Free Sunrise Calculator

What Is Sunrise and the Three Types of Dawn

Sunrise is defined as the instant when the upper edge of the solar disk first appears above the eastern horizon, marking the official start of day. However, the sky begins to brighten well before that moment. Astronomers, navigators, and geographers classify the pre‑sunrise period into three distinct phases based on the Sun’s angular distance below the horizon:

• Civil dawn – Sun between the horizon (0°) and 6° below. Outdoor activities can be carried out without artificial light.
• Nautical dawn – Sun between 6° and 12° below. The horizon is still visible, but only the brightest stars can be seen.
• Astronomical dawn – Sun between 12° and 18° below. The sky is dark enough for faint celestial objects (nebulas, galaxies) to be observed.

These thresholds apply symmetrically to dawn and dusk, making them essential for both Sunrise Time Calculator and Sunset Calculator functions.

Core Variables for Dawn Calculation

To compute the sunrise moment for any location on a given date, you need three inputs:

• $\phi$ – Latitude in degrees (positive for Northern Hemisphere, negative for Southern).
• $\lambda$ – Longitude in degrees (positive for Eastern, negative for Western). Longitude is later converted to hours.
• $n$ – Day of the year (1 = January 1).

The tool also relies on standard astronomical equations that model Earth’s orbit. The steps below outline the complete algorithm, which can be reused for Sunset Calculator and Day Length Calculator by adjusting the hour‑angle criteria.

Step‑by‑Step Formula for the Sunrise Hour

1. Approximate Sunrise Time

A rough guess for the local time of sunrise (in hours) is:

$$t = n + \frac{6 - \lambda_{\mathrm{h}}}{24}$$

where $\lambda_{\mathrm{h}} = \frac{\lambda}{15}$ (since the Earth rotates 15° per hour).

2. Mean Anomaly of the Sun

The mean anomaly $M$ (in degrees) accounts for the difference between a circular orbit and Earth’s actual elliptical orbit:

$$M = 0.9856 \times t - 3.289$$

3. True Longitude of the Sun

From $M$, compute the true solar longitude $L$:

$$L = M + 1.916 \sin(M) + 0.020 \sin(2M) + 282.634$$

Reduce $L$ to the range $[0°, 360°)$ by subtracting multiples of 360.

4. Right Ascension and Quadrant Alignment

The right ascension $RA$ (in degrees) is obtained from:

$$RA = \frac{360}{\pi} \arctan\bigl(0.91764 \tan(L)\bigr)$$

Bring $RA$ into the same quadrant as $L$ by setting:

$$L_{\text{quad}} = \left\lfloor\frac{L}{90}\right\rfloor \times 90,\qquad RA_{\text{quad}} = \left\lfloor\frac{RA}{90}\right\rfloor \times 90,$$ $$RA = RA + (L_{\text{quad}} - RA_{\text{quad}})$$

Finally, convert $RA$ from degrees to hours: $RA_{\mathrm{h}} = \dfrac{RA}{15}$.

5. Solar Declination

The declination $\delta$ (angle between the Sun and the equatorial plane) is:

$$\delta = \frac{180}{\pi} \arcsin\Bigl(0.39782 \sin(L)\Bigr)$$

6. Hour Angle for Sunrise

The hour angle $HA$ links the local time to the phenomenon. For the instant when the Sun’s upper limb touches the horizon (corrected for refraction and the Sun’s apparent radius), use a zenith angle $z = 90.833°$.

$$\cos(HA) = \frac{\cos(90.833) - \sin(\phi)\sin(\delta)}{\cos(\phi)\cos(\delta)}$$

If the argument falls outside $[-1,+1]$, the location is inside a polar circle where sunrise does not occur on that day.

For the other dawn phases, replace $90.833$ with:

• Civil dawn: $96°$
• Nautical dawn: $102°$
• Astronomical dawn: $108°$

A Dawn Calculator can use these values interchangeably.

7. Local Sunrise Time in UTC

The uncorrected sunrise time (in hours, UTC) is:

$$\text{UT}_{\text{rise}} = t - \frac{HA}{15} - \frac{\lambda_{\mathrm{h}}}{24}$$

Reduce this value to the range $(0,24]$ by adding or subtracting 24 if necessary.

8. Correcting for Time Zone and Daylight Saving

Finally, convert to local standard time and apply daylight‑saving adjustment (if active):

$$\text{Local Rise} = \text{UT}_{\text{rise}} + \text{tz} + \mathrm{ds}$$

where $\text{tz}$ is the time‑zone offset (e.g., +2 for Central European Summer Time) and $\mathrm{ds}$ is 1 if daylight saving is in effect, otherwise 0. Again, keep the result between 0 and 24.

The Effect of Atmospheric Refraction on Sunrise

Earth’s atmosphere bends sunlight as it enters, making the Sun appear higher than its geometric position. This effect is strongest at low elevation angles. At sunrise and sunset, refraction raises the apparent solar disk by about 0.58°. Adding the solar disk’s typical apparent radius (0.27°) gives a total correction of 0.833°. Consequently, the first rays appear several minutes earlier than a purely geometric calculation would predict. The Sun Position Calculator built into this tool automatically applies this correction, so you always get the moment when the uppermost rim becomes visible over an unobstructed horizon.

Worked Example: Sunrise in Reykjavík

Let’s test the calculation for Reykjavík, Iceland (64.164° N, 22.022° W) on June 21 (summer solstice, $n = 172$).
Using the above steps (with the tool’s built‑in logic):

• Declination reaches about 23.44°.
• The hour angle becomes extremely small (< 1°), giving a very early sunrise (around 02:55 UTC).
• Adding the local time‑zone (UTC+0) and no daylight saving yields a sunrise near 02:55 local time.

At such high latitudes, the day is nearly continuous around the solstice—a striking example of how Daylight Hours Calculator functions can differ dramatically by location.

Beyond Sunrise: Sunset, Day Length, and Solar Noon

The same mathematical framework serves to compute sunset, solar noon, and total day length. Simply alter the sign of the hour angle for sunset, and adjust the zenith angle accordingly. A Sunrise Sunset Times tool therefore becomes a full Solar Noon Calculator and Day Length Calculator with minimal additional code. Whether you need exact times for photography, gardening, navigation, or astronomy, this free online tool delivers reliable, location‑specific results.