Stellarium Lab: Kepler’s Third Law
Original by Paul McCudden, Colorado Mountain College – pmccudden@coloradomtn.edu
Revised by George Repicky, College of Saint Rose -- repickyg@strose.edu
Tycho Brahe and Johannes Kepler were two astronomers of the late sixteenth - early seventeenth
centuries whose study of the motions of objects in the sky contributed much to the final success of the
heliocentric model of the Solar System. Tycho, a Dane, led a team of skilled observers in making and
meticulously recording careful measurements of the locations of the planets in the sky. After his death
in 1601 Tycho's data was passed to his assistant Kepler, a German mathematician and astronomer, who
spent much of the following 20 years devising mathematically - based explanations of how the planets
moved in the sky. This was a huge task -- remember, at that time not all astronomers were even
convinced whether the planets orbited the Sun or vice versa -- but Kepler found a solution. He came
up with his famous three laws of planetary motion. Kepler’s laws are:
1.
Planetary orbits are ellipses, with the Sun at one focus.
2.
Planets move faster when they are closer to the Sun. The result is that the radius vector (the line
between the planet and the Sun) sweeps out equal areas during equal time periods.
3.
The ratio of the cube of the mean orbital radius (R) to the square of the orbital period (T) is a
constant. Mathematically,
R3
= constant
T2
Kepler’s Third Law is amazingly powerful. It tells us that the ratio of R3 to T2 is the same for any
planet or comet or asteroid or spacecraft that orbits the Sun. In other words, if we know a planet’s
distance from the Sun, we can use Kepler’s Third Law to give us its period, or vice versa. Newton
would eventually give Kepler's Third Law a theoretical underpinning based on his Law of Universal
Gravitation and his laws of motion. (In fact, Kepler's Third Law works for any orbital system, such as,
for example, the moons which orbit Jupiter; however, the value of the constant is dependent on the
masses of objects, and thus varies from system to system.)
To examine Kepler's Third Law we will need to to measure the periods of and distances from the Sun
for the eight planets, just as Tycho did.
Measuring the Periods of the Planets
Starting with measuring the periods first, the easiest way to do this would be to look down on the Solar
System from high above it and time how long it takes a planet to go around the Sun once. Of course,
this is impossible to do in the real world, but it is easy to do in Stellarium. (Tycho couldn’t have
looked down on the Solar System from above – he was stuck here on Earth -- so he used a more
complex, but very clever way of measuring the planets’ periods.) Since we do have Stellarium, we'll
(virtually) move out above the Solar System and, looking down on it, measure the planetary periods
from that vantage point.
Stellarium Lab - Kepler's Third Law.odt - 06/20/23
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Start Stellarium.
-Stop time [7].
-Use the Date/Time Window [F5] to set the date to September 1, 2023, time to midnight.
(9 - 1 - 2023 0 : 0 : 0). If you stop and restart the program, be sure to reset the date and time.
-If they are on, turn off the Ground, the Atmosphere, the Cardinal Points, and the Fog by
typing [G], [A], [Q], and [F], or, for the first three, using the icons at the bottom of the screen.
-Open the Search Window [F3] and search for Solar System Observer. Press Enter. You are
now looking at a point high above the Solar System, directly above the Ecliptic plane.
-Press [Ctrl][G] to go to that point. (This works for any selected point.) You are now floating at
a point high above the Solar System!
-Opening the Search window again, enter Sun and press Enter. Do not click on the Sun!
-Press [Alt][UpArrow] to move to a spot directly above the Sun. You are now looking down at
the Sun from your position high above it.
Now display the planets and their orbits.
-Open the Sky and Viewing Options window [F4] and, in the Sky section
-check the three boxes next to Solar System Objects, Show Planet Markers, and Show
Planet Orbits.
-check the Show Planet Only box under Show Planet Orbits.
-Click Colors, and select Separate Colors for Major Planets Only.
-put a check in the box next to Labels and Markers, and drag the slider next to Labels
and Markers about halfway to the right. (You may need to move this further to the right
as you study the outer planets.)
-in the Sky section of the View menu, un-check the Dynamic Eye Adaptation box.
-Close the Sky and Viewing Options window.
You should see the planets and their orbits on the screen.
