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.
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.
Witnessing a total solar eclipse could change your life. Unpredicted eclipses have changed the progression of historical events. The unexpected environmental effects of a sudden darkness at midday can be very unsettling to people and animals alike.
On July 29, 1878, Thomas Edison observed the total eclipse of the Sun as part of Henry Draper’s Expedition to Rawlins, Wyoming Territory. (Although this year’s total eclipse will pass through Wyoming, the City of Rawlins will be some distance south of the total path this time.) Edison was there to test a new invention that could detect infrared light and estimate the temperature of objects remotely, and he planned to try and estimate the heat of the Sun’s corona while the solar photosphere was blocked by the Moon.
Edison’s preparations for the eclipse were not as cautious as they should have been. Because total eclipses happen close to home so very rarely (2017 is the first year that the totality has appeared in US skies since 1979), and due to the practicalities of 19th century life, most astronomers arrived at the observing site early to uncrate their heavy observing equipment and build observing shacks, and mix concrete to make steady piers. Edison arrived with a few days left before the eclipse, without time to build a protective structure, so he used the existing shelter provided by a chicken coop. On eclipse day, as the sky darkened, the bewildered chickens literally came home to roost, disrupting his observations.
Illustration of Edison’s Tasimeter
Edison named the infrared-detecting device his “tasimeter”, and he was trying to break ground in a new scientific field in competition with the director of Pittsburgh’s Allegheny Observatory, Samuel P. Langley. Instead of letting Langley test the tasimeter’s eclipse performance alongside more proven devices such as the thermopile, Edison tried to “scoop” the competition with his solo work, and failed.
Although Langley also failed to estimate the heat coming from the solar corona with his thermopile, less than a year later he had invented the first bolometer, which is commonly used today in its most refined designs. Langley’s bolometer was so sensitive that it could detect the heat produced by a cow from a quarter of a mile away.
Although the temperature of the Sun’s corona has since been measured with bolometers, the reason why it is so hot – about 1.5 million Kelvin, considerably hotter than the Sun’s surface (5800K) – remains a theoretical problem in physics to this day.
Therefore, if you want to simulate an eclipse, or to prepare for the real thing, don’t neglect the auditory experience. Consider all possible distractions that could disrupt observations in advance, and avoid them. If you are standing near a farm in Wyoming, as the darkness approaches the sound of confused barnyard animals will reach a crescendo.
The new book American Eclipse offers a more in-depth view of the Solar Eclipse of 1878, following the trevails of Edison, planet hunter James Craig Watson, and astronomer Maria Mitchell.
Great Comet of 1811 as drawn by William Henry Smyth
The May issue of Sky and Telescope magazine has a timely item about “Napoleon’s Comets”. The most important of these was the Great Comet of 1811, which was the brightest comet with the longest duration of brightness on record (260 days) until Comet Hale-Bopp shattered that record in 1997.
It is referred to as “Napoleon’s Comet” because of the Napoleonic Wars and the impending War of 1812, in which the United States was allied with France, Germany and Austria against Britain, Spain, Portugal, and Russia. The wars are the backdrop for the novel War and Peace by Tolstoy, and also the newly Tony-award-nominated Broadway musical Natasha, Pierre and the Great Comet of 1812 based on a small part of the novel.
The Comet of 1811 was discovered in March of that year in what is now the constellation Puppis, and it was very bright in the evening sky in September and lingered for the rest of that year. The head and coma of the comet was reported to be wider than the diameter of the Sun and it had a very long, bright tail despite not coming very close to the Earth. The Comet was held to be responsible for unusually fine vintages of French wines harvested from the Autumn 1811 grape harvest, and it is possible that Napoleon was influenced in his decision to invade Russia in June 1812 if he thought of the comet as a portent of victory.
In the US midwest, the Comet was visible during the New Madrid Earthquakes in December 1811. The Shawnee leader Tecumseh, who was born in the year of the Comet of 1769 and was named accordingly, invoked the Comet of 1811 as he built a confederacy of tribes which allied with the British in the War of 1812.
The Comet of 1811 is only mentioned on one page at the conclusion of the first half of War and Peace, but it’s misnamed the Comet of 1812. Accordingly, although the musical is titled Natasha, Pierre & the Great Comet of 1812, the Comet only appears in the finalé and is not depicted in the publicity for the production. You have to go and see it for yourself. Dave Malloy, the creator of the show, says the Comet nevertheless got into the title of the show “for cosmic epicness”.
The Broadway production this year has been nominated for 12 Tony awards, so I can’t imagine it not being talked about in planetariums.
