ASTB01 Exoplanets Lab
Author: Anders Johansen
Reflected light
We first consider the stellar light reflected off a planet and estimate how hard it is to detect this light from Earth.
L p
L *
r
The figure shows the star with luminosity L? shining on a planet at a distance r away. The planet of radius Rp reflects the luminosity Lp. The stellar flux received by the planet is L F = ? . (1) ? 4πr2 The planet reflects the fraction A (the albedo) of the incoming light on its dayside,
2 Lref = AπRpF? . (2)
1 The ratio of planet light to stellar light received at Earth at a distance d from the system is
2 Fref (d) Lref /(2πd ) 2Lref = 2 = F?(d) L?/(4πd ) L? 2 2 2 2AπRpL?/(4πr ) A Rp = = 2 L? 2 r R !2 r −2 ≈ 4 × 10−9A p . (3) RJup 5 AU
Here the final equation is normalised to a planet of Jupiter radius RJup at 5 AU distance from the host star. The typical contrast of 2.5 × 108 corresponds to 21 in magnitude difference between planet and star. Considering a Sun-like star of absolute magnitude M = 5 at a distance of 10 pc gives a magnitude of m = 26 for a Jupiter-size planet at 5 AU from the host star, in principle observable with a good telescope (such as the Hubble Space Telescope). The problem is that the star, effectively a point source, seen by a telescope of diameter D will be imaged as an Airy pattern due to diffraction:
The angular diameter of central Airy disc is
λ λ ! D −1 θ ≈ 1.22 = 0.126” . (4) D 500 nm 1 m
Recall that 1 AU at 1 pc distance extends 1” in angle, so 5 AU (the distance between the Sun and Jupiter) extends 0.5” in angle at 10 pc. This means that planets within 5 AU from the host star will be lost in the Airy disc or in one the secondary peaks of the Airy pattern. Planets in wider orbits, on the other hand, have very low luminosities in their reflected light. That is the reason why astronomers resort to indirect methods to detect exoplanets.
Radial velocity method
The radial velocity method exploits that two gravitating bodies in fact orbit around their common centre of mass. The orbital motion of the star around the centre of mass can be detected from the periodic redshift and blueshift of its spectral lines. The amplitude of the radial velocity yields the semi-major axis of the planet as well as its minimum mass. Let us assume that the star and the planet interact only via gravity:
2 F* Fp
r* rp
We place the origin of the coordinate frame at an arbitrary position and denote the position vector of the star r? and the position vector of the planet rp. The force of gravity felt by the star is denoted F ? and the force of gravity felt by the planet is denoted F p. These two forces are given by Gm m r ¨ ? p ?p F ? = m?r? = 2 , (5) r?p r?p Gm m r ¨ ? p p? F p = mprp = 2 . (6) rp? rp?
Here we introduced the mass of the star m?, the mass of the planet mp, the gravity constant G, the vector pointing from star to planet r∗p = rp − r?, as well as the vector pointing from planet to star rp∗ = r? −rp. We immediately see that F p = −F ?. Gravity clearly should obey Newton’s third law!
Equations (5) and (6) can be subtracted to yield a single equation for r ≡ r∗p, G(m + m ) r r¨ = − ? p . (7) r2 r This equation corresponds to the simpler situation of a mass-less (non-gravitating) parti- cle orbiting around a central object with mass m? +mp. The solution is that the vector r follows the familiar circular or eccentric orbit. We were able to transform the complicated two-body problem into a much simpler one-body problem by choosing a reference frame fixed on the central star. Note that r is measured relative to the position of the star, but the star is of course moving as well.
