A Monte-Carlo Simulation Tool of Exoplanet Populations Based on the Results from the Kepler Mission

Home , HD 10647, HD 40307, Rho Centauri, Tau Centauri, Wolf 1061

Master Thesis Jens Kammerer March 20, 2017

Advisors: Prof. Dr. H. M. Schmid, Dr. S. P. Quanz Department of Physics, ETH Z¨urich

Abstract

The most recent planet occurrence statistics inferred from the transit measurements of the Kepler spacecraft are used to simulate exoplanetary systems around nearby main sequence stars. Using randomly drawn orbit parameters and planet albedos the equilibrium temperature, the emitted blackbody flux and the reflected host star flux of each exoplanet are calculated. These properties are assumed to be dependent on the reprocessed host star light only, formation heat is neglected. By applying different filter curves and taking into account the resolution limits and faint source detection limits of different telescopes the observability of all simulated exoplanets can be determined depending on the chosen instrument. Hence, statistics for the number and the properties of directly detectable exoplanets as well as for the most fruitful target stars can be derived. We analyse the expected exoplanet yield with the direct imaging technique for VLT/NACO, E-ELT/METIS and Darwin/MIRI and find that VLT/NACO is not suitable for the detection of old (terrestrial) exoplanets whereas E-ELT/METIS could +2.44 +115.9 detect 7.77−2.15 exoplanets and Darwin/MIRI could detect 329.4−81.3 exoplanets of which most have radii between 0.5 R⊕ and 4 R⊕ and equilibrium temperatures between 200 K and 700 K. We investigate the instrument performance and the impact of the generated random parameters and discuss the effect of zodiacal and exozodiacal light and the planet occurrence around binary stars. We find that the expected planet yield depends strongly on the underlying planet population and the Bond albedos, moderately on small variations in the instrument parameters and only weakly on the orbital eccentricities and the geometric albedos.

Contents

Contents iii

1 Introduction1

2 Methods3 2.1 Stellar Sample...... 5 2.2 Planet Population...... 7 2.2.1 Burke 2015...... 9 2.2.2 Dressing 2015...... 11 2.2.3 Fressin 2013...... 13 2.3 Planet Properties ...... 14 2.3.1 Randomly Oriented Orbits ...... 14 2.3.2 Randomly Distributed True Anomaly...... 16 2.3.3 Randomly Drawn Albedos ...... 18 2.3.4 Semi-Major Axis and Planet-Host Star Separation . . . 19 2.3.5 Equilibrium Temperature...... 21 2.3.6 Phase Angle and Phase Curve ...... 22 2.3.7 Emitted and Reﬂected Blackbody Flux ...... 22 2.4 Instruments ...... 25 2.4.1 VLT and E-ELT...... 26 2.4.2 Darwin...... 27 2.4.3 Observed Planet Flux ...... 28 2.4.4 Radial Velocity ...... 29

3 Results 31 3.1 VLT and E-ELT ...... 32 3.1.1 Exoplanets Detectable via Direct Imaging ...... 32 3.1.2 Host Star Characteristics...... 36 3.1.3 Instrument Performance...... 40 3.1.4 Exoplanets Detectable via Radial Velocity ...... 42

iii Contents

3.2 Darwin...... 45 3.2.1 Exoplanets Detectable via Direct Imaging ...... 45 3.2.2 Host Star Characteristics...... 49 3.2.3 Instrument Performance...... 53 3.2.4 Exoplanets Detectable via Radial Velocity ...... 54

4 Discussion 57 4.1 Comparison to Quanz et al.(2015)...... 57 4.2 Choice of Albedos...... 61 4.3 Choice of Eccentricity ...... 62 4.4 Underlying Planet Population...... 64 4.5 Zodiacal Light...... 68 4.6 Planet Occurrence Around Binary Stars...... 69 4.7 Background-Limited Observations...... 70 4.8 Properties of M Dwarf Stars...... 72 4.9 Target Stars with Known Companions...... 73 4.10 Exo-Earth Yield...... 76

5 Conclusions 77

6 Acknowledgements 81

A List of Units 83

B List of Parameters 85

C PlanetS General Assembly Poster 91

Bibliography 93

iv Chapter 1

Introduction

In the last years the impact of space-based telescopes on exoplanet science has grown significantly, most notably by the Kepler mission which was designed to continuously stare at a fixed region in the sky to search for transiting planets around hundreds of thousands of stars (Basri et al., 2005). During 47 months of observation Kepler discovered 4175 planet candidates among its 197329 target stars (Mullally et al., 2015) and to the current date 2331 confirmed planets were announced by the Kepler team and the commu- nity (NASA Exoplanet Archive1, 31. January 2017). The immense statistics provided by Kepler have been used frequently to estimate the occurrence of planets around stars from a broad range of spectral types (Howard et al., 2012; Dressing and Charbonneau, 2013; Fressin et al., 2013; Burke et al., 2015; Dressing and Charbonneau, 2015; Mulders et al., 2015; Gaidos et al., 2016). Most of these studies make use of different types of pipeline completeness models to estimate the distribution of exoplanets in the radius-orbital period space depending on the spectral type and give their results in either continuous probability density functions or binned occurrence rates. These statistics have been used by Crossfield(2013) and Quanz et al.(2015) to model exoplanetary systems around nearby stars with the goal to estimate the number and properties of planets directly detectable by different ground-based high-contrast imaging instruments (e.g. E-ELT/METIS). Un- like the transit method which is especially sensitive to exoplanets orbiting their host stars at very short separations (Beatty and Gaudi, 2008) the direct imaging method which aims on spatially separating the flux received from an exoplanet and the flux received from its host star is especially sensitive to widely separated planets (Quanz, 2016). Hence, as Quanz et al.(2015) show, the resolution and the sensitivity of even the next-generation ground- based telescopes basically rule out the detection of a significant sample of

1NASA Exoplanet Archive, Exoplanet and Candidate Statistics, 31.01.2017, http://exoplanetarchive.ipac.caltech.edu/docs/counts_detail.html

1 1. Introduction

exoplanets with sizes smaller than Neptune. Finding a terrestrial planet in the habitable zone of its parent star using these telescopes with the direct imaging technique would be a rather lucky strike. Already in the early 2000s it became clear that only a space telescope can overcome the limits of ground-based observations like atmospheric seeing and restricted degrees of freedom in mirror size and baseline. Back then space-based interferometers for the search of exoplanets were proposed to both the National Aeronautics and Space Administration (NASA, Terrestrial Planet Finder (TPF), Lawson(2001)) and the European Space Agency (ESA, Darwin, Leger and Herbst(2007)). Unfortunately, neither of the two missions made it to the launch for which one reason among many others was the uncertainty in their scientific yield. Kepler released its first data in 2010 only (Borucki et al., 2010) so that the occurrence of planets was largely un- constrained when TPF and Darwin were discarded. More recently several papers have investigated the capabilities of a space-based direct imaging mission for the search of exoplanets in terms of detecting Earth twins in environments appropriate for extrasolar life (Stark et al., 2014; Brown, 2015; Stark et al., 2015). Indeed these studies make use of a statistical occurrence of Earth twins inferred from Kepler data, however all of them solely focus on planets which are extraordinarily equal to our own Earth. This is reflected in their choice of physical planet parameters which they assign to their simulated exoplanet sample necessary to estimate the scientific yield of their hypothetical space observatories. It is beyond debate that the search for extrasolar life is one main goal of exoplanet science, but to understand the complicated processes of the formation and evolution of planets of various sizes and compositions a more diverse sample of exoplanets is needed. Furthermore, the occurrence of Earth-like planets in the habitable zone of their parent stars is still affected by large uncertainties, varying from a factor of ∼ 2 (e.g. Dressing and Charbonneau(2015) around M type stars) to a factor of ∼ 10 (e.g. Burke et al.(2015) around G and K type stars). In this paper we present an approach similar to the Monte-Carlo simulation described in Crossfield(2013), but using the most recent and complete planet occurrence statistics from Dressing and Charbonneau(2015), Burke et al.(2015) and Fressin et al.(2013) combined in a patchwork-like approach to obtain the most realistic synthetic planet population of close-in exoplanets achievable to date. Our population covers a broad range of planet radii from 0.5 R⊕ to 22 R⊕ and of planet orbital periods from 0.5 d to 418 d. We perform simulations for the VLT/NACO and the E-ELT/METIS instrument to update the results from Quanz et al.(2015). Therefore, we recon- sider and carefully adjust their astrophysical assumptions to present a more sophisticated expectation value of detectable exoplanets for E-ELT/METIS. Furthermore, we study a very specific space observatory by investigating the exoplanet yield of a formation-flying space interferometer like Darwin

2 and present extensive predictions for observations in three different ﬁlters at 5.6 µm, 10 µm and 15 µm.

Chapter 2

Methods

We estimate the exoplanet yield of the direct imaging technique with the existing NACO (Nasmyth Adaptive Optics System Near-Infrared Imager and Spectrograph1) instrument at the Very Large Telescope (VLT), the prospective METIS (Mid-Infrared E-ELT Imager and Spectrograph2) instrument at the European Extremely Large Telescope (E-ELT) and the proposed space interferometer Darwin (Cockell et al., 2009). Brieﬂy explained we model 2000 exoplanetary systems around each of 326 nearby main sequence stars (243 for the VLT and the E-ELT due to limited sky coverage) based on planet occurrence statistics from Kepler. All astrophysical and instrumental parameters for our baseline scenario can be found in Table 2.1.

1European Southern Observatory, NACO, 08.03.2017, http://www.eso.org/sci/facilities/paranal/instruments/naco.html 2European Southern Observatory, METIS, 08.03.2017, https://www.eso.org/public/teles-instr/e-elt/e-elt-instr/metis/

5 2. Methods

Table 2.1: Astrophysical and instrumental parameters for our baseline scenario. All values are distributed uniformly within the given ranges.

Parameter Value (range) Description

cos i [−1, 1) Cosine of inclination Ω [0, 2π) Longitude of ascending node ω [0, 2π) Argument of periapsis ϑ [0, 2π) True anomaly e 0 Eccentricity

AB [0, 0.8) Bond albedo

Ag [0, 0.1) Geometric albedo

VLT D 8.2 m Primary mirror diameter

θlim 2 λeff/D Resolution limit

Flim, L’ 24.42 µJy Faint source detection limit @ 3.77 µm

Flim, M’ 159.70 µJy Faint source detection limit @ 4.76 µm

E-ELT D 39 m Primary mirror diameter

θlim 2 λeff/D Resolution limit

Flim, L’ 0.27 µJy Faint source detection limit @ 3.77 µm

Flim, M’ 2.76 µJy Faint source detection limit @ 4.76 µm

Flim, PAH2 9.84 µJy Faint source detection limit @ 11.25 µm

Darwin D 500 m Imaging baseline

θlim 1.2 λeff/D Resolution limit

Flim, F560W 0.16 µJy Faint source detection limit @ 5.6 µm

Flim, F1000W 0.54 µJy Faint source detection limit @ 10.0 µm

Flim, F1500W 1.39 µJy Faint source detection limit @ 15.0 µm

6 2.1. Stellar Sample

2.1 Stellar Sample

The angular separation at which the Airy disk observed from a point source3 (e.g. a star) has its first minimum defines the diffractive resolution limit θdiff of all optical telescopes λ sin θ = 1.22 , (2.1) diff D where λ is the wavelength of the collected light and D is the diameter of the primary mirror or lens (Choudhuri, 2010)4. Therefore an exoplanet can only be spatially resolved from its host star if their apparent angular separation θ is larger than θdiff. Furthermore, the resolution of real ground-based telescopes is negatively affected by many other phenomena, primarily astronomical seeing caused by turbulences in the Earth’s atmosphere (Choudhuri, 2010)5. Although mod- ern telescopes are equipped with adaptive optic (AO) systems to mitigate the wavefront distortion caused by these turbulences they limit the resolution to an angular separation θlim > θdiff. In addition to astronomical seeing the light path aberrations of the instrument vary slowly with time and lead to quasi-static speckles in the final images (Snellen et al., 2015). Unfortu- nately, these speckles often have sizes comparable to potential planetary companions making it difficult to distinguish between an artefact and a real detection. In regard of all these physical limitations it seems reasonable to consider only the most nearby stars for the search for exoplanets using the direct imaging technique since they offer the largest apparent planet-host star separations rp, proj [AU] θ [00] ≈ , θ 1, (2.2) d [pc] where rp, proj is the projected physical separation between exoplanet and host star and d is the distance to the observer (Kutner, 2003)6 (i.e. Earth). While rp, proj is uncorrelated with the choice of the exoplanet system we can maximize θ by looking at close stars with minimal distance d. We thus adopt a star catalog from a similar study done by Quanz et al.(2015) containing 328 stars of spectral types A, F, G, K and M out to a distance of d = 21.4 pc. This star catalog was originally established by Kirkpatrick et al.(2012) and reached out to a distance of d = 8 pc. Then white dwarfs and stars with spectral type later than M7 as well as binaries with apparent

3Swinburne University of Technology, Airy Disk, 08.03.2017, http://astronomy.swin.edu.au/cosmos/A/Airy+Disk 4Section 1.7.1, Equation 1.13, Page 16 5Section 1.7.1, Pages 16 and 17 6Section 27.5, Page 544

7 2. Methods

angular separation θ < 500 were removed by Crossfield(2013). Stars with fainter companions were retained. The star catalog was finally filled by Quanz et al.(2015) with all dwarf stars with K-band magnitude Kmag < 7mag out to a distance of d = 10 pc and Kmag < 5mag out to a distance of d = 20 pc from the SIMBAD Astronomical Database7, again close binaries were removed. From this star catalog we use stellar properties like the parallax p, the spectral type S, the radius R∗, the effective temperature Teff, ∗ and the mass M∗. We further remove two stars with parallax p < 50 mas and therefore distance d > 20 pc so that our final sample contains 326 stars (8 A stars, 54 F stars, 72 G stars, 71 K stars and 121 M stars, Figures 2.1, 2.2 and 2.3). This star catalog is suitable for our Monte-Carlo analysis since it consists of nearby main sequence stars for which the planet occurrence statistics from Burke et al.(2015), Dressing and Charbonneau(2015) and Fressin et al.(2013) are valid.

7SIMBAD Astronomical Database, 08.03.2017, http://simbad.u-strasbg.fr/simbad/

8 2.2. Planet Population

0.5 ] ) ¯

M 0.0 ( 0 1 g o l [

s s a m

0.5 r a l l e t S

1.0

10000 9000 8000 7000 6000 5000 4000 3000 2000 Stellar effective temperature [K]

Figure 2.1: Hertzsprung-Russel diagram of our stellar sample. The stellar mass in log10(M ) is plotted against the reversed stellar eﬀective temperature Teﬀ, ∗ in K.

20 40

15 30

10 20

15 Number of stars Number of stars

5 10

0 0 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Stellar mass [M ] Stellar radius [R ] ¯ ¯

Figure 2.2: Histogram of the mass distribu- Figure 2.3: Histogram of the radius dis- tion of our stellar sample. The stellar mass tribution of our stellar sample. The stellar M∗ is plotted on the x-axis in M . radius R∗ is plotted on the x-axis in R .

2.2 Planet Population

To create a random planet population we use planet occurrence statistics from Burke et al.(2015), Dressing and Charbonneau(2015) and Fressin et al. (2013). The latter of these covers the broadest range of planet radii Rp from 0.8 R⊕ to 22 R⊕ and planet orbital periods Porb from 0.8 d to 418 d corresponding to 0.017 AU to 1.094 AU around a Sun-like star. We thus really focus on close-in companions with sizes down to terrestrial planets. As Fressin et al.(2013) do not ﬁnd a signiﬁcant dependence of the planet occurrence on the spectral type we apply these statistics to all AFGKM stars in our star catalog.

9 2. Methods

Table 2.3: Applied planet occurrence rates r with their planet radius Rp and planet orbital period Porb ranges for each spectral type S appearing in our star catalog. Note that planet radii Rp and planet orbital periods Porb are not simulated in the same range throughout all spectral types S. For G and K stars Rp goes down to 0.75 R⊕, for M stars Rp goes down to 0.5 R⊕ and Porb goes down to 0.5 d.

Spectral type Statistics Radius [R⊕] Orbital period [d]

A Fressin 0.8-22 0.8-418

F Fressin 0.8-22 0.8-418

Burke 0.75-2.5 50-300 Fressin 0.8-22 0.8-50 G Fressin 0.8-22 300-418 Fressin 2.5-22 50-300

Burke 0.75-2.5 50-300 Fressin 0.8-22 0.8-50 K Fressin 0.8-22 300-418 Fressin 2.5-22 50-300

Dressing 0.5-4 0.5-200 M Fressin 0.8-22 200-418 Fressin 4-22 0.8-200

Since we are especially interested in terrestrial exoplanets we decide to use the planet occurrence statistics from Burke et al.(2015) and Dressing and Charbonneau(2015) instead of the ones from Fressin et al.(2013) for small exoplanets in the planet radius and plane orbital period range where they are valid. In contrary to Fressin et al.(2013) who considers Q1-Q6 Kepler planet candidates only Burke et al.(2015) and Dressing and Charbonneau (2015) consider Q1-Q16 and Q0-Q17 Kepler planet candidates respectively. Therefore we expect their statistics to be more complete and closer to the real occurrence of exoplanets. Furthermore they focus on certain spectral types only so that the possibility of a spectral type dependence of the planet occurrence can be taken into account by only applying these statistics to the spectral types for which they are valid. This patchwork-like approach is illustrated in Table 2.3.

10 2.2. Planet Population

2.2.1 Burke 2015 The planet occurrence statistics from Burke et al.(2015) are given as continuous probability distribution function (PDF)  R α1 β 2  p Porb R < R , d f  R0 P0 p brk = F0Cn · (2.3) R α2 α1−α2 β dRpdPorb  p Rbrk Porb R ≥ R ,  R0 R0 P0 p brk where F0 is the average number of planets per star and Cn is a normalization constant so that Z Rp, max Z Porb, max f dRpdPorb = 1. (2.4) Rp, min Porb, min F0

We apply a Poisson distribution with mean value F0 for the number of planets per star (note that this allows for multi-planet systems) and separate the two-power law model into a planet radius and a planet orbital period part according to α  Rp 1 d f  R Rp < Rbrk, R = C · 0 R R α2 α1−α2 (2.5) dRp  p Rbrk R ≥ R , R0 R0 p brk β d fP Porb = CP , (2.6) dPorb P0 where F0 was divided out of the PDF and CR and CP are constants analogous to Cn normalizing the integrated PDFs to one. Now the fraction of planets with planet radii within [Rp, min, xR] and planet orbital periods within [Porb, min, xP] can be calculated analytically via

Z xR CDFR(xR) = fRdRp, (2.7) Rp, min Z xP CDFP(xP) = fPdPorb. (2.8) Porb, min

Here one must pay attention to the fact that fR has a different form for Rp < Rbrk and Rp ≥ Rbrk. Inverting CDFR(xR) and CDFP(xP) yields two functions (i.e. the inverse cumulative distribution functions normalized to one) for the planet radius and the planet orbital period

 1 α + (α +1)R 1 α1 1  1 0 α1+1  C x + Rp, min x < C, −1  R CDF (x) = 1 (2.9) R α α −α ( + ) 2 1 2 α2+1  α2 1 R0 R0 α2+1  · α −α (x − C) + R x ≥ C,  CR 1 2 brk Rbrk 1 β ! β+1 −1 (β + 1)P0 β+1 CDFP (x) = x + Porb, min , (2.10) CP

11 2. Methods

+ + C = CR ( α1 1 − α1 1 ) where α1 Rbrk Rp, min , into which a uniformly drawn random (α1+1)R0 variable x between 0 and 1 can be inserted as argument yielding a planet radius and a planet orbital period distribution according to the original PDF of Burke et al.(2015) (Figures 2.4 and 2.5).

14000 9000 max max 8000 12000 baseline baseline min 7000 min 10000 6000

8000 5000

6000 4000

3000 Number of planets 4000 Number of planets 2000 2000 1000

0 0 0.5 1.0 1.5 2.0 2.5 50 100 150 200 250 300 Planet radius [R ] Planet orbital period [d] ⊕

Figure 2.4: Simulated distribution of the Figure 2.5: Simulated distribution of the planet radius Rp for a sample of 100000 planet orbital period Porb for a sample of planets according to the PDF from Burke 100000 planets according to the PDF from et al.(2015) given in Equation 2.3. The Burke et al.(2015) given in Equation 2.3. green bars show the baseline scenario The green bars show the baseline scenario whereas the blue and the red bars show whereas the blue and the red bars show the maximal and the minimal scenario (pes- the maximal and the minimal scenario (pessimistic and optimistic eﬃciency in Table 7 simistic and optimistic eﬃciency in Table 7 of Burke et al.(2015)). of Burke et al.(2015)).

