TAU Chapter 2 --- Motion in the Sky

A very useful procedure in science is to construct a "toy" model in an attempt to explain some phenomenon and then modify it as needed once longer and/or more detailed observations are made. Let's do this in trying to explain the motion of objects in the sky -- in this way we'll also be (in a very simple sense) retracing the efforts of people over thousands of years to develop a cosmology, a robust and consistent explanation of how the Universe as a whole works.

Sun's apparent path on an "ideal" day.  Note that the angle
that the Sun's arc makes with the vertical is the
same as the angle of Polaris above the horizon.

For example, maybe the simplest observation you can make is that the Sun rises in the East and sets in the West.  Now, for simplicity, we'll assume that you're making your observation at a special time and place where sunrise is at exactly 6 am and sunset is at exactly 6 pm.  Of course, most days aren't like this (for reasons we'll discuss later), but usually in "toy" models we make many simplifying assumptions and then slowly generalize to the "real" case after we think we understand things.  Now let's suppose you're in the Northern Hemisphere in the middle of the United States, perhaps in Oklahoma City.  If you keep careful track of the Sun's position in the sky during the day, you'd see something very much like the diagram at left.  Notice, for example, that the Sun doesn't climb directly overhead (this overhead direction is called the zenith) but traces an arc across the sky that is always somewhat south of the zenith.  In fact, if you were very careful in your measurements, and if you were able to turn off the pesky blue glow of the atmosphere in order to be able to see stars during the day (yes, they're there all the time!) you'd see that the southerly angle of the sun at noon is the same as the "elevation" of a certain star above the horizon.

You would also notice that the stars move much as the Sun does, but their motion reveals an interesting pattern --- only some of them seem to rise and set.  There are others towards the North (for example, the Big Dipper in the Northern Hemisphere) that trace little circles around a certain stationary star mentioned earlier (actually, that star is a supergiant 50 times larger than our own sun (!) called Polaris, the "North Star", and makes a tiny little circle of its own).  After watching this motion for a while, you might convince yourself that the Sun, Moon, and stars in the heavens seem to be glued to a great dome that rotates around us, where the axis of the dome points very nearly at Polaris.  As this pattern seems to repeat itself daily, let's propose this to be a first cosmological model.  

The Sun's apparent path in the winter (lower arc) and the
summer (higher arc).

As with any scientific model, we should try to imagine consequences of the model, or predictions that it makes that we could check.  For example, if all the heavenly objects really are just firmly affixed to the rotating sphere, the Sun ought to rise and set at the same point every day.  Maybe this appears to be true to your naked eye for a few days, but if you carefully record the position of the rising and setting sun you'd notice that it changes slowly as the year goes by.  The Sun in the wintertime (for those of us in the Northern Hemisphere) rises noticeably further to the south than it does in summertime.  The stars do not follow the same pattern so it cannot be true that the Sun is "stuck" to the cosmic dome in the same way that the stars are.  We'll look at the consequences of the changing apparent position of the Sun in the next chapter.  It is also readily apparent that the Moon changes position (and phase!) from one night to another.

There are other lights in the sky that follow very strange patterns of movement relative to the stars if we carefully observe them over a long period of time:  these we'll call planets, from an ancient word meaning "wanderer".  As an example, look at the path of Mars above.  If you were to look at the same patch of sky and take a picture of the position of Mars every night from May 1 to Nov. 1 in 2018, you'd see Mars trace out a surprising loop as the weeks go by.  As the video shows, Mars initially appears to move across the sky relative to the stars steadily, and then start moving backwards for a time, then continue roughly in the original direction again.  This apparent backwards motion (which happens for each planet) is called retrograde motion.


