Astronomy ASTA01: The Sun and Planets
Department of Physical & Environmental Sciences,
UTSC
Fall 2018
Problem Set 4
DUE: Tuesday November 27, 2018,
**5+PM**
Where: Hand in your solutions in the mailbox marked for your tutorial, on the 5th floor corridor of the Science Wing, near office SW506C in the Physics & Astrophysics section.
Reminder: Write your name on your solutions. Also make sure you carefully read the entire problem set policy that was distributed on Quercus. It will help you avoid standard mistakes and score higher. We will assume that you have read this policy document by the time you return your solution.
How to write your solutions: Be precise and clear. Explain what you are calculating. The method of calculation you adopt and your reasoning are the most important. In case of a computational mistake, you will still get credits if your method is right, so explain it clearly.
1
1. The Structure of Super-Earth Planets.
(a) Let us assume rock is perfectly incompressible, meaning that its density is always constant. Calculate the radius of a “super- Earth” with 10 times the mass of Earth, using Earth mass/radius data as a starting point.
(b) Detailed calculations show that Super-Earth radii scale with their mass according to (
R
R⊕
)4 =
M
M⊕ . (1)
Evaluate the radius of the same super-Earth (with 10 times the mass of Earth) according to this formula.
(c) How does this prove that rock is in fact compressible? [NO MORE than 5 sentences]
2. Directly-imaged Planets. Any directly-imaged “planet” found in close proximity of a given star on the sky could in fact be another distant star far away in the back- ground, just located there by chance alignment. How do you propose astronomers can prove or disprove that such a companion is a planet associated with star or a distant background star? [NO MORE than 15 sentences]
3. Transit Geometry. Consult figure 1 to familiarize yourself with the notion that only some planetary systems show transits as seen from Earth (detected as a light reduction when the planet blocks some of the stellar light).
(a) Given the random orientation of planetary orbits relative to our line of sight, do you expect most or only a minority of all planetary systems to show transits? Explain briefly. [NO MORE than 10 sentences].
(b) Does the probability that a single-planet system shows transits increase, decrease or stay the same if the planet has a rather small or a rather large semi-major axis? Explain. [NO MORE than 10 sentences]
(c) If a system has multiple planets all in the exact same orbital plane, does the transit of one planet imply that all planets in the system must transit as well? Explain. [NO MORE than 10 sentences]
2
Figure 1: Transit Geometry. Depending on the orientation of a planet’s orbit relative to our line of sight to the system, some planets transit the disk of their star and some do not. [Figure Credit: slate.com]
(d) In a system with multiple planets, are we more likely to detect transits from several planets if all of the planets orbit in the exact same orbital plane or if the planets have randomly-oriented orbital planes? Explain. [NO MORE than 10 sentences]
(e) The Kepler satellite has found that in a large fraction of systems with one transiting planet, there are actually multiple planets transiting. Explain why this is interpreted as supporting evidence for the solar nebula theory of planet formation. [NO MORE than 10 sentences]
4. Dynamical Stability of Planetary Systems. SuperPlanetCrash is an online game based on the laws of physics that simulates the orbital dynamics of a planetary system. The goal is to pack a planetary system with as many planets as possible while avoiding an instability of the orbits. It can be accessed at: http://www.stefanom.org/spc/
The basic physical ingredient needed to understand the instability of orbits is the so called Hill radius:
RHill ‘ ( Mp M∗
)1/3 a, (2)
3
where Mp is the planet mass, M∗ is the stellar mass (typically 1 solar mass) and a is the semi-major axis of the planet. (The 1/3 exponent notation is equivalent to taking the cubic root of the planet-star mass ratio.)
You can assume that a strong perturbation/deformation of the orbits will occur if any planet ever gets as close as approximately 3 “Hill radius” from another planet (in 3D space) as it moves along its orbit. This will eventually lead to instability, which is caused by planet-planet gravitational perturbations at close proximity.
Read the help section to learn how the game is played. Note the fol- lowing capabilities (various buttons): accelerate/slow down time by up to a factor x128, pause, take an instantaneous snapshot of your system configuration, populate planets of various masses and the possibility to start from a template describing a known extrasolar planetary system. You can learn about additional properties of the template systems on- line (for example, the “open exoplanet catalogue”). All the planets that you will populate will have circular orbits and should be Super-Earths.
(a) Start with the Kepler 11 system template and comment on how “crowded” with planets it is or not, using the concept of Hill radius and stability defined above. Fill it with as many Super-Earth planets as you can. Explain your choice of strategy for populating the system. Report your best run/score, including a snapshot. Again the goal is for you to keep adding super-Earths without making the system go unstable. Snapshot the popup that is shown once the system goes unstable or at the end of simulation time. The highest score is achieved by adding more and more planets that do not destabilize the system. [NO MORE than 20 sentences]
(b) Do the same with the Kepler 18 system template. [NO MORE than 20 sentences]
(c) Do the same with the HD80606 system template. [NO MORE than 20 sentences]
4