Ever noticed how the summer days are long and the winter days are short? This isn't just a feeling; it's a fundamental consequence of the Earth's tilted axis. The length of daylight at any given location on Earth is determined by two main factors: your **latitude** and the **day of the year**. This tool uses a basic astronomical formula to estimate the length of a day for any location.

[Image of Earth's axial tilt]

The Declination of the Sun

The core of this approximation is the **declination of the sun** ($\delta$). This is the angle of the sun north or south of the Earth's equator. Because the Earth's axis is tilted, this angle changes throughout the year, varying from about $-23.45^\circ$ (the winter solstice) to $+23.45^\circ$ (the summer solstice). The formula for the declination is based on a simplified sine or cosine wave:

$$\delta = -23.45^\circ \times \cos\left(\frac360365 \times (N+10)\right)$$

Here, $N$ is the day of the year (e.g., January 1 is $N=1$). This formula helps us pinpoint the sun's position relative to the equator on any given day.

From Declination to Day Length

Once we have the sun's declination and your latitude, we can calculate the **hour angle** of the sun at sunrise. This is the angle the sun travels from its highest point in the sky to the horizon. The formula for the hour angle $H$ (in radians) is:

cos(H) = (sin(-0.83°) - sin(latitude) × sin(δ)) / (cos(latitude) × cos(δ))

The total length of the day is simply twice this angle, converted into hours. The result gives us a solid approximation, even handling the extreme cases in the polar regions where the sun never rises or never sets for part of the year.

Using the Calculator

Just enter your latitude (a negative number for the Southern Hemisphere) and the date you're interested in. The tool will do the math and provide a good estimate of the daylight hours for that location. It’s a great way to visualize the impact of the Earth’s tilt on our daily lives.