March 29, 2004

Saturn

To start with: let’s check the homepage of the Cassini spacecraft to see its progress towards its arrival at Saturn in July

http://saturn.jpl.nasa.gov/index.cfm

Announcements:

Observing session tonight, or next clear night, to see the five planets in the sky. Meet on 7^th floor stairway to roof at 6:50 PM. Mercury is best seen from 7 – 7:30 PM.
There is a new homework set on the web.

Today we move out in space to the next planet, Saturn. The semimajor axis of the orbit of Saturn is 9.54 astronomical units.

I mentioned earlier that the physics of the outer planets is determined by extreme cold. There are a number of illustrations of this in the case of Saturn. Since the intensity of light from the Sun (technically speaking the flux, in units of Watts/square meter) falls off according to the inverse square law, the power of sunlight at Saturn is weaker by a factor of (9.54)² = 91.

The increased cold causes the cloud layers to form deeper in the, so the clouds bands and other cloud deck features are less prominent on Saturn than on Jupiter.

As a planet, Saturn is similar to Jupiter, but smaller and less extreme. The two main features of Saturn that are worth emphasizing are its ring, and its moon Titan. I will spend time talking about both of these features of the planet.

The Rings…some quick preliminaries

The rings are seen above, and you can see that the ring is divided in “subrings”, the A, B, and C rings, and a hint of even more detail. The obvious dark gap (which can be clearly seen in a rather small telescope) is called Cassini’s Division.

The ring is in the equatorial plane of Saturn, and since the obliquity of Saturn is 26.7 degrees, we see different aspects of its ring during the course of its 29 year period of revolution (orbital period). This is shown in Figure 12.24 of the text.

We are presently near the optimum time for observing Saturn’s rings “fully opened up”. You can see that about ¼ of a Saturnian orbit earlier, in the mid-90’s the rings were turned edge on, and were difficult to see.

An important point to emphasize is that the ring is not solid material, but is composed of rubble; countless small pieces that orbit Saturn as independent little moons. An illustration of the nature of the ring can be seen in the following picture, which is an artist’s conception of the approach of the Cassini spacecraft to the ring.

You can see stars shining through the ring (you can see this from Earth as well).

How did the ring get there, and why is Saturn unique in having such a prominent ring. To understand that, we need to study the physics of gravitation.

Some Physics---Gravitation

Some topics in the physics of gravity will be important in understanding properties of the Saturn system, as well as later in the semester. A common situation in astronomy is to consider the gravitational force between a massive object M (say the Sun), and a smaller mass m (say a planet).

The force between the two is then given by

F = -GMm/r²

You can solve the problem of how the masses move in response to this force. It turns out that the masses will move on elliptical orbits that are described by Kepler’s laws. This is one of the triumphs of physics that we can explain Kepler’s laws in terms of Newton’s theory of gravitation. In this case of the two body problem, the elliptical orbit is constant and unchanging for all time.

Tides and Tidal Disruption

A tide is a differential force between the front side and the back side of an object of finite extent. Tides tend to deform an object. The primary manifestation on Earth is tides of the ocean, since the water in the ocean deforms more than the rock of the land.

If the tidal stress (differential force) is greater than the self-force holding an object together, tides can disrupt an object (tidal disruption) . The condition for this to occur was determined by the French mathematician Eduoard Roche around 1850.

It is called the Roche Limit.

The Roche limit is defined as the closest a small object with mass m can get to a massive object with mass M. The Roche limit is given as a distance

r_R = 2.44(d_M/d_m)^1/3R.

Where d gives the density of the massive and disrupted body, and R is the radius of the massive object. If a smaller object gets closer to a larger object than about 2.4 times the radius of the large object, the little object gets pulled apart.

Figure 12.36 of the book shows that the ring of Saturn lies just inside the Roche limit for Saturn, which pretty clearly identifies the Roche limit as the relevant concept here.

Gravitational Resonances

Kepler’s Laws are exactly true only if there are two objects in the universe. Since that is not the case, we have to consider them approximations, although very good approximations.

It is frequently the case that the gravitational force between two objects (say the Earth and the Sun) is stronger than other forces (say between the Earth and Jupiter). In that case, we speak of the extraneous force as a perturbation.

A diagram of an object in orbit, with another object exerting a perturbation is shown in Figure 12.33 of the textbook.

In many cases, the perturbing force is largely averaged out during the orbital motion of the two primary objects, so the effect of the perturbation is small. An example would be the precession of the Moon’s orbit due to the perturbing force of the Sun.

If the perturbing object moves in position, the net effect on the two primary objects is often still small.

There is, however, a case in which a perturbation can produce a very large effect over time. This is the case of resonance.

Resonance refers to a situation in which the perturbing force is periodic, and has the same period as that of the orbiting object. The basis of resonance is best illustrated with a ball-and-spring system called a harmonic oscillator (one of physicists’ favorite things). If you perturb a harmonic oscillator with a periodically-varying force which has a period equal to that of the natural oscillation, the amplitude of the oscillator will get bigger and bigger, and finally end up in wild oscillations. A good way to visualize this is to think of a swing (an excellent example of a harmonic oscillator) pushed periodically by someone.

>>>>>>> illustration of driven harmonic oscillator.

The same is true (although less dramatic development) if the period of the perturber is twice that of the oscillator (a so-called 2:1 resonance) or three times

(the 3:1 resonance), or the two periods stand in the ratio 3:2), etc.

In celestial mechanics, an object (let’s call it A) will have its orbit strongly perturbed if there is an object (B) further out from the central object that has a period such that the orbital periods TA and TB have a ratio which is equal to the ratio of two small integers. The smaller these integers (i.e 2:1 or 3:2) the stronger the perturbation will be. These A objects will then be moved out of their orbits into orbits which are not resonant.

These remarks probably sound pretty esoteric, but resonant perturbations are an extremely important phenomenon in the solar system, and we will see examples of them in action. The first example will be in the Saturn system.