Svensk Sammanfattning Vetenskap om exoplaneter är väldig kompetitiv. Teleskopobservationer är svåra och dyra. Korrekt selektion av objekt och observationsschema är viktiga för att detektera biomarkörer utan att missa dem. Att planera och schemalägga är där- för lika viktiga som observationerna. För att kunna detektera biomarkörer i en exoplanets atmosfär observeras planeten när den passerar framför stjärnan den cirkulerar runt om. Teleskopen och dess placering ger riktlinjer till observation- sprocessen, så väl som planetens och stjärnans egenskaper. I den första delen har vi diskuterat litteratur om exoplanetvetenskap, speciellt om hur man kan bestämma atmosfärens sammansättning och modellering av planeternas atmosfärer. I den an- dra delen presenterar vi en sådan programvara, utvecklad för CRIRES+. Vi visar några användningsexempel och presenterar förslag av planeter som kan observeras över året. Vi jämför resultaten med ett annat liknande verktyg och diskuterar skillnaderna.

1 Abstract

In recent decades, thousands of exoplanets have been discovered. The next step is to characterize the observed planets in terms of their radii, masses, density, physical conditions and composition of their atmospheres. Several space-based observatories such as TESS and CHEOPS have started determining the first three observables but characterization of exoplanetary atmospheres is waiting for observation campaigns with instruments like CRIRES+ at the VLT and NIRSpec on the JWST. To ensure the ef- ficiency of data acquisition, careful planning of observations is necessary. In this project we developed a software tool to select and rank candidates based on the feasibility of observations of atmospheric features during transits with CRIRES+. We also review different techniques to retrieve transmission spectra from transit observations and modeling of exoplanet atmospheres in order to clarify the re- quirements for the data. Our CRIRES-planning-tool is built on astronomical observation planning methods from astropy and astroplan and the exposure time calculator designed for CRIRES+ by ESO and UU. We conclude that observations for atmospheric characterization with CRIRES+ are fea- sible. However, we observed that for a robust candidate selection, careful iterative tuning of proposed constraints is required.

2 Contents

1 Introduction 4

2 Theory and literature review5 2.1 Exoplanet transit...... 5 2.1.1 Photometric lightcurve and transmission spectroscopy...... 10 2.2 Atmospheric retrieval methods for transmission spectroscopy...... 13 2.2.1 A novel planetary model-independent transmission spectrum recovery method.. 13 2.2.2 Cross correlation methods to retrieve the transmission spectrum of an exoplane- tary atmosphere...... 16 2.2.3 Optimal estimation and bayesian inference...... 17 2.3 Planetary atmospheric models...... 20 2.3.1 Parametric forward models...... 23 2.3.2 Scattering, albedos and the two-stream approximation...... 28 2.3.3 Self-consistent one-dimensional planetary models...... 36 2.3.4 General circulation models and three dimensional radiative hydrodynamics mod- els, detailed cloud and dust physics...... 44

3 Summary and conclusion of the literature review 51

4 Description and results of the CRIRES-planning-tool 54 4.1 Functionalities and methods of CRIRES-planning-tool...... 55 4.2 Constraints for candidates...... 59 4.3 Signal-to-noise ratio and the exposure time calculator...... 61 4.4 Input parameters to the ETC, restrictions and future improvements...... 62 4.5 Test and comparison of results...... 63 4.6 Results of detectable transits for one year...... 64

5 Discussion 69

6 Conclusion 70

3 1 Introduction

Today, the atmospheric composition of the planets in our Solar System has been roughly determined (de Pater and Lissauer, 2010). Besides, discoveries of exoplanets have revealed that the architecture of our Solar System is not typical at all (Wright et al., 2011). Therefore, we may expect very different planetary compositions, and atmospheric properties than can be found in our Solar System. On the other side, investigating their compositions will bring us closer to answer one of the most interesting questions about our own existence. Are we alone in the universe? However, we shall see, before we are able to answer this question we still have a long way to go. The next step will be to characterize planetary atmospheres and learn about the habitability of these remote and different worlds. To shed light onto the question of atmospheric composition of exoplanets, the CRyogenic high-resolution InfraRed Echelle Spectrograph (CRIRES)(Kaeufl et al., 2004) was developed. An upgraded version of CRIRES, now called CRIRES+, is mounted at Paranal observatory on the very large telescope, (VLT), unit telescope 3 (UT3). To plan observations with CRIRES+ we developed a software tool called (CRIRES-planning-tool). Although it was developed to plan transit observations, it may easily be extended to other kinds of observations with CRIRES+. With the upgrade CRIRES+ has now enhanced observation efficiency, increased wavelength coverage, an upgrade of the focal plane detector array to increase the number of accessible diffraction orders, new wavelength calibration methods to reach wavelength precision of 5 m/s and a new spectropolarimetric unit (Follert et al., 2014). Previous scientific highlights with CRIRES were measuring the length of an exoplanet day for the first time (Snellen et al., 2014), creating the first weather map for the nearest brown dwarf to Earth (Crossfield et al., 2014) or finding the first superstorm on an exoplanet, HD209458 b and measuring its mass (Snellen et al., 2010a) to name but a few.

To plan transit observations of exoplanets we used the python libraries astropy,(Price-Whelan et al., 2018) and astroplan,(Morris et al., 2018), which provide useful methods to assess times and coordi- nates for observability at observatories all over the world. To calculate the signal-to-noise ratio (SNR) for an observation we use the Exposure Time Calculator (ETC), developed by the European Southern Observatory (ESO) in Garching, Munich, Germany together with Uppsala University (UU)1.

CRIRES-planning-tool is written in python and selects candidates from the NASA Exoplanet Archive (Akeson et al., 2013) to assess which candidates can be observed when, what signal-to-noise ratio can be reached for CRIRES+, and ranks the planets after criteria deduced from these properties. The output of the software tool is used for decision making, which targets should be observed and when, and provides an important basis to plan for best efficiency in time and data quality. This manuscript provides an overview of the structure of the software tool and describes and discusses some of the used methods. The software comes with an extensive documentation. The README file provides information about the different functionalities, accessibility, structure, installation, and further development possibilities of the CRIRES-planning-tool. The tool is available on GitHub2.

The thesis contains two major parts. The first part (chapter2) gives an overview of the present state of exoplanetary astrophysical techniques, both in observation and theory. In section 2.1 we introduce the reader to the technique of transit photometry, highlighting its strengths and limitations, and derive the basic equations for transmission spectroscopy. In section 2.2 we present two important methods for transmission spectra retrieval, and two techniques to compare retrieved spectra with synthesized spectra

1https://etctestpub.eso.org/observing/etc/crires 2https://github.com/jonaszubindu/CRIRES-planning-tool

4 from planetary models. In section 2.3 we provide an extensive overview about the different physics treated by a planetary model, such as radiative transfer, thermodynamics, convection, chemistry, clouds, and hydrodynamics and present examples to the different kinds of planetary models developed to study exoplanetary atmospheres. At the end of this part we will give a short summary and conclusion of the reviewed topics.

In the second part (chapter4), we shall provide the reader with a brief introduction to the CRIRES-planning-tool. In sections 4.1, and 4.2, we explain the functionalities and used methods to select candidates for obser- vations and assess their observability from the Paranal observatory, in section 4.3 we give an overview about the ETC and the computation of signal-to-noise ratio, and in section 4.4 explain each parameter used for the ETC. In section 4.5 we compare our tool with two other transit observation planning tools, highlighting differences between these tools and verifying our own findings. Finally in 4.6 we present preliminary results of possible transit observations over the course of one year. A discussion of the CRIRES-planning-tool can be found in chapter5, and a short conclusion of the findings is given in6.

2 Theory and literature review

One idea to determine the composition of an exoplanet’s atmosphere is to retrieve the transmission spectrum of the star light passing through the atmosphere and investigate the spectrum for known molecular transitions. Since planetary atmospheres consist of molecules, and in case of hot Jupiters, atoms, one aims at observing only molecular bands. Subtracting the stellar spectrum, and if the observatory is based on Earth subtracting the telluric spectrum, the exoplanet’s transmission spectra should reveal its atmospheric composition and allow the deduction of the prevailing conditions at the surface of the planet. However, one is faced with several challenges. We will present these challenges in the following chapter one by one, and solutions or methods to circumnavigate these issues and retrieve as much information as possible about the composition and structure of an exoplanet’s atmosphere. We will discuss these solutions and point out where more research or improvement of the mentioned techniques is necessary.

2.1 Exoplanet transit Transits of exoplanets in front of different stars have been observed thousands of times by now and are increasing by the month. The technique is based on the photometric observation of a star and investigation of its lightcurve for a periodically occurring dip in the received brightness or flux. The number of exoplanets that can be observed this way is limited by the inclination of their orbit with respect to the observer. Assuming a planet of mass Mp and radius Rp transiting in front of its host star with M? and R? for stellar mass and radius the orbit can be projected onto the plane of the observer. Any point of the orbit of the exoplanet has a projected distance rsky to the barycenter of the orbit. The distance r of a planet from the barycenter is given by

a(1 − e2) r = , (1) 1 + e cos f and can be projected onto the plane of the observer, such that the projected distance rsky takes the form q 2 2 rsky = r 1 − sin (ω + f) · sin i . (2)

5 a is the semi-major axis of the planet’s orbit, e the eccentricity, f the true anomaly, i the orbital inclination with respect to the projected plane of the observer, and ω is the argument of periapse. The term under the square root represents the projection of the orbit onto the√ observer’s plane. The 2 2 projected distance expressed in Cartesian coordinates takes the form rsky = X + Y , where the X axis is pointing in the direction of the descending node of the planet’s orbit. Following the conventions of Winn(2010), X can be inferred directly from orbital mechanics and the projection term in rsky:

X = −r · cos (ω + f) (3)

π Applying X = 0 we get the condition for transit mid-time ftra = + 2 − ω. Similarly the condition for π occultation of the planet by the star is given by focc = − 2 − ω. From a photometric measurement of a transit and occultation one can constrain the parameter space for ω and the eccentricity e. If possible, the transit observations can be combined with the radial velocity method to measure e directly and in this way also infer ω. With the exception of a and the inclination i, the parameter space of an orbit is covered. Winn(2010) also deduce the probability of an observable transiting planet from the penumbra and antumbra cones of a transiting planet:

R ± R  1 + e sin ω  p = ? p . (4) tra a 1 − e2

The penumbra is represented here by the + sign and includes grazing orbits (not full planetary disk occulting the star) and − represents fully enclosed transits. The probability ptra to observe a transit is therefore determined by the orbital parameters a, e, and ω. Using population studies for e (Xie et al., 2016) and integrating out ω, we can estimate how many exoplanets we should be able to observe. For our purpose we do not want to have an estimate of how many exoplanets might be observable in the future (Brakensiek and Ragozzine, 2016), however, we would like to calculate the expectancy to find another Earth-twin around a Sun-like star, i. e. a G-dwarf. Plugging in the same values as for Earth in (4), we can compute the expected number of systems we must observe to find an Earth-sized planet at 1 AU around a G-dwarf. With the fraction η of systems with an Earth like planet we must observe N > 200 · η−1 systems to find a transiting Earth-twin. Using estimations for the relative abundance of G-dwarfs (Mamajek, 2016), the distribution of semi-major axis a around G-dwarfs (Han et al., 2014 (accessed July 20, 2020) (see Figure1), and the distribution of eccentricities e (Xie et al., 2016) of all exoplanets, we estimate η:

η = fa × fG × fe ≈ 0.00054, (5) 27 where fa = 1497 stands for the fraction of exoplanets found at 1 AU restricted to G-dwarf host stars, fG ≈ 0.06 the fraction of G-dwarf stars in our galactic neighbourhood and fe ≈ 0.5 the fraction of planets with eccentricity e ≈ 0. Plugging in the numbers from above we get N ≈ 3700000. This means on average out of 3700000 stars, 1 star is a G-dwarf with a transiting planet the size of Earth at 1 AU and a low eccentricity e below 0.1. Filtering the exoplanets archive (Han et al., 2014 (accessed July 20, 2020) for transiting planets around G-dwarfs with 0.8 < a < 1.2 and e < 0.1, we have not found any planets fulfilling the given constraints except 6 planets with undetermined eccentricity. Expecting to find an observable Earth twin in the near future might be ambitious to unrealistic. We will see later also that the required signal-to-noise ratio (S/N) to observe a Earth-twin around a G-dwarf is way beyond present detection limits. Concluding from this, we are forced to look for extraterrestrial life or hospitable environments for life elsewhere than what we are used to.

6 exoplanets.org | 7/20/2020

200

150

100 Distribution

50

0 0.01 0.1 1 10

Semi-Major Axis [Astronomical Units (AU)]

Figure 1: Distribution of the semi-major axis of exoplanets around G-dwarfs, pro- duced with (Han et al., 2014 (accessed July 20, 2020).

Measuring the lightcurve and observing the transit of an exoplanet we can infer two other important parameters of an eclipsing system, namely the period and the transit duration. Both are important to predict transit observability. We denote the period with P and the transit duration with Ttot. This should not be confused with the full transit duration Tfull, which stands for the time the complete planetary disk is occulting the star. The time difference from Ttot to the beginning of Tfull and vice versa is called the ingress and egress time during a transit (see Figure2). Using these properties we can predict future transits of an exoplanet in question:

tc(n) = tc(0) + P n, (6) where tc is the time of conjunction or for our purposes the transit mid-time. The uncertainty in tc(n) is mostly sensitive to the uncertainty in P. Using Gaussian error propagation we get

p 2 2 2 ∆tc(n) = ∆tc(0) + n ∆P ≈ n∆P. (7) Other errors that may influence the predictions and planning of transit observations are influences from other bodies in the observed system, which are hard to constrain. However, provide an excellent way of discovering the presence of other unobserved objects. If we also would like to observe occultations we can get another estimate for the eccentricity e and the observer’s celestial longitude ω from the time difference between transit and occultation ∆Tc P  4  ∆T ≈ 1 + e cos ω . (8) c 2 π

7 Additionally to observe exoplanets from an observatory on Earth, we must include the barycentric correction, which accounts for the light travel time from the observed system to the barycenter of the Solar System compared to the travel time to the observer. In the following expression UTC stands for universal time coordinated and if not mentioned otherwise, we always refer to UTC. BJD stands for the barycentric julian date and JD for the julian date. ~r is the position of the observer with respect to the barycenter of the Solar System, nˆ the unit vector from the observer to the observed system and c is the speed of light. Then the barycentric correction for objects outside of our Solar System can be written as (Eastman et al., 2010) ~r · nˆ BJD ' JD + . (9) UTC UTC c

The correction term is interchangeable and we can compute JDUTC , to compute our predicted transit times. This method was used to convert the transit observation times tc from the barycentric frame to the observers frame, the Paranal observatory.

If one measures the radial velocity of the host star and thereby knows the eccentricity e, the velocity semi-amplitude K? and the period P , one can estimate the mass of the planet Mp (Murray and Correia, 2010; Winn, 2010): √ M K 1 − e2  P 1/3 p = ? . (10) 2/3 (Mp + M?) sin i 2πG G is Newton’s gravitational constant. Using the transit method together with the radial velocity method the sin i degeneracy is approximately broken since in case of a transit sin i ≈ 1. However, normally 2/3 we are faced with the situation that Mp  M? and we can only measure Mp/M? . To measure the mass of the planet the mass of the star must be determined by other means, such as spectral type, luminosity, and spectroscopically determined properties, such as effective temperature Teff , surface gravity g?, and metallicity [Fe/H]. The surface gravity gp at the surface of the planet can be derived by using the expression for the radial velocity semi amplitude of the star K? (Murray and Correia, 2010)

Mp 2πa/P sin i Mp 2πa/P sin i K? = √ ≈ √ (11) Mp + M? 1 − e2 M? 1 − e2 and Kepler’s third law. We obtain the surface gravity in terms of Rp/a, P, K?, and i: √ 2 Mp 2π 1 − e K? gp = G 2 = 2 . (12) Rp P (Rp/a) sin i

Measuring the photometric lightcurve during a transit we can obtain the depth of the lightcurve δtra, which provides us with further information about the observed planet star system, such as the ratio between the radius of the planet and the star k = Rp/R?:   2 Ip(ttra) δtra ≈ k 1 − . (13) I?

In case of negligible nightside emission Ip = 0 compared to the disk averaged stellar intensity I?, δtra 2 can be set equal to k , and the stellar radius R? can be computed from stellar models or approximately −1/3 by R? ∝ M? . Thereby one can also solve the degeneracy between Rp and a, since by using Kepler’s third law again we can compute a using M?. Notice that without making assumptions about the stellar properties of the host star, the presented treatment to obtain Rp is always degenerate with either R?, M?, and a, or sin i.

8 Winn(2010) present another way to approximate the stellar radius and the orbital inclination from the shape of the transit lightcurve. However, in a real observation of a lightcurve we are faced with limb darkening, which means the blocked fraction of the stellar flux decreases at the edges of the stellar disk, resulting in an increase of the received flux. To obtain information about the shape of the lightcurve such that one may determine transit, ingress and egress time, one needs to correct the lightcurves for limb darkening.

Knutson et al.(2007) propose a way to fit the lightcurves in different bandpasses simoultanously for the shape of a limb darkening corrected lightcurve, using the non-linear limb darkening law presented by Mandel and Agol(2002): 4 X n/2 Iˆ(r) = 1 − cn(1 − µ ) (14) n=1 Iˆ(r) is the specific intensity at position r on the stellar disk, where Iˆ(0) = 1 and 0 ≤ r ≤ 1, and 0 stands for the center of the√ stellar disk, while 1 stands for the outer edge of the stellar disc. r is related to µ through µ = cos θ = 1 − r2 , where θ represents the angle between the line of sight and and the line of emergence of the flux, standing perpendicular on the stellar surface. cn are the four coefficients to fit the limb darkening from stellar models or from photometric transit lightcurves. A more frequently encountered limb darkening law is the quadratic limb darkening with c1 = c3 = 0, c2 = γ1 + 2γ2 and c4 = −γ2, and the new fitting coefficients γ1, 2. The choice of using the quadratic over the full treatment is due to higher efficiency and less complexity in fitting lightcurves. Mandel and Agol(2002) present the full treatment of possible transit scenarios and the following lightcurve geometry with application of limb darkening. Knutson et al.(2007) apply the non-linear limb darkening for small planets k ≤ 0.1 which returns a set of equations to fit the limb darkening coefficients simultaneously with the decoupled parameters k and i. The calculations are lengthy and complicated and we forgo presenting them here and only show the expressions derived by Winn(2010). Another technique developed to fit the specific intensity dependent on limb darkening was proposed by Aronson and Piskunov(2018) and basically follows a similar idea as described by Mandel and Agol(2002).

The impact parameter b is depicted in Figure2, and is fundamentally related to the geometry of the orbit of the transiting planet: a cos i  1 − e2  btra = , (15) R? 1 + e sin ω where the + in the denominator must be replaced with − for occultations bocc. If we can retrieve the limb darkening corrected shape of the lightcurve, we can decouple R?/a from b using the time of ingress and egress (Ttot − Tfull) and computing the scaled stellar radius R?/a for non-grazing orbits Rp  R?  a and small eccentricities in terms of Ttot and Tfull:

√ 2 2 √ 2 2 (1 − δtra ) − (Tfull/Ttot) (1 + δtra ) b = 2 (16) 1 − (Tfull/Ttot) q 2 2 R π Ttot − Tfull 1 + e sin ω  ? = √ (17) a 1/4 P 2 2δtra 1 − e

These expressions can be derived by computing Ttot and Tfull, also depicted in Figure2, in terms of the 2 shape of the lightcurve and the orbit and inverting for b and R?/a. Note that we have now independent expressions to compute i and R?/a. Still, to lift the degeneracy between Rp, R?, M? and a we need

9 Figure 2: Illustration of the different quantities introduced to describe the lightcurve of a transiting planet in front of its host star. T represents Tfull, Ttot would cover the times from tI until tIV , tI−II is the ingress time, and tIII−IV the egress time. b is called the impact parameter of the transit and is closely related to the orbital parameters of the transiting planet (Winn, 2010) . some input parameters for the star, either from fitted observational data, with stellar models, or from other theoretical assumptions for the star. Using photometric lightcurves together with radial velocity observations enables us to constrain the parameter space of the planet’s orbit, make good estimates for the radius of the planet Rp and the surface gravity gp and can be used as inputs for planetary climate and atmospheric models. However, the observed planetary radius Rp from a lightcurve is not well defined since cloudy or hazy atmospheres can alter the fraction of blocked stellar flux to total flux and defining the surface of a gas planet might be ambiguous. Nevertheless, the different values for Rp we may get by observing the lightcurve at different wavelengths is at the base of the idea of transmission spectroscopy of exoplanetary atmospheres, which is further elaborated in the next section.

2.1.1 Photometric lightcurve and transmission spectroscopy Using spectroscopy to probe transmission spectra of a planetary atmosphere needs some assumptions first. We define the radius of the planet Rp as the opaque part of the planet in all wavelengths, deduced from a photometric lightcurve as derived above. This means that the planetary radius Rp may be set at the top of an opaque cloud layer or at the optical depth τ ∼ 1, integrated over all wavelengths.

10 The photometric lightcurve, observed at frequency ν is defined by the fraction of the received flux to the stellar flux:

 2 k αtra(t), transits Fν(t) Ip(t)  f(t)ν ≡ = 1 + − 0, outside eclipse (18) Fν?(t) I? k2α (t) Iνp(t) , occultations occ Iν? where we have used the definition for the monochromatic flux (Pradhan and Nahar, 2011)

Z Z 1 Fν = Iν cos θdΩ = 2π Iνµdµ (19) ∂Σ −1

2 with Iν = Ip(t)+I? the received intensity in [ergs/(cm s str Hz)]. Here, the polar angle θ stands for the angle under which the light was emitted from the surface dA on the star. The monochromatic flux Fν 2 2 is given in [ergs/(cm s Hz)] or [W/(m s Hz)]. αtra and αocc describe the geometry of the area blocking the star light during a transit or the area adding to emitted light from the planet during occultations and out of transit. Mandel and Agol(2002) give the full analytic expressions for all possible cases that might be encountered doing transit observations. The different cases are derived from geometrical considerations.

Measuring the radius of the planet Rp at a certain wavelength might return a different radius than obtained from photometry. We rewrite the new radius as Rν = Rp +δRν depending on the wavelength, respectively the frequency ν. Then the part adding to the new occulting planetary disk can be described 2 by an annulus 2πRpδRν + πδRν with thickness δRν. The thickness of the annulus δRν can be related to the pressure scale height of the atmosphere observed at wavelength ν via the optical depth τν at ν. The optical depth is the integrated opacity along a path C weighted with the density ρ: Z τν = κν(s)ρ(s)ds. (20) C

The opacity can be related to the atomic absorption coefficient αν: κν = ανn, where n is the relative number density of a certain species. The expression for the absorption coefficient α is often seen to describe a single absorption line, whereas the opacity κ is used in general for radiative transfer with its continuous contributions from absorption and emission as well. Depending on the geometry and the physics we are investigating, κν or αν can be dependent on the integration path C. We will now show how the thickness of the annulus δRν can be related to the absorption at ν. For more details, see section 2.3.3 and (Gray, 2005).

Assume we have an atmopshere consisting of a single species. For a grazing light ray traveling through the annulus of the semi-transparent atmosphere at an impact parameter xν, we can compute the tangential optical depth τν according to Webb and Wormleaton(2001) and Brown(2001). The geometry is depcited in Figure3. In this derivation the geometry does not take into account any effects of refraction.

The tangential optical depth in this geometry is given by Z ∞ (r + Rp)dr τν(x) = 2 κνµn(r) . (21) p 2 2 x (r + Rp) − (x + Rp)

11 Figure 3: Geometry for a grazing light ray traveling through the semi-transparent atmosphere of a planet at impact parameter x, maximum atmospheric height H and planetary radius Rp (Webb and Wormleaton, 2001) . with the molecular mass µ of the species in question. In case of hydrostatic equilibrium we can use a barometric law for the number density of the species with height r: k T n(r) = n e−r/h, h = B , (22) 0 µg where the pressure scale height h is dependent on the surface gravity g, under the assumption of H  Rp, such that g does not change much with altitude, the temperature T of the atmosphere, and kB is the Boltzmann factor. This may be generalized to non-isothermal atmospheres with different temperature dependencies in different layers and non-homogeneous processes such as strong irradiation effects on one side of the planet in case of tidally locked planets. Assuming the absorption coefficient to be relatively constant along a tangential line of sight, the integral above can be evaluated using the modified Bessel function K1((x + Rp)/h) (Abramowitz and Stegun, 1965): (x + R ) τ (x) = 2n κ (x + R )eRp/hK p (23) ν 0 ν p 1 h In the absence of absorption lines we can estimate the overall pressure scale height h to one order of magnitude accuracy (Benneke and Seager, 2012; Miller-Ricci et al., 2008) by measuring the impact parameter x at two different wavelengths under the assumption τ(x1,2) = τ(δR1,2) = 1. Equating both expressions and solving for h, we get x1 − x2 h = q , (24) ln α1 x1 α2 x2 where κ1 and κ2 are the opacity or scattering cross section at the two different wavelengths λ1 and λ2 respectively.

12 To investigate spectral features, we need to relate the measured impact parameter x to the decrease of the stellar flux by the semi-transparent atmosphere: δF?,ν. The specific photometric light curve fν (18) in terms of the blocked stellar flux δF?,ν can be expressed as:

δF?,ν fν = 1 − , (25) F?,ν where the second term is given by (Brown, 2001) Z ∞ δF?,ν 1 0 0 0 = 2 2π(Rp + x )[1 − exp (−τν(x ))]dx , (26) F?,ν πR? 0 and one integrates over all impact parameters. τ is the integral along every line of sight at each impact parameter. The factor 2π arises from integrating around the whole annulus visible by the observer, neglecting effects of rotation of the planet, the star or wind in the atmosphere of the planet. Following the treatment of Brown(2001) and the method to measure the transmission spectrum of an exoplanetary atmosphere by Aronson and Waldén(2015), we may equate the expression for δFν with ia · P , where the latter describes the spectrum of the semi-transparent atmosphere expressed in terms of the measured impact parameter xν: Z ∞ 2 2 0 0 0 ip&a(·2πRpxν + xν + Rp) = 2π(Rp + x )Iν(µ(φ))[1 − exp (−τν(x ))]dx (27) 0 Z ip&a(φ) = Iν(µ(φ))dµ(φ) (28) Ap&a(φ) where ip&a denotes the specific intensity from the star covered by the planet and its atmosphere. For I(µ(φ)) one can use a quadratic Limb darkening law or spectral models. The dependency of µ on the orbital parameter φ can be looked up in Figure4. It describes the relation between the position of the planet on its orbit around the star to the observer’s view of the position of the planet. What we have not treated in our derivation of δF/F is the influence of the refractive index and the night side emission of the planet. Brown(2001) point out that the path of a light ray through the atmosphere is not perfectly straight. However to compute the complete obscurred area by the atmosphere of the planet they explain that the minimum height of a tangential light ray not altered by refraction is relevant. They do not discuss night side emission, and Aronson and Waldén(2015) do not discuss the influence of refraction or night side emission and neglect it. Using the previous arguments we may now present their novel retrieval method.

2.2 Atmospheric retrieval methods for transmission spectroscopy 2.2.1 A novel planetary model-independent transmission spectrum recovery method Aronson and Waldén(2015) describe a method to recover transmission spectra of exoplanetary atmo- spheres in the near-infrared from Earth based observatories with high-resolution spectrometry. The method should be applied on data from CRIRES+ in the future. The main idea is that even in presence of strong telluric absorption CRIRES+ will be able to measure in between the telluric lines and observe features shifting with time due to a time dependent Doppler shift, while the planet moves across the surface of the star. The full signal received by a high-resolution spectrograph can be expressed as received flux per exposure normalized to the exposure time:  2  ˜ π(Rp + xν) Sn(ν, t) = F (ν, t, vs) − ip&a(φ, vs) 2 · f(vs, vp) · T (ν, t), (29) πR?

