Historia de la Empresa Florícola Agroganadera

In Section1.5I summarized how clouds appear to be very common in exoplanets, and mod- els show that they can have a strong influence on an atmosphere’s absorption and scattering, and hence thermal properties. Nonetheless, on no exoplanet have the actual cloud species

yet been conclusively identified. The problem is actually even more severe: while qual- itatively the picture of cloud formation often seems straightforward, the theory and data needed for a quantitative description of clouds is still incomplete, which is symptomatically shown in the wide-spread use of parametrized, or strongly idealized cloud models (see, e.g.,

Tsuji et al. 1996;Ackerman & Marley 2001;Allard et al. 2001,2003a;Zsom et al. 2012and also my models,Mollière et al. 2017).

In what follows, I try to summarize the cloud formation processes which are thought to be most important and how they interact. This summary is based onRossow(1978), in which a much more detailed description can be found. An example for two different simple models of cloud parametrization can be found in the description of petitCODE in Section

3.5.

The basic processes and how they are related is shown in Figure 2.5, which has been taken fromRossow(1978). This schematic drawing shows how the cloud interacts with the sources which provide it with material. This sources themselves can also act as sinks.

FIGURE2.5: Schematic drawing showing the relations between various cloud microsphysics processes. Figure taken from

Rossow(1978).

A cloud always forms from the vapor phase which contains the cloud species, or the required chemical building blocks, in its not yet condensed form. If a packet of gas is cooled on a timescale ⌧⇤, until its vapor density exceeds the saturation vapor density, it becomes possible to form cloud particles.

The condensation of cloud particles out of the gas phase proceeds on a timescale ⌧cond. It gives rise to a so-called population of “small” particles. These “small” particles can also be brought to the location of the cloud on a timescale ⌧supply by winds. The winds could, e.g., be streaming over a particle covered surface. But winds can also deplete the cloud in small particles on a timescale ⌧removewhen streaming over a surface which is not covered by particles.

60 Chapter 2. Physical properties of planetary atmospheres

If the removal of “small” particles is faster than the process of collisional growth (oc- curring on a timescale ⌧growth) of the “small” particles, then the cloud will not contain any “large” particles and it will not be precipitating. If the collisional growth occurs on a shorter timescale than the removal of “small” particles, then the cloud will form “large” particles, which are defined by requiring that they fall through the atmosphere on a shorter timescale than they can be supplied or removed by winds.

As soon as the “large” particles have formed, they will start to fall and settle out of the cloud in a process called precipitation or rain. As they are defined by the fact that they cannot be supplied more quickly than they fall, “large” particles will always rain out. They will start to evaporate when they reach hotter regions, this happens on a timescale ⌧evap. They can make their way back up to the cloud if they are advected by winds on a timescale ⌧supply, but if ⌧supply > ⌧evapthey will evaporate further and go back to the vapor phase.

Note that there also exist so-called “chemical clouds”, in which the “small” particles are produced by chemical reactions on a timescale ⌧chem, rather then by cooling and condensa- tion.

All of the complex physics was hidden in simple timescales here, but they correspond to a plethora of physical processes and I will start with a short summary of the bottleneck for cloud formation to even start, which are the nucleation processes.

Nucleation processes

If equilibrium chemistry is considered (see Section 2.3.1), a species with a vapor (i.e. gas) pressure larger than its saturation vapor pressure will undergo condensation by forming condensation nuclei. It is useful to define the ‘saturation’ as

S = ⇢v

⇢s (2.87)

and the ‘supersaturation’ as

S1= S 1 , (2.88)

where ⇢vand ⇢sare the species’ vapor and saturation vapor densities, respectively. ⇢s cor- responds to the maximum density which the species can have in equilibrium, after which condensation must set in. Consequently, species with S1> 0will undergo condensation, if only equilibrium condensation is considered. In reality, however, multiple effects can come into play which either hamper or promote the formation of the first nuclei. In general, the condition for forming nuclei and starting condensation can be expressed as

