During the last class, we briefly discussed the A.M.E. and why it matters. Today, we will go over each part of the A.M.E. individually and discuss the contribution that each part makes to the amount of magic that a non-luminous celestial body reflects towards Earth. Should you be lucky enough to be given the chance to leave Earth’s atmosphere, you can apply the same concepts to your location at any place in the universe.
As I mentioned in Lesson Four, the A.M.E. was invented by Dr. Ayesha Mansour. She began her research into the planets’ magical effect with the information that was available to her through her studies at Cambridge University – the factors that determine the amount of light that a non-luminous celestial body reflects towards the Earth. For these calculations, the first thing that needs to be known is the amount of light that a celestial body receives from the Sun. That raw value is dependent upon how much light the Sun produces. However this is mostly constant, so for our purpose of merely comparing various bodies, it may be omitted. One thing that does vary from body to body, however, is its distance from the Sun. The farther a body is from the Sun, the less light it receives, which brings us to the first part of the A.M.E. Quotient.
The amount of light that an object receives from the Sun varies with the inverse of the square of its distance from the Sun; the word “inverse” means that it decreases with increasing distance. Why is this the case? The light from the Sun spreads out over a larger area the farther it travels from its source; so, the farther a body is from the Sun, the less of the Sun’s light it gets. For example, since Uranus is about twice as far from the Sun as Saturn, the same amount of sunlight has spread over four times as much area by the time it reaches Uranus as when it reaches Saturn. Consequently, Uranus receives one quarter as much sunlight as Saturn does. In the picture below, this concept is depicted with a flashlight, but it holds true for the Sun, or any other source of light, as well.
Inverse square law for light.
This is the size of an object as seen by an observer, who, for the purpose of this lesson, will be assumed to be on Earth. This size is not the actual physical size measured in miles or kilometers, but rather an angle. It is the greatest angular separation between two ends of the object, expressed in degrees, arcminutes, and arcseconds. The bigger the angular size of an object is, the more light it sends to the Earth. Strictly speaking, the amount of light it sends increases with the angular area of the object, but this depends on the square of the angular size.
The word “albedo” comes from the latin word “albus,” which means white. The albedo of a surface is the ratio of the light reflected off the surface to that incident on, or hitting, the surface. An ideal surface that reflected no light would have an albedo of 0, and one that reflected all the light that hit it would have an albedo of 1. But since no real material reflects either all or none of the light that falls upon it, the albedo of any surface is somewhere in the middle. For example, dry sand’s albedo is 0.4 because it reflects about 40% of the incident light, whereas fresh snow, which reflects 90% of the light that falls on it, has an albedo of 0.9, and charcoal, which reflects only 5%, has an albedo of 0.05. The substance with the highest known albedo (0.96) is magnesium oxide and the substance with the lowest albedo (0.01-0.03) is black velvet.
The same surface reflects some light and absorbs some light.
The phase of a non-luminous celestial body can be described by a number between zero and one: it is the proportion of the body that is lit in the sky. Logically, the more of the body that is lit, the more light it will reflect towards Earth. Why is this the case? Well, the Sun lights the half of a body that is facing it. The proportion of the body that is lit in the sky, as seen from Earth, is the proportion of its sunlit side that is facing Earth. Any body whose distance to the Sun is sometimes or always less than that of Earth will have a full set of phases, ranging from new (almost none of it is lit in the sky) to full (almost all of it is). While many of you have probably heard of the phases of the Moon (which will be discussed in more detail in Year Two), it is not the only body with phases. Since Mercury and Venus are closer to the Sun than Earth is, they too have a full set of phases, as seen in the picture below.
The phases of Venus.
There are two other ways in which the proportion of a body that is lit in the sky can be diminished. If some of the light reflected from the body is blocked from Earth by another body, it can’t be seen from Earth. Additionally, if part of the body is in the shadow of another body, for example, during a partial eclipse of the Moon, when part of the Moon is in Earth’s shadow, then the Sun’s light will not reach it and the body will not be lit in the sky (see the picture below, taken with three different exposure times).
Partial lunar eclipse.
