Epicycles and Discovering Bad Theories

Our ancestors were highly intelligent people who devised ingenious methods to model what they perceived to be reality in the skies. Unfortunately, they came at many of these observations with deep-rooted prejudices and a priori (preconceived) beliefs which shackled their creativity.

Figure 1: Close up of the Ptolemaic system out to the Sun’s sphere

The prevalent, far-reaching belief was that the Earth was immovable and at the center of the universe. Of course we know this is preposterous (even to the point that there is no such thing as a center to the universe), it is still a useful exercise to challenge students to prove, without leaving the Earth or using satellites, that the Earth does indeed rotate and that it revolves about the Sun.

Another a priori assumption was that celestial bodies never stopped moving, as opposed to “earthly” objects which eventually came to a halt. So, when the planets periodically went back and forth in the sky, this was unacceptable and Apollonius of Perga came up with a “solution” that allowed the wanderers to be always moving without stopping by coupling two motions at once. The planets were not simply attached to a mystical sphere (“deferent”) but they were actually attached to a mini-sphere (“epicycle”) which rotated on the larger one.

Figure 2: Mercury’s retrograde path in the Ptolemaic system

In this way planets could move around the sky but intersperse that generally easterly motion with apparent backwards motion (retrograde) when the transparent epicycle carried the planet backwards.  The ancients latched on to it and it was greatly preferred to having deferents slow down, stop, go backwards, stop, then resume their original direction.

My colleague David Steelman and I created a program called Epicycles for SciDome that illustrates the main characteristics of the Ptolemaic Geocentric Model.  It helps students discover the systemics of the model which can only be explained as “it just has to be that way”. Whenever that is the reasoning, it signals a problem with the theory/model.  This will become obvious as we go through this paper.

Let’s first take a close look at the bodies closest to the Earth in the geocentric model, as shown in Figure 1.

The Moon moves the fastest in the sky (and even changes shape!) so it was assumed to be closest to Earth.  Placement of Mercury and Venus closer to the Earth than the Sun was problematic.  The theory was based upon the idea that those that appeared to move the slowest must be farthest away from Earth.  The problem is that the epicycle containing Mercury, the epicycle containing Venus, and the Sun all orbited around the Earth in one year!  So their order was reluctantly agreed upon because Mercury moved fastest on it epicycle, Venus next fastest, and of course the Sun had no epicycle (because it never retrograded).

Figure 3: Venus and Mercury’s retrograde paths in the Ptolemaic system

The epicycle sizes are based on arbitrarily assumed distances from Earth. The angles had to match the size of the retrograde loops seen in the sky so, looking at Figure 1, Mercury’s epicycle is tiny compared to Venus’ because Mercury’s retrograde loop is about 52 degrees in extent whereas Venus’ is about 92 degrees! The fact that Venus is farther away than Mercury from the Earth in this model requires it to be considerably larger than one might expect, but these are to scale to create the properly sized retrograde patterns.

As time is progressed a trace can be turned on which shows the retrograding patterns of the planets. Figure 2 shows a close up of Mercury and Figure 3 that of Venus.

When I ask students if they see anything peculiar as time progresses, eventually someone notices that the centers of the epicycles of Mercury and Venus are exactly and always lined up with the line connecting the Earth and Sun (the Earth-Sun Line). What explanation would the ancients have given for this? “It just has to be this way for this model to work.” Red flag number 1 that there’s something wrong with this theory.

Figure 4: The planets beyond the Sun’s sphere

Of course we know that in the Copernican heliocentric model we don’t need epicycles to cause Mercury and Venus to wobble back and forth around the Sun because they are simply closer to the Sun than Earth and they orbit the Sun. In fact, Copernicus was the first to completely untangle the motions of Mercury and Venus from the Sun’s motion.

This confusion is one rarely-discussed reason why the Copernican heliocentric model was so appealing. It unambiguously separated the motions of Mercury and Venus and even established, for the first time, their orbital periods around the Sun (88 days and 225 days, respectively).