12 2.2. Planet Population

2.2.2 Dressing 2015 The planet occurrence statistics from Dressing and Charbonneau(2015) 8 are given in planet radius and planet orbital period bins. We assume that the planet radii are distributed uniformly in linear space inside a single bin whereas the planet orbital periods are distributed uniformly in logarithmic space inside a single bin in agreement to Petigura et al.(2013) and Silburt et al.(2015) who ﬁnd that exoplanets are distributed uniform with logarithmic period above 10.8 d and 20 d respectively. We conservatively assume that the planet occurrence rate r is zero inside these bins for which no statistics are provided. If the 1σ upper limits of the statistical occurrence of exoplanets r+1σ are given instead of r itself we proceed as follows: We calculate the missing r using the cumulative number of planets per star vs. orbital period from Table 5 of Dressing and Char- bonneau(2015). For 3.5 R⊕ ≤ Rp ≤ 4 R⊕ where r is missing for two bins we assume that the statistical occurrence of exoplanets is the same for the 3.5 R⊕ ≤ Rp ≤ 4 R⊕ and 0.5 d ≤ Porb ≤ 1.7 d bin and the 3 R⊕ ≤ Rp ≤ 3.5 R⊕ and 0.5 d ≤ Porb ≤ 1.7 d bin (including its errors).

In the cases where the 1σ upper limit r+1σ for the planet occurrence rate r is given we assume the upper error ∆r+ to be the difference between our calculated r and the given r+1σ. If ∆r+ ≤ r we also assume the lower error ∆r- = ∆r+ to be equal to the upper error. However, if ∆r+ > r we assume ∆r- = r to avoid negative planet occurrence rates. Table 2.5 shows the ﬁnal planet occurrence statistics which we use for our Monte-Carlo simulation (Figures 2.6 and 2.7).

8Table 4 (top panel, without revised stellar radii)

13 2. Methods

Table 2.5: Number of planets per star vs. orbital period r (in percentage) including upper errors ∆r+ and lower errors ∆r- from Dressing and Charbonneau(2015) with minor adaptions explained in Subsection 2.2.2 which we use for our Monte-Carlo simulation.

Radius [R⊕] 0.5-1.7 d 1.7-5.5 d 5.5-18.2 d 18.2-60.3 d 60.3-200 d

+1.01 +3.45 +8.29 +13.34 +5.34 0.5-1.0 1.92−0.64 9.88−2.46 23.06−5.57 17.75−6.54 7.10−2.62 +0.87 +2.62 +2.59 +9.38 +26.67 1.0-1.5 1.83−0.58 9.18−1.98 23.95−2.59 23.08−6.02 30.70−10.54 +0.46 +1.82 +5.45 +8.99 +14.67 1.5-2.0 0.33−0.16 3.70−1.18 20.06−4.04 26.73−6.08 18.90−6.99 +0.12 +1.57 +4.91 +8.81 +11.55 2.0-2.5 0.01−0.01 2.25−0.88 15.69−3.55 23.65−5.83 14.12−5.51 +0.11 +1.21 +3.51 +6.68 +7.85 2.5-3.0 0.00−0.00 1.19−0.54 6.54−2.18 10.42−3.74 5.30−2.52 +0.10 +0.91 +2.41 +4.14 +5.32 3.0-3.5 0.01−0.01 0.58−0.29 2.55−1.13 2.44−1.19 1.83−0.80 +0.10 +0.72 +1.65 +2.31 +2.21 3.5-4.0 0.01−0.01 0.36−0.18 0.87−0.43 0.53−0.17 0.50−0.50

100000 40000 max max baseline 35000 80000 baseline min 30000 min 60000 25000

20000 40000

15000 Number of planets Number of planets 10000 20000

5000 0 0 50 100 150 200 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Planet orbital period [d] Planet radius [R ] ⊕ Figure 2.7: Simulated distribution of the Figure 2.6: Simulated distribution of the planet orbital period P for a sample of planet radius R for a sample of 100000 orb p 100000 planets according to the planet planets according to the planet occurrence occurrence statistics from Dressing and statistics from Dressing and Charbonneau Charbonneau(2015) given in Table 2.5. (2015) given in Table 2.5. The green bars The green bars show the baseline scenario show the baseline scenario whereas the blue whereas the blue and the red bars show and the red bars show the maximal (r + the maximal (r + ∆r+) and the minimal ∆r+) and the minimal (r − ∆r-) scenario. (r − ∆r-) scenario.

14 2.2. Planet Population

2.2.3 Fressin 2013 The planet occurrence statistics from Fressin et al.(2013) 9 are given in planet radius and planet orbital period bins as well. We assume that the planet radii are distributed uniformly in linear space inside a single bin whereas the planet orbital periods are distributed uniformly in logarithmic space inside a single bin to be consistent with the planet occurrence statistics from Dressing and Charbonneau(2015). Note that Fressin et al.(2013) originally assume a logarithmic distribution inside the planet radius and the planet orbital period bins however. Again, we conservatively assume that the planet occurrence r is zero inside these bins for which no statistics are provided. Ta- ble 2.7 shows the ﬁnal planet occurrence statistics (including errors) which we use for our Monte-Carlo simulation (Figures 2.8 and 2.9).

50000 35000 max max baseline 30000 baseline 40000 min min 25000

30000 20000

15000 20000

Number of planets Number of planets 10000 10000 5000

0 0 0 5 10 15 20 25 0 50 100 150 200 250 300 350 400 450 Planet radius [R ] Planet orbital period [d] ⊕

Figure 2.8: Simulated distribution of the Figure 2.9: Simulated distribution of the planet radius Rp for a sample of 100000 planet orbital period Porb for a sample of planets according to the planet occurrence 100000 planets according to the planet oc- statistics from Fressin et al.(2013) given in currence statistics from Fressin et al.(2013) Table 2.7. The green bars show the base- given in Table 2.7. The green bars show the line scenario whereas the blue and the red baseline scenario whereas the blue and the bars show the maximal (r + ∆r+) and the red bars show the maximal (r + ∆r+) and minimal (r − ∆r-) scenario. the minimal (r − ∆r-) scenario.

9Table 2

15 2. Methods

Table 2.7: Average number of planets per star per period bin r (in percent) including upper errors ∆r+ and lower errors ∆r- from Fressin et al.(2013) which we use for our Monte-Carlo simulation.

Radius [R⊕] 0.8-2.0 d 2.0-3.4 d 3.4-5.9 d 5.9-10 d

0.8-1.25 0.18±0.04 0.61±0.15 1.72±0.43 2.70±0.60 1.25-2.0 0.17±0.03 0.74±0.13 1.49±0.23 2.90±0.56 2.0-4.0 0.035±0.011 0.18±0.03 0.73±0.09 1.93±0.19 4.0-6.0 0.004±0.003 0.006±0.006 0.11±0.03 0.091±0.030 6.0-22 0.015±0.007 0.067±0.018 0.17±0.03 0.18±0.04

Radius [R⊕] 10-17 d 17-29 d 29-50 d 50-85 d

0.8-1.25 2.70±0.83 2.93±1.05 4.08±1.88 3.46±2.81 1.25-2.0 4.30±0.73 4.49±1.00 5.29±1.48 3.66±1.21 2.0-4.0 3.67±0.39 5.29±0.64 6.45±1.01 5.25±1.05 4.0-6.0 0.29±0.07 0.32±0.08 0.49±0.12 0.66±0.16 6.0-22 0.27±0.06 0.23±0.06 0.35±0.10 0.71±0.17

Radius [R⊕] 85-145 d 145-245 d 245-418 d

0.8-1.25 0.0±0.0 0.0±0.0 0.0±0.0 1.25-2.0 6.54±2.20 0.0±0.0 0.0±0.0 2.0-4.0 4.31±1.03 3.09±0.90 0.0±0.0 4.0-6.0 0.43±0.17 0.53±0.21 0.24±0.15 6.0-22 1.25±0.29 0.94±0.28 1.05±0.30

2.3 Planet Properties

2.3.1 Randomly Oriented Orbits

For our Monte-Carlo analysis it is important to have randomly oriented orbits (i.e. randomly distributed inclination i, longitude of ascending node Ω and argument of periapsis ω) because the position of the exoplanet relative to its host star has a major impact on its observability. Randomly oriented orbits can be obtained by distributing the normal vector of the orbital plane

16 2.3. Planet Properties

~h uniformly on a sphere (Figure 2.10).

Figure 2.10: Illustration of an exoplanet’s orbital plane with inclination i and longitude of ascending node Ω. The host star is located in the center O = (0, 0, 0) of the coordinate system and the vector ~h is the normal vector of the exoplanet’s orbital plane10.

We assume spherical coordinates (r, θ, ϕ) according to

x = r · sin θ · cos ϕ, (2.11) y = r · sin θ · sin ϕ, (2.12) z = r · cos θ, (2.13) where r > 0, θ ∈ [0rad, πrad] and ϕ ∈ [0rad, 2πrad). For the purpose of randomly distributing orbits the radius r is irrelevant because the distance between the exoplanet and its host star is determined by a function f (a, e, ϑ) of the semi-major axis a, the eccentricity e and the true anomaly ϑ (Renner, 2014)11 anyways. For simplicity r = 1 is assumed in the following consider- ations.

To generate randomly oriented orbits the normal vector~h = (sin θ · cos ϕ, sin θ · sin ϕ, cos θ)T of the orbital plane is distributed uniformly on the sphere12

10Online image, 08.03.2017, http://what-when-how.com/wp-content/uploads/2012/02/tmp2A226_thumb.jpg 11Section 2.2.1, Page 24 12Wolfram MathWorld, Sphere Point Picking, 08.03.2017, http://mathworld.wolfram.com/SpherePointPicking.html

17 2. Methods

via

θ = arccos(2v − 1) v ∈ [0, 1) uniformly, (2.14) ϕ = u u ∈ [0rad, 2πrad) uniformly. (2.15)

Taking into account that the host star in the origin O = (0, 0, 0) must lie in the orbital plane E and using the normal vector ~h we can deﬁne E as

E : sin θ · cos ϕ · x1 + sin θ · sin ϕ · x2 + cos θ · x3 = 0. (2.16)

Now the inclination i is given as the angle between the orbital plane E and the xy-plane and can be determined via the planes’ normal vectors ~h and T ~nxy = (0, 0, 1) according to

~ ! h ·~nxy i = arccos = arccos(cos θ) = θ, (2.17) ~ |h| · |~nxy|

~ since |h| = |n~xy| = 1 by deﬁnition. The longitude of ascending node Ω is given as the angle between the intersection line of E and the xy-plane and the reference direction which is chosen as (1, 0, 0)T here. The intersection line ~g of E and the xy-plane can be easily calculated and turns out to be

0 − tan ϕ ~g = 0 + t ·  1  :=~sg + t ·~vg. (2.18) 0 0

T The angle between ~g and the reference direction ~ex = (1, 0, 0) is then   rad ~vg ·~ex − tan ϕ π Ω = arccos = arccos ≡ + ϕ (mod π), |~ | · |~ |  q  vg ex 1 + tan2 ϕ 2 (2.19) since Ω should be distributed between 0rad and πrad. Finally the argument of periapsis ω is distributed randomly on the orbit, therefore uniformly between 0rad and 2πrad.

2.3.2 Randomly Distributed True Anomaly Some exoplanet (candidates) in the immediate solar neighborhood are already known from radial velocity measurements (e.g. ε Eridani b (Benedict et al., 2006) or α Centauri Bb (Hatzes, 2013)) including constraints on their orbital parameters. In theory this knowledge could be used to optimize the instant of time at which the target is directly imaged (e.g. in order to achieve the maximal possible apparent angular separation θ between exoplanet and

18 2.3. Planet Properties host star). However, in the scope of this generic study we conservatively assume that each exoplanet system is observed at a randomly chosen time. This is reﬂected in the simulation by drawing a randomly distributed true anomaly ϑ for each exoplanet. If the simulated orbits are chosen to be perfectly circular (eccentricity e = 0), the true anomaly ϑ can simply be distributed uniformly between 0rad and 2πrad. However, if eccentric orbits are allowed, ϑ must be distributed in a way that it is more often around πrad than around 0rad since planets on eccentric orbits spend more time around their apoapsis (they move more slowly there). For an arbitrary (eccentric) orbit the mean anomaly M is the angle which a planet on an imaginary circular orbit with the same orbital period than the arbitrary eccentric orbit would enclose with the direction of periapsis. Therefore M is always distributed uniformly between 0rad and 2πrad. Now Kepler’s laws of planetary motion13 tell us that

M = E − e sin E, (2.20) e + cos ϑ cos E = . (2.21) 1 + e cos ϑ This set of equations can be used to solve for ϑ(M) numerically and yields a randomly distributed true anomaly ϑ (Figure 2.11).

16000

14000

12000

10000

8000

6000 Number of planets 4000

2000

0 0 1 2 3 4 5 6 True anomaly [rad]

Figure 2.11: Simulated distribution of the true anomaly ϑ for a sample of 100000 planets with eccentricities e uniformly distributed between 0 and 1. The distribution shows a clearly visible peak around the apoapsis where ϑ = πrad.

13Wikipedia, Kepler’s Laws of Planetary Motion, 08.03.2017 https://en.wikipedia.org/wiki/Kepler’s_laws_of_planetary_motion

19 2. Methods

2.3.3 Randomly Drawn Albedos We draw both Bond and geometric albedos from a linearly distributed random variable. Considering the Bond albedos of the Solar System planets and the Moon14 (confer Table 2.9) which range from 0.068 (Mercury) to 0.77 (Venus) we decide to distribute the Bond albedos between 0 and 0.8 for our baseline scenario (confer Table 2.1). The choice of the geometric albedos is even more difficult since only very few statistics about µm-range geometric albedos of planets are available. We decide to distribute the geometric albedos between 0 and 0.1 because many gases which dominate the atmospheres of our Solar System planets absorb strongly in the infrared (e.g. H2O, CO2,O3, CH4,N2O, Green et al.(1964)). Although we work with constant wavelength-independent geometric albedos we take into account wavelength dependent absorption and reflection effects by drawing albedos randomly and simulating a very large sample of exoplanets (typically of the order of 106 exoplanets). Note that Crossfield(2013) and Quanz et al.(2015) distribute the Bond and the geometric albedos linearly between 0 and 0.4 which might indeed be a reasonable choice for gas and ice giants considering the Bond albedos of Jupiter, Saturn, Uranus and Neptune which are all below 0.4. However, since we primarily focus on terrestrial planets with sizes smaller than Neptune we take into account the high Bond albedo of Venus and mention that higher Bond albedos translate into lower equilibrium temperatures and therefore less thermal emission from the planet. Furthermore, lower geometric albedos translate into less reflected host star light (confer Subsection 2.3.7). We thus regard our approach as conservative.

14NASA, Planetary Fact Sheet, 08.03.2017, http://nssdc.gsfc.nasa.gov/planetary/factsheet/

20 2.3. Planet Properties

Table 2.9: Bond albedos AB and geometric albedos Ag of the Solar System planets (and the Moon). Geometric albedos are taken from Seager(2010a) and given at ∼ 500 nm (note that this is not in the infrared).

Celestical object Bond albedo Geometric albedo

Mercury 0.068 0.106 Venus 0.77 0.65 Earth 0.306 0.367 Moon 0.11 - Mars 0.250 0.150 Jupiter 0.343 0.52 Saturn 0.342 0.47 Uranus 0.300 0.51 Neptune 0.290 0.42

2.3.4 Semi-Major Axis and Planet-Host Star Separation Semi-Major Axis

We calculate the exoplanet’s semi-major axis ap using Kepler’s third law s 2 3 GM∗P a = orb , (2.22) p 4π2

−11 3 −1 −2 where G ≈ 6.674 · 10 m kg s is the gravitational constant, M∗ is the 15 host star mass and Porb is the planet orbital period (Quanz, 2016) . Here we use the assumption that the mass of the exoplanet is negligible compared to the mass of its host star.

Considering the largest simulated planet (Rp = 22 R⊕) around the lightest host star (M∗ = 0.08 M ) we ﬁnd a deviation of roughly 10% from the actual semi-major axis using s 2 3 GPorb (a + a∗) = (M + M∗), (2.23) p 4π2 p

Mpap = M∗a∗, (2.24)

15Section 2.1.1, Equation 2.4, Page 13

21 2. Methods

where the subscript p denotes planet quantities and the subscript * denotes host star quantities (Quanz, 2016)16. For the calculation of the planet’s mass we assume a planet density comparable to Earth’s density (ρp = 5500 kg m−3)17. Firstly, such large planets must consist of a large fraction of gas (or ice) which has a density much lower than 5500 kg m−3. Secondly, we are primarily interested in sub-Neptunes with radii Rp < 4 R⊕ for which our simpliﬁed calculation deviates by less than 0.1% from the precise solu- −3 tion (assuming ρp = 5500 kg m ). Therefore the assumption Mp M∗, which is necessary regarding the lack of knowledge about planet mass or planet density, is valid for small planets. We keep in mind that there might be remarkable errors for the largest simulated exoplanets.

Physical Planet-Host Star Separation In the special case of non-eccentric circular orbits the physical separation of the exoplanet from its host star rp is simply equal to the exoplanet’s semi-major axis ap at any given time. On the other hand, if the orbit has a non-zero eccentricity 1 > e > 0 (trajectories with eccentricities e ≥ 1 are unbound) the planet-host star separation18 1 − e2 r = a (2.25) p 1 + e cos ϑ

depends on the total semi-major axis a = ap + a∗, the orbital eccentricity e and the true anomaly ϑ.

Projected Physical and Apparent Angular Planet-Host Star Separation Considering Figure 2.3.1 we choose the exoplanet system to be observed from the positive z-direction. In principle every other direction could be chosen as well leading to the same computational results since the orbits of the exoplanets are distributed randomly and uniformly on a sphere. Putting the host star into the center O = (0, 0, 0) of the coordinate system we can express the position of the exoplanet using Euler angles as   rp, x ~rp = rp rp, y (2.26) rp, z cos(Ω) cos(ω + ϑ) − sin(Ω) cos(i) sin(ω + ϑ) = rp sin(Ω) cos(ω + ϑ) + cos(Ω) cos(i) sin(ω + ϑ) , (2.27) sin(i) sin(ω + ϑ)

16Section 2.1.2, Equations 2.6 and 2.7, Page 14 17NASA, Earth Fact Sheet, 08.03.2017, http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html 18Wikipedia, Kepler’s Laws of Planetary Motion, 08.03.2017 https://en.wikipedia.org/wiki/Kepler’s_laws_of_planetary_motion

22 2.3. Planet Properties

where rp is the physical separation of the exoplanet from its host star (confer Equation 2.25) and ϑ is the true anomaly of the exoplanet in an orbit with inclination i, longitude of ascending node Ω and argument of periapsis ω. The length of this vector projected onto the xy-plane yields the projected physical separation q 2 2 rp, proj = rp (rp, x) + (rp, y) (2.28) q = cos2(ω + ϑ) + cos2(i) sin2(ω + ϑ) (2.29) as seen from an observer looking from the positive z-direction onto the system (Figure 2.12). The apparent angular planet-host star separation can ﬁnally be calculated using Equation 2.2.

1.0

0.8

0.6

0.4

0.2 Projected planet-host star separation [AU] 0.0 0 1 2 3 4 5 6 Orbital inclination [rad]

Figure 2.12: Projected planet-host star separation rp, proj in AU for an Earth-like exoplanet with argument of periapsis ω = 0rad and true anomaly θ = 0.5πrad for orbital inclinations between rad rad 0 and 2π . From Equation 2.29 it can be seen that rp, proj is independent of the longitude of ascending node Ω.

2.3.5 Equilibrium Temperature

In order to calculate the emitted thermal blackbody ﬂux Ftherm, p of the simulated exoplanets we need their effective temperature Teff, p. Like Crossﬁeld (2013) we assume that the exoplanet’s effective temperature is equal to its equilibrium temperature " # 1 R2(1 − A ) 4 = = ∗ B Teff, p Teq, p 2 Teff, ∗, (2.30) 4rp where R∗ is the radius of the host star, AB is the planet’s Bond albedo, rp is the physical separation of the planet from its host star and Teff, ∗ is the

23 2. Methods

effective temperature of the host star (Quanz, 2016)19. This equation can be easily obtained by considering the emitted and absorbed power of an idealistic spherical blackbody with atmospheric reﬂection proportional to the Bond albedo AB.

2.3.6 Phase Angle and Phase Curve

To calculate the host star flux which is reflected at the exoplanet’s surface Frefl, p we need to derive the exoplanet’s phase angle α first. The phase angle is defined as the angle between the direction of the observer (i.e. Earth) and the direction of the host star as seen from the exoplanet.

