1 ASTR 112 Lab 8: Exoplanets Your Name: Click or tap here to enter text. INTRODUCTION: In this lab you investigate how extrasolar planets are discovered using transit method. Learning Goals: Students will • Learn how transit method is used to discover extrasolar planets • Learn what are the strengths and limitations of this method Learning tools: Simulation: https://ccnmtl.github.io/astro-simulations/exoplanet-transit-simulator/ or from downloaded NAAP labs: select #12 Extrasolar planets and then Exoplanet Transit Simulator (see below) Background An extrasolar planet, or exoplanet, is a planet beyond the Solar System, orbiting around another star. The abundance and characteristics of exoplanets has long been a topic of great interest. It has direct implications on our understanding of our own solar system and planet. It is also directly linked with the broader philosophical question “are we alone in the universe?”. If other stars commonly have planets, this would greatly increase the likelihood of us discovering life elsewhere in the universe. 2 Planets around other stars proved very elusive to find. First detections go back to late 1980s and in 1990s, but there were few until better techniques and apparatus became available. Things picked up over the last decade and as of June 1st 2020 over 4200 exoplanets have been discovered. By far the most prolific method of exoplanet detection thus far is the transit method. Though this method has considerable limitations, because it only can be used to detect planets that have orbital planes edge on to our line of sight, about 75% of exoplanets found thus far were discovered using this method. What transit method entails is finding and measuring tiny dips in the host star’s light output that occur when a planet passes in front of it (please watch the simulation embedded on Blackboard in the introduction to this lab). The dimming is very small, in a few cases over 1.5%, but typically just a fraction of a percent. E.g. an Earth size planet in an earth sized orbit about a Sun-like star would dim it only by about 0.008 %. Watch video #5 embedded in the Blackboard introduction to this lab. In this animation, the green line tracing below the planet and star is called the "light curve." The light curve is a graph the brightness of the star over time. The dip in light that happens when the planet passes in front of the star is called the "transit." An individual dip in a light curve of a star does not necessarily indicate a transit, however, it could be due to other phenomena such as starspots. What astronomers look for is dimming that repeats in regular time intervals, every time a planet passes in front of the star that it orbits. The length of time between each transit is the planet's orbital period, or the length of a year on that particular planet. Not all planets have years as long as a year on the Earth. Some planets orbit around their stars so quickly that their years only last few hours! Figure 1shows a light curve from the Kepler mission for an exoplanet called Kepler 2b. First of all, notice the scale on the vertical axis of the plot above. The entire span of the graph is within 1% of the total brightness of the star (1 is equivalent to 100% that is the brightness of a star when it is nit eclipsed by a planet). From that you can surmise that searching for transiting planets requires highly precise brightness measurements. The light curve of Kepler 2b shows three very convincing Figure 1. transits - the three evenly spaced dips in brightness shown with red arrows. The amount of time between the consecutive dips gives you the orbital period of the planet. You can estimate it from the figure by reading the time elapsed between the dips to be about 2.3 days. Once we know orbital period, if we also know the mass of the orbited star, we can use generalized (to account for different stellar masses) Kepler’s 3rd law to find the average orbital distance (i.e. semi-major axis): 3 𝒑𝟐 𝑴 = 𝒂𝟑 (1) Here the period must be expressed in Earth years, orbital distance in AU and stellar mass in solar masses. For the planet Kepler 2b, it’s star, Kepler 2, has mass of about 1.47 solar masses, so 𝑀 = 1.47, and the period is 𝑝 = 2.3 days = 2.3/365 years = 0.0063 years Applying generalized Kepler’s 3rd law yields 𝑎3 = 𝑝2 𝑀 = (0.0063)2 × 1.47 = 5.8 × 10−5 3 So 𝑎 = √5.8 × 10−5 = 0.039 AU In addition to orbital period and distance, transit method allows you to measure planet's size. The amount of light a planet blocks in transit depends on the size of the planet. A large planet blocks more light than a small planet, so the large planet will have a deeper transit. The area of starlight blocked by a transiting planet depends on its size, specifically on the area of the disk of the planet. The ratio of area of planet’s disk to star’s disk determines the drop in star’s brightness during transit: i.e. the transit depth. 