How To Calculate The Distance To The Moon Using Trigonometry

How To Calculate The Distance To The Moon Using Trigonometry

Ever looked at the night sky and marveled at all the various celestial bodies that occupy it? Right from a very young age, we are taught that even though these heavenly bodies are several thousand times bigger than us compared to size, they still appear to be extremely tiny because they are very far away from us. Earth’s only natural satellite, the moon, appears no bigger than a soccer ball.  Over time, as we gain more knowledge, we get equipped with the necessary scientific information and logical reasoning to understand such phenomena properly. 

How does Trigonometry come into the picture?

With Math and Science subjects coming to their aid, individuals have found themselves unraveling the mysteries of the universe since time immemorial. Things, such as plotting the distance between us and the seemingly unattainable celestial bodies, including the sun and the moon, that was hitherto a puzzle, started becoming accessible by the day. However, at the heart of this plotting lies a subject that we are familiar with from a very young age – Trigonometry.

Ancient methods:

Originating from the Greek words trigonon and metron meaning “measurement of triangles,” Trigonometry is that branch of mathematics that examines the relationships between the three angles of a triangle and the length of its sides. As a field of study, it traces its origin to as far back as the 3rd century BC when it emerged from the applications of geometry that were exercised over astronomical studies. Ever since it has established itself as the foremost academic tool behind plotting distances to celestial bodies.

Study of the Moon:

As the Earth’s nearest celestial neighbour, the Moon has been a subject of our study since antiquity, with the first estimate of its distance made by Greek astronomer Hipparchus, dating back to the 2nd century BCE. Modern techniques of plotting the distance to the moon are an upgrade over the ancient Greek astronomer’s methods only in terms of the modern tools and equipment that it uses. However, in terms of theory, both rely on the technique of trigonometric parallax to achieve their objectives. The following section is a deep dive into this particular concept that will help us understand how one can use trigonometry to calculate the distance to the moon.

Introduction to Parallax:  

Using the method of trigonometric parallax that is otherwise also referred to as the stellar parallax, astronomers can successfully compute the distance of those objects nearby in space. They carry out this process by measuring the apparent movement of a star against the backdrop of those stars that are further away as Earth revolves around the Sun. In scientific circles, Parallax is widely regarded as the universally acknowledged standard for estimating celestial distances. It does not rely on or involve physics but rather concerns itself exclusively with the tenets of geometry.

How Parallax Works:

As a method, it is quite straightforward. It revolves around the measurement of the two angles and the included side of a triangle formed by Earth on one side of its orbit, and the Earth’s visualized location six months later on the other side of the orbit. For clarity, you can try out a simple activity. All you have to do is extend your right hand, shut your right eye, and carefully put your thumb on top of a faraway object. Upon switching eyes, you will notice that your thumb has shifted slightly across the background of your object. By simply measuring this tiny change and determining the distance between your eyes, you can compute the thumb’s distance from the eye with ease, thanks to parallax.  

Parallax in the context of stars:
Listed below is a breakdown of how parallax is used to measure distance in the context of stars.

  • Similar to the simple activity described above, to measure any particular star’s distance, astronomers primarily make use of a baseline of 1 astronomical unit (AU). 
  • Considered as the average distance between the Sun and the Earth, 1 AU, in other words, corresponds to 93 million in miles or a whopping 150 million in kilometers.
  • Further, they compute small angles in terms of arcseconds, visualized as tiny fractions of a degree on the night sky.   
  • Dividing this baseline of 1 AU by the tangent of a solitary arcsecond gives us a figure of approximately 19.2 trillion miles or 30.9 trillion kilometers.
  •  In terms of light-years, this corresponds to about 3.26 light-years. This particular unit of distance is referred to as a parallax second, or more popularly as parsec.

However, since even the closest star is at a distance that is greater than 1 parsec from our sun, astronomers are pushed to calculate stellar shifts that are lesser than one second if they wish to accurately determine the interval to the star in question. German astronomer Friedrich Wilhelm Bessel, in 1938, was the first person who succeeded in measuring the distance to a star using the parallax method.  

The cosmic distance ladder:

With reference to the cosmic distance ladder, the parallax method occupies quite an important step. By computing the length to a few nearby stars, cosmologists have been successful in establishing decisive correlations and relationships between the color of a star and that of its intrinsic brightness; simply put, the brightness that the star would seem to emanate when viewed from a standard distance. Consequently, these stars came to be known and addressed as “standard candles.” In such a situation, should a star happen to be too far away for the astronomers to determine its parallax, they can always match its spectrum and color to anyone of the above mentioned standard candles and note down its inherent brightness. 

Role of the Inverse Square Law:

A comparative study of this inherent brightness with the star’s apparent brightness can help them get an accurate estimation of its span, courtesy of the application of the inverse square law. The inverse square law states that the apparent brightness of a light source is inversely proportional to the square of its distance. Consequently, while employing this manner of measurement, if a star happens to be a part of a distant cluster, one can safely assume that all of those stars are situated at the same distance and thus can be added to the existing library of standard candles without any ambiguity whatsoever. Off late parallax has also come to be employed in the recreation and showcasing of 3 dimensional or 3D images. This is achieved by capturing the subject’s 2D images in consideration from two angles, which vary only slightly from one another. In the 19th century, the stereoscope was a popular device used to rely on this same philosophy to show photographs in 3D.

Now that you have learned how to use trigonometry to calculate the moon’s distance, you would have realized that a major chunk behind these computations relies on our practical understanding of things. 

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About the Author: Wayne Probert

Wayne Probert is a senior reporter at Zobuz, covering state and national politics, and he is a grantee with the Pulitzer Center on Crisis Reporting. Before joining Zobuz, he worked as a freelance journalist in Kentucky, having been published by dozens of outlets including NPR, the Center for Media.