Dust in the atmospheres of Brown Dwarfs

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Dust in the Atmospheres

of

Brown Dwarfs

von

Dipl.-Phys. Marcus L¨

uttke

aus Berlin

der Fakult¨

at II (Mathematik und Naturwissenschaften)

der Technischen Universit¨

at Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaft (Dr. rer. nat.)

genehmigte Dissertation

Berlin 2002 D 83

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Vorsitzender: Prof. Dr. P. Zimmermann

Berichter:

Prof. Dr. E. Sedlmayr

Priv. Doz. Dr. J. P. Kaufmann

Tag der m¨

undlichen Pr¨

ufung: 15.03.2002

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Zusammenfassung

Sterne verbringen einen Großteil ihres “Lebens” in einem metastabilen Gleichgewichtszus-tand zwischen der nach innen gerichteten Gravitationskraft einerseits und den nach aussen gerichteten Druckkräften des Gases und der Strahlung andererseits. Bereits in den 60er Jahren des zwanzigsten Jahrhunderts vermutete man, daß es eine Klasse von sternähnlichen Objekten geben müsse, deren Masse nicht ausreichen dürfte, um die thermodynamischen Bedingungen im Sterninneren zu erzeugen, die benötigt werden, “nukleares Brennen” zu zünden. Diese “verhinderten Sterne”, die erst durch die großen Fortschritte in den Beobach-tungstechniken während der letzten Dekade der Beobachtung zugänglich wurden, werden als Braune Zwerge bezeichnet.

In den Atmosphären Brauner Zwerge herrschen thermodynamische Bedingungen, die die Bildung von Staubteilchen, d.h. kleinen Festkörperpartikel ermöglichen (“tiefe” Tem-peraturen bei relativ hoher Dichte). Mit der Untersuchung dieser kühlen, staubbilden-den Atmosphären befasst sich die vorliegende Arbeit, die erstmalig die zeitabhängige Entwicklung des Staubkomplexes in der Beschreibung der Atmosphären Brauner Zwerge berücksichtigt.

Nach einer Einführung in das Themengebiet, in der die Eigenschaften und Charakter-istika Brauner Zwerge erläutert werden und eine Einordung dieser Objekte in den allge-meinen astrophysikalischen Kontext erfolgt, schließt sich die Vorstellung der physikalischen Grundgleichungen und Konzepte an. Neben der Beschreibung der Konvektion in Kom-bination mit der Strahlungshydrodynamik wird besonderes Gewicht auf die ausführliche Darstellung heterogenen Staubwachstums in astrophysikalischen Situationen gelegt. Bei der hier vorgestellten Methode handelt es sich um eine Weiterentwicklung bzw. Zusam-menführung bereits vorhandener Darstellungsweisen.

Darauf folgt die Beschreibung der verwendeten numerischen Methoden. Neben der detaillierten Darstellung der Diskretisierungs- und Lösungsvorschriften für die Gleichungen der konvektiven Strahlungshydrodynamik wird auch hier besonderes Augenmerk auf die numerische Umsetzung des Staubkomplexes gelegt. Hierzu wurde im Rahmen dieser Arbeit ein Algorithmus entwickelt, der auch den nicht kontinuierlichen Prozessen (Sedimentation, Koagulation u.ä.) Rechnung trägt.

Einem kurzen Einschub mit allgemeinen Überlegungen und Anmerkungen über hetero-genes Wachstum, der Form der Staubkörner und dem astrophysikalisch relevanten Prim¨ ar-kondensat ist ein weiterer Abschnitt gewidmet. Anschließend werden Ergebnisse der ersten zeitabhängigen Modellrechnungen für die Atmosphäre eines Braunen Zwerges präsentiert.

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List of Figures

1.1 The Brown Dwarf Gl229B . . . 10

1.2 Continuum of Objects . . . 12

1.3 Central temperature for selected masses as a function of age. . . 14

2.1 The equilibrium concentrations of some selected atoms and molecules over a temperature density plane. . . 28

2.2 The elemental abundances shown in a schematic view. . . 31

2.3 The bar chart shows the logarithm of the maximum relative error for the remaining elements (B) dependent on the removed element (A). . . 34

2.4 Sublimation temperature of various solids . . . 45

2.5 Nucleation rates for some astrophysical seed candidates . . . 48

2.6 Nucleation rates for some astrophysical seed candidates (contour plot) . . . 50

2.7 Energy, pressure and reciprocal value of the mean molecular weight over a density-temperature plane . . . 56

2.8 Energy, pressure, specific heat and adiabatic index over a restricted plane covering the typical conditions in Brown Dwarf atmospheres . . . 57

2.9 Rosseland Mean Opacity for the gas phase . . . 59

3.1 Staggered Mesh . . . 61

4.1 General quantities concerning the probing box. . . 70

4.2 The temporal development of the fraction of the condensed elements . . . . 71

4.3 The evolution of the particle densities for some selected species in the gas phase . . . 72

4.4 Resulting size distribution function for the heterogeneous growth calculation. 73 4.5 Temporal evolution of a bin . . . 74

4.6 Simple Molecular Dynamical Simulation, situation 1, A:B = 9:1 . . . 76

4.7 Simple Molecular Dynamical Simulation, situation 2, A:B = 1:9 . . . 76

4.8 Comparison of the Nucleation rates of TiO and TiO2 . . . 79

5.1 Temporal development of the luminosity L_∗ and the effective temperature Teff 82 5.2 Time evolution of the atmosphere . . . 83

5.3 Size distribution for a dusty atmosphere at an age of about 1.2 Gyr . . . . 85

5.4 Radial structure of the atmosphere . . . 87 5

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5.5 The radial distribution of the normalized number densities for a selection of chemical species in the dust free case. . . 88 5.6 The radial distribution of the normalized number densities for a selection of

chemical species in the depleted case. . . 89 5.7 The size distribution resulting from the enrichment due to the convection

process. . . 90 5.8 The size distribution in the artificial injected enrichment of the depleted

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List of Tables

2.1 Free parameters and standard values entering the model for time-dependent

turbulent convection . . . 25

2.2 Solar atomic abundances and molecular weights . . . 32

2.3 Resulting errors due to the reduction of the considered elements. . . 33

2.4 Dimers for whom thermodynamical data is available . . . 36

2.5 Data entering the nucleation rate. . . 51

4.1 Sticking coefficients α for the simple Molecular Dynamical Simulations . . 77

5.1 Stellar parameters for the initial hydrostatic model . . . 81

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Chapter 1 Introduction

“Twinkle little star, how I wonder what you are...”

Today, someone might just as easily find astronomers and astrophysicians humming this nursery rhyme as well as children. Rapid advances in telescope technology, e.g. adap-tive optics, space observatories, interferometry, and image processing techniques, enable researchers to see ever fainter and smaller companions to normal stars. As telescopic ca-pabilities sharpen, conventional definitions for planets and stars may seem to be getting blurry. In the search for other planetary systems, observers detect objects that straddle the dim twilight zone between planets and stars, and others that seem to contradict conven-tional wisdom, such as a planetary system accompanying a burned-out compact neutron star.

How brightly a star shines throughout its life is determined mainly by its birth weight: The greater its mass, the brighter it shines. But the relationship has its limits: At the upper end of the weight scale, some extremely massive stars will ultimately collapse due to their own weight to form a black hole. At the lower end of the scale, a would-be star may not be sufficiently massive to ignite the thermonuclear reactions that make a star shine. In these instances the low-mass object that forms never reaches stardom; instead it sits quietly and dimly throughout its life as a sub-stellar object, that astronomers now call a Brown Dwarf.

Originally called black dwarfs, these sub-stellar objects were first conceived in the early 1960s as dark bodies floating freely in space. Stellar models had suggested that a star must have a mass of at least 80 (todays model say 74) times that of Jupiter to kindle the stable fusion of hydrogen. Lighter objects were believed to exist, but it was recognized that they would be extremely difficult to detect because they would emit very little light.

