LOOK DADDY! LOOK HOW BIG THE MOON IS!

While they drive up the gently sloping road, his daughter screams in sudden surprise, ''look daddy, look. The Moon is so big!". He knows well enough that no matter how enormously huge the Moon rises behind the distant high hills, it will never get bigger than its actual size, just like it never turns pink or blue. Not being a killjoy, he decides to pull over and join with his little daughter's newfound discovery of the day. 

In all these years, ever since the days of our distant, tree-dwelling ancestors who were nowhere different than what we are today, every single child, their parents and their parent's parents have revered the unparalleled beauty of the Moon. For some subtle reasons, the She has always been a source of inspiration for numerous poets, painters, songwriters, philosophers, storytellers, other learned men and even the ordinary folks, transcending caste, creed, race and culture. She has to have something in Her; perhaps it is the silvery-white light against the contrasting dark background of the night sky or the mere fact that She is Earth's closest celestial neighbour, almost all of us, at some point or the other, have fallen for that divinely intoxicating romanticism, that which no other human consort can ever provide for. 

We all bear witness to the act that on some nights or evenings for the matter, as the Moon gradually creeps over the horizon, it seems to be quite large compared to its general appearance when high up in the sky. And not just the Moon, the rising and the setting Sun, similarly, seems to be a bit larger than what it appears to be at high noon. Further, the individual stars in the constellations seem to grow bigger and more dispersed near the horizon than that corresponding to their highest point in the sky. Although we do not get to see much of the constellations, if any, because of the menacing light pollution, the enlarged size of the rising Moon or the setting Sun is something we seldom miss. If we are keen observers, we will notice that the Moon seems to rises even bigger when near its full phase and then it becomes smaller as it climbs higher up. 

A rising Moon appears to be larger than ordinary as if it has come very close to Earth.
Image Credits: Photo By Pixahive

The enlarged appearance of the Moon, and similarly the Sun, is popularly referred to as the Moon Illusion and the Sun Illusion, respectively, which in addition to the enlargement of the constellations, falls under the umbrella term Celestial Illusion. It is nevertheless an illusion because using simple arguments we can easily establish that the apparent size (a term which will be explained shortly) of the Moon (including the Sun and the constellations) considering all things stays the same. No matter how loud any of us shouts saying, ''No sir, the Moon is really big tonight. It will get smaller after an hour or so'' the size of the Moon stays fixed and whatever we think are seeing is nothing other than a figment of our imagination. 

A setting Sun or a rising Sun for the matter appears to be larger than its normal appearance at high noon. Atmospheric effects cause a further distortion of its perceived size. 
Image Credits: Photo By Pxhere

Two Fundamental Questions: Initially, we would be tempted to think that at moonrise maybe the Moon is closer to Earth, and as it rises higher up, it moves farther away. Those with a piece of basic astronomical knowledge will know the sheer impossibility of such an occurrence. Some would say that maybe it is due to the atmosphere, as we all know that when light passes through the various layers of air, it refracts and makes the Sun and the Moon appear reddish near the horizon and ay also magnify them to some extent. However, when confounded with the fact that the [apparent] size of the Sun and the Moon stays nearly the same while the enlarged effect is all in our heads, most of us would dismiss such a thing, even in consideration of compelling evidence, and prefer to accept some physical basis for this. It is in our innate nature to explain all kinds of natural phenomena in terms of physical laws. But sometimes, there can be no violation of any such physical law, and the phenomena in question itself can be a faulty construct of our brain or a trick, if not an illusion. Before taking matters further, we need to address two fundamental questions; the size of the Moon, not the real size but the size it appears to occupy in the sky, and whether the size stays the same.

What Is The Size Of The Moon?

Contrary to our day-to-day objects, which we can easily measure by using a ruler or some length of a rope, celestial bodies such as the Sun or the Moon can not be measured physically (we need an impractical length of a rope for that). For this we require analytical, meaning, trigonometrical techniques and some specialised instruments. When it comes to measuring the size of a celestial object, astronomers invariably use the terms angular size, angular diameter, apparent size, apparent diameter, or angular distance, all of which means the same. Angular size is a way of measuring how wide a celestial body appears to us in the sky and is defined in terms of the angle subtended at our eyes. These measurements are performed using an instrument named theodolite. 