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 𝑑𝑒𝑝𝑡ℎ = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑙𝑎𝑛𝑒𝑡 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑡𝑎𝑟 (expressed as a fraction) (2) Since 𝑎𝑟𝑒𝑎 = 𝜋𝑅 2 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 𝑑𝑒𝑝𝑡ℎ = 2 𝜋𝑅𝑝𝑙𝑎𝑛𝑒𝑡 2 𝜋𝑅𝑠𝑡𝑎𝑟 or 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 𝑑𝑒𝑝𝑡ℎ = ( 𝑅𝑝𝑙𝑎𝑛𝑒𝑡 2 𝑅𝑠𝑡𝑎𝑟 ) Figure 2. (3) Thus 𝑹𝒑𝒍𝒂𝒏𝒆𝒕 = 𝑹𝒔𝒕𝒂𝒓 √𝒕𝒓𝒂𝒏𝒔𝒊𝒕 𝒅𝒆𝒑𝒕𝒉 (4) For Kepler 2, the transits decrease its brightness from 1 to about 0.993 - a decrease of about 0.7% or 0.007, if we express it as a fraction. Kepler 2 star has radius of about 2 solar radii. Using the formula (4) for the planetary radius we obtain: 𝑅𝑝𝑙𝑎𝑛𝑒𝑡 = 2 𝑅𝑆𝑢𝑛 √0.007 = 0.167 𝑅𝑆𝑢𝑛 Planetary radii are usually given in terms of Jupiter of Earth radii. Recall that Sun’s radius is about 10 times that of Jupiter, so we convert: 0.167 𝑅𝑆𝑢𝑛 = 0.167 × 10 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = 1.67 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 or recalling that about 11 Earths line up along Jupiter’s equator: 4 0.167 𝑅𝑆𝑢𝑛 = 1.67 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 =1.67 × 11 𝑅𝐸𝑎𝑟𝑡ℎ = 18.4 𝑅𝐸𝑎𝑟𝑡ℎ We can ask ourselves how hard would it be to use the transit method to detect planets like those in our solar system. For the moment, we’ll ignore the length of orbital period, which, in some cases would require many years if not centuries of observations to spot even three or four transits, and just focus our attention on the planetary sizes. We can use formula (3) to predict the depth of the transits of the solar system planets listed in table 1. First convert planet’s radii from being expressed in Earth radii, to being expressed in solar radii. For the purpose of this exercise you can approximate solar radius to be 110 Earth radii. Thus divide each planet’s radius expressed in Earth radii by 110 to get it in terms of solar radius. Then use formula (3) with the value for the planet’s radius and for sun’s (star’s) radius expressed in solar radii. Since, unsurprisingly, Sun has 1 solar radius radius ☺, i.e. 𝑅𝑠𝑡𝑎𝑟 = 𝑅𝑆𝑢𝑛 = 1 solar radius, you have: 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 𝑑𝑒𝑝𝑡ℎ = ( 𝑅𝑝𝑙𝑎𝑛𝑒𝑡 2 𝑅𝑠𝑡𝑎𝑟 𝑅𝑝𝑙𝑎𝑛𝑒𝑡 2 ) =( 𝑅𝑆𝑢𝑛 𝑅𝑝𝑙𝑎𝑛𝑒𝑡 2 ) =( 1 ) = (𝑅𝑝𝑙𝑎𝑛𝑒𝑡 ) 2 Table 1. planet planet’s radius (REarth) planet’s radius (RSun) transit depth Jupiter 11.2 Click or tap here to enter text. Click or tap here to enter text. Neptune 3.9 Click or tap here to enter text. Click or tap here to enter text. Earth 1 Click or tap here to enter text. Click or tap here to enter text. Mercury 0.38 Click or tap here to enter text. Click or tap here to enter text. Pluto 0.187 Click or tap here to enter text. Click or tap here to enter text. The size of the planet matters, because it gives us crucial information about planet’s possible habitability. If the planet is too small (like Mercury or Mars), it will not have enough gravity to hold on to a sufficiently thick atmosphere—gas molecules will escape the planet over a time-span of not many years in the lifetime of the planet-star system. If the planet is too large, it will retain a huge amount of atmosphere and have crushing atmospheric pressure, like the giant planets Jupiter and Saturn. In many cases astronomers found planetary systems i.e. systems of two or more planets orbiting stars. In such cases light curves may become quite complex and hard to read. Check figure 3 below which shows the light curve of small and cool star TRAPPIST-1 over a period of 20 days taken in 2016. On many occasions the brightness of the star drops for a short period and then returns to normal. These transits are of different duration and depth, and are a result of one or more of the star’s seven planets passing in front it. The lower part of the diagram shows which of the system’s planets are responsible for the transits. 5 Figure 3a. So how do we read the graph in figure 3 and extract information about TRAPPIST-1 planets from it? Let’s consider planet 1d marked with yellow diamonds. Notice that it transited on Sept. 23 rd, Sept. 27th, Oct. 1st, Oct 5th, and Oct. 9th. That tells us that its orbital period is about 4 days. We can now use Kepler’s 3rd Law to find this planets distance to its star. 