Issues in Earth Science
“Eww, There’s Some Geology in my Fiction!”
Issue 13, May 2020
Suggestions for Activities and Discussions to accompany Readings of
Standing in the Shadow of Phobos by Mary Alexandra Agner
Demetrius, the young boy excited by science in this issue's story "Standing in the Shadow of Phobos," describes the transit of the Sun by the moon Phobos this way
"Transits are pretty special because you get to see both of the objects at the same time. And, you get to sort of feel how the entire universe is moving."
How very true. Even before we humans had the ability to measure the distance to stars, we figured out the relative distances of objects in space and something about their motions by watching which objects pass in front of the others. We saw the Moon pass in front of the sun and the planets. We watched planets pass in front of distant stars. Galileo, with his newly-invented telescope, watched the moons of Jupiter pass in front of Jupiter, and then, a while later, watched Jupiter pass in front of the moons. With this observation, he proved that not all objects revolve around the Earth, the beginning of the end for the Earth-centric model for the universe.
As Demetrius points out, watching a transit can be particularly helpful in constructing a mental picture of what is going on in space. In science, we call that mental picture a model. Constructing models is particularly important in doing science, and is one of the important practices of science promoted by the Next Generation Science Standards (2013). For our classroom resources this issue, we are going to consider how various models for the movement of objects in our solar system can be supported or disproven by observations of the motions of bodies in space.
Consider the possible solar system models below. Which of these models does Demetrius' observation of the transit disprove (select all that apply)?
The answer, of course, is it disproves models 1 and 4. In those two models, Phobos would never come between Mars and the Sun, and Demetrius could never observe a transit of the Sun by Phobos.
Now, if you have already talked about the model for our solar system with your class, some of your students may have identified ALL of the models as being wrong. We do think they are all wrong! But they are not all proven wrong by Demetrius' observation of the transit.
Often in the classroom, instead of teaching students to identify evidence and construct models, we ask them to learn, explain, and apply the models that we have given them. They then just "know" which model is correct and don't always think about evidence. Sometimes, we even make the model the evidence for itself—An example of this type of circular reasoning is when a student offers an argument like "We know that model 2 is wrong because we know that the Earth orbits the Sun, not the other way around."
What one additional observation of Phobos would Demetrius have almost certainly made, living on Mars, that would disprove both models 2 and 3?
If you need a hint, think about watching the sky at night. What would Demetrius see in the night sky, when the Sun is on the complete opposite side of Mars? What would this observation imply?
He would have observed Phobos in the sky at night. Meaning that Phobos sometimes appears between Mars and the Sun (the transit) and Mars sometimes appears between the Sun and Phobos (when he sees Phobos in the night sky). From this, he can figure out that Phobos must orbit Mars, not only the Sun or the Earth.
As mentioned above, the observation that not everything in our solar system orbits the Earth was a key argument that Galileo offered against a geocentric model for our solar system. Of course, Galileo could not watch the moons of Jupiter from the surface of Jupiter, like Demetrius watches Phobos from Mars. He had to figure out that the moons orbit Jupiter by watching from Earth. Below is an illustration of the positions of the moons of Jupiter on nine sequential nights (With thanks to an unknown source for this image).
Can you follow the orbits of each moon? Can you figure out how long it takes each moon to orbit Jupiter?
Note: the distances from Jupiter are in millions of miles.
The modern orbital periods for the Galilean moons (in Earth days) are: Io= 1.77, Europa = 3.55, Ganymede = 7.15, and Callisto = 16.69
The following puzzle in celestial mechanics comes from Learning to Read the Earth and Sky (2016) by Russ and Mary Colson, published by NSTA Press.
Often people imagine that the geocentric model of Ptolemy was like that shown below. But this is not true. Even without telescopes, ancient people knew this model was not correct. Consider this model, and think about what it predicts that one could see in the night sky. What observation that ancient people could easily make disproves this model?
Need a hint? Think about the time of day and locations in the sky that Venus could appear.