The angular size of an object is determined by its actual size of the object and the distance to its centre from the point of observation. From the above figure, it is evident that the angular size is obtained by the angle subtended between the line of sight from the observer's eyes to one edge of the celestial object and its directly opposite edge (linearly). Using basic trigonometry, we can easily arrive at the given small-angle relation, where the angular size (𝜽, theta) is related to the actual linear size (d) of the celestial object and the distance (D) from the observer by the proportionality relation where the proportionality constant is 206,265. In astronomy, angular sizes of celestial objects are typically expressed in terms of arcseconds where one arcsecond is 1/3600th of a degree, while the surrounding sky represents 180 degrees (considering one hemispherical half of the whole sky) from the observer's position. From the above relation, it is evident that if any two of the three quantities, namely the angular size, the linear size and the actual distance is known, then the remaining quantity can be easily calculated. 

If we approximate the distance between the Earth and the Moon to be 384,000 km and the linear (edge to edge) diameter of the Moon to be 3500 km, then plugging the values into the above relation we get the angular size of the Moon to be near about 31.3 minutes of arc or 0.52, i.e., half a degree. Similarly, the Sun is approximately at a distance of 150 million km and with a diameter of 1.4 million km. Therefore, its angular size obtained is 0.53 degrees. 

Thus we see the Sun and the Moon to have nearly the same apparent size. This can be explained as follows. From the above small-angle relation it is found that the angular size (theta) of a celestial object varies directly with its actual diameter (linear size) and inversely with its distance. Accordingly, a large object or a closer object subtends a greater angle, whereas a smaller object or a more distant object subtends an equally smaller angle. Coincidentally, just as the Sun is 400 times farther away from the Moon, the latter is also 400 times smaller. As a result, both of them subtend more or less the same angle and appear to have the same size, even though they are completely unrelated to eachother. 

Does The Moon Grow In Size?

We all know that the Moon's orbit around the Earth is not a perfect circle but an ellipse, and therefore the distance between the two does not stay constant. At apogee, when the Moon is at its farthest distance of 405,500 km, it subtends an angle of 29.67 arcminutes in the sky, whereas, at perigee, when the Moon is at its nearest distance of 363,300 km, it subtends an angle of 33.1 arcminutes in the sky. From this data, it is evident that in course of a single orbital period, the angular size of the Moon varies from 33.1 arcminutes at maximum to 29.67 arcminutes at minimum. Thus, there is approximately a variation of 11% in the apparent size of the Moon. The Sun also undergoes a similar variation due to the elliptical shape of Earth's orbit as it movies through its perigee and apogee positions. 

During a total solar eclipse the Moon fully obscures the Sun
Image Credits: Photo By www.flickr.com/photos/nasahqphoto/

One way to take a note of this apogee-perigee variation is during a total solar eclipse and an annular solar eclipse. With regard to orbital dynamics and the respective apogee-perigee positions of the Sun, Moon and Earth, there are times when the Sun subtends a smaller apparent size than the Moon, and as a result, during a total solar eclipse, the Moon blocks all of the solar disc. Likewise there are times when the Moon subtends a smaller angle than the Sun, wherein we have an annular solar eclipse, where the Sun is not completely obscured and is seen as a ring around the Moon. 

During an annular solar eclipse, the Moon falls short of completely obscuring the solar disc.
Image Credits: Photo By www.flickr.com/photos/kevlar/

Even though we can easily see that a rising Moon appears quite large, the minute variation in apparent size due to the orbital dynamics is something beyond the limit of our eyes and can only be evident during a solar or a lunar eclipse or through photographs like these. For the mere sake of knowledge, we should bear in mind that our brains are incapable of perceiving such a tiny variation owing to orbital dynamics. Therefore this can not be the answer as to why the Moon appears so eerily large when it is just over the horizon. 

Apparent size of the Moon at perigee (closest distance to Earth) and apogee (farthest distance to Earth)
Image Credits: Photo By flickr.com/photos/suraky/6240265362

 Why Do We Call It An Illusion? 