𝑝 = 4 days = 4/365 years = 0.01096 years From Wikipedia I found that the mass of TRAPPIST-1 star is 0.0898 solar masses 𝑀 = 0.0898 𝑀𝑆𝑢𝑛 : Applying generalized Kepler’s 3rd law yields 𝑎3 = 𝑝2 𝑀 = (0.01096)2 × 0.0898 = 1.0785 × 10−5 3 So 𝑎 = √1.0785 × 10−5 = 0.0221 AU To find transit depth correctly, we need to find transits during which it is the only planet transiting at the time. Sept. 23rd, Oct 5th and Oct. 9th look like best candidates. For all of them, the curve of relative brightness dips down from 1 to about 0.9965. Thus the transit depth is about 0.0035. From this we can estimate this planets radius (from Wikipedia I found that the radius of TRAPPIST-1 star is 0.1192 solar radii 𝑅𝑠𝑡𝑎𝑟 = 0.1192 𝑅𝑆𝑢𝑛 ): 𝑹𝒑𝒍𝒂𝒏𝒆𝒕 = 𝑹𝒔𝒕𝒂𝒓 √𝒕𝒓𝒂𝒏𝒔𝒊𝒕 𝒅𝒆𝒑𝒕𝒉 = 0.1192 × √0.0035 = 0.007052 𝑅𝑆𝑢𝑛 = 0.007052 × 10 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = 0.07052 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 0.07052 × 11 𝑅𝐸𝑎𝑟𝑡ℎ ≈ 0.776 𝑅𝐸𝑎𝑟𝑡ℎ Try another TRAPPIST-1 planet for practice and compare your results with data in Wikipedia: https://en.wikipedia.org/wiki/TRAPPIST-1. Since that graphs have rough scales and are hard to read precisely, do not expect to get a perfectly matching result, but is should not be off by too much (more than 50%) either. 6 Procedure Part 1: Finding exoplanets STAR #1 Figure 4 below shows a light curve for a star. From the graph determine the orbital period, average orbital distance, and size of the planet that orbits this star. The orbited star has mass of 0.4 solar mass and radius of 0.374 solar radii. Figure 4. Planet b • • • Orbital period is Click or tap here to enter text. days = Click or tap here to enter text. years. Orbital distance is Click or tap here to enter text. AU Planet’s radius is Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ Note that the light curve shows a smaller dip at about 23 and 46 days. This shows a second planet transiting in front of this star. For this planet find the orbital period, average orbital distance, and size of the planet. Planet c • • • Orbital period is Click or tap here to enter text. days = Click or tap here to enter text. years. Orbital distance is Click or tap here to enter text. AU Planet’s radius is Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ Which planet is closer to the star? Click or tap here to enter text. Which one is bigger? Click or tap here to enter text. 7 STAR #2 Figure 5 shows the light curve of a star that has three transiting exoplanets. Their transits are marked by colored dots. From the graph determine the orbital period, average orbital distance, and size of each planet that orbits this star. The orbited star has mass of 0.99 solar masses and radius of 0.95 solar radii, so it is very similar to our Sun. Figure 5. Planet b (marked by yellow dots) • • • Orbital period is Click or tap here to enter text. days = Click or tap here to enter text. years. Orbital distance is Click or tap here to enter text. AU Planet’s radius is Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ Planet c (marked by red dots) • • • Orbital period is Click or tap here to enter text. days = Click or tap here to enter text. years. Orbital distance is Click or tap here to enter text. AU Planet’s radius is Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ Planet d (marked by blue dots) • • • Orbital period is Click or tap here to enter text. days = Click or tap here to enter text. years. Orbital distance is Click or tap here to enter text. AU Planet’s radius is Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ Which planet is closest to the star? Click or tap here to enter text. Which one is the largest? Click or tap here to enter text. Which planet is the smallest? Click or tap here to enter text. 8 STAR #3 Figure 6 shows single transits of five orbiting a star. From the graphs (the depth of the transit) determine the size of each planet that if the star has radius of 0.64 solar radii. Express each radius in terms of Jupiter and Earth radii. Use example of Kepler 2 and TRAPPIST-1d in the Background part of this document for sample calculations. • Planet b Transit depth = Click or tap here to enter text. Radius Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ • Planet c Transit depth = Click or tap here to enter text. Radius Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ • Planet d Transit depth = Click or tap here to enter text. Radius Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ • Planet e Transit depth = Click or tap here to enter text. Radius Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ • Planet f Transit depth = Click or tap here to enter text. Radius Click or tap here to enter text. 𝑅𝐽𝑢𝑝𝑖𝑡𝑒𝑟 = Click or tap here to enter text. 