Solar System Scale: Copernicus’ Method

Nicolaus Copernicus’ (1473 – 1543) paradigm-changing work de Revolutionibus Orbium Coelestium (On the Revolutions of the Celestial Spheres) famously laid the groundwork for the overthrow of the geocentric universe that had held sway for millennia. But what many people are not aware of is that Copernicus’ heliocentric system allowed for the first scale model of the entire known solar system in terms of the size of the Earth’s orbital radius.

What I find most incredible is that the values he determined, despite assuming incorrectly that the planets’ orbits must be circular with the planets traveling at constant orbital speed, are very close to the modern-day measurements (shown in Table 1).

In my never-ending quest to create meaningful and engaging planetarium curriculum, Clint Weisbrod and I have developed the ability to reproduce Copernicus’ method using SciDome and new features in Starry Night which allow us to do solar system geometry.

Table 1 – Results of Copernicus’ model versus modern values.

We begin by looking at the planets closer to the Sun than the Earth, the inferior planets Mercury and Venus (first denoted as such by Copernicus). Figure 1 shows the configuration for greatest elongation of Venus. By definition, this will occur when Venus appears to be the farthest away from the Sun as seen from Earth.

Via Euclid’s geometry we can prove that when Venus is at greatest elongation the angle at the position of Venus has to be exactly 90°. The line of sight from Earth to Venus must be tangent to Venus’ orbit, otherwise the line of sight would intersect the orbit in two places, both of which would display smaller angular separations from the Sun. The elongation angle θ is measured from the Earth as the angle between Venus and the Sun.

Knowing θ and that the angle at Venus is 90°, we can solve all sides of the triangle if we know one side of the triangle. Alas, we do not know any of the lengths, but if we define the distance from the Earth to the Sun (the hypotenuse) as 1 astronomical unit (1 AU), then we can immediately calculate the side of the triangle opposite the elongation angle θ as

Venus distance from Sun = (1 AU) sin θ

Figure 1 – The geometry (greatest elongation) of inferior planet distances from the Sun.

Figure 2 shows this configuration as seen from a top-down view of the Solar System in Starry Night using the new Copernican Method lines.

The value of the radius of the orbit of Venus (again assuming a circular orbit) is

Venus distance from Sun = (1 AU)sin(45.9°) = 0.72 AU

This is a remarkably accurate result, mostly due to Venus’ nearly circular orbit! The results for Mercury are not as accurate, but of course Mercury’s orbit is far from a circle. However, if enough measurements are made of multiple greatest elongations, the average will come out to be a fairly close estimate to the modern-day value.

Figure 2 – The Copernican Method lines for Venus in SciDome.

 

What’s awesome about the new feature in SciDome is that we can actually display the Copernican Method lines as seen from Earth as well as from space, as shown in Figure 3.

The line drawn between Venus and the Sun (just below the horizon) also displays the angular separation of the two bodies (45.9°) and the angle at Venus is the angle made between that line and the Earth’s line of sight (89.9° – close enough to 90° for government work).

The greatest elongation angle (45.9°) can be measured (in modern times) using a sextant (invented in 1715), so this new Copernican Method lines feature allows us to draw “sextant measurement lines” between the Sun and the planets.

Figure 3 – Venus’ greatest eastern elongation as seen from Philadelphia on August, 15, 2018.

Determining the sizes of the orbits of the planets further from the Sun than the Earth—the superior planets—is not quite so straightforward. The method is explained below, remembering again that we’re assuming the orbits are all circular and the planets are moving at constant orbital speeds.

Figure 4 shows the geometry of the situation for superior planets. We begin by noting the date (Julian Date) of an opposition of the planet, when the planet and the Sun are on opposite sides of the Earth—the superior planet would rise when the Sun sets, and be highest in the sky (on your local meridian) at midnight. We then wait and observe the planet until it is 90° from the Sun, a position Copernicus defined as quadrature.