-Zoom in [Page Up] or out [Page Down] until the orbit of Mars fills the screen. (You may need
to user the four arrow keys to keep the Sun and orbits centered.) You should see the first four
planets, Mercury, Venus, Earth and Mars, each with their orbit around the Sun. You may also see
the orbits of several of the largest Asteroids, for example Vesta.
-Press [L] several times, making time run faster and faster, until you can see the planets move
around in their orbits. Note how the planets move. You will have to zoom in and out to see all
the planets – not just the first four -- and you may have to press [L] a few more times to let time
go by faster. (Too fast? [J] will slow things down.) If you play with this a bit (and you should!)
you'll notice that by the time you get the Jovian planets moving appreciably the inner Terrestrial
planets are off to the races -- a vivid look at the relationship between distance from the Sun and
speed.
Q1: Which way do the planets move around the Sun when viewed from above the North pole clockwise or counter-clockwise?
Q2: Are there any exceptions?
Q3: Does Kepler’s Second law (which implies that the actual speed of a planet is inversely related to
its distance from the Sun) seem to hold true, at least qualitatively, from the way the planets
move? (Think: which planets would you expect to move fastest or slowest? What is the general
rule about how fast planets move around the Sun compared to how far they are from the Sun? Is
this what you see?)
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Now, we want to measure the orbital periods of the 8 planets. We'll start with Mercury:
-Press the [7] key. This key stops time.
-Zoom in [Page Up] until the orbit of Mercury fills the screen.
-Activate the Date/Time window [F5], switch to Julian Day, and record the Julian day, rounded to
the nearest whole number. Record this as the Start Date for Mercury on Table 1 on the Report
of Results. (You can either user the Julian Date (JD) or the Modified Julian Date (MJD) -- they
both count 24 hour days -- but be consistent!)
-Move your mouse pointer so that it is just touching Mercury. Don't click!
-Now press [L] about six times to speed up time. You should see Mercury moving in its orbit as it
goes around the Sun.
-As Mercury approaches its initial position use [J] to slow it's motion. Finally, press [7] to stop
Mercury when it returns to its original position, as indicated by the position of the mouse
pointer1. Use the [–] and [=] keys as needed to get Mercury back as close as possible to where it
started.
-Use the Day/Time to determine and record the End Date in the table.
-Calculate the number of days between Mercury's Start Date and Mercury's End Date -- just
simple subtraction -- and enter the result as Mercury's Orbital Period, T, measured in days.2
We we need to do the same for the other seven planets. Let's measure Venus next. The process is the
same as before:
-Zoom out until Venus’ orbit just fits in your screen.
-Move the mouse pointer to mark Venus’ position on the screen. Don't click!
-Start the clock and speed up time by pressing [L] a few times.
-Press [J] to slow time as Venus approaches the end of one orbit, then hit [7] to stop it when Venus
has completed one full orbit. (Remember, You can use [-] and [=] to go back and forth a bit to
fine tune the position.)
-Enter the End Date in Table 1, just as you did for Mercury, and subtract to get the orbital period.
Follow the same procedure for the other planets. As you can see, the further you get from the Sun, the
slower the speed of the planets – that's Kepler's Second Law! You may have to press [L] a few more
times with each planet to make time run faster, in order to get through a complete orbit in a reasonable
time. Enter all your data in Table 1.
You may find it difficult (or even impossible) to see some of the Jovian planets. However, the Show
Planet Markers and the Labels and Markers features places a small circle around and a label on
each planet. If some of the outermost planets are not labeled, go back to the Sky and Viewing
Options window [F4] and, in the Sky section, locate Labels and Markers, and drag the slider next to
Labels and Markers farther to the right.
1
2
Yes, I know that there'll be a bit of imprecision here, but we'll live with it -- even though it will make JK, with his
incredible dedication to precision, cringe.
Now you know why we astronomers use Julian Days. How'd you like to record the start and end dates in days, months,
and years, and subtract those? Different months have different numbers of days. And if any of the years are leap years,
that changes the number of days in February. (And thinking of leap year and February, I want words with the guy who
made that decision -- what would've been wrong with, say, July 32?)
Stellarium Lab - Kepler's Third Law.odt - 06/20/23
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Measuring the Radii of the Planets' Orbits
Now we need to measure each planet's orbital radius. To find each planet's orbital radius, we'll travel
to the Sun, and then measure the distance to each of the planets from the Sun's point of view.