13 with F (ν, t) the stellar flux and T (ν, t) the telluric transmission and f(vp, vs) a Doppler correction factor for the movement of the planet with respect to the star, the rotation of the star, the movement of the Earth, and the rotation of the Earth. vs and vp are the velocities of the star and the planet relative to Earth depending on their present position during the transit. They are dependent also on the orbital parameters φ at t and the shape of the orbit (a, ω, e, i). The geometry is depicted in Figure4. Additional Doppler shifts can be introduced by planetary winds across the terminator (the only part of the atmosphere observable with transmission spectroscopy) and the rotation of the planet, however, a reasonable assumption is that the planets are tidally locked and so the influence of the planets rotation is small. On the other hand, horizontal winds arise from the strong stellar irradiation onto a tidally locked planet and dissipate some of the energy to the night side. Snellen et al.(2010b) used CRIRES, the predecessor of CRIRES+, to measure the high altitude winds of the gas giant HD 209458 b with transmission spectroscopy and a cross correlation technique that we will present in the next section 2.2.2. They showed that these winds can reach speeds of several 100 ms−1, close to the sound speed of the transported gases and can therefore alter the spectrum through line broadening and overall wavelength shifts significantly. To obtain the complete received signal one needs to convolve the spectrum with the instrument profile Γ. It can be approximated by a Gaussian with full-width at half maximum equal to the spectral resolution of the spectrograph and a noise function for each pixel. The 1 noise function follows a Poisson distribution with a width of λ = S/N , where S/N is the signal-to-noise ratio. The signal is then finally multiplied with a continuum normalization η. Applying these factors we get h i Sn,syn(ν, t) = S˜n(ν, t) ⊗ Γ · (1 − N(ν, φ)) · η. (30)

The continuum normalization is necessary due to the limited wavelength range of high-precision spectrographs and generally to correct for any in-

� strumental changes in the signal over time. Con- � � ventionally the continuum is normalized via fit- ting of the recovered spectrum or removing all broadband signal contributions and computing the relative line strengths to the total stellar flux (de Kok et al., 2013). Aronson and Waldén (2015) use sensitivity curves, derived from mea- suring the Solar spectrum before and after the science exposure to compare with a theoretical Solar spectrum and normalize the measured spec-

�&(�) tra accordingly. Additionally they present a tech- nique to correct the measured spectra with a brightness correction parameter ν(t), to control Figure 4: In this figure we depict the geometry the continuum contribution to the retrieved spec- of an orbit in relation to the transit observation. trum. Computing this parameter gives not much The dependency of µ on φ is dependent on the insight into the physics at play and we forego its shape of the orbit given by the orbital param- presentation. To compute the impact parame- eters (a, e, i, ω) and the orientation of the orbit ter xν from Sn, which gives rise to the recovered observed from Earth. spectrum,

14 Aronson and Waldén(2015) formulate the recovery of the spectrum as a minimization problem, similar to a least square minimization between synthetic and measured signal, respectively spectra:

X 2 Φν = (ν(t) · Sn(ν, t) − Sn,syn(ν, t)) = min, (31) n where Sn,syn(ν, t) denotes the synthetic spectra according to (30). Notice, in Sn the parameter xν is free, while all the other contributions must be constrained prior to retrieving xν. The minimization problem is then rewritten in the form

Ω = Φ + Λ · R = min (32) !2 X dP R = (33) dν ν δRν · f(vs, vp) P = 2 (34) πR? where P denotes the part of the absorption by the atmosphere of the planet normalized to the projected stellar surface. The parameter Λ is used to balance the relative importance of the regularization term R, which balances the smoothness of the solution with the precision of the fit. By imposing the condition of all derivatives vanishing at the minimum and linearizing Ω in R, we get a system of equations for xν that is only dependent on the synthetic input spectra F (ν, t, vs), Iν(µ(φ), vs) and T (ν, t) of the star and the Earth. The flux spectrum for F (ν, t, vs) is generated using the stellar atmosphere model code MARCS (Gustafsson, B. et al., 2008) with the stellar parameters effective temperature Teff , surface gravity log g? and metallicity [F e/H] and is corrected with out of transit observations. The telluric absorption spectrum T (ν, t) is generated from the synthetic atmosphere spectrum algorithm to model transmission spectra and radiance, LinePak (Gordley et al., 1994). However, Aronson and Waldén (2015) recommend for the future to use tools for synthesizing high-precision telluric transmission spectra fitted to observation. Iν(µ(φ), vs) can be generated the same way as F or by using limb darkening equations, as mentioned earlier. However, Aronson and Waldén(2015) report higher accuracy using stellar models, although Aronson and Piskunov(2018) and Czesla et al.(2015) argue that using stellar models to compute the limb darkened specific intensity fails at reproducing center to limb variations even for the Sun. Aronson and Waldén(2015) propose to find the optimal value for Λ by comparing simulated exoplanet spectra to the synthetic one, generating a simulated spectra with the known parameters of the exoplanet as the one observed. The sky emission received by the telescope can either be included through models or can be corrected for by nodding (Chromey, 2010). To test their method, Aronson and Waldén(2015) generated synthetic planetary atmospheres with LinePak and used the transmission spectra to use their recovery method on. In this way, they could directly compare the recovered spectrum with the synthetic spectrum. They tested three types of exoplanetary atmospheres: super- Earth with Earth-like atmosphere, super-Earth with Venus-like atmosphere, both orbiting M-dwarfs and a hot Jupiter orbiting a Sun-like star. The transmission spectrum of the Venus-like atmosphere could be recovered within one transit of 75 min, the Earth-like atmosphere could be recovered by combining 4 transits of each 90 min and the hot Jupiter case could be recovered in a single transit. However, recent analysis including instrumental stability and instrumental drift from tests of CRIRES+ revealed that for the Earth like case about 10-15 transits, for the Venus-like case about 5-10 transits, and for the hot Jupiter case about 5 transits will be necessary, (private communication N. Piskunov). To test their method under different circumstances Aronson and Waldén(2015) introduced systematic errors, such as over- or under estimation of the telluric spectrum up to 2%, or variability in specific intensity by

15 inserting a Teff 5% too high. They also comment on stellar activity through star spots, the influence of clouds, Rayleigh scattering, or no atmosphere. Constant over- or underestimation of the telluric transmittance leads to complete loss of the transmission spectrum. Starspots and stellar flux/intensity variability increases the uncertainty in the received spectrum. Young active M-dwarfs with significant number of starspots are not well understood and may make spectrum recovery challenging, if not impossible. Tests with Solar like stars and including spectra of starspots showed the possibility of complete recovery of the transmission spectrum. However, they point out that star spots or stellar activity altering the flux of the observed star significantly could not be properly simulated by stellar atmospheric models and might lead to loss of the transmission spectral features. No atmosphere, a very thin atmosphere, or a Rayleigh scattering dominated atmosphere could not be distinguished from each other. Measuring the Rayleigh scattering slope using ground based spectroscopy turns out to be difficult to achieve, due to Rayleigh scattering in Earth’s atmosphere in the visual band. Also errors in planetary size can lead to over or underestimation of the atmospheric height, however, this can be corrected for by setting the continuum to zero, using the technique to compute ν(t) mentioned before. Finally they present the application on data from Snellen et al.(2010b) and showed the possibility to retrieve the same CO lines as Snellen et al.(2010b) have. To reach the specified recovery performance, a certain signal-to-noise ratio S/N is required. The requirements can be found in (Aronson and Waldén, 2015) and are applied in our planning tool to determine if the signal from a transit is strong enough for atmospheric spectra retrieval. The minimum S/N requirement can be different for other methods, such as the cross correlation method, described in section 2.2.2. What is important to notice with this method, is that it recovers the transmission spectrum prior to assuming any knowlegde about the atmospheric composition or structure of the exoplanet, however, at the cost of more observation time to obtain higher S/N and higher sensitivity to data quality, resolution and wavelength coverage than other methods.

2.2.2 Cross correlation methods to retrieve the transmission spectrum of an exoplanetary atmosphere The cross correlation between two continuous functions f and g is defined as: Z ∞ (f ? g)(τ) ≡ f(t)g(t + τ)dt, (35) −∞ where f(t) is the complex conjugate of f(t). In our case we can denote the signal fν(t) as the received signal Sn, adjusted for all the instrument and continuum contributions and gν(t) as the synthetic signal Sn,syn, generated from an atmospheric modeling code for transmission spectroscopy, such as LinePak (Gordley et al., 1994). What the cross correlation then measures is the similarity between Sn and ∆ν ∆v Sn,syn, as a function of velocity shift from the transit midpoint: ν ∼ v . This technique has been widely used for previous studies in transmission spectroscopy (Brogi et al., 2013, 2012; de Kok et al., 2013; Snellen et al., 2010b). Following the procedure used by de Kok et al.(2013), first the telluric and stellar contributions need to be removed. To align the spectra on a common wavelength grid, they corrected the spectra with a second order polynomial describing a wavelength shift for each pixel, obtained by the best cross correlated synthetic telluric and stellar spectra with the measured spectra. Next, they propose to use single value decomposition (Kalman, 1996). The single value decomposition of any matrix A containing all the spectra of a single detector is given by:

A = UWVT. (36)

16 The matrix U contains all the right singular vectors, which are the eigenvectors of A, V are the left singular vectors, which represent the eigenvectors scaled by the singular values of A. The matrix W is then a diagonal matrix containing the singular values of A. The main idea in applying singular value decomposition normally is that A can be approximated by the highest singular values of W. However, here the procedure is turned around and one is only interested in the lower singular values, while the higher singular values, containing the major contribution to the measured spectra, are set to zero. The major contribution comes from the telluric absorption lines and to some degree from the host star. The key to retrieve the planetary transmission spectra relies again on the fact that the shift of the absorption lines from the planetary atmosphere due to the change in radial velocity is greater than the shift in the telluric and stellar lines. Thus, the time varying information of A can be isolated to compare with model spectra by using cross correlation to obtain the best fit from a model grid. The model spectra used by de Kok et al.(2013) were generated with the parametric forward model from (Madhusudhan and Seager, 2009) (see chapter 2.3). Madhusudhan and Seager(2009) report constraints on the shape of the temperature-pressure profile (P-T) of HD 189733 b and the relative abundances of the major chemical constituents as well. However, they had not been able to successfully constrain the presence or abundance of CO. de Kok et al.(2013) report the presence of CO in the atmosphere of HD 189733 b at a 5σ significance. Still, due to the degeneracy of pressure, temperature and abundance, the abundance of CO could not be constrained from the line cores. Although constraints on P-T profiles exist from (Madhusudhan and Seager, 2009) and others, de Kok et al.(2013) report that there is still an ongoing discussion about the exact P-T profiles. Previous low-resolution observations of HD 189733 b have not provided strong enough constraints on P-T profiles at low pressure, which is where the observed spectral line cores are most sensitive. Nevertheless, they mention that with a wider wavelength coverage, observing other species such as H2O and CH4, and by measuring the H2 induced Rayleigh scattering, it would be possible to infer more exact P-T profiles and determine the absolute abundances of gas species. Benneke and Seager(2012) for instance present a retrieval method with their parametric forward model to distinguish hydrogen H2 dominated atmospheres from water vapour H2O dominated atmospheres and to constrain absolute abundance of the gas species assumed to dominate exoplanetary atmospheres. Brogi et al.(2012) use a similar method as de Kok et al.(2013) to prove the presence of CO in the atmosphere of τ Boötis b at a 6σ confidence level, and Brogi et al.(2013) find absorbtion by CO and water in the atmosphere of 51 Pegasi b at a 5.9σ confidence level. All the above mentioned observations applying the cross correlation method were conducted at the VLT with CRIRES (Kaeufl et al., 2004).

2.2.3 Optimal estimation and bayesian inference Traditionally for atmospheric retrieval a spectrum of parametric forward models, such as (Madhusudhan and Seager, 2009) would be generated with up to ∼ 107 models, (∼ 107 points in a ten-dimensional parameter space) and compared via least square fit or cross correlation, to find the model closest to the recovered spectrum. Madhusudhan and Seager(2009)’s model includes 6 parameters for the pressure- temperature profile P-T and 4 parameters for the chemical composition. Another model from Guillot (2010) used by Benneke and Seager(2012) uses one parameter less for the P-T profile. The models are both described in detail in chapter 2.3. On the other side, self-consistent planetary atmospheric models have emerged to understand the physics of the Solar System bodies’ atmospheres, but also, with the progress in high-resolution spectroscopy, to be used for exoplanetary atmosphere transmission spectra retrieval. One main difference between a parametric forward model and a self-consistent model is the following: The parametric model assumes a certain general P-T profile that can be adjusted

17 by parameters and does not necessarily require complete physical consistency. On the other hand self-consistent models give raise to their P-T profile naturally from the modelled physical processes. A standard method to compare recovered spectra to self-consistent models is the method of optimal Estimation, which is described next. Optimal estimation (OE) is suitable to apply in high-resolution spectroscopy with a wide wavelength coverage and high signal-to-noise ratio. High-resolution spectroscopy allows to identify single features in the transmission spectrum and compare to detailed self-consistent model atmospheres. Optimal estima- tion optimizes the likelihood function with least square minimization using the Levenberg-Marquardt algorithm. Shulyak, D. et al.(2019) developed their code for emission spectroscopy. Nevertheless, the method holds also for transmission spectroscopy and they present the OE method to be used with data from CRIRES+ and the τ-REx software package (Waldmann et al., 2015). (As it happens, τ-REx is also equipped with Bayesian inference methods in form of Markov-Chain Monte Carlo MCMC and nested sampling), The OE parameter fitting is used to obtain the global minimum of the cost function 2 φ containing the χ sum and the deviation from the initial guess state vector xi, which contains the parameters of the model fitted to observation (Rodgers, 2004). The deviation of a measured signal y to the synthetic signal yi = F (xi) at iteration i generated from the parameters in xi, and the deviation of xi from an initial guess xa are assumed to be Gaussian and follow a normal distribution with the co-variance matrices Sy and Sa:   1 1 T −1 P (y|x) = exp − (y − yi) Sy (y − yi) (37) n/2 1/2 (2π) mod Sy 2   1 1 T −1 P (x) = exp − (xi − xa) Sa (xi − xa) . (38) n/2 1/2 (2π) mod Sa 2 The OE method is then based on Bayes’ theorem and the above assumptions. Bayes’ theorem is also the key to Bayesian inference and therefore we quickly recall it here: P (x|y)P (y) P (y|x) = . (39) P (x) We can rewrite this expression in the form

− 2 ln P (y|x) = −2 ln P (x|y) − 2 ln P (y) − (−2 ln P (x)) (40) and using P(y|x), P(x) we can define the cost function φ

T −1 T −1 φ = (y − yi) Sy (y − yi) + (xi − xa) Sa (xi − xa). (41) P (y) is fixed by the measured signal y and can therefore simply be treated as a normalization constant. The co-variance matrix Sa is generally given by the expectation value E of the difference of xi from the initial guess xa: mn m m n n Sa = E((xi − xa )(xi − xa )). (42) For the application in planetary atmospheric forward models Shulyak, D. et al.(2019) assume the off-diagonal parameters to be correlated by the correlation length lcorr which depicts the number of layers between layer m and n and pm, pn represent the pressure in those layers

mn p mm nn −| ln (pm/pn)| Sa = Sa Sa exp ( ). (43) lcorr

18 Sy contains the measurement errors and the estimated errors for the parameters in x. The final errors after convergence can be calculated from the requirement of lim ∇xi φ = 0, xˆ is the vector containing xi→xˆ the parameters at the minimum of the cost function:

T −1 −1 lim −Ki Sy (y − yi) + Sa (xi − xa) = 0. (44) xi→xˆ

Ki = ∇xi F (xi) is the Jacobian of the forward model at iteration step xi and arises from acting with ∇ on (y − yi). Expressing yi as

yi ≈ F (xa) + ∇xi F (xi)(xi − xa), (45) where xa is assumed to be close to the minimum, (44) then becomes

T −1 T −1 T −1 T −1 −1 −1 lim −Ki Sy y + Ki Sy F (xa) + Ki Sy Kixi − Ki Sy Kixa + Sa xi − Sa xa = 0. (46) xi→xˆ

Collecting all the terms quadratic in xi in (41), all other terms remain constant varying xi and we can compare (46) with the alternative form of lim ∇xi φ = 0, xˆ: xi→xˆ

T ˆ−1 ˆ−1 ˆ−1 lim ∇xi φ = lim ∇xi (xi − xˆ) S (xi − xˆ) = lim (S xi − S xˆ) = 0, (47) xi→xˆ xi→xˆ xi→xˆ and thereby identify the co-variance matrix of the final result xˆ at the global minimum of φ as Sˆ, containing the co-variance of the model parameters, the measurement errors and the pre-assumed errors of the forward model: ˆ −1 T −1 −1 S = (Sa + Ki Sy Ki) . (48)

The construction of the iteration step from xi to xi+1 can be deduced from setting (44)=(47) and solving for xˆ, and is given by:

 −1 T −1 −1  T −1 −1  xi+1 = xi + (1 + γi)Sa + Ki Sy Ki Ki Sy (y − yi) − Sa (xi − xa) . (49)

γ is a fine-tuning parameter controlling the balance between the measurement and the initial guess. Notice that in case of xi+1 = xi =x ˆ and γ = 0 the first term in brackets becomes Sˆ and the second term becomes zero due to the minimization condition of the cost function φ. Although it might seam like a well-motivated assumption that the posterior probability distribution P (y|x) are following a normal distribution, Benneke and Seager(2012) criticise this assumption. Using a parametric forward model from (Guillot, 2010), they present the method of Bayesian inference, described next.

The method of Bayesian inference itself is also based on Bayes’theorem (39) as is the optimal Estimation method and combined with a suitable parameter iteration method investigates the full parameter distribution around the minimum of the likelihood function

2 L = L0 exp (−χn/2), (50) N n 2 X (yj − F (xj )) χ2 = , (51) n σ2 j=1 j where σj denotes the standard deviation of the j-th measured datapoint yj. Looking at the different terms in (39), we have P (y|x) = L the likelihood function, the probability of the measured data y under n n a set of parameters x = {xi }, P (x|y) the posterior probablility distribution of the model parameters

19 xn at iteration step n under the data set y. P (x) = Π is the prior parameter probability distribution and P (y) = Z is called the evidence, independent of n. Z is the likelihood of the data marginalized over the parameter space and is normally treated as a normalization factor for the employed model. It can be used to compare different models with each other (Madhusudhan, 2018). The evidence Z can be rewritten as the integral over the likelihood L and the prior parameter probability distribtution Π: Z Z P (y) = P (y|xn)P (xn)dN xn = LnΠndN xn = Z. (52)

This integral may be challenging to compute depending on the number of parameters N and the parameter range. The posterior probability distribution can be calculated nevertheless by combining the posterior probability distribution of the iteration step n + 1 and the step n, such that

P (xn+1|y) P (y|xn+1)P (xn+1) = , (53) P (xn|y) P (y|xn)P (xn) and we can eliminate P (y). The next step n + 1 is randomly chosen using for instance a Gaussian distribution around the current step n with a "jump-length" σ. The new step is accepted in case of the probability p is greater than a random number m between 0 and 1:

nP (xn+1|y) o p = min , 1 > m, m ∈ Uniform[0, 1]. (54) P (xn|y)

This is called the Markov Chain Monte Carlo (MCMC) algorithm, under the Metropolis-Hasting algo- rithm to generate the iteration steps xn. The method depends highly on the "jumping-length" σ and must be chosen such that the MCMC actually probes the parameter space towards higher values of the Likelihood function with increasing n. On the other hand σ needs to be chosen big enough such that a reasonable convergence speed can be reached (∼ 105 evaluations according to Benneke and Seager (2012)). To omit sampling of only one local minimum while there exist several, Benneke and Seager (2012) use the method of parallel tempering (Gregory, 2005). By replacing P (y|xn) with a tempered distribution profile P (y|xn)β and 0 < β < 1. Thereby, one can flatten the likelihood function and probe a much wider parameter space. Since the tempered profile leads to a higher acceptance of steps xn+1, the parameter space gets explored on a wider scale. β must be chosen as reasonalbe ladder βk increasing towards β = 1, which represents the original likelihood function. Modifying (54) to

nP (xn+1|y, βk)P (xn|y, βk+1) o p = min , 1 > m, m ∈ Uniform[0, 1]. (55) P (xn|y, βk)P (xn+1|y, βk+1) enables the cross influence of iteration step acceptance and the β-ladder, also called the "temperatures", to run in parallel. The behaviour of this modified MCMC algorithm depends strongly on the choice of the βk’s and must be chosen by probing. Additionally one can run several simulations with different initial parameter distributions x0. A proof why the Metropolis-Hastings works can be found in (Gregory, 2005).

2.3 Planetary atmospheric models One might assume that a good starting point in the age of digitalization and super computers would be to investigate planetary models of exotic unknown worlds by developing simulations containing as much and accurate physics as possible. Thereby covering all the aspects of 3D hydrodynamics,

20 radiative transfer, cloud physics, photochemistry, non-equilibrium chemistry, atmospheric escape, and many other processes, and calculating transmission spectra thereof. Following from that, the stellar input as energy input into the atmosphere as well as the light source to create transmission spectra would have to be as accurately and precisely modelled as the atmosphere of the transiting exoplanet itself. The reality shows that such models, at least for now, are not efficient to answer present scientific questions. Additionally, the instruments and telescopes available or under development could not satisfy the same need of accuracy in data retrieval to actually justify the effort in producing exclusively such models. Obviously, for this reason a great variety of ideas how to model exoplanetary atmospheres and how to retrieve transmission spectra have emerged and are specialized on certain science cases. One needs to be very careful in choosing which model to use to interpret their data. Naturally, that has lead to conflicts between retrieval methods and models, applied to the same data sets. See (de Kok et al.(2013) Discussion) and (Sotzen et al.(2019) Conclusion, and literature therein).

Historically, the first planetary atmospheric models were adapted from atmospheric models of brown dwarfs (Burrows et al., 1997) or irradiated stellar binary atmospheric models (Seager and Sasselov, 1998). Those were used to investigate close in extraSolar giant planets, highly irradiated by their host stars. The atmospheric models were based on the assumptions of local thermodynamic equilibrium (LTE) and convection was treated using the method of the mixing length theory. The radiative transfer equation is solved using Feautrier method or two-stream approximation (Gandhi and Madhusudhan, 2017; Heng et al., 2011a, 2014; Mihalas, 1978). Seager and Sasselov(1998) used an equation of state consisting of the elements up to two ionization stages H, He, C, N, O, Ne, Na, Mg, Al, Si, S, K, Ca, Cr, Fe, Ni, and Ti; the ions H2, H; and the molecules H2,H2O, OH, CH, CO, CN, C2,N2,O2, NO, NH, C2H2, HCN, C2H, HS, SiH, C3H, C3, CS, SiC, SiC2, NS, SiN, SiO, SO, S2, SiS, and TiO. Equation of states describe the thermodynamic balance of the different constituents in the atmosphere and can be encountered in two ways. The type of E. o. S. we might be more familiar with, is the mechanical equation of state (Mihalas, 1978):

P = P (ρ, T, chemical composition), (56) and accounts for the relation between pressure, temperature and density, whereas density can be related to the sum of the densities of the different constituents. The other type important in astrophysical simulations, especially allowing hydrodynamical motion, is the caloric equation of state:

e = e(ρ, T, chemical composition), (57) and accounts for the connection between the internal energy of the gas to the temperature and again the chemical constituents, for instance through the specific heat coefficients. In most cases for the equation of state we simply use the ideal gas law. However, this might not hold for instance in non-LTE cases or in case of adiabatic or dissipative heating or cooling and we need the caloric equation of state as well. Nevertheless, we will see that the ideal gas law is almost always valid for the internal processes in a planetary atmosphere.

Seager and Sasselov(1998) also included bound-free and free-free atomic transition opacities, Thomson scattering, and Rayleigh scattering by H2 and H, and straight means opacities of H2O and TiO. The important aspects of the model were the treatment of the incident flux, the angular dependence of the incident flux and the subsequent emergent flux, and therein solving the radiative transfer equation. Instead of total flux constancy they impose conservation of entropy, since total flux constancy is broken

21 locally. The incident flux is treated in the full manner of radiative transfer, computing the heat contribution from bound-bound, bound-free, and free-free transitions in different layers. Scattering does not add to the heat budget, however changes the radiation field towards a black body spectrum. The entropy conservation describes the redistribution of heat in the atmosphere. A parameter f describing a qualitative redistribution is defined such that f = 1 complete redistribution of heat and f = 2, night and day side are thermodynamically completely disconnected. Additionally, they incorporate the influence of dust as an extra parameter, without generally regarding processes of dust grain formation. The influence of scattering through dust is treated using Mie theory, to compute scattering opacities averaged over all angles. However, Seager and Sasselov(1998) do not treat cloud opacities, cloud formation, or winds in exoplanetary atmospheres. After Seager et al.(2000) found high dependency of the pressure temperature profile (P-T) and the emergent flux from the cloud formation depth, and the importance of the alkali metal Na I and K I resonance doublets and the He I 23S- 23P triplet, Seager and Sasselov(2000) update the code from Seager and Sasselov(1998) with Gibbs free energy minimization to calculate solids and gases in chemical equilibrium and condensate opacities for three solid species. This gives rise to cloud formation with cloud top and cloud base of different condensation species.

Complementary, others proposed other ways of modelling exoplanetary atmospheres, and focus on certain aspects differently. Brown(2001) for instance created a non-strict self-consistent model to describe the terminator region of an exoplanetary atmosphere. Thereby, restricting himself in the number of chemical species or the calculation of He opacities. However, he includes photoionization of the alkali metals Na and K, the dynamics of emergent horizontal winds due to the one sided heating at different incident angles, and differential rotation of the planet. Another model referred to before called LinePak, developed by Gordley et al.(1994) treats radiative transfer under similar assumptions as Seager and Sasselov(1998), however does not treat scattering, clouds, or dust. The atmospheric model can be generated in a customized way with different layers with different mass functions and accounts for refraction effects cell by cell of light rays passing through. At the same time Fortney et al. (2008, 2010) and references therein present their way of adopting and testing models for brown dwarfs as proxys for highly irradiated hot Jupiters. Gandhi and Madhusudhan(2017) review some of the work done until 2017. A table extracted from their work on the new self-consistent planetary model GENESIS (see section 2.3.3) is presented below:

22 Figure 5: The table is copied from Gandhi and Madhusudhan(2017) and presents the previous self consistent models and their major differences considering the treatment of convection, clouds and scattering. Some of the models presented here we have also covered in our text.

In this chapter we will give an outline of the different types of atmospheric models and present the most important physical inputs and assumptions these models are based on.

2.3.1 Parametric forward models Madhusudhan and Seager(2009) present a parametric forward model based on a simple atmospheric three-layer model with holistic assumptions for the general structure of the P-T profile, and chemical composition. The model is generated with 10 parameters in a computationally fast and simple manner, and has for instance been applied by de Kok et al.(2013). These kind of models are called parametric forward models, since they are generated based on parameters rather than physics. The method was designed for hydrogen-dominated hot Jupiter atmospheres. Each layer is described through a pressure- temperature relation approximated by

β α(T −T0) P = P0e (58) with the free parameters P0,T0, α and β for each layer. Layer number three is assumed to be convective and has a constant temperature T3. Eliminating two more parameters at the boundary of layer 2-3 and 1-2, 6 parameters remain to describe the P-T profile. The approach of a three-layer parametric P-T profile is motivated by studies of the atmosphere of Solar System bodies and self-consistent planetary models, in particular one described in Seager et al.(2005).

23 Figure 6: General temperature and pressure profile presented by Madhusudhan and Seager(2009) that can be adjusted by parameters for each layer. Layer 3 is isothermal. Notice that the profile allows temperature inversions to occur. Al- though it is a completely parameter dependent model, Madhusudhan and Seager (2009) enforce energy balance at the top of the atmosphere and discard any pa- rameter combinations violating this requirement.

The radiative transfer through the atmosphere is solved using a line-by-line code under the assumption of LTE, no scattering, and the requirement of energy balance at the top of the atmosphere. The density is given by the ideal gas law from the P-T profile and the radial dependence is determined by hydrostatic equilibrium. Stellar heating is incorporated into the model through the P-T profile, which in turn is indirectly dependent on the incident, absorbed, and scattered radiation. An important point to mention is that the emergent transmission spectrum in the infrared is decoupled from the opacities in the visual, responsible for some of the heating. Species in planetary atmospheres can have strong opacity contributions in the different wavelength regimes. Therefore, a parametric P-T profile can account for the overall heating, treating the underlying physics as a black box. On top of that the partial independence between the P-T profile and atmospheric composition allows for probing P-T profiles of atmospheres with loosely known chemical composition and does not need to be described self- consistently with the constituents in chemical equilibrium. This is called the dual-band approximation (Guillot, 2010; Heng et al., 2011a), which we will explain more later.

Molecules included in the model are H2,H2O, CO, CO2, CH4, and NH3, and it also incorporate H2 -H2 collision-induced absorption opacities. CO2 and NH3 are chosen to have a fiducial arbitrary concentration of Solar abundance, since NH3 has only minor absorption features in the bands considered by Madhusudhan and Seager(2009).

24 The concentrations and radiative transfer is calculated in 100 layers uniformely spaced in log P . The molecular abundances are controlled by the ratio between the concentration and a fiducial equilibrium i i concentration, equal to Solar abundances, of a molecule x in layer i: fx = cx/cx,eq. Although the con- centration of each molecule is different for each layer, dividing by the fiducial equilibrium concentration they get a constant abundance parameter fx. The abundance parameter for the main constituents H2O, CO, CH4, and CO2 are free parameters of the model. Thus, the parametric forward model of Madhusudhan and Seager(2009) comes with a total of 10 parameters, 6 parameters for the P-T profile and 4 parameters for the abundance of the main constituents of the atmosphere. Clouds are not treated in this version, however, were planned to be included in future versions. Since Madhusudhan and Seager(2009) focused on the study of hot Jupiters and previous literature suggested them to be cloud free, they had no need to include treatment of clouds initially. This assumption was shown to be invalid in more recent transmission observations in 2013, described in Barstow et al.(2014) and literature therein.