We calculate the number of days that have passed since opposition, and this will allow us to calculate how many degrees each planet has traversed in their respective orbits. For example, the Earth takes 365.2422 days to cover 360° (again, we’re assuming circular orbits and constant orbital velocity) so it will move

 

Figure 4 – The geometry of measuring the size of the orbits of the superior planets.

A similar calculation can be done for all the planets since Copernicus had calculated the sidereal periods of all the visible planets (see the Synodic Periods minilessons in Volume 3 of the Fulldome Curriculum). For example, Jupiter’s sidereal period is 4332.59 days yielding an angular rate of travel in its orbit of 0.0831 deg/day.

Since we know how many days it took for the planets to reach quadrature from opposition, we can immediately calculate how many degrees each planet traveled in their respective orbits. The Earth will travel a greater angle in its orbit in this time, and the difference between these two angles is the angle θ shown in Figure 5.

Assuming that the distance from Earth to the Sun is 1 AU, we can calculate the distance from the Sun to Jupiter (the hypotenuse of the triangle in Figure 5) as 

So, what Earth observers would need to do is to measure the number of days from the opposition of a planet to the next quadrature, calculate the difference in degrees traveled between the two planets, and then take the reciprocal of the cosine of that angle to calculate the superior planet’s distance from the Sun.

Figure 5 – The quadrature triangle to solve for Jupiter’s distance from the Sun.

In the case of Jupiter, one set of measurements placed opposition on May 9, 2018 (JD 2458247.75) and the following quadrature on August 6, 2018 (JD 2458337.492), for a difference in days of 89.74. This value, multiplied by the difference in angular velocities between Earth and Jupiter, yielded a θ = 81.0°. This resulted in an orbital radius for Jupiter of 6.41 AU, whereas the modern value is 5.20 AU (a 23% difference). The view from Earth of this quadrature is shown in Figure 6.

At first glance, this value seems to be significantly off from the modern value…and it is! Have we made a mistake, or is this yet another opportunity to encourage our students to think? What assumptions have we made that are probably not accurate? We (as did Copernicus) assumed circular orbits and constant orbital velocities, and neither of these assumptions is correct for any of the planets!

So how can we use this method and the (wrong) assumption of circular orbits and constant orbital velocities to arrive at relatively accurate values for the sizes of the superior planets’ orbits?

The answer is to take multiple measurements over at least one orbital cycle of the planet in order for the answers to average out to some median value which indeed will approach the modern-day value. I did this for Jupiter taking 11 successive opposition-quadrature pairs over one 11-year cycle of its orbit from 2007 to 2018. When I averaged these 11 determinations I obtained a value of 5.46 AU, an error of only 5% from the modern value.

Don’t see this as a problem but rather as a very teachable moment for your students. You might challenge them as a class to take multiple measurements of successive opposition and quadrature pairs and they can watch for themselves how the values average out to close to the modern-day value. They can see for themselves how the errors introduced by our assumptions of circular orbit and constant orbital velocities can be minimized (but not eliminated) by multiple observations.

It’s appropriately mind-blowing to see the genius of Copernicus through these observations that your students can now undertake for themselves in the SciDome planetarium! They will gain a much greater understanding of how Copernicus created his solar system scale model, as well as see how these measurements could actually be made from the ground.

Figure 6 – The quadrature of Jupiter as seen from Philadelphia on August 6, 2018.

Spitz Fulldome Curriculum Volume 3 Overview

I’m excited to announce that Volume 3 of the Spitz Fulldome Curriculum is being released to all SciDome users, and will of course be automatically incorporated into all future SciDome installations.  We thought that this would be an opportune time to give a very brief overview of what’s contained in this volume.  There are several revisions to previous minilessons as well as several all new offerings:


 

Galilean Moons

This minilesson gives 26 examples (in order of date) of Galileo’s first observations of the four major moons of Jupiter during the winter of 1610.  The actual configuration of each night is beautifully displayed on the dome by Starry Night and then Galileo’s sketch is presented directly underneath it so that your audience can compare the sketch to reality.  You will be astonished at Galileo’s accuracy, as well as the restrictions of his poor optics and resolution that confined his work.  My students enjoy these comparisons even more than I do!