Now observe the planets beyond the Sun, as shown in Figure 4. As we advance time another strange systematic displays itself, although this one is a lot more challenging to pick out. The Earth-Sun Line is always parallel to the planet’s epicycle radius!  You can easily see this in Figure 4 now that you know to look for it.

Again, the ancients noted this “coincidence” but could never explain it other than “it has to be this way for the model to work.” Another red flag has raised itself in the flawed Ptolemaic model!  The basic reason for this “coincidence” is because the retrograde motion of each planet is a function of its position relative to the Earth in its own orbit. Since we’re locking down the Earth and moving the Sun, it’s the orientation of the Earth-Sun Line that is the determining factor as to when planets exhibit their retrograde motions.

Figure 5 – The retrograde paths of the planets beyond the Sun’s sphere

When the planets leave breadcrumbs (see Figure 5) their retrograding paths become obvious. Again, the model has been carefully defined to accurately recreate the width of the retrograde loops as well as their frequency.

This is a fun and thought provoking lesson for my students because it demonstrates how intelligent and clever the ancients were in mimicking celestial motions, but it also shows how preconceived notions can weigh one down and severely complicate the model. It also clearly points out that when certain “features” of a model have no other explanation than “it has to be that way for the model to work” that the model is most likely flawed or incorrect at its core. But having the Earth move was a huge paradigm shift, and it took over 1500 years to overthrow it!

Exploring Roemer’s Method for Determining the Speed of Light

In the 17th century the speed of light was unknown, and scientists questioned whether it had a finite value. Descartes argued that if the speed of light was finite, when we looked out into space with telescopes we’d be looking into the past. That idea was so off-putting, he concluded the speed of light must be infinite.

We now know it’s not infinite. If it were, the universe couldn’t exist. Remember Einstein’s famous E=mc²: if the speed of light c was infinite, the amount of energy contained in any amount of matter would be infinite! Good luck with that…

Ole Roemer, a Danish astronomer in the 17th century, stumbled upon the speed of light during timing observations of the emergence of Jupiter’s closest Galilean moon, Io, from behind the planet’s limb. He noticed the moon’s appearances didn’t match his predicted times, and by studying them through the year realized it was ahead or behind the predicted time, depending upon how far away Jupiter was from Earth! He correctly reasoned that this variation was not due to some strange inconsistency with Io’s orbit but rather that he was observing what is now called the light-time effect.

Starry Night simulates the light time effect, so we can reproduce Roemer’s 1676 measurements of Io’s emergence from Jupiter’s limb to directly show the light time effect, and even measure the speed of light.

Figure 1: Io emerging from Jupiter’s eastern limb

Figure 1 shows Io emerging from Jupiter’s eastern limb. This view was measured by Roemer in Copenhagen on November 9, 1676. Using Starry Night’s ability to transport us anywhere in space, we can specify a direct route to Jupiter but hold the time constant.

In other words, if we could transport instantly to 0.20 AU from Jupiter (an arbitrarily distance for a nice view of the scene), what would we see? As we travel to Jupiter, we’ll see Io appear to move further and further eastward from the limb even though time has stopped and we’re traveling on a direct line to Jupiter, as shown in Figure 2.

Figure 2: Io appears farther from Jupiter as we reduce our distance

If the speed of light were infinite, when we transported to Jupiter Io would be emerging from the limb, the same view as from Copenhagen. Because the light time effect is built into Starry Night, we see Io has actually emerged significantly past the limb. In other words, on Earth we saw Io just emerging from Jupiter’s limb, but in the neighborhood of Jupiter it has already long since passed from behind the limb!

The difference occurs because of the light time effect.

We can measure the speed of light directly from these observations. We know the distance we covered in our journey from Earth (5.326 AU = 7.968 x 108 km). We can now step back time and return Io back to its emerging position from Jupiter’s limb as seen from this nearby location to Jupiter. If we place Io back on Jupiter’s eastern limb by reversing time in Starry Night, it takes 44m 20s to do so, or 2660 seconds. To estimate the speed of light, we simply take the distance we traveled and divide it by the time, as follows:

Thus we obtain a value for the speed of light only 0.08% different from the actual value of 299,792 km/s!

