Recently Spitz (thanks to Software Architect, Clint Weisbrod) implemented new guide lines in Starry Night, graphically linking the Sun and planets (see the fall 2018 Spitz Newsletter and the Copernican Method). This inspired me to consider an animated, user-controllable version depicting Kepler’s Second Law. I have already implemented static images created by Steve Sanders (Eastern University’s Observatory Administrator) for teaching Kepler’s Second Law in Volume 2 of the Spitz Fulldome Curriculum, but static images pale in comparison to animations depicting the same concept.
Figure 1 – Kepler’s Second Law from FDC Volume 2
Kepler’s Second Law states that a radius vector connecting a planet to the Sun will sweep out equal areas in equal intervals of time. This paradigm-shattering result enabled the understanding of how planets change velocities in an orderly and systematic fashion (they are actually conserving angular momentum, but that realization would have to wait for Newton). Historic models mimicking the changing speeds of planets were complex and impressive but completely unwieldy when it came to calculations. They were also inaccurate.
What we devised (to be included in Fulldome Curriculum Volume 4) is a new Starry Night feature allowing Kepler’s Second Law to be illustrated for bodies in the solar system (planets, moons around planets, asteroids, comets, etc.), as well as exoplanets around their parent star. The feature allows operators to make a real-time graphical version of the sweeping sections of orbits.
The controls for making Kepler Orbits are shown below:
The operator can enter time intervals for orbital segments (and we included a “Suggest Interval” function, which I almost always use, so beginning users have help in creating the segments). We can show individual orbital areas, and label the results.
Figure 2 – Kepler Orbit Section option dialog for Mercury
Figure 3a shows the result of running time for Mercury and Figure 3b displays the completed orbit. The numbers in each orbit segment are the numerically integrated areas of the segments, each of which is being calculated for the exact time interval set in the input box of Figure 2.
Figure 3a – Mercury beginning to draw Kepler Orbit sections in 7 day intervals
Figure 3b – Mercury completed Kepler Orbit sections in 7 day intervals
Note the Expected Area displayed at the bottom of the figures. This is the analytically calculated area for an orbital segment for Mercury given the time interval specified by the user. You may think, “Of course they are the same.” If you know anything about numerical analysis, it’s quite impressive that the numerical integration techniques implemented in this routine are accurate enough to reproduce the analytical prediction, i.e., validating Kepler’s Second Law.
We are so used to seeing equations predict outcomes that we’ve lost the astonishment of Kepler and Renaissance scientists for the precise nature of the universe, that it could be accurately modeled by mathematical equations!
Let’s take a look at this feature for Halley’s Comet. Because of the extreme eccentricity typical of comets, we’ve only displayed the perihelion passage of this body, as illustrated in Figure 4.
Figure 4 – Perihelion passage of Halley’s Comet in 1986
Figure 5 displays the eccentric (0.56) Venus orbit-crossing asteroid 27 Apollo around the Sun. Venus’ orbit is shown to scale.
Figure 5 – The orbit of the Venus orbit-crossing asteroid 27 Apollo
Figure 6 displays the very eccentric (0.75) orbit of Nereid, a moon of Neptune. I purposely didn’t display the numerically calculated areas to show what that looks like, i.e., to clean up the display.
Figure 6 Nereid’s orbit around Neptune, with an eccentricity of 0.75.
As a final example, Figure 7 displays the orbit of the exoplanet HD 87646b. It has an eccentricity of 0.500, an orbital period of 674 days and a semimajor axis of 1.580 AU.
Figure 7 – The orbit of exoplanet HD 87646b.
I sincerely hope that this upcoming new feature in SciDome excites audiences as much as it does all of us who have worked on implementing it! I also am extremely hopeful that when audiences see Kepler’s Second Law in action that they will finally be able to understand more fully what is meant by it.
Tomorrow, Friday, will be the 50th anniversary of the launch of Apollo 8, the first crewed space flight to orbit the Moon. You can simulate Apollo 8, and the other eight Apollo missions that went to the Moon, on your SciDome.
First, make sure that ‘Space Missions’ are checked to be visible in your View Options pane. If you type in ‘Apollo’ in the Starry Night search engine pane, each mission will come up, and you can break each one down by “Mission Path segments” that each describe a phase of flight, and look at the Command Module and Lunar Module separately at relevant points.
These missions can only be seen when Starry Night is displaying the right time between 1968 and 1972, which you can get by right-clicking on the mission you want and selecting “Set Time to Mission Event…” and picking “Launch”, for example. The best way to see the mission path of Apollo is to be looking at it from well above the Earth’s surface, with ‘Hover as Earth Rotates’ set so that the Earth’s surface can rotate underneath you and the fixed stars stay fixed on the dome.
Apollo 8 follows a curving path out from the Earth to the Moon, orbits around the Moon ten times, and then returns to the Earth. The different mission path segments are different colors.
