In Lesson Four, we introduced the A.M.E. components and showed why wizards and witches should care about them. In Lesson Five, we described these components in more detail and showed how to compare the amount of light and the amount of magic reflected to the Earth by different non-luminous bodies given the values of these components. In this lesson, we’ll show how to find these components experimentally.
In times gone by, witches and wizards relied heavily on magical tools, such as the von Rheticus telescope, to aid them in finding the components of the A.M.E. Quotient. These tools could do much more than their non-magical counterparts and were even useful for just finding the mundane components, particularly because the first non-magical telescope wasn’t invented until 1608, more than 70 years after von Rheticus invented his! Plus, the von Rheticus telescope has about as much resolving power as a 12-centimeter-wide Muggle-built telescope (about one arcsecond), but less than 1/20 the light-gathering power.
However, in time, Muggles have made so much more progress in technology, in particular telescope making, than we have, that by now, their best telescopes are far superior to ours. This technology enables them to make measurements of angular size, phase, optical albedo, and distance from the Sun more accurately than we can. The 100-inch-wide Hubble telescope has 20 times as much resolving power as the von Rheticus model, and the ground-based Keck telescope uses adaptive optics to equal or even better Hubble’s resolving power despite the wobbling of the image caused by the movement of the air. The picture below shows this amazing piece of technology. The beam of light is a laser beam, which is used for the aforementioned adaptive optics. The movement of the air makes the beam and the nearby stars seem to wobble, but the primary mirror follows the beam and adjusts to it by bending just enough to cancel the wobbling of the beam, thus cancelling the apparent wobbling of the nearby stars as well.
We in the magical world have made some progress in telescope design as well. An improvement has recently been made to the von Rheticus telescope, but the newer model is too expensive to be sold to First Year students. This improvement became possible only after Dr. Mansour had published her work on the A.M.E. You’ll learn all about it near the end of this lesson, but first, we need to discuss how to find the A.M.E. components using the original version, beginning with distance from the Sun.
Von Rheticus studied the motion of planets in the sky and calculated the distance of each of the planets from the Sun, including Uranus and Neptune, which he was the first to see through his telescope. He rigged up his telescope to be able to recognize a planet it was pointed at from the magic it reflected and entered into his telescope the distance of each of the planets from the Sun. For the Moon, he used Earth’s distance from the Sun. Unfortunately, like Copernicus, he assumed that all the planets travel around the Sun in a circle. As we will learn in the next lesson, a planet’s distance from the Sun isn’t quite constant, because it revolves around the Sun in an ellipse, not a circle. In addition, the Sun isn’t in the middle of the ellipse, so the distance he entered for each planet was the average one rather than the current one. The orbits of most of the planets are close to being circles, so the average distance is generally a good enough approximation to be used in calculating the planets’ A.M.E., but, as you can see from the image below, Mars and Mercury are exceptions. A planet’s distance from the Sun at any specific time can be calculated exactly using a method discovered by Isaac Newton, but that’s too advanced for this course. This is one of the many things you can learn about astronomy after obtaining your N.E.W.T. But you can always look up the current distance from the Sun via Muggle sources.
Orbits of the inner planets.
Having dealt with distance from the Sun, we now turn to the angular size, the phase, and the optical albedo of the planets and the Moon. Since Muggles can now find the optical albedo and angular size of these bodies more accurately than we can, we often use their values, presented in the table below, in our calculations. Keep in mind that the Moon and some of the planets have lighter and darker parts with different specific albedos; the albedo of each celestial body shown here is the average one, taken over its entire surface. You will notice that the albedo of each celestial body is constant, whereas its angular size depends upon how far the planet is from Earth, which varies throughout the day, season, and year. In the table, the smallest and largest values of the angular size, expressed in arcminutes (indicated by an apostrophe) and arcseconds (indicated by a quotation mark), are separated by a dash. Venus has the widest range of angular size, appearing about 6.8 times as large when it is closest to Earth as when it is farthest away.
While you can always obtain the albedo and angular size of a body by consulting a table of values, as a magical person, you also have another way of doing so. Do you recall from Lesson Three that the von Rheticus telescope has two additional buttons aside from the two zoom buttons and the focusing knob? If the telescope is held so that the focusing knob is on the left side, then those two buttons are situated on the top of the wider tube near the middle, where you might hold it to keep it balanced. That position was chosen so that an observer could press either one of the buttons without jarring the telescope, thus losing the object they are looking at.