Despite the appearances of some intriguing candidates over the years, it was not until 1995 that astronomers finally confirmed the existence of Brown Dwarfs with the discovery of two sub-stellar objects – christened Teide 1 and Gliese 229B (Rebolo et al. (1995) and Nakajima et al. (1995)) – which were reported within weeks of each other. The recent breakthrough has come from pushing the sensitivity of detectors together with a new gen-eration of large-format CCDs. Near-infrared detectors in ground-based telescopes have also

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Figure 1.1: The Brown Dwarf Gl229B is located almost in the center of both observations. He was first observed in far red light using the adaptive optics device and a 60-inch reflect-ing telescope on Palomar Mountain in California. Another year was required to confirm that the object was actually gravitationally bound to the companion star. The Hubble observations will be used to accurately measure the Brown Dwarf’s distance from Earth, and yield preliminary data on its orbital period, which may eventually offer clues to the dwarf’s origin.

been improved, and these devices will continue to add discoveries. Due to these develop-ments, previously unrecognized faint objects are now being discovered almost on a monthly basis since 1997 and more detailled observations have been performed (Zapatero Osorio et al. (1997), Ruiz et al. (1997), Schultz et al. (1998), Tinney (1998), Bouvier et al. (1998), Festin (1998), Martin et al. (1998), Oppenheimer et al. (1998), Zapertero Osorio et al. (1999), Delfosse et al. (1999), Lucas and Roche (2000), Burgasser et al. (2000), Moraux et al. (2001), Guinan and Ribas (2001), Barrado y Navascu´es et al. (2001) etc.).

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1.1. THE LIFE CYCLE OF BROWN DWARFS 11

1.1 The Life Cycle of Brown Dwarfs

The early lives of Brown Dwarfs and stars follow the same pattern. Both are believed to originate from the gravitational collapse of interstellar clouds. The collapsing clouds are primarily composed of hydrogen and helium, but they also contain initially small amounts of deuterium and lithium that are remnants of the nuclear reactions that took place a few minutes after the big bang. Additionally the cloud can consist of heavier elements.

Brown Dwarfs contract similar to young stars, their cores grow hotter and denser, and the deuterium nuclei fuse into 3He-nuclei. Deuterium fusion can occur in Brown Dwarfs because it requires a lower temperature – and resulting lower mass– than hydrogen fusion. The connected release of energy from these reactions temporarily halts the gravitational contraction and causes the objects to brighten. But the reservoir of deuterium lasts only for a few million years after which the contraction resumes. Lithium fusion occurs next in stars and in Brown Dwarfs more as 60 times as massive as Jupiter. Jupiters mass (1MÅ ≈ 9.5465 · 10−4M) is usually taken as the referring unit mass for the sub-stellar regime.

During the contraction phase the thermal pressure increases in its core and opposes the gravitational forces acting inwards. The interior temperatures are high enough to completely ionize all atoms and form a plasma. According to the Pauli principle no two electrons can occupy the same quantum state and in a very dense plasma all low-energy states are occupied, and many electrons are forced to fill very high energy states. This generates a form of pressure that is insensitive to the temperature, the so called degener-ation pressure. Objects supported in this manner are called degenerate. One consequence of this process is that all Brown Dwarfs are roughly the size of Jupiter, the heavier Brown Dwarfs are simply denser than the lighter ones.

In stars the core usually does not degenerate. Instead of this hydrogen fusion provides the pressure that supports the star against his own gravity. The star stops contracting and reaches a steady state, luminosity and temperature. In very heavy Brown Dwarfs (_≥ 60 MÅ) hydrogen fusion also begins but then sputters out. As degeneracy pressure slows the collapse of Brown Dwarfs, their luminosity from gravitational contraction declines. Although very low mass stars can shine for trillions of years, brown dwarfs fade steadily toward oblivion. This makes them increasingly difficult to find as they age. In the very distant future, when all stars have burned out, Brown Dwarfs will be the primary repository of hydrogen in the universe.

1.2 Planets vs. Brown Dwarfs

The discovery of Brown Dwarfs has occured almost in parallel with the discovery of Extra-solar planets. A comparison between Brown Dwarfs and the gasplanets within our Extra-solar system indicate some of the difficulties to tell them apart. For example, Jupiter has an outer atmosphere dominated by methane and water, which also have been observed in the Brown Dwarf Gliese 229B (Oppenheimer et al. (1996), Marley et al. (1996), Oppenheimer

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Figure 1.2: The continuum of objects from planets to stars shows that older Brown Dwarfs, such as Gliese 229B, are fairly similar to gas-giant planets in size and surface temperature. Younger Brown Dwarfs like Teide 1 are more closely to low-mass stars such as Gliese 229A. Brown Dwarfs and low-mass stars are fully convective. Thermonuclear reactions in the stellar cores destroy all their lithium, so its presence is a sign that the object may be a Brown Dwarf. (This image is by Bryan Christie and was found on the Homepage of Gibor Basri).

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1.3. STARS VS. BROWN DWARFS 13 et al. (1997)). But on the other hand, Teide 1 only has low water and methane abundancy (Zapatero Osorio et al. (1997a), Pavlenko (1997),...), but reveals abundant carbon monox-ide and the oxmonox-ides of titanium and vanadium. From this point of view Gliese 229B has more in common with Jupiter than with Teide 1. In this instance, Teide 1 stands out due to its tender age (_{≈ 10}8 yr) whereas Jupiter and Gliese 229B are about 10 to 50 times older. Evolutionary and atmospheric models of Brown Dwarfs and planets predict that both, methane and water, will accumulate in the atmosphere of Teide 1 as it ages and that the mentioned oxides will become depleted due to dust formation.

The classical view on the distinction between planets and Brown Dwarfs is that planets form in a different way than Brown Dwarfs or stars do. Giant gasplanets are thought to build up from planetesimals (small rocky or icy bodies) amid a protoplanetary disk of gas and dust orbiting around a star. Within a few million years these solid cores attract huge envelopes of gas. This theoretical model is based on our own solar system and predicts that all planets should be found in nearly circular orbits around stars and that gas-giants should travel in relatively distant orbits. These assumptions were proven not to be vaild in general by the discovery of the first extrasolar planets. Most of them have been found in close and eccentric orbits.

Another way to distinguish between Brown Dwarfs and planets is by their internal structure. Brown Dwarfs are believed to form out of a collapsing cloud of interstellar gas in contrast to the described agglomeration process. Therefore the internal structures should be different (cf. figure 1.2). Brown Dwarfs should be chemically nondifferentiated due to their fully convective interior, whereas planets would have a solid, highly metallic interior. It is not possible to observe these processes for any single celestial object due to the enormous disparity between the evolutionary time scales and the comparatively brief careers of human astronomers. This criterion is most difficult to implement. How can one probe the inside of Brown Dwarfs or planets?

Finally, the distinction between them suggest another borderline. The ability to ignite any nuclear fusion reaction imprints a dividing line at about 13 MÅ. Above this mass, deuterium fusion occurs in the object (cf. figure 1.3). To underline the difference between “ordinary” stars (cf. Section 1.3), Brown Dwarfs and Planets one might refer to Figures 1.3 and 1.2. Here the central temperature for selected masses as a function of age is plotted. Furthermore the hydrogen, lithium and deuterium burning temperatures are depicted.

But a binding definition of Planets and Brown Dwarfs is not at hand. There is only a “working definition” saying that everything orbiting around a star with less than 13 MÅ is a planet.

1.3 Stars vs. Brown Dwarfs

A Brown Dwarf is a failed star. A star shines due to the thermonuclear reactions in its core, which release enormous amounts of energy by fusing hydrogen into helium. For the fusion reactions to occur the temperature in the stellar core must reach at least 3 000 000 K. And because core temperature rises with gravitational pressure, the star must have a

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Figure 1.3: Central temperature for selected masses as a function of age. TH, TLi and

TD indicate the hydrogen, lithium and deuterium burning temperatures, respectively. The

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1.3. STARS VS. BROWN DWARFS 15 minimum mass of about 74MÅ, or about 7% of the mass of our sun. A Brown Dwarf misses that mark, it is heavier than a gas-giant planet but not quite massive enough to be a star.

Although Brown Dwarfs are dim objects compared to stars, their typical luminosities are about or less than one millionth of that of our sun, they do emit a steady warm glow, owing to an atmospheric temperature somewhere between 300 K and 3000 K. As a result, Brown Dwarfs are brightest at infrared wavelength, and it seems plausible that they would indeed appear to have a deep, ruddy brown color to the naked eye. Due to the relatively low atmospheric temperatures, the outer layers should contain certain molecules that are otherwise “cooked” out of the atmosphere of hot stars. Notable among these are the oxides of titanium (TiO) and vanadium (VO). These molecules are destroyed by energetic collisions in the hot gases of stars like our sun, but they should dominate the spectra of Brown Dwarfs. They are the signatures of cool objects.

