29:61 General Astronomy
Fall 2004
Lecture 7 ...September 17,2004
Eclipse Cycles and Precession

Just the facts, Ma'am

$\bullet$ Synodic period of the Moon: $P_{syn}=29.5306$ days.
$\bullet$ Nodal period of the Moon: $P_{nod}=27.2122$

$\bullet$ Condition for repetition of eclipses:

$$mP_{nod} = nP_{syn}$$ (1)

with $m$,$n$ integers

$\bullet$ Angular momentum of an object (with mass m) moving with velocity $\vec{v}$ a distance $\vec{r}$ from the origin of a coordinate system:

$$\vec{L} = m \vec{r} \times \vec{v}$$ (2)

$\bullet$ Definition of vector angular velocity $\vec{\omega}$, such that $\vert\omega\vert = \frac{d \theta}{dt}$,

$$\vec{\omega} = \frac{1}{r^2} \vec{r} \times \vec{v}$$ (3)

Then,

$$\vec{L} = r^2m \vec{\omega} = I \vec{\omega}$$ (4)

With $I$ the moment of inertia.

$\bullet$ Moment of inertia for uniform sphere of mass $M$ and radius $R$:

$$I = \frac{2}{5} M R^2$$ (5)

$\bullet$ Definition of torque: causes change in angular momentum with time:

$$\vec{\tau} = \frac{d \vec{L}}{dt}$$ (6)