Calculation of the Sun's Declination: Understanding the Formula and Its Derivation

Introduction

The position of the Sun in the sky changes throughout the year, which is due to the Earth's orbit around the Sun. One of the key parameters that describes this position is the Sun's declination, defined as the angle between the plane of the Earth's equator and the plane of the Earth's orbital plane (called the ecliptic). To understand how this angle changes over the year, a formula is used: Δ 23.45° × cos(360/365) × d10, where d is the day of the year, with January 1 being the first day. This article delves into the derivation and significance of this formula.

Understanding the Formula and Its Derivation

The formula for the Sun's declination, Δ 23.45° × cos(360/365) × d10, is based on several fundamental principles:

1. The Earth's Tilt

The Earth's spin axis forms a roughly 23.45° angle with the normal of its orbital plane. This tilt is a critical factor in defining the Sun's declination, as it causes the angle between the equator and the ecliptic to constantly change over the year. This angle, known as the obliquity of the ecliptic, is constant and is the first component in the formula.

2. Earth's Orbit Around the Sun

The Earth completes one orbit around the Sun in approximately 365 days. This means that the vector pointing from the center of the Sun to the center of the Earth rotates through an angle of 360°/365 in each day. The second component of the angle calculation, 360/365 × d10, is derived from this daily rotation. Note that the 10 in the formula accounts for the fact that January 1 (day 1) is 10 days after the winter solstice, which is when the Sun's declination reaches its minimum.

3. Spherical Trigonometry

The Sun's declination varies harmonically over the year due to the spherical geometry of the Earth's orbit. This phenomenon is understood through spherical trigonometry, which helps to describe the changing angles and distances between the Sun, Earth, and the equator. The cosine function in the formula reflects this harmonious variation, making the calculation more accurate for the given approximation.

4. Initializing the Day 10

The formula specifically defines January 1 as day 1, 10 days after the winter solstice. The winter solstice is the day of the year when the Sun's declination is at its minimum, and typically occurs around December 21 or 22. By starting the calculation from day 10 after the winter solstice, it aligns the reference point with the known lowest point of the Sun's declination, making the formula more coherent and applicable across different calendar systems.

Conclusion

The formula Δ 23.45° × cos(360/365) × d10 provides a useful and simplified method for calculating the Sun's declination, which is crucial for understanding the Sun's position in the sky throughout the year. While it is an approximation, it captures the essential aspects of the Earth's orbit and its tilt. This information is valuable for a variety of applications, including solar positioning, agriculture, and navigation.