Your Questions and Exoplanet Masses & Orbits

Astro 497, Week 7, Day 1

TableOfContents()

Questions (general)

question(md"What are some ways we can make sure a planet is transiting if the planet is only passing through one side of the star?")

Question

What are some ways we can make sure a planet is transiting if the planet is only passing through one side of the star?

LocalResource("../_assets/week2/circular_diagram.png")

question(md"How often do we see multiple exoplanet eclipses at once? Say, two planets passing in front of the host star at the same time? I'd think this to be very rare, but what could such a phenomena reveal about the host star's planetary system?")

Question

How often do we see multiple exoplanet eclipses at once? Say, two planets passing in front of the host star at the same time? I'd think this to be very rare, but what could such a phenomena reveal about the host star's planetary system?

Transit duration ($e=0$):

$$\begin{eqnarray} T & \simeq & 13\, \mathrm{hr} \left(\frac{P}{365\,\mathrm{d}}\right)^{1/3} \left(\frac{\rho_\odot}{\rho_\star}\right)^{1/3} \sqrt{1-b^2} \\ & \simeq & 4.5\, \mathrm{hr} \left(\frac{P}{4\,\mathrm{d}}\right)^{1/3} \left(\frac{\rho_\odot}{\rho_\star}\right)^{1/3} \sqrt{1-b^2}$ \end{eqnarray}$$

Fraction of time in transit:

$$\begin{eqnarray} \frac{T}{P} & \simeq{} & \simeq 0.0015 \left(\frac{P}{365\,\mathrm{d}}\right)^{-2/3} \left(\frac{\rho_\odot}{\rho_\star}\right)^{1/3} \sqrt{1-b^2} \\ & \simeq{} & 0.045 \left(\frac{P}{4\,\mathrm{d}}\right)^{-2/3} \left(\frac{\rho_\odot}{\rho_\star}\right)^{1/3} \sqrt{1-b^2} \end{eqnarray}$$

#= md"""
Duration of transiting another plannet relative to duration of transit of host star
$T_{pl} / T_{\star} \simeq \left(\frac{R_{p}}{R_\star}\right) \left(\frac{\sqrt{1-b_{pl}^2}}{\sqrt{1-b_\star^2}}\right)$
"""
=#

hint(md"""
Other factors that affect probability of mutual transits:
- Size of planets
- Mutual orbital inclination
- Relative orbital motion of the two planets 
- Ratio of orbital periods
""")

Hint

Other factors that affect probability of mutual transits:

Size of planets
Mutual orbital inclination
Relative orbital motion of the two planets
Ratio of orbital periods

question(md"""
What are the main differences between the maximum a posteriori and the maximum likelihood estimator?
""")

Question

What are the main differences between the maximum a posteriori and the maximum likelihood estimator?

$$\mathrm{MLE} = \mathrm{argmax}_\theta \; p(\mathrm{data} | \theta)$$

$$\mathrm{MAP} = \mathrm{argmax}_\theta \; p(\theta|\mathrm{data})$$

$$\mathrm{posterior} = \frac{(\mathrm{prior}) (\mathrm{likelihood)}}{(\mathrm{normalization\,constant})}$$

$$p(\theta | \mathrm{data}) = \frac{p(\theta) p(\mathrm{data} | \theta) }{\int \, p(\theta) p(\mathrm{data} | \theta) \, d\theta}$$

question(md"Are there external factors, such as dust clouds, gravitational lensing, or overlapping solar systems, that can significantly affect collected data?")

Question

Are there external factors, such as dust clouds, gravitational lensing, or overlapping solar systems, that can significantly affect collected data?

Questions (exam specific)

question(md"""Are there certain equations we should pay more attention to when studying?""")

Question

Are there certain equations we should pay more attention to when studying?

RV(t)
Depth, duration, and probability of transits
Magnitude of RM effect
TTVs: qualitative behavior and what variables the amplitude and timescale are most sensitive to

question(md"To what level of detail do we need to understand non-Keplerian motion? Qualitatively or Quantitatively?")

Question

To what level of detail do we need to understand non-Keplerian motion? Qualitatively or Quantitatively?

Mostly qualitatively
Could use TTVFaster calculator if need to.
Could compare precession rate due to GR (from reading) to another timescale (e.g., orbital period).

question(md"""How extensive are the numerical calculations going to be?""")

Question

How extensive are the numerical calculations going to be?

In most cases, there is a way to do the problem using arithmetic you could do in your head (e.g., if you remember the values for the Earth-Sun or a Hot-Jupiter and use scaling relations), perhaps using one simple calculation ($\sqrt{x}$ or $x^{1/3}$).
You're certainliy welcome to use scratch paper, a calculator, or Pluto notebook
Can also use interactive widges from labs or class (especially if you use expressions with mks/cgs units).

Kepler's Multiple Planet Systems

Architectures of systems with $\ge$4 known transiting planets.

Credit: Fig 1 from Weiss & Petigura (2020)

Upcoming figures from He, Ford & Ragozzine (2019) MNRAS, 490, 4575 and He, Ford, Ragozzine & Carrera (2020) AJ, 160, 276.

Number of Detections versus multiplicity

Marginal Distribution of Ratios of Transit Observables for Pairs of Planets in Same System

Forward model for generating simulated planetary systems

Drawing Periods & Radii

Drawing Eccentricities & Inclinations

Best-fit Model for Planetary Systems

Marginal Distribution of Transit Observables

Marginal Distribution of Ratios of Transit Observables for Pairs of Planets in Same System

LocalResource("../_assets/week5/Compare_models/Models_Compare_underlying_multiplicities.png", :width=>"75%")

Eccentricities & Inclinations of Multiple Planet Systems

LocalResource("../_assets/week5/Best_model/Clustered_P_R_Model_underlying_pratio_min_vs_amd_ecc_incl.png")

LocalResource("../_assets/week5/Best_model/Clustered_P_R_Model_underlying_mult_vs_amd_ecc_incl_dists_long.png")

#LocalResource("../_assets/week5/Best_model/Observed/Clustered_P_R_Model_logxi_per_mult.png")

#=
TwoColumn(
    LocalResource("../_assets/week5/Best_model/Observed/Clustered_P_R_Model_logxi_mmr_compare.png"),
    LocalResource("../_assets/week5/Best_model/Observed/Clustered_P_R_Model_logxi_nonmmr_compare.png") )
=#

Helper Code

ChooseDisplayMode()

Full Width Mode Present Mode

begin 
    using PlutoUI, PlutoTeachingTools
    #using CSV, DataFrames, Query
    using Downloads
    #using Distributions, Statistics
    #using Plots, LaTeXStrings, ColorSchemes, StatsPlots
end

question(str; invite="Question") = Markdown.MD(Markdown.Admonition("tip", invite, [str]))

question (generic function with 1 method)

Built with Julia 1.8.2 and

Downloads 1.6.0
PlutoTeachingTools 0.2.3
PlutoUI 0.7.43

To run this tutorial locally, download this file and open it with Pluto.jl.