Astronomy 100
Name(s):
Exercise 7: Orbital resonance
Prior to Isaac Newton’s theory of gravitation, astronomers collected data on
planetary motion without a theory to explain that motion. Johannes Kepler, a
German astronomer (1571 – 1630), published Mysterium Cosmographicum, in
which he described three mathematical relationships that seemed to predict
planetary orbits. We will concentrate on the third of these “laws”, which is stated
as a formula:
P2 = a3
Orbital resonance describes any phenomena that relates the orbital and
rotational periods of two different bodies with simple whole number
ratios. Tidal or gravitational lock is an extreme case of orbital resonance; the
Earth’s moon’s orbital period exactly matches its own rotational period (roughly
one month) — the ratio in this case is 1:1. The cause of resonance is gravitational;
when bodies are in simple whole number ratios the gravitational effects of a
larger body are synchronized – in other words, since the bodies are regularly
found in the same spatial orientation to each other in the same part of the orbit,
the gravitational effects are intensified. This can be stabilizing or
destabilizing, depending on the size of the bodies.
Simple cases of resonance
1. a. Mercury’s rotational period is given as 58.6 days; its sidereal orbital period
is given as 87.9 days. In Earth terms, what time units do these correspond to?
b. What is the ratio of Mercury’s rotational period to its sidereal orbital period?
Does this stabilize or destabilize Mercury?
2. Neptune’s orbital period is 163.7 years; Pluto’s orbital period is 248.0 years. To
a reasonable approximation, what is the ratio of Neptune’s period to Pluto’s
period?
Go to the University of Colorado site
http://lasp.colorado.edu/education/outerplanets/orbit_simulator/. Run the
program and observe Neptune and Pluto.
3. Is it at both planetary orbits’ aphelion or perihelion that the planets “line
up” on the same side of the Sun?
4. How many orbits does Neptune make before they line up again? How many
orbits does Pluto make before they line up again?
5. How does this relate to a 3:2 resonance?
A more complex case of resonance
Attached is a graph taken from the University of Washington astronomy tutorial
site at: www-hpcc.astro.washington.edu/stawarz/orbres.html
The graph plots a (the semi-major axis distance) of various asteroids versus the
number of asteroids found at that distance. Note that the distribution is not
uniform; there are some distances that have many asteroids and some distances
(called gaps) that have comparatively few or none. It is the placement of these
gaps that are governed by orbital resonance. The “gap letter” in the table below
refers to how I labeled each of the gaps on the graph.
The major gravitational influence on the asteroids, of course, is the sun. Jupiter is
the other major body that exerts some control over the asteroids, and it is with
respect to the orbital period of Jupiter that the simple whole number ratios will
be calculated.
All that is needed for the calculations is Kepler’s 3rd law (the simple version).
Recall that its form was: P2 = a3, where P is the period of the object’s orbit and a
is the length of its semi-major axis in AU.
gap letter
semi-major axis
(AU)
period (yr)
Whole number
ratio of
Jupiter’s period
to the period of
an object in the
gap
Jupiter
A
2.3
B
2.5
C
3.3
D
3.7
6. Using Kepler’s 3rd law, calculate the period of Jupiter’s orbit. Calculate the
period of an object in each of the four lettered gaps, based on the semi-major axis
given. Then divide Jupiter’s period by these calculated periods, and determine
what whole number ratio (expressed as “2:1”, “3:1”, etc.) best expresses the
relationship. Note that not all of the ratios will end in a “1”.
7. From the list of ratios above, notice that some simple ones are missing. For
instance, the 3:2 ratio is missing. In the table on the next page, calculate what the
period of these “missing ratio” objects ought to be, and their corresponding semimajor axis lengths.
“Missing ratio”
period (yr)
semi-major axis (AU)
3:2
4:3
8. Notice that the “missing ratio” gaps occur in a part of the belt that is a huge gap
anyway. Give a better reason (or fate) for why no asteroids are found at these
distances.
9. Given that other missing ratios cannot be as low as the ones you saw in
question 7, determine some other simple whole number ratios that are missing,
and determine where the corresponding gap should be in AU. Go back to the
graph, then, and find the letter that corresponds to the gap.
“Missing ratio”
period (yr)
semi-major axis
(AU)
gap letter
10. Why is there a clump of asteroids at the distance marked H?
An even more complex case of resonance
Consider Saturn, its moons and its rings. Mimas and Tethys are moons of Saturn
and the information about their orbital distances and periods are found in the
appropriate Appendix table. The information about the various ring distances are
found on the poster in the classroom.
11. Look up and fill in the first two open columns in the table on the next page.
Body
Orbital semimajor axis
(km)
Orbital period
(days)
Mimas
Orbital semimajor axis
(Mimas
distances)
Orbital period
(Mimas
revolutions)
1
1
Tethys
Cassini
Division
12. Now do our usual “redefine the units” trick: To calculate orbital periods, using
Kepler’s 3rd Law seems like a good idea. However, what is now the center of mass
of this system (hint: it’s not the Sun anymore!) So we will define everything in
terms of Mimas-center of Saturn distances (note how I filled in the rest of the row
for Mimas). Similarly, we will define orbital periods in terms of how long it takes
Mimas to complete one revolution. Use this method to fill in the rest of the table.
13. a. What is the resonance ratio between the orbital period of Mimas and the
orbital period of Tethys?
b. What is the resonance ratio between the orbital period of Mimas and the
orbital period of a particle in the Cassini Division?
Summary
Resonance is an important gravitational effect that determines where a body can
or cannot orbit another body, or what the relationship between a body’s orbital
period and its rotational period is.
14. Fill in the table below with either “stable” or “unstable” to describe whether
the situation mentioned is stable or unstable over a long period of time.
Situation
Whole number ratios between the
rotational and orbital periods of a given
body (e.g., Mercury, Moon)
Whole number ratios between the
orbital periods of a small body and a
larger body, both orbiting another
much larger body (e.g., asteroids, ring
Stable or unstable?
particles)
Whole number ratios between the
orbital periods of two similar sized
bodies orbiting the same larger body
(e.g., moons of Saturn)
Prediction
15. On the next page is an eccentricity versus semi-major axis graph for Kuiper
Belt Objects (KBOs). At how many AU’s out would a 2:1 resonance object with
Neptune’s orbital period be found? Does there seem to be a lot of objects at that
number of AUs on the graph?
16.At how many AU’s out would a 4:3 resonance object with Neptune’s orbital
period be found? Does there seem to be a lot of objects at that number of AUs on
the graph?
17. Why don’t all the Plutinos (I mean, look at the area around 39 AU on the
graph) collide with each other? (Hint: What’s the other axis on the graph?)
18. What might explain the cluster of low-eccentricity objects around 44 AU?
Hint: Is this necessarily a resonance phenomena? If it is, does it have to be with
Neptune’s orbit?