The energy balance is controlled via the incoming wavelength-integrated flux from the star F? and the emitted flux from the top of the atmosphere Fp. It is scaled via the Bond albedo AB, describing the ratio of the incident radiation flux reflected back into space, and a redistribution factor fr accounting for the redistribution of heat to the night side:

Fp = (1 − AB)(1 − fr)F?, (59) and the actual energy balance is constrained by η = (1 − AB)(1 − fr) ≤ 1. To use the model, they propose to calculate a goodness-of-fit dependent on the difference between the model calculated flux and the observed flux at each wavelength. The goodness-of-fit is then computed for each model on the parameter grid of about 107 grid points. Guillot(2010) followed a different approach compared to Madhusudhan and Seager(2009) and derived a T − τ model using the dual band approximation under the assumption of constant opacities in two wavelength bands. The radiation contributions are treated again as independent distinct bands of incoming shortwave or visual and outgoing longwave or thermal radiation. The radiative transfer equation to solve is (Heng et al., 2011a) ∂I κ I κ (1 − ξ)J µ ν = ν ν − κ B − ν ν . (60) ∂m ξ ν ν ξ The parameter ξ carries information about scattering, which is not treated by Guillot(2010). ξ will be taken up again in section 2.3.2. Using the three moments of the specific intensity (Guillot, 2010) Z 1 1 2 (Jν,Hν,Kν) ≡ Iµν(1, µ, µ )dµ (61) 2 −1 the radiative transfer equation (60) can be recast in the form: dH ν = κ (J − B ) (62) dm ν ν ν dK ν = κ H , (63) dm ν ν where m is the column mass and carries the information about the height profile z, through dm = ρ(z)dz. κν is the opacity and Jν is the mean intensity over all solid angles, 4πHν = Fν can be identified

25 with the flux, 4π/cKν represents the radiation pressure, and Bν is the Planck function or Black-body radiation. The upper boundary condition is given by

Hv(0) = −µ?Jv(0), (64) for the irradiation from the star. µ? = cos θ?, where θ? denotes the angle of the incoming radiation relative to the direction perpendicular to the atmosphere, and Hν is evaluated at m = 0. Guillot(2010) introduces the parameter γ ≡ κv , which is the ratio between the mean opacity in the visual range and κth the mean opacity of the infrared or thermal radiation. The mean opacities are defined as the average taken over the mean intensity Jν in the respective band: Z −1 κv = Jv κνJνdν (65) visual Z −1 κth = Jth κνJνdν. (66) thermal

However, we can approximate κv by using the Planck function Bν for the mean intensity Jν at the temperature Tirr, which is the temperature of the radiation field from the star, received at the top of the atmosphere. Then the upper boundary condition becomes µ H (0) = −µ J = − ? σT 4 (67) v ? v 4π irr The irradiation temperature can be computed by relating the stellar flux with the received stellar flux at the substellar point µ? = 0. Using the Stefan-Boltzmann law Z ∞ 4 2 4 L? σT? R? σTirr = 4π Hν(0)dν = 2 = 2 (68) 0 4πa a R 1/2 ⇒ T = T ? , (69) irr ? a where a is the average distance between the star and the planet and σ the Stefan-Boltzmann constant. Tirr can be related to the equilibrium temperature for complete redistribution of energy around the planet: σT 4 πR2 T R 1/2 4 irr p √irr ? σTeq = 2 ⇒ Teq = = T? . (70) 4πRp 2 2a

More accurately Teq would be dependent on the Bond albedo AB, carrying the information about the efficiency of the radiative redistribution of energy around the planet. κth is averaged the same way as in the visual band, but using the equilibrium temperature of the planet Teq for the Planck function Bν. Thereby, the mean opacities can be calculated from first principles and do not need to be iterated by computing Jν. Generally, the calculation of the equilibrium temperature is also dependent on the reflected energy back into space and we can not assume the complete uniform redestribution throughout the entire atmosphere. The Bond albedo and its relation to scattering properties in the atmosphere is treated later in section 2.3.2 as well. The argument for the approximation of the mean opacities from above is that the contribution in the visual is dominant at low τ  1 (optically thin), where the visual radiation field is dominated by the incoming stellar radiation, controlled by Tirr, and absorption effects are sparse. In the region of high absorption and high τ  1 the radiation is thermalized and dominated by the outgoing thermal

26 radiation, which can be approximated by Bν at Teq. This of course tailors the crude assumption of Bν ∼ 0 for ν in the visual, at high τ  1. Vice-versa the longwave stellar contribution to the radiation field must be negligible, which might not hold for instance for M-dwarfs. Similarly the moments of the specific intensity are integrated over the desired wavelength range in the visual v and thermal th regime. In the visual regime, following Eddington (Eddington, 1916) µ? can be approximated by µ2 = Kv , which is µ2 = 1 in case of an isotropic incoming radiation field. Thus, the radiative transfer ? Jv ? 3 equation in the visual becomes 2 2 d (Hv,Jv) κv 2 = 2 (Hv,Jv) (71) dm µ? and Bv, the thermal emission of the atmosphere in the visual at m can be neglected. A general solution for the visual band is then

(J(m)v,Hv(m)) = (Jv(0),Hv(0)) exp (−κvm/µ?). (72)

The optical depth τ is defined as the thermal opacity κth multiplied with m

τ = κthm. (73)

Guillot(2010) uses the first and second Eddington coefficients

Kth Hth(0) fKth = , fHth = , (74) Jth Jth(0) where Hth(0) means that Hth is evaluated at the outer boundary τ = 0 of the atmosphere. The set of equations for the thermal regime is then given by dH th = κ (J − B) (75) dm th th dK th = κ H (76) dm th th κth(Jth − B) + κvJv = 0 (77) R with B = thermal Bνdν. We can see through the radiative equilibrium condition (77) how the thermal κv and the visual band connect. The parameters of the Guillot(2010) profile are Tint,Teq, γ ≡ , fK , κth th and fHth . The equations (75), (76), and (77) can be combined and integrated in m and using the appropriate boundary conditions, such as Z ∞ Hνdν = H, (78) 0

Hth(0) = H + µ?Jv(0), (79) and (64), to express the source function B in terms of H,Hv(0), fHth , fKth , γ, µ?, and τ:       1 τ 1 µ? γ µ? B = H + − Hv(0) + + − exp (−γτ/µ?) . (80) fHth fKth fHth γfKth µ? γfKth

4 H can be identified with H = σTint, with Tint the temperature associated with the deeper layers, and internal heating and radiative equilibrium matches the sum of the flux in the visual and the thermal band. Hv(0) is expressed by the irradiation temperature Tirr according to (67). The equilibrium

27 temperature Teq can be calculated according to the assumptions that go into equation (70), and thus

Tirr and Teq are interchangeable. Guillot(2010) chose fKth = 1/3 due to the assumption of an isotropic radiation field even at low opacity, and fHth = 1/2 due to the same reasoning, but only integrating over half an arc in µ. Relating B to the Stefan-Boltzmann law, one can solve for T 4 and compute the horizontally averaged temperature profile T (Guillot, 2010):

3T 4 n2 o 3T 4 n2 2 h γτ  i 2γ  τ 2  o T 4 = int + τ + eq + 1 + − 1 e−γτ + 1 − E (γτ) . (81) 4 3 4 3 3γ 2 3 2 2

R ∞ −n E2(γτ) denotes the second exponential integral En(z) ≡ 1 t exp (−zt)dt. Through averaging over the whole planetary surface, heat flow through advection H q∇T dω = 0 is averaged out and the tem- perature profile becomes independent of q∇T in case of uniform advection.

2.3.2 Scattering, albedos and the two-stream approximation Heng et al.(2011a, 2014) generalized the parametric P-T profile of Guillot(2010) to include both isotropic scattering and anisotropic scattering, collision-induced absorption, and longwave absorption by an additional absorber, such as a cloud deck or haze layer. Notice that we treat only monochromatic or coherent scattering, namely there is no change in wavelength of the scattered photon during scattering. To include the influence of isotropic coherent scattering Heng et al.(2011a) introduce the scattering parameter ξ. To see the physical meaning of ξ, we first need to investigate its relation to the different definitions of albedos, which shall shortly be recast here. ξ can be related to the single scattering albedo ω0 given by Heng(2017) σν ω0,ν = (82) κν + σν with σν the scattering coefficient. The spherical albedo is given by

1 − β0 As = (83) 1 + β0 ω0 β0 = , (84) 1 − ω0g0 with the scattering asymmetry factor g0. The Bond albedo is the fraction of scattered radiation to the incident radiation on a planet. We can say the spherical albedo is the monochromatic version of the Bond albedo. In Heng(2017) the Bond albedo is described as the intensity normalized and weighted integral of the spherical albedo, R ∞ 0 AsIνdν AB = R ∞ , (85) 0 Iνdν however, one can also find the Bond albedo defined as simply the integral of the spherical albedo (Seager, 2010) over all possible frequencies ν.

ξ in the Heng et al.(2011a)-approximation is related to the Bond albedo such that √ 1 − ξ A (ξ) = √ (86) B 1 + ξ and therefore relates implicitly to the scattering coefficient σν. In practice one includes tabulated values for ω0 and g0 for different types of particles or calculates ω0 from the scattering and absorption cross

28 sections of the present elements. In case of isotropic scattering we have simply g0 = 0 and the scattering phase function is P(Ω) = 1/4π. The scattering phase function P(Ω) describes the ratio of incident radiation scattered into the solid angle Ω. g0 can either be used as an input parameter for anisotropic scattering to compute P(Ω) or can be computed from P(Ω), if the scattering phase function is known: Z 00 00 g0 ≡ µ PdΩ , (87) where µ = cos θ00, θ00 = (θ − θ0) depicts the difference between the scattered and the incident angle of a photon at a particle in the atmosphere. The scattering phase function and the asymmetry factor are the key inputs, when treating anisotropic scattering for instance in the two-stream approximation, applied by Heng et al.(2014). Depending on particle size and wavelength regime, different forms for P(Ω) need to be included. Figure7 gives an overview of scattering regimes dependent on wavelength and particle size (William H. Brune, 2019 (accessed February 3, 2014). Notice that these regimes apply to an atmosphere similar to Earth’s atmosphere and shall only give an overview about the different scattering regimes that might need to be taken into account.

Figure 7: Different regimes of scattering dependent on particle size and wavelength (William H. Brune, 2019 (accessed February 3, 2014). The particle size is compared to examples of type of particles on the right side. The parameter X depicts the ratio between the wavelength and the particle size multiplied with 2π.

29 An often encountered form of P is the Henyey-Greenstein scattering phase function 1 − g2 P = 0 , (88) 2 00 3/2 4π(1 + g0 − 2g0 cos θ ) dependent only on the scattering asymmetry factor g0. g0 is also an important proxy for the radiative energy redistribution over the planet in the absence of winds and can be included in flux distribution correction factors. One such factor appears in the relation between the Bond albedo and the equilibrium temperature Teq of an irradiated planet (Seager, 2010):

R 1/2 f 1/4 T = T ? (1 − A ) . (89) eq ? a 4 B

We can relate the scattering asymmetry factor g0 to f in the following way. Below, on the left side we have the ratio of the observable monochromatic flux scattered from the planet to the incident flux from the star and how they relate to the spherical albedo As and the scattering g0. On the right side we see the frequency integrated flux, only that this time, Teq is related to the absorbed and reemitted flux, instead of the scattered flux.  2 4  2   g0 Fν a Teq a f As(ν) = and 4 = (1 − AB) . (90) 4 Fν,? R? T? R? 4 Now to close this set of equations we can compare the spherical albedo, and the Bond albedo with the geometric albedo. Comparing (89) and (90) we can identify that f in the case of absolute flux plays a similar role as g0 in the monochromatic case and thus are relateable. The geometric albedo Ag stands for the ratio of the scattered radiation flux at phase angle zero to the incident stellar flux at the substellar point. The spherical albedo can thus be interpreted as the geometric albedo integrated over one hemisphere. Though, a more accurate expression is the spherical and the geometric albedo relate to each other through the phase function P(Ω) integrated over all possible scattering angles and all phase angles α: 2 Z π As = ψα(ν) sin αdα (91) π 0 Z π Z π 2 2 0 0 0 0 03 0 0 ψα(ν) = P(θ, φ, θ , φ ) cos (α − φ ) cos φ cos θ dθ dφ . (92) π π α− 2 − 2

ψα describes the scattered incident radiation into the phase angle α under which an observer would see the planet. Ag can then be expressed as Ag ≡ ψα=0(ν)π at phase angle α = 0. Likewise using g0 g 0 A (ν) = A (ν), (93) 4 s g and so we get the complete set of equations to calculate the different albedos dependent on intrinsic scattering properties of the planet’s surface and atmosphere and relate them to the energy budget of incident, scattered and reemitted radiation. Notice that these arguments only hold in absence of advective or convective heat redistribution. Now to get back to the treatment of scattering, Heng et al.(2011a) included the scattering parameter ξ in the approximation of the optical depth in the visual regime to scale the opacity in the visual: Z m κv 0 τv = dm (94) 0 ξ

30 and κv remains constant. For the thermal regime Heng et al.(2011a) argue that the assumption of a constant opacity breaks down at high pressures and introduce the approximative treatment of collision- induced absorption. They give κth a functional form, dependent on the column mass m and a correction factor , as a new parameter to the parametric P-T profile of Guillot(2010). The collision-induced absorption becomes important when P & P0/(−1) and mainly occurs between molecular species. Heng et al.(2011a) use the following functional form for κth to account for collision-induced absorption:

  m  κth(m) = κ0 1 + 2( − 1) . (95) m0

R m 0 0 Evaluating the integral τ = 0 κth(m )dm and expressing m by the pressure P using hydrostatic equilibrium, provides the optical depth in the thermal regime as a function of pressure:

 P  P  τth(P ) = τ0 + ( − 1) . (96) P0 P0 Additionally, they propose the treatment of an additional absorber, such as a cloud, or haze layer by introducing an ad-hoc opacity κc: " #  P 2 κc = κc0 exp −∆c 1 − (97) Pc and thereby introduce three new parameters: the opacity normalization κc0, the deck-thickness ∆c and the location of the deck in terms of pressure Pc. κc gives rise to the additional optical depth τc, which is added to the thermal optical depth τth. To bring the optical depth in the visual and the optical depth in the thermal regime together, Heng et al.(2011a) estimate the photon deposition depth as the √ −1 depth ξ τv = $√from the top of the atmosphere where the average visual flux inwards drops to e , which occurs at ξ τv ≈ 0.63. The deposition depth can be related to pressure and column density in the following way: √ p κvmp κvm $ ξ ξ τv = ξ = √ = $ ⇒ PD = mDg = , (98) ξ ξ κv where we have defined the deposition depth column density mD and the deposition pressure PD. Notice that the deposition depth $ is dimensionless as is τ . Below τ = √$ the optical depth takes v v ξ the functional form of the thermal regime, derived above. The complete expression of the corrected parametric pressure-temperature profile is lengthy and complicated and gives no useful insight into this method, we therefore forego presenting it here. It can be looked up in (Heng et al., 2011a) eq. 50 and 51.

We have presented some global aspects of scattering and how it might be treated in parametric forward models, and what different regimes of scattering must be considered. We have also briefly presented the parametric forward model of Guillot(2010) and the Markov Chain Monte Carlo (MCMC) technique, using the Metropolis-Hastings algorithm to retrieve atmospheric spectra through Bayesian inference (Benneke and Seager, 2012). One advantage of applying the MCMC technique to compare spectra with models is the possibility to calculate the complete distribution of the individual parameters and not just retrieve one local minimum of the χ2 sum of the measured spectra and the synthesized spectra. Here we want to give some more insight into Benneke and Seager(2012)’s work and its application. Benneke and Seager(2012) used the parametric temperature profile of Guillot(2010), namely the mean

31 temperature profile described by (81). While the ratio of the mean opacity in the visual band and the mean opacity in the thermal band γ = κv is calculated according to (65), the relation between the κth temperature and the pressure is controlled by hydrostatic equilibrium and the ideal gas law. Both κv and κth are calculated using the following principles. The absorption cross sections are calculated from the HITRAN (Rothman et al., 1998) and HITEMP (Rothman et al., 2010) databases for molecular line absorption and H2−H2 opacities from (Borysow, 2002). For a more detailed treatment of opacities see section 2.3.3. The scattering cross section is computed from Rayleigh scattering 4  2 2 24πν nν − 1 σR,i(ν) = 2 2 Fk,i(ν), (99) ni nν + 2 where n is the number density, nν the refractive index and Fk,i the King correction factor (Bates, 1984) of species i. The complete Rayleigh scattering cross section is then computed as the weighted average of each constituent times its volume mixing ratio X σR = XiσR,i (100) i and Benneke and Seager(2012) include the scattering cross sections of the species N 2, CO, CO2, CH4, N2O, and H2O. Xi is the mixing ratio of constituent i and is given by the ratio of the number density of the species in question to the number density of hydrogen Xi = ni/nH2 . Their method to perform atmospheric spectrum retrieval is intended to be used on observations in the visual and near infrared band. From Figure7 we know that scattering of the above mentioned molecules can be treated as Rayleigh scattering, except for condensates. To account for condensates they allow an altitude cutoff parameter at all observed wavelengths representing an opaque cloud layer. This is a method we have discussed earlier to define Rp and the "surface" of the planet. To include the possibility of convection, they use the convection criteria dT g − > Γ = , (101) dz Cp where g is the surface gravity and Cp is computed by the ideal gas law for the atmospheric constituents, and only the degrees of freedom of the individual molecules need to be considered. In the areas where the above criteria holds, they adopt an adiabatic temperature profile P 1−γT γ = const., (102) assuming fully convective layers. The transmission is then calculated following the same geometry as Brown(2001), presented in section 2.1.1. Their intention is to model planetary atmospheres for high-resolution spectroscopy (R> 105). In high-resolution spectroscopy features of single absorption lines become important and can alter the radiative transfer treatment significantly. However, for efficiency, the grid resolution follows the precision of the target instrument, such as the James Webb Space Telescope (JWST), at 1% of the errors of the observations. To get around the issue of not computing the line shapes of the observable absorption features accurately enough to directly deduce absolute abundances, they compute the absolute mixing ratios by 4 more constraining properties based on observables in the transmission spectrum, (see (Benneke and Seager, 2012), Fig. 3). First of all the relative abundances can be estimated from the relative depths of the absorption features in the spectrum. Then, taking up equation (24) again and substituting the α’s with the Rayleigh scattering −4 cross section σR and using σR ∝ λ , they estimate the scale height h of the atmosphere x − x h ≈ 2 1 (103) 4 ln λ1 λ2

32 and therefrom the mean molecular mass (22):

ln λ1 4kBT λ2 µmix ≈ , (104) gR x2 − x1 star R? R? which gives the first additional parameter. Another way is using the relative depth of lines of the same molecule. Similarly the mean molecular mass can also be determined from the depth of a single identified absorption feature, as Miller-Ricci et al.(2009) point out, however, Benneke and Seager (2012) discuss that this method is prone to inaccuracy since the depth can be influenced by clouds or haze layers. Secondly, we need that all mixing ratios sum up to 1. Thirdly, the offset of the impact parameter due to Rayleigh scattering is giving an estimate of the amount of spectrally inactive gas in the observed wavelength range, and finally, the lowest impact parameter x, gives an estimate of the surface or cloud top pressure. Only in combination of these parameters with the relative mixing ratios the absolute abundances can be uniquely constrained. Estimating the gas content of spectrally inactive gases is a crucial step to determine the absolute abundances of the gases with observable features. Vice- P versa from 1 − i Xi = Xinactive, the mixing ratios of the spectrally inactive gases can be precisely determined. The spectrally inactive gas components are assumed to be N2 and H2+He and using the mean molecular mass two components can be determined: X µmix = µN2 XN2 + µH2+He(Xinactive − XN2 ) + µiXi (105) i

XH2+He = Xinactive − XN2 (106)

However, not more than two mixing ratios of inactive gases can be determined and therefore H2 and He cannot be disentangled. One can see that Benneke and Seager(2012) do not depend on the measurement of single line shapes at high accuracy to compute their atmospheric model in general, but depend on the depth of individual absorption features in combination with other parameters such as the slope of the depth due to Rayleigh scattering in the near infrared to lift the degeneracy of different compositions leading to similar absorption depths. Nevertheless, in presence of haze layers, the atmospheric composition can not uniquely be determined. The offset of the Rayleigh scattering slope may be altered by molecular scattering outside of the Rayleigh regime and can affect the determination of the spectrally inactive gas contents significantly. Therefore, only the relative mixing ratios of the constituents are determinable in case of the presence of a haze layer.

The information of absolute gas abundances should in theory be present in the line shapes of the different features, if one properly constrains all the present broadening mechanisms at play. However, we will show in section 2.3.3 that this is not possible until now.

Another approximate way in comparison to the treatment of Guillot(2010), Heng et al.(2011a), and Heng et al.(2014) present the two-stream approximation to analytically solve the radiative transfer equation with isotropic and anisotropic scattering. The treatment and the name arise from the ne- glection of radiative transfer on the horizontal level and splitting up the energy transport into fluxes transporting the energy upwards or downwards through the atmosphere. Casting the full radiative transfer equation, we have (Heng, 2017):

Z 4π ∂Iν µ = Iν − ω0 PIνdΩ − (1 − ω0)Bν. (107) ∂τ 0

33 The individual terms were introduced in the previous section. Notice that any isotropic scattering contribution enters the equation via ω0 (82). The source function can be read off as

Z 4π Sν(µ) = ω0 PIνdΩ − (1 − ω0)Bν. (108) 0 Additionally, to the moments of the specific intensity (61), we define the upward and downward intensity and flux as: Z 2π Z 1 Z 2π Z 0 J↑(ν) ≡ Idµdφ, J↓(ν) ≡ Idµdφ (109) 0 0 0 −1 Z 2π Z 1 Z 2π Z 0 F↑(ν) ≡ µIdµdφ, F↓(ν) ≡ µIdµdφ (110) 0 0 0 −1 with the mean intensity J ≡ J↑ + J↓, the total flux F+ ≡ F↑ + F↓, and the net flux F− ≡ F↑ − F↓. Notice, that J and F+ are similar to the definitions in (61), up to factors of 4π. To enforce local energy conservation, the temperature has to be independent of time dT 1 dF = − − = 0, (111) dt ρCp dz R ∞ where F− = 0 F−dν is the frequency integrated net flux. This arises from the first law of ther- modynamics and represents the continuity equation for radiative heating. This is equivalent to the earlier mentioned flux constancy or the radiative equilibrium equation. In case of isotropic scattering the scattering phase integral simply becomes P = 1/4π. To recast the radiative transfer equation in 0 τ terms of the new quantities, we additionally define: The slant optical distance τ = µ¯ , where µ¯↑ is a characteristic value of µ for the outgoing hemisphere and for the incoming hemisphere we can define µ¯↓ = −µ¯↑, and the Eddington coefficients

F↑ F↓ ↑ = , , ↓ = (112) J↑ J↓ F  = + . (113) J The key assumption to make the radiative transfer equation analytically solvable in the two-stream approximation is that the Eddington coefficients are constant. Assuming that there is no asymmetry 0 between the two hemispheres ↑ and ↓, we adopt the symmetry µ¯ =µ ¯↑ = −µ¯↓ and  = ↑ = ↓. With these assumptions the new set of equations becomes: ∂F ↑ = γ F − γ F − γ B (114) ∂τ 0 a ↑ s ↓ B ∂F ↓ = −γ F + γ F + γ B (115) ∂τ 0 a ↓ s ↑ B  1 ω  γ ≡ µ¯ − 0 (116) a 0 2 µω¯ γ ≡ 0 (117) s 2 γB ≡ 2πµ¯(1 − ω0), (118)

34 and when we add and subtract the equations for the ↑ and ↓ fluxes, we get ∂F + = (γ + γ )F (119) ∂τ 0 a s − ∂F − = (γ − γ )F − 2γ B (120) ∂τ 0 a s + B (121)

µ¯ 1 ω0
with γa + γs = 0 and γa − γs =µ ¯ 0 −  . It can be shown that under the hemispheric closure of the two-stream approximation, which is 0 = 1/2, that in case of anisotropic scattering, (119) becomes

∂F + = 2¯µ(1 − ω g )F , (122) ∂τ 0 0 0 − 0 and g0 again the asymmetry factor from (87). One can easily show that  = 1/2 ⇒  = 1/2. The deriva- tion can be reviewed in (Heng, 2017) or (Heng et al., 2014). During the derivation of the anisotropic scattering case, the additional moment of the specific intensity K↑, K↓, and the second Eddington coefficient  = K− are introduced, the same way as in (61), but with the hemispheric definition and 2 F+ factors as in (109). Interestingly, if we compare the Eddington coefficients defined by Guillot(2010), one can show that fHth can be expressed by  and fKth . However, in this case, leaving fKth = 1/3 and  = 1/2, fHth cannot be 1/2 but rather 8/3, which was also pointed out by Heng et al.(2011a) and Heng(2017). The pressure temperature profile can be derived using a similar approach as Guillot (2010) used and their approach can be generalized to the treatment of any number of frequency bands (Heng et al., 2011a, 2014; Hubeny, 2017).

35 2.3.3 Self-consistent one-dimensional planetary models Laying the foundation for self-consistent planetary models, Gandhi and Madhusudhan(2017) present a new generation of planetary models to study exoplanetary atmospheres with spectroscopy and give an extensive review about previous work, (see Figure5). More information about the different techniques that were used to create novel atmopsheric models or adapt atmospheric models from other applications, such as brown dwarfs and binary stars, for the use in exoplanetary spectroscopy can be found in Hubeny (2017) and at the beginning of this chapter. They give an extensive review of the concurrent planetary models and techniques to solve radiative transfer, convection schemes, correct for radiative convective equilibrium and comment on the efficiency and accuracy of dedicated numerical solvers and the related computer model codes. We want to briefly lay out the basics for contemporary self-consistent models based on the work of Gandhi and Madhusudhan(2017); Heng and Workman(2014); Heng et al.(2014) and compare to the previously described parametric forward models. The basic equations to solve are dP = −ρg Hydrostatic Equilibrium (123) dz P = (ρ/m)kBT Ideal Gas (124) ∂Iν µ = Iν − Sν Radiative Transfer Equation (125) ∂τν dτν = −(κν + σν)ρ(s)ds Optical Depth (126) Z ∞ κν(Jν − Bν)dν = 0 Radiative Equilibrium (127) 0

Considering radiative transfer, we also have the source function Sν in local thermodynamic equilibrium (LTE) and the Planck function Bν(T ):

κνBν + σνJν Sν = (128) κν + σν 2hν3 1 B(T )ν = , (129) c2 exp ( hν ) − 1 kB T whereas κνB(T )ν is the emissivity. An extensive treatment of absorption and emission from atomic transitions, ionization and recombination can be found in chapters 8, 11 and 13 of (Gray, 2005). Although the arguments presented here are not related to stellar atmospheres, many of those arguments hold for planetary atmospheres likewise. In order to construct the source function, one needs the integrated line strength, which can be looked up from line lists, such as from HITRAN (Rothman et al., 1998) or HITEMP (Rothman et al., 2010). The integrated line strength is given at a certain reference temperature and is scaled to the temperature of the atmospheric layer in question, using

Q(Tref ) exp (−Elower/kBT ) 1 − exp (−hν0/kBT ) S(T ) ≡ S0 , (130) Q(T ) exp (−Elower/kBTref ) 1 − exp (−hν0/kBTref ) where we have, the lower level of the energy transition Elower, the energy of the transition, Eupper − Elower = hν0 and Q(T ) the partition function of the species in question, giving the statistical distribu- tion of each energy level at temperature T : X Q(T ) = gj exp (−Ej/kBT ), (131) j

36 with the degeneracy of the state j given by gj. Q(T ) is usually tabulated in the line lists. Tref in case of the HITRAN database is at Tref = 298 K and in case of the HITEMP database at Tref = 1000 K or Tref = 2000 K. The transitions in HITEMP are listed according to the more relevant reference temperature for each transition. An extensive treatment of the computation of integrated line strengths and line shapes can for instance be found in Heng(2017); Pradhan and Nahar(2011). Gandhi and Madhusudhan(2017) use the treatment of Hedges and Madhusudhan(2016) to describe the line shapes in their radiative transfer code. The line shape f is usually described by a Voigt profile, which is the convolution of a Gaussian with a Lorentzian profile. The Gaussian arises from Doppler shifts by a Maxwell-Boltzmann velocity distribution in the gas and microturbulence. Microturbulence are mass motions on a scale smaller than τ < 1 resulting in a velocity distribution affecting the line profile. The Lorentzian, also called the natural broadening, comes from the uncertainty of the energies of the states. It is related to the lifetime of a state and thereby to the Einstein coefficients relevant for the transition in a given radiation field. Z ∞ 0 0 0 fV (ν − ν0) = fD(ν − ν0)fL(ν − ν )dν (132) −∞ An additional broadening mechanism is collisional or pressure broadening, which is more delicate to treat, and can be described through another Lorentzian line shape, convolved with the natural broad- ening Lorentzian, and the half-width at half-maximum HWHM of both Lorentzians are added together to build one single Lorentzian: 1 ΓL fL(ν − ν0) = 2 2 (133) π (ν − ν0) + ΓL with ΓL defined as  −n " # A21 T X ΓL = + P ΓL,bpb , (134) 4πc T0 b where the first term stands for the natural broadening and the second one for the contribution of all collisional perturbers b, described by the partial pressure and their Lorentzian damping factor ΓL,b. P is the pressure of the gas at which the collisions take place. (Hedges and Madhusudhan, 2016) compute the ΓL,b from air- and self-broadening parameters provided by HITRAN and HITEMP, and the PS 1997 list (Partridge and Schwenke, 1997) for broadening of water spectral lines. Additionally, some examples for pressure broadening parameters relevant for exoplanetary atmospheres can be found in the following work:

Li et al.(2015) computed the line broadening of nine isotopologues of CO, by rovibrational transitions. They construct a dipole-moment function, fitting rovibration matrix elements from measurements car- ried out by the authors and measurements from elsewhere. They also compute contributions from CO2 and H2, based on extrapolation of measurement results for broadening effects from for instance (Sung and Varanasi, 2004, 2005) and other sources cited within (Li et al., 2015). Faure et al.(2013) report H2O and CO pressure broadening parameters from H2, standing in good agreement with (Sung and Varanasi, 2004). Treating pressure broadening adequately is challenging, one can overcomplicate the problem by computing each contribution separately from measurement results, from approximations as the one mentioned above or from approximation techniques mentioned in the literature we have cited here. Hedges and Madhusudhan(2016) report that tabulated broadening parameters are still sparse. Also Grimm and Heng(2015) report that the treatment of pressure broadening especially regarding the line wings in contemporary planetary models is either done using ad-hoc cutoff methods focused on