The coolest true (hydrogen-burning) star, a M dwarf star (or Red Dwarf), is a low-mass object with temperatures below 3500 K. To the naked eye they would appear to have a deep red color, and their diameters may not be larger than that of a Brown Dwarf. Because they are so cool, red dwarfs also form titanium oxide and vanadium oxide in their atmospheres.

The important difference between Brown Dwarfs and Red Dwarfs arises from the ab-sence of sustained thermonuclear reactions in the sub-stellar objects. This distinction permits the existence of an even more fragile substance – the element Li – in the Brown Dwarf’s atmosphere. The high temperatures in a star’s core promote high-energy collisions between a7Li nucleus and a proton, producing two4He nuclei. Since even the coolest stars attain sufficient temperatures to destroy lithium through the burning of hydrogen, all true stars lack this element. Because M dwarf stars are fully convective, they efficiently mix the matter on their surfaces with that of their interiors, even a very old Red Dwarf with an atmospheric temperature resembling that of a Brown Dwarf will have destroyed all of its lithium. Brown Dwarfs, on the other hand, never attain interior temperatures that can destroy lithium, provided their masses are less then 60MÅ. Brown Dwarfs of 60 to 74 MÅ do destroy lithium, and even burn some hydrogen, but these “transition objects” never become stable stars. Consequently, if lithium is observed in a very cool dwarf, it guarantees that it is a Brown Dwarf. In contrast, the absence of lithium is an ambiguous indicator of status since the object could be a star or a transition object. This so called lithium test was proposed by Rebolo et al. (1992).

The depletion by dust formation has required the extension of the well established stellar OBAFGKM classification sequence with a proposed L-type, to include those objects cooler than Teff < 2000K, which do not fit into the existing M-type.

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1.4 Remark: Missing Mass – still missing

Dynamical studies of the rotation of our galaxy indicate that the disk is embedded within a massive dark halo, which outranges the observable (baryonic) mass by many orders of magnitudes. Similar studies in other spiral galaxies show the same behavior (Ashman (1992)). As a possible solution for this missing mass problem the existence of a population of dark halo objects with a mass function dominated by Brown Dwarfs formed in the early evolutionary stages of the galaxy has been considered. But taking a closer look to the only possible observations that can probe the dark halo directly, the gravitational microlensing events, negates this consideration. The principles of such observations are reviewed in Paczynski (1996) and references therein. Early results were interpreted as providing evidence for a dark halo at least partially composed of objects with a mass similar to that of a Brown Dwarf, but a more recent work by Alcock et al. (1997) indicate that, if the observed events are due to members of a standard isothermal halo, than about 50% of the halo’s mass must be in objects with preferred masses around 0.5 M1. This is somewhat larger than the typical mass of a Brown Dwarf. Although the total halo mass observed is well constrained, the preferred mass is quite model dependent (Alcock et al. (1997)), this makes it problematic to determine the preferred mass. Additionally alternative interpretations do exist (Zaritsky and Lin (1997)).

Summarizing it can be said, that only a small subset of the likely halo models permit a significant dark halo mass in Brown Dwarfs, and that microlensing results strongly argue against a dark halo composed of sub-stellar-mass objects (Alcock et al. (1998)). Those models which do permit a Brown Dwarf halo require a mass function strongly peaked at Brown Dwarf masses, which seems to be inconsistent to the fact that every other observed population shows a more or less smooth mass function for both the lowest-mass stars and Brown Dwarfs.

The universe is not stuffed with Brown Dwarfs and they might not solve the missing mass problem, but they are one of the most numerous objects and their study will reveal interesting problems and solutions by bridging the twilight zone between stars and planets.

1.5 Drift ’N’ Weather – The relative motion of dust

Bailer-Jones and Mundt (2001b,a, 1999) found that eleven out of a sample of 21 Brown Dwarfs to be variable on time scales of a few minutes up to a few tens of hours with a confidence level of 99%. Eight of these objects show small amplitude variations between 0.01 and 0.03 mag not correlated with a rotation period. It was suggested by the authors that these variations might well result from an inhomogeneous distribution of dust in the atmosphere. Mart´ın et al. (2001) follow this interpretation for their observations of the ultracool, very inactive dwarf BRI 0021–0214. Oppenheimer et al. (1998) found evidence for variability in the brown dwarf Gl 229B of a maximum of 10% on time scales from

1_{Maybe they are Black Dwarfs, dead, burned-out White Dwarfs, but this hypothesis leads to other}

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1.5. DRIFT ’N’ WEATHER – THE RELATIVE MOTION OF DUST 17 several minutes over a few days to one year in the 1.6µm CH4 absorption line. Gelino et al.

(2001) argue that magnetic spots are unlikely to be responsible for such variability since the magnetic Reynolds number of the atmospheric gas is very small on the mesoscopic scales (lref = HP) considered.

The inhomogeneous distribution of dust might be the result of the interior’s convective motion and/or the induced turbulent motion. Helling et al. (2001) started to investigate the influence of turbulence on the dust complex on microscopic scales. The inhomogeneous dust distribution is strongly associated with the formation of clouds and resulting weather-like features of Brown Dwarfs and Giant Gas Planets weather-like the observed vivid motion on the surface of Jupiter.

The discussion of weather-like features in Brown Dwarf atmospheres leads inevitably to the question about sedimentation processes (gravitational settling), which means a non-zero relative velocity between solid dust particles or fluid droplets and the surrounding gas. This “drift problem” of the dust has been investigated by various authors (e.g. Berruyer and Frisch (1983), MacGregor and Stencel (1992), Kr¨uger and Sedlmayr (1997), Kr¨uger et al. (1994)).

To solve this drift problem mostly, two-fluid (dust and gas) approaches have been adopted assuming a constant gain size. Krüger and Sedlmayr (1997), Krüger et al. (1994) and Simis et al. (2001) e.g. adopted a mean grain size radius which results from the time-dependent description of dust formation according to Gail et al. (1984), Gail and Sedlmayr (1988). The authors assume an instantaneous dust nucleation. Assuming further stationarity, Krüger et al. (1995) developed a multi-component method for the 1D drift problem including a full time-dependent dust formation description and allowing explicitly for a size-dependent, directed drift velocity (i.e. grains of different size may move with different velocities) including effects such as the dynamical dilution of the dust component. This powerful approach has unfortunately the drawback that only one independent spatial coordinate can be handled so far.

Current models for the atmospheres of Brown Dwarfs and Extra-solar gas planets follow pursue a much simpler approach to study the effect of dust sedimentation by gravitational settling in the frame work of hydrostatic but frequency-dependent model calculations. The principle effect applied is to remove heavy elements like Ti, Fe, Mg, . . . from the star’s atmosphere. The substantiation is founded on the assumption that the depleted elements have been consumed by dust formation (Burrows et al. (1997), Burrows and Sharp (1999), Seager and Sasselov (2000)). An extensive time-scale study of dust sedimentation for the atmospheres of Jupiter, Venus, and Mars has been presented by Rossow (1978).

The analysis of the drift problem in the atmospheres of Low-Mass stars and substel-lar objects is of high interest for the interpretation and understanding of many of the observational data accumulated so far.

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1.6 On Actual Research And This Work

Brown Dwarfs are very faint objects and the radiation pressure is negligible in their atmo-spheres. The interior of brown dwarfs is entirely convective. Chabrier and Baraffe (1997) found the upper limit of fully convective stars to be 0.35M for metallicities that are solar or less solar based on the mixing length theory. The interior convective zone ends only a few pressure scale heights (Hp) below the photosphere (Chabrier et al. (2000)). The

convective transport of energy, momentum, and material reaches into the Brown Dwarf at-mosphere (Chabrier and Baraffe (1997), Burrows et al. (1997), Burrows and Sharp (1999), Allrad et al. (2000), Tsuji et al. (1999)) and can be important for the replenishment of the atmosphere by heavy elements which might have been previously depleted by incorporation into dust particles and their gravitational settling (Chabrier et al. (2000), Helling et al. (2000)).