𝑅𝐸𝑎𝑟𝑡ℎ Which planet a, b, c, d, e or f is the largest? Click or tap here to enter text. Which planet is the smallest? Click or tap here to enter text. 9 Part 2. Properties of transits Click on this link to access the simulation: https://ccnmtl.github.io/astro-simulations/exoplanet-transitsimulator/. You should see a site that looks like what’s shown in figure 7. Figure 7. This simulator allows us to examine how different properties of the star, the planet and its orbit affect light curves. The upper left panel shows the star and planet as they would be seen from earth if we had an extremely powerful telescope. In fact, this privileged view is impossible with any existing telescope. The light curve is shown in the upper right panel. You have the option of showing simulated noisy measurements and hiding the theoretical curve — this gives a better idea of what it is like working with real data. The planet, star, and system properties can be set in the lower panels. These parameters can also be set by selecting one of the presets from the dropdown menu in the Presets panel. Select Option A and click set. This option configures the simulator for Jupiter in a circular orbit of 1 AU with an inclination of 90°. Determine how increasing each of the following variables would affect the depth and duration of the transit. (Note: the transit duration is shown underneath the flux plot.) Vary one quantity at a time and the return to default setting before you start observing the variation in the next quantity. I’m not looking for any numerical values here, but rather trends. E.g. If I increase the mass of the planet, does the transit depth 10 change? If so, how? Does it increase or decrease? Does transit duration (eclipse time) change? How? Does the frequency of transits change (i.e. planet’s orbital period which is listed below the graph of the transit in the statement below the transit graph: “eclipse takes 2.93 hour of 3.56 day orbit” – values will vary, these appear on figure 7 – with 2.93 hours denoting transit duration and 3.56 days denoting orbital period) Planet properties: • Mass of the planet: Click or tap here to enter text. • Radius of the planet: Click or tap here to enter text. • Semi-major axis: Click or tap here to enter text. • Eccentricity: Click or tap here to enter text. Star properties: • Mass (and thus, temperature and radius) of the star: Click or tap here to enter text. What does the information you gathered above tell you about what types of exoplanets are most easily detected via the transit method? Click or tap here to enter text. Select Option B and click set. This preset is very similar to the Earth in its orbit. How long does the transit of an Earth-like planet take? Click or tap here to enter text. How much time passes between eclipses? Click or tap here to enter text. How deep is the transit of an Earth-like planet around a sun-like star? Click or tap here to enter text. Return to Option A and click set. Now try changing the inclination of the system under “System Orientation and Phase”. What happens as you move the slider from right to left? Click or tap here to enter text. Can you make the planet transit disappear? Click or tap here to enter text. If so, over what range of inclinations is transit visible? Click or tap here to enter text. Over what range of inclinations is the transit invisible? Click or tap here to enter text. What does this tell you about the alignment of transiting exoplanetary systems and the likelihood of our detecting planets via this method? Click or tap here to enter text. Conclusion questions: In table 1, you calculated transit depths for planets in our solar system. How easy or hard do you think it will be to detect them based solely on their sizes? Click or tap here to enter text. A single dip in star’s brightness does not usually suffice to be proof enough of transit, since other factors can cause star’s brightness to vary. Usually multiple transits must be observed before a claim of discovery is made. Go to the planetary data worksheet for lab 2 and check the orbital periods. If we decided that just three transits are enough, how long would you need to observe to make a firm claim of discovery of such a planet? (think of how many orbital periods would it take). You’d need to observe 11 • • • • • Jupiter for Click or tap here to enter text. years. Neptune for Click or tap here to enter text. years. Earth for Click or tap here to enter text. years. Mercury for Click or tap here to enter text. years. Pluto for Click or tap here to enter text. years. Submission details: Submit into this lab’s drobox on Blackboard: • MS Word report (this document with your entries) only 
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