37 computation time optimization (Amundsen et al., 2014; Sharp and Burrows, 2007) or are simply left unmentioned (Benneke and Seager, 2012; Madhusudhan and Seager, 2009). Generally pressure broad- ening parameters in exoplanet atmospheric models are mostly included for broadening by H2 and He for giant planets and hot Jupiters, since those are the most abundant broadening species. The problem with line broadening is on one hand that one might under estimate the contribution of absorption in the wings by choosing a too narrow cutoff νc or overestimating the absorption in the wings due to collisional narrowing of the Doppler profile at higher pressures (Dicke, 1953). Another issue with computing lines from line lists is where to perform the cutoff for the regarded integrated line strengths. Some lines might be too weak to be resolved by observations, or do not have a significant influence on the radiative transfer scheme. Amundsen et al.(2014) propose a cutoff criteria which takes both issues into account, and is computed on the fly. Grimm and Heng(2015) propose a cutoff value at a fixed multiple of Lorentz width ΓL, used to compute the Voigt profile. Hedges and Madhusudhan (2016) comment on Grimm and Heng(2015) that this strategy underestimates the line wing absorption, after the convolution with the Gaussian profile. They propose to use a fixed multiple of 500 widths of the Voigt profile as a cutoff value instead. Sharp and Burrows(2007) on the other hand make their cutoff value linearly dependent on the pressure, however, are critized by Amundsen et al.(2014) and (Grimm and Heng, 2015), that their approach underestimates the absorption in the line wings even more, and Grimm and Heng(2015) show errors of > 10% at higher pressures P > 1atm and Hedges and Madhusudhan(2016) report errors of up to 50% for pressures of the order of P = 0.001 atm. Nevertheless, none of these approaches are from physical (first) principles and which one is the most accurate approach remains disputable. Line-by-line calculations can be computationally expensive and one alternative to line-by-line calculations is presented by Grimm and Heng(2015). They review the k-distribution method and the correlated-k approximation and present a possible implementation for fast computation based on using GPU’s. In the k-distribution method the opacity κ(ν) is resampled and reordered as a new function running from 0 to 1 in y, such that κ(y) is a smooth and monotonically increasing function: Z ∞ Z 1 κ(ν)dν = κ(y)dy. (135) 0 0 This can be generalized to functions F of a function f(x) such that the integral of F over x becomes Z ∞ Z 1 F dx = HdF, dF = F 0(f(x))df (136) 0 0 Z f0 dy ≡ Hdf ⇒ y(f0) = Hdf, (137) 0 where H is called the fractional cumulative distribution function and y the cumulative sum of intervals. In other words H carries the information how to redistribute the intervals of x ∈ [0, ∞), such that A(f0) ≡ {x ∈ [0, ∞) | f0 −  ≤ f(x) ≤ f0,  > 0} and dy is the measure µ of the set A. Hence, y is the fractional area over all x for which f(x) ≤ f0. Mathematically speaking, this is equivalent to calculating Lebesgue integrals. Comparing the k-distribution applied in different layers or for different species, the criteria to resample the opacity will be different in each layer and for each species. Consider the transmission function for two adjacent layers 1 and 2 with opacities κ1, κ2 and column density m1, m2. Using the theory from above we have F = exp (−κ1m1 − κ2m2) and dF = −(m1dκ1 +m2dκ2)F . Thus, the integral of F over x becomes Z ∞ Z 1 exp (−κ1m1 − κ2m2)dx = − exp (−κ1m1 − κ2m2)Hm1dκ1 − exp (−κ1m1 − κ2m2)Hm2dκ2. 0 0 (138)

38 Identifying dy1 = Hm1dκ1 and dy2 = Hm2dκ2 we see that we get two different integrals where x is resampled once according to κ1 and in the other integral according to κ2: Z 1 Z 1 exp (−κ1(y1)m1 − κ2(y1)m2)dy1 + exp (−κ1(y2)m1 − κ2(y2)m2)dy2 (139) 0 0 Z 1 6= 2 exp (−κ1(y)m1 − κ2(y)m2)dy. (140) 0 To somehow still be able to combine these integrals to one single resampled integral in y we construct a total opacity function for all species in each layer before we change to the resampled integral and assume that the integration sampling of κ from one layer to the other does not change significantly. We take y = yn, n denoting the layer and N the total number of layers considered, κ0 the value of κ(ν) up to which the cumulative sum of intervals is computed. The integral becomes

" ( N )# Z ∞  Z ∞  Z 1 X Z 1 T = exp κ(m, ν)dm dν ' exp κn(y)mn dy = exp κdy, (141) 0 0 0 n=0 0 with

( N ) X κ ≡ κn(y)mn (142) n=0 Z κ0 dy = Hdκ, y = Hdκ. (143) 0 This is called the correlated-k approximation. For an accurate result discussion, it must always ex- plicitly be stated, if the correlated-k approximation or the k-distribution method has been used. The correlated-k approximation allows to compute a high number of parameter ensembles, requiring much less computation power than for instance line-by-line computations. However, paying the price of scrambling the information of the frequency ν, at which the opacity of each layer was computed. Amundsen et al.(2014) compare a two-stream approximation scheme combined with a correlated-k approximation to a radiative transfer code using line-by-line computation and solving the radiative transfer equation with the discrete ordinates method (Thomas and Stamnes, 2002). The differences are quantified in atmospheric heating in the different layers for the two models. The tests are performed for the molecules H2O, CH4, CO, and NH3, and for TiO and VO for hotter atmospheres. As mentioned earlier, they also included H2 and He pressure broadening parameters and collision induced absorption opacities for H2-H2 and H2-He collisions. The line data were combined from different sources to cover most of the spectrum of interest and temperature range of interest, and can be found in table 1, in (Amundsen et al., 2014). The opacities were then computed according to the outline at the beginning of this chapter and the correlated-k approximation. Amundsen et al.(2014) report total differences between the two radiation schemes of up to 10%, and locally of up to 30%. They also test the approach of using averaged opacities, such as in the dual band approximation or used by Dobbs-Dixon and Agol (2013), calculated from the same line data, and show inaccuracies of 1 order of magnitude. On top of that, special care should be taken when computing harmonic mean opacities, since the integral may be divergent in case of contributions equal to 0. Therefore, harmonic mean or Rosseland mean opacity (see (178)) integrals are very sensitive to small κν and may be completely dominated around κν ∼ 0.

39 Falling back on our introduction of line-by-line computation, we can compute the opacity κν by summing over each line or transition in the spectrum:

X X j n κν = nj Sif (T ) · fV,if (ν), (144) j if

n and nj is the number density of species j, normalized to the abundance of hydrogen, and fV,if (ν) is nor- malized over a range of frequency for which the frequency cut off value has to be chosen. The statistical weight of each transition is controlled by gj, (see (131)). Scattering is included via Rayleigh scattering of H2 and collision-induced absorption opacities from H2-H2 and H2-He collisions are calculated from the HITRAN database. Gandhi and Madhusudhan(2017) use a fixed number of frequency grid points to compute κ for, and choose the grid resolution such that changes from increased resolution remain below a relative tolerance of 10−4. Grimm and Heng(2015) discuss that in any case of computing the opacity line-by-line, the following requirements must be fulfilled: The number of sampling points must be far greater than the number of lines considered. Only in areas where lines are dominated by broadening or continuum, the number of sampling points can be of the order of the number of lines.

Further, Sharp and Burrows(2007) discuss other contributions that might need to be considered in radiative transfer such as Mie scattering due to condensates (see section 2.3.2), in case of non-negligible flux contributions in the ultra violet: bound-free transitions such as photoionization and photodissoci- ation of molecules and of course transition lines in the ultra violet, and in case of ionization of alkali metals, the free-free transitions of H−.

The relative abundances of the different elements are given by the radiative chemical equilibrium. The chemical processes included into the model of Gandhi and Madhusudhan(2017) are

CH4 + H2O  CO + 3H2, (145) CO2 + H2  CO + H20, (146) 2CH4  C2H2 + 3H2, (147) C2H4  C2H2 + H2, (148) 2NH3  N2 + 3H2, (149) NH3 + CH4  HCN + 3H2. (150) and each equilibrium constant of each reaction is given by Gibbs free energy for each reaction i:

 2   Πni,left P0 −∆G0,i Ki = = exp (151) Πni,right P RT

∆G0,i is the standard Gibbs free energy of reaction i, taken from the NIST-JANAF database (Heng and Lyons, 2016), and for any P-T profile the chemical equilibrium can be solved for the number densities ni of each species. The chemical constituents of the atmosphere are chosen to represent atmospheres of hot Jupiters. If one wants to study terrestrial planets, eventually different chemical reactions need to be considered. One question obviously is then, are atmospheric models conceptualized for Earth a good alternative? We take this question up again in section 2.3.4.

Not surprisingly we re-encounter assumptions for planetary models mentioned also earlier. However, what is novel about the approach of Gandhi and Madhusudhan(2017) is their combination of solving

40 the radiative transfer equation with Feautrier method (Mihalas, 1978), and temperature corrections for radiative-convective equilibrium using a Rybicki linearisation scheme. To apply Feautrier method, the monochromatic intensity of the incoming radiation and the outgoing radiation are decoupled, and in contrast to the two-stream approximation, µ remains a free variable with µ ∈ (0, 1):

∂Iν(µ) µ = Iν(µ) − Sν(µ) (152) ∂τν ∂Iν(−µ) −µ = Iν(−µ) − Sν(−µ) (153) ∂τν = Iν(−µ) − Sν(µ), (154) and the source function is assumed to be symmetric in µ. Additionally, similar to the total and the net flux in the two-stream approximation, we define the total and the net intensities as follows: 1 j = (I (µ) + I (−µ)) (155) µ,ν 2 ν ν 1 h = (I (µ) − I (−µ)), (156) µ,ν 2 ν ν and thus we can redefine the radiative transfer equation as a second order differential equation in jµ,ν:

∂2j µ2 µ,ν = j − S (µ). (157) ∂τ 2 µ,ν ν Using the definition of the first and the second Eddington coefficient the same way as Guillot(2010), (74), and integrating over µ ∈ (−1, 1), using the definitions in (61), we reconstitute the radiative transfer equation dependent on Jν: ∂2(f J ) Kν ν = J − S . (158) ∂τ 2 ν ν This equation is sometimes called the Feautrier equation. Additionally we have the boundary conditions

∂(fKν Jν) = fHν Jν(0) − Hirr (159) ∂τ τ=0

∂(fKν Jν) 1 1 ∂Bν = (Bν − Jν) + , (160) ∂τ 2 3 ∂τ τ=τmax ν where Hirr = Firr/4π is the irradiation flux and the second equation arises from the internal heating flux plus the flux from the diffusion approximation (Heng et al., 2014), since the lower atmosphere is assumed optically thick. This does not necessarily hold in case of terrestrial planets though. One great advantage of solving the radiative transfer equation using Feautrier method in comparison to the two-stream approximation is the iterative computation of the Eddington coefficients. Thereby we do not need to make a pre-assumption of the value of the Eddington coefficients fKν and fHν . Another advantage is that the radiative transfer is accurate up to introduced errors through the linearization scheme, exact in µ and does not involve averages over hemispheres. However, in the presence of condensates, such as cloud formation, dust or haze on intermediate altitudes, the source function is not symmetric in these layers. Hence, the source function and the equations to solve have to be adjusted. Revisiting equations (107), (108), and (87) we can derive the adjustments of the Feautrier equation

41 (158) and the µ−integrated radiative transfer equation as follows. The µ−integrated radiative transfer equation becomes: Z 4π ∂Kν = Hν − µSν(µ)dµ. (161) ∂τν 0 One can show that (108) becomes Z 4π Z 4π Z 4π  Z 4π 0 µSν(µ)dΩ = ω0 Iν µPdΩ dΩ − (1 − ω0) µBνdΩ (162) 0 0 0 0 Z 4π 0 0 = ω0g0 µ IνdΩ , (163) 0 with g0 from (87). Thereby, (161) becomes

∂Kν = Hν − ω0βJν (164) ∂τν Z 4π 1 0 0 β = g0 µ IνdΩ . (165) Jν 0

β is a form factor describing the asymmetry of the intensity with µ and the asymmetry factor g0. The details of the derivation are presented in chapter 3.5 of Heng(2017) and in (Hubeny, 2017). Following the derivation of (158) in (Hubeny, 2017), the Feautrier equation becomes 2 ∂ (fKν Jν) ∂ 2 = Jν − [ω0βJν] . (166) ∂τ ∂τν Hubeny(2017) point out that the linearisation scheme used to solve the radiative transfer equation in the Feautrier form can be adapted to the additional terms and can be solved similarly to the case of isotropic scattering. A more thorough treatment of Feautrier method and the different linearization and iteration schemes and the discussion of their convergence and computation times can be found in (Hubeny and Mihalas, 2014) and again in (Hubeny, 2017). In Gandhi and Madhusudhan(2017), convection is included via Mixing Length Theory (Kippenhahn et al., 2012) and the radiative equilibrium equation (127) becomes Z ∞ ρg dFconv ρκν(Jν − Bν)dν + = 0 (167) 0 4π dP Z ∞ fKν ∂Jν Fconv σ 4 dν + = Tint. (168) 0 ∂τν 4π 4π After a formal solution with a pre-assumed pressure temperature profile of the basic radiative transfer equation with Feautrier method in a Rybicki linearisation scheme, convection is included as a correction to the temperature profile in the layers where the criteria for convection is fulfilled. The quantities J, T, δτν, κν, σν,B,P are perturbed in T and the new linearized terms can be combined in two invertible matrix equations to solve for J and T . Gandhi and Madhusudhan(2017) also linearize the temperature correction for the radiative convective euqilibrium in the Rybicki linearization scheme and iteratively correct the P-T profile until the radiative-convective equilibrium equation is satisfied. Below, we briefly lay out the treatment of convection in the Mixing Length Theory following the treatment of(Hubeny and Mihalas, 2014) and (Kippenhahn et al., 2012). There may be few other schemes how convection can be treated, which we are not presenting here.

42 Fconv in equation (167) is the additional heat flux due to convection. Deriving Fconv in full detail would go beyond the scope of this chapter and we will restrict ourselfs on the introduction of the basic idea behind the Mixing length theory and the diffusion approximation. The Mixing Length theory describes a gas element of volume V with temperature Tel, that starts rising under the condition ∇ > ∇ad, with ∂ ln T ∂ ln T ∇ = ∂ ln P and ∇ad = ∂ ln P |ad = γ − 1, the adiabatic gas law. In this way the gas element will always be warmer than the surrounding air until it traveled a distance lm, the mixing length. The assumption is, that after moving lm the gas element has mixed with the surrounding gas enough, such that the convective movement of the parcel stops and T ≈ Teq. Since it is assumed that the movement can start anywhere throughout the atmosphere where ∇ > ∇ad is fulfilled, a convective heat flux Fconv forms, dependent on the ratio between the mixing length and the atmospheric scale height lm/Hp, the temperature T and the temperature gradients ∇, ∇ad, and ∇el. ∇el is the actual temperature gradient of the gas element and is given by λHp ∇el = ∇ad + . (169) ρV CP vT v can be eliminated by computing the work done by the buoyancy force on the parcel, under the assumption that half of it is converted into kinetic energy v2/2 per unit mass:

2 2 lm v = gδ(∇ − ∇el) . (170) 8Hp

λ = SFν,el depicts the loss of energy per unit time through the surface S of V , and we may write the frequency integrated flux of the element Fel by

3 16σT ∂T Fel = . (171) 3κRρ ∂n

∂T Combining the arguments from above and using ∂n ≈ 2DT/d with the element diameter d and DT = (∇ − ∇ ) lmT , the resulting equation can be solved for ∇ − ∇ in terms of ∇ − ∇ ,T,H , g, ρ and el 2Hp el ad p lmS the specific heat CP , the Rosseland mean opacity κR, and the form factor V d . Thereby one gets all the necessary terms to compute Fconv. The Rosseland mean opacity can be derived from the diffusion approximation, in which the radiation field is assumed to be completely thermalized Jν = 4πBν and scattering is isotropic g0 = 0. The basic law of diffusion is given by

j = −D∇n (172) where ∇n is the gradient of the number density of particles and j the particle flux j = nv with the diffusion coefficient D, dependent on the mean free path lp of particles in the medium and their mean velocity v 1 D = vl . (173) 3 p σ 4 To obtain the radiative diffusion flux, replace n with the energy density of radiation U = 4 c T and differentiate in the radial direction r. We can identify lp with the photon mean free path and relate it 1 to the Rosseland mean opacity κR such that lp = . Then the derivative of U times D becomes the κRρ radiative diffusion flux F : 16σ T 3 ∂T F = − . (174) 3 κRρ ∂r

43 The Rosseland mean opacity can also be derived from the diffusion flux using the frequency dependent expression for D and U: 1 c Dν = clν = (175) 3 3κνρ 4π U = B (176) ν c ν where taking the gradient of Uν, proceeding as with (174) and integrating over ν returns 4π Z ∞ 1 ∂B  F = − dν ∇T, (177) 3ρ 0 κν ∂T and comparing with (174) we can identify the Rosseland mean opacity

4σT 3 Z ∞ 1 ∂B −1 κR = dν , (178) π 0 κν ∂T

∂B which is the harmonic mean of the opacity, weighted with ∂T . If one integrates (174) in P and uses dP dz = −ρg to relate the Rosseland opacity κR to the Rosseland optical depth τR, one gets the following simple P-T profile: 3 1/4 T = T (τ + const.) , (179) int 4 R which is called Milne solution (Heng, 2017). const. is an integration constant, such that if τR ∼ 1, T = Tint. In other words, Tint is defined as the temperature observed at τR = 1, in case of the temperature profile of a grey atmosphere (Gray, 2005). This expression for the internal temperature Tint was defined for self-luminous objects.

2.3.4 General circulation models and three dimensional radiative hydrodynamics models, detailed cloud and dust physics So far we have introduced different techniques to model pressure temperature structure and radiative transfer of 1-dimensional static planetary models. However, we are aware that planetary atmospheres have internal motion which redistributes heat additionally to isotropic and non-isotropic scattering. We have seen that these motions were partially treated in one way or another. For example, Guillot (2010) include an advection scheme, Gandhi and Madhusudhan(2017) however, neglect motion due to advection but include temperature corrections due to convection. If one wants to go a step further, and include general motion, we end up with the equations of fluid dynamics. The conservation of momentum of a fluid element can generally be written as: ∂ρu i + δ ∂ [ρu u − S ] = −ρ∇Φ (180) ∂t ij j j k jk where u = uk is the velocity of the fluid element and ρ the density of the fluid element. Here we have used Einstein notation, repeating indices are summed over. We have written the equation in components of u, since it is the cleaner way of deriving the different terms important in the fluid equations of motion. Naturally we have the Kronecker delta δij, which is 1 in case of i = j and 0 otherwise. On the right hand side we have the gradient of the potential Φ, which in our case is the gravitational potential Φ = gr, with r the distance above the surface. Care must be taken here and we

44 choose |r| = Rp + r for the position vector of the fluid element. g is the gravitational acceleration and we take the value at the surface of the planet since r  Rp, the mass of the atmosphere compared to the core is negligible, and g does not change much with altitude. Multiplying out the components on the left we can separate two equations. The first one arises naturally if we collect all derivatives in ρ and we retrieve the continuity equation: ∂ρ ∂ρ u + u δ ∂ (ρu ) = + u δ u (∂ ρ) + u δ ρ(∂ u ) = 0 (181) k ∂t k ij i j ∂t k ij j i k ij i j ∂ρ ∂ρ + ∇(ρu) = + ρ(∇u) + (u∇)ρ = 0. (182) ∂t ∂t

D ∂ Dρ We can see that with the material derivative Dt = ∂t + u · ∇, the statement Dt = 0 is equivalent to ∇ · u = 0, which is the incompressibility of the fluid element. The remaining terms from (180) describe the change in velocity of the fluid element Du ρ = ρg + δ ∂ S , (183) Dt ij i jk which is sometimes also called Cauchy’s equation (Heng, 2017). Sjk is the viscous stress tensor and is defined as 2 S = −P δ + µ{∂ v + ∂ v } − µδ ∂ u . (184) ij ij i j j i ij 3 k k Rewriting (183) in vector notation and converting to the corotating frame with Ω the angular velocity of the rotation of the planet, we obtain the full Navier Stokes equation (Dobbs-Dixon et al., 2010):

Du ∇P ν = g − − 2Ω × u − Ω × (Ω × r) + ν∇2u + ∇(∇u). (185) Dt ρ 3

µ ν = ρ is the viscosity and µ is called the dynamical viscosity. The centrifugal term −Ω × (Ω × r) can also be absorbed in the gravitational potential as a constant correction to g. The other rotational term is called the Coriolis term.

The internal energy of the fluid can be derived using similar considerations as in (180) and consists of three energy equations. The first one can be derived by multiplying (185) with u, ignoring the viscous terms and evaluating the equation in the inertial frame. Dropping the rotational terms, one obtains 1 2 the kinematic energy density ekin = 2 ρu . The potential energy density is simply epot = ρΦ and is rewritten under the material derivative. Lastly, the first law of thermodynamics and the ideal gas law give the relation between internal energy e, the viscous terms, and the compressional heating term. ∂e + ∇(eu) = ∇[S u + F ] (186) ∂t jk − 1 d(r2F ) = −(P ∇) · u − − + S + D , (187) r2 dr ? ν where e(r) = ρCV T (r) is the internal energy density of the fluid element. S?(r) denotes the heating from stellar irradiation at optical depth τ, F− the wavelength-integrated radiative net flux, and Dν the heating from dissipative forces or viscous heating. The total energy density is then etot = ekin + epot + e. It may also be desirable to derive an equation for the conservation of the total energy instead, nevertheless both governing equations for the energy are equivalent and arise from energy conservation. A derivation for the governing equation of the total energy can for instance be found in chapter 9.4 of

45 Heng(2017). Since Dobbs-Dixon et al.(2010); Lee et al.(2016) use a two-stream approximation the gradient of the net flux F− only contributes in radial direction. The radiative heating from the star S? is constructed using Beer’s law for stellar irradiation:

 2 Z νmax R? X dτb S = exp (−τ /µ ) πB(T ) dν (188) ? a dr b ? ? b b νmin

2 where b ≡ [νmin, νmax], the frequency interval of bandwidth b, the scaling factor (R?/a) denotes the R? 2 scaling of the stellar temperature to the irradiation temperature Tirr = ( a ) T? (see (68)). Furthermore the factor exp (−τb/µ?) denotes the absorption under the irradiation angle µ? down to the level of τb, while the derivative in r comes from the fact that the change in thermal energy due to radiation can be R R written as ρCpdT/dt = −dF−/dr = ρκν(Jν −cER) = ρκν(Bν −cER)+S? (see Eq. (111)). ER is the radiative energy density. It can be shown that ER can be treated independently of the other heating sources and sinks in the energy equation. B(T ) is the radiation field of the fluid element at temperature T under the assumption of (LTE). Thus, B(T ) connects the flux F− and the radiative energy density ER to the rest of the internal energy equation (186). The expression above is also equivalent to (167). The last term in (186) is the viscous heating rate and is the product of the bilinear form of the gradient and the velocity vector under the viscous heat tensor:

Dν = ∂iSijuj. (189)

The advantage of this approach is its generality. It can be easily extended to include effects such as magnetic fields, shocks, clouds, and dust. The purpose of such models is to study certain physics in depth. With the rise of super computers and overall increasing computation power, RHD models provide a great direct insight into some of the physics at play, and provide a useful tool to study atmospheres of Giant-planets. For instance Dobbs-Dixon and Agol(2013); Dobbs-Dixon et al.(2010, 2012) use their model to study radiative transfer and hydrodynamic motion in highly irradiated atmospheres such as the hot Jupiters HD 189733 b and HD 209458 b. Without specifying a wavelength band, the equation for the radiative energy density is ∂E R + ∇F = ρκ [B(T ) − cE ] + S , (190) ∂t − R R ? where κR is the Rosseland mean opacity. Now to connect the radiated flux F− to the radiative energy density ER, they use the flux-limited diffusion approximation (Dobbs-Dixon et al., 2010; Levermore and Pomraning, 1981): c F− = −λ ∇ER, (191) ρκR and λ is a variable parameter controlled by the "thickness" of the atmosphere. The two limits are the diffusion approximation (177) in case of a thick atmosphere and

|F−| = cER (192) in case of a thin atmopshere. Dobbs-Dixon et al.(2010); Lee et al.(2016) then use the dual band approximation, additionally with the Rosseland mean opacity and the two stream approximation to solve the radiative transfer equation in their models. Dobbs-Dixon et al.(2010) solve the 3D-RHD equations using the successive overrelaxation method (SOR) to evolve the ∇F− part, dependent on ER.

46 To solve the general hydrodynamics equations they use a so called "mono-scheme", a version of a finite difference scheme, which is a well known and applied method to solve partial differential equations. The scheme gives a semi-second-order temporal accuracy, allowing accurate resolution of discontinuities, that could for instance arise from shocks and instabilities. They use the same functional form to implement viscosity terms as was used in the ZEUS-2D code (Stone and Norman, 1992), who apply a finite-difference sheme as well. How the equations are solved and how the conservation of mass, momentum, energy, and radiative equilibrium if applicable are accounted for is important to understand how the model deals with effects such as shocks, rarefactions, turbulence, waves, instabilities in the fluid flow and other formations of discontinuous effects. Another way to account for discontinuities that can not be resolved by the finite-difference scheme, is to use an artificial viscosity to smooth out the discontinuities (Stone and Norman, 1992). Thereby, one gets the Rankine-Hugoniot relations (look up for instance (Penner, 2003)) and correct shock velocity. Although Dobbs-Dixon et al.(2010) use spherical coordinates, they introduce artificial slip-free impenetrable boundaries at the poles at > | ± 70◦|. The reason is the choice of grid symmetry and the decreasing grid size and thereby the Courant time step constraint limits the computational efficiency near the poles. This was solved subsequently in Dobbs-Dixon et al.(2012) through grid adaptations for the polar regions and applied by Lee et al.(2016). The initial pressure temperature profiles are used from the results of Burrows et al.(2007).

The model was run according to the known properties of HD 209458 b and under the assumption of being tidally locked. Although Dobbs-Dixon and Agol(2013)’s model has a great advantage over 1D hydrostatic models by accounting for the motion in the atmosphere and dissipative heating, they point out that including the physics of clouds and condensates is a next crucial step in the understanding of hot Jupiters atmospheres, since condensates form the major opacity contribution. In 1D hydrostatic models this contribution was not neglected, however, to account for the detailed physics of condensates the "surface" was chosen to be at τ ∼ 1 with appropriate boundary conditions. This makes sense, since observations are limited to the atmosphere above a τ ∼ 1 surface and Dobbs-Dixon et al.(2010) show contributions for the redistribution of energy by motion is dominant below the photosphere and therefore must rather be accounted for in the lower layers of a planetary atmosphere. Nevertheless, Dobbs-Dixon et al.(2010, 2012) show non negligible effects on the temperature profile in the photosphere especially on the night side of a hot Jupiter. Additionally, Dobbs-Dixon et al.(2012) report major contributions to the shape of the wavelength dependent transit depths at the terminator, and variation of transit times up to 6 s for HD 209458 b with increasing strength of circumplanetary jets. The circumplanetary jets form as a consequence of heat advection from the irradiated side of the planet to the night side, eastwards and westwards along the equator and meeting in a subduction point on the night side of the planet. They investigated the effect of viscosity on the jets and reported formation of supersonic velocities, turbulence, waves, and shocks. They also report and quantify a non negligible effect of viscous heating on the temperature structure of the atmosphere, even though for most of the atmopsphere the compressional heating dominates. The effects are most prominent in the terminator region and at the subduction point, where the flow to the nightside converges.

Lee et al.(2016) combine the code from Dobbs-Dixon et al.(2010) with a mineral based cloud formation and dust grain formation and transport model, and account for the thermo-chemical evolution of the dust grains following the treatment of Woitke and Helling (references in (Lee et al., 2016)). They compute Mie scattering properties and wavelength dependent opacities to account for their influence on the radiative hydrodynamical nature of the model. Lee et al.(2016) report differences of several magnitudes in density dependent on latitude, longitude, and depth as well as non-uniform element

47 distribution throughout the atmosphere. These findings hint towards that cloud properties significantly affect transit spectroscopy and can be further constrained by phase dependent eclipsing spectroscopic observations. In a subsequent paper Lee et al.(2017) replace the dual band approximation from Dobbs- Dixon et al.(2010) by a Monte Carlo radiative transfer code and make observational predictions for the two ongoing missions TESS and CHEOPS.

In comparison, solving the radiative hydrodynamics equations and implementing additional physics comes at high computational cost, one needs to limit him or herself in realism by neglecting some effects and treating others in more detail. This leads for instance to a very narrow timeframe to simulate atmospheric dynamics for, typically on the order of 100 Earth days. Now one might argue that this is still orders of magnitude greater than timescales on which hot Jupiters orbit their host star, and their rotational period. However, if we would like to study the evolution of a hot Jupiter over large timescales and for instance vary the irradiation or activity of the host star, or if we want to study the stability of the climate of Earth or super-Earth-sized exoplanet to investigate the question about habitability, one needs to be able to perform greater timesteps and mask some of the complicated intrinsic physics of the atmosphere. One way to achieve this are General Circulation Models (GCM).

In a GCM the atmospheric circulation is parameterized by the primitive equations of meteorology. The horizontal and vertical flows are separated and the fluid element is considered as inviscid. The motivation for this approach is the shallow nature of the atmosphere in comparison to the core of a planet Rp  h, holding for terrestrial planets as well as Jupiter-sized planets. However, so called deep Jupiters such as some exo-Jupiters, these assumptions might break down and large scale convective motion becomes non-negligible (Heng et al., 2011b). In comparison to the full hydrodynamics analysis, in a GCM we find the following approximations: The centrifugal term is absorbed into the gravitational acceleration g and the vertical structure of the atmosphere is assumed to be hydrostatic. For the vertical coordinate one uses pressure p rather than altitude and horizontal motion follows constant pressure heights. Thus, large scale vertical flows can be described by ∂ω = −∇ · v, (193) ∂p p

dp where ω = dt is the vertical velocity of a fluid element and v = (v, u), where v is the horizontal velocity in east-west direction φ = const. and u the horizontal direction in north-south direction λ = const. in the co-rotating frame of latitude φ and longitude λ. Convection can be included through an additional parameterization such as the mixing length theory. Although the gas is assumed to be inviscid, diabatic cooling and heating are masked by quantities entering the redefined quantities vorticity ζ, divergence D, and temperature T . We will not introduce all the new governing equations here, though, we will shortly explain the two terms often encountered with climate models and GCMs: Relaxation and Forcing. To ensure consistency when performing time steps of several orders greater than short term effects such as turbulence, one uses relaxation schemes. Newtonian relaxation enforces the right balance between excess temperature and equilibrium temperature from radiative heating and the act of accounting for the whole physics of radiative transfer through relaxation is called forcing. Vorticity and divergence use the so called Rayleigh friction or Rayleigh drag, which follows a similar idea as the Newtonian relaxation and has the same functional form. The relaxation schemes are presented in Menou and

48 Rauscher(2009) Heng et al.(2011b) as follows:

Teq − T QT = (194) τrad ζ − 2µ Qvor = − (195) τfric D Qdiv = − (196) τfric where τrad and τfric are the characteristic radiative and friction time scales, a parameter that needs to be adjusted and one can use studies such as detailed radiative transfer calculations in 1D for the same modelled object to adjust for instance the radiative time scale τrad. µ = sin (φ) is related to the equilibrium value for the vorticity ζ at latitude φ 1 ∂v 1 ∂u cos (φ) ζ = 2µ + − (197) Rp cos (φ) ∂λ Rp cos (φ) ∂φ 1 ∂u 1 ∂v cos (φ) D = − . (198) Rp cos (φ) ∂λ Rp cos (φ) ∂φ D = 0 stands for no horizontal divergence. Of course, the mass, momentum, and energy conservation equations are expressed in the new terms as well and must be consistent with the solutions of the dynamical equations.