As before we choose the direction of the Earth as seen from the exoplanet as T ~ez = (0, 0, 1) . The direction of the host star as seen from the exoplanet is given by

cos(Ω) cos(ω + ϑ) − sin(Ω) cos(i) sin(ω + ϑ) ~rp − = − sin(Ω) cos(ω + ϑ) + cos(Ω) cos(i) sin(ω + ϑ) := ~ep (2.31) |~r | p sin(i) sin(ω + ϑ)

(confer Equation 2.26). The phase angle α then turns out to be

~ez ·~ep α = arccos = arccos(− sin(i) sin(ω + ϑ)), (2.32) |~ez| · |~ep|

since |~ez| = 1 and |~ep| = 1 by definition. The phase curve f (α) now expresses the dependence of the reflected host rad star flux Frefl, p on the phase angle α. For example a phase angle of α = 0 (α = πrad) means that the exoplanet is exactly behind (before) its host star so that the whole observed planet surface is illuminated (not illuminated). This position is also called opposition (conjunction). For our simulation we assume that all exoplanets can be approximated by Lambert spheres which reflect light uniformly into all directions and obey

1 f (α) = (sin α + (π − α) cos α) (2.33) π

analogously to the approach of Crossﬁeld(2013), where we use the absolute value to avoid negative values for the reﬂectance (Quanz, 2016)20 (Fig- ure 2.13).

19 2 4 Section 4.1, Equation 4.3, Page 78 (substituting L = 4πR∗σTeff, ∗) 20Section 4.3, Equation 4.16, Page 86

24 2.3. Planet Properties

1.0

0.8

0.6

0.4 Lambertian reflectance 0.2

0.0 0 1 2 3 4 5 6 Phase angle [rad]

Figure 2.13: Phase curve of a Lambert sphere f (α) for phase angles α between 0rad and 2πrad.

2.3.7 Emitted and Reflected Blackbody Flux Most important for our Monte-Carlo simulation is the observed flux from the exoplanet (we always assume that our observation is background limited and the observability of an exoplanet is thus independent of the planet- host star flux contrast). The observed flux from an exoplanet is composed of thermal radiation from the planet itself and of host star radiation which is reflected at the planet’s atmosphere. The temperature of young planets is determined by the self-contraction of the protoplanetary nebula whereas the temperature of old planets is determined by the absorbed host star radiation (Quanz, 2016)21. Since we only consider stars out to a distance of d = 20 pc and the nearest star forming regions are located at a distance of ∼ 140 pc (Schmid, 2016)22 we assume that the thermal radiation of all simulated exoplanets is dominated by the absorbed host star radiation and therefore their equilibrium temperature Teq, p (confer Section 2.3.5). We approximate the radiation which is emitted by the exoplanets and their host stars as perfect blackbody radiation and take wavelength-dependent atmospheric absorption and reflection effects into account by drawing randomly distributed Bond and geometric albedos (confer Subsection 2.3.3) and simulating a very large sample of exoplanets (typically of the order of 106 exoplanets). The flux observed from a perfect spherical blackbody

2πhc2 1 R2 Fbb = , (2.34) λ5 ehc/λkBTeff − 1 d2 where h ≈ 6.626 · 10−34 m2 kg s−1 is the Planck constant, c ≈ 2.998 · 108 m s−1

21Section 6.1, Pages 105 and 106 22Section 5.1, Pages 135 and 136

25 2. Methods

−23 2 −2 −1 is the speed of light and kB ≈ 1.381 · 10 m kg s K is the Boltzmann constant, depends on the wavelength λ, the effective temperature of the blackbody Teff, the radius of the blackbody R and its distance to Earth d (Seager, 2010a)23.

Furthermore, the reflected host star flux from the planet Frefl, p can be calculated using 2 Rp F = A f (α) F ∗, (2.35) refl, p g d2 inc,

where Ag is the geometric albedo of the planet, f (α) is the phase curve of the planet (confer Section 2.3.6), Rp is the planet radius, d is its distance to Earth and Finc, ∗ is the ﬂux from the host star incident on the planet (Quanz, 24 2016) . Finc, ∗ follows from Equation 2.34 with d replaced by the physical planet-host star separation rp (confer Section 2.3.4). Figure 2.14 shows what the spectra of the Sun, Earth and Jupiter would look like according to our simulation if they were observed from a distance of d = 10 pc.

102 101 Sun 100 Earth 10-1 Jupiter 10-2 10-3 10-4 -5 Flux [Jy] 10 10-6 10-7 10-8 10-9 10-10 10-1 100 101 102 Wavelength [um]

Figure 2.14: Spectra of the Sun, Earth and Jupiter in the wavelength range λ = 0.1 µm to λ = 100 µm observed from a distance of d = 10 pc according to our simulation. For the Sun we 25 assume an effective temperature of Teff, = 5772 K . For Earth and Jupiter we assume Bond albedos of AB = 0.306 and AB = 0.343 as well as (constant) geometric albedos of Ag = 0.367 and Ag = 0.52 which correspond to the geometric albedos measured at ∼ 0.5 µm from Table 3.1 of Seager(2010a). The effective temperatures follow from Equations 2.30, 2.25 and 2.22 using planet orbital periods of Porb = 365 d and Porb = 4330 d as well as zero eccentricity e = 0.

23Section 2.5, Equation 2.19, Page 16 and Section 2.8, Equations 2.36 and 2.37, Page 19 24Section 4.1, Equation 4.17, Page 86 25NASA, Sun Fact Sheet, 08.03.2017, http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html

26 2.4. Instruments

2.4 Instruments

We run our Monte-Carlo simulation for the existing NACO instrument at the VLT, the prospective METIS instrument at the E-ELT and the proposed space observatory Darwin. All resolution limits θlim and faint source detection limits Flim are presented in Table 2.1. Whether an exoplanet is observable or not is determined by two parameters: The apparent angular separation between exoplanet and host star θ and the observed flux Fobs, p = Ftherm, p + Frefl, p from the exoplanet itself. Therefore, each simulated instrument has its resolution limit θlim and its faint source detection limit Flim. Resolution limits are given in units of λeff/D, where λeff is the effective wavelength of the corresponding filter and D is the diameter of the primary mirror or lens. Faint source detection limits are given in µJy and in terms of the detection consciousness in a particular observation time (e.g. 5σ in 8640 s). Detailed specifications for all filters used in this work are presented in Table 2.11.

Table 2.11: Effective wavelengths λeff [µm], effective widths Weff [µm], resolution limits θlim [λeff/D] and faint source detection limits Flim [µJy] for all filters used in this work.

Telescope Filter λeff Weff θlim Flim

NACO L’03.770 0.624 2.0024.42 (5 σ in 3600 s) VLT NACO M’04.755 0.592 2.0 159.70 (5 σ in 3600 s)

NACO L’03.770 0.624 2.0 000.27 (5 σ in 8640 s) E-ELT NACO M’04.755 0.592 2.0 002.76 (5 σ in 8640 s) VISIR PAH2 11.252 0.607 2.0 009.84 (5σ in 8640 s)

MIRI F560W05.600 1.200 1.2 000.16 (10 σ in 10000 s) Darwin MIRI F1000W 10.000 2.000 1.2 000.54 (10σ in 10000 s) MIRI F1500W 15.000 3.000 1.2 001.39 (10σ in 10000 s)

27 2. Methods

2.4.1 VLT and E-ELT The Very Large Telescope (VLT) of the European Southern Observatory (ESO) is located in Chile on top of Cerro Paranal at an altitude of 2635 m26. ◦ 27 The site at Cerro Paranal has a geographic latitude of δVLT ≈ 25 S and we assume that the telescope can access celestial objects with declinations ◦ ◦ ◦ ranging from δVLT ± 55 = [80 S, 30 N]. We perform simulations for the filters L’ and M’ which are currently installed at the Nasmyth Adaptive Optics System Near-Infrared Imager and Spectrograph (NACO)28. All significant filter properties can be found in Table 2.11 and the diameter of the primary mirror (of one of the unit telescopes) is D = 8.2 m26. The European Extremely Large Telescope (E-ELT) is ESO’s next generation 40 m-class telescope which is currently under construction at Cerro Arma- zones, only about 20 km away from Cerro Paranal29. We therefore assume that the E-ELT will have the same sky coverage as the VLT. We perform simulations for the filters L’ and M’ as well as PAH2 which is currently installed at the VLT Imager and Spectrometer for Mid Infrared (VISIR)30. Quanz et al. (2015) predict the faint source detection limits Flim of the Mid-Infrared E-ELT Imager and Spectrograph (METIS)31 at wavelengths which are comparable to the effective wavelengths λeff of these filters (confer Table 2.11). We further assume a diameter of D = 39 m for the segmented primary mirror32. The VLT/NACO instrument is in operation already so that we can use its real filter curves which we obtain from the Filter Profile Service of the Span- 33 ish Virtual Observatory (including effective wavelengths λeff and effective widths Weff). The resolution limit θlim of the VLT is assumed to be 2 λeff/D (analogously to Quanz et al.(2015) for the E-ELT) and the faint source detection limits Flim for NACO L’ and NACO M’ are calculated via F0 ∆mag = 2.5 log10 , (2.36) Flim 26European Southern Observatory, Very Large Telescope, 08.03.2017, http://www.eso.org/public/teles-instr/paranal/ 27Wikipedia, Cerro Paranal, 08.03.2017, https://en.wikipedia.org/wiki/Cerro_Paranal 28European Southern Observatory, NACO, 08.03.2017, http://www.eso.org/sci/facilities/paranal/instruments/naco.html 29Wikipedia, Cerro Armazones, 08.03.2017, https://de.wikipedia.org/wiki/Cerro_Armazones 30European Southern Observatory, VISIR, 08.03.2017, http://www.eso.org/sci/facilities/paranal/instruments/visir.html 31European Southern Observatory, METIS, 08.03.2017, https://www.eso.org/public/teles-instr/e-elt/e-elt-instr/metis/ 32European Southern Observatory, European Extremely Large Telescope, 08.03.2017, https://www.eso.org/sci/facilities/eelt/ 33Spanish Virtual Observatory, Filter Profile Service, 08.03.2017, http://svo2.cab.inta-csic.es/theory/fps3/index.php?mode=browse

28 2.4. Instruments

where F0 = 244.17 Jy (NACO L’) and F0 = 159.70 Jy (NACO M’) are the ﬁlter zero points in the Vega system (also taken from the Spanish Virtual Obser- vatory) and ∆mag = 17.5mag (NACO L’) and ∆mag = 15mag (NACO M’) are the limiting magnitudes of the ﬁlters for the detection of faint companions. These limits are currently achieved at apparent angular separations of θ ≈ 5 λeff/D but they shall be pushed to θ ≈ 2 λeff/D in the next years.

The predicted resolution limit θlim and faint source detection limits Flim for the E-ELT are taken from Quanz et al.(2015). We apply the limits given for the L-, M- and N-band filters to the L’, M’ and PAH2 filters as both types of filters operate at roughly the same wavelengths (PAH2 is a filter currently installed at the VLT/VISIR instrument34, we also obtain its specifications from the Filter Profile Service of the Spanish Virtual Observatory, Figure 2.15).

VLT/E-ELT Filter Curves 1.0 Lp 0.8 Mp PAH2 0.6

0.4 Transmission

0.2

0.0 2 4 6 8 10 12 14 Wavelength [um]

Figure 2.15: Filter curves T(λ) (relative transmission vs. wavelength) of the ﬁlters which we simulate for the VLT and the E-ELT, i.e. NACO L’, NACO M’ and VISIR PAH2, as provided by the Filter Proﬁle Service of the Spanish Virtual Observatory.

2.4.2 Darwin Unlike other studies (Stark et al., 2014; Brown, 2015; Stark et al., 2015) which investigate the planet yield of a space mission for a broad range of instrument parameters we focus on a single, very specific configuration which was proposed to the European Space Agency (ESA) in 2007 and is commonly known as Darwin. We take all key properties of this space-based nulling interferometer from an updated Darwin mission proposal which was pub- lished to public in 2009 (Cockell et al., 2009) to demonstrate the capabilities of this discarded milestone project. Cockell et al.(2009) suggested an instrument consisting of four formation- flying mirrors which focus their reflected light onto a fifth beam combiner spacecraft (BCS) where detectors and communication devices are located. The spacecraft would be arranged in an X-like architecture with the four mirrors flying in a single plane and forming the tips of the X and the BCS

34European Southern Observatory, VISIR, 08.03.2017, http://www.eso.org/sci/facilities/paranal/instruments/visir.html

29 2. Methods

flying ∼ 1 km above this plane in the center of the X. The whole setup could be launched to the Earth-Sun L2 point in a single Ariane 5 rocket and would be able to observe an annular region between 46◦ and 83◦ from anti- solar direction and thus reach complete sky-coverage (> 99%) throughout a whole year. The advantage of this configuration also known as EMMA X-array is that nulling baseline and imaging baseline can be varied independently. Besides a nulling baseline from 7 m to 168 m an imaging baseline of up to 500 m would be possible resulting in a diffraction-limited resolution of 5 mas at 10 µm. Darwin would further achieve a sensitivity comparable to the James Webb Space Telescope (JWST) if there is a bright source (Kmag < 13mag) in the field of view to stabilize the array. This is typically the host star for the direct imaging of exoplanets (all of our host stars have K-band magnitudes Kmag brighter than 9.3mag). We therefore equip our space-based nulling interferometer with the F560W (5.6 µm), F1000W (10 µm) and F1500W (15 µm) filters from the mid-infrared instrument (MIRI) of the JWST. We use the MIRI filter curves and the faint source low background detection limits from Glasse et al.(2015) (Figure 2.16). They perform an extensive study on the detector sensitivity taking into account zodiacal background at the L2 point, thermal instrument background, photon conversion efficiency, image broad- ening, scattering and crosstalk to provide realistic 10σ in 10000 s observation time detection limits (confer Table 2.11).

Darwin Filter Curves 1.0 F560W 0.8 F1000W F1500W 0.6

0.4 Transmission

0.2

0.0 4 6 8 10 12 14 16 18 20 22 Wavelength [um]

Figure 2.16: Filter curves T(λ) (relative transmission vs. wavelength) of the filters which we simulate for the proposed space interferometer Darwin, i.e. MIRI F560W, F1000W and F1500W. For all three filters we trim the obtained data to the wavelength range (λeff − 2Weff, λeff + 2Weff).

2.4.3 Observed Planet Flux

To calculate the observed flux from an exoplanet Fobs, p we fold its incident flux Finc, p(λ) with the filter transmission T(λ), integrate over the wavelength λ and divide by the effective width Weff of the corresponding filter according to 1 Z λmax Fobs, p = Finc, p(λ)T(λ)dλ. (2.37) Weff λmin

30 2.4. Instruments

For the integration bounds λmin and λmax we use the limits given by the data provided by the Spanish Virtual Observatory for NACO L’, NACO M’ and VISIR PAH2. The ﬁlter curves of MIRI F560W and MIRI F1000W are recorded for wavelengths from λ = 1.3 µm to λ = 27.5 µm, the ﬁlter curve of MIRI F1500W from λ = 1.43 µm to λ = 30.00 µm. We trim all of them to λmin = λeff − 2Weff and λmax = λeff + 2Weff and report that the transmission drops to substantially less than 0.1% for lim λ → λmin, λ > λmin and lim λ → λmax, λ < λmax.

2.4.4 Radial Velocity In addition to direct imaging we also present the capabilities of radial velocity measurements. The radial velocity semi-amplitude K∗ which an exoplanet induces to its host star is given as r G −1/2 −1/2 K∗ = M | sin(i)|(M∗ + M ) a , (2.38) 1 − e2 p p

−11 3 −1 −2 where G ≈ 6.674 · 10 m kg s is the gravitational constant, M∗ is the host star mass, Mp is the planet mass and e and i are the planet’s orbital eccentricity and the planet’s orbital inclination (Seager, 2010b)35. Here the total semi-major axis a of the planet-host star system follows from

a3 G = ( + ) 2 2 M∗ Mp , (2.39) Porb 4π

4 3 36 where Mp = 3 πRpρp is the planet mass (Quanz, 2016) . We assume a −3 planet density of ρp = 5000 kg m for all planets with radii Rp ≤ 2 R⊕ considering the mean value of the densities of the four terrestrial planets of our own Solar System37: Mercury, Venus, Earth and Mars; which is 5029 kg m−3. However, the assumption that all planets with radii below two Earth radii are rocky is a very strong simplification considering the findings of Wolf- gang and Lopez(2015) and Rogers(2015) who use hierarchical Bayesian modelling to show that the transition from rocky planets to gas planets must lie between 1.2 R⊕ and 1.8 R⊕ as well as that most planets with radii above 1.6 R⊕ are not rocky considering a Kepler subsample of planets with radial velocity mass constraints. We thus keep in mind that our analysis is only a rough simplification and might overestimate the radial velocity signal for a significant fraction of exoplanets between approximately 1.5 R⊕ and 2 R⊕. −1 As detection threshold we adopt a value of K∗ ≥ 0.1 m s accounting for

35Section 2.1, Equation 12, Page 29 36Section 2.1, Equation 2.6, Page 14 37NASA, Planetary Fact Sheet, 08.03.2017, http://nssdc.gsfc.nasa.gov/planetary/factsheet/

31 2. Methods

the aimed precision of the new ESPRESSO spectrograph currently installed at the Very Large Telescope of the European Southern Observatory in Chile (Pepe et al., 2010).

32 Chapter 3

Results

We present and analyse the results of our Monte-Carlo simulation. We consider the existing NACO instrument at the VLT (Figure 3.1), the prospective METIS instrument at the E-ELT (Figure 3.3) and the proposed space interferometer Darwin (Figure 3.12). The presented expectation values of detectable exoplanets η can be translated into a Poisson-like probability mass function

ηke−η PMF(η, k) = k ∈ N ∪ 0. (3.1) k! Performing a real observation can then be modelled by drawing a single value from this PMF. The chance P(N) to detect at least N exoplanets can thus be calculated by summing up all discrete probability weights from N to ∞, ∞ P(N) = ∑ PMF(η, k). (3.2) k=N All results are further affected by statistical Poisson errors ∆η. Since we simulate 2000 exoplanetary systems around each star these errors can be calculated for each expectation value of detectable exoplanets η via

pη · 2000 √ ∆η = ≈ 0.022 · η (3.3) 2000 (error of a Poisson distribution).

33 3. Results

3.1 VLT and E-ELT

3.1.1 Exoplanets Detectable via Direct Imaging Table 3.1 states the total expected number of detectable exoplanets η for VLT/NACO (L band and M band) and E-ELT/METIS (L band, M band and N band). For VLT/NACO we expect to ﬁnd 0.21 exoplanets in total, for E-ELT/METIS we expect to ﬁnd 7.77 exoplanets in total. The major fraction of the detectable exoplanets (even all for VLT/NACO) can be observed in the L band.

Table 3.1: Expected number of detectable exoplanets η (planet yield) for VLT/NACO and E-ELT/METIS.

Telescope Filter Planet yield

VLT NACO L’ 0.21 NACO M’ 0.07 total 0.21 E-ELT NACO L’ 7.28 NACO M’ 4.09 VISIR PAH2 2.16 total 7.77

For VLT/NACO we find an expected number of detectable exoplanets of η = 0.21 in the L-band and η = 0.07 in the M-band by integrating the numbers in all individual bins of Figure 3.1. Detecting a small exoplanet (Rp ≤ 4 R⊕) is possible but very unlikely while detecting a terrestrial habitable exoplanet (Rp ≤ 2 R⊕ and Teq, p ≈ 300 K) does not seem to be possible at all. Therefore the VLT is not the appropriate tool for directly imaging old exoplanets whose temperature is rather determined by their reprocessed host star light than their formation heat. In the past the VLT succeeded in directly imaging several planets (e.g. Schmidt et al.(2016)), however these planets are all widely separated gas giants still glowing from their formation. Anyway, the first filter is superior to the second one in terms of planet yield since it has the better resolution limit θlim (scales with λeff) and the better faint source detection limit Flim. The chance is highest to detect a sub- Neptune (Rp ≤ 4 R⊕) with an equilibrium temperature between 400 K and 900 K. Cooler exoplanets cannot be detected since their thermal blackbody flux decreases sharply at shorter wavelengths and is too faint at 3.77 µm.

34 3.1. VLT and E-ELT

0.12

3.77 um 0.06 3 5000 kg/m 0.0 22.0

0.00 0.01 0.02 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 6.0 ]

⊕ 0.00 0.00 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 R [

s 4.0 u i

d 0.00 0.00 0.01 0.03 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 a r

t 2.0 e n 0.00 0.00 0.00 0.02 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 a l nan% nan% nan% 100% 79% 91% 80% 76% 100% nan% nan% nan%

P 1.25

0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 nan% nan% nan% 67% 73% nan% nan% nan% nan% nan% nan% nan% 0.5 200 400 600 800 1000 1200 0.0 0.06 0.12 Equilibrium temperature [K]

0.12

4.76 um 0.06 3 5000 kg/m 0.0 22.0

0.00 0.00 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 6.0 ]

⊕ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 R [

s 4.0 u i

d 0.00 0.00 0.00 0.01 0.00 0.01 0.01 0.00 0.00 0.00 0.00 0.00 a r

t 2.0 e n 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 a l nan% nan% nan% 100% nan% 100% 100% nan% nan% nan% nan% nan%

P 1.25

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% 0.5 200 400 600 800 1000 1200 0.0 0.06 0.12 Equilibrium temperature [K] Figure 3.1: VLT/NACO. Expected number of detectable exoplanets η for VLT/NACO observing in the L-band (top) and the M-band (bottom) depicted as two-dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p. The two one-dimensional histograms at the top and at the right of each subﬁgure show the projected equilibrium temperature distribution (top) and the projected radius distribution (right) of the detectable exoplanets. The color shade and the axes are scaled equally in both subﬁgures for better comparison. For all exoplanets with radii Rp ≤ 2 R⊕ we calculate the fraction of exoplanets which would be detectable via radial velocity in addition to direct imaging (number in percent), assuming a detection limit −1 −3 of K∗ ≥ 0.1 m s and a mean density of ρp = 5000 kg m . If not a single simulated exoplanet is detectable with direct imaging in an individual bin this fraction is not a number (nan).