The atmospheres of Brown Dwarfs provide favorable conditions for the gas–solid phase transition due to their low temperatures and high densities. Stevenson (1986) and Lunine et al. (1989) were the first to point out the importance of dust as an opacity source in the evolution of a Brown Dwarf. Tsuji et al. (1996b) suggested dust containing photospheres as the only reasonable fit to the observed spectral energy distribution (SED) of the Brown Dwarf GD 165B observed by Tinney et al. (1993) and to the infrared H2O bands observed

by Jones et al. (1994). Kirkpatrick et al. (1999) provide a comparable fit by hydrostatic atmospheres considering the existence of solid particles with a power-law size distribution in order to account for the unexpectedly weak molecular absorption bands. The absorption bands of TiO and VO predicted by these modells are still too strong.

The present interpretation of the observed spectra of Brown Dwarfs relies exclusively on hydrostatic model atmospheres with an elaborate treatment of frequency dependent radiative transport including a large body of atomic and molecular line data. Encouraged by extensive theoretical investigations of possible compounds in the cool and dense envi-ronment of Brown Dwarfs and gas planets (Burrows and Sharp (1999), Lodders (1999)), dust has been included in the static model atmospheres as an opacity source. The dust caused a strong backwarming and element depletion, resulting in weaker molecular ab-sorption (Tsuji et al. (1996a,b), Jones and Tsuji (1997), Pavlenko et al. (2000), Chabrier et al. (2000), Leinert et al. (2000), Basri et al. (2000), Allard et al. (2001)) and a better agreement with observation could be achieved. Depending on the purpose of the models, dust has been unconsidered or taken into account as opacity source. The easy access of the dust complex in the static model atmospheres must be payed for by various simplifications: 1. The compounds are assumed to be in phase equilibrium with the gas (e.g. Burrows and Sharp (1999), Chabrier et al. (2000), Leinert et al. (2000), Allard et al. (2001)) as introduced by Grossman (1972) and Lunine et al. (1989). If, however, the su-persaturation ratio _{S equals 1 (phase equilibrium), and therefore the formation and} destruction rates equilibrate, no solid particles can form. A sufficient supersaturation is needed to form solids (cf. Section 2.3.2), therefore, a much lower temperature than T (S = 1) is required (see e.g. Rossow (1978), Gail et al. (1984)).

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1.6. ON ACTUAL RESEARCH AND THIS WORK 19 2. Adopting phase equilibrium, the description of the formation of the compounds is omitted. Therefore it appears the inconsistence that the monomer molecules from which the compounds are meant to form are often not present in the gas phase or only in very small amounts (e.g. Al2O3, CaTiO3, MgAl2O4, CaMgSi2O7 . . .).

3. The grain size distribution has to be prescribed. The interstellar power law has been adopted for grain sizes of 0.025 . . . 0.25µm in order to derive the spectral character-istics of the dust for the opacity calculation.

Lunine et al. (1989) were the first to point out the influence of the grain size on the resulting grey atmosphere structure in brown dwarfs following time-scale arguments of Rossow (1978).

The gas–solid phase transition is in fact a non-equilibrium process and the grain size distribution is not known a priori. Furthermore, the formation of dust particles is a time dependent process and a consistent treatment with hydro- and thermodynamics is therefore required. Congruously Allard et al. (1997) stated that

“The effect of grain formation and of its opacity on the atmospheric structure of M dwarf atmospheres – and also of the cooler Brown Dwarfs – will, therefore, not be fully understood until grain formation and time-dependent grain growth calculations incorporating the effects of sedimentation, diffusion, coagulation and coalescence are included. ... But, as yet, no results habe been obtained for oxygen rich dwarf atmospheres.”

During the last years no significant progress was achieved. Only the hydrostatic models be-came more sphisticated by considering further refinements in the model, e.g. a parametric description for the formation of clouds.

This work represents the first step towards the demanded complete time-dependent approach to the dusty atmospheres of Brown Dwarfs. In order to perform a time-dependent approach to dust formation one has to treat the surrounding physical description also in a time-dependent way to take care of the effects and feedbacks on the resulting atmospheric structure. The second chapter presents the basic equations and physical concepts that are relevant to formulate a time-dependent model for a dust processing atmosphere of a fully convective star. The presented approach is kept as general as possible, so the concepts might easily migrated to other (astro-)physical situations like the atmospheres of stars on the Asymptotic Giant Branch (AGB) or extrasolar planets. First the equations of Convective Radiation Hydrodynamics (CRHD) are given, followed by the description of chemical equilibrium. Thereupon a description for the formation of heterogeneous dust grains is developed. This theoretical building finds its basement in the previous work of the group of Prof. Sedlmayr (Gail and Sedlmayr (1988), Gauger et al. (1990), Dominik (1992), Patzer et al. (1998a),...). This chapter ends with the presentation of the concept of the equation of state and discusses shortly the used opacities.

The numerical methods applied to solve the arising CRHD equation system follow a nowadays state of the art approach. Only for the description of the dust complex an algorithm is developed that is capable to include effects of sedimentation (“rainout”) of larger dust grains, other discontinous processes and even different symmetries. After some

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applications and remarks to heterogeneous growth, shape and first atmospheric nuclei the first time-dependent modelled atmosphere of Brown Dwarf is presented. The work closes with the discussion of the results and some proposals for further projects based on this thesis.

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Chapter 2 Basic Equations

In this chapter the basic equations and physical concepts are briefly reviewed that are re-quired to model astrophysical objects. The focus lies thereby on Brown Dwarf atmospheres, but the concepts are also applicable to a great variety of astrophysical situations ranging from the pulsating atmospheres of AGB stars over the outer parts of Extra-solar Giant Planets to the description of the stellar interiors. In principle even Atmospheric science in general might also serve as “model host” for the concepts presented in this chapter.

On the following pages the state of the art approach to Convective Radiation Hydrody-namics (CRHD) in one spatial dimension is described followed by the concept of chemical equilibrium. After that the focus is laid on the description of dust processing. Finally, the Eqation of State necessary for the “closure” of the non-linear equation system and the radiative transport coefficients are presented.

2.1 The Equations of Convective Radiation

Hydrody-namics

One of the first detailed and complete descriptions of the application of the radiation hydrodynamics to astrophysical situations in one dimension was given by Mihalas and Weibel Mihalas (1984). Since then many successful applications (some also including the description of a dust component) has been realized (e.g. Ensman (1994), Gehmeyr and Mihalas (1994b), Gehmeyr and Mihalas (1994a), Feuchtinger et al. (1993), Dorfi and Feuchtinger (1999), Winters et al. (1994a), the series following Dorfi and Feuchtinger (1991) and Dorfi and H¨ofner (1991), and the series “Circumstellar dust shells around long-period variables” from the local group and many, many others... 1

Radiation and self-gravity dominate many astrophysical situations. The relative in-efficiency of heat conducting and radiative transfer in cooler stars (e.g. M-Dwarfs and Brown Dwarfs) and planets rapidly produce convective unstable situations. Thus, convec-tion plays often the role of the dominating heat transfer process. When the evoluconvec-tion of

1_{(my apologies for the hundreds of publications I didn’t mention here)}

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the flow in a star becomes so rapid that the evolutionary time scale is comparable to the rise time of buoyancy driving convective eddies over a distance where the flow quantities change considerably, then a model for time-dependent turbulent convection is needed to calculate the physical properties of the flow correctly. In most astrophysical situations a convective instability occurs in flows which are non hydrostatic, and therefore requiring the solution of the equation of motion to determine their temporal evolution. In hydro-static models convection is regarded in principle on the base of two descriptions, namely the static mixing length theory and the model by Canuto and Mazzitelli (1991). One of the first time-dependent approaches of convection was given by Kuhfuß (1986). He intro-duces a one-equation model for convection. Wuchterl and Feuchtinger (1998) improved and extended this model leading to a description which is capable to account for the most qualitative requirements of star and planet formation and evolution calculations and also of the quantitative demands of stellar pulsation calculations. One time-dependent effect in convective energy transfer is the energy exchange between the gas and radiation one the one hand and between gas and convective eddies on the other. This is an intrinsically time-dependent effect, where convection plays the role of an additional degree of freedom, with an additional energy equation in the dynamics. This model was successfully applied in Feuchtinger (1999).

