The Synodic and Orbit Periods of the Planets - Astrophysics 480
Objective: to see the development of a paradigm shift in our understanding of the motions of the
planets in the solar system
Equipment: Voyager planetarium software, Excel
Background: For at least three thousand years people have watched the planets move against the
background formed by the constellations of stars. Sometimes a planet would be seen directly
opposite the sun in the sky, at other times it would be lost in the glare of the sun. We defined the
synodic period (S) of a planet as the number of days between successive line-ups of the planet
relative to the sun. When Copernicus proposed that the planets and the Earth orbit the sun rather
than the Earth, he said that the innermost planets moved faster than the outermost planets. Let's
see how he figured out their speeds, and then convinced himself that this was the more
reasonable of the two models.
Define:
Paradigm shift:
Synodic period:
Sidereal period:
Elongation angle:
Inferior planet:
Superior planet:
Monotonic:
Circumference of a circle:
1.Synodic Period Observations
We want to find out the synodic periods by “observing” the motions of the planets relative to the
sun over a span of several years. Go to Start, All Programs, Carina Software, Voyager 4.5,
Voyager 4.5.
First, set the time to be January 1, 2000 by using the top tool bar’s Chart, Set Date and Time.
Next, go to the nighttime view by Chart, Coordinates, Star Atlas- Equatorial.
Use Center, Find and Center, the Sun. Click Lock to keep the sun in the center of the screen so
that we can watch the planets move relative to the sun rather than relative to the constellations.
Set the time step to 1 day.
.
Hold down the Play triangle to find one time of alignment of the planet Mercury with the sun.
(Be ready to Stop the action quickly.) Then use the BarTriangle to step forwards or backwards
one day at a time to find the day of best alignment. Record this date.
2
Alignment date __________________
Now move ahead in time until the alignment repeats itself. Record this date also.
Next Alignment date __________________
Enter these dates in a spreadsheet and subtract them to find your
observed synodic period _______________
Repeat the observations for each of the planets out to Saturn.
Check your observed dates by using Tools, Conjunction Search, and clicking the Sun and
Mercury (etc). The separation angle might have to be increased from one degree to two degrees.
Were there any changes?
2. Synodic Period Analysis
Here are the observed values for synodic period S for the planets known in Copernicus’ time:
Mercury = 116 days, Venus = 584 d, Mars = 780 d, Jupiter = 399 d, Saturn = 378 d. (Table 4-1 in
Freedman)
Calculate the percent difference between your Voyager observations and these accepted S values.
Make a graph of accepted S vs. planet number (Mercury = 1, Venus = 2, etc.).
Is this relationship monotonic? ___________
Is this relationship linear? _____________
Describe the curve in words. (Do not try to fit the curve with Excel.) (This is what Copernicus
had to explain with his new model.):
3. Copernicus’ Model of Planetary Motion
Copernicus thought of the planets as orbiting the sun rather than the Earth. Well then, what
should he do with the Earth? Well, have it orbit the sun also!
In Copernicus' model, if the planet has an elongation angle which is always smaller than 180
degrees, then it is an inferior planet (orbit radius is smaller than Earth's). If it is observed to line
up with the sun, then after S more days it will line up with the sun again as seen from the Earth.
Since the Earth's orbit yearly period is 365 days, the Earth will have gone S/365 of the way
around the sun in its orbit in those same S days. If we define the inferior planet's true orbit period
to be P days, then it will have gone S/P of the way around the sun. However, since it is moving
faster than the Earth it will have already lapped the Earth when they line up again, so this full
circle will have to be subtracted. Thus, when the planet lines up with the sun as seen from the
Earth, the fraction of a circle gone will be the same for both the planet and the Earth and is
Earth’s fraction of a circle = S/365 = S/P - 1 = Planet’s fraction of a circle
3
Freedman Box 4-1
Dividing each term in this equation by S gives
1/365 = 1/P - 1/S
or
1/P = 1/365 + 1/S
= (S + 365)/(365 * S)
or
P = (365 * S)/(S + 365)
What is the formula for P if the planet is a superior planet?
4. Copernicus’ Orbit Periods
Calculate the planets' true orbit periods P from the synodic periods S above.
Make a graph of true period P vs. planet number.
Is this relationship monotonic? ___________
Is this relationship linear? _____________
Describe this curve in words.
Why did this make Copernicus think he was on the right track?
4
Plot S vs. planet number, and P vs. planet number on the same graph. Where do the curves cross?
ie at what value does S = P? _________________
Which is the nearest planet to this equality? ______________
This is not significant, but just interesting.
5. Orbit Radius and Speed
Look up each planet's mean orbit radius (semimajor axis length) in millions of kilometers.
Calculate the speed of each planet in km/hour from this radius and the time period P, assuming
that its orbit is a circle in shape.
Plot speed vs. planet number. Comment on the shape of the plot and why Copernicus would have
been happy with it.
Plot speed vs. radius. He would have been even happier with this plot. Why?
6. Newton thought that the best plot would be speed vs. the inverse of the square root of the
radius. Plot this. Comment on it.
7. Extend your table by looking up data for the orbit periods and orbit radii of Uranus and
Neptune. Comment on the magnitudes of the speed numbers, especially for the extreme values,
Mercury and Neptune.
8. Conclusion: How can the shape of a curve drive a paradigm shift?
Reference:
Freedman, RA, Geller, RM, Kaufmann, WJ III, Universe 9th Ed, WH Freeman & Co, 2011, p 71