S > 0 , (2.89)

where

S = S1 Scurv+Sgas+Sion+Schem+SCCN, (2.90) is the ‘effective supersaturation’. The individual terms in the above equation each describe a process affecting the nucleation efficiency. Some of the above term strongly depend on

the conditions in which the nuclei form and some also depend on the nuclei’s size. The condition S = 0 therefore also defines a minimum nucleii radius. If S < 0 the particles will start to evaporate, which then represents the thermodynamically favored state. The individual processes controlling the nucleation of cloud particles are

• Homogeneous homomolecular nucleation of liquid droplets: here nucleii form from “sponta- neous condensation” of a single species (hence homomolecular) from the gas phase. A nucleus has to have a minimum size in order to overcome its own surface tension, thus the vapor density needs to be high enough. This lowers the effective supersaturation by an amount Scurv.

• Heterogeneous homomolecular nucleation of liquid droplets: spontaneous incorporation of chemically inert gas molecules (Sgas) or ions (Sion) of another species energetically favor the nucleation process.

• Chemical nucleation: here the nuclei form directly out of the gas phase by means of chemical reactions, which is different from “simple” condensation, and captured by introducing Schem.

• Heterogeneous nucleation: if there is a pre-existing population of small particles, the resulting intermolecular forces arising at the various interfaces (gas–surface, liquid– surface, gas–liquid) will lower the surface tension of the nucleus considerably, which favors nucleation. This is included through the term SCCN, where CCN stands for ‘cloud condensation nuclei’. Water droplets in Earth’s atmosphere more or less exclu- sively form through this pathway, as they have a large Scurv of 5 to 8. (Rossow 1978). Thus, in an complete nucleation theory, all of the above processes need to be included, and their modeling is necessary to determine the nucleation rates. If the nucleation occurs in presence of a CCN population, which is by far the most efficient nucleation catalyst, it can be shown that the nucleation rate ⌧ 1

nucis very large compared to the cooling rate ⌧⇤ 1and can therefore be neglected in the timescale comparison. In any case, if nucleation occurs then it is usually faster than the subsequent growth of large particles (Rossow 1978).

After the particles have nucleated, they will grow by either collisions or condensation, both processes are summarized below.

Condensational growth

For a nucleus to grow further one needs to provide it with an influx of vapor molecules. This flux competes with the release of latent heat from the nucleus’ surface. In order to model the nucleus–gas interaction one requires treatments for both the hydrodynamical or ballistical interaction regime, with the former including a treatment of whether or not the flow is laminar or turbulent. An a-priori unknown free parameter is the sticking efficiency ↵, which determines the efficiency with which the vapor molecules are captured. For solid nuclei, the sticking efficiency furthermore depends on the shape of the nucleus (Rossow 1978).

62 Chapter 2. Physical properties of planetary atmospheres

Collisional growth

Collisions between the cloud particles can lead to further growth. The collisions can occur either because of Brownian motions of because of gravitational coalescence, where more massive particles will settle through the atmosphere and sweep up the smaller ones. Again, different hydrodynamical regimes need to be modeled, and the cross-sections are veloc- ity dependent, as collisions can lead to different outcomes, such as bouncing, sticking, or fragmentation. The growth processes are typically locally described by the ‘Smoluchowski equation’ @n(a) @t = 1 2 Z a 0 K(a b, b)n(a b)n(b)db Z ₁ 0 K(a, b)n(a)n(b)db , (2.91) where n(a) is the differential number density of particles of mass a, and K(a, b) is the ‘coag- ulation kernel’, which contains all the physics, and for which K(a b, b)n(a b)n(b) is the number of particles of differential mass a produced by coagulation of the particles of mass a band b.

