Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Sep. 20, 2021
 
 

TA James Ternullo

Last Time

  • Tides and Offsets
  • Impact on Earth and Moon's Angular Momentum
  • Tidal Locking
  • Librations
  • Extreme Tides
  • Tidal Disruption
  • Roche Limit
  • Hill Radius

Today

  • Eclipses

Phases of the Moon

  • As the most visible change from night to night in the night sky, the cycle of lunar phases held great interest for our ancestors.
  • The lunar (from Latin) month (from Old English) is the time to complete the cycle of lunar phases (29.53 days).

Sidereal Month

  • As with the Earth’s day, the period of the moon’s rotation with respect to the fixed stars is less than the perceived period. $$ \frac{1}{P_{\mathrm{sid}}}=\frac{1}{P_{\mathrm{syn}}} + \frac{1}{P_{\oplus}} $$ $$ \frac{1}{27.32~\mathrm{d}}=\frac{1}{29.53~\mathrm{d}} + \frac{1}{365.24~\mathrm{d}} $$
  • The synodic month is the period from new moon to new moon.

Lunar Eclipse

  • Eclipses of moons by their much larger planets are very common, as planets cast relatively large shadows.
  • A lunar eclipse occurs at opposition, meaning that a full moon disappears.
  • Because of the Sun's \( 0.5^{\circ} \) angular size, the Earth’s shadow on the moon is slightly smaller (9380 km) than \( D_{\oplus} \) and 2.7 \( D_{\mathrm{Moon}} \).
  • With the Moon orbiting at 1.0 km/s, the total lunar eclipse lasts ~ 100 minutes, with 60 minutes of partial shadowing preceding and following.

Solar Eclipse

  • During Solar Eclipses, both a dark inner full shadow (Umbra) but also a outer partial shadow (Penumbra) appear.
  • Depending on the orbital orientation and the observers location on Earth, partial, total and annular eclipses are possible.
  • These phenomena occur because of the coincidence of the Sun’s and Moon’s angular size.

Length of the Umbra

  • For most moons, the umbra falls well short of the planet, but the Moon is a relatively large moon.
  • $$ \tan \alpha = \frac{R_{\odot}}{(a_{\oplus}-r_0)+ \ell_{\mathrm{umbra}}} = \frac{R_{\mathrm{Moon}}} {\ell_{\mathrm{umbra}}} $$ $$ \ell_{\mathrm{umbra}} = \frac{R_{\mathrm{Moon}}} {R_{\odot}-R_{\mathrm{Moon}}}(a_{\oplus}-r_0) = 3.73 \times 10^5~\mathrm{km}=0.97 r_0 $$
  • The Moon’s orbital eccentricity is 0.055, making perigee = 0.945 \( r_0 \) and apogee = 1.055 \( r_0 \), thus the umbra only reaches the Earth when the Moon is near perigee.

Orbital Orientation

  • The plane of the Moon’s orbit around the Earth crossed the plane of the Earth’s orbit around the Sun at \( 5^{\circ} \).
  • Since solar and lunar eclipses only occur when the Sun, Earth and Moon form a line, eclipses can only occur when the new/ full moon lies in the ecliptic.

Crossing Ellipses

  • The line of nodes is the line formed by intersection of the plane of the Moon’s orbit and the Ecliptic. It connect the ascending node ☊ and descending node ☋.
  • The line of nodes precesses with an 18.6 year period as a result of the Sun gravitational attraction on the Moon.

Finite Sizes

  • Because of the finite angular sizes of the Sun and the moon $$ \theta_{\odot} = \frac{R_{\odot}}{a_{\oplus}} = 0.27^{\circ} \quad \mathrm{and} \quad \theta_{\mathrm{Moon}} = \frac{R_{\mathrm{Moon}}}{r_0} = 0.26^{\circ} $$ eclipses can occur in a small region around the nodes.
  • An eclipse can occur if the centers of the Sun and Moon appear within \( \theta_{\mathrm{eclipse}} \leq \theta_{\mathrm{Moon}} + \theta_{\odot} = 0.53^{\circ} \) to an observer on the Earth’s surface.
  • But this angle changes over the course of the day because of the Earth’s rotation. To remove the diurnal effects, the angle between the Earth-Moon line and the Earth-Sun line can be calculated relative to the Earth’s center.

Angle From Center

  • To derive the angle between the Earth-Moon line and the Earth-Sun line from the Earth's center.
  • Law of sines $$ \frac{R_{\oplus}}{\sin \theta_{\oplus}} = \frac{r_0}{\sin(90^{\circ}+\theta)} $$ but \( \theta_{\mathrm{eclipse}} \approx 0.5^{\circ} \) so \( \sin(90^{\circ}+\theta) = 1 \) $$ \sin \theta_{\oplus} = \frac{R_{\oplus}}{r_0} \Rightarrow \theta_{\oplus} \approx 0.95^{\circ} $$ $$ (90^{\circ}+\theta)+(90^{\circ}-\Delta)+\theta_{\oplus} = 180^{\circ} $$
$$ \Rightarrow \Delta_{\mathrm{partial}} \approx \theta_{\oplus} + \theta = \theta_{\oplus} + \theta_{\odot} + \theta_{\mathrm{Moon}} = 1.48^{\circ} $$ and $$ \Delta_{\mathrm{total}} \approx \theta_{\oplus} + \theta_{\odot} - \theta_{\mathrm{Moon}}= 0.96^{\circ} $$

Eclipse Window

  • Because of the small \(5^{\circ}\) of the Moon's orbit relative to the Ecliptic, the Moon spends significant time within \(1.5^{\circ}\) of the Ecliptic. This time is called the eclipse window.
  • $$ \delta \theta_{\mathrm{partial}} = \Delta_{\mathrm{partial}}/\sin 5.1^{\circ} = 11 \Delta_{\mathrm{Partial}} = 16.7^{\circ} $$ $$ \delta \theta_{\mathrm{total}} = \Delta_{\mathrm{total}}/\sin 5.1^{\circ} = 11 \Delta_{\mathrm{Total}} = 10.8^{\circ} $$
  • The Sun moves \(360^{\circ}/\mathrm{year} = 0.986^{\circ}/\mathrm{day} = 29.1^{\circ}/\mathrm{syn~month} \)
  • Comparing this to the eclipse windows, per passage of the nodes, there will be 1-2 partial solar eclipses \( (2 \delta \theta_{\mathrm{partial}} = 33.3^{\circ} > 29.1^{\circ} ) \) and 0-1 total solar eclipses \( ( 2 \delta \theta_{\mathrm{total}} = 21.6^{\circ} < 29.1^{\circ} ) \) and similar for lunar eclipses