 


 

North Celestial Pole (NCP) Altitude

My students always scratch their heads when presented with the idea that the North Celestial Pole is always the same number of degrees above your horizon as your latitude.  This series of overlaying diagrams attempts to clearly lay out exactly why this is the case.

 

 


 

Planetary Tilts

Steve Sanders, Observatory Administrator at Eastern University and my right hand man, came up with this idea to beautifully illustrate the various planetary axis tilts side by side as well as their rotation periods.  This animation is so impactful that the folks at ViewSpace used it in one of their presentations last year!


 

Quasars Fulldome

This is one of my all time favorite mind-blowing demonstrations!  In a series of overlaying fulldome illustrations (again created by Steve Sanders), the second cosmological principle of the universe looking the same everywhere is demonstrated by using the appearance of quasars as seen from any galaxy, starting from the Milky Way.  Your audience will be left awestruck when they discover that the Milky Way is a quasar as seen by a distant galaxy which to us looks like a quasar!


 

Roemer’s Method Revised

One of my favorite minilessons from Volume 1, we’ve revised this presentation with a new animation by Steve Sanders which very clearly shows the concept behind the light time effect and how Roemer was the first to demonstrate that the speed of light was finite and approximate its value.  You can not only show this effect to your audience but make an incredibly precise and straightforward measurement from it of the speed of light!


 

Solar System Scale Revised

I still use this minilesson in nearly every one of my presentations and for all ages.  We have greatly improved the graphics used in this minilesson and I know you will like the results!

 

 


 

Stellar Sizes Revised

Like Solar System Scale, I use this minilesson frequently in most of my presentations, and we’ve revised it by adding a final graphic at the end which shows VY Canis Majoris in its entirety on the dome in one final scale shift.

 

 


 

Synodic Periods of Mercury, Venus, Mars and Jupiter

These are my favorite new additions in Volume 3! Each is a separate minilesson and carefully steps the audience through how Copernicus disentangled synodic periods of the planets into their sidereal periods around the Sun! Although very few people have ever been taught this concept, it’s very straightforward and illuminating when you see it on the dome. Test one out for yourself and you’ll be hooked!

 


 

Titius-Bode Rule

We often mention this infamous “Law” in our astronomy classes, so I wanted to present it in a historical fashion to demonstrate what effect it had on astronomer’s thinking when the Solar System was being explored and new planets being discovered.  It’s the perfect example of a mathematical oddity that may or may not be scientifically meaningful.  I think you will find it a fascinating subject as presented on the dome in this minilesson!


 

Watery Constellations

This little minilesson playfully depicts the fact that the region of the sky known as “The Sea” by the ancients has water-related constellations residing in it for a specific reason, namely that the Sun traversed this part of the sky during the rainy season in the Mediterranean. You will also be able to show your audience in a natural way that the position of the winter solstice used to be in Capricorn around 1000 BC, and hence that latitude parallel is called the Tropic of Capricorn.

 


Perhaps the greatest contribution to the official contents of Volume 3 is the availability of three unique fulldome interactive programs: Epicycles, Newton’s Mountain, and Tides.  These three programs allow you to clearly demonstrate subjects which I have found extremely challenging for my students:

  • Epicycles shows many of the intricacies and systematics of the simplified Ptolemaic geocentric system and will alert your audiences to the vagaries of “saving the model at any cost.”
  • Newton’s Mountain is a 21st century interactive version of Newton’s attempt to explain exactly what an orbit is allowing you to show your audience in real time different orbits as a cannonball literally falls around the Earth.
  • Tides shows exactly why the Moon causes the water to bulge on either side of the Earth via differential gravitational forces as well as demonstrating that the bulge is not the same on both sides!