Going instead to the (inertial) centre-of-mass frame will illuminate the actual orbit of the star. In the centre-of-mass frame the relative positions must obey
m?r? + mprp = 0 . (8)
Here we redefined the coordinate system so that r? and rp are measured relative to the centre of mass. Using r ≡ r?p = rp − r? (which follows a Keplerian orbit) we get
m?r? + mp(r? + r) = 0 . (9)
3 The positions of star and planet follow as
mp mp r? = − r ≈ − r for mp m? , (10) m? + mp m? m? rp = + r ≈ r for mp m? . (11) m? + mp We see here how both the planet and the star follow the Keplerian orbit vector r, but with different coefficients in front, which depend on their relative masses. The star is always on the opposite side of the planet, in order to conserve the centre of mass at the origin of the coordinate frame. In the limit where the star is much more massive than the planet, the planet’s orbit will be approximately rp ≈ r and the star’s orbit will be the much smaller r? ≈ −(mp/m?)r. The centre of mass between Sun and the planets in the Solar System is shown in the following table:
Planet CM with Sun [km] CM with Sun [R ] Mercury 9.6 km 1.4 × 10−5 Venus 265 km 3.8 × 10−4 Earth 449 km 6.4 × 10−4 Mars 74 km 1.1 × 10−4 Jupiter 743,000 km 1.07 Saturn 408,000 km 0.59 Uranus 125,000 km 0.18 Neptune 232,000 km 0.33
Jupiter has by far the largest effect on the Sun. In the presence of only Jupiter the Sun would orbit around a point just outside the surface of the Sun.
The orbital frequency of the planet Ωp ≡ Ω must equal that of the star Ω? in order to maintain the centre of mass. Therefore we will know the orbital frequency of the planet by measuring the orbital frequency of the star from the periodic redshift and blueshift of the spectral lines. The period of the planet, P = 2π/Ω, then directly yields the semi-major axis of the planet a, provided that the star’s mass is known, from v u 3 s 3 2π u a a P = = 2πt ≈ 2π . (12) Ω G(m? + mp) Gm?
In order to get the mass of the planet we need to measure the speed of the star around the centre of mass. The star’s speed is given by 2πr v = ? . (13) ? P
The centre of mass of the system is located at a distance r? = (mp/m?)a from the centre of the star, so knowing v? from the spectrum we can infer r? and hence mp (provided that the star’s mass is known). The speed of the Sun around the centre of mass with the planets of the Solar System is shown in the following table.
4 Planet RV of Sun Mercury 0.008 m/s Venus 0.09 m/s Earth 0.09 m/s Mars 0.008 m/s Jupiter 12.5 m/s Saturn 2.8 m/s Uranus 0.3 m/s Neptune 0.3 m/s
Jupiter again gives the largest effect, of approximately 12.5 m/s. This is nevertheless an extremely low speed in astronomical terms. The Earth gives a velocity signal of only 10 cm/s. The problem is that only the component of the star’s velocity that is perpendicular to the plane of the sky can be measured. This radial velocity is obtained directly from the spectrum. The result is that we measure RV? = v? sin i ∝ mp sin i. Thus we see only the 0 effect of a planet with mass mp = mp sin i. Here i is the angle between the plane of the sky and the plane of the orbit.
The above sketch shows three examples of how the star’s orbit around the centre of mass could be oriented on the plane of the sky. In the first case the unseen companion could be very massive, but the orbital plane is almost parallel to the sky, so only a small fraction of the star’s speed is measured in the radial velocity. Only if the inclination is close to 90◦ (rightmost sketch) would we be able to measure the true companion mass from the radial velocity. Since the inclination i is generally unknown from the spectrum, the radial velocity method measures the minimum mass of the planet
In 1995 astronomers Michel Mayor and Didier Queloz from the Observatory of Geneva reported that they had detected the radial velocity signal of a star orbited by an exoplanet.