One of those buttons is labeled “S” for angular size. When you press it, the angular size of the object, expressed in degrees, arcminutes, and arcseconds, appears at the bottom of the field of view in pure red characters. To minimize the number of characters displayed, the word degrees is replaced by this symbol o, and standard notation is used for minutes and seconds. If the object isn’t round, then the greatest dimension is chosen as its angular size. For example, if you’re looking at a comet, whose angular length is two degrees, 35 arcminutes, and 28 arcseconds, then, since a comet’s length is much greater than its width, you’ll see 2o35'28", whatever its width happens to be.
If the angular size is less than one degree, only the number of minutes and seconds are shown. If it is less than one minute, only the number of seconds is shown. If the object is too long to fit into the field of view of the telescope even at minimum magnification, press the angular size button while one end of the object is in the field of view and keep holding the button while scanning the object until you can see the other end, and then release the button. The telescope calculates the angular size by measuring the angular size of the magnified object and dividing by the magnification.
Turning to phase, the von Rheticus telescope calculates the phase of an object by measuring its angular length and width and dividing the width by the length. This is the true value if none of the light from the object is blocked from the Earth. Otherwise, the formula for the phase is more complicated - too much so for von Rheticus to include in his telescope - and is another subject of post-Hogwarts astronomical education.
The other button on your telescope is labeled “A”. In the original version of the von Rheticus telescope, A stands for albedo (what it stands for in the latest version will be explained later on). When you press the A button, the (optical) albedo of the object in the telescope’s field of view, a number between zero and one, appears in red characters below the object for as long as the button is pressed. If the whole object is in view, you will see the average albedo of the whole object; otherwise you’ll see the average albedo of the part of the object that you can see. If you want to find the albedo of a specific light or dark part of the Moon, you will need to increase the magnifying power until only that part can be seen. You’ll have to do the same thing if more than one object is visible, because otherwise the telescope will average the albedos.
As you can probably imagine, albedo is much harder to measure than angular size. Here is how the original von Rheticus telescope does so. It first identifies a familiar celestial object (a planet or the Moon) from the magic it reflects. It uses the information it has about the object’s (average) distance from the Sun and measures its angular size, its phase, and the amount of light it reflects to Earth. Then it calculates the albedo from the formula below, which we reproduced from the previous lesson.
But the value it calculates is a good approximation only if the body is a familiar one (a planet or the Moon), the orbit of the planet is nearly circular, and the phase isn’t a result of part of the light being blocked. That was nevertheless better than Muggles could do when von Rheticus invented his telescope and for a couple of centuries thereafter, but now we have access to the more accurate albedos that they have measured with their superior instruments. For calculating an object’s A.M.E., however, we need information inaccessible to Muggles, and this is the topic of the next section.
Having dealt with the non-magical components of the A.M.E., we now turn to the magical ones. To calculate the interference, you need to measure the angular separation between the object in question (the target) and each of the other objects in the sky, which can be estimated accurately enough from viewing angles. Additionally, you need the relative strength of the magic reflected by each of the other bodies compared to the magic reflected by the target itself. How these two pieces are brought together will be discussed more in depth in later years.
To calculate the magical albedo from the optical albedo, you need to know what the surface is made of. By careful observation, Dr. Mansour discovered that the magical albedo of most surfaces is the same as the optical albedo, but a rocky surface reflects only about half as much magic as light, and water, including ice, cuts the magical albedo in half again, as it tames and absorbs the Sun’s magic.
With all this information, you can calculate the amount of magic that an object reflects towards the Earth. But, as you can see by now, this calculation is both difficult and approximate. Fortunately, while we will be working more with these in-depth calculations in the future, for now, we do not need to worry about them because of a further modification of the von Rheticus telescope. The new version of the telescope can detect the amount of magic coming from a celestial object, so when you press the A button on the newest model, it actually calculates the A.M.E. Quotient from the intensity of the magic it detects and displays it in pure red light, giving a new meaning to the letter A on the button.
There are two assignments today: a quiz and an essay. For the essay, you will need to do some observation with your telescope and report on your experience. Choose a night when the skies are clear and the Moon isn’t full, and have fun!