In this activity, we will explore the role of resonances in planetary science, focusing particularly on the Saturn system.
First, let us consider the eight classical moons of Saturn. Here is a table summarizing the orbital properties of these moons:
|Moon||Mass||Orbit Period||Semi-Major Axis||Eccentiricty||Inclination|
|Mimas||0.4*1020 kg||0.942 days||185,520 km||0.0202||1.53|
|Enceladus||1.1*1020 kg||1.370 days||238,020 km||0.0045||0.02|
|Tethys||6.2*1020 kg||1.888 days||294,660 km||0.0000||1.09|
|Dione||11*1020 kg||2.737 days||377,400 km||0.0022||0.02|
|Rhea||23*1020 kg||4.518 days||527,040 km||0.0010||0.35|
|Titan||1350*1020 kg||15.945 days||1,221,850 km||0.0292||0.33|
|Hyperion||0.05*1020 kg||21.277 days||1,481,100 km||0.1042||0.43|
|Iapetus||18*1020 kg||79.330 days||3,561,300 km||0.0283||7.52|
Resonances occur between two objects when the orbit period of one object is a simple whole number ratio times the orbit period of the other object. The following table gives the period ratios for all possible combinations of Saturn's moons. a) Find three pairs of moons where the period ratio is close to a ratio of whole numbers.
|(0.942 days)||(1.370 days)||(1.888 days)||(2.737 days)||(4.518 days)||(15.945 days)||(21.277 days)||(79.330 days)|
Resonances are not just numerology, they actually reflect a physical interaction between the two moons. The mutual gravitational attraction between the moons depends on the distance between them and the distance between the moons varies with time. If the moon 1 has an orbit period P1 and moon 2 has an orbit period P2, then the distance between the moons varies over time with a period P12=(1/P1+1/P2)-1=P1P2/(P1+P2). If this period s a multiple of the period of one of the moons, then the gravitational tugs from one moon on the other occur at the same point in the moons' orbits, allowing small perturbations to build up. Thus strong resonances correspond to cases where:
n can be any non-zero whole number. This corresponds to the condition:
(n-1)P1=nP2 Hence the strongest resonances occur where the period ratio is 2/1, 3/2, 4/3, 5/4 etc. These are known as first order resonances, and are usually represented as follows: 2:1, 3:2, 4:3 etc.
b) Look at the above table of period ratios again, which moon should be in the strongest resonance with Titan? Is there anything in the orbital elements for this moon that suggest that its orbit could be perturbed by Titan's gravity?
Resonances also influence the structures of Saturn's rings. For example, the image below shows the full ring system, both as an image and as a profile showing the density (or brightness) of the ring as a function of radius. a) How are the image and the brightness profile related to each other?
Since the rings cover a broad range of radii, they consist of particles with a wide range of orbit periods. You can compute the orbit period at a given radius using Kepler's Third Law: radius3/Period2= a constant.
One of the most powerful resonances in Saturn's rings is the 2:1 Mimas resonance, where the ring-particles' orbit periods are 1/2 Mimas' orbit period. b) Using Kepler's Third Law, and Mimas' semi-major axis (given above), compute the location of this resonance in the rings. Is there any obvious feature in the rings near this location?
Many other features in the rings are produced by resonances with a group of smaller moons that are found closer to the rings. The orbital properties of these moons are given below:
|Moon||Orbit Period||Semi-Major Axis|
|Pan||0.575 days||133,583 km|
|Daphnis||0.594 days||136,506 km|
|Atlas||0.602 days||137,640 km|
|Prometheus||0.613 days||139,350 km|
|Pandora||0.629 days||141,700 km|
|Janus/Epimetheus||0.695 days||151,472 km|
Janus, being the largest of these moons, is responsible for some of the most prominent structure in Saturn's A ring. Below is an image and brightness profile of this ring. c) How are the image and plot shown here related to the ones above?
d) Compute the locations of the various first order resonances with Janus and identify which features could be produced by these resonances.
