- Imagine that you have discovered two new exoplanets: Planet A is orbiting around Star A, and Planet B orbits around Star B. You measure the angular size of the semi-major axis of each planet’s orbit, and you find that they are exactly the same. However, Star B is 5 times farther away from the Sun as Star A, which means that the physical sizes of the semi-major axes (which are what we care about) are different.
(a) Which of the planet’s orbit has the larger physical semi-major axis? Just answer “Planet A” or “Planet B”, and explain your answer without doing any math.
(b) Calculate the ratio of the semi-major axes of these two systems, putting the larger semi-major axis on the top of the ratio. Remember that the relationship between physical size and angular size is:
x = d θ [2π / 360◦]
where x is the physical size,θ is the angular size in degrees, and d is the distance of the system away from you. If Planet A’s orbit has the larger semi-major axis, you would start by taking xA/xB. If Planet B’s orbit has the larger semi-major axis, start by taking xB/xA.
2. Which of the main planets in the Solar System (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, or Neptune) has the largest orbital eccentricity?
3. We can use the semi-major axis (a) and the eccentricity (e) to calculate the distance of each planet from the Sun at its closest (perihelion) and most distant (aphelion) points in its orbit. The formula for the aphelion is a × (1 + e) and for perihelion it is a × (1 − e). Calculate these distances for the Earth’s orbit. Give your answer in AU.
4. Perhaps the most famous comet in the Solar System is Halley’s comet, which has a orbital period of around 75 years and an orbit with a high eccentricity. Using your knowledge of Kepler’s laws, calculate the semimajor axis distance of the orbit of this comet. Start out by writing down the law that you are using and then do the calculation. Give your answer in AU.