REQUIRES WINDOWS 7 ON THE RENDERBOX COMPUTER. Multi-projector systems must be based on Scaleable – not compatible with EasyBlend.

These three programs require purchase because of the many years of work which went into their development and implementation. They are now available for online purchase and immediate download:

Purchase Astrophyics Apps

I hope that you and your audiences thoroughly enjoy this latest addition to the Fulldome Curriculum, and that they will be helpful as you continue to strive to educate people in the subjects that we all love.

Astrophysics Apps for SciDome

Three long-awaited fulldome astrophysics apps created by Dr. David H. Bradstreet are now available for purchase and immediate download for installation on SciDome systems. Tides, Newton’s Mountain and Epicycles are selling for $200 individually or $500 for all three.

REQUIRES WINDOWS 7 ON THE RENDERBOX COMPUTER. Multi-projector systems must be based on Scaleable – not compatible with EasyBlend.

Purchase Astrophyics Apps

These programs teach the difficult concepts of tides, orbital motion, and epicycles in unique ways on your dome.  All three programs are completely controllable via an intuitive interface on your Preflight computer.  In addition, Tides is controllable via SciTouch for a seamless teaching experience for your audience.

The respective At A Glance teaching guides are all available for free download if you’d like to preview what you can do with these unique Fulldome interactive programs:

Upcoming Fulldome Curriculum Lesson: Titius Bode Rule

Volume 3 of the Fulldome Curriculum includes a lesson based on the Titius-Bode “Rule.” In this new teaching module we present the orbits predicted by the Titius-Bode relation in a historical timeline compared to the actual planetary orbits to show students why this apparent rule was important in 18th and 19th century astronomy.

The Titius-Bode “Rule” purports to describe an apparent mathematical correspondence in the sizes of the orbits of the classic planets in our Solar System. Although the idea of some kind of relationship had been hypothesized before Johann Daniel Titius and Johann Elert Bode, their publications in 1766 and 1772, respectively, brought this relation into the limelight of astronomical thought, and hence it is named after them.

The idea is that there is a mathematical relationship between each of the orbits of the classic planets. Usually it is presented in the following form:

d=0.4+0.3x2m

… where m = -∞, 0, 1, 2, 3,… and d is the semi major axis of the planet in astronomical units.

Historically, this relationship was believed to be revealing something intrinsic about the positioning of the planets in the Solar System, that there might have been some type of resonance phenomenon within the formation of the planets within the solar nebula. The reason for this belief came out of the astronomical discoveries which were made subsequent to its popularization in the 18th century. To see this in its historical context, let’s set up a table the way it would have been constructed in the late 1700’s:

Interesting results, but the huge gap between Mars and Jupiter posed a real problem!

SciDome view showing Uranus’ orbit
compared to the Titius-Bode prediction

Shortly after the Titius-Bode “Law” became publicized, William Herschel in 1781 discovered a new planet, Uranus! This was a paradigm changing discovery, but what was just as incredible was that its semi major axis was calculated to be 19.2 AU, nearly doubling the size of the Solar System! Just as remarkable, the next predicted semi major axis from the Titius-Bode “Law” was 19.6 AU, only 2.1% different from the measured size!

This discovery started astronomers thinking that perhaps there was more to the Titius-Bode “Law” than they once thought, that perhaps it wasn’t coincidence but was revealing a yet undiscovered physical relationship within the Solar System. Twenty years later, on the first night of the new century, 1801, Father Giuseppe Piazzi discovered a new “planet,” later named Ceres.

What was truly remarkable about this new planet was that it’s semi major axes was eventually calculated with a new mathematical method by Carl Friedrich Gauss to be 2.8 AU, nearly exactly what the Titius-Bode “Law” had predicted for a planetary body residing in the gap between Mars and Jupiter!  Of course soon thereafter many more bodies were discovered to reside within the gap, and by the 1850’s these objects were renamed asteroids.