5 They used the special-purpose spectrograph ELODIE. The star 51 Peg’s radial velocity has a period of 4.23 days and an amplitude 59 m/s (above plot). The inferred planet has an orbital distance of 0.053 AU and a minimum mass of 0.428 MJup. They compared the projected rotation speed of 51 Peg to the magnetic activity of the star to get a lower limit to the inclination i. Their reasoning is that if the motion of the star would be almost parallel to the sky, then the actual rotation speed of the star would also be much higher than the measured rotation speed. Such a fast rotation would show up in the magnetic activity of the star, but they found no signs of the stellar activity caused by strong magnetic fields. Hence they concluded that i could not be very low and obtained an upper limit to the planet’s mass of 1.2 MJup. The spectrograph used to detect 51 Peg b was built for ultra-high-precision measure- ments of stellar spectra. Several thousand spectral lines are considered simultaneously to measure the star’s velocity from the tiny Doppler shift, ∆v ∆ν = . (14) c ν Furthermore a reference spectrum (Thorium) is superimposed on the stellar light as it enters the spectrograph to eliminate systematic errors within the instrument. The ELODIE spectrograph used to find 51 Peg b achieved a precision of 13 m/s. This would not be precise enough to detect Jupiter orbiting around the Sun. However, 51 Peg b is much closer to the host star than Jupiter is to the Sun, and hence the radial velocity signal is much higher. Exoplanets are named after their host star with an additional lower case letter (b for first discovered, c for second, etc.). The following table gives a list of the first exoplanets discovered:
Planet mp [MJup] a [AU] Reference 51 Peg b 0.468 0.052 Mayor & Queloz (1995) 47 Uma b 2.53 2.1 Butler & Marcy (1996) 70 Vir b 7.44 0.44 Marcy & Butler (1996) 55 Cnc b 0.824 0.115 Butler, Marcy, et al. (1997) tau Boo b 3.9 0.046 Butler, Marcy, et al. (1997) ups And b 0.69 0.059 Butler, Marcy, et al. (1997) 16 Cyg B b 1.68 1.68 Cochran et al. (1997)
Some of these, like 51 Peg b, are very close to the host star. This led to the definition of a new class of planets, namely the hot Jupiters. The presence of gas giants like Jupiter so close to the host star was a surprise, since the Solar System contains no such planet.
Transit method
The radial velocity method gives no information about the physical properties of the planet and only yields the minimum mass, although stellar rotation and astrometric measurements can give good upper limits to the mass. The radius of the planet Rp can be obtained if the planet transits its host star. Combined with the mass known from the radial velocity method (since the transit yields also the inclination of the orbit) one then 3 obtains the planet’s mass density ρp = mp/[(4π/3)Rp].
6 A planet transiting its host star blocks out the stellar light fraction
2 2 ∆L πRp Rp = 2 = . (15) L πR? R?
Measurement of ∆L/L and knowledge of R? thus yields directly the radius of the planet. Jupiter’s radius is roughly 10% of the radius of the Sun and would give a transit depth of 1%, a precision that is easily obtainable with ground-based photometry.
Tangent plane to sky
i To Earth a
A planet transits its host star when the orbital inclination i is close to 90◦ (edge on). It is required that R + R a cos i < R + R ⇒ cos i < ? p . (16) ? p a The geometric probability for such an alignment is found by integration over all possible angles on the unit sphere
R 2π R π/2 −1 sin ididθ R + R R P = 0 cos (R?+Rp)/a = ? p ≈ ? . (17) geom 2π a a The factor sin ididθ represents an infinitesimal area element on the unit sphere. The probability that the orbit is aligned right to see a transit is
! −1 R? R? a Pgeom = = 0.46% . (18) a R AU The transit probabilities for the planets in the Solar System and 51 Peg b are shown in the following table:
Planet Pgeom Mercury 1.15% Venus 0.62% Earth 0.45% Mars 0.29% Jupiter 0.094% Saturn 0.051% Uranus 0.024% Neptune 0.015% 51 Peg b 9.95%
7 Mercury has the highest transit probability of the planets in the Solar System (approxi- mately 1%), for a randomly placed alien observer. The transit probability for 51 Peg b is much higher, due to its proximity to the host star, at almost 10%. Unfortunately 51 Peg b does not transit when seen from the Earth, but HD 209458 b does (Charbonneau et al. 2000). The following plot shows the transit light curve obtained from a ground-based telescope:
The scatter towards the end of the transit is due to a weather-related worsening of the seeing conditions. The radial velocity signal – independently discovered with the Keck telescope (HIRES spectrograph) and Observatoire de Haute Provence (ELODIE spectrograph) – gives a minimum mass of mp sin i = 0.64MJup. The measured transit period is 3.52 d, in good agreement with the period of the radial velocity signal. This gives a semi-major axis of 0.047 AU. The transit depth of 1.6% gives a planet radius of Rp = 1.38RJup. The inclination is also known if the planet transits, from the shape and duration of the lightcurve, yielding i = 86.7◦. This breaks the degeneracy of the radial velocity mass measurement and gives a planetary mass of mp = 0.64MJup. The mass density of HD 209458 b can then be calculated to be ρ = 0.37 g cm−3, less than that of −3 Jupiter (ρJup = 1.326 g cm ). This was the first proof that Jupiter-mass exoplanets are gas giants like Jupiter and Saturn. Transiting planets are not considered confirmed without radial velocity follow-up to de- termine the mass, since there are many kinds of false positives that can mimic a transit signal. One example is a distant eclipsing binary star which blends with the signal from the measured star. The mutual eclipses of the primary and secondary of the background binary can mimic the signal from a planet transiting the star. Hence radial velocity measurements are crucial for determining the planetary nature of companions and to get the mass density. Currently the best instruments for radial velocity measurements have a precision of around 1 m/s (HARPS). The ESPRESSO instrument at the VLT aims to reach cm/s precision in 2016. This precision is high enough to determine the mass of Earth-mass exoplanets.
Direct imaging
Indirect methods have been very successful in finding exoplanets during the last two decades. Direct imaging of exoplanets is nevertheless still an important goal of exoplanet studies, since that provides direct access to the spectrum of the planet, and in the future to resolved images of exoplanets.
8 The best way to look for the direct light from exoplanet is to image very young stars at infrared wavelengths. Gas giants like Jupiter are very hot when they form, due to the gravitational energy released when they accrete their gaseous envelope. The left plot above shows the planetary system orbiting the young star HR 8799 (Marois et al. 2008). The four planets shine brightly in the infrared. The image of the central star has been suppressed by subtraction of several rotated images of the planetary system. The planets are inferred to be between 7 and 10 Jupiter masses, by comparing to thermal evolution models. The exoplanet orbiting the star Fomalhaut (right plot above) has actually been detected at visible wavelengths (Kalas et al. 2008). This is an extraordinary discovery in many ways. The planet orbits at 110 AU from the host star and should not be detectable in reflected light at those distances. A possible explanation is that the planet is orbited by an extensive ring system, like Saturn’s but much larger, which reflects large portions of the incoming stellar light. The planet has not been detected in the infrared, so mass estimates are not possible. The orbit has now been followed for a few years. It appears that the planet’s orbit is very eccentric (e = 0.8±0.1) and that the planet passes regularly above and below the ring of dust orbiting Fomalhaut.
Lab exercises
The lab exercises consist of two components:
1. Finding the potentially habitable super-Earth Kepler-22 in the Kepler data
2. Learning about exoplanet statistics with the interactive catalogue of the Extrasolar Planets Encyclopedia
General instructions for the lab and the report:
• You should work in a group of up to four people to get most out of the teamwork and the discussions. Write on the report who was in your group. Reports must be written individually.
• Hand in the report at the latest two weeks after the lab.
• Remember to answer all questions and subquestions in the report and to include all the relevant plots.
9 • Clear plots are key to illustrating scientific results. Carefully consider your axis ranges and whether to use linear or logarithmic axis.
Before the lab session you should:
• Read this overview of exoplanet detection methods carefully.
• Read about the Kepler mission, the CHEOPS mission and the PLATO mission on Wikipedia (or elsewhere).
At the beginning of the lab you should:
• Describe the two most successful ways to detect exoplanets around other stars. Discuss advantages and disadvantages of the two methods. Why is radial velocity detection necessary for confirmation of an exoplanet? Discuss this both in the group and in the report.
