Condensation. Lecture 8. Lecture Universität Heidelberg WS 11/12 Dr. C. Mordasini. Based partially on script of Prof. W. Benz. Mentor Prof. T.

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1 Lecture 8 Condensation Bond et al Lecture Universität Heidelberg WS 11/12 Dr. C. Mordasini Based partially on script of Prof. W. Benz Mentor Prof. T. Henning

2 Lecture 8 overview 1. Condensation 1.1 Carbonaceous chondrites 1.2 The thermodynamic of condensation 1.3 Examples 1.4 The full sequence 1.5 Water ice condensation 1.6 Condensation in extrasolar systems

3 1.1 Carbonaceous chondrites

4 Carbonaceous chondrites The condensation of dust grains out of the gas phase represents the very first phase of planet growth. While the Earth, the Moon and many other planetary bodies show clear signs of chemical differentiation and fractionation, the most primitive meteorites, the so-called carbonaceous chondrites, seem to present an unaltered image of the chemical composition of the nebula as it was 4.6 billion years ago. The most primitive sub-class are so called CI-chondrites. In their appearance, these mostly small, black, and very friable rocks remind more of a piece of tar or charcoal than of a stone. They contain a large fraction of water (bound in silicates) of 17-22%. The iron content (in form of iron oxides) is about 25% in mass. Carbon makes about 3-5%. Interestingly, amino acids are also present. The chemical/mineralogical composition shows that the origin of CI-chondrites is in the outer part of the solar system (>4 AU) since they never have been heated above 50 C during their formation and subsequent evolution. Holweger Except for some volatile elements like hydrogen or oxygen are the relative elemental abundances in CI-chondrites nearly identical to those measured in the photosphere of the sun. Lithium is depleted in the sun relative to CI-chondrites, as it is destroyed by nucleosynthesis.

5 1.2 The thermodynamic of condensation

6 Thermodynamic equilibrium The collapse of the interstellar gas cloud that leads to the formation of the protoplanetary nebula is a relatively violent process during which temperatures high enough to vaporize most (but not all) solids are reached. Therefore, the dust grains originally contained in the gas will mostly get vaporized. Solids which survived the collapse (so-called presolar grains) are tiny, very refractory grains like nano-diamonds, graphite particles or silicon carbide (SiC) grains. After the disk has formed, it cools, and new dust grains condense out (the assumption that first a disk forms, and then condensation happens is clearly an idealization, In reality these processes would occur partially concurrently). The formation of the early dust grains proceeds therefore along a condensation sequence in which the most refractory elements condense in the inner regions of the nebula while volatile elements condense only at larger distance (outside the icelines). In order to compute which elements condense where, we assume that changes in temperature and density occur on a relatively long timescale compared to the chemical reaction timescale. This is a reasonable assumption at least for the inner part of the disk where temperatures and gas densities are high. Under this assumption, we can presume that these changes always occur at constant temperature and pressure (which will however by different as function of the distance from the star) and in thermodynamical equilibrium. The first model that computed the sequence of solid which emerge from the gas phase when we let the gas slowly cool was Grossman & Larimer Here we reproduce some aspects of this work as well as some later improvement to this simple minded approach.

7 Thermodynamic potentials In a thermodynamical system, processes will spontaneously continue until the relevant thermodynamical potential is minimized. Examples are: 1) In a isothermal-isochor system in equilibrium, the Helmoltz free energy F will be minimal. F = U - TS 2) In a isothermal-isobar system in equilibrium, the Gibbs free energy (also called free enthalpy) G will be minimal. G = F + pv = U - TS + pv = H - TS H is the enthalpy U + pv. Note that the unit of these potentials is erg.

8 Thermodynamic potentials II In the situation that the chemical reactions happen at constant temperature and pressure, the free enthalpy or the Gibbs energy G is the natural choice for the thermodynamical description of the changes. G = H T S dg = dh T ds SdT with H the enthalpy H = U + pv dh = du + pdv + V dp = δq + V dp We have made use of the first principle of thermodynamics: du = δq pdv For a reversible change we must therefore have dg = dh T ds SdT = δq + V dp δq SdT = V dp SdT where we have used the definition of the entropy: ds = δq T Clearly, for a process that takes place at constant temperature and constant pressure, we have dg=0 in the final state (equilibrium). The free enthalpy just as the entropy are thermodynamic potentials defined to within a constant. It is therefore useful to define standard conditions to be used as reference point. The standard conditions are generally set to be T=298 K and p=1 atm.

