Blog #20 - What is TRAPPIST?

What is TRAPPIST? 

A lot of excitement was generated on February 22, 2017 when NASA released a press statement announcing the founding of 7 exoplanets orbiting a star 12 parsecs from Earth. Using transit photometry, as discussed in earlier blog posts, the Transiting Planets and Planetesimals Small Telescope (TRAPPIST) was able to record the presence of 7 exoplanets around a single star, 5 of which appear to be in or around the habitable zone while 3 have roughly Earth-sized radii.

TRAPPIST is actually a system of two 60-cm reflective, optical telescopes. The first, TRAPPIST-South is located in the Chilean mountains at La Silla Observatory while TRAPPIST-North is located in the Atlas Mountains of Morocco. Despite their locations, both telescopes are controlled out of Liege, Belgium and operate under fully autonomous, robotic systems. The team of telescopes creates photometry data by observing when planets transit across their host stars, causing an observed drop in incoming flux. The plot below shows the process for determining the transit of a planet which helps determine its period, radius, and location around a star.

When it came to the 7 planets orbiting TRAPPIST-1, the photometry data was truling startling as the 7 planets all created well defined transit drops. Below is the aligned data for all 7 planets as well as the brightness plots and a schematic of the orbits for the star system. 

The bottom left image is the aligned 7 photometry plots with clear ingress and egress dips and corresponding transit depths. These images in conjunction with the TRAPPIST telescopes were imaged by the Spitzer Space Telescope. 

Blog #19 - Transiting Exoplanet pt. 2

Transiting Exoplanet pt. 2

2. Now draw the star projected on the sky, with a dark planet passing in front of the star along the star’s equator.

The picture above depicts a planet transiting in front of its host star as observed from Earth

(a) How does the depth of the transit depend on the physical properties of the star and planet? What is the depth of a Jupiter-sized planet transiting a Sun-like star?

The optical depth or, \(\delta F\), is solely dependent on the cross sectional areas of the planet and the star, thus the radius is the only physical property we care about when considering depth of transit. \[\delta F = \frac{A_p}{A_{\star}} = \frac{\pi R_p^2}{\pi R_{\star}^2} = \frac{R_p^2}{R_{\star}^2} \] From the previous problem we know for a Jupiter-sized planet and a Sun-sized star, \[R_{\star} \approx 10*R_p\]Substituting into the equation for optical depth, we get \[\delta F = \frac{R_p}{10 R_p} = 10 \% \space transit \space depth\]

(b) In terms of the physical properties of the planetary system, what is the transit duration, defined as the time for the planet’s center to pass from one limb of the star to the other?

We know that for an full transit to occur, the planet's center must travel a distance of \(2 R_{\star}\) if observing the planet as traveling along a line at the equator of the star. We can use the simple relation between distance, velocity, and time to get the total transit time. \[t = \frac{d}{v}\]and \[v = \frac{2 \pi a}{p} \]This gives us \[t = \frac{2 R_{\star} P}{2 \pi a}\]To get the answer in terms of solely physical properties we can convert the orbit period, P, to \[P = 2 \pi \sqrt{\frac{a^3}{MG}}\] leading to a final answer of \[t = 2 R_{\star} \sqrt{\frac{a}{M_{\star}G}} \]Where \(a\) is the semi-major axis, M is the mass of the star, and G is the universal gravity constant.

(c) What is the duration of “ingress” (from no transit to full transit) and “egress” (from full transit to no transit) in terms of the physical parameters of the planetary system?


We know that given a spherical star and planet, the time of ingress must equal the time of egress\[t_{ingress} = t_{egress} \]Following the same method as part (b) we can solve for the time for the planet to move across the star's surface the distance of one planetary radius equating to the ingress transit time as seen in the diagram below.

From this diagram we can solve the time of transit using the same relation of distance, velocity, and time. \[t = \frac{d}{v} = \frac{R_p}{\frac{2 \pi a}{P}} = R_p \sqrt{\frac{a}{M_{\star}G}}\]

Blog #18 - Transiting Exoplanet

Transiting Exoplanet 



1. Draw a planet passing in front of its star, with the star on the left and much larger than the planet on the right, with the observer far to the right of the planet. The planet’s semi-major axis is a.