There are several other artificial parameterizations used to make solving of the governing equations more efficient, which we will not present here and only give some comments on how the equations are solved and how such models can be tested for consistency. The equations derived by Menou and Rauscher(2009) and Heng et al.(2011b) basically are the same as derived by (Hoskins and Simmons, 1975) and are well established to build GCMs with additional features. These equations are solved using dynamical cores. A dynamical core is a solver only dealing with the essential dynamics of an atmosphere, disregarding details such as radiative transfer, which are accounted for by forcings. Menou and Rauscher(2009) use the semi-implicit pseudospectral method, also used by Hoskins and Simmons(1975), and the equations are solved horizontally in spectral space and vertically with a finite difference scheme. One advantage of this method is its spherical symmetry, in comparison to Dobbs-Dixon et al.(2010) who had to treat the poles seperately. Time integration is performed with a semi-implicit leapfrog scheme. Menou and Rauscher(2009) also report that their model is comparable to the classic benchmark calculations of Held and Suarez(1994). Those benchmark calculations allow an intercomparison of GCM models with different parameterizations and different solvers through statistically averaged output quantities, a test to validate a new model or a new solver. It was intended to be applied to GCM Earth models.

Heng et al.(2011b) applied these test for the first time to tidally locked exoplanets using the dynamical cores of the Flexible Modelling System (FMS). The FMS contains spectral and finite difference cores and is described in Washington and Parkinson(2005), chapter 5. Heng et al.(2011b) compared and tested an Earth like model, a tidally locked Earth, shallow Jupiter and deep Jupiter. Modeling of horizontal dissipation comes down to two parameters used in the two different types of cores. Since spectral codes are intrinsically non-dissipative (Stephenson, 1994), dissipation has to be included using so called hyperviscosity terms (Heng et al., 2011b; Menou and Rauscher, 2009), with the hyperviscosity ν and the hyperviscocity damping parameter nν. In case of the finite difference method, the dissipation is

49 controlled by the horizontal mixing coefficient K, which takes vales between 0 and 1, and the strength of the horizontal mixing denoted by ∆i with the numerical restriction of ∆iK ≤ 1/8 (Heng et al., 2011b). Both the parameters for the relaxation and forcings, and the hyperviscosity do not arise from first principles and must be treated as free parameters with certain sensitivities. However, the sensitivity of the parameters should not dominate the longterm averaged model outputs, since the parameters for hyperviscosity are directly related to the choice of grid resolution. While Heng et al.(2011b) found good agreement between the different solvers and previous studies for the Earth, tidally locked Earth, and shallow Jupiter, in case of the deep Jupiter model for HD 209458 b, Heng et al.(2011b) found dynamical uncertainties at the level of 10%, both between different solvers in the same model and between different choices of resolution and magnitude of the horizontal dissipation under the same solver for the same model. They conclude that in case of simulations of "deep" Jupiter like planets care must be taken when modeling the horizontal dissipation, either the treatment of horizontal dissipation must be based on a more rigourous physical approach or the parameters must be constrained by observations and model intercomparison.

Moving away from hot Jupiters to terrestrial planets such as super-Earths, and Earth-sized planets with atmospheres of the order of Mars to Venus, GCMs adobted from Earth climate simulations such as Way et al.(2017) did, allow a marvellous approach to include additional physics, that has so far not been covered in other models presented here. Examples are ocean dynamics, coupled dynamics between ocean surface and atmospheric winds, water cycle as moisture source in the atmosphere, radiative transfer, convection and influence from condensates such as clouds, chemistry, such as photochemistry, but also simple chemical balance schemes for the used chemical species, the cryosphere, land ice, and sea ice, and snow coverage, land and water coverage with different distributions for the variability in shorelines, continental drainage, connecting the water cycle in land to the water cycle of the sea, and even different types of vegetation, which may have a significant effect on the planetary albedo. The model can be combined with transmission spectroscopy calculations, to predict observational constraints of such exoplanets. Such models contain a myriad of detail about different interplaying processes, can be used for extensive studies of planets in habitable zone, and refine the definition of the habitable zone.

Additionally one can study the formation of biosignatures in more detail, dependent on realistic as- sumptions of abiotic sources and sinks for certain candidate biosignatures. Where such insights can make a great difference for the interpretation of observed potential biosignutares, is for instance the recently detected phosphine in the upper atmosphere of Venus (Greaves et al., 2020). A remarkable finding in any case, since the authors report a phosphine content several magnitudes above any pos- sible abiotic pathway. Nevertheless, further research needs to be conducted to confirm the observed phosphine feature, both for its robust identification as phosphine as well as its strength and related to that the abundance of phosphine in Venus atmosphere. Supplementary, Way et al.(2016) showed that Venus could have been a planet hospitable to life, covered with oceans until about 715 million years ago, before it underwent a runaway greenhouse effect, which lead to its present day thick atmosphere.

Opening up the discussion of extraterrestrial life, many aspects other than the ones presented in this work have to be considered as well. Kopparapu et al.(2019) give an extensive summary about what other factors must be accounted for to constrain the habitable zone around a star and the chances to find extraterrestrial life. One particular question that can only be answered by including these effects, is the question about a planet’s evolutionary path. The main factors that need to be considered to answer this question come down to investigating the star planet interaction, beyond the stellar irradiation

50 budget and the redistribution of heat around the planet. Another important factor they point out, is the architecture of the orbits, presence of Giant-planets, and to some degree the initial composition of the planet. However, comparing the history of Venus, Mars and Earth, initial composition can not straight- forwardly explain the observed differences. Studying the star planet interaction of course also opens up a completely new field of research.

3 Summary and conclusion of the literature review

For the detection of exoplanets, there are several methods very different in their nature. We focused on the transit photometry method since it is a prerequisite to understand transmission spectroscopy, which is the method applied by CRIRES+ to determine the composition of exoplanetary atmospheres. As pointed out in the Introduction, probably the biggest driver of curiosity about the composition of exoplanetary atmospheres is the question about the presence of life. To answer this question a long journey is necessary of which we have covered part in our literature review. We headed off by reviewing transit photometry Winn(2010). Transit photometry allows one to discover the presence of exoplanets and together with radial velocity observations infer the planets mass and radius, as well as the distance to the host star, the presence of other planets, the eccentricity of the orbit and the orbital period. However, the mass of the planet and the star, or the radius of the planet and the star, and the semi- major axis of the planet’s orbit can not be determined independently only by observation. One stellar parameter must be determined by other means. Using the knowledge about the distribution of the orbital parameters of discovered exoplanets, we have estimated the chance to find an Earth-twin to be 1 in 370’000 planets. The photometric lightcurve is inherently related to the blocking of star light by the atmosphere of the transiting planet. With a simple approximation of the geometry, the photometric lightcurve can be related to the atmospheric scale height and by examining it at different wavelengths, one can relate the blockage of stellar light via different atmospheric heights to the presence of certain absorption features. Hence, investigating the blockage of the star light over a certain wavelength range provides a transmission spectrum of the top of the atmosphere of the transiting exoplanet (Brown, 2001).

Two methods have been presented to infer from the observed spectrum the transmitted signal from the exoplanet’s atmosphere. To differentiate the faint transmission signal embedded in the background signal of the star light and the absorption of the Earth’s atmosphere, those methods use two different ways. Aronson and Waldén(2015) remove the contribution from the host star and the Earth by simulated or observed out-of-transit spectra by optimizing a cost function defined as the difference between the observed and synthetic spectra. Solving for the parameter describing the extra atmospheric height blocking the star light at a given wavelength, they reconstruct the full transmitted spectra. Snellen et al.(2010b) remove the background contributions by decomposing the observed spectra with a singular value decomposition and investigating the lower singular values for a time varying signal, coming from the Doppler shift by the movement of the planet in front of the host star. The prepared signal is then compared with synthetic signals from planetary models via cross correlation to directly compare to atmospheric parameters used by the planetary atmospheric model. However, it needs to be pointed out that the actual transmission spectrum is never fully retrieved. The cross correlation method only allows to find statistically significant correlations between a synthetic signal from a planetary atmospheric model and the prepared signal from the observation. The prepared signal does not show the spectrum itself. Therefore, one could argue that the employed atmospheric model could introduce some bias to the spectrum retrieval. For the purpose of comparing retrieved spectra to planetary models,

51 we have reviewed two important techniques based on Bayes’ theorem of conditional probabilities.

The Optimal Estimation method tries to find the best fit between observation and model by assuming the difference between the two follows a normal distribution. It gives the parameters minimizing a cost function between observed spectrum and synthetic spectrum using Bayes’ theorem. The second method called Bayesian Inference tries to map the whole parameter space by computing the posterior probability distribution of the parameters used in the model, using the Metropolis-Hasting algorithm to choose the stepsize from one set of parameters to the next, moving to the minimum of the cost function, mapping out the space around the minimum with high precision. There exist techniques to smooth out the prior-probability distribution and map a wider set of the parameter space to not miss other minima. Both methods can be used to compare any retrieved transmission spectrum to modeling, however, Bayesian Inference was especially designed for low-resolution spectra and simpler parametric forward models with less prior assumptions about the planet’s atmosphere, while Optimal Estimation was developed more for high resolution spectra and detailed self-consistent planetary models with many prior assumptions about the planetary atmosphere (Madhusudhan, 2018), (Benneke and Seager, 2012).

The parametric forward model is based on a parametric form of the pressure temperature profile of the atmosphere. Prominent examples can be found in (Guillot, 2010) and (Madhusudhan and Seager, 2009). Heng(2017) explain how the profile in Guillot(2010) can be derived using the two-stream approximation, in the dual band approximation. The profile used by Madhusudhan and Seager(2009) is derived from atmospheric characterization of our Solar System gas-planets. Both tools require a set of ∼ 10 parameters. The dual band approximation assumes the radiation to travel through the atmosphere in two channels, namely the shortwave incoming radiation and the outgoing longwave radiation. Thereby, the opacity κ is averaged over each band and the radiative transfer equation is used separately for each band, only connected by radiative equilibrium of the incoming and outgoing radiation. The dual band approximation should not be confused with the two-stream approximation. The main difference lies in the approximation of the opacity κ. The two-stream approximation can be solved for frequency dependent forms of κ, while in the dual band approximation the spectral information about κ is absorbed into the shortwave and longwave approximations.

Madhusudhan and Seager(2009) solve the radiative transfer problem by a line-by-line calculation. Such calculations are usually done with the Feautrier method. It allows to solve the radiative transfer equation by redefining the problem in terms of total and net intensities, dependent on the optical depth τ and the direction µ. The problem is further simplified by defining the first and second Eddington coefficient in the same way as defined by Guillot(2010). The purpose of the Feautrier method is the same as of the two-stream approximation: to solve the radiative transfer equation in a closed form. However, the main differences between these two methods are the following. The Feautrier method solves for the mean intensity J, while the two-stream approximation for the flux F , thus any lateral information about the distribution of heat is excluded and µ is fixed at a characteristic value of the incoming and outgoing radiation. In the two-stream approximation one has three Eddington coefficients, set to predefined constants and constraining the solutions further, whereas in the Feautrier method the two Eddington coefficients remain free. Both methods can account for coherent scattering through the source function and non-coherent scattering through the scattering phase function and the scattering asymmetry parameter. The temperature profile can be adjusted for convection for instance using mixing length theory. Additionally in self-consistent planetary models the pressure temperature profile is derived from hydrostatic equilibrium and radiative equilibrium, accounting for heating of the atmosphere. Via the equation of state, where usually the ideal gas law applies, the hydrostatic

52 equilibrium can be corrected for the temperature profile, derived from radiative equilibrium. The linearized problem cast with the Feautrier method is then reiterated until both equilibrium equations are satisfied (Gandhi and Madhusudhan, 2017).

These methods have been used to model stellar atmospheres before they were applied to model exo- planetary atmospheres. However, two major differences have to be pointed out. The first one is that all calculations for atmospheric models can be performed in local thermodynamic equilibrium LTE. Secondly, in contrast to stellar atmospheres we deal with molecular absorption bands. Since the re- search field of exoplanetary atmospheric characterization is fairly new, parameters for molecules such as pressure broadening parameters or certain molecular transition parameters are sometimes scarce, making it harder to study individual spectral features in detail Hedges and Madhusudhan(2016). Ad- ditionally, line-by-line calculations can become computationally expensive with increasing number of spectral lines. An alternative method to deal with the integration over many spectral lines is to bin the spectral information using the k-distribution method or the correlated-k approximation, thereby adjusting the computational cost with the number of bins to recast the opacity κν (Grimm and Heng, 2015).

One-dimensional planetary atmospheric models have been widely used to constrain atmospheric prop- erties of hydrogen-rich atmospheres of Giant-planets. A logical step is to extend atmospheric models to more dynamical models. Dobbs-Dixon et al.(2010) propose a 3D model including hydrodynamics by approximately solving the Navier-Stokes equation. Such models can give insight into the influence of strong winds and turbulent dynamics on the transmission spectrum, especially in case of highly irra- diated tidally-locked exoplanets. One can also study detailed cloud physics by including condensation physics models and detailed scattering properties of clouds. For instance Lee et al.(2016) point out that clouds can have a major impact on the transmission spectra. Despite the detail-rich treatment of the physics in 3D-hydrodynamics models, they fall short by the timescales they can be run for and the amount of processes that can be included. For such questions one can use general circulation models, GCMs. They are based on solving the primitive equations of meteorology and treat horizontal motion and vertical motion in different manners. The atmosphere is generally assumed to be in hydrostatic equlibrium with small scale convection and horizontal flows parameterized for efficient solving and ra- diative transfer is accounted for by forcings. The advantage of GCMs is the vast amount of processes that can be included. From water cycle, to vegetation, even to the emerge of biosignatures to name a few examples.

To come back to our discussion about the characterization of exoplanets especially for habitability, GCMs are important tools, where 3D-hydrodynamical models are rather useful to better understand atmospheres of more extreme objects and extreme physics, such as hot Jupiters and magnetic field interactions between a planet and the host star. Thus, to answer the question of finding habitable worlds with the presently available tools, we must further study the climate and longterm evolution of these exotic worlds, and answer the questions about seasonality, atmospheric escape, magnetic fields and star-planet couplings, irradiation, stellar activity, stellar wind and finally the type and abundance of biosignatures we should be looking for Kopparapu et al.(2019). In combination with the limited knowledge about M-type stars, around which we find most of the exoplanets observable for spectroscopy, we can look forward to many interesting observations and findings in the near future, both from Earth based observatories like VLT with the high-resolution spectrograph CRIRES+, as well as from space based observatories such as the upcoming James Webb Space Telescope JWST, at mid-resolution.

53 4 Description and results of the CRIRES-planning-tool

CRIRES-planning-tool is a software tool developed over the course of my Master’s project under the supervision of Prof. Dr. Nikolai Piskunov and Dr. Andreas Korn, in collaboration with Dr. Alexis Lavail. The tool can be accessed on GitHub3

The CRIRES-planning-tool is intended to be used to plan transit observations of exoplanets for CRIRES+, the new cross-dispersed high-resolution infrared spectrograph for the ESO VLT (Follert et al., 2014). Observation of exoplanets can be planned in two ways. 1.) For a single planet candidate, referenced to it via its name during a userdefined timespan 2.) by finding all planet candidates fulfilling constraints, provided by the user (see section: 4.2). The known exoplanets fulfilling these constraints are downloaded from NASA Exoplanet Archive (Akeson et al., 2013) and each candidate is checked for its observability according to observational constraints from the Paranal observatory, Chile during the given time frame. Other catalogs or custom targets can be added. Each observable transit is checked for reaching a default minimum signal-to-noise ratio (S/N) per single exposure, allowing 20 exposures during a single transit. The S/N constraint will be updated in the future to distinguish between observations of super-Earths and Giant-planets on the fly. Each single exposure has a total exposure time Texp, calculated from the detector integration time DIT multiplied with the number of detector integrations NDIT : Texp = DIT × NDIT . Therefore, the transit length must be longer than 20 × Texp. DIT is initialized at DIT = 10 s and then adjusted to DIT = Texp/24 during each iteration until NDIT 16≤NDIT≤32. Candidates where the complete transit is fulfilling the constraints provided by the user (regardless of reaching 20 single exposures) are added to the list of observations, and further information can be found in the output files. The methods used for astronomical calculations are used from the astropy library (Price-Whelan et al., 2018), and the astroplan library (Morris et al., 2018). The tool comes with plotting tools and a command line window to access its functionalities.

The tool is initialized by running the file Transit_List.py and comes with a command line menu presenting the following options to the user:

1. Run a complete check of all available candidates from the NASA Exoplanet Archive fulfilling the constraints provided by the user for a certain time frame. The tool asks for the starting date and the number of days to look for observable transits and asks if the ETC part should also be run. Final results can only be obtained by running the ETC part as well.

2. Run only the ETC part, where each transit of each observable candidate is checked for the possibility of 20 exposures with each one of them reaching a minimum S/N. The transits are loaded from a stored file. This can be used for instance if something during the ETC part running option 1 goes wrong, and one wants to continue from where the problem occurred during the first run.

3. Check the observability of a single candidate referenced by its name for a given time frame and determines the necessary exposure time to reach 20 single exposures.

4. Other targets of interest to observe with CRIRES+ can be run in the same way as exoplanetary candidates. However, this feature is not included yet.

5. Make plots from a stored file, this option is also presented at the end of running 1, 2 or 3.

3https://github.com/jonaszubindu/CRIRES-planning-tool

54 The following plots can be produced:

1. Schedule for entire period, depicted in Figure8. If the estimated uncertainty of the predicted transit mid-time is greater than one hour, the transit is shown in orange. If the candidate is not reaching the minimum signal-to-noise ratio S/N with 20 exposures, the transit is shown in red. The error is estimated from the uncertainty in the measurement of the period of the exoplanet’s orbit and the uncertainty in the transit mid-time used to calculate the future transit times, see eq. (7).

2. Visibility plot of single night, depicted in Figure9.

3. Target finder image, target referenced by name, example is depicted in Figure 10.

These options will most probably be subject to future updates, as the tool is developed further and its application grows. In what follows, we will present more details about the functionality in section 4.1 of the tool and explain the most important methods, the parameters used, how these parameters are obtained and discuss areas of future improvement. In section 4.2 we present the applied constraints to the candidates from the NASA Exoplanet Archive and the observational constraints to be able to observe a transit from the Paranal observatory. How the exposure time calculator computes S/N for each transit is explained in detail in section 4.3 and what input parameters were used for the exposure time calculator is summarized in 4.4. How the tool was tested and verified is shortly explained in 4.5 and we present preliminary results in section 4.6.

4.1 Functionalities and methods of CRIRES-planning-tool The CRIRES-planning-tool consists of three necessary modules and one optional module misc, which controls the operation of the tool through the command line menu and could be replaced by some other operating menu or left out completely. The three necessary modules are:

• classes: Contains all the important data structures to store planet data, night data, and provide the methods to compute the observability of transits.

• Helper_fun: Contains all the functions not inherently connected to any of the class structures, such as plotting functions, excel writing and storing functions, and functions to operate the ETC part.

• Etc_form_class: Initializes the data structure with all input parameters to call the ETC, and automatically debug the input parameters if the ETC is returning an error.

The dependency tree for the different methods and modules is depicted in Figure 11. It provides a overview of how the different modules relate to each other and should guide any user who would like to make changes to the tool, to adjust all methods linked to the changed method or module. The three types of plots that can be generated with the tool are presented in the following figures. Figure 8 is a graphical depiction of the observable transits during a certain time-span, Transits that fulfill the minimum S/N criteria with 20 exposures and observation time error below 1 hour are depicted in blue.

55 WASP-77 A b : 2.16

HIP 65 A b : 0.786

HD 3167 b : 1.65

WASP-50 b : 1.83

WASP-140 b : 1.51

WASP-126 b : 3.42

TRAPPIST-1 d : 0.819 Planet name : : name Planet Transit duration [h] duration Transit TOI-540 b : 0.346

LTT 9779 b : 0.671

LP 714-47 b : 1.29

LHS 3844 b : 0.521

K2-141 b : 0.94

2020-11-16 2020-11-17 2020-11-18 2020-11-19 2020-11-20 Date

Figure 8: Schedule-like presentation of transits. Times are in UTC. Blue symbols are the transits that fulfill the minimum signal-to-noise ratio S/N criteria and the observational time uncertainty is less than one hour. All transits shown in red do not reach the minimum S/N criteria. Transits which were shown in orange would reach the minimum S/N, but would have an uncertainty in the observation time greater than one hour.

The transits not reaching 20 exposures are depicted in red, and transits with an observation time error above one hour are depicted in orange. In Figure9 an example of the visibility plot of a single night is given. The altitude of the targets appearing on the night sky during the selected night are plotted as a function of time. The red highlighted part is the time when the transit happens. It is only plotted, if the transit is detectable according to the minimum S/N criteria. The grey dashed line and solid line

56 stand for the Moon and Sun’s altitude. If a transit is getting closer than 5o to the zenith, the title of the plot contains a warning. This is due to the fact that the telescope is mounted in alt-azimuth and can not move through the zenith but has to turn by 90o at zenith. In Figure 10 a target finder image of a single target is presented. The target finder image is from the Digital Sky Survey (DSS). It can be used to identify a target on the night sky before observation.

Sun Moon -11-17, WARNING: Target close to Zenith! LHS1 3844 b HD 3167 b WASP-141 0 b

1

1

1

1

0 1

1

1

1 − 2 2 2 2 2 0−−

Figure 9: Plot which depicts the observable targets during a single night. Times are in UTC. The red part of the lines show the transit times that reached the minimum S/N.

Figure 10: Target finder image.

57 class class misc pickled_items class Exoplanets etc_form

__init__ Etc_calculator_Texp user_menu __init__ fun connect

Planet_finder aks_for_value Etc_calculator_SN update_etc_form misc.user_menu hasproperties wait_for_enter calculate_SN_ratio write_etc_format_file

1: Run full transit calculation class Nights run_etc_calculator 2: Run call ETC part extract_out_data for a list of transits

3: Run single __init__ etc_debugger transit planning airmass_moon_sep_obj_altaz

4: Run single target calculate_nights_paranal planning pickle_dumper_objects 58 CallETC 5: Plotting data of some result file class

Eclipses SN_Transit_Observation_Optimaization

__init__ SN_estimate_num_of_exp

Observability data_sorting_and_storing

load_Eclipses_from_file plotting_transit_data

find_target_image

plot_night

xlsx_writer

postprocessing_events Figure 11: Class inheritance structure, before and after the exposure time calculator processing. 4.2 Constraints for candidates For this thesis we used the constraints listed in Table1.

Property Value Planetary mass [MJup] ≤ 1 Declination [o] ≤ 10 Star J-mag ≤ 10 Star H-mag ≤ 10 Star Teff [K] ≤ 6000

Table 1: Constraints after which candidates were selected from the NASA Exoplanet Archive for observation with CRIRES+.

The constraints in Table1 were chosen by the CRIRES+ collaboration to limit the observations to planets in the regime of Earth, super-Earth, Mini-Neptune and Jupiter like planets around M-type, K-type, and G-type stars. However, the focus will be set on Earth and super-Earth-sized planets, since those are below previous observational limits of other high-resolution spectrometers. From the NASA Exoplanet Archive the Planetary Systems Composite Data are downloaded and filtered for the constraints in Table1 with the module Request_Table_NasaExoplanetArchive and the data is stored to the file PlanetList.csv. In Transit_List.py one can define the catalog from where the planetary data should be loaded. The standard operation is to load the filtered data from PlanetList.csv, however, users can provide their own planetary list, as long as it has the same structure and parameters as defined in PlanetList.csv. To check each candidate for its observability, the constraints are defined in Table2.

Constraint Value Altitude [o] ≥ 30 Airmass ≤ 1.7 Night Time (Alt Sun) [o] ≤ −18 Moon-Target Separation [o] ≥ 45

Table 2: Constraints after which candidates were selected from the NASA Exoplanet Archive for observation with CRIRES+.

Here, we use the definition for astronomical twilight to constrain the times when observations can be performed at the Paranal observatory, Chile. The constraints can be modified and are found in the file Transit_List.py. The coordinates of the observatory are optained using functionality from the astroplan library. The airmass is computed using functionality from the astropy library and involves a transformation of the coordinates of the object to observe from right ascension (ra) and declination (dec) to azimuth (az) and altitude (alt). The time is always taken in UTC. The attribute .secz of the transformed frame object is the plane-parallel approximation of the airmass. The coordinates for the targets in (ra, dec) and the transit times are provided by the CSV file PlanetList.csv. The transit times extracted from the NASA Exoplanet Archive are assumed to be given in the barycentric frame and are converted to the geocentric frame using the astropy.time library (see also Eq. (9)). The Moon-target separation and the altitude of the Sun and the coordinates for Moon and Sun are computed using functionalities from astropy and astroplan. To determine if a transit event is observable or not, it takes the constraints from Table2, the time to observe the transit, the coordinates of the observatory,

59 and the coordinates of the target in (ra, dec) as arguments and returns a Boolean. The function also takes lists of times to observe the target. The coordinates (ra, dec) of the Earth, Sun and Moon are updated with the IERS ephemeris data (Seidelmann, 1982) at the beginning of every execution of the planning tool. We have treated each transit separately by first determining the number of transits occurring during the given time frame for each object and then checking each transit separately by creating a list with the times for begin of the transit, transit mid-time and the end of the transit. The time span from the begin of the transit to the end of the transit is described in section 2.1, Figure 2, as the time between t1 and t4, covering Ttot. So far, we have not included any additional buffer before or after the transit or time buffer between each exposure. If the full transit is observable, the observational coordinates (az, alt), airmass, moon-target separation, and moon phase, together with the observation time are stored in a dictionary and added to a list called eclipse_observable. This list is an attribute of each candidate for which a class is generated called Eclipses. The structure of this class is presented in Figure 12.

Class inheritance Diagram: Eclipses

Name class Eclipses: time Obs time Attributes airmass Obs time error class method: observability moon sep __init__ attributes: Primary eclipse observable and moon phase name Transit Length Helper_fun-function: airmass_moon_sep_obj_altaz az epoch Effective Temperature period alt J-magnitude period_err Eclipse Begin transit_duration Eclipse Mid eccentricity Eclipse End star_Teff star_jmag Number of exposures possible Planets_eclipse Time necessary to reach 20 exposures [s] num_eclipses S/N overall median eclipse_observable Minimum S/N target_observable Helper_fun-function: Maximum S/N SN_estimate_num_of_exp/ Coordinates SN_Transit_Observation_Optimization Minimum Exposure Time Maximum Exposure Time List of Exposure Times

Name Effective Temperature J-magnitude Object w. o. primary eclipse observable? Obs Data

Figure 12: Class inheritance structure, before and after the Exposure Time Calculator processing.

60 It includes the necessary information to plan an observation. Time to reach 20 exposures is meant as how long it takes to reach 20 exposures with a minimum S/N, no matter how long the total transit is. Nevertheless, the goal will be to observe the complete transit. The Minimum SN and Maximum SN stand for the highest pixel S/N pixel value and the lowest S/N pixel value reached during all the exposures. In the smaller uncolored box, one can find the names of the corresponding methods to compute the values of the different attributes. The blue box on the left contains all information obtained from PlanetList.csv and creates the empty lists eclipse_observable and target_observable. The purpose of eclipse_observable has been explained above. target_observable is not configured yet, but was intended as a configurable proxy for observations of the target out of transit, to for instance obtain information about the host star with CRIRES+.

4.3 Signal-to-noise ratio and the exposure time calculator The ETC computes the signal-to-noise ratio S/N for each spectral resolution element corresponding to what the instrument receives, no matter if a transit is occurring or not. The decrease in flux in the presence of a transiting planet or the ratio between the transmitted signal through the exoplanet’s atmosphere and the total signal is not respected. The default minimum S/N per exposure is set at S/N = 100 median over all pixel in our planning tool. The wavelength range selected is λmin = 2291 nm, λmax = 2349 nm. The signal-to-noise ratio S/N in the infrared can be calculated in the following way (A.Modigliani, 1999). √ N · NDIT S/N = obj (199) p 2 Nobj + 2 · nbinsky + npix × RON + 2 · nbin × Dark · T and F · ∆ · T · E · S N = s (200) obj P −1 2 with Nx the number of electrons per bin [e /bin], F the incident flux in [W/m /µm], ∆s the spectral bin in [µm/bin], E the efficiency, S the telescope surface in m2, and P the energy of a single photon. RON is the read out noise of the CCD on-chip amplifier and npix is the number of integration pixels to evaluate the RON contribution to S/N. DARK represents the dark current, the read out current from the detector in absence of any incident light and nbin denotes the number of integration bins to evaluate the DARK contribution to S/N. T stands for the pre-selected integration time, which is denoted by DIT in our tool. To compute the exposure time Texp necessary for a certain S/N, one needs to solve for NDIT and multiply with T . For this purpose we are using the CRIRES+ Exposure Time Calculator4. Our tool optimizes DIT , respectively T such that NDIT lies between 16 and 32, under the given constraint for S/N. DIT is initialized at DIT = 10 s and then adjusted to DIT = Texp/24 during each iteration until NDIT 16≤NDIT≤32. If a minimum of 20 exposures is possible during a transit, the transit is regarded as detectable. This means it is possible to constrain the transmission spectra for atmospheric features at the chosen minimum S/N. The software tool has two ways of computing the number of exposures possible. One is the quick way by optimizing DIT and NDIT at transit mid-time and calculating simply: Ttot Nexp = ,Texp = NDIT × DIT, (201) Texp with Ttot the transit duration including ingress and egress. The other mode can be used if only the number of exposures possible Nexp for a single candidate is requested. Thereby, the software tool uses

4https://etctestpub.eso.org/observing/etc/crires

61 0 00 Texp at transit mid-time tc(n) and computes the next set of exposures Texp, Texp at

i=10 i=100 tc (n), tc (n) = tc(n) ± Texp/2, (202)

0 00 0 00 where stands for the (+) and the (−) sign. With the new Texp at time and Texp the next set of exposures at:

i=20 0 tc (n) = tc(n) + Texp/2 + Texp (203) i=200 00 tc (n) = tc(n) − Texp/2 − Texp (204) and so forth for i ∈ {−10, ..., −1, 0, 1, ...10}. Subsequently the overall median S/N for the selected wavelength range over the full transit is computed and is tabulated together with the number of exposures possible Nexp throughout the transit. Comparison of the two methods have turned out to produce similar exposure times Texp, with differences of less than 1 % for the candidate selection list we use the first method and for single candidate observation planning we use the second method.