For the equations of convective radiation hydrodynamics no general analytical solu-tion exists, therefore one dependents on a numerical approach. The system of equasolu-tions represent a non linear “mostly” hyperbolic system of partial differential equations. Only the Poisson equation that describes the “own-gravity” of the studied system is of elliptical type. A short definition of the hyperbolic type can be found in e.g. L¨uttke (1995) and references therein. The dominating hyperbolic equation type permits non continuous so-lutions of the equation system, and according to Courant and Friedrichs (1948) even out of continuous initial values a non continuous solution might develop. Therefore it is useful to formulate the conservation laws in the conservative (or integral) form for an arbitrary moving coordinate system. This can be achieved by integration of the equations over a (temporal variable) control volume.

As a consequence, every conservation law for a physical quantity X can be written in the unique basic form of an equation of continuity:

d dt Z V X dV + Z ∂V XureldA− Z V (X+− X−) dV = 0 (2.1)

The first term describes the temporal change of the quantity X within the volume V , the second term represents the variation due to the transport (flux) of the quantity X through the surface of the volume ∂V , and finally the last integral stands for the change within the volume due to constructive (X+) and destructive (X−) processes. The relative

velocity (urel = u− v) is given by the difference of the velocities of the gas u and of the

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2.1. THE EQUATIONS OF CONVECTIVE RADIATION HYDRODYNAMICS 23

2.1.1 Gas Dynamics

The gas dynamics are governed by three conservation laws for mass (continuity equation, Eq. 2.2), momentum (equation of motion, Eq. 2.3) and internal energy (gas energy equation, Eq. 2.4). The corresponding independent variables are the gas density %g, the

gas velocity u and the internal energy e of the gas. d dt Z V ρ dV + Z ∂V ρureldA = 0 (2.2) d dt Z V ρu dV + Z ∂V ρu ureldA = − Z V ρ 1₋ µs 2 gext+ µs 2 Gmr rµs dV (2.3) − Z V ∂p ∂rdV − Z V ∂pt ∂r dV + 4π c Z V κρH dV + Z V UQdV

The considered pressures are the gas pressure p and the turbulent pressure pt, H stands

for the radiative flux, the first moment of the radiation field, κ for the opacity and UQ

for the turbulent viscous momentum generation. The symmetry parameter µs adapts the

equations to the geometry of the underlying system. Thus, gext stands for the gravitational

acceleration in slab symmetry (µs = 0), whereas in spherical systems (µs = 2) it is given

by the second term of the right hand side of (2.3) caused by the inner mass mr.

d dt Z V ρe dV + Z ∂V ρe ureldA = − Z V p∂ (r µs_u) rµs_∂r dV + 4π Z V κρ (J_{− S) dV} (2.4) − Z V ∂ (rµs_j w) rµs∂r dV − Z V ρSω¯ − ˜Sω¯ − Drad dV + Z V ρ dV

In equation (2.4) the radiation energy is denoted with J and the radiative source function with S. The convective enthalpy flux is represented by jw. Sω¯, ˜Sω¯, and Drad are the

turbulent driving function, the turbulent dissipation function and the radiative cooling rate. In the case of energy production due to thermonuclear reactions or other processes an additional term containing the energy production rate has to be introduced.

Finally in spherical symmetry the mass mr within a shell of radius r is given via the

Poisson Equation (2.5) by integration over the density structure. mr =

Z r

0

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2.1.2 Radiative Transfer

One of the simplest ways of taking time-dependent radiative transfer into account is the grey approximation yielding two frequency-integrated moment equations of the radiation field representing energy conservation (radiation energy equation (2.6)) and momentum conservation (radiation momentum equation (2.7)) for the first three moments, the mean specific intensity erad = 4π/cJ , the radiative flux Frad = 4πH and the radiation pressure

Prad = 4π/cK. d dt Z V J dV + Z ∂V J ureldA = −c Z V ∂ (rµs_H) rµs_∂r dV − c Z V κρ(J_{− S) dV} (2.6) − Z V K∂ (r µs_u) rµs∂r + µs 2(3K− J) u r ! dV d dt Z V H dV + Z ∂V H ureldA = −c Z V ∂K ∂r dV − Z V µs 2(3K− J) c r (2.7) − Z V H∂u ∂r dV − c Z V κρH dV

The coupling terms containing a mean opacity κ describe the momentum and energy exchange between matter and radiation. A more detailed discussion of the used opacity is given in section 2.5.

For the two radiation moment equations a closure condition is required which gives a relation between the radiative grey moments. Therefore a “equation of state for the radiation field” by means of the variable Eddington factor fedd= J/K should be specified.

Normally the Eddington factor has to be calculated by solving the time-independent grey radiative transfer equation (e.g. Yorke (1980), Balluch (1988)). Due to the negligible radiation fields in Brown Dwarf atmospheres the Eddington approximation (fedd = 1/3) is

applied within this thesis.

2.1.3 Convective Transfer

Due to the lack of a self-consistent convection theory the inclusion of turbulent convec-tion into the radiaconvec-tion hydrodynamical model introduces a number of free parameters. In principle these parameters cannot be determined from theoretical considerations, but have to be chosen in order to fit observational constraints. A self-consistent modeling of as-trophysical situations including convection must await considerable progress in turbulence theory.

In the past decades a number of time-dependent one-dimensional models of turbulent convection have been developed. Turbulent motion often is assumed as fluctuations of the mean hydrodynamic field and this assumption is used to linearize the equations in the fluctuating parts (cf. Balluch (1988)). In this work a one-equation model of time-dependent

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2.1. THE EQUATIONS OF CONVECTIVE RADIATION HYDRODYNAMICS 25

parameter standard value physical meaning

αML 1.5 mixing length αs 1₂ q 2 3 turbulent driving cD 8₃ q 2 3 turbulent dissipation γR 2 √ 3 radiative cooling αµs – turbulent viscosity αt 0.6093αs overshooting distance

βr 0.1 weak gravity mixing length

Table 2.1: Free parameters and standard values entering the model for time-dependent turbulent convection by Wuchterl and Feuchtinger (1998).

convection proposed by Kuhfuß (1986) is used in the improved version of Wuchterl and Feuchtinger (1998). d dt Z V ρ¯ω dV + Z ∂V ρ¯ω ureldA = Z V ρSω¯ − ˜Sω¯ − Drad dV (2.8) − Z V ∂ (rµs_j t) rµs∂r dV − Z V pt ∂ (rµs_u) rµs∂r dV + Z V EQdV

This equation represents a conservation law for the turbulent kinetic energy density ¯ω. The essential term is the turbulent driving through buoyancy forces (Sω¯):

Sω¯ =

T p∇S

∂p

∂rΠ, (2.9)

where _∇S stands for the isentropic temperature gradient and Π describes the velocity

entropy correlation and is given via: Π =q2/3ω Tω¯ 1/2_F L " −q3/2αsΛT ω ∂s ∂r # (2.10) Here ω = e + p/ρ is the specific enthalpy and αadenotes a free parameter. Λ represents the

mixing length composed of the usual mixing length parameter αML times the hydrostatic

pressure scale high Hstat

p and a mixing length limiting term containing another parameter

βr that becomes important in regions of weak gravity were Hpstat/r > 1. Λ is than given

by: 1 Λ = 1 αMLHpstat + 1 βrr (2.11)

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The gradient of the specific entropy relates the turbulent driving function to the well known Schwarzschild criterion: ∂s ∂r =− cP Hp (_{∇ − ∇}S) (2.12)

Additionally a flux limiting function FL is introduced:

FL(x) =

(

x _{kxk < 1}

1 _{kxk ≥ 1} (2.13)

The dissipation of turbulent kinetic energy ˜Sω¯ through conversion of the turbulent

mo-tions into thermal energy was gained by dimensional arguments introducing an additional parameter cD: ˜ Sω¯ = cD ¯ ω3/2 Λ (2.14)