Now we will let you search for high-resolution images of resonant features in Saturn's rings using the OPUS search tool, which is available here:http://pds-rings.seti.org/search/
Open up OPUS in a separate browser window, and you should see this:
Click on the "Planet", "Nominal Target Name" and "Instrument Name" tabs, and select "Saturn", "S RINGS" and "Cassini ISS", respectively. This selects all images of Saturn's rings taken by Cassini's main camera:
You should now see a tally of all images that meet these criteria in the upper right. Note that Cassini has now taken over 60,000 images of Saturn's rings!
This is far too many to look through, so let's focus our attention on a small part of the A ring, say between 131,500 km and 132,500 km from Saturn's center. Look back at the image and brightness profile of the A ring. a) Why have we chosen to focus on this particular region?
We can search for images of this part of the A ring by clicking the "Ring Geometry Constraints" tab at the left, which produces this screen:
Under the "Ring Radius" tab, we can select the range 131,500 to 132,500 km, and further select "all" to be only in this range. This gives over 15,000 images, which is still too many to look at, so we can also look at images with resolutions better than 2 km/pixel.
This gives only 23 images, so now we can click on View results, and get some close-up images of this disturbance.
The first few are false positives, but the highlighted image is a very nice image of these features, showing them to be waves.Which waves are they? These two features actually have the same resonant relationship with the same moon. The inner wave (which propagates inward) is a bending wave, which only arises when the perturbing moon has an orbit that is significantly inclined with respect to Saturn's equator. The outer wave (which propagates outward) is a density wave, of a kind that is more common throughout the rings. b) Using the information above, make a guess as to which moon is close enough to the rings and massive enough to create these major features, as well as having a significant inclination; once you have made your guess, click here for the answer.
Bending waves can only be generated by a resonance that is at least second-order (for reasons we won't get into). Above, you already located the 2:1 resonance (that's first-order, since 2-1=1) with the moon responsible for these features, and you shouldn't be surprised to learn that there is an adjacent 4:2 resonance (that's second order, since 4-2=2) that drives a bending wave. c) Does that resonance account for the features we see here? If not, what resonance would you try next? Calculate the location of that resonance and compare it to the location of these features. Don't belabor this question; if you haven't gotten it after a couple tries, then move on.
BONUS: You can use OPUS to look at other resonant features in the rings close up. For example, derive the locations of some resonances, Say the Janus 2:1 or Prometheus 9:8, and see what you find!
Two of Saturn's small moons actually orbit within gaps in the rings, as shown below.
The moon on the left is Pan, and it orbits in the Encke Gap; the moon on the right is Daphnis, which orbits in the Keeler Gap. Notice both these moons are smaller than the gaps they inhabit, so these gaps are not just "snow-plowed" out by actual collisions of material with the moons. Instead, the moon's gravity must be somehow keeping ring material from getting too close to the moon's orbit. This phenomenon, known as gravitational shepherding, is a bit surprising, since gravity is an attractive force.
To see how this works, print out the following picture of Daphnis in the Keeler gap:
In this image, Daphnis and all the ring mateiral are moving to the lower left, but they will move at different speeds due to Kepler's Third Law. a) Imagine we are sitting on the moon; in which direction would material in the two gap edges appear to move relative to us? Draw arrows on the image showing this relative motion. Does this make sense given the locations of the disturbed regions on the gap edges?
Next, imagine you are one of the ring particles on the edge of the gap. b) Mark on this image where you would feel the strongest gravitational force from the moon, and draw a vector indicating the direction of that force. Would a force acting in this direction affect how fast the particle moved around the planet?
Let's take a closer look at the interactions between the moon and the ring material. In the close up below we mark two points on the outer gap edge. The red point is just before the particle encounters the moon and the blue point is just after encounter. Draw a diagram showing the force exerted by the moon on the ring at both these positions. c) Will these two forces yield a net force in the direction of motion? How will such a force affect the orbit of the ring particles?