Cloud opacities

If the size and shape of a cloud particle are known, the particle’s corresponding opacity can be calculated. For this one needs to know its optical properties, i.e. its real and imaginary refractive indices, n( ) and k( ). Most often one assumes a spherical, homogeneous particle. Then the solution is obtained by expanding the incoming and scattered radiation field in vector spherical harmonics, an by imposing the surface boundary conditions known from the Maxwell equations on the particle’s surface. The absorption and scattering cross-sections can then be found as expansions in sums of terms which contain the refractive indices and Bessel functions, and for which the effective variable is x = 2⇡r/ , where r is the particle radius (see, e.g., Bohren & Huffman 1998). The corresponding theory is also called ‘Mie scattering’, named after its inventor Gustav Mie. The larger x is, the more terms need to be considered for the cross-sections, such that their calculation becomes more and more cumbersome. For x ⌧ 1 one recovers that the particle scattering becomes Rayleigh-like, with scat / 4. This means that one recovers the Rayleigh scattering criterion derived in Equation2.82.

Due to the comparative ease with which they can be calculated, cloud opacities in at- mospheric calculations are usually obtained from Mie theory (see, e.g.,Helling et al. 2008;

Madhusudhan et al. 2011a;Morley et al. 2012;Benneke 2015;Baudino et al. 2015). For solid particles however, it is uncertain if this is the correct approach: silicate grains have been ob- served in multiple contexts in astronomy: in disks around Herbig Ae/Be stars (Bouwman et al. 2001;Juhász et al. 2010), also in AGB stars, post-AGB stars, planetary nebulae, and massive stars (Molster et al. 2002). In these cases a good fit to the data could only be ob- tained when non-spherical, non-homogeneous shapes for the particles were assumed, such as in the case of ‘Distribution of Hollow Spheres’ (DHS) (Min et al. 2003), where the particles

are modeled as spheres with a spherical cavity in the middle, which is still comparatively easy to treat with a modified Mie theory.

2.3 Chemistry

In order to know the total opacity of a given atmospheric layer it is necessary to know its composition, i.e. its molecular and atomic abundances. In order to obtain the composi- tion, the chemical reactions between the individual species, at the atmospheric pressure– temperature conditions, need to be modeled. In petitCODE I solve this problem by assum- ing that the atmosphere is in chemical equilibrium, which speeds up the computation of the composition significantly. Both equilibrium and non-equilibrium chemistry are summa- rized in the sections below.

2.3.1 Equilibrium chemistry

Consider a gas at pressure P , temperature T , and known atomic composition. Equilibrium chemistry corresponds to the answer to the question which ionic, atomic and molecular abundances the system would attain, due to chemical reactions, in the limit of t ! 1, where t is the time. To answer this one can make use of the Gibbs free energy G, which is a thermodynamic state function, and its properties. The change in Gibbs free energy can be expressed as (see, e.g.,Schwabl 2006)

dG = V dP SdT + Nspecies_X

j=1

µjdNj , (2.92)

where V is the volume, P the pressure, S the entropy, T the temperature, µj(P, T )the chem- ical potential, and Nj the number of particles of species j.

From statistical mechanics it can be shown that for an isolated thermodynamical system evolving with time, it holds that the Gibbs free energy G decreases

G < 0, (2.93)

which follows from the second law of thermodynamics, S > 0. Because systems that equi- librate evolve towards the state of maximum entropy, a system in equilibrium (i.e. all state functions are constant in time) has evolved towards minimum Gibbs free energy G.

From Equation2.92one sees that at constant pressure and temperature, the Gibbs free en- ergy may be expressed as

G = Nspecies_X j=1 µjNj+ cst , with dG = Nspecies_X j=1 µjdNj. (2.94)

Thus, in order to find the equilibrium chemical abundances of molecules and atoms of a given system one has to minimize Equation2.94, while keeping the number of elemental

64 Chapter 2. Physical properties of planetary atmospheres

building blocks of all molecular species (i.e. atoms) constant. The latter condition corre- sponds to the Natomconstraints

