Saturn

>>>>>>>> Mention of lab section for those who took this course

Today we move out in space to the next planet, Saturn. Look at pictures on p214, 225. 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 atmosphere (see Figure 10.11) , so the clouds bands and other cloud deck features are less prominent on Saturn than on Jupiter.

The main features of the Saturnian system we will discuss will be the ring system and the moon Titan. Titan is discussed on p245 of the book. I will have some additional things to say about it, based on recent findings.

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). >>>>>> diagram with two gravitating masses. 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.

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.

>>>>>>> Diagram of elliptical orbit with perturbing object.

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. >>>>> Blackboard illustration.

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 ration 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.