## Last Time

- Phases of the Moon
- Eclipses
- Radius of the Earth
- Parallax
- Retrograde
- Telescopes
- Solar Orbits
- Python

TA James Ternullo

- Phases of the Moon
- Eclipses
- Radius of the Earth
- Parallax
- Retrograde
- Telescopes
- Solar Orbits
- Python

- Planetary Periods and Orbital Distances
- Ellipses
- Kepler's laws of planetary motion

- Measure planetary orbital periods relative to the Earth-Sun line, this is the "synodic period"
- To get the planet's sidereal period, we need to compensate for the motion of the Earth $$ \omega_{P} = \omega_E + \omega_{\mathrm{syn}} $$ or since \( \omega=2 \pi/P \) $$ \frac{1}{P_P} = \frac{1}{P_E} + \frac{1}{P_{\mathrm{syn}}} $$
- For Mercury, \( P_{\mathrm{syn}} = 116~\mathrm{days} \), what is \( P_{P,\mathrm{Mercury}} \)?

- For a superior planet: $$ \omega_{E} = \omega_P + \omega_{\mathrm{syn}} $$ or since \( \omega=2 \pi/P \) $$ \frac{1}{P_P} = \frac{1}{P_E} - \frac{1}{P_{\mathrm{syn}}} $$
- For Jupiter, \( P_{\mathrm{syn}} = 399~\mathrm{days} \), what is \( P_{P,\mathrm{Jupiter}} \)?

- You can measure the distance from an inferior planet and the sun by measuring the angle of the greatest elongation
- For Venus, \( \theta=46^{\circ} \), and \( \sin \theta=0.72 \) thus Venus is 0.72 AU from the sun.
- For superior planets, you have to measure the time from opposition to quadrature, \( t_{\mathrm{opp}} \) $$ \alpha = 360^{\circ}~\left( \frac{t_{\mathrm{opp}}}{P_E} - \frac{t_{\mathrm{opp}}}{P_P} \right) $$ then, $$ \cos \alpha = \frac{D_{\mathrm{Earth-Sun}}}{D_{\mathrm{Planet-Sun}}} $$
- For Mars, \( P_{\mathrm{Mars}}=1.88~\mathrm{yr} \), \( t_{\mathrm{opp}} = 107~\mathrm{d} \), \( \alpha = 49^{\circ} \), thus Mars is 1.52 AU from the Sun

- Based on observations by Tycho Brahe, Kepler was unable to explain the heliocentric models using circle
- He found ellipses worked instead
- Ellipses have a major axis, a minor axis and an eccentricity
- Zero eccentricity gives a circle, eccentricity of 1 implies a line segment

- The ellipse which describes the Earth-Sun distance (in polar coordinates) is $$ r = \frac{a (1-e^2)}{1+e \cos \theta} $$
- The perihelion, at \( \theta=0 \) is the point closest to the Sun.
- The aphelion, at \( \theta=\pi \) is the point furthest from the Sun.
- For the Earth, \( e=0.017 \)

- The observation that equal areas are swept out per unit of time implies that the planet moves fastest at perihelion.
- This effect is larger the more eccentric the orbit. For the Earth, the ratio of velocities ~1.03, for Mercury it is > 1.50.

- Kepler found that the square of the orbital periods of the planets was proportional to the cube of their semi-major axis.
- This relationship we now call Keplerâ€™s third law of planetary motion.
- Law applies to all orbital systems, with a different constant of proportionality, suggesting a universal law.