Γ— (0.000075 + 0.000927 + 0.027843 + 0.007424 + 0.035211) = 229.18 Γ— 0.07148 β‰ˆ +16.38 minutes

Result:

Around November 3, the Equation of Time is approximately +16 minutes β€” the annual maximum. A sundial reads about 16 minutes ahead of clock time.

Equation of Time on February 12

Problem:

What is the approximate EoT around day 43 (February 12), the annual minimum?

Solution Steps:

  1. 1Ξ³ = (2Ο€/365) Γ— (43 βˆ’ 1) β‰ˆ 0.723 radians
  2. 2The eccentricity and obliquity effects are both near their most negative alignment on this date
  3. 3EoT β‰ˆ βˆ’14.3 minutes at the annual minimum (exact value depends on the year)

Result:

Around February 12, the Equation of Time reaches its annual minimum of about βˆ’14 minutes. A sundial reads approximately 14 minutes behind clock time.

Zero-Crossing Near Christmas

Problem:

On approximately what date does the EoT cross zero in late December?

Solution Steps:

  1. 1The EoT crosses zero four times per year: ~April 15, ~June 13, ~September 1, ~December 25
  2. 2Near December 25, the positive eccentricity effect and negative obliquity effect cancel each other exactly
  3. 3On this day, a sundial shows exactly the same time as a clock

Result:

Around December 25, the Equation of Time equals zero β€” one of four annual dates when sundials perfectly match clock time.

Tips & Best Practices

  • βœ“On the four zero-crossing dates (~April 15, June 13, September 1, December 25), a sundial reads exactly the same as a clock.
  • βœ“The largest correction occurs in early November (+16 min) and mid-February (βˆ’14 min) β€” useful for calibrating fixed-mount solar panels.
  • βœ“Solar noon in your local time zone is NOT 12:00 PM β€” it differs by up to 16 minutes (EoT) plus your longitude offset from the time zone center.
  • βœ“The eccentricity component peaks in early February and early August; the obliquity component peaks near the solstices (June, December).
  • βœ“Sundial correction = subtract EoT if positive, add if negative β€” then also adjust for your local longitude and DST.
  • βœ“The Equation of Time is factored into solar tracker algorithms in concentrated solar power (CSP) plants to maximize energy yield.

Related Calculators

Analemma Calculator

Visualize the analemma and sun position throughout the year

Sunrise/Sunset Calculator

Find sunrise and sunset times for any date and location

Solstice Calculator

Calculate solstice dates and times for any year

Frequently Asked Questions

In older mathematical usage, 'equation' meant a correction or adjustment β€” not an algebraic equation in the modern sense. The Equation of Time is the correction needed to equate (reconcile) sundial time with mean clock time. The terminology dates from 17th-century astronomy.
To convert a sundial reading to clock time: if the EoT is positive (sundial fast), subtract the EoT value from the sundial reading. If the EoT is negative (sundial slow), add the absolute value. Also remember to account for your longitude offset from the center of your time zone and for daylight saving time.
The analemma is the figure-eight curve traced by the sun's position in the sky at the same clock time each day throughout the year. Its east-west component reflects the Equation of Time (the sun's time offset from mean noon), while its north-south component reflects the solar declination (the sun's height above the equator). You can photograph the analemma by taking a picture of the sky at exactly the same clock time every week for a year.
The Equation of Time is nearly identical from year to year, varying by only a few seconds due to leap year adjustments and small long-term changes in Earth's orbital parameters. For most practical purposes the same EoT value can be used for a given calendar date regardless of year.
This calculator provides both. The NOAA Fourier series formula (using Ξ³) is derived empirically from observation data and is accurate to within a few seconds for dates within a few centuries of J2000. The orbital elements formula (using angle B referenced to March 21) gives a simpler approximation but is slightly less accurate, typically off by 20–30 seconds. Both are far more than sufficient for solar energy and navigation applications.

Sources & References

Last updated: 2026-06-06

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: Standard Mathematical References

by Various

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.

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