Solar System Scale: Copernicus’ Method

Nicolaus Copernicus’ (1473 – 1543) paradigm-changing workde 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.

Roundness of the Earth

48 years ago last week Apollo 11 landed on the Moon. There is another anniversary last week that seems appropriate to mention at this point: On July 20th of 1925 the greatest scene in American legal history took place, and it was an astronomy lesson.

You’re probably familiar with the play Inherit the Wind, which was based on the Scopes Monkey Trial. In the summer of 1925, more specifically on July 20th, on the courthouse lawn in Dayton, TN, Clarence Darrow had William Jennings Bryan on the witness stand to respectively challenge and defend the state’s Butler Act that prohibited public school teachers from denying the Biblical account of the origin of humanity.

Darrow and Bryan were agreed on the terms of the Earth being a sphere, and that the Earth orbits around the Sun and not the other way round. Therefore it was necessary for them to interpret the biblical passages that seemed to indicate that the Earth was flat and that the Sun stopped at midday for Joshua.

Illustration of Erastothenes’ method by CMG Lee. CC BY-SA 4.0

That the Earth was round, and that the Earth was turning and the Sun was at the axis of the solar system was not difficult to accept in 1925. People were familiar with Eratosthenes’ 3rd-Century-BC experiment in Egypt to estimate the circumference of the Earth (252,000 stadia.) They were also familiar with the great American novelist Washington Irving’s biography of Christopher Columbus, which laid out Columbus’ theory of the roundness of the Earth and his discovery of America obstructing the route to India.

That the Earth was round was also not difficult to accept in the 1480s when Columbus solicited the crowned heads of Europe to fund his voyage to India. It’s just a simplification of Washington Irving’s biography of Columbus to say that Columbus was trying to prove that the Earth was round and that his opposites held that it was flat. In the 4th chapter of the biography, the author puts Columbus in front of the School of Salamanca where he is criticized for the way he contradicts classical dogma from Saint Augustine in the 4th Century AD concerning the “Doctrine of Antipodes“.

In modernity, the antipodes are the geographic point opposite one’s position on the globe, but these medieval Antipodes were the mythical people supposed to inhabit the southern hemisphere who walked upside down (antipode meaning “reversed feet.”) but Saint Augustine did not dispute that the Earth was round:

“As to the fable that there are Antipodes, that is to say, men on the opposite side of the earth, where the sun rises when it sets to us, that is on no ground credible. And, indeed, it is not affirmed that this has been learned by historical knowledge, but by scientific conjecture, on the ground that the earth is suspended within the concavity of the sky, and that it has as much room on the one side of it as on the other: hence they say that the part which is beneath must also be inhabited. But they do not remark that, although it be supposed or scientifically demonstrated that the world is of a round and spherical form, yet it does not follow that the other side of the earth is bare of water; nor even, though it be bare, does it immediately follow that it is peopled.”

Columbus’ critics in the Inquisition, if any, subscribed to dogma that the Earth was round but that human civilization was limited to the temperate zone of the northern hemisphere by the Torrid Zone at the equator. That there was a corresponding southern temperate zone in the southern hemisphere, but that humans created in Genesis could not exist there because the Garden of Eden was in the north and the Torrid Zone was impassable or nearly so. That navigation to get there wasn’t easy because there was no North Star in the south, and the Doldrum Belt made headway under sail to the opposite end of the Earth impossible. The 1st-Century-BC Roman writer Cicero had written about the impassable Torrid Zone in an item called the Dream of Scipio, which is a good basis for an old-timey planetarium show in itself.

“Moreover you see that this earth is girdled and surrounded by certain belts, as it were; of which two, the most remote from each other, and which rest upon the poles of the heaven at either end, have become rigid with frost; while that one in the middle, which is also the largest, is scorched by the burning heat of the sun. Two are habitable; of these, that one in the South—men standing in which have their feet planted right opposite to yours—has no connection with your race: moreover this other, in the Northern hemisphere which you inhabit, see in how small a measure it concerns you! For all the earth, which you inhabit, being narrow in the direction of the poles, broader East and West, is a kind of little island surrounded by the waters of that sea, which you on earth call the Atlantic, the Great Sea, the Ocean; and yet though it has such a grand name, see how small it really is!”

It is true that Columbus was trying to sail around the world to reach India, and that he had underestimated the circumference of the Earth due to a conversion error from Eratosthenes: by the 15th Century, the value of 252,000 stadia was remembered, but the value of a stadion was uncertain, and Columbus used the wrong value. Therefore the Earth seemed smaller, and globes of the Earth from that period show the East Indies on the western edge of the Atlantic Ocean.

Columbus was convinced that the Torrid Zone was not a barrier to travel. Earlier in his career he had sailed to West Africa, almost to the Equator. The first European transit of the Cape of Good Hope (which is in the southern temperate zone) into the Indian Ocean was by the Portuguese navigator Bartolomeu Dias in 1488, two years after Columbus’s first unsuccessful examination at Salamanca.

This 1492 globe of the Earth is under a Creative Commons licence, so feel free to demonstrate it via its own API. It could be converted and wrapped around the Earth in Starry Night, but I don’t feel ready make the final product available for SciDome at this time due to the rights.

However, there are lots of ways to use SciDome to demonstrate that the Earth is round. The upcoming total solar eclipse is one event that is not easy for flat-earth believers to explain, when its occurrence is so accurately predicted with established science. Performing Eratosthenes’ experiment in SciDome is not difficult, by displaying the sky above his two observing stations in Alexandria and Aswan at local noon on June 21st with the Local Meridian switched on with graduations.