However, the belief in the Titius-Bode “Law” was gaining new proponents, since it seemed to have predicted positions in which Solar System objects were subsequently discovered! The next predicted orbit would lie at 38.8 AU, and the search was on for yet another planet! Sure enough, Neptune was discovered with the aid of Newtonian physics in 1846, but its semi major axis was 30.1 AU, not the 38.8 AU expected from the Titius-Bode relationship.

SciDome display showing the large discrepancy between Neptune’s orbit (30.1 AU) and the predicted Titius-Bode orbit of 38.8 AU

This large discrepancy led to the virtual abandonment of the Titius-Bode relationship as a physical law. However, it’s interesting to note that when Pluto was discovered in 1930 its semi major axis was determined to be 39.5 AU, very close to the previously expected distance. Of course Pluto has now been relegated to dwarf planet status because of the myriad of new objects which have been discovered in the Kuiper Belt.

The next expected semi major axis from the Titius-Bode relationship is 77.2 AU. And isn’t it interesting that Sedna’s perihelion distance is 76.1 AU, although its semi major axis is a whopping 506.8 AU!

The moral of the story seems to be that although the Titius-Bode relationship has never been convincingly proven to come from physical laws, it is noteworthy historically but also serves to perhaps warn us about jumping to conclusions even though the initial evidence may seem inviting. The Titius-Bode relationship is today such a controversial topic that Icarus, the main professional journal for presenting papers on Solar System dynamics, refuses to publish any articles on the subject!

Messier Mischief

What’s Up with the Pleiades Being M45?

Hubble image of the Pleiades (M45)

During a recent planetarium conference session, an interesting question came up about why the Pleiades is listed as M45 in Messier’s catalog. Few people know the reason for it.

Charles Messier is best known for his list of some of the best deep sky objects in the sky, and most everyone knows that he ostensibly put this list together to alert other sky watchers so that they wouldn’t mistake any of these objects for comets. Of course discovering comets was the big thing in those days because the comet was then named after the discoverer!

This reasoning begs the question as to why the Pleiades, the bright and nearby Seven Sisters open cluster (which has been well known since antiquity), was designated as M45! No one is going to mistake this for a comet, and everyone knew of its existence! What gives?

In reality, there’s more to this mystery than just M45.  Messier accidentally discovered M41 (an open cluster SW of Sirius) in 1765 – so at that time his list contained 41 objects.  He decided to publish the list in 1771, but that list had 45 objects.

Note the last 4 are well-known objects, objects that had been detected by the naked eye for many centuries:

Hubble image of the Orion Nebula (M42 and M43)

None of these objects could possibly be mistaken for a comet! Although no one knows for certain, it seems that Messier wanted to have a longer list with a more “rounded” number of objects in it than 41, hence the addition of four well-known objects for this first publication by measuring their positions himself.

My suspicion (and that of some others as well, see references) is that he wanted to have more objects in it than a well-known list published by Lacaille in 1755 which had 42 objects in it. While this is only speculation, it certainly makes sense from an egotistical point of view. After all, why else did people want to discover comets so badly?

M42 through M45 are all up in the late winter-early spring sky so markers could be placed on all four of them at once to emphasize this. This could make an interesting little side note planetarium lesson for your audiences.

Spitz is developing a Fulldome Curriculum Mini-lesson based on this idea in the future, but I thought I’d relay this interesting hypothesis beforehand in case you want to steal it for your own use.

References:

  • http://messier.seds.org/m-q&a.html#why_M42-45
  • Messier’s Nebulae and Star Clusters, Kenneth Glyn Jones (1968; 1991), p. 352
  • The Messier Album, John H. Mallas and Evered Kreimer (1978), pp. 1-16 (historical introduction written by Owen Gingerich, originally published in Sky and Telescope, August 1953 and October 1960)