We therefore modify the definition of phase to mean the proportion of the body that is lit in the sky and can be seen from Earth.
As a note, these four quantities (distance from the Sun, angular size, albedo, and phase) are independent of one another. For example, the albedo of the Moon does not depend on its phase – a new Moon reflects just as much light as a full Moon, although that light doesn’t reach Earth. The albedo also doesn’t depend on its distance from Earth, but its angular size does.
We now have all the information needed to present a formula for the amount of light an observer on Earth sees from a non-light-producing astronomical body:
There is a constant term, which depends on the brightness of the Sun and the units used, but this will be discussed in later years.
Dr. Mansour already knew the above formula for the amount of sunlight reflected to Earth, but she wanted to modify it to create a similar formula for the amount of the Sun’s magic that a planet or moon would reflect. To do so, she began by making a careful study of the amount of magic that a body reflects towards Earth. She found that in general, light-coloured objects reflect more magic than dark-coloured objects, but there are exceptions. For example, magic can pass through clouds and walls relatively easily, whereas light cannot. Also, as you will learn in Year Three, while some metals reflect light, stone and metal absorb magic. She defined the magical albedo of a non-luminous celestial body as the ratio of the magic reflected from the body to that incident on, or hitting, the body. This value tends to be similar to light albedo, but again, there are exceptions. In addition, she discovered through careful measurement that the magic reflected by various bodies interact with each other, leading to the fifth part of the A.M.E., which she called interference.
This is the only component of the A.M.E. that is unique to magic. When magic is reflected by an astronomical object, it is influenced by the magic reflected by other objects in the sky. There are two types of interference. Constructive interference occurs when reflections from two or more magical sources enhance each other so that the magic reflected from each of those objects is stronger than it would be normally. Destructive interference occurs when reflections from two or more magical sources partly or completely cancel each other out so that there is less or no magic coming from either of them. Total constructive interference refers to a situation in which the reflected magic from one of the bodies is doubled, putting it at maximum strength, while total destructive interference refers to a complete cancellation of a body’s reflected magic.
Objects that appear to be 90 degrees apart in the sky do not interfere magically with each other. Objects that appear to be closer together than 90 degrees interfere constructively, and the closer together they appear, the stronger the constructive interference is. Objects that appear to be farther apart than 90 degrees interfere destructively, and the farther apart they appear, the stronger the destructive interference is. In addition, the more magic a body reflects towards the Earth, the more it will interfere with the magic reflected by other bodies and the less its magic will be interfered with. At most, the magic reflected by a body can be almost doubled or almost eliminated entirely. While interference ranges from -1 to +1, a 1 is added to it in the equation for the A.M.E. Quotient to make this doubling and cancellation happen. A full explanation of how this works will be provided in an upper year; for now it is sufficient to know that the interference causes this change to the A.M.E. Total constructive interference (doubling the magic) and total destructive interference (eliminating the magic entirely) can never quite be achieved, but the full Moon’s magic is so strong that it can very nearly do so to other bodies. The calculation of this quantity will be discussed in an upper year.
Jupiter is aligned with Mars.
In the picture above, Jupiter is aligned with Mars. Jupiter tends to increase the power of spells and other magic, and Mars tends to make people more likely to commit violence. The magic reflected from each of them is made stronger by the alignment, so when this happens, you’d be best to avoid a bully or someone who personally dislikes you, but just in case you can’t, keep your broom with you at all times!
We can now give a formula for the amount of magic reflected to the Earth by a non-luminous body. Here, the interference component encompasses the total interference from all other visible bodies.
It is because of this division that Dr. Mansour called her formula the Astronomical Magical Effect Quotient.
The Sun’s direct magic also interferes with the magic reflected by other objects in the solar system, including the Moon. An interesting consequence of this fact will be presented when we discuss werewolves in Year Two.
This lesson is shorter than the previous ones … when you’ve stopped cheering, I’ll continue … but there will be more mandatory homework to be done. There will be no essay for this lesson, but there will be a midterm exam as well as a ten-question quiz. Calculators are permitted on both of these tests.
Good luck on the midterm!
Original lesson written by Professor Brad Turing
Part of the lesson written by Professor Robert Plumb