Stellarium will tell us the distance to the planet, which, since we're measuring from the Sun, is the
orbital radius at that moment3.
To get started:
-Open the Sky and Viewing Options window [F4] and, in the Sky section
-Un-check the boxes next to Show Planet Orbits.
-In the Sky section of the View menu, check the Dynamic Eye Adaptation box.
-Close the Sky and Viewing Options window.
-Click on the Sun, and then press [Ctrl][G] to travel to it. (Yes, the Sun will disappear from view
-- you're on it!)
-Zoom out.
-If they are on, turn off the Ground, the Atmosphere, the Cardinal Points, and the Fog by
typing [G], [A], [Q], and [F], or, for the first three, using the icons at the bottom of the screen.
Now,
-Open the Search window [F3] and type in Mercury and press Enter. Mercury is centered and
selected. Look for the distance from the Sun to Mercury (in AU’s) – it’s listed in the information
at left of the screen, about half way down, in the line that says “Distance from Sun”.
-Enter the distance to Mercury in the second column (Orbital Radius) of Table 2 on the Report of
Results, rounded to three decimal places.
-Repeat the previous two steps for each of the other seven planets.
Processing the Data:
To use our data to examine Kepler's Third Law,
-Convert each planet's Orbital Period from days into years, by dividing the Orbital Period from
Table 1 by 365. Round to three decimal places, and enter the result for each planet in the 3rd
column of Table 2, Orbital Period (years).
-Calculate R3 for each planet, by cubing the Orbital Radius (R3 = R x R x R). Round to three
decimal places, and enter this result for each planet in the 4th column of Table 2, R3 (AU3)
-Calculate T2 for each planet by squaring the Orbital Period (in years) that you entered for each
planet in Table 2. Round the result to three decimal places, and record the result in the fifth
column in Table 2, T2 (years2) .
-For each planet, calculate R3/T2 by dividing R3 by T2. Round to three decimal places, and record
in the sixth column, R3/ T2 (AU3/years2).
-Calculate the average value of R3/T2 by adding up the eight values and dividing by 8. enter your
result in Table 2.
3
Yes, the orbital radius varies as a planet travels in its elliptical orbit. Kepler's Third Law uses the mean (i.e., average)
orbital radius. This will be a source of error in our lab (especially for Mercury, whose orbit is quite eccentric), but it's
the best we can do in a reasonable amount of time. The alternative would be to measure periodically -- say, daily -- day
after day after day after day after .....
Stellarium Lab - Kepler's Third Law.odt - 06/20/23
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Analysis:
Q4: Precision is a measure of how reliable a result is. If the result of the R3 / T2 calculations are
similar we can infer that the result is likely reliable. Comment on whether or not your
calculations generally produce similar results.
Q5: Are there any outliers -- values of R3 / T2 which strongly deviate from the majority? If so, can
you offer an explanation?
Q6: Overall, what do your answers to the preceding two questions imply about the precision -reliability -- of your data?
Newton would go on to show that if the mean orbital radius (R) is in astronomical units, and the orbital
period (T) is in Earth years, the result of the R3/T2 calculation is numerically equal to the sum of the
masses of the two objects involved in the orbit measured in solar masses. Mathematically:
R3
= ( M 1 + M 2)
T2
where M1 and M2 are the masses of the two objects orbiting each other.
In the case of the Sun and a planet, since the masses of the planets are very small when compared to
the Sun, this implies that the result of the R3/T2 calculations should be very close to mass of the Sun,
which is, of course, 1 solar mass. In other words, for the Solar System we can rewrite it:
R3
= 1
T2
or
T2 = R3
Q7: Accuracy is a measure of how close a measurement is the the accepted value; in our case, how
close the result of the average R3/T2 calculation is to 1. Comment on the accuracy of your result.
And Now a Little Challenge
Remembering that for the Solar System Kepler's Third Law, as modified by Newton, can be written:
T2 = R3
Q8: Suppose you wanted to look for an asteroid. Asteroids are clustered in a band between Mars and
Jupiter, at around 4 AU from the Sun. How long would it take an asteroid at this location to orbit
the Sun? (Note the answer will be in years.)
Q9: If we found a planet that took 1,000 Earth years to orbit the Sun, how far would it be from the
Sun? (Note that the answer will be in AU.)
Stellarium Lab - Kepler's Third Law.odt - 06/20/23
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