The radiative cooling of convective elements is given by Drad =

¯ ω τrad

, (2.15)

where the radiative cooling time scale of convective elements can be calculated via τrad =

cPκρ2Λ2

4σT3_γ2 R

(2.16) and adjusted by an additional free parameter γR. The turbulent kinetic energy flux

ap-proximated by the diffusion of ¯ω is given by jt= αtρΛ¯ω1/2

∂ ¯ω

∂r (2.17)

and is responsible for the non local character of the convection model (overshooting). The overshooting distance is adjusted by means of the free parameter αt. The dissipation

through turbulent Reynolds stresses is treated analogous to the molecular viscosity. The resulting Reynolds tensor is divided into a trace part yielding the turbulent pressure

pt=

2

3ρ¯ω (2.18)

and a trace-free part leading to the turbulent viscosity. The resulting viscous energy dissipation can be written as

EQ= 4 3µQ ∂u ∂r − µs 2 u r !2 (2.19) where the kinetic turbulent viscosity

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2.2. GAS PHASE CHEMISTRY 27

µQ = αµρΛ¯ω1/2 (2.20)

contains a free parameter αµ. The additional term in the equation of motion (2.3) taking

the momentum transfer through viscous forces into account can be written as UQ= 1 r3µs/2 ∂ ∂r " 4 3µQr 3µs/2 ∂u ∂r − µs 2 u r !# (2.21)

2.1.4 Total Energy Equation

An equation for the total energy can be obtained by addition of the Eqs. (2.4), (2.6), (2.8) concerning energies. The application of the resulting equation to the overall problem might be more useful for numerical treatment.

d dt Z V ρe +4π c J + ρ¯ω dV + Z ∂V ρe + 4π c J + ρ¯ω ureldA (2.22) = ₋ Z V p + pt+ 4π c K _{∂ (r}µs_u) rµs∂r dV − Z V 4π∂ (r µs_H) rµs_∂r + ∂ (rµs_j w) rµs_∂r + ∂ (rµs_j t) rµs_∂r ! dV − Z V 2πµs c (3K− J) u r dV + Z V EQdV

This set of equations creates the complete equation system of convective radiation hydrodynamics (CRHD) in slab or spherical symmetry.

2.2 Gas Phase Chemistry

2.2.1 Chemical Equilibrium

The composition of the gas determines both, the below described dust complex and the opacity entering the CRHD-Equations. Within this work the gas is assumed to be in local thermodynamical equilibrium (LTE), and the concentration of atoms, ions and molecules are calculated by assuming chemical equilibrium (CE). The chemical equilibrium is a con-sequence of the thermodynamical principle of the minimization of the Gibbs free energy G = E +pV_{−T S for a thermodynamical system at constant temperature T and pressure p.} It expresses a congruence between the minimization of the energy E and the maximization of the entropy S.

Although the assumption of chemical equilibrium is debatable in several astrophysical situations depending on the characteristical time scales, it is often the only possibility

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Figure 2.1: The equilibrium concentrations of some selected atoms and molecules over a temperature density plane.

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2.2. GAS PHASE CHEMISTRY 29 to calculate the abundances of atoms and molecules. The reasons for this are twofold. First, most of the necessary reaction rates needed for non-LTE calculations are still lack-ing, especially for inorganic compounds which play an important role in most oxygen-rich situations. Second, the numerical effort to solve the full rate network of chemical reac-tions is enormous and makes a self-consistent treatment of the different coupled physical and chemical processes in astrophysical environments often impracticable. Of course, the advent of even faster computer facilities might change this within the next years.

The following equation system describes the chemical equilibrium situation. It is solved with a damped Newton algorithm (for numerical details refer to section 3.3).

The conservation of charge and the vanishing resulting charge of each ensemble leads to the number density of free electrons ne, which is given by summation over the particle

densities of all ions with respect to the degree of ionization Z.

ne = Nions

X

j=1

Zj nj (2.23)

Assuming a given elemental abundance in the gas phase the conservation of elements yields e.g. for element i to:

n<i> = in<H> = ni+

Nmolecules X

m=1

ai,m nm (2.24)

where i is the relative abundance of the element i with respect to the total number density

of Hydrogen n<H>. n<i> denotes the total number density of the element i respectively.

The right and side of equation (2.24) is given by the element i’s number density of free atoms and the summation over all Nmoleculesmolecules in the gas phase. The stoichiometric

factor is represented by ai,m ∈ N0.

According to the law of mass action the partial pressure pAaBb of molecules containing

a atoms of element A and b atoms of element B is given by: pAaBb pa ApbB = 1 (p−◦₎a+b−1 exp − ∆fG−◦(T ) RT ! ≡ KP(T ) , (2.25)

where pA,B denotes the partial pressure of the free atoms of the corresponding elements, R

the gas constant, and T the temperature of the gas phase. The standard reference pressure is given by p−◦ = 1bar = 106 dyn/cm2. KP(T ) denotes the equilibrium constant.

The Gibbs free energy (or free enthalpy) for the molecule at the reference pressure is determined via

∆fG−◦ = ∆fG−◦(AaB (gas)

b )− a · ∆fG−◦(A(gas))− b · ∆fG−◦(B(gas)). (2.26)

The combination of equation (2.25) with the ideal equation of state pi = nikBT leads

to the number densities of the corresponding molecules. Furthermore it is most useful for the numerical solution to express the equations with the logarithms of the densities:

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log nAaBb = log KP + a log nA+ b log nB+ (a + b− 1) log kBT (2.27)

In general the partial pressure for all polyatomic molecules m ˆ= A1_a₁A2_a₂. . . ANelements

a_Nelements

consisting of ai atoms of element Ai is given by the following equation:

pm = KP(m) Nelements Y i=1 pai i (2.28)

Analogous to equation (2.27) the logarithm of the particle density of the molecule m is given by: log nm = log KP(m) + Nelements X i=1 ai,mlog ni+     Nelements X i=1 ai,m   − 1  log k_BT (2.29)

The generalized equilibrium constant KP is defined by

KP(m)≡ 1 (p−◦₎NA(m)−1 exp − ∆fG−◦ RT ! (2.30) where the number of atoms in the molecule NA is given by

NA(m) =

Nelements X

i=1

ai,m (2.31)

and the Gibbs free energy for the molecule at the reference pressure is analogous determined via ∆fG−◦ = ∆fG−◦(m)− Nelements X i=1 ai,m· ∆fG−◦(i) (2.32)

It is beyond the scope of this thesis to give a complete discussion of the molecular data entering the chemical equilibrium calculations. However, the electronic version of JANAF tables (Chase et al. 1985) is used being almost identically to the last printed version from 1998 (Chase 1998). The set of molecules given in the JANAF tables is extended with some species, that are be important for the completeness of the chemical equilibrium calculation (e.g. TiH, TiS, TiC, etc.).

Tsuji (1973) introduced a 4-th order polynomial fit over Θ = 5040K_T for the calculation of the constants. In this thesis a rational polynomial fit of order two is applied (see equation (2.33)). This leads to more accurate fits than a 4-th order polynomial. Despite this, the (experimental) data entering the JANAF-tables have errors up to several percent. The calculated fitting coefficients according to the newest available data are given in appendix A.

log KP(m)≡

a0+ a1T + a2T2

b0+ b1T + b2T2

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Atom log₁₀i mi /AMU Atom log10i mi /AMU Atom log10i mi /AMU H 0.00 1.00794 Cl -6.50 35.4527 Rb -9.40 85.4678 He -1.01 4.0026 P -6.55 30.9738 B -9.40 10.8110 O -3.07 15.9994 Mn -6.61 54.9381 Xe -9.84 131.2900 C -3.40 12.0110 K -6.88 39.0983 Ba -9.87 137.3270 Ne -3.91 20.1797 Ti -7.01 47.8670 Mo -10.08 95.9400 N -4.00 14.0067 Co -7.08 58.9332 Pb -10.15 207.1900 Mg -4.42 24.3050 Zn -7.40 65.3900 I -10.50 126.9045 Si -4.45 28.0855 F -7.44 18.9984 Nb -10.58 92.9064 Fe -4.50 55.8450 Cu -7.79 63.5460 Li -10.84 6.9410 S -4.73 32.0660 V -8.00 50.9415 Hg -10.84 200.5900 Ar -5.44 39.9480 Kr -8.70 83.8000 Be -10.85 9.0122 Al -5.53 26.9815 Sr -9.10 87.6200 Cs -10.88 132.9054 Ca -5.64 40.0780 Ga -9.12 69.7230 W -10.89 183.8400 Na -5.67 22.9898 Br -9.39 79.9040 Hf -11.12 178.4900 Ni -5.75 58.6934 Zr -9.40 91.2240 Ta -12.14 180.9479 Cr -6.33 51.9961

Table 2.2: Solar atomic abundances, Data taken from Anders and Grevesse (1989), except Fe is taken from (Grevesse and Sauval (1999)). Additionally the molecular weights in

atomic mass units are given.