Hotter exoplanets cannot be detected since the equilibrium temperature −1/2 Teq, p ∝ rp scales with the inverse square root of the physical separation between exoplanet and host star rp, therefore these exoplanets are too close to their host stars to be resolved. Several larger exoplanets (Rp ≥ 6 R⊕) also pass the detection threshold even though they are slightly cooler than the detectable sub-Neptunes. This is possible because the thermal black- 2 body ﬂux Ftherm, p ∝ Rp is proportional to the square of the planet radius Rp so that larger exoplanets emit more thermal radiation. A small fraction

35 3. Results

of the detectable exoplanets is also visible due to their reflected host star 2 flux Frefl, p ∝ Rp only, which is also proportional to the square of the planet radius Rp. However, Figure 3.2 reveals that this fraction is negligibly small.

0.12

3.77 um 0.06 3 5000 kg/m 0.0 22.0

0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 6.0 ]

⊕ 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 R [

s 4.0 u i

d 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 a r

t 2.0 e n 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 a l nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% nan%

P 1.25

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% nan% 0.5 200 400 600 800 1000 1200 0.0 0.06 0.12 Equilibrium temperature [K]

Figure 3.2: VLT/NACO L-band. Expected number of detectable exoplanets η for VLT/NACO observing in the L-band like in Figure 3.1 but only showing these exoplanets which can be detected due to their reflected host star flux Frefl, p.

For E-ELT/METIS in contrast we find an expected number of detectable exoplanets of η = 7.28 in the L-band, η = 4.09 in the M-band and η = 2.16 in the N-band. Again the first filter is superior to the other two in terms of planet yield (for the same reasons as for VLT/NACO). The resolution limit θlim and the faint source detection limits Flim are greatly enhanced with respect to VLT/NACO (mainly due to the larger primary mirror) so that both the cool (i.e. faint) and widely separated exoplanets as well as the closely separated and hot (i.e. bright) exoplanets can be detected with E-ELT/METIS. However, since the occurrence rate r of these planets is very small, the majority of the detectable exoplanets are still expected to be sub- Neptunes (Rp ≤ 4 R⊕) with equilibrium temperatures between 400 K and 900 K. Although the N-band filter has the lowest planet yield we report an increased chance to detect a small and cold exoplanet (Rp ≤ 4 R⊕ and Teq, p ≤ 300 K). The reason is that the PAH2 filter operates at the longest wavelength (11.25 µm) where the thermal blackbody spectra of these cool planets peak.

36 3.1. VLT and E-ELT

3.0

3.77 um 1.5 3 5000 kg/m 0.0 22.0

0.18 0.35 0.49 0.14 0.05 0.04 0.02 0.01 0.01 0.01 0.00 0.00 6.0 ]

⊕ 0.01 0.03 0.13 0.07 0.04 0.03 0.02 0.01 0.01 0.00 0.00 0.00 R [

s 4.0 u i

d 0.03 0.14 0.83 0.66 0.28 0.21 0.14 0.12 0.08 0.04 0.01 0.02 a r

t 2.0 e n 0.00 0.06 0.48 0.56 0.23 0.16 0.12 0.11 0.06 0.04 0.02 0.02 a l 100% 99% 97% 97% 96% 98% 95% 96% 98% 97% 100% 95%

P 1.25

0.00 0.01 0.28 0.32 0.13 0.08 0.06 0.05 0.05 0.03 0.02 0.02 nan% 80% 64% 64% 67% 54% 53% 50% 40% 47% 62% 50% 0.5 200 400 600 800 1000 1200 0.0 1.5 3.0 Equilibrium temperature [K]

3.0

4.76 um 1.5 3 5000 kg/m 0.0 22.0

0.00 0.14 0.26 0.08 0.04 0.03 0.02 0.01 0.00 0.00 0.00 0.00 6.0 ]

⊕ 0.00 0.01 0.06 0.04 0.03 0.02 0.01 0.01 0.00 0.00 0.00 0.00 R [

s 4.0 u i

d 0.00 0.04 0.42 0.34 0.21 0.17 0.12 0.09 0.05 0.03 0.01 0.02 a r

t 2.0 e n 0.00 0.00 0.25 0.27 0.17 0.13 0.10 0.08 0.04 0.03 0.01 0.02 a l nan% 100% 98% 97% 96% 97% 96% 95% 97% 98% 100% 94%

P 1.25

0.00 0.00 0.15 0.15 0.09 0.07 0.05 0.04 0.03 0.02 0.02 0.02 nan% nan% 63% 67% 69% 50% 51% 53% 42% 41% 56% 47% 0.5 200 400 600 800 1000 1200 0.0 1.5 3.0 Equilibrium temperature [K]

3.0

11.25 um 1.5 3 5000 kg/m 0.0 22.0

0.04 0.09 0.04 0.03 0.02 0.01 0.00 0.01 0.00 0.00 0.00 0.00 6.0 ]

⊕ 0.00 0.02 0.02 0.02 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 R [

s 4.0 u i

d 0.01 0.20 0.20 0.12 0.07 0.03 0.02 0.02 0.01 0.01 0.00 0.00 a r

t 2.0 e n 0.00 0.21 0.22 0.11 0.04 0.03 0.02 0.03 0.01 0.00 0.00 0.00 a l nan% 95% 97% 94% 91% 94% 92% 89% 91% 100% nan% nan%

P 1.25

0.00 0.13 0.21 0.09 0.03 0.01 0.00 0.01 0.01 0.01 0.00 0.00 nan% 51% 51% 58% 65% 56% 0% 18% 15% 13% 50% nan% 0.5 200 400 600 800 1000 1200 0.0 1.5 3.0 Equilibrium temperature [K] Figure 3.3: E-ELT/METIS. Expected number of detectable exoplanets η for E-ELT/METIS observing in the L-band (top), the M-band (middle) and the N-band (bottom) depicted as two- dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Figure 3.1).

37 3. Results

Altogether we expect to detect 7.77 exoplanets with the E-ELT/METIS instrument within our simulated planet radii and planet orbital period space (Figure 3.4), 2.94 of these only in the L-band, 0.48 of these only in the N- band, 2.66 of these in the L- and the M-band together, 0.25 of these in the L- and the N-band together and 1.43 of these in all three bands together (the sum of these values is only 7.76 due to rounding errors). Exoplanets which can be detected in all three ﬁlters together are observable over a broad range of wavelengths and can therefore be analyzed spectroscopically.

2.94

0.00 3.77 4.76 0.48 11.25 3.77 & 4.76 3.77 & 11.25 4.76 & 11.25 3.77 & 4.76 & 11.25

1.43

2.66

0.00 0.25

Figure 3.4: E-ELT/METIS. Pie chart showing the expected number of detectable exoplanets η for each individual filter and all possible combinations of filters for E-ELT/METIS. The numbers in the legend state the effective filter wavelengths λeff in µm.

3.1.2 Host Star Characteristics To identify the most fruitful targets for directly imaging small and old exoplanets we present the expected number of detectable exoplanets η for each individual host star in our star catalog for VLT/NACO (Figure 3.5) and for E-ELT/METIS (Figure 3.6). For VLT/NACO we ﬁnd that all simulated detectable exoplanets are concen- trated around ﬁve host stars only (Figure 3.5). Alpha Cen A and alpha Cen B are (besides Proxima Cen) the closest neighbours to our Sun and thus offer the largest apparent angular separations θ between simulated exoplanet and host star. Furthermore they are G2V and K1V type stars and therefore brighter than Proxima Cen which is an M5.5V type star. Exoplanets orbiting brighter (i.e. hotter) stars tend to have higher equilibrium temperatures Teq, p (confer Equation 2.30). Thus they emit stronger thermal blackbody radiation peaking at shorter wavelengths so that the NACO instrument is

38 3.1. VLT and E-ELT

0.12 0.11 - alpha Cen A 0.09 - Sirius A 0.01 - Procyon A 0.10 0.00 - alpha Cen B 0.00 - Altair 0.00 - c Boo 0.00 - Gl 66 A 0.08 0.00 - Fomalhaut 0.00 - gj 2005 A 0.00 - L 991-14 0.00 - G 109-35 0.06 0.00 - Gl 480.1 0.00 - BD-03 2870 0.00 - Gl 66 B 0.00 - Ross 490 0.04 0.00 - LP 876-10 0.00 - L 674-15 0.00 - LP 467-16 0.00 - Wolf 718 0.02 0.00 - G 154-44 Expected number of detectable planets 0.00 0 50 100 150 200 Star index from star catalog

0.12 0.04 - Sirius A 0.03 - alpha Cen A 0.00 - Procyon A 0.10 0.00 - alpha Cen B 0.00 - c Boo 0.00 - Fomalhaut 0.00 - gj 2005 A 0.08 0.00 - L 991-14 0.00 - G 109-35 0.00 - Gl 480.1 0.00 - Gl 66 B 0.06 0.00 - Gl 66 A 0.00 - Ross 490 0.00 - BD-03 2870 0.00 - LP 876-10 0.04 0.00 - L 674-15 0.00 - LP 467-16 0.00 - Wolf 718 0.00 - 107 Psc 0.02 0.00 - LTT 9283 Expected number of detectable planets 0.00 0 50 100 150 200 Star index from star catalog

Figure 3.5: VLT/NACO. Expected number of detectable exoplanets η for each individual host star in our star catalog (accessible from Cerro Paranal, 243 targets in total) for VLT/NACO observing in the L-band (top) and the M-band (bottom). The list on the right side of each subfigure states the 20 host stars with the highest planet yield. Since the stars in our star catalog are sorted by distance d histogram bins with smaller star indexes (left side) represent stars which are closer to the observer (i.e. Earth). The axes are scaled equally in both subfigures for better comparison. sensitive enough to detect them. In addition they are brighter in their reflected host star light simply because their host stars are brighter. Sirius A, Procyon A and Altair are the other three very close and bright A1V, F5IV-V and A7V type stars around which it would be theoretically possible to detect exoplanets with VLT/NACO1.

1Spectral types taken from the SIMBAD Astronomical Database, 08.03.2017 http://simbad.u-strasbg.fr/simbad/

39 3. Results

1.4 1.20 - alpha Cen A 0.74 - alpha Cen B 1.2 0.63 - Sirius A 0.52 - Procyon A 0.42 - delta Pav 0.39 - Altair 1.0 0.38 - tau Cet 0.33 - epsilon Eri 0.25 - Fomalhaut 0.8 0.22 - beta Hyi 0.18 - epsilon Ind A 0.16 - Gl 166 A 0.15 - GJ 139 0.6 0.14 - LTT 2364 0.09 - LHS 2465 0.08 - LHS 348 0.07 - zet Tuc 0.4 0.07 - chi01 Ori 0.06 - iot Peg 0.05 - gam Pav 0.2 Expected number of detectable planets 0.0 0 50 100 150 200 Star index from star catalog

1.4 1.05 - alpha Cen A 0.57 - Sirius A 1.2 0.56 - alpha Cen B 0.45 - Procyon A 0.30 - Altair 0.18 - tau Cet 1.0 0.16 - Fomalhaut 0.16 - delta Pav 0.16 - epsilon Eri 0.8 0.07 - LTT 2364 0.05 - beta Hyi 0.05 - epsilon Ind A 0.05 - Gl 166 A 0.6 0.03 - GJ 139 0.02 - LHS 2465 0.02 - zet Tuc 0.01 - LHS 348 0.4 0.01 - gam Pav 0.01 - iot Peg 0.01 - 1 Eri 0.2 Expected number of detectable planets 0.0 0 50 100 150 200 Star index from star catalog

1.4 0.93 - alpha Cen A 0.63 - alpha Cen B 1.2 0.23 - Sirius A 0.12 - Procyon A 0.06 - tau Cet 0.06 - epsilon Eri 1.0 0.04 - Altair 0.02 - epsilon Ind A 0.01 - delta Pav 0.8 0.01 - Gl 166 A 0.01 - Fomalhaut 0.01 - GJ 139 0.00 - beta Hyi 0.6 0.00 - Gl 570 A 0.00 - Gl 825 0.00 - Gl 887 0.00 - 36 Oph C 0.4 0.00 - chi01 Ori 0.00 - 107 Psc 0.00 - LTT 2364 0.2 Expected number of detectable planets 0.0 0 50 100 150 200 Star index from star catalog

Figure 3.6: E-ELT/METIS. Expected number of detectable exoplanets η for each individual host star in our star catalog (accessible from Cerro Armazones, 243 targets in total) for E- ELT/METIS observing in the L-band (top), the M-band (middle) and the N-band (bottom, like in Figure 3.5).

40 3.1. VLT and E-ELT

For E-ELT/METIS we find that exoplanets can be detected around a broad range of stars up to a distance of d = 20 pc (our star catalog is sorted by distance so that the rightmost stars in the histograms are also the furthest ones, Figure 3.6). Since the resolution limit θlim scales with the wavelength λeff, filters operating at longer wavelengths can only detect exoplanets orbiting closer stars. Companions around stars further afar cannot be resolved any- more. This effect is clearly visible comparing the histograms of the L-band (3.77 µm) and the M-band (4.76 µm) to the N-band (11.25 µm). Similar to VLT/NACO the most fruitful targets are the closest and brightest stars. The five targets around which exoplanets could be detected with VLT/NACO are also on the top of the list for E-ELT/METIS. Figure 3.7 shows a scatter plot of all simulated exoplanets around the most fruitful star for E-ELT/METIS: Alpha Cen A. Normalized to the 2000 repetitions of our simulation we obtain on average 1.44 simulated exoplanets orbiting alpha Cen A of which 1.20 can be detected with the direct imaging technique. Using Equation 3.2 this corresponds to a chance of 70% to find at least one exoplanet around alpha Cen A via direct imaging. These directly detectable planets are located in the upper right quadrant defined by the vertical and the horizontal black line which indicate the resolution limit θlim = 2 λeff/D and the faint source detection limit Flim, L’ = 0.27 µJy. High- lighted in red are these simulated exoplanets which are rocky (Rp ≤ 2 R⊕) −1 and can be detected with the radial velocity technique (K∗ ≥ 0.1 m s ). This are 0.74 planets on average of which the majority can be detected via direct imaging as well. This means that in case an exoplanet is discovered around alpha Cen A using the radial velocity method, there is a high chance that it can be followed-up with E-ELT/METIS using direct imaging.

41 3. Results

107

106 0.74 of 1.44 obs. via radial velocity

105

104

103

102

101

100

10-1

-2 Observed flux [uJy] - limit = 0.27 uJy 10

10-3 0 5 10 15 20 25 30 35 40 Angular separation [λ/D] - limit = 2.0 λ/D

Figure 3.7: E-ELT/METIS L-band. Scatter plot showing each simulated exoplanet around the star alpha Cen A. The x-axis represents the apparent angular separation θ between exoplanet and host star and the y-axis represents the observed planet ﬂux Fobs, p. The resolution limit θlim = 2 λeﬀ/D and the faint source detection limit Flim, L’ = 0.27 µJy are indicated by the vertical and the horizontal black line. All exoplanets which are rocky (Rp ≤ 2 R⊕) and can −1 be detected with the radial velocity technique (K∗ ≥ 0.1 m s ) are highlighted in red (ρp = 5000 kg m−3). If normalized to the 2000 repetitions of our simulation this are on average 0.74 of the 1.44 exoplanets which are simulated around alpha Cen A in total.

3.1.3 Instrument Performance

In order to investigate the stability of the scientiﬁc output with respect to variations in the instrument’s performance we calculate the expected number of detectable exoplanets η in dependence of the faint source detection limits Flim for ﬁve different resolution limits θlim from 1.2 λeff/D (diffraction- limited) to 5 λeff/D (Figure 3.8). For this analysis we consider E-ELT/METIS in the L-band since it has the highest predicted exoplanet yield in our baseline scenario.

As anticipated the exoplanet yield is increasing towards better resolution limits and better faint source detection limits. All ﬁve curves saturate at small faint source detection limits Flim as they allow to detect every single simulated exoplanet with projected separation θ ≥ θlim.

The ratios of the diffraction-limited 1.2 λeff/D curve to the other four curves are plotted as dashed lines in Figure 3.8. They are a measure for the stability of the exoplanet yield with respect to the instrument’s resolution limit θlim. For all resolution limits θlim these ratios increase towards the left. This increase is more dramatic for worse resolution limits θlim. This means that the instrument is more sensitive to variations in its resolution limit θlim if operated at better faint source detection limits Flim and worse resolution limits θlim. The green and the red dashed curve which compare our baseline

42 3.1. VLT and E-ELT

160 20 res = 1.2 λ/D 140 res = 2.0 λ/D res = 3.0 λ/D res = 4.0 λ/D Ratio to diffraction limit 120 res = 5.0 λ/D 15

100

80 10

40 5

20 Expected number of detectable planets 0 0 10-4 10-3 10-2 10-1 100 101 Faint source detection limit [uJy]

Figure 3.8: E-ELT/METIS L-band. Expected number of detectable exoplanets η (left axis) in dependence of the faint source detection limits Flim for five different resolution limits θlim from 1.2 λeff/D to 5 λeff/D (different colors). The baseline scenario with Flim, L’ = 0.27 µJy is indicated by the black dashed vertical line. The coloured dashed curves are the ratios of the diffraction-limited curve to the other four curves (right axis). Note that the x-axis is scaled logarithmically.

scenario of θlim = 2 λeff/D and the next resolution step of 3 λeff/D to the diffraction limit are approximately constant meaning that the METIS instrument (if operated at our baseline scenario) has a constant sensitivity with respect to small variations in its resolution limit θlim for a broad range of faint source detection limits Flim. The slopes of the individual (solid) curves are a measure for the stability of the exoplanet yield with respect to the instrument’s faint source detection limit Flim. For all presented resolution limits θlim the curves are steepest at roughly two orders of magnitude better faint source detection limits Flim if compared to our baseline scenario of Flim, L’ = 0.27 µJy (black dashed vertical line). Therefore the METIS instrument (if operated at our baseline scenario) is only moderately sensitive with respect to small variations in its faint source detection limit Flim. Figure 3.9 illustrates the stability of the METIS instrument in a discrete way by quantifying the variation in the expected exoplanet yield η for different resolution limits θlim and different faint source detection limits Flim with respect to our baseline scenario. Since the two-dimensional histogram tends to have red cells on the left side and blue cells on the right side one might conclude that the METIS instrument is more sensitive to variations in the resolution limit θlim than in the faint source detection limit Flim. Nevertheless, one must keep in mind that the bin steps are chosen completely arbitrary. There is no reason why a step in resolution space should be achieved with

43 3. Results

the same likelihood than a step in faint source detection space.

0.045 +29.1 +12.3 +5.1 +1.5 -0.8 -2.1 -3.1 24 Planet gain w.r.t. baseline scenario

18 0.068 +24.5 +9.9 +3.7 +0.5 -1.4 -2.6 -3.5 12 0.135 +17.9 +6.4 +1.6 -0.9 -2.4 -3.3 -4.0 6

0.270 +12.6 +3.7 +0.0 -2.0 -3.1 -3.9 -4.5 0

6 0.540 +8.4 +1.6 -1.2 -2.8 -3.7 -4.4 -4.8 12 1.080 +5.0 -0.1 -2.3 -3.4 -4.2 -4.7 -5.1 18 Max. faint source detection [uJy]

1.620 +3.4 -0.9 -2.7 -3.8 -4.4 -4.9 -5.3 24

1.0 1.5 2.0 2.5 3.0 3.5 4.0 Max. resolution [λ/D]

Figure 3.9: E-ELT/METIS L-band. Absolute variations in the expected number of detectable exoplanets η for different resolution limits θlim and different faint source detection limits Flim with respect to our baseline scenario (which is highlighted by a black frame). The resolution steps are 0.5 λeff/D and the faint source detection steps are 1/6, 1/4, 1/2, 1, 2, 4, 6 times the baseline value of Flim, L’ = 0.27 µJy. The color shade is scaled symmetrically around 0.

3.1.4 Exoplanets Detectable via Radial Velocity

Observation time at the E-ELT will be very limited since such a unique instrument will be used for various other astrophysical purposes apart from the search for exoplanets. It is thus very unlikely that it will be possible to conduct large planet searching surveys with the E-ELT. Therefore it is crucial to identify promising targets for directly imaging extrasolar planets in advance. We identify radial velocity measurements as the ideal technique to support the direct imaging of nearby exoplanets. The transit technique is not suitable for this purpose since it requires an edge-on planetary system which is very rare simply due to geometrical reasons.