• Describe the main characteristics of the Kepler mission, the CHEOPS mission and the PLATO mission as well as the most important differences between these mis- sions. Discuss this both in the group and in the report.
The Mikulski Archive for Space Telescopes (MAST)
In this exercise you will look at data from the Kepler satellite to “discover” the exoplanet Kepler-22b. This is the first discovered super-earth near the habitable zone of its host star, and it has a very clean signal. Data from the Kepler satellite can be found at The Mikulski Archive for Space Telescopes (MAST) at the URL http://archive.stsci.edu/kepler/data_search/search.php. Instructions for the Kepler-22b exercise:
• Choose “Kepler-22” in the “Target Name” box. Many links appear.
• Click on the 2nd, 11th and 23rd links.
• Zoom-in to produce plots that show clearly the transiting planet.
• Save the plots as PNG and include the plots in your report.
• Use the time of the transits to figure out the orbital period.
• Use the star mass (e.g. from the Extrasolar Planets Encyclopedia) to find its semi- major axis.
• Measure the depth of the transit.
• Use the star radius to estimate the radius of the planet.
Describe the procedure and include results and plots in the report.
10 The Extrasolar Planets Encyclopedia
The Extrasolar Planets Encyclopaedia (http://exoplanet.eu/) is an online database for exoplanets. It contains the physical characteristics of exoplanets and their host stars for basically all confirmed exoplanets. The mass of an exoplanet must be known before it is considered confirmed (an exoplanet for which only the radius is known is labelled a can- didate exoplanet); masses are generally obtained either by the radial velocity method or by detection of tiny variations in the transit signals caused by mutual perturbations be- tween planets in a planetary system (transit timing variation or TTV). The encyclopedia is maintained by Jean Schneider at Paris Observatory. The Extrasolar Planets Encyclopedia is an excellent tool for professional astronomers as well as for students who wish to become acquainted with the vast catalogue of exoplan- ets. Start by going to the Extrasolar Planets Encyclopedia and follow the link to the ‘Diagrams’ page.
Question 1: Plot the planetary mass versus semi-major axis. Identify the planets known as “hot Jupiters”. Identify the planets known as “super-Earths” or “mini-Neptunes” (hint: Neptune is 0.05 Jupiter masses, Earth is 0.003 Jupiter masses). Explain how these planets are different from the planets in the Solar System.
Question 2: Use Extrasolar Planets Encyclopedia to plot the mass of detected exoplan- ets on the y-axis versus the year of discovery on the x-axis. When was the first exoplanet discovered? Why do you think that the planets found in 1989 and in 1992 are generally not credited with being the first exoplanets? You can click on a planet to get information about the discovery and a list of all relevant papers.
Question 3: Limit the plot to planets discovered by the radial velocity method, by entering the detection method in the field above the x-axis. Click the question mark for instructions. Describe the trend that you see for the mass of the heaviest and the lightest exoplanets found as a function of time.
Question 4: Plot now the radius of planets detected by the transit method. Compare the result to the radial velocity exoplanets (use the same axis range for best comparison). What is the trend for transiting planets before 2009 and what is the trend after 2009? Why is there a change in the trend?
Question 5: Plot the eccentricity of the planet orbit versus the orbital period of the planet. How is the eccentricity of exoplanet orbits compared to the planets in the Solar System? What happens to the eccentricity for planets in very short periods? Some planets have very long periods. How are those detected?
Question 6: Plot now the eccentricity of the planet orbit versus planet mass. What trends do you see? Next limit the exoplanets to those for which a transit has been detected. Comment on the difference. Place the Solar System planets on the plots and comment.
11 Question 7: Plot the planetary radius versus the planetary mass. Do you see two regimes? Make separate power-law fits for masses below and above 0.2 Jupiter masses. What is the mass-radius relationship for low-mass planets and for high-mass planets? Note that individual mass measurements can be very uncertain, so you can ignore outliers when you make the fit.
Question 8: Plot the metallicity of the host star versus the planetary radius. Do you see two regimes? Discuss implications for the search for life in the Universe.
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