Well, obviously our simple initial model needs to be revised (this is exactly the process of science!).  Sometimes we are lucky and a beautiful spark of imagination occurs to someone whereby a truly new and simpler model is able to explain the new as well as the old data.  The usual way of modifying an existing model, though, is to add just enough new complexity to explain the new observations while still remaining consistent with all prior observations.  For example, if our first simple model had all the heavenly lights stuck to a single rotating cosmic dome, an easy modification would be to suppose that each object that moved in a "weird" way might move on its own independent Celestial Sphere (and all of these spheres obviously still rotate around the Earth).  But how to explain the retrograde motion?  The prejudice of the time was to assume all heavenly motion was perfect, which meant to them that all motions must be circular and uniform (not changing in speed).  It's difficult to account for the backwards motion of the planets if their speeds are constant!  The solution, ultimately formulated in an approximately final model by Ptolemy in about 150 AD, was very clever indeed.  What if Mars, for instance, traveled on a sphere (an epicycle) that rides along on another sphere (called the deferent)!  This combined motion can be made to reproduce the observed loop as shown above without violating the above assumptions --- all motion is circular and uniform, as long as the added complexity of more spheres is still palatable.  Note in the little movie that the arrow representing our sight line to Mars will temporarily move backwards from time to time just like the "real" motion as seen from Earth.

The classical geocentric Ptolemaic model

The eventual model had, in outgoing order, the Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, and the "fixed" stars, each on their own sphere rotating around the Earth.  Additionally, as we've seen, it's necessary to put all but the Moon, Sun, and stars on additional epicycles to account for the observed retrograde motion.

You can see the importance of imagination in science --- it's difficult to create some new model that still can account for all known observations.  Science is rarely a process of deduction, where a conclusion logically follows from some set of premises; instead, it is often inductive, where we try to take a creative leap to a new idea and concoct experiments that may provide evidence for or against it.  This Earth-centered (geocentric) model was a successful explanation of how the Universe worked for about 1,500 years until significant pieces of evidence obtained through careful observation overruled it in favor of a new heliocentric (Sun-centered) model.

By the way, it's commonly thought that only during the Middle Ages was it shown that the Earth was "round" (spherical).  This had actually been demonstrated a number of ways back in ancient Greek times by Aristotle, among others, but the most precise demonstration and measurement of the size of the Earth was done by Eratosthenes using a fairly simple method (see image).

At the same time on the particular day, a stick in one location casts no shadow,
but a stick in another location does; this is direct evidence of a spherical Earth.

He had read accounts of the observation in a southern town that, on the longest day of the year at noon, sticks happened to cast no shadows and the Sun shone into the bottoms of wells.  That would mean that the Sun was directly overhead.  What set him apart from the myriad of people that knew this fact was that he had the curiosity to ask if that was true in his city of Alexandria.  When the experiment was done, it turned out that at this exact time, sticks do cast shadows.  This immediately convinced him that the Earth must then be spherical, and being an accomplished mathematician, a careful measure of the length of the shadow along with a knowledge of the distance to the southern village allowed him to calculate the size of the Earth to high precision (the shadow made an angle of about 7 degrees with the top of the stick, so the two places must be about 7 degrees apart along the Earth's surface; knowing there are 360 degrees around the sphere meant that they were about 1/50 of a full circumference apart, so knowing that the actual distance between the sticks was about 500 miles and multiplying by 50 yields the right answer of about 25,000 miles around).

Many people knew of this curious observation, but it is truly only the curious people that change the world.

Hot Streaks

Chance of having a 6-make streak out of 200 chances as a function of shooting percentage

I was playing around with streak probabilities, and it's remarkable how fast the probability of a certain streak rises with even a moderate increase in percentage per try.  Here, for example, is a toy model of a basketball player who takes 200 3-point shots in a given season.  Suppose we're interested in the chances that, at some point, the player has a "hot streak" of 6 makes in a row --- something that fans are bound to remember for some time.  If the player is a 30% shooter, there is only a 10% chance of such a streak occurring.  But the chances double for a 35% shooter, and for a very good shooter (45%) we'd expect such a streak more often than not.  I suspect these relatively small differences in skill are responsible for attributes of a so-called "streaky" shooter (which are almost never borne out by the statistics).  I chose a 6-shot streak arbitrarily, here is the likelihood for a 4-shot streak over the course of 50 attempts (a few games, so that this might happen a few times during the season, reinforcing the notion of "streaky"):

Chance of having a 4-make streak out of 50 chances as a function of shooting percentage

Here, you can see that for an average shooter (35%) you might expect a "hot streak" 40% of the time over a sample of 50 attempts.