This model implies that Venus should sometimes be seen on the complete opposite side of the Earth from the Sun. For example, we should be able to see Venus high in the sky at midnight during times when Venus has orbited to the opposite side of Earth from the Sun. In fact, Venus—the morning and evening star—is never seen at midnight, nor is it ever seen high in the sky at night. Thus, ancient people knew this model was wrong. A better illustration of Ptolemy's model, also from Learning the Read the Earth and Sky (2016) is shown below. In this model, Venus and the Sun both orbit the Earth in lock-step. Notice that this model does not allow Venus to be on the opposite side of the Earth from the Sun.
Galileo disproved this model by observing the phases of Venus through a telescope, that is, he observed how much of the lit side of Venus can be seen from Earth at different times. Only a tiny sliver of the lit side of Venus could ever be seen if Ptolemy's model were right. Yet Galileo observed a nearly fully-lit Venus, proving that Venus must orbit the Sun. Try sketching models for orbits to show that this is so. Explain your reasoning!
More on this investigative problem is found in Learning to Read the Earth and Sky (2016).
Think about the model for the solar system shown below, showing Phobos orbiting Mars and Mars and Earth orbiting the Sun. What might the phases of Earth look like as seen from Mars? How much of the Earth would appear to be "lit" as viewed from Mars at each of the stages of Earth shown?
Your students might model the appearance of phases with reference to the illustration below.
You students might make a prediction of phases like that shown below.
You might encourage your students to also draw what the phases would look like if Earth and the Sun both orbited Mars (a Mars-centric solar system, much like the geocentric model of Ptolemy, except putting Mars in the center—Use the Ptolemaic geocentric model illustration above for reference in predicting the phases, but imagine Mars for the Earth and Earth for Venus). The phases in this case will look quite different—never being more than a crescent Earth. Thus, a Martian could watch the phases of Earth and see that there are gibbous and full-Earth phases—proving that the Sun, and not Mars, is the center of Earth's orbit.
How long do you think it would take the Earth to go through its set of phases if you viewed Earth from a fixed point out in space, at about the distance of Mars? Yes, you can figure this out!
Since it takes Earth 1 year to orbit the Sun, it will take one year to go through its set of phases.
What about the duration of the cycle of phases as seen from Mars? Would it be the same or different?
Since Mars is also moving as it orbits the sun, the duration of the cycle of phases of Earth as seen from Mars would be longer than 1 Earth year. Convince yourself by sketching out what the Earth phases would look like as Mars also orbits the Sun but not as fast as the Earth.
What would the phases of Mars look like as viewed from the Earth? Would they go through the same cycle of phases as shown above for the phases of Earth as seen from Mars, or would they be different? Sketch out what these phases would look like.
A little bit of thought reveals that since Mars' orbit is outside Earth's orbit, Mars will never appear less than half lit. There will be no "new" or "crescent" Mars! That is one way that we can confirm that Mars' orbit is outside of Earth's orbit.
What is another way that someone living on Mars could confirm this?
Think about transits!
Only planets closer than you to the Sun will be seen to transit the Sun. From Earth, it only happens with Mercury and Venus. From Mars, you could also see the Earth transit the Sun. Imagine living on Mars and watching your own home world cross the face of your sun!
Kepler described the motion of the planets with empirical laws. Later Newton's laws of gravity provided a deeper theoretical understanding of why planetary motions are as they are and expanded the relationships to include moons and other satellites.
If one body orbits another, and if one body is much, much larger than the other, then Kepler's third law*, with Newton's modifications, can be expressed as the following:
Where P = the period of orbit, G = the gravitational constant, a = the distance of orbit, and M1 = the mass of the larger object (or, more exactly, the sum of the mass of both objects).
From the story, Phobos was 9300 km from Mars. The mass of Mars is about 6.39 x 1023 kg. G = 6.674 × 10-11 m3 kg-1 s-2. Based on this, how long will it take Phobos to orbit Mars once (in Earth days)? This will be within a few seconds of the time it takes for Phobos to go through all of its phases.