Figure 3: Line of sight from Earth to Jupiter’s limb

Steve Sanders (Eastern University Observatory Administrator) and I have created a simulation which clearly illustrates this phenomenon, as seen in the figures below. This simulation will be included in the release of Volume 3 of the Fulldome Curriculum and you can preview it on Youtube.

Figure 3 shows the line of sight from Earth to Io as it has emerged past Jupiter’s limb, the event that Roemer was measuring. We depict the event close to opposition with Jupiter (Earth’s closest approach to the planet) and the distance between the bodies is approximately 3.95 AU = 3.67 x 108 million miles.

The same event is shown in Figure 4 when Earth and Jupiter are nearing conjunction (Jupiter nearing a syzygy with the Earth and Sun in between). Note that the distance separating the planets is now 5.75 AU = 5.34 x 108 million miles.

Figure 4: Earth and Jupiter nearing conjunction

Roemer measured the emergence of Io as being about 15 minutes later than when this emergence occurred close to opposition and attributed the lateness (correctly) to the extra distance that the light had to travel across the Earth’s orbit.

If you assume that this tardiness is entirely due to the extra time required for light to travel the extra distance, you can estimate the speed of light as follows:

 

The value of the astronomical unit at that time was very crudely known, so Roemer’s value for the speed of light was not nearly this accurate, but nonetheless he demonstrated that the speed of light was finite, and its value was of this order.

We encourage SciDome operators to use the Roemer Speed of Light minilesson in Volume 1 of the Fulldome Curriculum, along with our new simulation. The discovery of the finite speed of light forever changed our view of the universe, turning our distance-shrinking telescopes into literal time machines as we explore back into our cosmic past.

Recreating Jupiter’s Galilean Moons in SciDome

Figure 1: Page from the original printing of Sidereus Nunicius showing Galileo’s sketches of the Medicean Moons

In many of our astronomy classes, we discuss the importance of Galileo’s first telescopic observations in eventually overthrowing the Ptolemaic geocentric system. His first observations were relayed to the public in his short book Sidereus Nuncius, which is Latin for The Starry Messenger (or arguably, The Starry Message). In it he relates his observations of the Moon, the myriad of new stars he observed (with sketches of the Pleiades and Praesepe regions), and the Moons of Jupiter.

He originally called these the Medicean Stars, a call out to his potential benefactors, the four Medici brothers (the book itself was dedicated to one of them who had been a former pupil). Seeking for funds for your science… things really haven’t changed very much in 400 years…

With Starry Night, SciDome can easily reproduce the date and situations of Galileo’s observations. Others have done this in the past, and I refer you to the excellent article by Enrico Bernieri called “Learning from Galileo’s Errors” published in the Journal of the British Astronomical Association, 122, 3 (2012) which goes through his observations in detail and discusses the errors which Galileo made.

Another excellent article is by Michael Mendillo in the Proceedings of the IAU Symposium No. 269 (2010) called “The Appearance of the Medicean Moons in 17th Century Charts and Books – How Long Did It Take?” which gives rich background on some of the aftermath of Galileo’s revelations.

I also highly recommend the Wikipedia article on Sidereus Nuncius as an excellent starting point in building your background information on Galileo’s first telescopic observations. In addition, Ernie Wright has graphically reproduced Galileo’s observations and placed them online in an excellent web presentation.

With the incredible talent of Steve Sanders (Eastern University Observatory Administrator), I have created a minilesson for Volume 3 of the Fulldome Curriculum which reproduces all of Galileo’s published observations of the Medicean moons.

Figure 2: SciDome presentation of Galileo’s first Jupiter observations from January 7, 1610 from Padua, Italy.

Using Padua, Italy as our observation location and the approximate times given for each observation in Sidereus Nuncius, we begin with the close-up view of Jupiter on the dome as seen on January 7, 1610 at approximately 6 PM local time. Next we place a slide of the view as drawn by Galileo in Sidereus Nuncius below the view to show just how accurate Galileo was in his sketches. The labels of the Galilean moons are then displayed.

