Question

The formula \(\sin \sigma=\tan \delta \tan \phi\) only makes sense if the right- hand side is less than or equal to I. Since the maximum value of \(\delta\) is \(23 \frac{1}{2}^{\circ}\), show that the right-hand side will be greater than 1 whenever \(\phi>66 \frac{1^{\circ}}{2} .\) Interpret the formula in this case in terms of the length of daylight.

Short answer

Answer: When the observer's latitude exceeds 66.5 degrees, the given formula cannot be used to calculate the solar zenith angle, and the regular relations between the observer's latitude, solar declination angle, and solar zenith angle do not apply. This scenario occurs within the Arctic and Antarctic Circle regions, where the length of daylight can become continuous for multiple days during the summer solstice.

Step-by-step solution

Understand the given formula

The given formula is \(\sin \sigma = \tan \delta \tan \phi\), where \(\sigma\) is the solar zenith angle, \(\delta\) is the solar declination angle, and \(\phi\) is the observer's latitude. The formula relates the solar zenith angle (the angle between the sun's position and the observer's vertical) to the solar declination angle and the observer's latitude.

Confirm the maximum value of \(\delta\)

It is given that the maximum value of the solar declination angle is \(23\frac{1}{2}^\circ\). This is a known fact due to Earth's axial tilt with respect to the plane of its orbit around the Sun.

Analyze the right-hand side of the formula for \(\phi > 66\frac{1}{2}^\circ\)

To understand the implications of \(\phi > 66\frac{1}{2}^\circ\), let's substitute the maximum value of \(\delta\) and find the minimum value of \(\phi\) that gives a right-hand side greater than 1. $$\tan \delta \tan \phi > 1$$ With \(\delta = 23\frac{1}{2}^\circ\), we have: $$\tan(23\frac{1}{2}^\circ) \tan \phi > 1$$

Solve for \(\phi\)

To find the minimum value of \(\phi\) that satisfies the inequality, we isolate \(\phi\) by dividing both sides by \(\tan(23\frac{1}{2}^\circ)\), and then apply the inverse tangent function to both sides: $$\phi > \tan^{-1}\left(\frac{1}{\tan(23\frac{1}{2}^\circ)}\right)$$ Using a calculator, we have: $$\phi > 66\frac{1}{2}^\circ$$

Interpret the result

Since the right-hand side of the given formula is greater than 1 when \(\phi > 66\frac{1}{2}^\circ\), this means that the formula no longer holds in this case. It can be interpreted in terms of the length of daylight as follows: when the observer's latitude exceeds \(66\frac{1}{2}^\circ\), the formula cannot be used to calculate the solar zenith angle, and as such, the regular relations between the observer's latitude, solar declination angle, and solar zenith angle do not apply. This scenario occurs within the Arctic and Antarctic Circle regions, where the length of daylight can become continuous for multiple days during the summer solstice.

Key concepts

Solar Declination Angle

The solar declination angle is an important concept in understanding the position of the Sun in the sky. It is the angle between the Sun's rays and the plane of the Earth's equator. This angle varies throughout the year and is influenced by the tilt of the Earth's axis. The maximum solar declination angle is approximately 23.5° , correlating with the Earth's axial tilt. This maximum angle occurs during the summer and winter solstices, when the Sun is at its highest and lowest points in the sky respectively.
The solar declination angle is crucial for calculating solar angles, which help determine things like shadow length and daylight duration. These calculations are important for activities such as agriculture, solar energy deployment, and navigational astronomy. Understanding this angle allows us to gain insights into seasonal changes and their effects on the Earth.

Solar Zenith Angle

The solar zenith angle is the angle formed between the zenith (directly overhead point) and the center of the Sun. It provides a measure of how high the Sun is in the sky at any given time and location. At solar noon, when the Sun is at its highest point, the solar zenith angle is smallest. Conversely, just after sunrise and before sunset, the solar zenith angle is largest.
Mathematically, the solar zenith angle can be calculated using the formula \(\sin \sigma = \tan \delta \tan \phi\), where \(\sigma\) is the solar zenith angle, \(\delta\) is the solar declination angle, and \(\phi\) is the observer's latitude. This formula links the observer's position on Earth and the time of year to the Sun's position in the sky. Understanding the solar zenith angle is essential in predicting the intensity of sunlight received at a particular location, which impacts everything from climate and temperature to solar panel efficiency.

Latitude and Daylight Variation

Latitude plays a significant role in the variation of daylight hours. As one moves closer to the poles, the variation in daylight from season to season becomes more pronounced. For example, at higher latitudes beyond 66.5° , which includes the Arctic and Antarctic Circles, the phenomenon of polar day and polar night can occur.
During summer, areas near the poles can experience continuous daylight, known as the Midnight Sun, where the Sun does not set for several days or even weeks. Conversely, during winter, these areas can undergo polar night, where the Sun does not rise above the horizon for extended periods. These effects are due to the extreme angles at which the sunlight reaches these regions, influenced heavily by the Earth's axial tilt and the observer's latitude. Understanding how latitude affects daylight helps in predicting seasonal changes and preparing for challenges related to lighting, agriculture, and daily life in these unique environments.