The radial velocity technique makes use of the small host star wobbling induced by an exoplanet as exoplanet and host star orbit their common center of mass. Of course the radial velocity signal of the host star is several orders of magnitude weaker than the radial velocity signal of its exoplanet (due to the typically large ratio of stellar mass to planet mass M∗/Mp) but the most advanced instruments (e.g. the Echelle Spectrograph for Rocky Exoplanet and Stable Spectroscopic Observations2) achieve detection limits

2European Southern Observatory, ESPRESSO, 08.03.2017, https://www.eso.org/sci/facilities/develop/instruments/espresso.html

44 3.1. VLT and E-ELT

−1 of K∗ ≥ 0.1 m s which would be sufﬁcient to detect an Earth twin around a Solar-like star3. Figure 3.10 shows the expected number of exoplanets which induce a certain radial velocity semi-amplitude K∗ to their host star for all rocky exoplanets −1 −1 (Rp ≤ 2 R⊕). We observe a peak for 0.1 m s ≤ K∗ ≤ 0.2 m s and that the majority of the rocky planets passes the detection threshold of ESPRESSO (indicated by the red vertical line). Figure 3.3 shows that a large fraction of the exoplanets which are detectable via direct imaging are also detectable via radial velocity, mostly above 90% for exoplanets with radii Rp ≥ 1.25 R⊕ and above 50% for exoplanets with radii Rp ≥ 0.5 R⊕. Nevertheless, the fraction of exoplanets which can be detected via direct imaging in addition to radial velocity is extremely small (almost invisible in Figure 3.10) simply since very few exoplanets can be directly imaged with E-ELT/METIS only. However, for this small fraction it would be possible to determine the exoplanet’s orbit from direct imaging and therefore its true mass from radial velocity as well as a constraint on its radius from its observed ﬂux.

50 All rocky planets (R 2R , ρ = 5000kg/m3 ) ⊕ + detectable via direct imaging 40

Expected number of planets 10

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial velocity [m/s]

Figure 3.10: E-ELT/METIS. Expected number of exoplanets which induce a certain radial velocity semi-amplitude K∗ to their host star for all rocky exoplanets (Rp ≤ 2 R⊕) assuming −3 a planet density of ρp = 5000 kg m (blue histogram). The fraction of exoplanets which is detectable via direct imaging (in at least one of the three ﬁlters which we simulate for the E- ELT/METIS instrument) in addition to radial velocity is shown by the green histogram. The red −1 vertical line indicates our baseline radial velocity detection threshold of K∗ ≥ 0.1 m s .

Since ESPRESSO will be available in the near future it would be possible to scan nearby stars for exoplanets to ﬁnd ideal targets for direct imaging follow-up observations with E-ELT/METIS. This would also have the advantage that constraints on the exoplanets’ orbits could be imposed in ad-

3Wikipedia, Doppler Spectroscopy, 08.03.2017, https://en.wikipedia.org/wiki/Doppler_spectroscopy

45 3. Results

vance and one could directly image them at this instant of time when the apparent angular separation θ between the exoplanets and their host stars is largest (at quadrature). If we assume circular orbits like in our baseline scenario this maximal apparent angular separation θmax = a will be equal to the semi-major axis a. Figure 3.11 shows the expected number of detectable exoplanets η if all observations were performed at quadrature. The total planet yield could be increased from 7.77 to 10.28, furthermore 1.94 of the detectable exoplanets could be analyzed spectroscopically over a broad range of wavelengths.

4.11

3.77 0.00 4.76 11.25 0.61 3.77 & 4.76 3.77 & 11.25 4.76 & 11.25 3.77 & 4.76 & 11.25

1.94

3.31

0.00 0.31

Figure 3.11: E-ELT/METIS. Pie chart showing the expected number of detectable exoplanets η for each individual filter and all possible combinations of filters for E-ELT/METIS. In contrary to Figure 3.4 all planets are assumed to be observed at quadrature. The numbers in the legend state the effective filter wavelengths λeff in µm.

46 3.2. Darwin

3.2 Darwin

3.2.1 Exoplanets Detectable via Direct Imaging The expected numbers of detectable exoplanets η for each individual ﬁlter are presented as two-dimensional histogram showing the planet radii Rp on the y-axis and the planet equilibrium temperatures Teq, p on the x-axis (Figure 3.12). Integrating over the whole radius-equilibrium temperature plane results in 256.3 expected planets at 5.6 µm, 274.6 expected planets at 10 µm and 206.6 expected planets at 15 µm (confer Table 3.2).

Table 3.2: Expected number of detectable exoplanets η (planet yield) for the space interferometer Darwin. Filter Planet yield

MIRI F560W 256.3 MIRI F1000W 274.6 MIRI F1500W 206.6 total 329.4

47 3. Results

80.0

5.60 um 40.0 3 5000 kg/m 0.0 22.0

0.77 3.54 2.97 1.63 0.88 0.47 0.33 0.23 0.17 0.09 0.06 0.04 6.0 ]

⊕ 0.04 1.33 1.76 1.24 0.77 0.49 0.30 0.19 0.12 0.08 0.05 0.03 R [

s 4.0 u i

d 0.14 16.3528.8817.7210.74 6.82 4.33 2.59 1.61 0.94 0.56 0.37 a r

t 2.0 e n 0.03 8.65 27.4818.9510.73 6.96 4.65 3.15 2.01 1.28 0.84 0.51 a l 100% 100% 99% 99% 99% 99% 99% 99% 99% 99% 98% 98%

P 1.25

0.00 1.96 16.4616.47 9.11 5.62 3.73 2.39 1.61 1.11 0.72 0.47 100% 89% 87% 79% 77% 79% 80% 82% 83% 83% 82% 82% 0.5 200 400 600 800 1000 1200 0.0 60.0 120.0 Equilibrium temperature [K]

80.0

10.00 um 40.0 3 5000 kg/m 0.0 22.0

2.45 3.82 2.85 1.54 0.81 0.40 0.25 0.15 0.09 0.05 0.03 0.02 6.0 ]

⊕ 0.58 2.02 1.69 1.17 0.70 0.42 0.22 0.14 0.07 0.05 0.03 0.02 R [

s 4.0 u i

d 3.58 35.2027.2415.79 9.41 5.77 3.41 1.90 1.08 0.59 0.33 0.21 a r

t 2.0 e n 1.09 26.4727.4816.15 8.84 5.53 3.38 2.02 1.18 0.70 0.43 0.25 a l 100% 99% 99% 98% 98% 98% 98% 98% 98% 98% 97% 97%

P 1.25

0.18 8.94 17.5212.24 6.75 4.04 2.46 1.38 0.86 0.55 0.33 0.21 90% 89% 83% 78% 73% 74% 74% 74% 76% 73% 70% 71% 0.5 200 400 600 800 1000 1200 0.0 60.0 120.0 Equilibrium temperature [K]

80.0

15.00 um 40.0 3 5000 kg/m 0.0 22.0

3.57 3.71 2.71 1.42 0.70 0.30 0.15 0.08 0.05 0.02 0.02 0.01 6.0 ]

⊕ 1.03 1.94 1.59 1.08 0.60 0.31 0.15 0.09 0.04 0.03 0.02 0.01 R [

s 4.0 u i

d 7.64 30.5921.9913.27 7.60 4.23 2.23 1.14 0.61 0.31 0.19 0.14 a r

t 2.0 e n 2.75 21.4716.6511.08 6.78 3.77 2.05 1.11 0.62 0.34 0.22 0.14 a l 100% 99% 99% 98% 98% 98% 98% 98% 98% 97% 97% 96%

P 1.25

0.41 6.41 8.85 5.10 3.01 2.13 1.34 0.73 0.43 0.25 0.15 0.11 93% 88% 85% 80% 79% 77% 71% 66% 67% 63% 60% 61% 0.5 200 400 600 800 1000 1200 0.0 60.0 120.0 Equilibrium temperature [K] Figure 3.12: Darwin/MIRI. Expected number of detectable exoplanets η for Darwin/MIRI observing at 5.6 µm (top), at 10 µm (middle) and at 15 µm (bottom) depicted as two-dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Fig- ure 3.1).

48 3.2. Darwin

The majority of the directly detectable exoplanets is smaller than Neptune (Rp ≤ 4 R⊕) and has equilibrium temperatures Teq, p between 200 K and 700 K. Most of the exoplanets can be detected using the 10 µm filter although it is inferior to the 5.6 µm filter in terms of resolution (scales with λeff) and sensitivity. However, the underlying planet population consists of more planets with equilibrium temperatures Teq, p around 300 K which emit thermal blackbody radiation peaking at approximately 10 µm. These planets are just warm enough to be detected with the 10 µm filter, but their thermal blackbody flux Ftherm, p decreases sharply at shorter wavelengths so that the 5.6 µm filter, although more sensitive, is not able to detect their thermal emission. Going from shorter to longer filter effective wavelengths λeff it can further be observed that the equilibrium temperatures Teq, p of the detectable exoplanets shift slightly towards the left (i.e. towards cooler planets). This is similar to what is observed for the VLT and the E-ELT (confer Subsection 3.1.1). The fraction of exoplanets which can be detected in addition to direct imaging by measuring the radial velocity signal induced to their host stars is very high for planets between 1.25 R⊕ and 2 R⊕, typically close to 100%. For these planets the true mass and the orbit could be reconstructed by combining data from both observation techniques including a constraint on the planets radius from its observed flux. Due to their lower mass planets with radii between 0.5 R⊕ and 1.25 R⊕ are less frequently detectable with radial velocity and direct imaging together, however the fraction is mostly still above 70%. Exoplanets which can only be detected due to their reflected host star light are again very rare. For illustrative purposes Figure 3.13 shows the expected number of detectable exoplanets η for the 5.6 µm filter (like in Figure 3.12), but only taking into account the planets’ reflected host star flux Frefl, p. In this filter one would expect the most exoplanets detectable due to their reflected light. The reason for this is that the hot host stars emit blackbody radiation peaking at wavelengths much shorter than the effective wavelengths λeff of all of our simulated filters. Nevertheless, there is only a small fraction of exoplanets having very cool equilibrium temperatures Teq, p / 300 K which can be detected due to their reflected host star light. These planets’ own thermal blackbody emission is too weak at 5.6 µm to be detected.

49 3. Results

80.0

5.60 um 40.0 3 5000 kg/m 0.0 22.0

0.49 0.74 0.83 0.69 0.48 0.33 0.26 0.19 0.15 0.08 0.06 0.04 6.0 ]

⊕ 0.02 0.11 0.15 0.14 0.13 0.14 0.12 0.09 0.07 0.05 0.03 0.02 R [

s 4.0 u i

d 0.08 0.76 1.62 1.65 1.18 0.86 0.74 0.63 0.55 0.40 0.26 0.21 a r

t 2.0 e n 0.01 0.18 0.53 0.76 0.62 0.46 0.35 0.31 0.26 0.22 0.19 0.15 a l 100% 100% 100% 100% 100% 100% 100% 99% 100% 98% 99% 99%

P 1.25

0.00 0.02 0.11 0.23 0.26 0.24 0.17 0.12 0.10 0.08 0.08 0.06 nan% 98% 96% 95% 95% 96% 93% 93% 89% 85% 85% 88% 0.5 200 400 600 800 1000 1200 0.0 60.0 120.0 Equilibrium temperature [K]

Figure 3.13: Darwin/MIRI F560W. Expected number of detectable exoplanets η for Dar- win/MIRI observing at 5.6 µm like in Figure 3.12 but only showing these exoplanets which can be detected due to their reflected host star flux Frefl, p.

If the exoplanets which Darwin detects are supposed to be analyzed spectroscopically, they must be visible over a broad range of wavelengths. To estimate the fraction of exoplanets for which such a broadband characterization would be possible, Figure 3.14 shows the expected number of exoplanets which can be detected in only one filter, two filters or all three filters together in a pie chart. According to the Darwin mission proposal (Leger and Herbst, 2007) a low-resolution spectrum from 6 µm to 20 µm could be taken allowing for the search of O3 (@9.6 µm), CO2 (@15 µm) and H2O (@6.3 µm & ≥ 20 µm) in the exoplanets’ atmospheres. Considering the wavelengths of these atmospheric molecular absorptions we assume that exoplanets which can be detected in all three of our simulated filters together can be observed spectroscopically. This is the case for ∼ 153 exoplanets, slightly less than the half of the expected ∼ 329 exoplanets which can be detected in at least one filter. Note that we do not take into account the time which is necessary to perform spectroscopic observations precise enough to detect molecules and biosignatures in the exoplanets’ atmospheres which can be of the order of months for Earth-twins (Lunine et al., 2008). We refer toL eger´ et al.(2015) who estimate the capabilities of a formation-flying space interferometer with 4 × 0.75 m collecting area in terms of atmosphere characterization and find that ∼ 14 Earths and super-Earths (which are identified beforehand) could be analyzed spectroscopically during a five year mission.

50 3.2. Darwin

10.02 58.92 19.07

0.04

44.81

44.08

5.6 10 15 5.6 & 10 5.6 & 15 10 & 15 5.6 & 10 & 15

152.50

Figure 3.14: Darwin/MIRI. Pie chart showing the expected number of detectable exoplanets η for each individual filter and all possible combinations of filters for Darwin/MIRI. The numbers in the legend state the effective filter wavelengths λeff in µm.

3.2.2 Host Star Characteristics From the observer’s point of view it is not only interesting to know how many exoplanets one can expect to detect with Darwin, but also around which targets one could find them. For this purpose we present the expected number of detectable exoplanets η for each individual host star in our star catalog in Figure 3.15. We sort the stars by planet yield to show that most of them have expectation values of η ≥ 0.5 directly detectable planet(s). For roughly 15% of the stars in our star catalog our simulation predicts outstandingly high planet yields of up to 2.52 expected detections (Gl 887 @15µm). Such a big number of planets can only be found around M type stars since the underlying planet occurrence statistics predict more planets around these stars (confer Table 3.3). M type stars are very cool so that the planets orbiting them are also cool (typically Teq, p ≈ 300 K). Since such cool planets emit thermal blackbody radiation peaking around ∼ 10 − 15 µm and decreasing sharply at shorter wavelengths the expected planet yield per star is higher for M dwarfs in the 10 µm and the 15 µm filter compared to the 5.6 µm filter . Due to the strong impact of the underlying planet population Figure 3.15 can indeed tell us which targets are most fruitful, however it does not quantify how well suited a target is for detecting exoplanets using the direct imaging technique.

51 3. Results

2.5 2.03 (0.64) - Gl 887 1.96 (0.62) - Gl 15 A 1.94 (0.60) - CD-37 15492 1.89 (0.59) - BD+53 1320 1.88 (0.59) - LPM 730 2.0 1.88 (0.59) - GJ 832 1.78 (0.57) - Lalande 21185 1.75 (0.54) - L 991-14 1.71 (0.54) - Lalande 25372 1.5 1.69 (0.53) - Gl 725 A 1.65 (0.52) - G 202-48 1.64 (0.51) - Wolf 1453 1.62 (0.52) - BD-12 4523 1.52 (0.48) - BD+68 946 1.0 1.51 (0.47) - Ross 154

0.5 Expected number of observable planets 0.0 0 50 100 150 200 250 300

2.5 2.49 (0.79) - Gl 887 2.40 (0.76) - Gl 15 A 2.34 (0.75) - Lalande 21185 2.33 (0.72) - CD-37 15492 2.17 (0.69) - GJ 832 2.0 2.15 (0.67) - Gl 725 A 2.11 (0.66) - BD+53 1320 2.10 (0.66) - LPM 730 2.04 (0.63) - Ross 154 1.5 2.03 (0.64) - Lalande 25372 1.98 (0.64) - BD-12 4523 1.94 (0.61) - Wolf 1453 1.89 (0.60) - BD+68 946 1.85 (0.57) - L 991-14 1.0 1.82 (0.58) - Barnard's Star

0.5 Expected number of observable planets 0.0 0 50 100 150 200 250 300

2.5 2.52 (0.80) - Gl 887 2.46 (0.78) - Lalande 21185 2.38 (0.75) - Gl 15 A 2.21 (0.68) - CD-37 15492 2.11 (0.66) - Gl 725 A 2.0 2.06 (0.65) - Barnard's Star 2.01 (0.62) - Ross 154 1.96 (0.63) - Proxima Cen 1.96 (0.62) - GJ 832 1.5 1.85 (0.60) - BD-12 4523 1.83 (0.57) - LPM 730 1.83 (0.57) - BD+53 1320 1.80 (0.57) - Lalande 25372 1.77 (0.56) - BD+68 946 1.0 1.74 (0.53) - Gl 725 B

0.5 Expected number of observable planets 0.0 0 50 100 150 200 250 300

Figure 3.15: Darwin/MIRI. Expected number of detectable exoplanets η for each individual host star in our star catalog (326 targets in total) for Darwin/MIRI observing at 5.6 µm (top), at 10 µm (middle) and at 15 µm (bottom). Each of the 326 host stars is represented by one bin of the histogram. In contrary to Figure 3.5 the stars are sorted by planet yield η and no longer by distance d. The list on the right side of each subﬁgure states the 15 host stars with the highest planet yield η and their completeness c (number in brackets).

52 3.2. Darwin

To decouple the underlying planet population from our analysis we calculate the completeness c for each individual host star as the fraction of all generated exoplanets which can be directly detected. Figure 3.16 shows the expected number of detectable exoplanets η for each target (like in Fig- ure 3.15), but this time the stars are sorted by their completeness c rather than their planet yield η. The completeness is close to 100% for the brightest and closest stars translating into a very high chance to ﬁnd an exoplanet if it then exists. This trend of higher completeness towards higher spectral type can be observed since planets with same physical separations from their host stars tend to have higher equilibrium temperatures and therefore emit stronger thermal radiation compared to cooler M type stars. Brighter stars are therefore more suitable targets for directly imaging small and old exoplanets meaning that almost every possibly existing planet could be detected around them. This conclusion can also be drawn considering Table 3.3 which shows that especially the bright A and F type stars have complete- nesses above 90%, although not the highest planet yields per star.

Table 3.3: Expected number of detectable exoplanets η in total (planet yield), per star (planet yield/star) and completeness c by spectral type for the ﬁlters MIRI F560W, MIRI F1000W and MIRI F1500W combined. The completeness is calculated as the expected number of detectable exoplanets divided by the total number of simulated exoplanets.

Spectral type Planet yield Planet yield/star Completeness

A 6.73 0.88 96% F 43.43 0.80 92% G 69.72 0.97 67% K 58.67 0.83 58% M 150.89 1.25 39%

53 3. Results

1.0 2.5 0.87 (1.00) - Procyon A 0.89 (1.00) - Sirius A 1.43 (1.00) - alpha Cen A 0.86 (0.99) - Altair 0.88 (0.99) - eta Cas A 0.8 2.0 0.84 (0.99) - Fomalhaut Relative completeness 0.88 (0.99) - Vega 0.86 (0.97) - LTT 2364 0.89 (0.97) - zet Tuc 0.6 1.5 0.83 (0.97) - gam Ser

1.0 0.4

0.5 0.2 Expected number of observable planets 0.0 0.0 0 50 100 150 200 250 300

1.0 2.5 1.43 (1.00) - alpha Cen A 0.89 (1.00) - Sirius A 0.86 (0.99) - Procyon A 1.44 (0.99) - alpha Cen B 0.84 (0.98) - Altair 0.8 2.0 1.41 (0.97) - delta Pav Relative completeness 0.85 (0.96) - eta Cas A 0.81 (0.96) - Fomalhaut 0.85 (0.96) - Vega 0.6 1.5 1.43 (0.95) - tau Cet

1.0 0.4

0.5 0.2 Expected number of observable planets 0.0 0.0 0 50 100 150 200 250 300

1.0 2.5 1.43 (1.00) - alpha Cen A 1.45 (0.99) - alpha Cen B 0.89 (0.99) - Sirius A 0.85 (0.98) - Procyon A 1.43 (0.95) - tau Cet 0.8 2.0 0.82 (0.95) - Altair Relative completeness 1.35 (0.94) - epsilon Eri 1.35 (0.93) - delta Pav 0.81 (0.92) - eta Cas A 0.6 1.5 0.77 (0.91) - Fomalhaut

1.0 0.4

0.5 0.2 Expected number of observable planets 0.0 0.0 0 50 100 150 200 250 300

Figure 3.16: Darwin/MIRI. Expected number of detectable exoplanets η (histogram, left axis) and completeness c (orange curve, right axis) for each individual host star in our star catalog (326 targets in total) for Darwin/MIRI observing at 5.6 µm (top), at 10 µm (middle) and at 15 µm (bottom, like in Figure 3.15). The stars are sorted by completeness c and no longer by planet yield η. The list on the right side of each subﬁgure states the 10 host stars with the highest completeness c (number in brackets) and their expected number of detectable exoplanets η.