Note that although all four moons presented themselves, Io and Europa were too close together to be resolved by Galileo’s homemade 20X telescope which suffered also from chromatic and spherical aberration. This is an important fact to remember, because essentially all of the “errors” which we will find in comparing his sketches to the actual viewing circumstances were because of his lack of resolution.

We proceed by advancing time in Starry Night so that the audience can watch the dance of the moons around Jupiter and stop at the next observations of Jupiter as recorded by Galileo, on January 8, 1610. Then his sketch of this configuration is displayed, and again we note the accuracy of his rough sketches.

The minilesson continues in this fashion, showing the moons moving from date to date and then presenting 21 successive sketches by Galileo as presented in Sidereus Nuncius. Galileo concluded after four nights of observations that these tiny “stars” were indeed most likely satellites of Jupiter, which was of momentous importance because it was the first time that moons had been discovered around another body.

It also indicated that a planet could move and moons “stay up with it” despite its motion, an Aristotelian argument once offered to discount that the Earth could be moving because, if it did, how could the Moon know enough to keep up with it? Obviously Jupiter had at least four moons and they had no problem staying with it!

I have found that going through many of these configurations with my students greatly enhances their appreciation of Galileo and the great discoveries that he made despite the limitations of his equipment. Presenting this minilesson engages students in the realization that Galileo was both an excellent and honest observer as well as a genius. His observations helped to lead to the downfall of the geocentric universe and the eventual acceptance of the heliocentric model

Tides Revisited and Explained

A historical lesson in Volume 2 of the Fulldome Curriculum is the astronomical aspect of the Boston Tea Party. As we learn, Colonists, furious at the tea tax, tossed chests of tea into Boston Harbor on the night of December 16, 1773. Eyewitnesses noted that the chests became stuck in the mud adjacent to the ship and piled up in such a way that the Colonists feared the chests wouldn’t float out to sea and might be partially recoverable. The low tide that night (between 6-9 PM local time), which occurred (coincidentally) simultaneously with the raid, was especially low that night.

Although our lesson concentrates on discovering the Moon’s phase that night, a key element is why both high and low tides were so extreme. It turns out that the Moon was close to New Moon phase and perigee that night, making the tides about as extreme as they ever get. So I’d like to concentrate on explaining why tides occur and why the Moon is the major cause of tides and not the Sun.

tides1The ancients knew that the Moon caused the tides because the tides roughly coincided with the position of the Moon in the sky. The interval between successive high and low tides is ~13 hours, and the high and low tides occurring the next day were approximately one hour later, mimicking the position of the Moon which moved about 13 degrees to the east each day.

The ancients also noted that the greatest tides occurred during New and Full Moons (spring tides) and the least extreme tides occurred at 1st and 3rd quarter (neap tides). But how the Moon caused the tides had to wait until Isaac Newton. I often ask students: “What causes tides on the Earth?” Their answer is always: “The Moon’s gravity pulling on the water of the Earth.” I then ask them, “Then why doesn’t the Sun cause the tides, since its gravitational pull on the Earth is 180 times stronger than the Moon’s (which is why we orbit the Sun and not the Moon!)?”

tides2The answer is that it’s the Moon’s differential gravitational force on the Earth that causes the tides. The Earth is close enough to the Moon that the Moon pulls considerably harder on the Earth’s near side compared to its far side. The percentage difference between the Moon’s gravitational pull on the Earth’s near side compared to the far side is 6.9%. The Sun, which is so far away from the Earth that it sees the Earth as essentially a point mass, pulls differently on each side of the Earth by a percentage difference of only 0.017%! Ok, most people (students) will buy this, but do NOT understand why the water bulges on both sides of the Earth.

Figure 1 illustrates the strength and direction of the Moon’s gravitational pull on the Earth at New Moon, i.e., the Sun and Moon are in the same direction. This was essentially the phase of the Moon during the Tea Party, i.e., a day or so past New Moon (thin waxing Crescent).

tides3Determining the differential force experienced by the Earth is equivalent to asking, “What does the Earth actually ‘feel’ due to the Moon’s uneven tugging on it?