4.4 Input parameters to the ETC, restrictions and future improvements The CRIRES+ ETC5 is divided into 5 sections.

The first section contains information about the target. We use "Point Source" and "Spectrum" for the spectral energy distribution SED. For the type of SED we tested the options "template" and "blackbody". The difference between templates and blackbody as SED can be up to 20 %, considering the exposure time Texp to reach S/N ≥ 100. The SED templates can be loaded from different catalogs, where we have used the MARCS plane-parallel geometry templates with Solar metallicity [Fe/H] = 0. The MARCS templates are available in effective temperatures Teff between 4000 K and 8000 K in steps of 500 K (Gustafsson, B. et al., 2008). However, the candidates we are interested in might go below Teff of 4000 K. The ETC is being updated with new templates to cover lower temperature objects down to 2500 K. For now, the CRIRES-planning-tool uses Teff = 4000 K for every candidate below 4000 K. This has of course an influence on the accuracy on the exposure time Texp estimate. We estimated the size of the effect as follows. Comparing the difference between 4000 K and 8000 K the exposure time changes by a factor of ≈ 2. Same comparison was done for blackbody spectrum with a temperature of Teff = 3000 K compared with 4000 K. We found a maximum decrease in Texp of up to 30 %, keeping the J-magnitude at 10. Assuming a precision of the blackbody spectrum of 20 %, Texp could decrease by ≈ 36 % from 4000 K to 3000 K using spectral templates. When the new templates become available the tool will be updated to cover the extended range of Teff , however, we do not expect the number of detectable planets to change much. We performed a test by looking for all detectable planets over a period of 30 days, comparing blackbody spectrum with accurate Teff to spectral templates with the lower limit of the effective temperature at Teff = 4000 K. We found 14 detectable transits and no difference in which transits are detectable between the two methods. We found a difference in exposure time Texp of 24 % for the candidate with the shortest exposure time. However, for the other targets Texp differs by ≈ 10 % from blackbody to spectral templates. The metallicity has effects on the order of a few percent on the exposure time Texp and Solar metallicity seams to be a robust compromise. For future applications the tool could be updated to include metallicity of the target object or other types of templates than MARCS plane-parallel approximation. Next, the brightness is set at the J-magnitude value of the target object in the Vega system.

5https://etctestpub.eso.org/observing/etc/crires

62 The second section takes the parameters airmass, Moon phase and distance between Moon and target. How these parameters are obtained is explained in section 4.2. The last input parameter in that section is the precipitable water vapour in mm and is unchanged at the typical value of 2.50. Testing the influence of this parameter with the ETC web-interface, we have not found relevant sensitivity. In the third section, called Seeing and IQ, we use the adaptive optics mode (AO), which requires information about the guide star (GS), the host star of the target planet in our case. The AO is based on the original CRIRES and requires the R-magnitude of the guide star and the spectral type. Due to restrictions from original CRIRES, the GS magnitude is restricted to 9.3 ≤ gsmag ≤ 16. Since the AO function is not implemented for CRIRES+ yet, the spectral type is unchanged and the R-magnitude is set equal to the J-magnitude of the target. We have performed a few calculations to estimate the impact and found differences on the order of a few seconds. Additionally, it is noted for the original CRIRES that only extreme colours of the GS have a considerable influence on the results of the ETC. It was not the idea to use the J-magnitude here for precision, but only to have a section in the planning tool for the guide star that can be updated as soon as the new AO module will become available. The fourth section includes information about the instrument, where we use the spectral setting K2217 with center wavelength λ = 2217 nm, a slit width of 0.2 arcseconds, and polarimetry disabled. The fifth section is the important part for the optimization of DIT and NDIT , mentioned in the previous section 4.3. The standard mode we use is "S/N per pixel" and the DIT is initially set at 10 s. The minimum and maximum wavelength were fixed at λmin = 2291 nm and λmax = 2349 nm. The input value for the mode "S/N per pixel" is the median of S/N over all pixels and we used 100 during all our tests. All the input data are written to a JSON file named etc-form.json using the functions update_etc_form and write_etc_format_file from the module Etc_form_class. The method run_etc_calculator calls the ETC and the result is stored in the JSON file etc-data.json. If calling the ETC returns some error, the function etc_debugger tries to find the input value that makes the ETC fail to compute the exposure time and returns the faulty input value to the user. With increasing amount of information that is returned by the ETC, in case a faulty input was used, the etc_debugger can be updated to handle the error in a more adequate way than simply cycling through all input values. If the reason for an error can not be found, the data is stored in a file and the user can continue running the ETC part using option 2 of the user menu, after fixing the error source. This is particularly useful, since it happens that the ETC returns an error without reason. By saving the present state of the processed data, this issue can be circumnavigated and the user can continue the ETC data processing from where it stopped. The ETC is regularly updated. Hence, it is necessary to regularly check if the above mentioned constraints to determine the input parameters to the ETC are still valid.

4.5 Test and comparison of results The tool was tested by comparing the results to a tool developed by Fabio Lesjak, a master student at the University of Göttingen. We both used the exoplanet data from the NASA Exoplanet Archive. To compare our results we computed the observable transits for the time span from the 2020-5-28 until the 2020-06-26. The differences we found between the two tools were missing transits due to missing parameters in the NASA Exoplanet Archive catalog data, which were required by the tool of Lesjak. In addition, the ETC part of Lesjak’s tool is fundamentally different to our tool. Lesjak used a formula from the old CRIRES ETC with a slit width of 0.4 arcseconds, compared to 0.2 arcseconds in our tool, one single wavelength instead of the new range of wavelengths of CRIRES+ and dependent only on

63 K-band magnitude, but not dependent on spectral type or effective temperature of the host star. Since the ETC provided by ESO returns generally more accurate and precise results, it is hard to compare both tools in that aspect.

To make sure that the computed airmass, moon-target separation, and moon-phase parameters are ro- bust, we also checked our results for those parameters with the Sky Model Calculator6 provided by ESO. The observational times were compared with results from the freely available Transit Finder7(Jensen, 2013), who use the NASA Exoplanet Archive and ExoFOP-TESS catalog8. The findings of our com- parison are as follows:

The main difference of our tool to the tool of Lesjak and the Transit Finder is the remote access of the ETC, and thus our tool provides the highest precision to constrain the detectability of exoplanets with CRIRES+. Other than for the ETC, the three tools do not differ much in functionalities. The transit times found by our tool and the tool of Lesjak match up to differences on the order of 1-2 minutes, and differ in the transits they found due to missing parameters or different selection criteria for some parameters. Also the Transit Finder predicts the same transits as our tool up to an accuracy of a few minutes for most targets. Other targets the Transit Finder has found, and our tool could not find, can be explained either because those candidates are not listed in the NASA Exoplanet Archive, or because they do not match our criteria to select candidates. However, for some transits predicted by all three tools we found discrepancies between the times of our and Lesjak’s tool, and the times of the Transit Finder of up to one hour. Comparing the planetary data, the only source for the discrepancy we found for some was the transit mid-time tc[0] (see eq. (6)). However, we also found similar discrepancies for candidates using the same parameters in our tool and found in the Transit Finder. Despite such great discrepancies, we trust our results since they agree with the Transit Finder results in most cases and our results nevertheless match Lesjak’s results in all cases, although we do not use the same methods to compute the observation times.

4.6 Results of detectable transits for one year The ranking and sorting of the results is performed by the method data_sorting_and_storing and postprocessing_events in the module Helper_fun. One could for instance wish to include the plan- etary mass into the ranking and build a custom ranking metric. The metric we use here is constructed as follows:

From all transits of a planet, take {Nexp | Nexp ≥ 20} : (205) X Nexp = Nexp/number_of_transits (206)

number_of_transits = #{Nexp | Nexp ≥ 20} (207) 2 rank = Nexp · number_of_transits. (208)

For planets which have no transits with Nexp ≥ 20: (209) rank = 0. (210)

The fully processed files get stored as CSV files and Excel files. The CSV files contain two lists. The first list contains all the detectable transits ranked after the previously mentioned method in (208).

6https://www.eso.org/observing/etc/bin/gen/form?INS.MODE=swspectr+INS.NAME=SKYCALC 7https://astro.swarthmore.edu/transits/transits.cgi 8https://exofop.ipac.caltech.edu/tess/

64 The second list contains the observational data at the begin, mid, and end of each transit. In the CSV file the observation times are chronologically sorted. The Excel files have the same lists as the CSV files, however, in the second list in the Excel file called Ranked Observations, the nights are grouped together in the following way:

1. Sort list of observations for number_of_exposures_possible

2. Make groups with observations during the same night

3. Merge groups of adjacent nights

4. Sum number_of_exposures_possible > 20 for each night

5. Rank according to the summed up number_of_exposures_possible per night

Notice that, if there are overlapping observations, they are counted both into the ranking, since it is not clear which observation should be selected. For comparison we ran three different scenarios: S/N = 100, 150, and 200 per exposure with 20 exposures per transit, over the course of one year. These values for S/N correspond to a cumulative S/Ntot computed from

p 2 S/Ntot = (number_of_transits) · 20 · S/N (211) and result in S/Ntot = 447, 671, and 894, number_of_transits set equal to 1. The minimum S/N values above correspond approximately to the minimum required cumulative S/Ntot for CRIRES+ to observe atmospheric absorption features of a hot Jupiter around a G-dwarf, a hot Neptune around a K-dwarf, or a super-Earth (2.5 R⊕) around an M-dwarf. The minimum cumulative S/Ntot increases with stellar spectral class, respectively the effective temperature of the host star. For comparison, the minimum cumulative S/Ntot of an Earth-twin around a G-dwarf is 150’000. Therefore, we can be sure not to determine the atmospheric composition of an Earth-twin in the near future with the methods at hand (Aronson and Waldén, 2015). The minimum cumulative S/Ntot is estimated from the ratio between the area of the planet’s expected atmospheric height to the unblocked area of the host star, and the flux of the host star.

65 Table 3: Results for minimum cumulative S/Ntot ≥ 447.

Name Planetary Number Effective Transit Radius [R⊕] of transits Temperature [K] Length [h] AU Mic b 4.2 4 3700 3.5 HD 21749 c 0.896 3 4640 2.52 GJ 357 b 1.2208 7 3505 1.53 HD 213885 b 1.7472 29 5978 1.53 LTT 1445 A b 1.3776 8 3337 1.38 HD 15337 b 1.6352 4 5125 2.49 HD 15337 c 2.3856 1 5125 2.25 HD 23472 b 1.8704 1 4813 2.70 HIP 67522 b 10.0576 3 5675 4.82 WASP-69 b 12.432 4 4700 2.23 HD 3167 b 1.7024 12 5528 1.65 GJ 9827 c 1.2432 6 4340 1.83 L 168-9 b 1.3888 20 3800 1.27 L 98-59 c 1.344 9 3412 1.25 K2-233 d 2.6432 1 4950 3.81 TOI-763 c 2.632 1 5450 3.66 GJ 9827 b 1.5792 20 4340 1.27 L 98-59 b 0.7952 13 3412 1.02 GJ 9827 d 2.016 4 4340 1.22 TOI-421 c 5.0848 1 5325 2.71 WASP-127 b 15.344 3 5620 4.3

Table 4: Results for minimum cumulative S/Ntot ≥ 671.

Name Planetary Number Effective Transit Radius [R⊕] of transits Temperature [K] Length [h] AU Mic b 4.2 4 3700 3.5 HD 21749 c 0.896 3 4640 2.52 GJ 357 b 1.2208 7 3505 1.53 HD 15337 b 1.6352 5 5125 2.49 HD 213885 b 1.7472 29 5978 1.52 LTT 1445 A b 1.3776 7 3337 1.38 HD 15337 c 2.3856 1 5125 2.25 HD 23472 b 1.8704 2 4813 2.70

66 Table 5: Results for minimum cumulative S/Ntot ≥ 894.

Name Planetary Number Effective Transit Radius [R⊕] of transits Temperature [K] Length [h] AU Mic b 4.2 4 3700 3.5 HD 21749 c 0.896 3 4640 2.52

The planets we found having transits reaching the minimum S/N criteria are presented in Tables3,4, and5 and in Figures 13, and 14. In Figure 13, we present the number of planets binned after planetary radius and the number of detectable transits, and in Figure 14, the planets binned after the effective temperature of the host star.

minimum SN = 447.2 minimum SN = 447.2 minimum SN = 670.8 minimum SN = 670.8 minimum SN = 894.4 minimum SN = 894.4 10 5

8 4

6 3

4 2 Number of Planets of Number Planets of Number

2 1

0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 5 10 15 20 25 30 Planet Radius [Earth radii] Number of observable transits

Figure 13: On the left side we have presented planets binned after radius in Earth radii and on the right after number of detectable transits for the three different minimum cumulative S/Ntot = 447, 671, and 894.

Only two planets reached a higher S/N than 200. One with a radius of about 0.9 R⊕, corresponding to the super-Earth HD 21749 c, with 3 transits, and a Neptune-sized planet called AU Mic b with a radius of 4.2 R⊕, with 4 transits. The latter even reached S/N ≥ 500 in three transits. Thus, planets with similar parameters as AU Mic b and HD 21749 c could be well detectable at higher S/N ≥ 200. In total we found 21 planets detectable with S/N ≥ 100 in 154 transits and 8 planets detectable with S/N ≥ 150 in 58 transits. We found many more transits occurring during the selected time frame, however, those are not detectable according to our choices for S/N with a single transit.

67 minimum SN = 447.2 minimum SN = 670.8 minimum SN = 894.4 3.0

2.5

2.0

1.5

Number of Planets of Number 1.0

0.5

0.0 3500 4000 4500 5000 5500 6000 Stellar Effective Temperature

Figure 14: Planets binned after stellar effective temperature of the host star for the three different minimum cumulative S/Ntot = 447, 671, and 894.

In Figure 14 we can see the distribution of the number of detectable planets according to the host star temperature. The host stars of AU Mic b and HD 21749 c correspond to an M-dwarf and a K-dwarf. Since S/N ≥ 200 is the predicted minimum cumulative S/Ntot to observe a Neptune-sized planet around a K-dwarf, we can say AU Mic b is detectable during one transit. HD 21749 c has a radius of 0.896R⊕. To be able to detect atmospheric feature in the atmosphere of HD 21749 c we would need a minimum cumulative S/Ntot ≥ 5700. Subsequent analysis has shown that the maximum S/Ntot reached by HD 21749 c is ∼ 1600. Thus, HD 21749 c is not detectable in one transit and we would need to combine transits to reach the minimum cumulative S/Ntot. Generally we can say, candidates residing in the super-Earth radius regime, a S/N ≥ 100 per exposure will not be enough to observe their atmospheric features. We also found three Jupiter-sized planets with S/N ≥ 100: HIP 67522 b, WASP-69 b, and WASP-127 b. The corresponding effective temperatures of their host stars are: Teff = 5675.0 K, 4700.0 K, and 5620.0 K. Thus, comparing to Aronson and Waldén(2015) minimum cumulative S/N, WASP-69 b, HIP 67522 b and WASP-127 b are detectable planets. The planets found to reach S/N ≥ 150 per exposure are all in the radius regime of super-Earths and are not detectable during a single transit. There is one more planet with a radius of 5.0848 R⊕ and a stellar effective temperature of 5325.0 K of the host star, which corresponds to a Neptune-sized planet around a star at the border between K-dwarf and G-dwarf and therefore does not reach the minimum S/N to be detectable in one transit. The complete data output for this analysis is presented in the Appendix:6. The corresponding minimum cumulative S/N per exposure reached is depicted in the first column of each table.

68 5 Discussion

We have presented a tool to plan future observations of transits with CRIRES+ and elaborated on its functionalities, and limits. The ETC has been tested for the sensitivity to each input parameter. Especially since the lower limit of the stellar spectral templates’ effective temperature is 4000 K, we have tested the influence of using lower temperatures by comparing to using a blackbody spectrum with the exact effective temperatures of the host stars. We compared the exposure times of an effective temperature of 4000 K and 8000 K, at a J-magnitude of 10 and keeping all other parameters fixed. Our analysis has shown that exposure times can differ up to 20 % comparing blackbody spectrum to stellar spectral templates with the same effective temperature. Comparing the exposure times using blackbody spectrum and an effective temperature of 4000 K and 3000 K, the exposure time could decrease by 36%. The decrease was highest for a J-magnitude of 10. Therefore, under fixed minimum cumulative S/Ntot, faint targets with a lower effective temperature than 4000 K currently rejected by our tool might turn out to be detectable. Nevertheless, comparison of results using blackbody or spectral templates for targets over the course of 30 days has not shown a difference in detectability of transits. A comparison over the course of one year could maybe show an increased number of detectable planets than what we have found yet. However, for detections of a planetary atmospheres around faint, cool stars a higher minimum S/N than for stars with Teff > 4000 K is required. Therefore, the extended temperature range of the spectral templates most likely will enlarge the list of detectable gas-planets, but not of super-Earth or Earth sized planets. All other parameters that might change in the future, such as the guide star parameters for the adaptive optics have been tested for their influence by hand and we found a maximum variation of up to 10 %. However, since the guide star module still corresponds to the old CRIRES, we can not expect to get more accurate results by using more accurate input parameters for the guide star. Hence, although some parts of the ETC will change in the future, our tests have shown that the results from the ETC are not expected to change much.

We tested our tool in different fashions to make sure the computation of our input parameters and transit times are correct. In particular we compared our tool with the a tool developed for the same purpose by Fabio Lesjak, and found good agreement in the predicted observable transits and observation times. However, we have found discrepancies between the times computed by Lesjak and me in comparison to the tool Transit Finder(Jensen, 2013). Some of the transit times mismatch by up to one hour. Despite cross checking the used parameters for planetary candidates, we could not identify the source of the discrepancies. The only difference found for some of these mismatching candidates is the transit mid- time, which the future transits were estimated from. The transit mid-time comes from observations itself and different transit mid-times between different exoplanet catalogs leading to such big errors would mean that the determination of the transit mid-time is still very imprecise and have greater errors than currently reported. However, we have also found candidates with mismatching transit observation times of up to one hour with the same planetary parameters, used in the Transit Finder, the tool of Lesjak, and our tool. Nonetheless, we have also found many transits agreeing with our findings. Furthermore, since the results found by Lesjak and us agree well in all cases, we trust our method and our results.

We presented a list of detectable planets according to constraints for our tool defined in Tables1 and 2, and minimum S/N for detection according to Aronson and Waldén(2015). Four planets were found to be indisputably detectable in a single transit, namely the Neptune-sized planet AU Mic b, and the Jupiter-sized planets HIP 67522b, WASP-69 b, and WASP-127 b. The super-Earth HD 21749 c was found not being detectable during a single transit due to the high minimum cumulative S/Ntot ≥ 5700

69 despite reaching S/Ntot = 894. More planets found by TESS and CHEOPS in the future could add to the NASA Exoplanet Archive and enlarge the set of detectable planets. The other planets presented in the appendix6, do reach certain minimum S/N, however, the analysis in comparison to the minimum cumulative S/Ntot provided by Aronson and Waldén(2015) showed, that most of these planets are not detectable during a single transit. It needs to be pointed out, that the minimum cumulative S/Ntot from Aronson and Waldén(2015) is related to the method described in their paper and other methods might be able to observe atmospheric features at a lower minimum cumulative S/Ntot, such as cross- correlation methods described in section 2.2.2 and used for example by de Kok et al.(2013). We found that the set of detectable planets is quite sensitive the regime of minimum S/N we imposed and a more careful analysis will be necessary to constrain each planets detectability in detail. This will be subject to future work. However, we can already tell now that finding a second Earth around a Sun-like star will not be possible with CRIRES+ and unlikely with observational campaigns in the near future.

6 Conclusion

The results presented in this work are preliminary findings, subject to future investigation and careful analysis, as the work with CRIRES+ will continue. The purpose of our work here was exclusively to provide a planning tool for future transit observations with CRIRES+. Our tests and comparisons have shown that our tool is fulfilling its purpose and we have pointed out the ways to customize the constraints, the candidate catalog, and the parameters used. Preliminary results have shown that the detection of exoplanets for characterization of the atmospheric composition with CRIRES+ will be feasable and characterization of atmospheres of exoplanets below the limit of 5 Earth masses (i.e. reaching into the regime of sub-Neptunes/super-Earths) will be possible.

On top of that, we have presented an extensive literature analysis of the currently most promising methods and models for exoplanet atmosphere characterization. We discussed the difficulties related to the different methods and observational limitations. We also explained the differences between different approaches to model a planetary atmosphere and distinguished between the application purposes. Together with the planning tool, this manuscript should give the attentive reader a brief introduction to the field of transit spectroscopy of exoplanets and atmospheric characterization, and provide a robust and customizable tool to plan future observations with CRIRES+.

Acknowledgements

I would like to thank my supervisors Prof. Dr. Nikolai Piskunov and Dr. Andreas Korn for their support and patience during my work. I would also like to thank Dr. Alexis Lavail for the discussions and feedback he provided and Dr. Ulrike Heiter for examining the manuscript on scientific correctness. A special thank goes to Jakob Vinther, at ESO Garching who maintains the exposure time calculator ETC and who was of great support during the process of writing the CRIRES-planning-tool, debugging, and testing the interaction with the ETC. Finally, I would like to thank Fabio Lesjak for his collaboration in testing our tools and result comparison.

70 Appendix D137c22-00 65:8000 .552 .5 0140 6.11020 1 1 8 8 8 4 4 2500 2116 4 4 25312 25312 29 25312 12544 12544 29 160 210 12544 8 29 12544 29 8 162.31 160.65 90 8 90 94221 8 160 90 163.80 8 160 94221 163.85 0 0 160 94221 162.30 163.92 25312 160 94221 162.05 29 25312 0 162.31 29 104.07 25312 101.60 0 162.30 29 96 25312 0 0 29 25312 101.87 0 96 29 158.47 102.11 0 96 29 86 104.04 102.50 0 94221 96 29 158.45 103.06 86 94221 29 158.42 160.61 50 104.09 86 94221 46 29 158.39 0 104.03 86 94221 160.63 29 86 94221 160.63 29 0 55 29 94221 160.51 7.553 0 55 101.61 7.865 0 29 94221 160.55 96 56 0 55 94221 0 101.54 96 29 56 7.294 94221 0 101.45 96 29 56 158.36 101.24 7.294 94221 0 101.32 96 29 56 158.49 7.553 7.294 101.32 94221 0 94221 96 29 5125 158.48 4813 7.553 101.39 96 29 158.28 94221 7.553 100.61 29 57 2.695974 96 0 158.32 7.553 100.88 3337 94221 96 29 0 2.25 158.36 57 3337 94221 0.426 96 29 0 158.39 7 57 5125 3337 94221 57 04:13:39 0.0304 1.3776 101.17 96 29 0 6.806 2020-12-25 158.42 57 5125 94221 06:50:08 1.3776 101.67 2021-10-09 96 29 0 57 158.45 96 7 0.0012 5125 94221 b 6.806 1.3776 101.62 29 0 23472 57 158.27 7 94221 HD 05:36:07 0.0036 5125 96 2.49 2020-12-06 c 6.806 100.90 29 0 15337 57 7.294 158.50 7 HD 158.51 02:34:08 0.0195 b 2.49 94221 2021-01-18 6.806 A 0.01037 101.03 96 0 57 1445 7 LTT 02:53:27 7.294 b 2.49 94221 2021-10-29 158.22 07:29:05 7 A 0.00068 101.16 96 0 2021-08-31 1445 5978 LTT 26047 7.294 b 2.49 94221 05:04:36 A 0.01105 101.32 96 0 2020-12-03 1445 158.52 7 LTT 57 7.294 1.534232 94221 b 08:03:18 0.01258 101.45 5978 96 0 15337 2021-09-19 158.23 0 26047 HD 57 7.294 94221 b 03:22:38 101.54 5978 96 15337 2021-11-01 158.52 3337 26047 0.0042 HD 57 1.534232 96 94221 0 b 100.87 5978 15337 158.33 08:58:56 26047 3 0.004179 HD 57 1.534232 2021-06-06 3337 b 6.806 101.73 96 0 15337 101.78 158.37 08:47:27 26047 3 0.004158 HD 57 1.534232 2021-06-05 1.3776 3337 158.40 b 6.806 26047 96 0 213885 08:35:58 4 3 0.004137 HD 57 2021-06-04 100.71 3337 b 90 6.806 1.3776 96 0 213885 158.51 08:24:29 26047 4 HD 57 2021-06-03 0.018 3337 b 6.806 1.3776 101.80 96 0 213885 158.50 4 HD 57 07:50:43 4 0.0204 90 2021-10-02 b 6.806 1.3776 100.76 96 160.74 0 213885 158.43 HD 57 04:43:47 0.0003 b 90 0 2021-11-14 6.806 A 1.3776 101.81 199692 5978 96 1445 158.46 LTT 57 03:45:03 0.0171 b 90 2020-11-20 160.71 6.806 A 101.07 199692 5978 0 1445 158.49 LTT 57 1.534232 06:01:44 2528100 0.0147 b 90 57 2021-09-16 160.87 6.806 A 101.21 199692 5978 0 1445 158.52 90 0.004116 LTT 1.534232 09:12:38 2528100 101.34 b 2021-08-04 160.76 6.806 A 0 5978 57 0 1445 08:13:00 0.004221 LTT 1.534232 2021-06-02 2528100 b 90 2528100 160.75 6.806 A 101.78 5978 0 1445 09:10:26 0.006153 LTT 160.87 57 1.534232 2021-06-07 0 b 6.806 101.74 5978 0 6.806 213885 02:53:17 0.006027 HD 57 1.534232 2021-09-08 0 100.70 b 160.85 101.47 5978 0 213885 01:43:34 35 0.006468 HD 57 1.534232 6.806 2021-09-02 0 b 101.56 5978 213885 05:47:49 35 0.006447 HD 57 1.534232 2021-09-23 100.51 0 b 6.806 16 101.69 5978 213885 05:36:10 0 35 0.006426 HD 57 1.534232 2021-09-22 162.13 101.53 b 6.806 16 101.80 5978 57 213885 05:24:32 0.006405 HD 1.534232 2021-09-21 162.03 100.85 0 b 6.806 16 5978 213885 161.59 16 5978 05:12:53 0.006384 HD 57 1.534232 2021-09-20 162.07 100.79 b 6.806 213885 161.65 101.50 05:01:15 0.006489 HD 57 1.534232 2021-09-19 5978 1.534232 b 6.806 213885 161.69 05:59:28 61 161.71 0.006321 HD 0 57 6.806 2021-09-24 101.40 b 5978 1.534232 213885 04:26:20 0.0063 HD 0 57 2021-09-16 b 6.806 0 5978 61 213885 04:14:42 0.00651 HD 0 57 1.534232 2021-09-15 b 6.806 0 5978 61 213885 06:11:06 102.05 0.006279 HD 57 1.534232 2021-09-25 7.337 b 6.806 0 5978 61 213885 0 04:03:04 101.57 0.006006 HD 1.534232 2021-09-14 b 101.09 6.806 5978 61 213885 01:31:57 101.72 0.006258 HD 1.534232 2021-09-01 7.337 5978 61 b 101.45 6.806 213885 03:51:26 0.006048 HD 1.534232 2021-09-13 7.337 number b 101.71 6.806 Total 5978 1.534232 101.83 61 213885 01:55:11 0.006069 HD 2021-09-03 7.337 b 5978 213885 02:06:48 0.00609 HD 1.534232 2021-09-04 7.337 3505 b 5978 rank 213885 7.337 02:18:25 0.006216 HD 258 1.534232 2021-09-05 b Exposure 5978 213885 Average 03:28:10 0.006195 HD 258 1.534232 2021-09-11 7.337 3505 b 795 5978 Maximum 213885 03:16:32 0.006111 HD 258 1.534232 2021-09-10 3505 b Minimum 1.53 795 5978 213885 02:30:02 0.006132 HD 1.534232 2021-09-06 3505 median 6.081 b 795 S/N 213885 795 02:41:40 0.006174 HD 1.534232 2021-09-07 3505 1.53 6.081 b 0.002 213885 3505 03:04:55 0.006237 HD 5.436 2021-09-09 1.53 6.081 b 06:16:53 0.00336 213885 2021-02-23 03:39:48 HD 5.436 2021-09-12 3505 1.53 b 02:04:54 0.00216 213885 2021-05-01 HD 5.436 of 5.436 1.53 Number b 02:57:24 0.00224 213885 2021-03-03 1.53 HD J-magnitude b 01:17:43 4640 0.00088 357 2021-03-07 GJ 1.53 b 05:36:37 4640 0.0008 357 2020-12-30 GJ 3700 b 07:16:47 4640 0.00344 357 2020-12-26 GJ 3700 2.515 b 00:25:34 357 2021-05-05 GJ Effective 3700 2.515 3700 b 0.02193 357 GJ 2.515 06:25:06 b 0.00051 3.5 2021-10- 0 ro h egh[]Tmeaue[]epsrspsil / / ie[]o transits of [s] Time S/N S/N possible exposures [K] Temperature [h] Length [h] error 100 =