54 3.2. Darwin

3.2.3 Instrument Performance

As for VLT/NACO and E-ELT/METIS we investigate the instrument performance by calculating the expected planet yield η for ﬁve different resolution limits θlim dependent on the faint source low background detection limits Flim. We consider the MIRI F1000W ﬁlter operating at 10 µm since it has the highest predicted exoplanet yield in our baseline scenario.

The ratios of the 1.2 λeff/D curve to the other four curves are plotted as dashed lines and are a measure for the stability of the exoplanet yield with respect to the instrument’s resolution. These dashed lines are approximately constant (in particular the green one which compares the exoplanet yield of our baseline scenario of 1.2 λeff/D to the next resolution step of 2 λeff/D), but all four show a small peak towards the right side of the presented parameter space. This implies that the instrument is a little bit more sensitive with respect to variations in its resolution towards faint source detection limits roughly one order of magnitude worse than our baseline parameter of Flim, F1000W = 0.54 µJy. The slopes of the individual curves are a measure for the stability of the exoplanet yield with respect to the instrument’s faint source detection limit Flim. For all presented instrument resolutions θlim the curves are steepest at approximately our baseline scenario of Flim, F1000W = 0.54 µJy implying that the instrument is operated in a regime where it is particularly sensitive to variations in its faint source detection limit.

600 4.0 res = 1.2 λ/D res = 2.0 λ/D 500 res = 3.0 λ/D 3.5 res = 4.0 λ/D Ratio to diffraction limit res = 5.0 λ/D 400 3.0

300 2.5

200 2.0

100 1.5 Expected number of detectable planets 0 1.0 10-4 10-3 10-2 10-1 100 101 Faint source detection limit [uJy]

Figure 3.17: Darwin/MIRI F1000W. Expected number of detectable exoplanets η (left axis) in dependence of the faint source low background detection limits Flim for five different resolution limits θlim from 1.2 λeff/D to 5 λeff/D (different colors). The baseline scenario with Flim, F1000W = 0.54 µJy is indicated by the black dashed vertical line. The coloured dashed curves are the ratios of the diffraction-limited curve to the other four curves (right axis). Note that the x-axis is scaled logarithmically.

55 3. Results

Figure 3.18 illustrates the stability of the MIRI instrument in a discrete way by quantifying the variation in the exoplanet yield η for different resolution limits θlim and different faint source low background detection limits Flim with respect to our baseline scenario (similar to E-ELT/METIS, Subsec- tion 3.1.3). Increasing the imaging baseline by a factor of ∼ 1.7 (with respect to our baseline scenario) would lead to a resolution limit of θlim ≈ 0.7 λeff/D and an increase in planet yield by ∼ 47. However, decreasing the imaging baseline by a factor of ∼ 1.4 would lead to a resolution limit of θlim ≈ 1.7 λeff/D and a decrease in planet yield by ∼ 45.

200 0.090 +206.7 +170.0 +115.9 +61.8 +16.4 -21.5 -53.2 Planet gain w.r.t. baseline scenario 150 0.135 +182.6 +146.3 +93.2 +40.5 -3.4 -39.8 -70.1 100

0.270 +135.0 +99.8 +49.1 -0.3 -41.0 -74.2 -101.6 50

0.540 +80.6 +47.1 +0.0 -45.0 -81.5 -110.8 -134.7 0

50 1.080 +21.9 -9.1 -51.2 -90.5 -121.7 -146.4 -166.0

100 2.160 -39.2 -66.2 -101.7 -133.9 -158.9 -178.4 -193.6

Max. faint source detection [uJy] 150 3.240 -75.5 -99.4 -130.2 -157.7 -178.9 -195.3 -208.0 200 0.2 0.7 1.2 1.7 2.2 2.7 3.2 Max. resolution [λ/D]

Figure 3.18: Darwin/MIRI F1000W. Absolute variations in the expected number of detectable exoplanets η for different resolution limits θlim and different faint source low background detection limits Flim with respect to our baseline scenario (which is highlighted by a black frame). The resolution steps are 0.5 λeff/D and the faint source detection steps are 1/6, 1/4, 1/2, 1, 2, 4, 6 times the baseline value of Flim, F1000W = 0.54 µJy. The color shade is scaled symmetrically around 0.

3.2.4 Exoplanets Detectable via Radial Velocity

Figure 3.19 shows the expected number of exoplanets which induce a certain radial velocity semi-amplitude K∗ to their host star for all rocky exoplanets −1 −1 (Rp ≤ 2 R⊕). We observe a peak for 0.1 m s ≤ K∗ ≤ 0.2 m s and that the majority of the rocky planets passes the detection threshold of ESPRESSO (indicated by the red vertical line). Figure 3.12 shows that a large fraction of the exoplanets which are detectable via direct imaging are also detectable via radial velocity. Typically this fraction is close to 100% for exoplanets with radii Rp ≥ 1.25 R⊕ and around 80% for exoplanets with radii Rp ≥ 0.5 R⊕. Since Darwin would be able to discover such a large sample of nearby exoplanets, the fraction of planets which can be detected via direct imaging in addition to radial velocity is signiﬁcant (Figure 3.19). Similar to

56 3.2. Darwin

E-ELT/METIS it would be possible to follow-up this fraction of exoplanets which were directly imaged by Darwin with radial velocity measurements in order to determine the exoplanets’ orbits, their true masses as well as a constraint on their radii from their observed ﬂuxes Fobs, p.

70 All rocky planets (R 2R , ρ = 5000kg/m3 ) ⊕ 60 + detectable via direct imaging

20 Expected number of planets 10

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Radial velocity [m/s]

Figure 3.19: Darwin/MIRI. Expected number of exoplanets which induce a certain radial velocity semi-amplitude K∗ to their host star for all rocky exoplanets (Rp ≤ 2 R⊕) assuming −3 a planet density of ρp = 5000 kg m (blue histogram). The fraction of exoplanets which is detectable via direct imaging (in at least one of the three ﬁlters which we simulate for the Darwin/MIRI instrument) in addition to radial velocity is shown by the green histogram. The −1 red vertical line indicates our baseline radial velocity detection threshold of K∗ ≥ 0.1 m s .

Chapter 4

Discussion

4.1 Comparison to Quanz et al.(2015)

Similar to this work Quanz et al.(2015) predict the exoplanet yield of the E-ELT/METIS instrument on the basis of the Monte-Carlo simulation from Crossfield(2013). They find expectation values for the number of detectable exoplanets of roughly 5 in the L-band and the M-band and roughly 2 in the N-band using the same resolution limit and the same faint source detection limits as in our baseline scenario (confer Table 2.1). We find significantly more detections in the L-band (7.28), slightly less detections in the M-band (4.09) and an equal number of detections in the N-band (2.16). However, where Quanz et al.(2015) predict roughly 10 exoplanets detectable in all bands together we expect only 7.77 exoplanets since many of them can be seen already in the L-band and the M-band together (confer Table 4.1).

There are several differences between our Monte-Carlo simulation and the one used by Quanz et al.(2015) including the different planet occurrence statistics. Since the statistics from Burke et al.(2015), Dressing and Char- bonneau(2015) and Fressin et al.(2013) are more recent than the statistics from Howard et al.(2012) which are used by Crossﬁeld(2013) and Quanz et al.(2015) we consider our results to be an update of their work. However, we need to take into account that Quanz et al.(2015) observe all simulated exoplanets at quadrature (when the apparent angular separation between exoplanet and host star is largest) and that they distribute Bond and geometric albedos uniformly in [0, 0.4) instead of [0, 0.8) (Bond albedo) and [0, 0.1) (geometric albedo). These changes lead to an increase in the expected exoplanet yield. Firstly, lower Bond albedos translate into higher planet equilibrium temperatures and therefore stronger thermal emission from the exoplanets. Secondly, higher geometric albedos translate into more reﬂected host star light from the exoplanets. We run our simulation again observing all exoplanets at quadrature and distributing both planetary albe-

59 4. Discussion

dos uniformly in [0, 0.4) in order to ﬁnd expected numbers of detectable exoplanets of η = 18.90 (L-band), η = 7.69 (M-band) and η = 3.15 (N-band, confer Figure 4.1). These results are up to ∼ 3.8 times larger compared to Quanz et al.(2015) what was expected due to the higher planet occurrence predicted by our more recent statistics (confer Table 4.1).

10.81

3.77 4.76 11.25 3.77 & 4.76 3.77 & 11.25 4.76 & 11.25 3.77 & 4.76 & 11.25

0.00 2.21 0.54

0.00 0.40

5.48

Figure 4.1: E-ELT/METIS. Pie chart showing the expected number of detectable exoplanets η for each individual filter and all possible combinations of filters for E-ELT/METIS. All exoplanets are observed at quadrature and their Bond and geometric albedos are distributed uniformly in [0, 0.4). The numbers in the legend state the effective filter wavelengths λeff in µm.

Table 4.1: Expected number of detectable exoplanets η for E-ELT/METIS.

Filter η (this worka) η (this workb) η (Quanz et al., 2015)

L-band 7.28 18.90 ∼ 5 M-band 4.0907.69 ∼ 5 N-band 2.1603.15 ∼ 2 Total 7.77 19.44 ∼ 10

a) Baseline scenario. b) Observation @ quadrature and planetary albedos in [0, 0.4).

If we compare the planet radii and the planet equilibrium temperatures of the detectable exoplanets predicted by Quanz et al.(2015) to this work (Fig- ure 4.2) we ﬁnd that the major difference occurs for the smallest (1 R⊕ ≤

60 4.1. Comparison to Quanz et al.(2015)

Rp ≤ 2 R⊕) and coolest (100 K ≤ Teq, p ≤ 200 K) planets (primarily in the L-band and the M-band). For the other part of the simulated parameter space we get similar results for the projected planet radius distribution (one- dimensional histogram on the right of each subfigure) and the projected planet equilibrium temperature distribution (one-dimensional histogram on the top of each subfigure). The significant increase in detectable small and cold exoplanets can be explained considering that our underlying planet population consists of more small exoplanets. Furthermore, cool planets around some close M dwarfs like Ross 154 and YZ Ceti (which have a higher occurrence of small exoplanets) can be resolved if all observations are performed at quadrature.

61 4. Discussion

8.0 3.77 um 4.0

0.0 16

0.43 0.34 0.48 0.16 0.04 0.03 0.02 0.02 0.01

8 ] ⊕ R [

0.24 0.11 0.40 0.21 0.06 0.03 0.03 0.01 0.02 s u i

d 4 a r

e 1.71 0.65 1.59 1.42 0.45 0.26 0.18 0.16 0.14 n a l

P 2

1.77 0.53 1.07 1.59 0.56 0.25 0.21 0.22 0.19

1 100 300 500 700 900 0.0 4.0 8.0 Equilibrium temperature [K]

2.0 4.76 um 1.0

0.0 16

0.08 0.12 0.29 0.09 0.03 0.02 0.02 0.01 0.00

8 ] ⊕ R [

0.04 0.02 0.18 0.11 0.04 0.03 0.03 0.01 0.01 s u i

d 4 a r

e 0.30 0.06 0.61 0.71 0.28 0.20 0.17 0.16 0.09 n a l

P 2

0.31 0.04 0.46 0.61 0.34 0.20 0.20 0.22 0.12

1 100 300 500 700 900 0.0 2.0 4.0 Equilibrium temperature [K]

1.0 11.25 um 0.5

0.0 16

0.04 0.08 0.02 0.02 0.02 0.01 0.00 0.00 0.00

8 ] ⊕ R [

0.01 0.07 0.03 0.02 0.02 0.01 0.01 0.00 0.00 s u i

d 4 a r

e 0.00 0.28 0.29 0.17 0.14 0.06 0.03 0.03 0.03 n a l

P 2

0.00 0.29 0.48 0.21 0.15 0.07 0.01 0.04 0.04

1 100 300 500 700 900 0.0 1.0 2.0 Equilibrium temperature [K] Figure 4.2: E-ELT/METIS. Expected number of detectable exoplanets η for E-ELT/METIS observing in the L-band (top), the M-band (middle) and the N-band (bottom) depicted as two- dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Figure 3 of Quanz et al.(2015)). All exoplanets are observed at quadrature and their Bond and geometric albedos are distributed uniformly in [0, 0.4).

62 4.2. Choice of Albedos

4.2 Choice of Albedos

The expected number of detectable exoplanets turns out to be heavily dependent on the choice of the planetary albedos, particularly on the choice of the Bond albedos. Considering Section 4.1 where we compare our predictions for E-ELT/METIS to those of Quanz et al.(2015) a decrease in the Bond albedos by a factor of two leads to an increase in the planet yield also by a factor of ∼ 2 (the impact of the increase in the geometric albedos can be neglected since the fraction of exoplanets which can be detected due to their reflected host star light only is extremely small, confer Figure 3.2). Crossfield(2013) and Quanz et al.(2015) assume a uniform distribution of the Bond and the geometric albedos between 0 and 0.4. Unfortunately, there are only few statistics available on the albedos of exoplanets. However, considering the Bond albedos of the Solar System planets, we find it more reasonable to distribute those of our simulated planets uniformly between 0 and 0.8 (confer Subsection 2.3.3). Increasing the Bond albedos obviously reduces the scientific yield since higher Bond albedos translate into lower equilibrium temperatures and therefore less thermal emission from the exoplanets. A reasonable choice of the geometric albedos is even more difficult since they are wavelength dependent and contribute linearly to the reflected host star light. We decide to distribute the geometric albedos uniformly between 0 and 0.1 because many gases which dominate the atmospheres of our Solar System planets strongly absorb in the infrared (confer Subsec- tion 2.3.3). However, as we adapt filters operating in the infrared where the planets’ thermal emission is superior to their reflected host star light (confer Figures 3.2 and 3.13) the choice of the geometric albedos has only a minor impact on the expected number of detectable exoplanets. Even if we distribute the geometric albedos uniformly between 0 and 1 instead of 0 and 0.1 the predicted planet yield of Darwin would increase by no more than 1.4% (confer Figure 4.3). More specifically the expected number of detectable exoplanets would increase by 4.7. The planets which could be additionally discovered would be cool (Teq, p ≤ 300 K) and orbit their host stars at the widest separations covered by our simulation.

63 4. Discussion

100.0

50.0 5000 kg/m3

0.0 22.0

3.62 3.92 2.97 1.63 0.88 0.47 0.33 0.23 0.17 0.09 0.06 0.04 6.0 ]

⊕ 1.05 2.11 1.76 1.24 0.77 0.49 0.30 0.19 0.12 0.08 0.05 0.03 R [

s 4.0 u i

d 7.76 37.8930.1017.7210.74 6.82 4.33 2.59 1.61 0.94 0.56 0.37 a r

t 2.0 e n 2.77 28.1131.2918.9510.73 6.96 4.65 3.15 2.01 1.28 0.84 0.51 a l 100% 99% 99% 99% 99% 99% 99% 99% 99% 99% 98% 98%

P 1.25

0.41 9.26 20.4516.47 9.11 5.62 3.73 2.39 1.61 1.11 0.72 0.47 93% 89% 85% 79% 77% 79% 80% 82% 83% 83% 82% 82% 0.5 200 400 600 800 1000 1200 0.0 70.0 140.0 Equilibrium temperature [K]

100.0

50.0 5000 kg/m3

0.0 22.0

3.78 3.90 2.90 1.58 0.88 0.50 0.32 0.23 0.16 0.09 0.06 0.05 6.0 ]

⊕ 1.17 2.06 1.79 1.24 0.80 0.53 0.31 0.20 0.10 0.08 0.04 0.02 R [

s 4.0 u i

d 8.57 38.9529.9517.5910.65 6.80 4.36 2.67 1.67 0.95 0.56 0.36 a r

t 2.0 e n 3.06 28.8731.6718.8610.79 7.01 4.75 3.15 2.01 1.25 0.83 0.53 a l 100% 99% 99% 99% 99% 99% 99% 99% 99% 99% 99% 99%

P 1.25

0.43 9.65 20.9416.27 9.09 5.72 3.72 2.52 1.71 1.10 0.74 0.47 93% 89% 85% 78% 78% 80% 81% 82% 83% 81% 79% 79% 0.5 200 400 600 800 1000 1200 0.0 70.0 140.0 Equilibrium temperature [K] Figure 4.3: Darwin/MIRI. Expected number of detectable exoplanets η for Darwin/MIRI combined for all three simulated ﬁlters assuming a geometric albedo uniformly distributed between [0, 0.1) (top) and between [0, 1) (bottom) depicted as two-dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Figure 3.12).

4.3 Choice of Eccentricity

For our baseline scenario we assume circular orbits (eccentricity e = 0), however all equations are also derived for eccentric orbits. On the one hand planets on eccentric orbits spend more time around their apoapsis where the separation between exoplanet and host star is largest. Therefore one would expect more detections because (statistically) more exoplanets can be spatially resolved from their host stars. On the other hand an exoplanet which is located further away from its host star will be cooler and emit less thermal blackbody radiation and less reﬂected host star light. Therefore one would expect less detections because (statistically) less exoplanets pass the instrument’s detection threshold. Here we assume that the timescales of the thermodynamic processes in the exoplanets’ atmospheres are short

64 4.3. Choice of Eccentricity compared to their orbital periods so that the exoplanets’ equilibrium temperatures are always determined by the instantaneous physical separations between exoplanet and host star rp in Equation 2.30. Considering Figure 4.4 which compares the expected number of detectable exoplanets for Darwin assuming circular orbits (e = 0) and highly eccentric orbits (e distributed uniformly in [0, 1)) it can be concluded that the second effect has a larger impact since a smaller planet yield is expected in the case of highly eccentric orbits. The upper subﬁgure which shows the non-eccentric case predicts 329.4 detectable exoplanets whereas the lower subﬁgure which shows the case with highly eccentric orbits predicts 306.8 detectable exoplanets. However, the difference is very small (roughly -7%) so that we can regard our results as quite robust with respect to variations in the orbital eccentricity. Note that if we assume a delay in the cooling (or heating) of the exoplanets whilst moving from smaller to larger (or from larger to smaller) physical separations to their host stars the exoplanets still had a higher temperature and emitted more thermal radiation when they reach the apoapsis. This would certainly lead to an increase in the expected planet yield (but of the same order of magnitude like the decrease assuming instantaneous cooling and heating).

65 4. Discussion

100.0

50.0 5000 kg/m3

0.0 22.0

3.62 3.92 2.97 1.63 0.88 0.47 0.33 0.23 0.17 0.09 0.06 0.04 6.0 ]

⊕ 1.05 2.11 1.76 1.24 0.77 0.49 0.30 0.19 0.12 0.08 0.05 0.03 R [

s 4.0 u i

d 7.76 37.8930.1017.7210.74 6.82 4.33 2.59 1.61 0.94 0.56 0.37 a r

t 2.0 e n 2.77 28.1131.2918.9510.73 6.96 4.65 3.15 2.01 1.28 0.84 0.51 a l 100% 99% 99% 99% 99% 99% 99% 99% 99% 99% 98% 98%

P 1.25

0.41 9.26 20.4516.47 9.11 5.62 3.73 2.39 1.61 1.11 0.72 0.47 93% 89% 85% 79% 77% 79% 80% 82% 83% 83% 82% 82% 0.5 200 400 600 800 1000 1200 0.0 70.0 140.0 Equilibrium temperature [K]

100.0

50.0 5000 kg/m3

0.0 22.0

3.68 3.95 2.58 1.49 0.83 0.48 0.34 0.22 0.13 0.09 0.07 0.03 6.0 ]

⊕ 1.02 2.08 1.69 1.07 0.71 0.45 0.31 0.19 0.09 0.06 0.04 0.03 R [

s 4.0 u i

d 7.71 35.7428.0616.36 9.85 6.16 3.83 2.36 1.47 0.86 0.54 0.34 a r

t 2.0 e n 2.64 25.7529.0617.7910.43 6.49 4.34 2.84 1.78 1.18 0.72 0.47 a l 100% 100% 99% 99% 99% 99% 99% 99% 99% 99% 99% 99%

P 1.25

0.44 8.57 18.3715.22 8.83 5.38 3.36 2.26 1.46 0.99 0.61 0.40 94% 93% 90% 85% 85% 86% 86% 87% 88% 87% 88% 84% 0.5 200 400 600 800 1000 1200 0.0 70.0 140.0 Equilibrium temperature [K] Figure 4.4: Darwin/MIRI. Expected number of detectable exoplanets η for Darwin/MIRI combined for all three simulated ﬁlters assuming an eccentricity equal to 0 (top) and uniformly distributed between [0, 1) (bottom) depicted as two-dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Figure 3.12).

