Equation of Time
Difference between sundial (apparent) and clock (mean) time.
Formula
Variables
Example
Around Nov 3 sundials run ~16 min ahead of clocks
How it works
The equation of time estimates how far the Sun’s "apparent" solar time, read from a sundial, drifts from uniform mean time kept by clocks. The mismatch happens because Earth’s orbit is slightly elliptical and its axis is tilted, so the Sun does not move across the sky at a perfectly steady rate throughout the year. This formula combines those two effects into a practical approximation in minutes.
When to use it
Use it when converting between sundial time and clock time, or when studying why local solar noon does not always happen at exactly 12:00 on a clock. It is also useful in astronomy, navigation history, and climate/solar-energy calculations that depend on the Sun’s apparent position.
Worked example, step by step
Find the equation of time on day N = 200. 1) Compute B: B = 360°(N - 81)/364 B = 360°(200 - 81)/364 B = 360°(119)/364 B ≈ 117.74° 2) Substitute into the formula: E ≈ 9.87 sin(2B) - 7.53 cos(B) - 1.5 sin(B) E ≈ 9.87 sin(235.48°) - 7.53 cos(117.74°) - 1.5 sin(117.74°) 3) Evaluate the trig terms: sin(235.48°) ≈ -0.822 cos(117.74°) ≈ -0.467 sin(117.74°) ≈ 0.884 4) Multiply: E ≈ 9.87(-0.822) - 7.53(-0.467) - 1.5(0.884) E ≈ -8.11 + 3.51 - 1.33 E ≈ -5.93 minutes 5) Interpret the result: E ≈ -5.9 min means apparent solar time is about 5.9 minutes behind mean clock time on day 200.
Common mistakes to avoid
- Using the day number from the wrong year or counting from zero instead of from 1
- Forgetting that E is in minutes, not hours
- Using radians in the formula when B is intended in degrees
Did You Know?
Plotting the Sun at the same clock time all year traces a figure-eight in the sky called the analemma.
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