29:61 General Astronomy
Fall 2004
Lecture 12 ...November 30, 2004
Planetary Magnetism

Just the facts, Ma'am

The Roche Limit

$\bullet$ The closest that a satellite can come to a planet (or small mass to a much larger mass) before it is torn apart by tidal stresses. This means that the tides (differential ``stretching'' force across the object becomes larger than the gravitational force holding the object together. The Roche limit is defined as a distance $r_R$, such that

$$r_R = 2.44 \left( \frac{\rho_M}{\rho_m} \right)^{\frac{1}{3}} R$$ (1)

where $\rho_M$ is the density of the massive object, $\rho_m$ is the density of the smaller object (i.e. satellite), and $R$ is the radius of the massive object.

Resonant Perturbations

$\bullet$ A periodic perturbation is said to be resonant with a periodic system when

$$nP_n = mP_p$$ (2)

where $P_n$ is the natural period of the unperturbed system (think of the orbital period of a satellite around a planet), and $P_p$ is the period of the periodic perturbing force (think of the orbital period of a satellite further out) and $n$ and $m$ are any two integers. When equation (2) is satisfied, the perturbation produces a large change in the orbital properties of the object being acted on. For example, n=2, m=1 corresponds to the so-called 2:1 resonance, n=5, m=2 is the 5:2 resonance, etc. All of these can be seen in the form of ``holes'' in the rings of Saturn and Kirkwood's Gaps in the asteroid belt.