4.4 Underlying Planet Population

The underlying planet population has a high impact on the properties of the detectable exoplanets. For VLT/NACO and E-ELT/METIS as well as for Darwin/MIRI the peak observed at planet radii of 1.25 R⊕ ≤ Rp ≤ 4 R⊕ and planet equilibrium temperatures of 200 K ≤ Teq, p ≤ 500 K is simply a ﬁnger- print of the radius and orbital period distribution which we use to generate our artiﬁcial exoplanetary systems. Moreover, Darwin’s superior resolution and sensitivity allow the detection of a broad range of planets from widely separated but faint to bright but tiny separated ones, so that the underlying planet population itself is the major limiting factor. This also explaines why it is not possible to detected more cooler or hotter exoplanets. Another important point is our patchwork-like approach to merge planet occurrence

66 4.4. Underlying Planet Population statistics from different authors. Of course each statistic rests on different assumptions and models to estimate Kepler’s pipeline completeness and infer the true population of exoplanets from the highly biased sample of transiting exoplanets. We implicitly assume a spectral type dependence of the planet occurrence by applying the statistics from Dressing and Charbon- neau(2015) to M type stars only and those from Burke et al.(2015) to G and K type stars only. This choice reappears in our results (see Table 3.3) as we obtain that all the most fruitful targets for Darwin/MIRI are M type stars. We try to bypass the impact of the underlying planet population by in- troducing the completeness as a more sophisticated parameter to determine the suitability of a star for directly imaging exoplanets. Moreover, the underlying planet population is affected by statistical errors. Dressing and Charbonneau(2015) and Fressin et al.(2013) present these errors as upper and lower bounds for the planet occurrence rate within each individual radius-orbital period bin. Burke et al.(2015) present an optimistic and a pessimistic efficiency scenario for their two-power law model. We perform error analyses for E-ELT/METIS and Darwin/MIRI comparing the expected planet yield of our baseline scenario with planet occurrence rates r to a maximal and a minimal scenario with planet occurrence rates r + ∆r and r − ∆r respectively (confer Figures 4.5 and 4.6). All findings are summarized in Table 4.2 and reveal that the Poisson errors from our Monte- Carlo simulation ∆η are negligible compared to the statistical errors in the underlying planet population ∆r. We finally state the expected number of +2.44 detectable exoplanets for E-ELT/METIS as 7.77−2.15 and for Darwin/MIRI +115.9 as 329.4−81.3 .

67 4. Discussion

4.0

2.0 5000 kg/m3

0.0 22.0

0.26 0.47 0.59 0.20 0.09 0.05 0.02 0.01 0.01 0.00 0.00 0.00 6.0 ]

⊕ 0.01 0.07 0.18 0.10 0.04 0.03 0.02 0.01 0.00 0.01 0.00 0.00 R [

s 4.0 u i

d 0.06 0.32 0.97 0.79 0.32 0.25 0.17 0.13 0.08 0.04 0.03 0.03 a r

t 2.0 e n 0.00 0.32 0.71 0.74 0.31 0.20 0.16 0.14 0.07 0.04 0.03 0.02 a l 100% 96% 96% 96% 94% 96% 97% 96% 97% 96% 89% 98%

P 1.25

0.00 0.21 0.48 0.50 0.19 0.12 0.12 0.09 0.08 0.04 0.03 0.02 nan% 54% 58% 65% 61% 58% 52% 47% 44% 37% 44% 54% 0.5 200 400 600 800 1000 1200 0.0 2.0 4.0 Equilibrium temperature [K]

4.0

2.0 5000 kg/m3

0.0 22.0

0.20 0.38 0.49 0.14 0.05 0.04 0.02 0.01 0.01 0.01 0.00 0.00 6.0 ]

⊕ 0.01 0.05 0.13 0.07 0.04 0.03 0.02 0.01 0.01 0.00 0.00 0.00 R [

s 4.0 u i

d 0.03 0.25 0.84 0.66 0.28 0.21 0.14 0.12 0.08 0.04 0.01 0.02 a r

t 2.0 e n 0.00 0.21 0.49 0.56 0.23 0.16 0.12 0.11 0.06 0.04 0.02 0.02 a l 100% 96% 97% 97% 96% 98% 95% 96% 98% 97% 100% 95%

P 1.25

0.00 0.13 0.30 0.32 0.13 0.08 0.06 0.05 0.05 0.03 0.02 0.02 nan% 52% 61% 64% 67% 54% 53% 50% 40% 47% 62% 50% 0.5 200 400 600 800 1000 1200 0.0 2.0 4.0 Equilibrium temperature [K]

4.0

2.0 5000 kg/m3

0.0 22.0

0.14 0.27 0.36 0.11 0.04 0.03 0.01 0.01 0.01 0.00 0.00 0.00 6.0 ]

⊕ 0.01 0.03 0.07 0.05 0.02 0.01 0.01 0.01 0.00 0.00 0.00 0.00 R [

s 4.0 u i

d 0.02 0.20 0.64 0.50 0.24 0.16 0.12 0.10 0.06 0.03 0.03 0.02 a r

t 2.0 e n 0.00 0.17 0.37 0.42 0.17 0.11 0.09 0.08 0.04 0.02 0.02 0.02 a l nan% 96% 98% 96% 96% 96% 95% 97% 96% 98% 100% 97%

P 1.25

0.00 0.08 0.16 0.19 0.06 0.04 0.03 0.03 0.02 0.02 0.01 0.01 nan% 58% 63% 68% 62% 71% 63% 65% 62% 52% 55% 67% 0.5 200 400 600 800 1000 1200 0.0 2.0 4.0 Equilibrium temperature [K] Figure 4.5: E-ELT/METIS. Expected number of detectable exoplanets η for E-ELT/METIS combined for all three simulated ﬁlters for our maximal scenario (top, r + ∆r), for our baseline scenario (middle) and for our minimal scenario (bottom, r − ∆r) depicted as two-dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Fig- ure 3.3).

68 4.4. Underlying Planet Population

120.0

60.0 5000 kg/m3

0.0 22.0

4.44 4.97 3.67 2.00 1.08 0.63 0.40 0.27 0.19 0.12 0.07 0.05 6.0 ]

⊕ 1.38 2.84 2.34 1.58 0.99 0.61 0.38 0.23 0.16 0.09 0.05 0.04 R [

s 4.0 u i

d 14.9358.4041.1322.5813.21 7.94 4.93 3.04 1.82 1.07 0.65 0.40 a r

t 2.0 e n 3.99 37.3641.4124.6713.69 8.57 5.77 3.77 2.44 1.56 0.93 0.56 a l 99% 99% 99% 98% 98% 99% 99% 99% 99% 99% 99% 99%

P 1.25

0.56 13.0928.1522.9212.97 7.73 4.92 3.25 2.15 1.38 0.88 0.60 91% 88% 84% 78% 76% 77% 80% 82% 80% 80% 81% 79% 0.5 200 400 600 800 1000 1200 0.0 90.0 180.0 Equilibrium temperature [K]

120.0

60.0 5000 kg/m3

0.0 22.0

3.62 3.92 2.97 1.63 0.88 0.47 0.33 0.23 0.17 0.09 0.06 0.04 6.0 ]

⊕ 1.05 2.11 1.76 1.24 0.77 0.49 0.30 0.19 0.12 0.08 0.05 0.03 R [

s 4.0 u i

d 7.76 37.8930.1017.7210.74 6.82 4.33 2.59 1.61 0.94 0.56 0.37 a r

t 2.0 e n 2.77 28.1131.2918.9510.73 6.96 4.65 3.15 2.01 1.28 0.84 0.51 a l 100% 99% 99% 99% 99% 99% 99% 99% 99% 99% 98% 98%

P 1.25

0.41 9.26 20.4516.47 9.11 5.62 3.73 2.39 1.61 1.11 0.72 0.47 93% 89% 85% 79% 77% 79% 80% 82% 83% 83% 82% 82% 0.5 200 400 600 800 1000 1200 0.0 90.0 180.0 Equilibrium temperature [K]

120.0

60.0 5000 kg/m3

0.0 22.0

2.67 2.86 2.15 1.28 0.63 0.36 0.25 0.17 0.11 0.07 0.04 0.04 6.0 ]

⊕ 0.73 1.36 1.18 0.90 0.59 0.35 0.23 0.15 0.09 0.06 0.03 0.02 R [

s 4.0 u i

d 5.01 27.3223.0314.15 8.88 5.78 3.74 2.30 1.33 0.84 0.51 0.30 a r

t 2.0 e n 2.06 21.5824.7114.43 8.16 5.48 3.74 2.44 1.65 0.99 0.67 0.43 a l 100% 99% 99% 99% 99% 99% 99% 99% 99% 99% 99% 99%

P 1.25

0.31 6.74 15.2111.30 5.78 3.62 2.53 1.70 1.17 0.79 0.51 0.35 96% 91% 87% 82% 81% 81% 83% 83% 85% 84% 81% 83% 0.5 200 400 600 800 1000 1200 0.0 90.0 180.0 Equilibrium temperature [K] Figure 4.6: Darwin/MIRI. Expected number of detectable exoplanets η for Darwin/MIRI combined for all three simulated ﬁlters for our maximal scenario (top, r + ∆r), for our baseline scenario (middle) and for our minimal scenario (bottom, r − ∆r) depicted as two-dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Figure 3.12).

69 4. Discussion

Table 4.2: Expected number of detectable exoplanets η including errors for E-ELT/METIS and Darwin/MIRI.

Telescope Instrument Planet yield Errora (∆r) Errorb (∆η)

+2.44 E-ELT METIS 007.77 −2.15 ±0.06 +115.9 Darwin MIRI 329.4 −81.3 ±0.4 a) From underlying planet population. b) From Monte-Carlo simulation (Poisson error, confer Equation 3.3).

4.5 Zodiacal Light

Stellar environments are traversed by thin dust clouds which do not only reflect star light but also emit thermal light when heated up by the radiation of their host stars. The radiation emitted by these clouds is called zodiacal light (around our Sun) or exozodiacal light (around other stars) and its thermal emission gives rise to a significant radiation background in the infrared (Reach et al., 2003). The scattered and emissive components of the zodiacal light from the inter- planetary dust cloud in our own Solar System are modelled by Glasse et al. (2015) for the MIRI instrument of the JWST. They predict that it is the dom- inant background source shortward of ∼ 17 µm and that the instrument’s reflected and thermal emission becomes superior longward of ∼ 17 µm only. The zodiacal light is certainly dependent on the observatory’s location in the Solar System. Since both the JWST and Darwin are supposed to be operated at the Earth-Sun L2 point we can assume that the contamination caused by zodiacal light is absorbed by adopting the faint source low background detection limits from the MIRI instrument of the JWST (Glasse et al., 2015). Regarding the existing VLT/NACO instrument we choose faint source detection limits which can already be achieved in measurements today (∆mag = 17.5mag for NACO L’ and ∆mag = 15mag for NACO M’). These limits clearly take into account the contamination by zodiacal light. For the prospective E-ELT/METIS instrument Quanz et al.(2015) model the sky background noise (besides the instrument throughput and the light path aberrations) which also includes the contamination by zodiacal light. We do not put any additional effort into modelling the impact of exozodiacal light on the observations of exoplanets in our simulation. Firstly, quantita- tive measurements of exozodiacal light are very rare so that it would be difficult to estimate and model its light contamination on a statistical basis. We note that an excellent and extensive analysis of the impact of exozodi-

70 4.6. Planet Occurrence Around Binary Stars acal light is presented in Stark et al.(2014) who take into account possible disk geometries and stellar type dependences in order to investigate how the expected exoplanet yield changes with the mean exozodiacal light level around other stars. Secondly, the angular differential imaging (ADI) (Marois et al., 2006) and the nulling interferometry (Cockell et al., 2009) data reduction can cancel out exozodiacal light at least partially. The success is highest if the exozodiacal dust cloud is observed face-on and has a radially symmetric shape. In order to model and subtract radially symmetric on-axis sources both techniques, ADI and nulling interferometry, make use of either the rotation of the sky with respect to the telescope or the rotation of the interferometric fringe pattern with respect to the sky. The basic idea (for ADI) is to take thousands of short exposures while the sky itself is rotating so that a potential faint companion is located at a different position on each individual image. However, radially symmetric sources like the host star (or a face-on exozodiacal dust cloud) look similar on each individual image. By calculating the average of all images a potential faint companion is cancelled out almost completely while radially symmetric sources remain. This average image can then be decomposed into principal components (Amara and Quanz, 2012) to model and subtract radially symmetric on-axis sources in each individual image (leaving only the faint signal of a possible companion behind). Finally all images are de-rotated and added together so that all the faint signals of a potential companion sum up to a clearly visible off-axis point source.

4.6 Planet Occurrence Around Binary Stars

Our simulation predicts the number of detectable exoplanets within a distance of 20 pc only, excluding close binaries due to the fact that removing a second off-axis stellar PSF (point-spread function) from the data to recover a planets signal below it is extremely difficult. However, many processes like disc truncation (Jang-Condell, 2015), enhanced accretion (Jensen et al., 2007) and enhanced photoevaporation (Alexander, 2012) can influence the formation and evolution of planets in binary star systems. Kraus et al.(2016) show that the occurrence of planets around close-in binaries with separations be- +59 low acut = 47−23 AU differs from that around single stars or wider separated +0.14 binaries by a suppression factor of S = 0.34−0.15. They use high-resolution follow-up observations of a subsample of 382 KOIs (Kepler Objects of In- terest) to distinguish single star hosts from multiple star hosts. Since the Kepler target stars are chosen almost blindly with respect to stellar multiplicity (Kraus et al., 2016) their findings translate into a higher planet occurrence for a stellar sample which systematically excludes close-in binaries. Assuming a Gaussian distribution of binary companions in log(P)-space +9 according to Raghavan et al.(2010) we find that f<47 AU = 49−7% of all bi-

71 4. Discussion

nary stars have separations below acut. Moreover, we ﬁnd a scaling factor of +0.07 rsingle/r = 1.14−0.07 for the planet occurrence around single stars or wider separated binaries via

S · fbin · f<47 AU · rsingle + (1 − fbin · f<47 AU) · rsingle = r, (4.1)

where fbin is the fraction of binary stars among Kepler targets and r is the occurrence of planets predicted by the Kepler data. We take fbin from Table 16 of Raghavan et al.(2010) who investigate a sample of 454 Solar-type stars for stellar companions ( fbin = Nbin/(Nsingle + Nbin)). However, we decide not to scale our planet occurrence rates with this factor of rsingle/r = +0.07 1.14−0.07 and to regard our results as conservative.

4.7 Background-Limited Observations

In all simulations we assume our observations to be background-limited. This means that an exoplanet is considered to be detectable as soon as its observed flux exceeds the instrument’s faint source detection limit (and its apparent angular separation from the host star exceeds the instrument’s resolution limit). We do not take into account the planet-host star flux contrast at any point in our calculations. The faint source (low background) detection limits for the VLT/NACO instrument (from the Spanish Virtual Observatory) and the E-ELT/METIS instrument (from Quanz et al.(2015)) are given for a 5σ detection significance in 8640 s observation time, the ones for the Darwin/MIRI instrument (from Glasse et al.(2015)) for a 10 σ detection significance in 10000 s observation time. Considering our assumption of background-limited observations fainter exoplanets can be detected by increasing the observation time since thereby the number of detected photons from the exoplanet (which simply scales linearly with the observation time) can be increased. Here the instrument noise and the sky background noise which scale roughly proportional to the square root of the observation time are neglected. Certainly, time is a valuable factor, especially for a space mission like Dar- win. For the 326 target stars in our star catalog the 10000 s observation time per target would translate into a total mission duration of ∼ 38 d which is only a fraction of the nominal one year assumed by Stark et al. (2014) and Stark et al.(2015) (without observational overheads). Therefore the exoplanet yield of Darwin could be increased significantly with respect to our baseline scenario by either observing more targets or increasing the observation time per target in order to achieve a better sensitivity. This would then allow to detect for instance more Earth twins. If we assume that the faint source low background detection limits scale with the inverse of the observation time, ten times fainter planets could be detected with ten

72 4.7. Background-Limited Observations times longer observations. This translates into a mission duration of roughly one year (∼ 380 d) and an expected number of 47.5 detectable Earth twins (0.5 R⊕ ≤ Rp ≤ 1.25 R⊕ and 200 K ≤ Teq, p ≤ 300 K) combined in all three simulated ﬁlters instead of only 9.3 detectable Earth twins in our baseline scenario. The total number of detectable exoplanets would increase from 329.4 to 500.8 and the planets which could be additionally discovered would be cool (Teq, p ≤ 400 K) and therefore faint (confer Figure 4.7).

180.0

90.0 5000 kg/m3

0.0 22.0

3.62 3.92 2.97 1.63 0.88 0.47 0.33 0.23 0.17 0.09 0.06 0.04 6.0 ]

⊕ 1.05 2.11 1.76 1.24 0.77 0.49 0.30 0.19 0.12 0.08 0.05 0.03 R [

s 4.0 u i

d 7.76 37.8930.1017.7210.74 6.82 4.33 2.59 1.61 0.94 0.56 0.37 a r

t 2.0 e n 2.77 28.1131.2918.9510.73 6.96 4.65 3.15 2.01 1.28 0.84 0.51 a l 100% 99% 99% 99% 99% 99% 99% 99% 99% 99% 98% 98%

P 1.25

0.41 9.26 20.4516.47 9.11 5.62 3.73 2.39 1.61 1.11 0.72 0.47 93% 89% 85% 79% 77% 79% 80% 82% 83% 83% 82% 82% 0.5 200 400 600 800 1000 1200 0.0 90.0 180.0 Equilibrium temperature [K]

180.0

90.0 5000 kg/m3

0.0 22.0

4.44 3.93 2.97 1.63 0.88 0.47 0.33 0.23 0.17 0.09 0.06 0.04 6.0 ]

⊕ 2.08 2.16 1.76 1.24 0.77 0.49 0.30 0.19 0.12 0.08 0.05 0.03 R [

s 4.0 u i

d 35.3348.0830.1717.7210.74 6.82 4.33 2.59 1.61 0.94 0.56 0.37 a r

t 2.0 e n 32.2860.4335.8018.9510.73 6.96 4.65 3.15 2.01 1.28 0.84 0.51 a l 99% 98% 98% 99% 99% 99% 99% 99% 99% 99% 98% 98%

P 1.25

11.4247.5035.1617.16 9.11 5.62 3.73 2.39 1.61 1.11 0.72 0.47 89% 75% 73% 78% 77% 79% 80% 82% 83% 83% 82% 82% 0.5 200 400 600 800 1000 1200 0.0 90.0 180.0 Equilibrium temperature [K] Figure 4.7: Darwin/MIRI. Expected number of detectable exoplanets η for Darwin/MIRI combined for all three simulated ﬁlters for our baseline scenario (top) and ten times longer observations (bottom) depicted as two-dimensional histograms for the planet radii Rp and the planet equilibrium temperatures Teq, p (like in Figure 3.12).

73 4. Discussion

4.8 Properties of M Dwarf Stars

The properties of cool M dwarf stars from the Kepler Input Catalog (KIC) show significant errors as e.g. Dressing and Charbonneau(2013) illustrate by calculating revised stellar parameters from the observed colors using the Dartmouth Stellar Evolution Database (Dotter et al., 2008). They find that the stellar effective temperatures Teff, ∗ and the stellar radii R∗ are overestimated in the KIC. This translates into overestimated planet radii Rp since the 2 2 measured transit depth is proportional to Rp/R∗ and a smaller R∗ implies a smaller Rp so that the (measured) transit depth remains constant (Quanz, 2016)1. In a very recent work Dressing et al.(2017) present inferred stellar parameters for low-mass dwarfs identified as candidate planet hosts during the Kepler K2 mission. We compare their parameters to these of stars of similar spectral types (M1V, M2V, M3V and M4V) from our star catalog to verify the properties of the cool M dwarf stars used in our simulation. We find that the mean value of our stellar effective temperatures Teff, ∗ deviates by maximally −7.65% for M1V stars and minimally −0.30% for M2V stars and that the mean value of our stellar radii R∗ deviates by maximally −23.01% for M4V stars and minimally +4.00% for M3V stars. In general we always un- derestimate the stellar effective temperatures what translates into underestimated planet equilibirum temperatures and lower thermal blackbody emission from the exoplanets. The stellar radii are significantly underestimated for M1V (−18.97%) and M4V (−23.01%) stars but slightly overestimated for M2V (+12.07%) and M3V (+4.00%) stars. Considering Equations 2.30 and 2.34 smaller stellar radii translate into lower planet equilibrium temperatures and lower thermal blackbody emission from the host stars resulting in less reflected host star light from the exoplanets (confer Equation 2.35). Therefore our simulation sets a lower limit for the expected number of detectable exoplanets around M dwarfs. Revising the stellar parameters for the M dwarfs in our star catalog would lead to an increase in the fluxes observed from the exoplanets and thus to an increase in the expected exoplanet yield.

1Section 3.4.1, Equation 3.14, Page 65

74 4.9. Target Stars with Known Companions

4.9 Target Stars with Known Companions

There are several exoplanets which were already discovered within a distance of 20 pc. We use the database of conﬁrmed exoplanets from the NASA Exoplanet Archive2 to identify stars in our star catalog with known companions (Table 4.3). Planet radius and planet density are unknown for most of these exoplanets because most of them were discovered with the radial velocity technique. Therefore we do not determine if these exoplanets could be directly imaged with our simulated instruments. We refer to Table 2 of Quanz et al.(2015) for a list of exoplanets detected with the radial velocity technique which can be directly imaged with the prospective E-ELT/METIS instrument.

