Solartime

Solartime is a web application that displays key solar timing information based on your current location. Specifically, it provides:

• Local Solar Time – the actual time based on the Sun’s position in your sky (as a sundial would read).
• Sunrise and Sunset Times – the exact local clock times when the Sun rises and sets each day.
• Day Length – how long the Sun stays above the horizon each day.
• Day Length Change – the difference in daylight duration from one day to the next.

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How Solar Time Is Calculated

Solar time (apparent solar time) is computed from the current UTC time by applying two corrections:

1. Equation of Time (EOT)
2. Longitudinal Offset

The combined formula is:

$$\text{Solar Time} = \text{UTC} + \Delta t_{EOT} + \Delta t_{longitude}$$

Equation of Time

Solartime calculates the Equation of Time from the Julian Day (JD) using:

$$D = 0.01720197\,(JD - JD_{2000}) + 6.24004077$$

where $JD_{2000}$ is the Julian Day of 2000‑01‑01 (UTC).

Then the EOT in hours is:

$$\Delta t_{EOT} = \frac{-7.659\,\sin(D) \;+\; 9.863\,\sin\bigl(2D + 3.5932\bigr)}{60}$$

• The numerator is in minutes, so division by 60 converts to hours.
• Positive $\Delta t_{EOT}$ means solar time runs ahead of mean time; negative means it lags.

EOT on Wikipedia

Longitudinal Offset

The longitudinal offset corrects for your position east or west of the prime meridian (UTC reference). Earth rotates 360° in 24 h (15°/h), so:

$$\Delta t_{longitude} = \frac{\text{Longitude}_{local}}{15}\quad[\text{hours}]$$

Longitudes east are positive (solar time ahead of UTC); west are negative.

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How Sunrise and Sunset Are Calculated

Solartime finds sunrise and sunset by solving for the Sun’s hour angle when its center reaches an altitude of (-0.833°) (accounts for refraction and solar radius) via:

$$\cos\omega_0 = \frac{\sin(a) - \sin(\phi)\sin(\delta)}{\cos(\phi)\cos(\delta)}$$

• $\phi$ = observer’s latitude
• $\delta$ = solar declination (see below)
• $a = -0.833°$

Solar Declination

Solar declination $\delta$ is computed from $JD$ as follows:

1. Compute days since J2000:
   $n = JD - JD_{2000}$
2. Mean anomaly in radians:
   $g = \bigl(357.529 + 0.98560028\,n\bigr)\,\times\frac{\pi}{180}$
3. Ecliptic longitude in radians:
   $L = \bigl(280.459 + 0.98564736\,n + 1.915\sin g + 0.020\sin(2g)\bigr)\,\times\frac{\pi}{180}$
4. Obliquity:
   $\varepsilon = 23.439°\times\frac{\pi}{180}$
5. Declination:
   $\delta = \arcsin\bigl(\sin\varepsilon\,\sin L\bigr)$

This yields $\delta$ in radians for accurate hour‑angle solution.

Declination of the Sun on Wikipedia

Iterative Refinement

To achieve sub-second accuracy, Solartime:

1. Initial guess uses $\delta$ at midnight UTC to solve $\omega_0$ and get approximate rise/set times.
2. Loop: at each approximate time $t$, recompute $JD$ as $JD + t/24$, recalc $\delta$, resolve $\omega_0$, update $t$.
3. Repeat until changes in (t) are below a tolerance (~0.0001 h ≈ 0.36 s) or after 6 iterations.

This method accounts for the declination’s daily change and yields precise sunrise and sunset times.

Sunrise equation on Wikipedia

Why 90.833° is Used

The standard angle of 90.833° from the zenith (i.e. (-0.833°) altitude) corrects for:

• Atmospheric Refraction (~0.57°)
• Solar Disk Radius (~0.27°)

Ensures calculated times match observed events at sea level.

Atmospheric Refraction on Wikipedia

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References

• Equation of Time
• Sunrise Equation
• Position of the Sun
• Atmospheric Refraction

Enjoy exploring daylight dynamics with Solartime!