We can calculate that by subtracting out the force vector that the Earth experiences at its center from all the other vectors. This is, by definition, the meaning of differential gravitational force.

To illustrate this, Figure 2 shows the inverse center vector’s tail (drawn in orange) placed at the head of each force vector. We will add this inverse vector to all five vectors and illustrate their resultants by white vectors in Figure 3.

For clarity, let’s just look at the resultant differential force vectors in Figure 4.

tides4As already noted, there is no vector at the center of the Earth because we added its exact inverse so that it canceled out. (That was the point – asking what does the Earth “feel” is a center of Earth force transformation.  Sounds cool anyway…) The remaining vectors make sense if you think about each one of them.

The resultant vector on the right occurs because the Moon pulls hardest on the near side, so subtracting the weaker central force resulted in a vector pointing to the right. The resultant vector on the far side to the left will also make sense because the Moon pulls least on that side, so subtracting the larger central force vector resulted in a force pointing to the left! The resultants on the top and bottom occur because of the slightly different directions that the Moon’s force pulls on those points compared to the direct line at the Earth’s center.

If we continued this exercise at 15-degree intervals (instead of 90-degree intervals) around the surface of the Earth, we would have the resultant vectors shown in Figure 5. The locus of all the vector heads has also been drawn in and this egg shape represents the gravitational equipotential of all the tidal forces, i.e., it’s this shape that the water will attempt to emulate.

tides5Ok, now what? Well, let’s label Boston in the diagram and talk about it – see Figure 6.

Remember that the Moon and Sun are off in the direction towards the right in all the figures. What approximate time is it in Boston in Figure 6? With the Sun highest in the sky, it would be noon. (I hesitate to say “directly over your head,” but you know what I mean.) What tide would the folks in Boston be experiencing at this moment?

The highest bulge is in the direction of the Sun and Moon, so they must be experiencing high tide. Now let’s advance 6 hours to sunset, i.e., let’s turn the Earth counterclockwise by 90 degrees, as depicted in Figure 7. (Pretend that the Moon hasn’t moved in those six hours, so that the tidal bulge is still in the same direction.)

This is essentially the situation during the Boston Tea Party! The Sun had set and the thin crescent Moon was setting. Boston was experiencing low tide at this time (as shown in Figure 7) and it was an especially extreme low tide because the Moon was (coincidentally) close to perigee, so its tidal effects were maximized. So, when would Boston experience its next high tide?

tides6Approximately 6 hours later, near midnight, when Boston was on the opposite side of the Earth relative to the Sun and Moon. Figure 7 implies that the high tide it experiences on the far side is not as high as the one on the near side, and this is exactly the case with the tides! The one facing the Moon is somewhat higher (by a foot or so) than the one on the far side! Most people don’t know this, since diagrams in books always draw the tidal bulge in equal amounts on both sides of the Earth, but (as you can see in our accurately drawn diagrams) they are wrong!

How do you implement this knowledge in the planetarium? If you understand the above analysis, you can understand that indeed the tides will be in synchronization with the position of the Moon relative to your location. So, if you have New Moon, then high tides at your location are going to occur (approximately) at noon and midnight as in our Boston Tea Party example. This is because New Moon is highest and lowest (near your nadir) in your sky near noon and midnight, respectively.

tides7Actually the tides lag by about an hour because the water is trying to keep up with the Moon and lags behind its position. Let’s not get too technical! We haven’t even worried about the irregular landforms of the continents and how they complicate the actual tide heights, like at the Bay of Fundy!

What about the tides at Full Moon? Full Moon will behave similarly to New Moon, i.e., high tides will occur when Full Moon is (approximately) highest and lowest in the sky, i.e., midnight and noon.

What about 1st quarter? Again, the principle is exactly the same! When 1st quarter is highest and lowest in the sky, you should be experiencing high tides. 1st quarter is highest at approximately sunset and lowest (near your nadir) at sunrise. Likewise, 3rd quarter is highest and lowest at sunrise and sunset (respectively).