2NASA Exoplanet Archive, 31.01.2017, http://exoplanetarchive.ipac.caltech.edu/cgi-bin/TblView/nph-tblView?app= ExoTbls&config=planets

75 4. Discussion

Table 4.3: List of conﬁrmed exoplanets (from the NASA Exoplanet Archive) which orbit host stars from our stellar sample (semi-major axis a in AU, lower limit on planet mass Mp sin(i) in MJupiter and distance to Earth d in pc).

Host star Planet a Mp sin(i) d Discovery method

Proxima Cen b 0.049 0.004 1.437 Radial Velocity eps Eri b 3.390 1.550 3.268 Radial Velocity GJ 15 A b 0.072 0.017 3.634 Radial Velocity Wolf 1061 b 0.036 0.004 4.331 Radial Velocity Wolf 1061 c 0.084 0.013 4.331 Radial Velocity Wolf 1061 d 0.204 0.016 4.331 Radial Velocity GJ 687 b 0.164 0.058 4.565 Radial Velocity GJ 674 b 0.039 0.035 4.577 Radial Velocity GJ 876 b 0.208 2.276 4.724 Radial Velocity GJ 876 c 0.130 0.714 4.724 Radial Velocity GJ 876 d 0.021 0.021 4.724 Radial Velocity GJ 876 e 0.334 0.046 4.724 Radial Velocity GJ 832 b 3.560 0.680 4.987 Radial Velocity GJ 832 c 0.163 0.017 4.987 Radial Velocity HD 20794 b 0.121 0.009 6.071 Radial Velocity HD 20794 c 0.204 0.008 6.071 Radial Velocity HD 20794 d 0.350 0.015 6.071 Radial Velocity GJ 581 b 0.041 0.050 6.492 Radial Velocity GJ 581 c 0.072 0.017 6.492 Radial Velocity GJ 581 e 0.028 0.005 6.492 Radial Velocity HD 219134 b 0.038 0.012 6.572 Radial Velocity HD 219134 c 0.065 0.011 6.572 Radial Velocity HD 219134 d 0.235 0.067 6.572 Radial Velocity HD 219134 f 0.146 0.028 6.572 Radial Velocity

76 4.9. Target Stars with Known Companions

HD 219134 g 0.375 0.034 6.572 Radial Velocity HD 219134 h 3.110 0.340 6.572 Radial Velocity Fomalhaut b 160.000 0.000 7.725 Imaging 61 Vir b 0.050 0.016 8.545 Radial Velocity 61 Vir c 0.217 0.057 8.545 Radial Velocity 61 Vir d 0.476 0.072 8.545 Radial Velocity GJ 849 b 2.320 0.910 8.793 Radial Velocity HD 102365 b 0.460 0.050 9.258 Radial Velocity GJ 86 b 0.113 3.910 10.929 Radial Velocity HD 3651 b 0.295 0.229 11.122 Radial Velocity HD 85512 b 0.260 0.011 11.167 Radial Velocity 55 Cnc b 0.115 0.831 12.545 Radial Velocity 55 Cnc c 0.241 0.171 12.545 Radial Velocity 55 Cnc d 5.503 3.878 12.545 Radial Velocity 55 Cnc e 0.015 0.025 12.545 Radial Velocity 55 Cnc f 0.788 0.141 12.545 Radial Velocity HD 40307 b 0.047 0.013 12.842 Radial Velocity HD 40307 c 0.080 0.021 12.842 Radial Velocity HD 40307 d 0.132 0.030 12.842 Radial Velocity HD 40307 f 0.247 0.016 12.842 Radial Velocity HD 40307 g 0.600 0.022 12.842 Radial Velocity HD 147513 b 1.320 1.210 12.885 Radial Velocity ups And b 0.059 0.688 13.480 Radial Velocity ups And c 0.828 1.981 13.480 Radial Velocity ups And d 2.513 4.132 13.480 Radial Velocity 47 UMa b 2.100 2.530 14.088 Radial Velocity 47 UMa c 3.600 0.540 14.088 Radial Velocity

77 4. Discussion

47 UMa d 11.600 1.640 14.088 Radial Velocity tau Boo b 0.049 4.320 15.606 Radial Velocity HR 810 b 0.910 2.130 17.251 Radial Velocity rho CrB b 0.220 1.045 17.437 Radial Velocity rho CrB c 0.412 0.079 17.437 Radial Velocity HD 10647 b 2.015 0.940 17.362 Radial Velocity GJ 3021 b 0.490 3.370 17.628 Radial Velocity GJ 504 b 43.500 4.000 17.959 Imaging 70 Vir b 0.481 7.400 18.119 Radial Velocity 14 Her b 2.770 4.640 18.155 Radial Velocity HD 39091 b 3.380 10.270 18.217 Radial Velocity HN Peg b 773.000 21.999 18.402 Imaging bet Pic b 9.100 0.000 19.288 Imaging

4.10 Exo-Earth Yield

As already discussed in Section 3.1 the VLT/NACO and the E-ELT/METIS instrument are not suitable for the detection of Earth twins (planet radius between 0.5 R⊕ and 1.25 R⊕ and planet equilibrium temperature between 200 K and 300 K). However, considering our baseline scenario for the space interferometer Darwin (confer Figure 3.12) we expect that ∼ 9.3 Earth twins could be discovered within 20 pc. Using Equation 3.2 this translates into a 99.99%, 95.44%, 45.21% chance to find at least 1, 5, 10 Earth-like exoplanets around stars from our stellar sample. As we revealed in Section 4.7 the instrument’s faint source detection limits could be enhanced by increasing the observation time per target in order to detect more Earth twins (e.g. ∼ 47.5 Earth twins for ten times longer observations and therefore ten times better faint source low background detection limits with Darwin/MIRI). In the work of Seager et al.(2013) it is shown that life could be possible on planets with surface temperatures up to ∼ 400 K by modelling the atmospheric photochemistry and biomass. Considering this findings we move away from an Earth-focussed perspective and put a less stringent constraint on a potentially habitable exo-Earth (0.5 R⊕ ≤ Rp ≤ 1.25 R⊕ and 200 K ≤ Teq, p ≤ 400 K). We thereby find that Darwin/MIRI could detect up to ∼ 30 exoplanets which might harbour life, even in our baseline scenario.

78 Chapter 5

Conclusions

Our Monte-Carlo simulation predicts the expected number of directly detectable small and old exoplanets for VLT/NACO, E-ELT/METIS and for the first time Darwin/MIRI on the basis of planet occurrence statistics from NASA’s Kepler mission. We find total expected planet yields of 0.21 for +2.44 +115.9 VLT/NACO, 7.77−2.15 for E-ELT/METIS and 329.4−81.3 for Darwin/MIRI. The uncertainties are due to statistical errors in the underlying planet population. They are only presented for E-ELT/METIS and Darwin/MIRI since we interpret the tiny expected planet yield of VLT/NACO as not being suitable for directly detecting small and old exoplanets anyway. Exploring the capabilities of preliminary radial velocity measurements (for E-ELT/METIS) and longer total exposure times per target star (for Darwin/MIRI) we show that the exoplanet yield could be increased significantly by ∼ 30% (for E- ELT/METIS) and ∼ 50% (for Darwin/MIRI).

We model all exoplanets and their host stars as spherical blackbodies and take into account wavelength dependent absorption and reflection effects. Therefore we draw randomly distributed Bond and geometric albedos from a conservative distribution constrained by the albedos of the Solar System planets. We find that the expected planet yield depends strongly on the underlying planet population and the Bond albedos. This is shown by performing an error analysis by investigating the impact of the statistical errors in the underlying planet population predicted by Kepler (Burke et al., 2015; Dressing and Charbonneau, 2015; Fressin et al., 2013) and of the Poisson errors arising from our Monte-Carlo simulation. We find that the former could affect the expected planet yield by ∼ 30% for E-ELT/METIS and ∼ 35% for Darwin/MIRI while the latter are negligible. Compared to Crossfield(2013) and Quanz et al.(2015) we choose a very conservative and uniform distribution of Bond (and geometric) albedos in [0, 0.8) (in [0, 0.1)) translating into lower planet equilibrium temperatures and therefore less thermal emission (less reflected host star light) from the exoplanets. We further find

79 5. Conclusions

that the expected planet yield depends moderately on small variations in the instrument parameters. We investigate and quantify the instrument performance in detail in Subsection 3.1.3 for E-ELT/METIS and 3.2.3 for Dar- win/MIRI. The instrument parameters are adopted from works of Quanz et al.(2015) and Glasse et al.(2015) and can be regarded as reasonable but optimistic. Especially observations with VLT/NACO and E-ELT/METIS being background-limited at 2 λeff/D will be difficult to achieve considering the current limit of ∼ 5 λeff/D. Finally, we find that our results are quite robust with respect to the orbital eccentricities and the geometric albedos. Highly eccentric orbits for example would decrease the expected planet yield by ∼ 7% only (confer Section 4.3), assuming an instantaneous adaptation of the planets’ equilibrium temperatures with changes in the physical planet- host star separations. The impact of the choice of the geometric albedos is limited by the tiny fraction (less than 1%) of detectable exoplanets which can be seen only due to their reflected host star light.

Taking into account that the planet occurrence is suppressed around close- in binaries (Kraus et al., 2016) we note that the expected exoplanet yield of our simulated instruments should be increased compared to our predictions. The reason for this is that the KIC is chosen almost blindly with respect to stellar multiplicity while our stellar sample systematically excludes close-in binaries . Furthermore, correcting the properties of our M dwarf stars which yield underestimated planet equilibrium temperatures and therefore underestimated thermal emission from the exoplanets would also increase the expected exoplanet yield. Therefore, we regard our results as conservative. We ﬁnd that the contamination arising from zodiacal light is absorbed in our faint source detection limits already and that exozodiacal light can be cancelled out at least partially using advanced ADI and nulling interferometry data reduction techniques.

We conclude that VLT/NACO is not suitable for the direct detection of small terrestrial exoplanets whose equilibrium temperature is determined by the reprocessed host star light. E-ELT/METIS will be able to directly image a small sample of ∼ 10 exoplanets the majority of which will be smaller than Neptune (Rp ≤ 4 R⊕) and have equilibrium temperatures between 200 K and 700 K. However, the only instrument which will have the capabilities to discover a statistically valuable sample of (terrestrial) exoplanets in our immediate Solar neighbourhood with the direct imaging technique is a space interferometer akin to Darwin/MIRI. Its outstandingly high predicted exoplanet yield (329.4 exoplanets in total), having properties similar to these detectable with E-ELT/METIS, could even be improved (to 500.8 exoplanets in total) by increasing the total exposure time per target from 10000 s to 100000 s resulting in a reasonable mission duration of ∼ 1 year (excluding observational overheads).

80 Such a huge sample of directly imaged exoplanets would offer manifold opportunities. The atmospheric composition could be estimated from Dar- win’s multi-band observations and biosignatures could be detected on extrasolar planets for the ﬁrst time. Furthermore, the composition distribution of planets could be analyzed if both planet radius and planet mass were known from direct imaging and radial velocity measurements. Also, the transition regime from rocky planets to ice and gas giants could be explored. The formation of planets could be studied for such systems where a protoplanetary disk is present in addition to an exoplanet. Finally, a wealth of other quantities like the mean exozodiacal light intensity around stars in our immediate neighborhood and the distribution of Bond and geometric albedos among the sample of detectable exoplanets could be inferred. From our work we conclude that the Darwin mission would be of tremendous importance for exoplanet science in general.

Chapter 6

Acknowledgements

This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. This research has made use of the SIMBAD database, operated at CDS, Stras- bourg, France. I thank my supervisor SPQ for his great support and his huge efforts, the helpful team meetings, the many inspiring discussions and the fruitful co- operation. I thank my teammates AJB and GC for sharing their knowledge and their passion with me. I thank my professor HMS for organizing the exciting group meetings and for useful material about planetary albedos. I thank my family SAN, SRK and AK for mentally and ﬁnancially supporting me and my dreams. I thank my grandparents DMN and PHN for maintain- ing my work-life balance by travelling and doing sports together. I thank ISB for proofreading, helpful comments and her support in stressful times. I am glad that great people like you shaped a part of my life and therefore a part of me.

Appendix A

List of Units

85 A. List of Units

Table A.1: List of units used in this work.

Unit Symbol Value

Arcsecond 00 2.778 · 10−4 deg Astronomical unit AU 149597870700 m −23 2 −2 −1 Boltzmann constant kB 1.381 · 10 m kg s K Day d 8.640 · 104 s Degree deg - 24 Earth mass M⊕ 5.972 · 10 kg 6 Earth radius R⊕ 6.371 · 10 m Gravitational constant G 6.674 · 10−11 m3 kg−1 s−2 Jansky Jy 10−26 J m−2 Joule J - 27 Jupiter mass MJupiter 1.898 · 10 kg Kelvin K - Kilogram kg - Magnitude mag - Meter m - Milliarcsecond mas 1.000 · 10−300 Parsec pc 3.086 · 1016 m Planck constant h 6.626 · 10−34 m2 kg s−1 Radians rad - Second s -

Solar effective temperature Teff, 5772 K 30 Solar mass M 1.989 · 10 kg 8 Solar radius R 6.957 · 10 m Speed of light c 2.998 · 108 m s−1 Stefan-Boltzmann constant σ 5.670 · 10−8 W m−2 K−4 Watt W -

86 Appendix B

List of Parameters

87 B. List of Parameters

Table B.1: List of parameters used in this work.

Parameter Unit Description

AB - Bond albedo

Ag - Geometric albedo a AU Total semi-major axis

a∗ AU Stellar semi-major axis

acut AU Binary star semi-major axis cut-off

ap AU Planet semi-major axis

Cn - Normalization constant

CP - Normalization constant for planet orbital period

CR - Normalization constant for planet radius c - Stellar completeness D m Primary mirror diameter m Imaging baseline d pc Distance to observer E - Orbital plane e - Orbital eccentricity

F0 - Average number of planets per star Jy Filter zero point

Fbb µJy Blackbody ﬂux

Finc, ∗ µJy Incident stellar ﬂux

Finc, p µJy Incident planet ﬂux

Flim µJy Faint source detection limit

Flim, F560W µJy Faint source detection limit (MIRI F560W)

Flim, F1000W µJy Faint source detection limit (MIRI F1000W)

Flim, F1500W µJy Faint source detection limit (MIRI F1500W)

Flim, L’ µJy Faint source detection limit (NACO L’)

Flim, M’ µJy Faint source detection limit (NACO M’)

88 Flim, PAH2 µJy Faint source detection limit (VISIR PAH2)

Fobs, p µJy Planet observed ﬂux

Frefl, p µJy Planet reflected host star flux

Ftherm, p µJy Planet thermal blackbody ﬂux f (α) - Relative reﬂectance of Lambert sphere f<47 AU - Fraction of binary stars with a < 47 AU fbin - Fraction of binary stars ~h - Normal vector of planet’s orbital plane i rad Orbital inclination −1 K∗ m s Radial velocity semi-amplitude Kmag mag K-band magnitude

M∗ M Stellar mass

Mp M⊕ Planet mass

Nbin - Number of binary stars

Nsingle - Number of single stars O - Center of coordinate system

P0 d Normalization constant for planet orbital period

Porb d Planet orbital periods p mas parallax R m Radius

R0 R⊕ Normalization constant for planet radius

R∗ R Stellar radius

Rbrk R⊕ Power-law break for planet radius

Rp R⊕ Planet radius r - Planet occurrence rate rsingle - Planet occurrence rate around single stars

89 B. List of Parameters

r+1σ - 1σ upper limit of planet occurrence rate

rp AU Physical planet-host star separation

rp, proj AU Projected physical planet-host star separation S - Spectral type - Suppression factor for binary stars

Teff K Effective temperature

Teff, * K Stellar effective temperature

Teff, p K Planet effective temperature

Teq, p K Planet equilibrium temperature T(λ) - Filter transmission curve

Weff µm Filter effective width α rad Planet phase angle

α1 - Exponent for planet radius power-law

α2 - Exponent for planet radius power-law β - Exponent for planet orbital period power-law

∆r+ - Upper error of planet occurrence rate

∆r- - Lower error of planet occurrence rate ∆η - Error of expected number of detectable exoplanets

δVLT deg Geographic latitude of Cerro Paranal η - Expected number of detectable exoplanets

θ λeff/D Apparent angular separation

θdiff λeff/D Diffractive resolution limit

θlim λeff/D Resolution limit

θmax λeff/D Maximal apparent angular separation ϑ rad True anomaly λ µm Wavelength

λeff µm Filter effective wavelength

90 λmax µm Upper ﬁlter wavelength bound

λmin µm Lower ﬁlter wavelength bound −3 ρp kg m Planet density Ω rad Longitude of ascending node ω rad Argument of periapsis

Appendix C

PlanetS General Assembly Poster

For the NCCR PlanetS General Assembly1 which took place from 23-25 Jan- uary 2017 we designed a poster to promote the space interferometer Darwin for the search of exoplanets. Title: Darwin 2.0: Detecting & Characterising Hundreds of Exoplanets. Abstract: We present the first study to quantify the planet yield of a Darwin- like space telescope based on planet occurrence rates from Kepler. We simulate 2000 exoplanet systems around nearby main-sequence stars to show that such an instrument could detect > 300 planets. Our baseline scenario consists of a formation flying space telescope having specifications similar to Darwin (proposed to ESA in 2007) equipped with the sensitivity of the MIRI instrument of the James Webb Space Telescope.

1NCCR PlanetS, General Assembly in Grindelwald, 08.03.2017, http://nccr-planets.ch/general-assembly-grindelwald/

93 Darwin 2.0: Detecting & Characterising Hundreds of Exoplanets Jens Kammerer & Sascha P. Quanz

We present the first study to quantify the planet yield of a Darwin-like space telescope based on planet occurrence rates from Kepler. We simulate 2000 exoplanet systems around nearby main-sequence stars to show that such an instrument could detect >300 planets. Our baseline scenario consists of a formation flying space telescope having specifications similar to Darwin (proposed to ESA in 2007) equipped with the sensitivity of the MIRI instrument of the James Webb Space Telescope. Assumptions for our MC simulation 326 nearby main-sequence stars Stellar 1 500m imaging baseline Instrument 8 A stars, 54 F stars, 72 G stars, 71 K stars, 121 M stars sample 1 5mas @ 10μm resolution parameters Distance ≤ 20pc

1 4 x 2.82m diameter formation flying mirrors Culled for binaries closer than 5’’ Original MIRI filter curves Apparent brightness <7mag out to 10pc and <5mag out to 20pc Original MIRI faint source detection limits2 • 0.16μJy @ 5.6μm, 0.54μJy @ 10μm, 1.39μJy @ 15μm Planet occurrence rates from Kepler3 • 10σ in 10000s Planet • Planet radius in [0.5, 22] Rearth, orbital period in [0.5, 418] d Complete sky coverage1 (> 99%) population Randomly distributed circular planet orbits

Randomly drawn wavelength-independent planet albedos • Bond albedo distributed linearly in [0, 0.8] • Geometric albedo distributed linearly in [0, 0.1]

1 from Darwin mission proposal, Cockell et al. 2007 Planets & host stars are spherical blackbodies 2 from Glasse et al. 2015 3 from Burke et al. 2015, Dressing et al. 2015, Fressin et al. 2013 Planet effective temperature = planet equilibrium temperature 4 from Liège University Artist’s concept of Darwin4 How many planets can we detect?

Expected number of observable exoplanets in the radius-temperature plane using filters with λeff = 5.6μm (F560W, left), 10μm (F1000W, center) and 15μm (F1500W, right)

Relative amount (number in percent) of exoplanets which would be detectable via radial velocity (vrv ≥0.1m/s) for all planets with radii below 2 Earth radii (mean density = 5000kg/m³) Characterisation of detected planets Radial velocity

Stellar Expected number of Completeness type observable planets 383 rocky planets 171 rocky planets total per star detectable via RV RV + dir. imaging A 6.79 0.85 96% 3.22 around A stars 3.02 around A stars F 43.31 0.80 92% 22.53 around F stars 19.99 around F stars G 69.79 0.97 67% 55.13 around G stars 34.88 around G stars K 58.75 0.83 58% 58.36 around K stars 29.42 around K stars M 150.46 1.24 39% 243.68 around M stars 83.35 around M stars Expected number of observable exoplanets for each Expected number of exoplanets which are detectable in The blue histogram shows the radial velocity (RV) of all individual star (histogram) only one, two or all three filters simulated planets with radii R ≤ 2Rearth • The list shows the 15 host stars with the highest planet yield and • The numbers in the legend state the effective wavelengths of the • The vertical red line indicates our baseline RV sensitivity limit of their completeness (number in brackets) filters in μm vRV ≥ 0.1m/s Completeness = expected number of observable planets Roughly half of all detectable planets can be observed in all The green histogram shows the RV of only these planets divided by total number of simulated planets filters and therefore be analysed spectroscopically which are detectable via direct imaging, too

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