Linear and Angular Size

It is a well-known phenomenon that an observer will perceive the moon closer to the horizon as being larger than one high up in the sky.  However, when one photographs both the horizon and zenith moons and compares them, they turn out to be the same size.  This is the famous moon illusion.  

Psychologists have been studying the moon illusion for more than a hundred years, and there are many conflicting theories.  A popular one suggests that we visualize the celestial dome around us as a rectangular “roof” rather than a hemispherical one.  This is called the “flat sky” theory.  Hence, looking at the horizon moon is akin to looking at a Ponzo illusion (Fig. 63) upside down. 

The most widely accepted theory is by American psychologist Don McCready.  Along with a few other prominent researchers, McCready believes that the moon illusion is due to a size illusion known as oculomotor micropsia.  However, before we examine this theory, we need to differentiate between the various types of moon illusions.

When one says the moon is of a certain “size”, it is important to clearly define what that means.  In Fig. 57, the three spheres shown subtend the same angle θ to the observer.  Hence, they are the same angular size.  However, spheres farther away from the observer are of a larger linear size. 

Fig. 57 – The three spheres all subtend the same angle to the observer. 

In Fig. 58, the spheres are of the same linear size, but the one nearer to the observer subtends a larger angle.

Fig. 58 – The spheres are the same linear size, but
subtend different angles to the observer.

These two concepts of size are intimately related to the distance between the observer and an object, which is known as the linear distance.  It is important to note that angular size is measured in degrees, while linear size is measured in metres.  

To complicate matters, there is a difference between an object’s absolute linear size and its perceived linear size.  For instance, the moon has a diameter of about 2160 miles and this is known as its absolute diameter or absolute linear size.  It is very difficult, if not impossible, for the unaided eye to know the moon’s absolute linear size.  Instead, an observer would make a subjective approximation known as the perceived linear size.  Similarly, an approximation for the moon’s angular size is known as the perceived angular size.  Note that the moon always subtends 0.5° to an observer.  Finally, the distance from an observer to the moon is about 238,800 miles and this is its absolute linear distance.  The corresponding approximation would be the perceived linear distance.