You can add the orbit of the Great Comet of 1811 to your SciDome by right-clicking on the Sun in Starry Night Dome Preflight and selecting “New Comet…” In the orbit specification window that pops up, enter the following values:
Name: Great Comet of 1811
Pericentre distance: 1.0354120
Ascending node: 143.0497000
Arg of pericentre: 65.4097000
Pericentre time: 2382768.2562000
Elements epoch: 2382760.5
And in the “Other Settings” tab, change the Diameter to 40 km and change the Absolute magnitude to 0.
“X” out of the new orbit window and confirm you want the changes to be saved. Then quit out of Starry Night properly.
If you are using Starry Night Dome version 6, the comet will be loaded on to the Renderbox when Starry Night is properly exited and will be available the next time the application is started. Because it is a user-created object, though, it will be automatically “hidden” until you uncheck it in the “Hide” column of the Find Pane. Then you can save some favourites showing the sky in the year 1811 featuring it for later playback.
If you are using Starry Night Dome 7, the comet will be saved into a file named User Planets.ssd in the Preflight folder:
C:\Users\Spitz\AppData\Local\Simulation Curriculum\Starry Night Prefs\Preflight
And that file will need to be manually ported over to the corresponding location on the Renderbox. Future versions of Starry Night Dome V7 will make this process automatic.
Today is Benjamin Franklin‘s birthday under the calendar we use today, although he was born on the 6th of January of 1706. He was born before the Gregorian calendar reform was implemented in the English-speaking world.
The Gregorian calendar reform adjusted the way that leap years are counted. Instead of observing an intercalary day in February once every four years, the Gregorian observes one such day every four years except when the year is divisible by 100, except when the year is also divisible by 400.
The exact time it takes the Earth to go around the Sun 365.2422 days, not an integer number of days. During the Julian period the remainder was reduced from the 365-day year by adding a leap day every four years. However, the remaining error compounded. The Gregorian calendar uses 97 leap days every 400 years instead of 100 leap days, so the average length of the Gregorian year is 365.2425 days. The Gregorian change also ran a correction to delete the accumulated lag, which had grown to 10 or 11 days.
The new calendar came into force in Roman Catholic states in 1582.
Denmark switched to the Gregorian calendar in mid-February 1700.
The British Empire made the change in 1752.
For any celebrity birthdates you want to celebrate that are older than a certain limit and come from a certain country, it may be important to see if they should be read out as a Julian or a Gregorian date.
If you bring up Starry Night with a date of October 4th, 1582, and advance by one day, you can observe the 10-day correction when the Gregorian calendar was assumed.
The only alternative to observing the ten-day gap that followed in 1582 when looking at earlier dates is to use the proleptic Gregorian calendar, which eliminates the need for a Julian calendar correction when observing past dates. I wouldn’t recommend using the proleptic Gregorian calendar for earlier dates because the people of the time did not use it either, and Starry Night will read out those dates using the Julian.
By 1752, when the British Empire adopted the Gregorian, the accumulated error had grown to 11 days, and the change was reflected in the British colonies that later became the United States. Benjamin Franklin had already been publishing Poor Richard’s Almanac since 1733, and he included a long explanation of the calendar reform in the 1752 edition.
(When Abraham Lincoln used an almanac to show the phase of the moon during the Trial by Moonlight in 1858, as in our Fulldome Curriculum, he was taking a page from one of the most popular kinds of document in the English language other than the Bible.)
The calendar reform of 1752 didn’t catch everyone by surprise, and although the correction was run in September 1752, Franklin had adequate notice before his publication deadline the previous year. The British Parliament passed the new rule as the Calendar (New Style) Act 1750, although the code used for that legislation was “24 Geo. 2 c.XXIII”, meaning it was the 23rd piece of legislation that received royal assent in the 24th year of the reign of King George II. King George had commenced his reign in 1727.
The first point of the new law, before the Julian correction, was to correct the date of the beginning of the legal new year. Although different cultures have strong traditions to begin the new year on January 1st, even now it is impractical for all of our traditions to line up on a single start date: the school year and the NFL season being a couple of examples. The British Empire up to 1752 had observed the start of its legal year on March 25th. The law passed in 1751 corrected the New Year to January 1st at the beginning of 1752, so the official year 1751 was only 282 days long.
Starry Night does not incorporate any of the other weirdness around calendar reform, except that the new year always starts on January 1st, and there is a year Zero in between the BC and CE periods. (ATM-4 does not calculate a Year Zero).
Happy Birthday to Ben Franklin, who was not born on Blue Monday (It was a Sunday in both calendars, and the days of the week have never accumulated an error).
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.
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