Confused? It’s ok. What you need to do is understand that the tidal bulge (high tide) is always trying to point to the Moon. So high tide will follow the Moon and the Earth “rotates” within the tidal bulging water so to speak. If you carefully work through the explanation and figures above, you should be able to comprehend when to expect high and low tides! If you can convey this to your audiences, they will have re-discovered what the ancients knew and why knowing the position of the Moon in the sky was important to them!

Spitz Fulldome Curriculum – Volume 2

After years of development, Volume 2 of the Spitz Fulldome Curriculum has been released as a free supplement to all SciDome sites. The curriculum covers a wide gamut of subjects (see contents in sidebar) and gives educators a new library of slides, animations, and scene files to convey both straightforward and complex subjects.

Analemmas

pluto_analemma

As many of you know, I’m a fan of analemmas because understanding them leads to a wealth of knowledge about orbital dynamics. With Volume 2, we’ve added the ability to draw to scale analemmas on the planet, just like they were drawn on all quality Earth globes in the past.

The analemma is a tracing of the center of the Sun’s specular reflection on a planet as seen from the center of the Sun at the same time each solar day for that planet.

Time and Timekeeping

There are six mini-lessons which can be combined into one super-class on Time and Timekeeping. Breaking these topics into self-contained subprogram cue files allows the user much more flexibility to adapt the curriculum into their own presentations if they so desire.

Time Zones

Actual Time Zones - fulldome image

One of our favourite sections is the one on Time Zones which makes use of custom made slides showing time zones on the dome as they would be theoretically drawn, if no humans interfered. Then we crossfade into the real time zones. The Sun can also be moved along the zones to illustrate how time changes relative to universal time – a really cool effect!

New Astronomical Simulations

Steve Sanders, Observatory Administrator at Eastern University, has produced numerous original simulations for Volume 2. These include accurate three-dimensional rotating eclipsing binary stars synchronized with their actual light curves; three-dimensional constellations to clearly show their 3D aspects as seen from other places in the Milky Way; and original depictions of the regression of lunar nodes, and precession of the Earth. One of the most compelling simulations shows exactly why we only have eclipse seasons twice a year separated by approximately six months.

Lincoln Almanac Trial

regression_of_nodes

The regression of lunar nodes animation is used within the Lincoln Almanac Trial lesson to illustrate why the Moon’s high/low seasons were crucial in vindicating Lincoln’s defense of a client in 1857. Lincoln produced an almanac that said that the Moon was setting only three hours after transiting the local meridian – contradicting the witness’s claim that he saw the murder by the light of the moon.

Precession

The Precession video depicts what the precession of the Earth looks like from space and emphasizes that the motion of the equinoxes along the ecliptic is a simple consequence of this gyroscopic motion. The Eclipse video depicts the Earth-Moon system relative to the Sun through the year and wonderfully illustrates how the inclination of the Moon’s orbit causes its New and Full Moon shadows to only lie along the ecliptic during the two eclipse seasons.

You have to see these animations to truly understand how excellent they are! And since they are just mpg files they can also be shown in a classroom setting through a digital video projector.

Planet Locations

jupiter_line_of_sight

One of my primary purposes in planetarium education is to convey to my students straightforward ways to translate what they see in the sky into a fundamental understanding of why they are seeing what they see. This new class exploits the setting Sun’s position to then plot the positions of the other planets onto a curtate orbit chart which each student has on a clipboard.

I have had tremendous success using this method to convey the planets’ positions along the ecliptic to where they actually are in their orbits around the Sun, and the students never look at the skies the same way once they have worked through this class!

I have only touched upon some of the material contained within Volume 2. If you will take the time to peruse the numerous At A Glance scripts you will find an incredibly rich source of new astronomy topics for your audiences.

I’m very excited to have you explore the potential teaching features of Volume 2 with your students! Now it’s time to begin working on Volume 3. If you have any ideas that you’d like to see me create, just email me at dbradstr@eastern.edu.