Remember, you have to get units correct.
(did you convert kilometers to meters to match the units in G? Did you convert seconds to days to match the question request?)
Period (in seconds) = sqrt(4*3.14159*3.14159*9300000*9300000*9300000)/(6.674E-11 * 6.39E23))
Period (in days) = (Period (in seconds) /(60*60*24)
You should get about 0.316 days. Pretty fast, huh!? That little moon is really zipping around. That transit didn't last long, that's for sure!
The actual time for Phobos to go through its cycle of phases (called the synodic period) is 0.3191 days. That compares to Earth's moon at 29.5 days, nearly 100 times longer.
* Note: Kepler's three empirical laws describe the motions of the planets of our solar system. Kepler's third law relates the distance of a planet from the sun to the time it takes for the planet to orbit the sun, according to the relationship P2 ∝ a3.
In the story, small Phobos does not cover the entire Sun, making it very clear that Phobos must be smaller than the Sun. That is not so obvious on Earth where the Moon can completely cover the Sun during an eclipse. Which must be larger?
Over 2000 years ago, Aristarchus measured the distances to the Sun and Moon, and measured their relative sizes compared to the size of the Earth. It was quite astonishing, at the time, to discover that the Earth was really a tiny little place in a very vast universe.
Let's reproduce one part of Aristarchus' calculation—how big is the Sun?
Aristarchus started with the observation that the Sun and the Moon appear to be about the same size in the sky (thus, the Moon just covers the Sun during an eclipse of the Sun). He first figured out the size and distance to the Moon and then the distance to the Sun (which we aren't going to do here),. With this information, he calculated the size of the Sun using similar triangles.
Here is his concept based on the Moon and Sun appearing to be the same size in the sky:
Notice that triangles A-B-C and A-D-E are similar, meaning the ratios of their sides will be equal. Knowing the size of the Moon (radius is about 0.135 Earth Diameters), the distance to the Moon (about 30.1 Earth Diameters), and the distance to the Sun (about 11679 Earth diameters). How much bigger is the Sun than the Earth (don't forget to go from radius to diameter!)
Using similar triangles, we have:
0.135/30.1 = X/11679 where X = the radius of the Sun.
X ~ 52.4 so the Sun is about 105 times bigger than the Earth! This was big news at the time of Aristarchus, when many people thought not only that the Earth was the center of the universe, but that it took up most of the space in the universe. (Artistarchus estimate was a bit smaller than this, and today it is thought that the Sun is about 109 times bigger that Earth).
Connecting to the Next Generation Science Standards.
Students can exercise skills in the practices of science including:
1) Developing and Using Models in conjunction with Arguing from Evidence—in Puzzles 1, 2, 3, 4 and 5, students test various models against observational evidence. In Puzzles 6 and 7, students use their model of motion in the solar system to make predictions.
2) Using Mathematics—in Puzzle 8, students use Kepler’s 3rd Law and Newton’s work on gravity to calculate the orbital period of Phobos. In Puzzle 9, students use similar triangles to find the distance to the Sun.
Students use the crosscutting concepts of
1) Patterns — Students describe patterns of change in celestial objects as predicted by various models of motion in the solar system (geocentric, heliocentric)
2) Scale, proportion and quantity— Students use similar triangles and the unit of ‘earth diameter’ to calculate the distance to the sun.
The investigative activities above support the following NGSS performance expectations:
MS-ESS1-1 Develop and use a model of the Earth-sun-moon system to describe the cyclic patterns of lunar phases, eclipses of the sun and moon, and seasons.
MS-ESS1-2 Develop and use a model to describe the role of gravity in the motions within galaxies and the solar system.
HS-ESS1-4 Use mathematical or computational representations to predict the motion of orbiting objects in the solar system.
The Teacher Resources for Standing in the Shadow of Phobos are written by Russ and Mary Colson, authors of .
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