Now that we have established that the roundness of the Earth was accepted by both sides in the 1925 Scopes Trial, and that the roundness of the Earth was accepted by both Columbus and his critics (admitting serious gaps in the knowledge of both sides) and by the ancient Greeks, I hope that we can help elevate current concerns about the Earth being flat. I understand that a large billboard was recently used in suburban Philadelphia next to the freeway to state “Research Flat Earth”. And when we argue against modern flat-earth believers, we should not compare their belief to Columbus’s critics, and commit another simplification of the actual story.

Simulating Juno’s Buzz by the Great Red Spot

Great Red Spot cloud detail from Juno’s extreme close approach. Credit : NASA / SwRI / MSSS / Gerald Eichstädt / Seán Doran © cc nc sa

One of the astronomical highlights of last week was the pictures returned by the Juno spacecraft orbiting Jupiter when it zipped over the Great Red Spot at an extremely low altitude (8000 km.) Although the JunoCam camera on this mission was an afterthought for public outreach purposes and not a research experiment, the camera has returned some data that can be amazing when processed, and shows no signs of stopping yet, despite Jupiter’s harsh radiation environment.

To simulate this mission in Starry Night Version 7 on a Spitz SciDome planetarium, a couple of changes need to be made, even with recent updates. But with those changes made, you can simulate this flypast in Starry Night, and also think about using some of the real camera images from Juno on your dome as slides with ATM-4.

Firstly, we need to update the Space Missions file Juno.xyz. Starry Night V7 may already have a version of the mission path, but that is the *planned* mission. An anomaly in Juno’s rocket engine led to a revised mission plan with a different path. The original path does not include a periapsis over the Great Red Spot on the date in question, July 10th. To update the mission path, download this zipped folder, unzip it, and move the contained file Juno.xyz to the following location:

C:\ProgramData\Simulation Curriculum\Starry Night Prefs\Sky Data\Space Missions\Juno.xyz

This change is only made in one networked location to affect both computers, to avoid tediously installing it on Preflight and Renderbox in two steps. Files added to the “ProgramData” Sky Data folder will override files with the same names added to the old-fashioned Sky Data folder in the folder “Program Files (x86)”. The “ProgramData” structure exists so that V7 users no longer need to tediously make changes to Program Files on either computer.

Secondly, the position of the Great Red Spot needs to be updated. Jupiter is not a solid body, and the Great Red Spot has a tendency to drift, and its drift rate has a tendency to change, generating an accumulating error. So it’s not practical to just use the GRS as the index for the fixed period of rotation of Jupiter that is mapped out by the surface texture in Starry Night. The value of the drift is currently about +5° per month, and the current value of the drift is about 271° in Jupiter System II longitude. (Last week I was using a value of 269° and that also came out pretty good: 269° represents the value during the Juno encounter.)

To edit the value in Starry Night V7 for SciDome, locate the following file and open it using Wordpad (not Notepad:)

C:\Program Files (x86)\Starry Night Preflight\Sky Data\JupiterGRS.txt

You may recognize that the code inside is a little odd: Double slashes in odd places. If you are familiar with the coding, these slashes take on added significance. They should each represent the beginning of a new line of code that should be ignored by the program.

The only part of the file that is read by the program is the line that does not begin with two slashes. Please edit the file if necessary so the text is as follows, and the value is updated:

// Enter the mean longitude of the Great Red Spot on the following line. Visit
// the Starry Night Pro website at http://www.starrynightpro.com to get the
// latest value.
269.0

Then save the file into the ProgramData folder as follows, in a single step:

C:\ProgramData\Simulation Curriculum\Starry Night Prefs\Sky Data\JupiterGRS.txt

Once again, saving in this location means it’s not necessary to save changes on the other computer as well.

Artist’s rendering of the Juno spacecraft.

There is a 3D model of the Juno spacecraft in SciDome version 7, so you ought to be able to simulate its swooping down on the Great Red Spot in different ways: A long view of Jupiter with the Juno “Mission Path” turned on and the spacecraft labelled as a dot, or also flying alongside the spacecraft 3D model as the GRS looms on the limb of Jupiter overhead.

If you are using Starry Night Version 6 for Scidome, you can still place the attached Juno.xyz in the Space Missions folder of the original Sky Data folder on both computers and chart the updated path, but there is no 3D model of the spacecraft available. There is a separate 3D model that represents the asteroid (3) Juno, and they could get mixed up.

Because the GRS will continue to drift, you may wish to return to make subsequent edits to JupiterGRS.txt. The drift currently accumulates +5° per month, but because the drift rate can change, I recommend doing one of two things:

1) Now that you have the Juno simulation of what will probably be the best and closest images of the Great Red Spot for our lifetime, don’t make any further changes to the GRS value. Further edits to the drifting value will start to “break” the position of the GRS during the Juno flyby on July 10th, if you have built an ATM-4 automation out of it.

2) Continue to update the GRS position to represent reality based on observations, not predictions to avoid accumulating drift error. The current System II longitude of the GRS is kept up to date in a couple of places on the Internet, such as CalSky.

It is possible that Juno will have another encounter with the Great Red Spot on one of its remaining orbits, but the period of its orbit around Jupiter is 53 days. In multiples of 53 days the GRS position value will change by multiples of 9°, with some accumulation of error, and the orbital period of the spacecraft is not an integer multiple of the rotation period of Jupiter. Let’s wait and see.

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!