Since we have answered both of our questions, we can finally delve deeper into the Moon illusion. For the last 2500 years, a great many of the brilliant minds, starting from Aristotle, Ptolemy, Ibn al-Haytham (Latinized as Alhazen), Xenophanes, Confucius, da Vinci, Galileo, Descartes, Kant, Schopenhauer, to name a few, have devoted significant years of their lives to find a solution to the Moon Illusion. Initially, every one of them tried to explain this variation in size in terms of physical effects like atmospheric refraction or some other optical phenomenon. But in vain. After generations of heavy debate, it is now an established fact that the perceived variation in lunar size is nothing other than a faulty construct of our brains and as of this date, no one has a clear idea why it happens in the first place. Here we are using the term perceived variation because we have seen that the apparent size stays the same, and the enlarged appearance of the Moon is nothing other than a psychological effect. 

Now, whether it is really an illusion or not can be easily proved with some simple experiments. When the Moon is high up in the sky (zenith), we should raise our thumb or hold a coin at arms length and try to block the lunar disc. Even though we know that the Moon is very far away and very much bigger yet it is a surprise that we can still block such a large object with our bare thumb. This is a brilliant conclusion of the above small angle-relation. The reason why we can block out the lunar disc is that two objects (in this case, the Moon and the coin or the thumb) can subtend the same angle at the eye even if they have different linear size or distance. Again when the Moon is just over the horizon, appearing so large as if it is just a few clicks away, to our dismay, we will find that the same thumb or the coin can once again block out the lunar disc. Had the Moon really increased in size the way it seems to be when it rises, then the coin would not be able to cover it. Instead of the coin or the thumb, one can also hold a ruler and try to measure the sizes of the horizon and the zenith Moon, which will remain the same. Thus, once again, it is proved that the Moon has neither increased in size nor shrunk but subtends the same apparent diameter. 

Another way to verify the Moon illusion is to roll up a piece of cardboard into a small tube and see the ''big'' horizon Moon through it. We need to roll it in a way so that the horizon Moon fits perfectly on the other end of the tube. Now keeping this tube unchanged we need to see the zenith Moon. Again we will find that the zenith Moon too fills the other end of the tube. Had the zenith Moon shrunk in size then it would not fill the other end of the tube which has already been fixed with the horizon Moon. Similarly, for those who have a fit physique, they could bend down and try to see the Moon through the space between their legs. In these two occasions the Moon illusion disappears completely. Now if they look normally they will find the Moon illusion resurfaces once again. 

The final blow to the Moon illusion can be served as follows. All we need to do is set up a camera so that it snaps a picture at an interval of 2.5 minutes (it is the time during which the Moon rises approximately 0.5 degrees) with the same technical settings until the Moon reaches zenith. Now from the final photograph of the series of Moon clicked over a suitable period we will find from elementary calculations that the apparent size has remained constant at all times. 

What The Moon? 

From the above tasks we can all agree that the angular size of the Moon always stays the same. Thus we need we to look at other physical explanations. At this point we can think of only one thing, i.e., the atmosphere. During the classical times, Aristotle who was well aware of the baffling Moon illusion argued that just as an object immersed in a column of water appears to be magnified, a similar thing happens to the Moon as its light passes through the atmosphere. But now we know that the atmosphere does nothing of the sort. This is because when an object (take for example a drinking straw) is submerged in a glass of water, light rays travel from water to air before reaching our eyes. Since water is a denser media compared to air, the light rays are refracted (bent) to such an extent to make the straw appear larger than ordinary. This is also why fishes in a pond appear to be closer than they really are. In these two situations, the water column acts as a magnifying lens thereby bringing the image of the submerged object closer. 

Earth's atmosphere has different layers corresponding to varying degrees of temperature and pressure. At lower altitudes, the air is denser, and consequently, the pressure is higher, and the converse is true for higher altitudes. When light from the Sun or the Moon passes through these layers of the atmosphere, they are bent (refract) towards the surface of the Earth, and as a result, the celestial objects (Sun, Moon or the constellations) becomes visible near the horizon before they actually rise above the horizon (this is evident from the following figure). Even if the atmosphere magnifies the Moon to some extent, it is too small to be perceived with our bare eyes. Quite interestingly, the horizon Moon is supposed to look smaller than the zenith Moon by a very small amount since the former is actually farther away than the latter and we are looking at the horizon Moon through an additional distance of the Earth's radius. How tis happens is evident from the following figure.

