Problem 11 - A History of Mathematics: An Introduction

Question

Calculate the declination and right ascension of the sun when it is at longitude \(90^{\circ}\) (summer solstice) and longitude \(45^{\circ}\). By symmetry, find the declination at longitudes \(270^{\circ}\) and \(315^{\circ}\).

Step-by-Step Solution

Answer

To summarize, the declination and right ascension of the Sun at the given longitudes are as follows: 1. Longitude \(90^{\circ}\): Declination = \(23.44^{\circ}\), Right Ascension = Undefined 2. Longitude \(45^{\circ}\): Declination = \(16.30^{\circ}\), Right Ascension = \(49.93^{\circ}\) 3. Longitude \(270^{\circ}\): Declination = \(23.44^{\circ}\) (by symmetry) 4. Longitude \(315^{\circ}\): Declination = \(16.30^{\circ}\) (by symmetry)

Step 1: Calculate the declination at \(90^{\circ}\) and \(45^{\circ}\) sun longitude

Using the formula for declination, we can calculate the declination at longitudes \(90^{\circ}\) and \(45^{\circ}\). \(L=90^{\circ}\): \(\delta = \arcsin(\sin (23.44^{\circ}) \cdot \sin(90^{\circ}))\) \(\delta = \arcsin(\sin (23.44^{\circ}) \cdot 1)\) \(\delta = 23.44^{\circ}\) \(L=45^{\circ}\): \(\delta = \arcsin(\sin(23.44^{\circ}) \cdot \sin(45^{\circ}))\) \(\delta = \arcsin(\sin(23.44^{\circ}) \cdot 0.7071) \) \(\delta \approx 16.30^{\circ}\) So, the declination at sun longitude \(90^{\circ}\) is \(23.44^{\circ}\), and at longitude \(45^{\circ}\) is \(16.30^{\circ}\).

Step 2: Calculate right ascension at \(90^{\circ}\) and \(45^{\circ}\) sun longitude

To calculate the right ascension at longitudes \(90^{\circ}\) and \(45^{\circ}\), we can use the following formula: \(\alpha = \arctan(\frac{\cos\epsilon \cdot \sin L}{\cos L})\) \(L=90^{\circ}\): \(\alpha = \arctan(\frac{\cos(23.44^{\circ}) \cdot \sin(90^{\circ})}{\cos(90^{\circ})})\) \(\alpha = \arctan(\frac{\cos(23.44^{\circ})}{0})\) Right ascension is undefined at sun longitude \(90^{\circ}\). \(L=45^{\circ}\): \(\alpha = \arctan(\frac{\cos(23.44^{\circ}) \cdot \sin(45^{\circ})}{\cos(45^{\circ})})\) \(\alpha \approx 49.93^{\circ}\) So, the right ascension is undefined at sun longitude \(90^{\circ}\), and at longitude \(45^{\circ}\) is \(49.93^{\circ}\).

Step 3: Find the declination at longitudes \(270^{\circ}\) and \(315^{\circ}\) by symmetry

By symmetry, we know that the declination at longitude \(270^{\circ}\) is equal to the declination at longitude \(90^{\circ}\), and the declination at longitude \(315^{\circ}\) is equal to the declination at longitude \(45^{\circ}\). Therefore, the declination at longitude \(270^{\circ}\) is \(23.44^{\circ}\), and the declination at longitude \(315^{\circ}\) is \(16.30^{\circ}\).

Key Concepts

Declination

Declination is a crucial aspect of understanding celestial coordinates. It measures how far north or south an object is from the celestial equator. In simpler terms, think of it like latitude on Earth, but for the sky. The declination is expressed in degrees and can range from +90° at the celestial north pole to -90° at the celestial south pole. This measure helps astronomers determine where an object is situated in the sky.

• Positive declination means the object is north of the celestial equator.
• Negative declination means it's to the south.

When calculating declination based on the sun's longitude, such as during different times of the year, we use the position of the ecliptic - the path the sun appears to trace through the stars. For example, at the summer solstice, the sun reaches its maximum declination of approximately 23.44°. This happens because Earth's axial tilt causes the sun to appear furthest north of the celestial equator.

Right Ascension

Right ascension is another fundamental component of celestial coordinates, functioning similarly to longitude on Earth but projected onto the sky. It helps determine the east-west position of celestial objects and is usually measured in hours, minutes, and seconds. With 24 hours in a full circle, each hour represents 15°.

• Like time zones on Earth, right ascension increases from west to east.
• This measurement is independent of the observer's location.

During calculations, right ascension can become undefined, as seen at a sun longitude of 90°. This happens because the mathematical operations can't resolve a valid position when the sun is directly overhead of the ecliptic. However, at positions like 45°, right ascension yields a numeric value, helping us pinpoint the sun's eastward location in the sky.

Sun Longitude

The concept of sun longitude refers to the position of the sun along the ecliptic, an apparent path it follows over the course of a year due to Earth's orbit. Sun longitude is measured from the vernal equinox, which is the point of 0° on this path.

• A sun longitude of 0° marks the start of spring in the Northern Hemisphere.
• At 90°, it aligns with the summer solstice, a crucial astronomical event.

Astrologically speaking, the sun's longitude determines its position in the zodiac signs. For example, at 90° longitude, the sun is in the constellation of Cancer. This movement affects seasonal changes, playing a significant role in climactic fluctuations and daylight patterns throughout the year.

Summer Solstice

The summer solstice is one of the most significant astronomical events of the year. It marks the day when the sun reaches its highest point in the sky at noon, resulting in the longest period of daylight for the Northern Hemisphere. The sun during the summer solstice has a declination of approximately 23.44° north, hence why this specific angular measure is fundamental in many solar calculations.

• The summer solstice occurs around June 21st each year.
• It is the point where the North Pole is tilted closest toward the sun.

This solstice not only signals the official start of summer but also has cultural significance across various societies, often celebrated as a time of renewal and growth. Understanding when and why the summer solstice occurs is essential for comprehending the seasonal dynamics and how they affect both natural and human activities.