71 Name obs time obs time Transit Effective J-magnitude Number of S/N median Minimum Maximum Average Exposure rank Total number S/N = 100 error [h] Length [h] Temperature [K] exposures possible S/N S/N Time [s] of transits HIP 67522 b 2021-05-28 02:31:24 0.000448 4.822 5675 8.587 39 101.27 0 156.99 440 4563 3 HIP 67522 b 2021-05-14 04:27:28 0.000416 4.822 5675 8.587 39 101.04 0 156.94 440 4563 3 HIP 67522 b 2021-05-21 03:29:23 0.000432 4.822 5675 8.587 39 101.19 0 156.97 440 4563 3 WASP-69 b 2021-06-23 05:58:02 0.000114 2.2296 4700 8.032 38 102.38 0 161.84 210 5776 4 WASP-69 b 2021-07-20 07:46:57 0.000128 2.2296 4700 8.032 38 102.17 0 161.79 210 5776 4 WASP-69 b 2021-09-24 02:01:06 0.000162 2.2296 4700 8.032 38 102.91 0 161.94 210 5776 4 WASP-69 b 2021-05-23 07:20:42 9.8E-05 2.2296 4700 8.032 38 101.80 0 161.72 210 5776 4 HD 3167 b 2021-08-02 06:35:24 0.00756 1.65 5528 7.548 34 100.39 0 157.16 170 13804 12 HD 3167 b 2020-11-17 03:13:53 2.8E-05 1.65 5528 7.548 34 100.59 0 157.21 170 13804 12 HD 3167 b 2020-11-18 02:15:50 5.6E-05 1.65 5528 7.548 34 101.02 0 157.32 170 13804 12 HD 3167 b 2021-07-07 08:48:39 0.006804 1.65 5528 7.548 34 100.74 0 157.25 170 13804 12 HD 3167 b 2021-11-08 03:42:33 0.010416 1.65 5528 7.548 34 100.67 0 157.23 170 13804 12 HD 3167 b 2021-09-18 07:02:37 0.008932 1.65 5528 7.548 34 100.68 0 157.23 170 13804 12 HD 3167 b 2021-10-13 05:50:52 0.00966 1.65 5528 7.548 34 100.36 0 157.14 170 13804 12 HD 3167 b 2021-10-14 04:52:45 0.009688 1.65 5528 7.548 34 100.90 0 157.30 170 13804 12 HD 3167 b 2021-10-15 03:54:38 0.009716 1.65 5528 7.548 34 101.13 0 157.35 170 13804 12 HD 3167 b 2021-11-09 02:44:29 0.010444 1.65 5528 7.548 34 101.06 0 157.33 170 13804 12 HD 3167 b 2021-11-10 01:46:26 0.010472 1.65 5528 7.548 34 101.13 0 157.35 170 13804 12 HD 3167 b 2020-12-12 02:05:15 0.000756 1.65 5528 7.548 33 103.11 0 161.66 180 13804 12 GJ 9827 c 2021-08-30 07:09:40 0.0050007 1.82496 4340 7.984 32 102.03 0 161.44 200 6144 6 GJ 9827 c 2021-09-10 05:49:12 0.0051906 1.82496 4340 7.984 32 102.32 0 161.51 200 6144 6 GJ 9827 c 2021-10-02 03:09:08 0.0055704 1.82496 4340 7.984 32 102.55 0 161.55 200 6144 6 GJ 9827 c 2021-10-13 01:49:32 0.0057603 1.82496 4340 7.984 32 102.35 0 161.52 200 6144 6 GJ 9827 c 2021-06-18 08:11:52 0.0037347 1.82496 4340 7.984 32 101.49 0 161.33 200 6144 6 GJ 9827 c 2021-08-19 08:30:25 0.0048108 1.82496 4340 7.984 32 101.45 0 161.32 200 6144 6 L 168-9 b 2021-09-02 08:13:54 0.03726 1.264937 3800 7.941 28 100.23 0 162.00 160 15680 20 L 168-9 b 2021-09-26 04:03:49 0.04032 1.264937 3800 7.941 28 101.24 0 162.21 160 15680 20 L 98-59 c 2020-11-18 06:47:56 0.0003 1.252186 3412 7.933 28 100.40 0 162.55 160 7056 9 L 168-9 b 2021-08-05 07:30:22 0.03366 1.264937 3800 7.941 28 101.24 0 162.21 160 15680 20

L 168-9 b 2021-08-26 08:02:54 0.03636 1.264937 3800 7.941 28 100.68 0 162.09 160 15680 20 72 L 168-9 b 2021-08-12 07:41:08 0.03456 1.264937 3800 7.941 28 101.14 0 162.19 160 15680 20 L 168-9 b 2021-08-19 07:51:59 0.03546 1.264937 3800 7.941 28 100.97 0 162.15 160 15680 20 L 168-9 b 2021-09-19 03:52:35 0.03942 1.264937 3800 7.941 28 101.25 0 162.21 160 15680 20 L 168-9 b 2021-08-29 03:19:18 0.03672 1.264937 3800 7.941 28 100.72 0 162.10 160 15680 20 L 168-9 b 2021-07-29 07:19:40 0.03276 1.264937 3800 7.941 28 101.26 0 162.21 160 15680 20 L 168-9 b 2021-09-05 03:30:19 0.03762 1.264937 3800 7.941 28 101.00 0 162.16 160 15680 20 L 98-59 c 2020-12-25 04:28:45 0.0033 1.252186 3412 7.933 28 100.45 0 162.56 160 7056 9 L 168-9 b 2021-07-01 06:37:35 0.02916 1.264937 3800 7.941 28 100.35 0 162.03 160 15680 20 L 98-59 c 2021-01-05 06:10:55 0.0042 1.252186 3412 7.933 28 100.93 0 162.67 160 7056 9 L 168-9 b 2021-07-22 07:09:02 0.03186 1.264937 3800 7.941 28 101.16 0 162.20 160 15680 20 L 168-9 b 2021-10-24 04:49:13 0.04392 1.264937 3800 7.941 28 100.23 0 162.00 160 15680 20 L 168-9 b 2021-10-17 04:37:48 0.04302 1.264937 3800 7.941 28 100.69 0 162.09 160 15680 20 L 168-9 b 2021-10-10 04:26:26 0.04212 1.264937 3800 7.941 28 100.98 0 162.15 160 15680 20 L 168-9 b 2021-09-12 03:41:25 0.03852 1.264937 3800 7.941 28 101.16 0 162.20 160 15680 20 L 168-9 b 2021-10-03 04:15:06 0.04122 1.264937 3800 7.941 28 101.15 0 162.19 160 15680 20 L 168-9 b 2021-07-08 06:48:00 0.03006 1.264937 3800 7.941 28 100.75 0 162.11 160 15680 20 L 168-9 b 2021-07-15 06:58:29 0.03096 1.264937 3800 7.941 28 101.01 0 162.16 160 15680 20 L 98-59 c 2021-03-20 01:32:26 0.0102 1.252186 3412 7.933 28 100.91 0 162.66 160 7056 9 L 98-59 c 2021-02-11 03:51:30 0.0072 1.252186 3412 7.933 28 100.92 0 162.67 160 7056 9 L 98-59 c 2021-01-16 07:53:05 0.0051 1.252186 3412 7.933 28 100.25 0 162.52 160 7056 9 L 98-59 c 2021-03-31 03:14:50 0.0111 1.252186 3412 7.933 28 100.14 0 162.49 160 7056 9 L 168-9 b 2020-12-10 01:31:10 0.00306 1.264937 3800 7.941 28 100.37 0 162.03 160 15680 20 L 98-59 c 2021-01-31 02:09:18 0.0063 1.252186 3412 7.933 28 100.48 0 162.57 160 7056 9 L 98-59 c 2021-02-22 05:33:43 0.0081 1.252186 3412 7.933 28 100.21 0 162.50 160 7056 9 K2-233 d 2021-05-11 05:09:51 0.0168 3.808 4950 8.968 24 101.48 0 158.26 560 576 1 TOI-763 c 2021-04-17 03:45:39 0.0689 3.656382 5450 8.858 23 101.77 0 157.73 570 529 1 GJ 9827 b 2021-10-09 04:19:50 0.0018711 1.2648 4340 7.984 22 102.14 0 161.47 200 9680 20 GJ 9827 b 2021-10-03 03:14:51 0.00183645 1.2648 4340 7.984 22 102.57 0 161.55 200 9680 20 L 98-59 b 2020-12-07 07:21:12 0.004 1.02 3412 7.933 22 100.89 0 162.66 160 6292 13 Name obs time obs time Transit Effective J-magnitude Number of S/N median Minimum Maximum Average Exposure rank Total number S/N = 100 error [h] Length [h] Temperature [K] exposures possible S/N S/N Time [s] of transits GJ 9827 b 2021-11-01 03:39:31 0.00200277 1.2648 4340 7.984 22 101.45 0 161.32 200 9680 20 GJ 9827 b 2021-09-27 02:09:56 0.0018018 1.2648 4340 7.984 22 102.02 0 161.44 200 9680 20 GJ 9827 b 2021-09-16 05:01:20 0.00173943 1.2648 4340 7.984 22 102.50 0 161.54 200 9680 20 GJ 9827 b 2021-09-10 03:56:40 0.00170478 1.2648 4340 7.984 22 102.33 0 161.52 200 9680 20 GJ 9827 b 2021-09-05 07:53:00 0.00167706 1.2648 4340 7.984 22 100.78 0 161.18 200 9680 20 GJ 9827 b 2021-10-26 02:34:14 0.00196812 1.2648 4340 7.984 22 102.42 0 161.53 200 9680 20 GJ 9827 d 2021-09-01 06:18:00 0.0029422 1.2228 4340 7.984 22 102.36 0 161.52 200 1936 4 GJ 9827 d 2021-10-27 01:51:02 0.0035056 1.2228 4340 7.984 22 102.58 0 161.55 200 1936 4 GJ 9827 d 2021-09-26 01:38:27 0.0031926 1.2228 4340 7.984 22 101.48 0 161.33 200 1936 4 L 98-59 b 2020-11-28 07:03:34 0.0024 1.02 3412 7.933 22 100.72 0 162.63 160 6292 13 GJ 9827 d 2021-08-01 06:09:24 0.0026292 1.2228 4340 7.984 22 102.15 0 161.47 200 1936 4 GJ 9827 b 2021-08-30 06:48:30 0.00164241 1.2648 4340 7.984 22 102.20 0 161.49 200 9680 20 TOI-421 c 2021-02-14 02:35:11 0.0021 2.71 5325 8.547 22 101.42 0 157.95 430 484 1 L 98-59 b 2020-12-16 07:38:48 0.0056 1.02 3412 7.933 22 100.92 0 162.67 160 6292 13 L 98-59 b 2020-11-19 06:45:54 0.0008 1.02 3412 7.933 22 100.41 0 162.55 160 6292 13 GJ 9827 b 2021-09-04 02:52:06 0.00167013 1.2648 4340 7.984 22 101.19 0 161.27 200 9680 20 GJ 9827 b 2021-07-15 08:16:47 0.00137907 1.2648 4340 7.984 22 102.55 0 161.55 200 9680 20 GJ 9827 b 2021-08-18 04:39:45 0.00157311 1.2648 4340 7.984 22 101.94 0 161.42 200 9680 20

L 98-59 b 2021-03-05 04:12:41 0.0196 1.02 3412 7.933 22 100.51 0 162.57 160 6292 13 73 L 98-59 b 2021-02-15 03:37:25 0.0164 1.02 3412 7.933 22 100.92 0 162.67 160 6292 13 L 98-59 b 2021-02-06 03:19:49 0.0148 1.02 3412 7.933 22 100.89 0 162.66 160 6292 13 L 98-59 b 2021-03-14 04:30:22 0.0212 1.02 3412 7.933 22 100.04 0 162.47 160 6292 13 L 98-59 b 2021-05-05 00:12:53 0.0304 1.02 3412 7.933 22 100.50 0 162.57 160 6292 13 WASP-127 b 2021-02-21 05:44:42 4.8E-05 4.308 5620 9.092 22 101.04 0 156.89 700 1408 3 WASP-127 b 2021-03-14 03:06:41 5.8E-05 4.308 5620 9.092 22 100.96 0 156.86 700 1408 3 GJ 9827 b 2021-08-13 08:36:21 0.00154539 1.2648 4340 7.984 22 101.79 0 161.39 200 9680 20 L 98-59 b 2021-05-14 00:30:52 0.032 1.02 3412 7.933 22 100.04 0 162.47 160 6292 13 GJ 9827 b 2020-11-30 01:23:33 7.623E-05 1.2648 4340 7.984 22 101.83 0 161.40 200 9680 20 GJ 9827 b 2021-06-16 07:58:14 0.00121275 1.2648 4340 7.984 22 101.04 0 161.23 200 9680 20 GJ 9827 b 2021-06-22 09:02:04 0.0012474 1.2648 4340 7.984 22 102.26 0 161.50 200 9680 20 GJ 9827 b 2021-07-09 07:12:52 0.00134442 1.2648 4340 7.984 22 101.90 0 161.41 200 9680 20 GJ 9827 b 2021-07-21 09:20:45 0.00141372 1.2648 4340 7.984 22 102.23 0 161.50 200 9680 20 GJ 9827 b 2021-08-01 06:28:01 0.00147609 1.2648 4340 7.984 22 102.29 0 161.51 200 9680 20 L 98-59 b 2021-01-28 03:02:15 0.0132 1.02 3412 7.933 22 100.71 0 162.62 160 6292 13 L 98-59 b 2021-01-19 02:44:41 0.0116 1.02 3412 7.933 22 100.40 0 162.55 160 6292 13 GJ 9827 b 2021-08-07 07:32:09 0.00151074 1.2648 4340 7.984 22 102.54 0 161.54 200 9680 20 L 98-59 b 2021-02-24 03:55:02 0.018 1.02 3412 7.933 22 100.77 0 162.64 160 6292 13 WASP-127 b 2021-04-29 02:10:29 8E-05 4.308 5620 9.092 21 101.53 0 157.95 710 1408 3 Name obs time obs time Transit Effective J-magnitude Number of S/N median Minimum Maximum Average Exposure rank Total number S/N = 150 error [h] Length [h] Temperature [K] exposures possible S/N S/N Time [s] of transits AU Mic b 2021-07-01 06:01:57 0.00104 3.5 3700 5.436 360 152.18 0 240.95 35 518400 4 AU Mic b 2021-08-04 02:29:16 0.0012 3.5 3700 5.436 360 151.25 0 240.78 35 518400 4 AU Mic b 2021-07-18 04:15:17 0.00112 3.5 3700 5.436 360 151.79 0 240.89 35 518400 4 AU Mic b 2021-06-14 07:49:12 0.00096 3.5 3700 5.436 360 152.35 0 240.97 35 518400 4 HD 21749 c 2021-10-13 06:25:06 0.02091 2.515 4640 6.081 118 151.37 0 240.15 77 41772 3 HD 21749 c 2021-11-21 05:13:43 0.02346 2.515 4640 6.081 118 151.18 0 240.10 77 41772 3 HD 21749 c 2021-09-04 07:37:22 0.01836 2.515 4640 6.081 118 150.88 0 240.05 77 41772 3 GJ 357 b 2021-03-03 02:57:24 0.00192 1.53 3505 7.337 27 151.68 0 240.01 200 4889 7 GJ 357 b 2020-12-26 07:16:47 0.00056 1.53 3505 7.337 27 151.66 0 240.00 200 4889 7 GJ 357 b 2021-05-05 00:25:34 0.0032 1.53 3505 7.337 27 151.51 0 239.98 200 4889 7 HD 15337 b 2021-08-31 07:29:05 0.00986 2.49 5125 7.553 26 151.97 0 236.77 340 3328.2 5 HD 15337 b 2021-09-19 08:03:18 0.01054 2.49 5125 7.553 26 152.05 0 236.79 340 3328.2 5 GJ 357 b 2020-12-30 05:36:37 0.00064 1.53 3505 7.337 26 154.34 0 245.76 210 4889 7 GJ 357 b 2021-02-23 06:16:53 0.00176 1.53 3505 7.337 26 154.19 0 245.73 210 4889 7 HD 15337 b 2021-11-01 03:22:38 0.01207 2.49 5125 7.553 26 151.96 0 236.77 340 3328.2 5 GJ 357 b 2021-03-07 01:17:43 0.002 1.53 3505 7.337 26 154.44 0 245.77 210 4889 7 GJ 357 b 2021-05-01 02:04:54 0.00312 1.53 3505 7.337 26 153.90 0 245.69 210 4889 7 HD 15337 b 2021-11-20 03:58:58 0.01275 2.49 5125 7.553 26 152.05 0 236.79 340 3328.2 5 HD 213885 b 2021-09-16 04:26:20 0.006069 1.534232 5978 6.806 25 154.49 0 240.36 220 18125 29 HD 213885 b 2021-09-01 01:31:57 0.005754 1.534232 5978 6.806 25 153.02 0 239.95 220 18125 29 HD 15337 b 2020-12-03 05:04:36 0.00017 2.49 5125 7.553 25 152.75 0 239.86 350 3328.2 5 HD 213885 b 2021-09-25 06:11:06 0.006258 1.534232 5978 6.806 25 152.95 0 239.93 220 18125 29 HD 213885 b 2021-09-24 05:59:28 0.006237 1.534232 5978 6.806 25 153.19 0 240.01 220 18125 29 HD 213885 b 2021-09-23 05:47:49 0.006216 1.534232 5978 6.806 25 153.43 0 240.08 220 18125 29 HD 213885 b 2021-09-22 05:36:10 0.006195 1.534232 5978 6.806 25 153.61 0 240.14 220 18125 29 HD 213885 b 2021-09-21 05:24:32 0.006174 1.534232 5978 6.806 25 153.87 0 240.19 220 18125 29 HD 213885 b 2021-09-20 05:12:53 0.006153 1.534232 5978 6.806 25 154.06 0 240.24 220 18125 29 HD 213885 b 2021-09-19 05:01:15 0.006132 1.534232 5978 6.806 25 154.20 0 240.28 220 18125 29

HD 213885 b 2021-06-02 08:13:00 0.003864 1.534232 5978 6.806 25 153.62 0 240.14 220 18125 29 74 HD 213885 b 2021-09-15 04:14:42 0.006048 1.534232 5978 6.806 25 154.56 0 240.37 220 18125 29 HD 213885 b 2021-06-04 08:35:58 0.003906 1.534232 5978 6.806 25 154.06 0 240.24 220 18125 29 HD 213885 b 2021-06-05 08:47:27 0.003927 1.534232 5978 6.806 25 154.20 0 240.28 220 18125 29 HD 213885 b 2021-06-06 08:58:56 0.003948 1.534232 5978 6.806 25 154.29 0 240.31 220 18125 29 HD 213885 b 2021-06-07 09:10:26 0.003969 1.534232 5978 6.806 25 154.40 0 240.34 220 18125 29 HD 213885 b 2021-06-03 08:24:29 0.003885 1.534232 5978 6.806 25 153.87 0 240.20 220 18125 29 HD 213885 b 2021-09-02 01:43:34 0.005775 1.534232 5978 6.806 25 153.23 0 240.03 220 18125 29 HD 213885 b 2021-09-03 01:55:11 0.005796 1.534232 5978 6.806 25 153.49 0 240.10 220 18125 29 HD 213885 b 2021-09-14 04:03:04 0.006027 1.534232 5978 6.806 25 154.58 0 240.38 220 18125 29 HD 213885 b 2021-09-13 03:51:26 0.006006 1.534232 5978 6.806 25 154.61 0 240.38 220 18125 29 HD 213885 b 2021-09-12 03:39:48 0.005985 1.534232 5978 6.806 25 154.59 0 240.38 220 18125 29 HD 213885 b 2021-09-11 03:28:10 0.005964 1.534232 5978 6.806 25 154.57 0 240.37 220 18125 29 HD 213885 b 2021-09-10 03:16:32 0.005943 1.534232 5978 6.806 25 154.50 0 240.36 220 18125 29 HD 213885 b 2021-09-09 03:04:55 0.005922 1.534232 5978 6.806 25 154.41 0 240.34 220 18125 29 HD 213885 b 2021-09-05 02:18:25 0.005838 1.534232 5978 6.806 25 153.91 0 240.21 220 18125 29 HD 213885 b 2021-09-08 02:53:17 0.005901 1.534232 5978 6.806 25 154.31 0 240.32 220 18125 29 HD 213885 b 2021-09-04 02:06:48 0.005817 1.534232 5978 6.806 25 153.71 0 240.15 220 18125 29 HD 213885 b 2021-09-06 02:30:02 0.005859 1.534232 5978 6.806 25 154.09 0 240.25 220 18125 29 HD 213885 b 2021-09-07 02:41:40 0.00588 1.534232 5978 6.806 25 154.22 0 240.29 220 18125 29 LTT 1445 A b 2021-10-02 07:50:43 0.0174 1.3776 3337 7.294 24 154.52 0 244.78 200 4032 7 LTT 1445 A b 2021-10-29 02:53:27 0.0189 1.3776 3337 7.294 24 153.13 0 244.55 200 4032 7 LTT 1445 A b 2021-11-14 04:43:47 0.0198 1.3776 3337 7.294 24 154.65 0 244.80 200 4032 7 LTT 1445 A b 2020-12-06 05:36:07 0.0006 1.3776 3337 7.294 24 152.18 0 244.37 200 4032 7 LTT 1445 A b 2021-01-18 02:34:08 0.003 1.3776 3337 7.294 24 152.57 0 244.45 200 4032 7 LTT 1445 A b 2021-08-04 09:12:38 0.0141 1.3776 3337 7.294 24 153.99 0 244.69 200 4032 7 LTT 1445 A b 2021-09-16 06:01:44 0.0165 1.3776 3337 7.294 24 153.57 0 244.62 200 4032 7 HD 15337 c 2021-10-09 06:50:08 0.0304 2.25 5125 7.553 23 152.01 0 236.79 340 529 1 HD 23472 b 2020-12-25 04:13:38 0.284 2.695974 4813 7.865 21 150.70 0 237.96 460 882 2 HD 23472 b 2021-11-08 04:20:39 2.84 2.695974 4813 7.865 21 151.42 0 238.11 460 882 2 Name obs time obs time Transit Effective J-magnitude Number of S/N median Minimum Maximum Average Exposure rank Total number S/N = 200 error [h] Length [h] Temperature [K] exposures possible S/N S/N Time [s] of transits AU Mic b 2021-07-01 06:01:16 0.00108 3.5 3700 5.436 196 206.21 0 326.34 64 153664 4 AU Mic b 2021-07-18 04:14:37 0.00116 3.5 3700 5.436 196 205.69 0 326.26 64 153664 4 AU Mic b 2021-08-04 02:28:36 0.00124 3.5 3700 5.436 196 204.95 0 326.12 64 153664 4 AU Mic b 2021-06-14 07:48:32 0.001 3.5 3700 5.436 196 206.45 0 326.37 64 153664 4 75 HD 21749 c 2021-10-13 06:24:40 0.02193 2.515 4640 6.081 64 204.71 0 324.62 140 12288 3 HD 21749 c 2021-09-04 07:36:56 0.01938 2.515 4640 6.081 64 204.06 0 324.48 140 12288 3 HD 21749 c 2020-11-20 02:10:47 0.00051 2.515 4640 6.081 64 203.74 0 324.41 140 12288 3 References

Digital Sky Survey, ESO. URL https://archive.eso.org/dss/dss.

Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Books on Mathematics, 1965. URL https://ui.adsabs.harvard.edu/abs/ 1965hmfw.book.....A.

R. L. Akeson, X. Chen, D. Ciardi, M. Crane, J. Good, M. Harbut, E. Jackson, S. R. Kane, A. C. Laity, S. Leifer, and et al. The nasa exoplanet archive: Data and tools for exoplanet research. Publications of the Astronomical Society of the Pacific, 125(930):989–999, Aug 2013. ISSN 1538-3873. doi: 10.1086/672273. URL http://dx.doi.org/10.1086/672273.

Pascal Ballester A.Modigliani. Exposure Time Calculators Formula Book 29.07.99, 1999. URL https://www. eso.org/observing/etc/doc/formulabook/index.html.

David S. Amundsen, Isabelle Baraffe, Pascal Tremblin, James Manners, Wolfgang Hayek, Nathan J. Mayne, and David M. Acreman. Accuracy tests of radiation schemes used in hot Jupiter global circulation models. , 564:A59, April 2014. doi: 10.1051/0004-6361/201323169. URL https://doi.org/10.1051/0004-6361/ 201323169.

E. Aronson and P. Waldén. Using near-infrared spectroscopy for characterization of transiting exoplanets. Astronomy Astrophysics, 578:A133, Jun 2015. ISSN 1432-0746. doi: 10.1051/0004-6361/201424058. URL http://dx.doi.org/10.1051/0004-6361/201424058.

Erik Aronson and Nikolai Piskunov. Model-independent exoplanet transit spectroscopy. The Astronomical Journal, 155(5):208, Apr 2018. ISSN 1538-3881. doi: 10.3847/1538-3881/aaa3fe. URL http://dx.doi.org/ 10.3847/1538-3881/aaa3fe.

J. K. Barstow, S. Aigrain, P. G. J. Irwin, T. Hackler, L. N. Fletcher, J. M. Lee, and N. P. Gibson. Clouds on the hot jupiter hd189733b: Constraints from the reflection spectrum. The Astrophysical Journal, 786(2):154, Apr 2014. ISSN 1538-4357. doi: 10.1088/0004-637x/786/2/154. URL http://dx.doi.org/10.1088/0004-637X/ 786/2/154.

D. R. Bates. Rayleigh scattering by air. , 32(6):785–790, June 1984. doi: 10.1016/0032-0633(84)90102-8. URL https://doi.org/10.1016/0032-0633(84)90102-8.

Bjoern Benneke and Sara Seager. Atmospheric retrieval for super-earths: Uniquely constraining the atmospheric composition with transmission spectroscopy. The Astrophysical Journal, 753(2):100, jun 2012. doi: 10.1088/ 0004-637x/753/2/100. URL https://doi.org/10.1088%2F0004-637x%2F753%2F2%2F100.

A. Borysow. Collision-induced absorption coefficients of H2 pairs at temperatures from 60 K to 1000 K. , 390:779–782, August 2002. doi: 10.1051/0004-6361:20020555. URL https://doi.org/10.1051/0004-6361: 20020555.

Joshua Brakensiek and Darin Ragozzine. Efficient geometric probabilities of multi-transiting exoplanetary sys- tems from corbits. The Astrophysical Journal, 821(1), apr 2016. doi: 10.3847/0004-637x/821/1/47. URL https://doi.org/10.3847%2F0004-637x%2F821%2F1%2F47.

M. Brogi, I. A. G. Snellen, R. J. de Kok, S. Albrecht, J. L. Birkby, and E. J. W. de Mooij. Detection of molecular absorption in the dayside of exoplanet 51 pegasi b? The Astrophysical Journal, 767(1):27, Mar 2013. ISSN 1538-4357. doi: 10.1088/0004-637x/767/1/27. URL http://dx.doi.org/10.1088/0004-637X/767/1/27.

Matteo Brogi, Ignas A. G. Snellen, Remco J. de Kok, Simon Albrecht, Jayne Birkby, and Ernst J. W. de Mooij. The signature of orbital motion from the dayside of the planet boötis b. Nature, 486(7404):502–504, Jun 2012. ISSN 1476-4687. doi: 10.1038/nature11161. URL http://dx.doi.org/10.1038/nature11161.

76 Timothy M. Brown. Transmission spectra as diagnostics of extrasolar giant planet atmospheres. The As- trophysical Journal, 553(2):1006–1026, Jun 2001. ISSN 1538-4357. doi: 10.1086/320950. URL http: //dx.doi.org/10.1086/320950.

A. Burrows, M. Marley, W. B. Hubbard, J. I. Lunine, T. Guillot, D. Saumon, R. Freedman, D. Sudarsky, and C. Sharp. A nongray theory of extrasolar giant planets and brown dwarfs. The Astrophysical Journal, 491 (2):856–875, dec 1997. doi: 10.1086/305002. URL https://doi.org/10.1086%2F305002.

A. Burrows, I. Hubeny, J. Budaj, H. A. Knutson, and D. Charbonneau. Theoretical spectral models of the planet hd 209458b with a thermal inversion and water emission bands. The Astrophysical Journal, 668(2): L171–L174, Sep 2007. ISSN 1538-4357. doi: 10.1086/522834. URL http://dx.doi.org/10.1086/522834.

F. R. Chromey. To measure the sky: An introduction to observational astronomy. Cambridge University Press, 01 2010. doi: 10.1017/CBO9780511794810. URL https://doi.org/10.1017/CBO9780511794810.

I. J. M. Crossfield, B. Biller, J. E. Schlieder, N. R. Deacon, M. Bonnefoy, D. Homeier, F. Allard, E. Buenzli, Th. Henning, W. Brandner, and et al. A global cloud map of the nearest known brown dwarf. Nature, 505 (7485):654–656, Jan 2014. ISSN 1476-4687. doi: 10.1038/nature12955. URL http://dx.doi.org/10.1038/ nature12955.

S. Czesla, T. Klocová, S. Khalafinejad, U. Wolter, and J. H. M. M. Schmitt. The center-to-limb variation across the fraunhofer lines of hd 189733. Astronomy Astrophysics, 582:A51, Oct 2015. ISSN 1432-0746. doi: 10.1051/0004-6361/201526386. URL http://dx.doi.org/10.1051/0004-6361/201526386.