Atmospheric refraction only distorts the perceived shape and size of the celestial objects and has nothing to do with the enlarged appearance of the rising Moon near the horizon. In fact, if we take a series of successive images with a camera (as mentioned two paragraphs before) then to our surprise we would find that the Moon or the Sun for that matter appears flattened and squashed near the horizon due to the effects of atmospheric refraction. Once again, if the atmosphere had any magnifying effects then both the rising and the setting Moon would appear larger and their magnification would at least be detected by a camera. But that is contrary to observed facts and experimental data.

An Inescapable Illusion 

At last, we come to the conclusion that physical effects can not be the reason behind the enlarged size of the Moon and is indeed an illusion or more precisely in the words of Immanuel Kant, as he writes in his influential work Critique Of Pure Reason (1781) , ''...the astronomer can not prevent the moon from appearing larger at its rising, although he is not deceived by it''. Another luminary of similar stature, Arthur Schopenhauer in his work On The Fourfold Root Of Sufficient Reason opined of the Moon illusion to be something ''purely intellectual or cerebral and not optical or sensuous''. 

There are numerous competing theories regarding the Moon Illusion, and even after all these years, not a single theory has come up with a complete explanation. Because nothing physical is behind the Moon illusion, it has become more of a study of psychology than physical sciences. It is now evident that our brains are not good at perceiving the world around us. One such among the many theories, known as the relative size hypothesis or the apparent distance theory proposes that when the Moon is near the horizon, there are a lot of foreground objects such as buildings, trees, mountains and etc. Since we know how big a mountain or a tree can be, our brains try to approximate the size of the Moon with respect to the known size of the foreground objects which also provide finer details to our visual field. But since the Moon is much bigger than the foreground objects our brains adjust the size of the Moon and makes it appear eerily large near the horizon. On the other hand when the Moon is high up (zenith) there is nothing to compare its size to but the vast expanse of the open sky, because of which the brains adjusts and shrinks the Moon's perceived size. This is best understood in terms of the famous Ebbinghaus Illusion. However, this theory has its share of inconsistencies such as the intervening foreground objects does not necessarily result in a large looking Moon as the same can be seen when viewed across the vast expanse of open ocean. 

Even though both the orange circles are of the same size, the right one seems smaller than the left one
Image Credits: Author's Computer

There is another theory, the flattened dome hypothesis, which states that the sky above our heads appears to have the shape of a flattened dome contrary to its spherical shape. This comes from the fact that we are accustomed to viewing a flat horizon wherein objects get smaller as they recede farther into the distance. Similarly, objects directly overhead in the sky (birds, airplane or clouds) seems closer while the same object shrinks in size when seen at the horizon. A bird or an airplane exists within the atmosphere and consequently follows the the flattening curve. Again, when these objects recede into the distance, their retinal image, i.e., the size of the image falling onto our retina, becomes smaller. When it comes to the Moon, first of all, its retinal image stays fixed at 0.15 mm, and secondly, it is located not within the atmosphere but far away from Earth itself. Since the retinal image stays the same but the brain knows that objects must become smaller when they recede to the horizon the brain assumes that the Moon must be very large. thereby we are given the impression of seeing an extraordinarily large Moon. 

There are more than a dozen theories regarding the Moon illusion, and many are yet to come. Since this article has been going on for quite a length and further, as this topic is of more interest to psychologists than physicists, it is time for us to call in the curtains. As a concluding remark, it is worth mentioning that for some unknown reasons a fraction of people do not even see the Moon illusion. Again, no one is sure why and this is thus an open question for now. However, we should keep looking up.

References:  

1. https://en.wikipedia.org/wiki/Moon_illusion
2. https://www.scientificamerican.com/article/why-do-the-moon-and-the-s/
3. https://www.vox.com/2015/5/11/8584779/moon-illusion
4. https://www.lockhaven.edu/~dsimanek/3d/moonillu.htm
5. Ross, Helen E.; Plug, Cornelis. The Mystery Of The Moon Illusion. Oxford University Press.