For the numerical solution of the resulting equation system it is useful to employ the renaming convention yi ≡ log ni and to define the index of the particle density of

the free electrons as 0. Furthermore a convenient norm for each element i, e.g. Yi =

maxNmolecules

m=1 (ym− yi), is introduced to avoid numerical over- and underflows. The resulting

system of equations which has to be solved numerical is than given by:

y0 = log   Nmolecules X m=1 Zm exp (ym)   (2.34) log (in<H>) = yi+ Yi+ log  exp(−Yi) + Nmolecules X m=1 ai,m exp (ym− yi− Yi)   (2.35)

Where the densities of the molecules in logarithmic form are given by:

ym = log KP(T ) + Nelements X j=0 aj,m yj +     Nelements X j=0 aj,m  − 1   log kBT (2.36)

In figure 2.1 the concentrations (number density per total number density hydrogen) of some selected atoms and molecules is pictured.

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2.2. GAS PHASE CHEMISTRY 33 El. log₁₀εrel El. log10εrel El. log10εrel

Ne 6.6868e-11 N 1.3358e-01 Mg 4.6493e+01

Si 2.9339e+01 Fe 3.4030e+01 S 4.1047e+01

Ar 2.2648e-06 Al 3.5791e+01 Ca 4.1780e+01

Na 2.5647e+02 Ni 1.1483e+01 Cr 2.5345e+00

Cl 2.0014e+02 P 2.3148e-02 Mn 2.9507e-01

K 1.0482e+02 Ti 6.1125e-01 Co 2.2759e-01

Zn 1.2988e-02 F 2.7757e-01 Cu 1.4195e-01

V 5.7039e-02 Kr 4.4361e-06 Sr 1.2107e-02

Ga 5.3624e-03 Br 8.9902e-01 Zr 3.0085e-03

Rb 6.6295e-02 B 1.1299e-01 Xe 4.4361e-06

Ba 1.3435e-03 Mo 2.3790e-04 Pb 1.3296e-04

I 3.3282e-03 Nb 1.0691e-04 Li 1.3875e-03

Hg 5.8378e-06 Be 1.7551e-05 Cs 3.3407e-04

W 3.1759e-05 Hf 3.3675e-05 Ta 4.4361e-06

Table 2.3: Resulting errors due to the reduction of the considered elements.

2.2.2 Elemental Abundances

Crucial parameters entering the calculation of the equilibrium densities are the elemental abundances (cf. Eq. 2.24). In many astrophysical situations the atomic abundances are manifold, e.g. there are lots of objects with low-metallicity like in the Small and Large Magellanic Clouds, and on the other hand there are many objects with higher metallicity like the giant gas planets. Table 2.2 shows the solar abundances mainly taken from Anders and Grevesse (1989). In figure 2.2 a schematic view of this data is given.

The relative elemental abundances are changed by (thermo-)nuclear reactions in the inner parts of stars where high temperatures and densities dwell and by radioactive pro-cesses. In the temperature regime of stellar atmospheres these reactions do not take place, but the elemental abundances in the gas phase are affected by the dust complex. Forma-tion and growth of dust particles reduce the relative abundance of the elements in the gas phase involved in the process, on the other hand evaporation and destruction lead to an increase. So that due to the movement of dust particles, like sedimentation, a deferral of the elemental abundances might occur in e.g. different parts of Brown Dwarf atmospheres. Neglecting nuclear reactions and radioactivity the elemental abundance for element i follows equation d dt Z V n<i>dV + Z ∂V n<i>ureldA = Z

V {consumption or release of the dust complex} dV

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Figure 2.3: The bar chart sho ws the logarithm of the maxim um relativ e error for the remaining elemen ts (B) dep enden t on the remo v ed elemen t (A).

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2.2.3 A numerical Remark

As one can see, the final system describing the chemical equilibrium situation scales with the number of considered elements. The more elements are taken into account the more computational time is required, so it is obvious to reduce this number for practical purposes in the numerical calculations. One way is to eliminate all elements with an elemental abun-dance below some selectable value, another way is to use chemical or physical arguments to include or exclude elements. In this work another approach is applied. According to the available data and with respect to the metallicity the influence of each element upon the chemical system is investigated by analyzing the resulting deviations caused by excluding each element one by one.

For a given set of elements the one with the least influence in a given temperature-density range is located and removed from the system. Within the resulting system again the element with the smallest deviation is searched and removed, and so on. The resulting densities of the species of the reduced system are compared to the ones of the “complete” system. As starting set the 46 elements are taken (cf. table 2.2). In table 2.3 the resulting maximum relative deviation for solar abundances is given. Some elements like e.g. Mg or Si have a very strong influence on the remaining system. Consequently, a more detailed analysis of the distribution of the deviations is necessary to advance in tighten up the system. Figure 2.3 shows the individual relative error for each element depending on the removed element. The elements colored in red are the elements that will not be excluded, due to abundance arguments or, as in the case of V and Ti, the specific elements serve as important dust species. The yellow ones would be desirable and against the removing of the green ones no other than the error argumentation is present. The figure reveals that the strongest influence of possible candidates affects other possible candidates. After intensive investigations including the simultaneous removal of two or more elements leads to a set of 20 elements that have to be considered to achieve an accuracy of less than 1% in the resulting particle densities of atoms and molecules for solar abundances. The set of molecules includes H, He, O, C, N, Mg, Si, Fe, S, Al, Ca, Na, Ni, Cr, Cl, P, K, Ti, F, and V.

Even the so called “complete” system is incomplete due to the lack of available data. For example, in table 2.4 the problem of missing data becomes clear. Only for the dimers denoted with an X data are available; the situation becomes more worse for the trimers. At the other hand many possible dimers are very unlikely, as, for instance, a molecule consisting of Na and Rb seems to be very improbable, but for many molecules like ZnO or MnO the data should be recognized. For some dimers containing Ti (Ti2, TiH, TiC, TiS)

the data were calculated together with John (2002) in order to provide an approximation of the equilibrium constants.

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Ba B Rb Zr Br Sr V Cu F Co Ti K Mn P Cl Cr Ni Na Ca Al S F e Si Mg N C O H H X X X X N X X X X X X X X X X X X O X X X X X X X X X X X X X X X X X X X X X X C X X X N X X X X X X X N X X X X X X X X X X X X Mg X X X X X Si X X X X F e X X X S X X X X N X X X X X X Al X X X X Ca X X X X Na X X X X Ni X Cr Cl X X X X X X X X X X X P X X X Mn K X X X Ti X X N Co F X X X X X X X Cu X V Sr X Br X X X X Zr Rb X B X Ba T able 2.4: The a v ailable thermo dynamical data for the dimers are lab eled with an X. The one’s lab eled with an N are added to the set of dimers. The lac king n um b er of dimer data still has the ma jorit y and should b e reduced in the future in order to gain more accurate particle densities in chemical equilibrium calculations.

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2.3. DUST 37

2.3 Dust

The formation of dust grains is a two-step process. The nucleation of seed particles is followed up by their growth towards macroscopic particles. On the other hand the dust complex has also to include the reversal processes, the evaporation and extinction. In the following a concept of heterogenous growth and evaporation is outlined taking all efefcts into account.

Taking a look on oxygen-rich astrophysical dust producing situations (like the atmo-spheres of oxygen-rich stars on the Asymptotic Giant Branch or the here studied Brown Dwarfs) high temperature compounds like CaTiO3, Mg2xFe2−2xSiO4, Al2O3 or TiO2 may

be expected to form first, if only thermal stability arguments are considered (see also Burrows and Sharp (1999), Lodders (1999)). The description of the formation of such heterogeneous particles, however, is still a matter of debate, since the nominal molecules of these solid compounds are often only present in negligible amounts (e.g. Fe2SiO4, Al2O3)

or even completely absent (e.g. CaTiO3) in the gas phase. Nevertheless, the rich variety of

molecular species of comparable abundances in an oxygen-rich gas suggests the formation of such heterogeneous dust.