The picture depicts the setup of the problem with the exoplanet transiting in front of the spectator at a circular orbit with semi-major axis, a. 

(a) Show that the probability that a planet transits its star is R/a, assuming Rp<<R<<a. What types of planets are most likely to transit their stars?

The probability that the star passes in front of our view is simply a matter of proportions of the surface areas of the stars entire sphere of movement and the cylindrical area taken up by the far field view of our observer. To "transit" the star, the exoplanet must move within either side of the planet, within the cylindrical gaze of the viewer. The surface area of the entire potential orbit of the planet is simply the spherical surface area at a distance of the semi-major axis: \[A_{sphere} = 4 \pi a^2\]and \[A_{view} = (2 \pi R_{\star}) * (2a) = 4 \pi R_{\star} a\]To find the probability we simply see how likely it is for the planet (on the full spherical surface area) falls within the cylindrical area of the viewpoint. \[Probability = \frac{A_{view}}{A_{sphere}} = \frac{4 \pi R_{\star} a}{4 \pi a^2} = \frac{R_{\star}}{a}\]Which gives us the expected relation of \(\frac{R_{\star}}{a}\) for the probability. 

(b) If 1% of Sun-like stars in the Galaxy have a Jupiter-sized planet in a 3-day orbit, what fraction of Sun-like stars have a transiting planet? How many stars would you need to monitor for transits if you want to detect ten transiting planets?

We begin by solving for the semi-major axis of the mentioned planet using the relation between period and semi-major axis: \[P^2 \approx a^3 \]. Using the mention of a 3-day planetary orbit we can solve for an approximate semi-major axis as \[(\frac{3 day}{365 days}) = (\frac{a}{1 AU})\]Giving us \[a \approx .04 AU \]We also know that Jupiter's radius is roughly: \[R_{\star} = .0046 AU\]Giving us a probability relation of \[\frac{R_{\star}}{a} = \frac{.0046}{.04} = .114 = 11.4 \% \] However only 1% of all Sun-like stars have the possibility of containing these Jupiter-sized planets so the total probability of observing one of these exoplanets is \[.114*.01 = .001143 = .1143 \% \]To observe 10 transiting planets, this means you would need to observe \[\frac{10}{.001143} \approx 8,748 \space planets\]

Blog #17 - Orders of Magnitude

Orders of Magnitude and Scaling

In this worksheet, we’re going to do some “order of magnitude differentiation.” Let’s start of with a simple example that is completely unrelated to stellar structure:

(a) The velocity of a particle is v = αt2, where α is a constant, and we want to find the scaling of position with time. First, write down the equation in the form of a differential equation for x,the position. Next, we are going to say that dx ≈ ∆x ∼ x and dt ≈ ∆t ∼ t. In English: “dx is approximately the change in x, which scales as x.” Now it should be easy to show the scaling of x with t. What is the form of this scaling relationship?

We begin with the information we are given. \[v = \alpha t^2 \]From there we know the differential version of velocity is simply the change in distance over the change in time. \[v = \frac{dx}{dt} = \alpha t^2 \] Which gives us \[\frac{\Delta x}{\Delta t} \approx \frac{x}{t} \approx \alpha t^2\]Now we simply solve for x to give the scaling relation \[x = \alpha t^3\]

(b) So you are probably saying to yourself, “This doesn’t feel right mathematically. How can you treat differential quantities with such disdain?!” But this is a simple differential equation, so you can actually integrate it. What do you get? How does it compare to your scaling relationship?

To prove through integration that the scaling relation holds, we have to show the integral of velocity is the position scaling relation that we expect. We start with \[\int v = x\]Thus we have the integral for velocity as \[\int v = \int \alpha t^2 dt\]Which leads to \[\int \alpha t^2 dt = \frac{1}{3} \alpha t^3 + C_1\]Which gives us the same scaling relation as part a). \[\frac{1}{3} \alpha t^3 \approx \alpha t^3\]