R.J. de Kok, M. Brogi, I.A.G. Snellen, J. Birkby, S. Albrecht, and E.J.W. de Mooij. Detection of carbon monoxide in the high-resolution day-side spectrum of the exoplanet hd 189733b. Astronomy Astrophysics, 554:A82, Jun 2013. ISSN 1432-0746. doi: 10.1051/0004-6361/201321381. URL http://dx.doi.org/10. 1051/0004-6361/201321381.

Imke de Pater and Jack J. Lissauer. Planetary Sciences. Cambridge University Press, 2010. doi: 10.1017/ CBO9780511780561. URL https://doi.org/10.1017/CBO9780511780561.

R. H. Dicke. The effect of collisions upon the doppler width of spectral lines. Phys. Rev., 89:472–473, Jan 1953. doi: 10.1103/PhysRev.89.472. URL https://link.aps.org/doi/10.1103/PhysRev.89.472.

Ian Dobbs-Dixon and Eric Agol. Three-dimensional radiative-hydrodynamical simulations of the highly irradi- ated short-period exoplanet HD 189733b. , 435(4):3159–3168, November 2013. doi: 10.1093/mnras/stt1509. URL https://doi.org/10.1093/mnras/stt1509.

Ian Dobbs-Dixon, Andrew Cumming, and D. N. C. Lin. Radiative hydrodynamic simulations of hd209458b: Temporal variability. The Astrophysical Journal, 710(2):1395–1407, Jan 2010. ISSN 1538-4357. doi: 10.1088/ 0004-637x/710/2/1395. URL https://dx.doi.org/10.1088/0004-637X/710/2/1395.

Ian Dobbs-Dixon, Eric Agol, and Adam Burrows. The impact of circumplanetary jets on transit spectra and timing offset for hot jupiters. The Astrophysical Journal, 751(2):87, May 2012. ISSN 1538-4357. doi: 10. 1088/0004-637x/751/2/87. URL http://dx.doi.org/10.1088/0004-637X/751/2/87.

Jason Eastman, Robert Siverd, and B. Scott Gaudi. Achieving better than 1 minute accuracy in the heliocentric and barycentric julian dates. Publications of the Astronomical Society of the Pacific, 122(894):935–946, Aug 2010. ISSN 1538-3873. doi: 10.1086/655938. URL http://dx.doi.org/10.1086/655938.

A. S. Eddington. On the radiative equilibrium of the stars. , 77:16–35, November 1916. doi: 10.1093/mnras/ 77.1.16. URL https://doi.org/10.1093/mnras/77.1.16.

77 A. Faure, L. Wiesenfeld, B.J. Drouin, and J. Tennyson. Pressure broadening of water and carbon monoxide transitions by molecular hydrogen at high temperatures. Journal of Quantitative Spectroscopy and Radiative Transfer, 116:79 – 86, 2013. ISSN 0022-4073. doi: https://doi.org/10.1016/j.jqsrt.2012.09.015. URL http: //www.sciencedirect.com/science/article/pii/S0022407312004268.

R. Follert, R. J. Dorn, E. Oliva, J. L. Lizon, A. Hatzes, N. Piskunov, A. Reiners, U. Seemann, E. Stempels, U. Heiter, T. Marquart, M. Lockhart, G. Anglada-Escude, T. Löwinger, D. Baade, J. Grunhut, P. Bristow, B. Klein, Y. Jung, D. J. Ives, F. Kerber, Eszter Pozna, J. Paufique, H. U. Kaeufl, L. Origlia, E. Valenti, D. Gojak, M. Hilker, L. Pasquini, A. Smette, and J. Smoker. Crires+: a cross-dispersed high-resolution infrared spectrograph for the eso vlt. In Suzanne K. Ramsay, Ian S. McLean, and Hideki Takami, editors, Ground-based and Airborne Instrumentation for Astronomy V, volume 9147. International Society for Optics and Photonics, SPIE, 2014. doi: 10.1117/12.2054197. URL https://doi.org/10.1117/12.2054197.

J. J. Fortney, K. Lodders, M. S. Marley, and R. S. Freedman. A unified theory for the atmospheres of the hot and very hot jupiters: Two classes of irradiated atmospheres. The Astrophysical Journal, 678(2):1419–1435, May 2008. ISSN 1538-4357. doi: 10.1086/528370. URL http://dx.doi.org/10.1086/528370.

J. J. Fortney, M. Shabram, A. P. Showman, Y. Lian, R. S. Freedman, M. S. Marley, and N. K. Lewis. Trans- mission spectra of three-dimensional hot jupiter model atmospheres. The Astrophysical Journal, 709(2): 1396–1406, Jan 2010. ISSN 1538-4357. doi: 10.1088/0004-637x/709/2/1396. URL http://dx.doi.org/10. 1088/0004-637X/709/2/1396.

Siddharth Gandhi and Nikku Madhusudhan. genesis: new self-consistent models of exoplanetary spectra. Monthly Notices of the Royal Astronomical Society, 472(2):2334–2355, Jun 2017. ISSN 1365-2966. doi: 10.1093/mnras/stx1601. URL http://dx.doi.org/10.1093/mnras/stx1601.

Larry L. Gordley, Benjamin T. Marshall, and D. Allen Chu. Linepak: Algorithms for modeling spectral trans- mittance and radiance. Journal of Quantitative Spectroscopy and Radiative Transfer, 52(5):563 – 580, 1994. ISSN 0022-4073. doi: https://doi.org/10.1016/0022-4073(94)90025-6. URL http://www.sciencedirect. com/science/article/pii/0022407394900256.

David F. Gray. The Observation and Analysis of Stellar Photospheres. Cambridge University Press, 3 edition, 2005. doi: 10.1017/CBO9781316036570. URL https://doi.org/10.1017/CBO9781316036570.

Jane S. Greaves, Anita M. S. Richards, William Bains, Paul B. Rimmer, Hideo Sagawa, David L. Clements, Sara Seager, Janusz J. Petkowski, Clara Sousa-Silva, Sukrit Ranjan, Emily Drabek-Maunder, Helen J. Fraser, Annabel Cartwright, Ingo Mueller-Wodarg, Zhuchang Zhan, Per Friberg, Iain Coulson, E’lisa Lee, and Jim Hoge. Phosphine gas in the cloud decks of venus. Nat Astron, 2020. doi: 10.1038/s41550-020-1174-4. URL https://doi.org/10.1038/s41550-020-1174-4.

Phil Gregory. Markov chain Monte Carlo, chapter 12, page 312 – 351. Cambridge University Press, 2005. doi: 10.1017/CBO9780511791277.013. URL https://doi.org/10.1017/CBO9780511791277.013.

Simon L. Grimm and Kevin Heng. Helios-k: An ultrafast, open-source opacity calculator for radiative transfer. The Astrophysical Journal, 808(2):182, aug 2015. doi: 10.1088/0004-637x/808/2/182. URL https://doi. org/10.1088%2F0004-637x%2F808%2F2%2F182.

T. Guillot. On the radiative equilibrium of irradiated planetary atmospheres. Astronomy and Astrophysics, 520: A27, Sep 2010. ISSN 1432-0746. doi: 10.1051/0004-6361/200913396. URL http://dx.doi.org/10.1051/ 0004-6361/200913396.

Gustafsson, B., Edvardsson, B., Eriksson, K., Jørgensen, U. G., Nordlund, Å., and Plez, B. A grid of marcs model atmospheres for late-type stars - i. methods and general properties. A&A, 486(3):951–970, 2008. doi: 10.1051/0004-6361:200809724. URL https://doi.org/10.1051/0004-6361:200809724.

78 Eunkyu Han, Sharon X. Wang, Jason T. Wright, Y. Katherina Feng, Ming Zhao, Onsi Fakhouri, Jacob I. Brown, and Colin Hancock. Exoplanet Orbit Database, 2014 (accessed July 20, 2020). URL http://exoplanets.org/.

Christina Hedges and Nikku Madhusudhan. Effect of pressure broadening on molecular absorption cross sections in exoplanetary atmospheres. , 458(2):1427–1449, May 2016. doi: 10.1093/mnras/stw278. URL https: //doi.org/10.1093/mnras/stw278.

Isaac M. Held and Max J. Suarez. A Proposal for the Intercomparison of the Dynamical Cores of Atmo- spheric General Circulation Models. Bulletin of the American Meteorological Society, 75(10):1825–1830, Oc- tober 1994. doi: 10.1175/1520-0477(1994)075<1825:APFTIO>2.0.CO;2. URL https://doi.org/10.1175/ 1520-0477(1994)075<1825:APFTIO>2.0.CO;2.

Kevin Heng. Exoplanetary Atmospheres: Theoretical Concepts and Foundations. Princeton University Press, 2017. URL https://ui.adsabs.harvard.edu/abs/2017eatc.book.....H.

Kevin Heng and James R. Lyons. Carbon dioxide in exoplanetary atmospheres: Rarely dominant compared to carbon monoxide and water in hot, hydrogen-dominated atmospheres. The Astrophysical Journal, 817(2): 149, Jan 2016. ISSN 1538-4357. doi: 10.3847/0004-637x/817/2/149. URL http://dx.doi.org/10.3847/ 0004-637X/817/2/149.

Kevin Heng and Jared Workman. Analytical models of exoplanetary atmospheres. i. atmospheric dynamics via the shallow water system. The Astrophysical Journal Supplement Series, 213(2):27, Jul 2014. ISSN 1538-4365. doi: 10.1088/0067-0049/213/2/27. URL http://dx.doi.org/10.1088/0067-0049/213/2/27.

Kevin Heng, Wolfgang Hayek, Frédéric Pont, and David K. Sing. On the effects of clouds and hazes in the atmospheres of hot jupiters: semi-analytical temperature-pressure profiles. Monthly Notices of the Royal Astronomical Society, 420(1):20–36, Dec 2011a. ISSN 0035-8711. doi: 10.1111/j.1365-2966.2011.19943.x. URL http://dx.doi.org/10.1111/j.1365-2966.2011.19943.x.

Kevin Heng, Kristen Menou, and Peter J. Phillipps. Atmospheric circulation of tidally locked exoplanets: a suite of benchmark tests for dynamical solvers. Monthly Notices of the Royal Astronomical Society, 413(4): 2380–2402, Mar 2011b. ISSN 0035-8711. doi: 10.1111/j.1365-2966.2011.18315.x. URL http://dx.doi.org/ 10.1111/j.1365-2966.2011.18315.x.

Kevin Heng, João M. Mendonça, and Jae-Min Lee. Analytical models of exoplanetary atmospheres. ii. radiative transfer via the two-stream approximation. The Astrophysical Journal Supplement Series, 215(1):4, Oct 2014. ISSN 1538-4365. doi: 10.1088/0067-0049/215/1/4. URL http://dx.doi.org/10.1088/0067-0049/215/1/4.

B. J. Hoskins and A. J. Simmons. A multi-layer spectral model and the semi-implicit method. Quarterly Journal of the Royal Meteorological Society, 101(429):637–655, July 1975. doi: 10.1002/qj.49710142918. URL https://doi.org/10.1002/qj.49710142918.

Ivan Hubeny. Model atmospheres of sub-stellar mass objects. , 469(1):841–869, July 2017. doi: 10.1093/mnras/ stx758. URL https://doi.org/10.1093/mnras/stx758.

Ivan Hubeny and Dimitri Mihalas. Theory of Stellar Atmospheres. Princeton University Press, 2014. URL https://ui.adsabs.harvard.edu/abs/2014tsa..book.....H.

Eric Jensen. Tapir: A web interface for transit/eclipse observability. Astrophysics Source Code Library, record ascl:1306.007, art. ascl:1306.007, June 2013. URL https://astro.swarthmore.edu/transits/transits. cgi.

Hans-Ulrich Kaeufl, Pascal Ballester, Peter Biereichel, Bernard Delabre, Rob Donaldson, Reinhold Dorn, Enrico Fedrigo, Gert Finger, Gerhard Fischer, Francis Franza, Domingo Gojak, Gotthard Huster, Yves Jung, Jean- Louis Lizon, Leander Mehrgan, Manfred Meyer, Alan Moorwood, Jean-Francois Pirard, Jerome Paufique,

79 Eszter Pozna, Ralf Siebenmorgen, Armin Silber, Joerg Stegmeier, and Stefan Wegerer. Crires: a high- resolution infrared spectrograph for eso’s vlt. In Alan F. M. Moorwood and Masanori Iye, editors, Ground- based Instrumentation for Astronomy, volume 5492. International Society for Optics and Photonics, SPIE, 2004. doi: 10.1117/12.551480. URL https://doi.org/10.1117/12.551480.

Dan Kalman. A singularly valuable decomposition: The svd of a matrix. The College Mathematics Journal, 27(1):2–23, 1996. doi: 10.1080/07468342.1996.11973744. URL https://doi.org/10.1080/07468342.1996. 11973744.

Rudolf Kippenhahn, Alfred Weigert, and Achim Weiss. Stellar Structure and Evolution. Springer, 2012. doi: 10.1007/978-3-642-30304-3. URL https://ui.adsabs.harvard.edu/abs/2012sse..book.....K.

Heather A. Knutson, David Charbonneau, Robert W. Noyes, Timothy M. Brown, and Ronald L. Gilliland. Using stellar limb-darkening to refine the properties of hd 209458b. The Astrophysical Journal, 655(1):564–575, Jan 2007. ISSN 1538-4357. doi: 10.1086/510111. URL http://dx.doi.org/10.1086/510111.

Ravi Kumar Kopparapu, Eric T. Wolf, and Victoria S. Meadows. Characterizing Exoplanet Habitability. The University of Arizona Press, 2019.

G. Lee, I. Dobbs-Dixon, Ch. Helling, K. Bognar, and P. Woitke. Dynamic mineral clouds on hd 189733b. Astronomy Astrophysics, 594:A48, Oct 2016. ISSN 1432-0746. doi: 10.1051/0004-6361/201628606. URL http://dx.doi.org/10.1051/0004-6361/201628606.

G. K. H. Lee, K. Wood, I. Dobbs-Dixon, A. Rice, and Ch. Helling. Dynamic mineral clouds on hd 189733b. Astronomy Astrophysics, 601:A22, Apr 2017. ISSN 1432-0746. doi: 10.1051/0004-6361/201629804. URL http://dx.doi.org/10.1051/0004-6361/201629804.

C. D. Levermore and G. C. Pomraning. A flux-limited diffusion theory. , 248:321–334, August 1981. doi: 10.1086/159157. URL https://doi.org/10.1086/159157.

Gang Li, Iouli E. Gordon, Laurence S. Rothman, Yan Tan, Shui-Ming Hu, Samir Kassi, Alain Campargue, and Emile S. Medvedev. Rovibrational line lists for nine isotopologues of the co molecule in the x1+ ground electronic state. The Astrophysical Journal Supplement Series, 216(1):15, jan 2015. doi: 10.1088/0067-0049/ 216/1/15. URL https://doi.org/10.1088%2F0067-0049%2F216%2F1%2F15.

N. Madhusudhan and S. Seager. A temperature and abundance retrieval method for exoplanet atmospheres. The Astrophysical Journal, 707(1):24–39, nov 2009. doi: 10.1088/0004-637x/707/1/24. URL https://doi. org/10.1088%2F0004-637x%2F707%2F1%2F24.

Nikku Madhusudhan. Atmospheric retrieval of exoplanets. Handbook of Exoplanets, page 2153–2182, 2018. doi: 10.1007/978-3-319-55333-7_104. URL http://dx.doi.org/10.1007/978-3-319-55333-7_104.

Eric Mamajek. Fraction of Stars by Spectral Type in the Solar Vicinity, 4 2016. URL https://figshare.com/ articles/figure/Fraction_of_Stars_by_Spectral_Type_in_the_Solar_Vicinity/3206527.

Kaisey Mandel and Eric Agol. Analytic light curves for planetary transit searches. The Astrophysical Journal, 580 (2):L171–L175, Dec 2002. ISSN 1538-4357. doi: 10.1086/345520. URL http://dx.doi.org/10.1086/345520.

Kristen Menou and Emily Rauscher. Atmospheric circulation of hot jupiters: A shallow three-dimensional model. The Astrophysical Journal, 700(1):887–897, jul 2009. doi: 10.1088/0004-637x/700/1/887. URL https://doi.org/10.1088%2F0004-637x%2F700%2F1%2F887.

Dimitri Mihalas. Stellar atmospheres. Princeton series in astrophysics. W. H. Freeman, 1978, Princeton, NJ, 1978. URL https://cds.cern.ch/record/2005250.

80 Eliza Miller-Ricci, Sara Seager, and Dimitar Sasselov. The atmospheric signatures of super-earths: How to distinguish between hydrogen-rich and hydrogen-poor atmospheres. The Astrophysical Journal, 690(2):1056– 1067, dec 2008. doi: 10.1088/0004-637x/690/2/1056. URL https://doi.org/10.1088%2F0004-637x%2F690% 2F2%2F1056. Eliza Miller-Ricci, Sara Seager, and Dimitar Sasselov. The Atmospheric Signatures of Super-Earths: How to Distinguish Between Hydrogen-Rich and Hydrogen-Poor Atmospheres. , 690(2):1056–1067, January 2009. doi: 10.1088/0004-637X/690/2/1056. URL https://doi.org/10.1088/0004-637X/690/2/1056. B. M. Morris, E. Tollerud, B. Sipőcz, C. Deil, S. T. Douglas, J. Berlanga Medina, K. Vyhmeister, T. R. Smith, S. Littlefair, A. M. Price-Whelan, W. T. Gee, and E. Jeschke. astroplan: An Open Source Observation Planning Package in Python. Astronomical Journal, 155, Mar 2018. doi: 10.3847/1538-3881/aaa47e. URL http://adsabs.harvard.edu/abs/2018AJ....155..128M. Carl D. Murray and Alexandre C. M. Correia. Keplerian Orbits and Dynamics of Exoplanets, chapter 2, pages 15 – 23. University of Arizona Press; In collaboration with Lunar and Planetary Institute, 2010. URL https://arxiv.org/abs/1009.1738. Harry Partridge and David W. Schwenke. The determination of an accurate isotope dependent potential energy surface for water from extensive ab initio calculations and experimental data. The Journal of Chemical Physics, 106(11):4618–4639, 1997. doi: 10.1063/1.473987. URL https://doi.org/10.1063/1.473987. S. S. Penner. Physics of shock waves and high-temperature hydrodynamic phenomena - Ya.B. Zeldovich and Yu.P. Raizer (translated from the Russian and then edited by Wallace D. Hayes and Ronald F. Probstein); Dover Publications, New York, 2002, 944 pp., $34. , 76:235–236, January 2003. doi: 10.1016/S0022-4073(02)00065-1. URL https://doi.org/10.1016/S0022-4073(02)00065-1. Anil K. Pradhan and Sultana N. Nahar. Atomic Astrophysics and Spectroscopy. Cambridge University Press, 2011. doi: 10.1017/CBO9780511975349. URL https://doi.org/10.1017/CBO9780511975349. A. M. Price-Whelan, B. M. Sipőcz, H. M. Günther, P. L. Lim, S. M. Crawford, S. Conseil, D. L. Shupe, M. W. Craig, N. Dencheva, A. Ginsburg, J. T. VanderPlas, L. D. Bradley, D. Pérez-Suárez, and M. de Val-Borro. The Astropy Project: Building an Open-science Project and Status of the v2.0 Core Package. Astronomical Journal, 156:123, September 2018. doi: 10.3847/1538-3881/aabc4f. URL https://doi.org/ 10.3847/1538-3881/aabc4f. Clive D. Rodgers. Inverse Methods for Atmospheric Sounding: Theory and Practice. Series on Atmospheric Oceanic and Planetary Physics. World Scientific, 2004. ISBN 981022740X,9789810227401. doi: 10.1142/3171. URL https://doi.org/10.1142/3171. L.S. Rothman, C.P. Rinsland, A. Goldman, S.T. Massie, D.P. Edwards, J-M. Flaud, A. Perrin, C. Camy- Peyret, V. Dana, J.-Y. Mandin, J. Schroeder, A. Mccann, R.R. Gamache, R.B. Wattson, K. Yoschino, K.V. Chance, K.W. Jucks, L.R. Brown, V. Nemtchinov, and P. Varanasi. The hitran molecular spectroscopic database and hawks (hitran atmospheric workstation): 1996 edition. Journal of Quantitative Spectroscopy and Radiative Transfer, 60(5):665 – 710, 1998. ISSN 0022-4073. doi: 10.1016/S0022-4073(98)00078-8. URL http://www.sciencedirect.com/science/article/pii/S0022407398000788. L.S. Rothman, I.E. Gordon, R.J. Barber, H. Dothe, R.R. Gamache, A. Goldman, V.I. Perevalov, S.A. Tashkun, and J. Tennyson. Hitemp, the high-temperature molecular spectroscopic database. Journal of Quantitative Spectroscopy and Radiative Transfer, 111(15):2139 – 2150, 2010. ISSN 0022-4073. doi: 10.1016/j.jqsrt. 2010.05.001. URL http://www.sciencedirect.com/science/article/pii/S002240731000169X. XVIth Symposium on High Resolution Molecular Spectroscopy (HighRus-2009).

S. Seager and D. D. Sasselov. Extrasolar giant planets under strong stellar irradiation. The Astrophysical Journal, 502(2):L157–L161, Aug 1998. ISSN 0004-637X. doi: 10.1086/311498. URL http://dx.doi.org/ 10.1086/311498.

81 S. Seager and D. D. Sasselov. Theoretical transmission spectra during extrasolar giant planet transits. The Astrophysical Journal, 537(2):916–921, Jul 2000. ISSN 1538-4357. doi: 10.1086/309088. URL http://dx. doi.org/10.1086/309088.

S. Seager, B. A. Whitney, and D. D. Sasselov. Photometric light curves and polarization of close-in extrasolar giant planets. The Astrophysical Journal, 540(1):504–520, Sep 2000. ISSN 1538-4357. doi: 10.1086/309292. URL http://dx.doi.org/10.1086/309292.

S. Seager, L. J. Richardson, B. M. S. Hansen, K. Menou, J. Y.-K. Cho, and D. Deming. On the dayside thermal emission of hot jupiters. The Astrophysical Journal, 632(2):1122–1131, Oct 2005. ISSN 1538-4357. doi: 10.1086/444411. URL http://dx.doi.org/10.1086/444411.

Sara Seager. Exoplanet Atmospheres: Physical Processes. Princeton University Press, 2010. URL https: //ui.adsabs.harvard.edu/abs/2010eapp.book.....S.

P. K. Seidelmann. 1980 I.A.U. Theory of Nutation - the Final Report of the I.A.U. Working Group on Nutation. Celestial Mechanics, 27(1):79–106, May 1982. doi: 10.1007/BF01228952. URL https://doi.org/10.1007/ BF01228952.

C. M. Sharp and A. Burrows. Atomic and Molecular Opacities for Brown Dwarf and Giant Planet Atmospheres. , 168(1):140–166, January 2007. doi: 10.1086/508708. URL https://doi.org/10.1086/508708.

Shulyak, D., Rengel, M., Reiners, A., Seemann, U., and Yan, F. Remote sensing of exoplanetary atmospheres with ground-based high-resolution near-infrared spectroscopy. A&A, 629:A109, 2019. doi: 10.1051/0004-6361/ 201935691. URL https://doi.org/10.1051/0004-6361/201935691.

Ignas A. G. Snellen, Remco J. de Kok, Ernst J. W. de Mooij, and Simon Albrecht. The orbital motion, absolute mass and high-altitude winds of exoplanet hd209458b. Nature, 465(7301):1049–1051, Jun 2010a. ISSN 1476- 4687. doi: 10.1038/nature09111. URL http://dx.doi.org/10.1038/nature09111.

Ignas A. G. Snellen, Remco J. de Kok, Ernst J. W. de Mooij, and Simon Albrecht. The orbital motion, absolute mass and high-altitude winds of exoplanet hd209458b. Nature, 465(7301):1049–1051, Jun 2010b. ISSN 1476- 4687. doi: 10.1038/nature09111. URL http://dx.doi.org/10.1038/nature09111.

Ignas A. G. Snellen, Bernhard R. Brandl, Remco J. de Kok, Matteo Brogi, Jayne Birkby, and Henriette Schwarz. Fast spin of the young extrasolar planet pictorisb. Nature, 509(7498):63–65, Apr 2014. ISSN 1476-4687. doi: 10.1038/nature13253. URL http://dx.doi.org/10.1038/nature13253.

Kristin S. Sotzen, Kevin B. Stevenson, David K. Sing, Brian M. Kilpatrick, Hannah R. Wakeford, Joseph C. Filippazzo, Nikole K. Lewis, Sarah M. Hörst, Mercedes López-Morales, Gregory W. Henry, and et al. Trans- mission spectroscopy of wasp-79b from 0.6 to 5.0 m. The Astronomical Journal, 159(1):5, Dec 2019. ISSN 1538-3881. doi: 10.3847/1538-3881/ab5442. URL http://dx.doi.org/10.3847/1538-3881/ab5442.

David B. Stephenson. The northern hemisphere tropospheric response to changes in the gravity-wave drag scheme in a perpetual january gcm. Quarterly Journal of the Royal Meteorological Society, 120(517):699–712, 1994. doi: 10.1002/qj.49712051709. URL https://rmets.onlinelibrary.wiley.com/doi/abs/10.1002/ qj.49712051709.

James M. Stone and Michael L. Norman. ZEUS-2D: A Radiation Magnetohydrodynamics Code for Astrophysical Flows in Two Space Dimensions. I. The Hydrodynamic Algorithms and Tests. , 80:753, June 1992. doi: 10.1086/191680. URL https://doi.org/10.1086/191680.

Keeyoon Sung and Prasad Varanasi. Hydrogen-broadened half-widths and hydrogen-induced line shifts of 12c16o relevant to the jovian atmospheric spectra. Journal of Quantitative Spectroscopy and Radiative Transfer, 85 (2):165 – 182, 2004. ISSN 0022-4073. doi: https://doi.org/10.1016/S0022-4073(03)00202-4. URL http: //www.sciencedirect.com/science/article/pii/S0022407303002024.

82 Keeyoon Sung and Prasad Varanasi. Co2-broadened half-widths and co2-induced line shifts of 12c16o relevant to the atmospheric spectra of venus and mars. Journal of Quantitative Spectroscopy and Radiative Transfer, 91(3):319 – 332, 2005. ISSN 0022-4073. doi: https://doi.org/10.1016/j.jqsrt.2004.05.063. URL http://www. sciencedirect.com/science/article/pii/S0022407304002225.

Gary E. Thomas and Knut Stamnes. Radiative Transfer in the Atmosphere and Ocean. Cambridge University Press, 2002. URL https://ui.adsabs.harvard.edu/abs/2002rtao.book.....T.

I. P. Waldmann, G. Tinetti, M. Rocchetto, E. J. Barton, S. N. Yurchenko, and J. Tennyson. Tau-rex i: A next generation retrieval code for exoplanetary atmospheres. The Astrophysical Journal, 802(2):107, Mar 2015. ISSN 1538-4357. doi: 10.1088/0004-637x/802/2/107. URL http://dx.doi.org/10.1088/0004-637X/802/ 2/107.

Warren Washington and Claire Parkinson. An Introduction to Three-Dimensional Climate Modeling. University Science Books, 02 2005. URL http://n2t.net/ark:/85065/d7348mm0.

M. J. Way, Anthony D. Del Genio, Nancy Y. Kiang, Linda E. Sohl, David H. Grinspoon, Igor Aleinov, Maxwell Kelley, and Thomas Clune. Was venus the first habitable world of our solar system? Geophysical Research Letters, 43(16):8376–8383, Aug 2016. ISSN 1944-8007. doi: 10.1002/2016gl069790. URL http://dx.doi. org/10.1002/2016GL069790.

M. J. Way, I. Aleinov, David S. Amundsen, M. A. Chandler, T. L. Clune, A. D. Del Genio, Y. Fujii, M. Kelley, N. Y. Kiang, L. Sohl, and et al. Resolving orbital and climate keys of earth and extraterrestrial environments with dynamics (rocke-3d) 1.0: A general circulation model for simulating the climates of rocky planets. The Astrophysical Journal Supplement Series, 231(1):12, Jul 2017. ISSN 1538-4365. doi: 10.3847/1538-4365/ aa7a06. URL http://dx.doi.org/10.3847/1538-4365/aa7a06.

John K. Webb and Imma Wormleaton. Could we detect o2 in the atmosphere of a transiting extra-solar earth- like planet? Publications of the Astronomical Society of Australia, 18(3):252–258, 2001. ISSN 1448-6083. doi: 10.1071/as01037. URL http://dx.doi.org/10.1071/AS01037.

Penn State University William H. Brune, Raymond Najjar. METEO 300 Fundamentals of Atmospheric Science : Lesson 6: Atmospheric Radiation, 2019 (accessed February 3, 2014). URL https://www.e-education. psu.edu/meteo300/print/book/export/html/785.

Joshua N. Winn. Exoplanet Transits and Occultations, chapter 4, pages 55 – 77. University of Arizona Press; In collaboration with Lunar and Planetary Institute, January 2010. URL https://ui.adsabs.harvard.edu/ abs/2010arXiv1001.2010W.

J. T. Wright, O. Fakhouri, G. W. Marcy, E. Han, Y. Feng, John Asher Johnson, A. W. Howard, D. A. Fischer, J. A. Valenti, J. Anderson, and et al. The exoplanet orbit database. Publications of the Astronomical Society of the Pacific, 123(902):412–422, Apr 2011. ISSN 1538-3873. doi: 10.1086/659427. URL http: //dx.doi.org/10.1086/659427.

Ji-Wei Xie, Subo Dong, Zhaohuan Zhu, Daniel Huber, Zheng Zheng, Peter De Cat, Jianning Fu, Hui-Gen Liu, Ali Luo, Yue Wu, Haotong Zhang, Hui Zhang, Ji-Lin Zhou, Zihuang Cao, Yonghui Hou, Yuefei Wang, and Yong Zhang. Exoplanet orbital eccentricities derived from lamost–kepler analysis. Proceedings of the National Academy of Sciences, 113(41), 2016. ISSN 0027-8424. doi: 10.1073/pnas.1604692113. URL https: //www.pnas.org/content/113/41/11431.

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