You can see that the spacecraft orbits around the Moon from lunar west to lunar east. However, when we look up at Apollo’s path around the Moon it appears to be opposite the path that spacecraft orbit the Earth, even though everything launched from Cape Canaveral also goes towards the east. The Apollo spacecraft were launched into a figure-eight trajectory, so the “patching of conics” that reverses the frame of reference is like when two people shake hands on their right side. From one person’s point of view the other person is shaking their left hand, even though both participants are using their right.
The Apollo spacecraft and the Saturn V rocket are rendered in 3D in Starry Night if you go to them and look at them up close. The Apollo 8 spacecraft is pointed at the Earth by default.
The famous “Earthrise” photo was taken at the beginning of the fourth orbit, on December 24th, 1968, at about 16:25 Universal Time, as shown in SciDome. There is some question of which of the three astronauts – Commander Frank Borman, Command Module Pilot Jim Lovell, or “Lunar Module” Pilot Bill Anders – took the photo, and the question was resolved by Apollo historian Andrew Chaikin, who recounts his investigations in Smithsonian Magazine.
In Starry Night Preflight’s ‘SkyGuide’ pane there is a section on the Apollo missions, and Apollo 8 has 13 sub-headings that go into phases of flight like the Earthrise photo in some detail. Each subheading calls up a Starry Night application favourite scene that describes that phase of flight, with some text and images that appear in the SkyGuide pane.
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.
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:
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.
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!
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!
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!
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:
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.
For the last couple of days, there has been some news coverage of another small asteroid that’s going to fly close to the Earth tonight. This happens fairly often, although it is a little unsettling when it does. We have just passed the 9th anniversary of the discovery of a small asteroid that was on a collision course with Earth in October 2008. That object produced a fresh field of meteorites in Sudan.
The new asteroid 2012 TC4, was discovered on October 4th, 2012. It flew past Earth on October 12th of that year, 96,000km away. It’s only 13 meters across, and small asteroids like it and the other one tend to be discovered only when they are close enough to the Earth to be visible in large telescopes. Because the orbit of 2012 TC4 is tangent to the orbit of the Earth in October, the only time during the year when Earth and TC4 can be close together is in October. Since that 2012 encounter, the period of TC4’s orbit has been 1.67 years, so we are bound to be close together again after a five-year interval.
After this evening’s encounter, when only 43,000km will separate Earth and TC4, the period of its orbit will be lengthened to 2.06 years. The asteroid orbit will still be tangent to Earth’s orbit in October, and that extra 0.06 of a year will accumulate and may put the asteroid back close to Earth in October 2033 or 2050.
The way Starry Night simulates asteroid orbits in a conventional way depends on those orbits not changing very much. The Keplerian orbital elements model just requires six numbers that describe the orbit of an asteroid around its parent body. Starry Night models these numbers with an accuracy of up to 7 decimal places, and that’s accurate enough to describe asteroids in most of interplanetary space. Keplerian orbits do not simulate the way the Earth’s gravity can deflect the orbit of an asteroid around the Sun, and any asteroid that passes really close to the Earth will be deflected in that way. So the best way for us to simulate TC4’s encounter with Earth in SciDome is unconventional.
Simulating TC4 on SciDome
I have prepared a “Space Missions” file, composed of a set of state vectors from the JPL website that describe TC4’s path for the 8-day period centered on this week’s encounter. It accurately models the way TC4 will sneak as close to Earth as the belt of geosynchronous satellites, and the orbit should be accurate to 0.1km and about 10 seconds in time.
The space missions dataset, and the JPL data format that can make data for it, are originally meant to simulate spacecraft, not asteroids, but the way the data is presented is mostly the same. Although if you “fly to” TC4 in Starry Night, it will look like a space probe and not an asteroid.
If you have a recently updated SciDome, you can get 2012 TC4 on your dome by downloading this zip file, opening it up, then installing “2012 TC4.xyz” on your Preflight computer at the following folder location:
It may be necessary to create a “Space Missions” folder in Sky Data here. If it is, it should be named just so.
If you have an older SciDome, the destination folder for the new file is a little different. Please contact me for directions.
With that done, the next time you run SciDome V7, you ought to be able to find 2012 TC4 by typing it into the search engine field at the top of the pane you choose to use as your “Find” pane. The “Celestial Path” or “Local Path” can be highlighted to show its course across Earth skies, and its “Mission Path” can be highlighted to represent its three-dimensional path through space around the Earth. The position of TC4 will only be simulated during the current 8-day period. If interest in it persists, its new orbit ought to be stable enough to represent with Keplerian elements after everything settles down.
TC4 will be passing through the constellations Aquarius, Capricornus and Sagittarius this evening. I used the JPL data to set up a prediction and make a reservation to use a 150mm online telescope in New Mexico tonight to try and take some images of TC4 as it passes by. We’ll see how it goes. Please feel free to contact me if you would like a little more guidance on installing 2012 TC4 on your SciDome.
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:
… 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, notthe 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!