# Astronomy 217

## Prof. Andrew W. Steiner

Sep. 1, 2021

TA James Ternullo

## Last Time

• Phases of the Moon
• Eclipses
• Parallax
• Telescopes
• Solar Orbits
• Python

## Today

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

## Inferior Periods

• 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 $P_{P,\mathrm{Mercury}}$?

## Superior Periods

• 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 $P_{P,\mathrm{Jupiter}}$?

## Orbital Distances

• 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

## Ellipse

• 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

## Earth-Sun Distance

• 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, $e=0.017$

## Equal Areas

• 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's Third Law

• 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 Kepler’s third law of planetary motion.
• Law applies to all orbital systems, with a different constant of proportionality, suggesting a universal law.