2.3.1 The Size Distribution Function

The number density of grains in the volume interval [_{V, V + dV] is given by the size} distribution function f (_{V, t)dV. The temporal change of the size distribution function is} given via (stoichiometric) growth and/or evaporation reactions on the surface of the grains according to equation (2.38). d dtf (V, t)dV = X solids s    X reactions r(s) " f (_{V − m(r(s))∆V}s, t)dV τgr(r(s),V − m(r(s))∆Vs, t) − f (_{V, t)dV} τev(r(s),V, t) ! − f (V, t)dV τgr(r(s),V, t) − f (_{V + m(r(s))∆V}s, t)dV τev(r(s),V + m(r(s))∆Vs, t) !#) +qcoag(V) + . . . (2.38)

In general the growth process is given by the sum of all possible reactions r(s) leading to the solidification of m(r) monomers of solid s on the grains surface. Each monomer increases the grain’s volume by ∆_Vs. Otherwise the evaporation process is the sum of all

destructive reactions removing m(r(s)) monomers of solid s from the surface of the grain leading to a decrease of the grain’s volume by ∆_Vs per evaporated monomer.

Of course, other processes like coagulation and mechanical sputtering also affect the size distribution function.

The notation of the explicit dependency of the reactions r(s) to the considered solid s will be presumed from this point on to improve clarity.

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2.3.2 Growth and Evaporation

The growth and the formation of the dust particles confronts a large necessary condition: The resulting dust grains must be thermally stable!

Non-thermal influences like mechanical sputtering or (UV) radiation fields may affect (usu-ally reduce) the dust stability. But it is difficult to assume any formation process if the dust grains are thermally unstable.

The thermal stability of the dust grains are determined by the net effect of chemical reactions on the surface of the dust grains. The possible reactions are manyfold (the accreting grain is omitted for clearness):

TiO2 *) TiO2,solid (2.39a)

TiO + O *) TiO2,solid (2.39b)

TiO + H2O *) TiO2,solid+ H2 (2.39c)

Let Msolid denote one unit (monomer) of the solid which may vaporize into a free

molecule (2.39a), may decompose into constituting molecules or atoms (2.39b) or may be transformed by means of a more complex chemical surface reaction (2.39c) (chemical sputtering). The reactions directed to the right and to the left will be hencefore referred to growth and evaporation reactions, respectively. The explicit time dependency X(. . . , t) will be omitted for simplicity of the description.

The growth time scales can be derived directly from the consideration of the micro-scopic processes based on gaskinetic theory. Normally the gas density in astrophysical dust producing regions is low enough to account only for two-body reactions. Then the groeth time scale can be written as

τ_gr−1(r,_{V) = A(V) n}f(r, Tgas) vrel(nf(r, Tgas),V) α(r, V, Tgas) , (2.40)

where nf is the particle density of the molecule of the growth (or forward) reaction in the

gas phase, vrel is the average relative velocity between the molecule and the dust grain,

α is the reaction efficiency (often also called sticking probability) on a grain with volume V and temperature Tgas and A(V) is a suitable defined reaction surface of the cluster of

volume_V.

The relative velocity between the molecule with mass mf and the dust grain vrel is

given by the maximum value of the mean local thermal velocity vth =

q

2kBTgas/mf and

the relative velocity (drift velocity v_V,d, cf. section 2.3.7) of the grains with volume_V.

vrel(nf(r, Tgas),V) = max (vth, vV,d) (2.41)

The evaporation time scale of the reverse reaction corresponding to r can be written formally in the same way (cf. Patzer et al. (1998a))

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2.3. DUST 39 where nb is the particle density of the molecule in the reverse reaction and β(r) is the

respective reaction efficiency. If there is no molecule of the gas phase involved in the evaporation process, like in reaction (2.39a) of (2.39b) nb can be omitted (nb = 1). If

the reaction efficiencies were all known, the growth and evaporation time scales could be calculated directly from these equations. However, the growth efficiencies are often uncertain and the evaporation coefficients are mostly unknown.

In local thermodynamical equilibrium between the gas phase and the dust component, the principle of detailed balance holds for the single microscopic growth process and their respective reverse actions. This can be used to express the evaporation time scale of the reverse reaction in terms of the growth time scale of the corresponding growth reaction. The equilibrium state is characterized by the phase equilibrium between monomers, clusters and the bulk solid, by simultaneous chemical equilibrium in the gas phase, and by thermal equilibrium, with equal temperatures of the solid, the clusters and the gas, i.e. it is a state of local thermodynamic equilibrium (LTE-state). Then the equilibrium temperature is given by the internal temperature Tdust(V) of the grain with volume V. It is obvious that,

if the grain temperature depends on the grain size, one has to refer to different LTE-states for each volume, which is determined by the respective cluster temperature.

In the following all quantities referring to the LTE-state were denoted with an “˚” and the explicit dependence on Tdust will be omitted. The condition of detailed balance leads

to a vanishing net transition rate of the chemical reaction r, which results in the following equation: ˚ f (_{V − m(r)∆V}s)dV ˚τgr(r,V − m(r)∆Vs) = ˚ f (_V)dV ˚τev(r,V) (2.43) Inserting equation (2.40) and (2.42) into equation (2.43) leads to

˚ β(r,_{V, T}dust) = ˚ f (_{V − m(r)∆V}s)dV ˚ f (_V)dV A(_{V − m(r)∆V}s) A(_V) (2.44) ·˚nf(r) ˚nb(r) ˚vrel(˚nf(r),V − m(r)∆Vs) ˚vrel(˚nb(r),V) ˚ α(r,_{V − m(r)∆V}s)

Then the evaporation time scale takes the form

τ_ev−1(r,_{V) =} ˚ f (_{V − m(r)∆V}s)dV ˚ f (_V)dV A(V − m(r)∆Vs) (2.45) ·˚nf(r) ˚nb(r) ˚vrel(˚nf(r),V − m(r)∆Vs) ˚vrel(˚nb(r),V) ˚α(r,_{V − m(r)∆V}s) β(r,_{V, T}dust) ˚ β(r,_V) · vrel(nb(r, Tgas),V) nb(r, Tgas) .

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τ_ev−1(r,_{V) = τ}_gr−1(r,_{V − m(r)∆V}s) ˚ f (_{V − m(r)∆V}s)dV ˚ f (_V)dV ˚nf(r) nf(r, Tgas) nb(r, Tgas) ˚nb(r) (2.46) · ˚vrel(˚nf(r),V − m(r)∆Vs) vrel(nf(r, Tgas),V − m(r)∆Vs) vrel(nb(r, Tgas),V) ˚vrel(˚nb(r),V) · ˚α(r,V − m(r)∆Vs) α(r,_{V − m(r)∆V}s, Tgas) β(r,_{V, T}dust) ˚ β(r,_V)

In order to describe the destructive processes in the case of homogeneous growth and destruction it is necessary to determine the fraction of the equilibrium particle densities and the fraction of the particle densities of the molecule of the forward and backward reaction. A analogous contemplation for homogeneous dust grains can be found in appendix A of Gauger et al. (1990).

According to the rules of chemical thermodynamics the equilibrium partial pressure of a heterogenous cluster of size _{V containing D solids d, each contributing M(d) monomers} to the grain, in LTE at the temperature T is determined by

˚p(_{V) =} Y solids d (˚p1(d))M (d) exp ( −∆Gf(V) RT ) (2.47) where ∆Gf(V) denotes the free enthalpy of formation and ˚p1(d) is the partial pressure of

the monomers. ∆Gf(V) can be expressed in terms of the enthalpies ∆Hf of formation of

the substances from the elements in their standard states and in terms of the entropies S by ∆Gf(V) = ∆Hf(V) − X d M (d)∆Hf,1(d)− T " S(_{V) −}X d M (d)S1(d) # . (2.48)

Inserting equation (2.48) in equation (2.47) results in

˚p(_{V) =} Y d (˚p1(d)) M (d) (2.49) · exp      − ∆Hf(V) −P d M (d)∆Hf,1(d) RTdust + S(_{V) −}P d n(d)S1(d) R      .

Introducing the thermodynamic functions of the solid, ∆Hf, solid and Ssolid, the following

Dust in the atmospheres of Brown Dwarfs