Nspecies_X j=1

aijNj = bi (2.95)

bi = cst8 i 2 {1, . . . , Natom}, (2.96) where the aij are the so-called stoichiometric factors, giving the number of atoms of species i in a given molecule of species j. Natom is the number of the atomic species that need to be included for the system under consideration. This problem formally corresponds to the task of minimizing a function G(N1, . . . , NN_species)while having to fulfill Natomdifferent constraints. This can be done using the method of ‘Lagrange multipliers’. For this one defines and minimizes the function

L(N1, . . . , NNspecies, 1, . . . , Natom) = Nspecies_X j=1 µjNj NXatom i=1 i 0 @ Nspecies_X j=1 aijNj bi 1 A , (2.97)

where iare the so-called Lagrange multipliers. This is approach leads to the desired result because the minimum value will satisfy

rL = 0 , (2.98) with r = ✓ @ @N1 , . . . , @ @NNspecies , @ @ 1 , . . . , @ @ Natom ◆T , (2.99)

which is the mathematical description of minimizing G while conserving the atom numbers. Such equilibrium chemistry models are commonly known as ‘Gibbs minimizers’, and a good introduction can be found inGordon & McBride(1994). Note that one does not require any knowledge about the chemical reactions involved in driving the system to equilibrium, considering the individual species’ chemical potentials, and their stoichiometric factors, is sufficient.

2.3.2 Non-equilibrium effects

A crucial assumption for equilibrium chemistry is that it yields the chemical abundances of an isolated parcel of gas at constant P and T , in the limit of t ! 1. If ⌧chemis the timescale of the slowest chemical reaction in a chemical network, this means that equilibrium chemistry will break down if

⌧chem> ⌧change, (2.100)

where ⌧change is the timescale of any process which changes the state of the parcel of gas under consideration. Such processes can be the transport of the gas to regions of different P or T , or the dissociation and ionization of molecules and atoms by incident radiation, and the resulting reactions of the ions and dissociation products, which drive the system

further away from equilibrium. If equilibrium breaks down, then the chemical abundances can be obtained by bookkeeping of the available number of atomic building blocks, and by modeling the individual chemical reactions in a so-called chemical network. The equations describing the change in number density niof a given species i then take the form (see, e.g.,

Venot et al. 2012)

@ni

@t = Pi niLi r · Fadv, (2.101)

where Pi is the production rate of species i, arising from reactions and photolytic dissocia- tion and ionization processes, Li is the corresponding loss term due to the same processes, and Fadv is the flux of particles, arising from atmospheric circulation, convection, or other vertical diffusion processes. If production and loss terms are negligible, the equation takes the form of the continuity equation Equation2.1, as expected. Because the production and loss terms depend on the densities of other chemical species, the chemical reaction network, described by equations like Equation2.101, corresponds to a set of coupled, non-linear dif- ferential equations. For the full solutions one requires knowledge of the spatial temperature and velocity distribution within the atmosphere, which is in itself challenging.

That being said, there exist numerous studies which tackle reduced versions of this prob- lem, an we now know that diffusive processes in the radiative zones of planets may trans- port abundances from the hot, deep atmospheric layers to the photosphere (see, e.g.Zahnle & Marley 2014). This is called ‘quenching’, and may affect planets with Tequ . 1000 1500 K, depending on the strength of diffusion (Miguel & Kaltenegger 2014;Zahnle & Marley 2014). For hot jupiters with strong horizontal circulation the eastward blowing jet may distribute the hot spot abundances around the whole planet (Agúndez et al. 2014). In both of the above scenarios cooler regions are overwhelmed by the equilibrium abundances of hotter regions, because the chemical timescale is temperature-dependent. The effect of photodis- sociation and ionization by stars and flares has been studied in, e.g.,Venot et al.(2012,2015,

2016). This so-called ‘photochemistry’ usually only affects the molecular abundances in the high, low-pressure regions of the atmospheres, such that the observational signatures may be challenging to see.

66 Chapter 2. Physical properties of planetary atmospheres