9 Example To illustrate this concept, let us compute the change in free enthalpy for the reaction taking place at standard conditions: H O 2 H 2 O g Let us define the free enthalpy change (final minus initial) at these standard conditions as G 00 = G 00 (H 2 O) G 00 (H 2 ) 1 2 G 00(O 2 ) where the double 0 subscript indicates standard p and T. The free enthalpy of the individual components can be interpreted as the free enthalpy of formation of the substance. From tables we can the following values: G 00 = = k J /mole Note that by convention the free enthalpy of the most stable form of a substance is taken to be zero. The change of the free enthalpy is negative, which means the reaction is exergonic and thus a favored reaction (spontaneous).

10 Condensation in the nebula Since the changes in the nebula do not happen at reference temperature and pressure, we need to be able to compute the change in free enthalpy for other thermodynamical conditions. 1) changes at constant temperature From the definition of the free enthalpy change, we have in this case (dt=0) for an ideal gas dg = V dp = nrt dp p p G(p, T ) G 0(T ) = nrt ln where G0(T) stands for G(p0,T) the free enthalpy at standard pressure but at temperature T. Clearly p0 is the reference pressure (1 atm). We can apply this formalism to describe a chemical reaction that occurs at different pressures but constant temperature. If the reaction involves i components, i=1,...,n, each with different concentrations ni, we have p 0 where the Δ represents the difference operator ''after'' ''before'' of the chemical reaction. The pi are the partial pressures. In equilibrium, ΔG(p,T) = 0, and (1)

11 Condensation in the nebula II To understand this formalism, we consider the simple example aa + bb cc + dd In equilibrium, we must have: = ln pc c d pd p 0 p0 a b pa pb p 0 p0 = G 0(T ) RT = ln K p For the reaction constant Kp(T) we have 2) changes at constant pressure From the definition of the free enthalpy change, we have in this case (dp=0) dg = SdT G(p, T ) G 0 (p) = T S(T )dt T 0 where T0 is the reference temperature 298 K. To compute S(T) we need to recall the definition of the entropy:

12 Condensation in the nebula III ds = δq T S(T ) S 0 = T where S0 is the entropy at standard condition. T 0 δq T = T T 0 c p dt T = c p ln T T 0 In case of a chemical reaction taking place at standard pressure but varying temperature, we can write (integral of S over T): T G(p, T ) G 0 (p) = S 0 (T T 0 ) c p T ln (T T 0 ) where ΔS0 is the difference of standard entropy of the reaction (''after'' ''before'') and Δcp is the difference in specific heat at constant pressure taking place as a consequence of the reaction (cp is assumed to be independent of temperature). At equilibrium, we will again have ΔG(p,T) = 0, so that T G 0 (p) = S 0 (T T 0 ) c p T ln Finally, in our isothermal-isobar situation, we can combine (1) and (2), taking into account the enthalpy of formation of the substances: T 0 T 0 (T T 0 ) (2) (3)

13 1.3 Examples

14 Example 1: Dissociation of hydrogen For the reaction H2 H + H, at equilibrium, we must have from equation (1) We have taken into account that one H2 becomes two H. In order to deal with the partial pressures, it is convenient to define the dissociated fraction α, so that α=0 means H2 only, while α=1 means complete dissociation. We assume that we start with n moles of undissociated H2. We can then write the following table: H2 H total nb of moles (1-α)n 2 αn (1+α)n molar fraction (1-α)/(1+α) 2α/(1+α) 1 partial pressure (1-α)ptot/(1+α) 2αptot/(1+α) ptot Inserting these partial pressures in the expression above yields (ptot will be given by our nebula model while p0 is the reference pressure of 1 atm):

15 Dissociation of hydrogen II Solving for α yields To determine Kp(T), we can use the results for the change at constant pressure. The entropy change is (end minus beginning): Finally we write with eq. 3 for the change of the free enthalpy From lookup tables we find the following numerical data: so

16 Dissociation of hydrogen III For the specific heats we assume an ideal gas, therefore so Grouping all terms yields: With this equation, we can calculate the quantity in which we are interested. and thus finally α(t,ptot) which is Numerically we find for four different nebular pressures ptot (typical will be 10-4 atm):

17 Dissociation of hydrogen IV Fraction α of dissociated H ptot [atm] Fraction α of dissociated H ptot [atm] Notes: Temperature [K] Temperature [K] A high total pressure inhibits dissociation. One dissociation begins, it is a very steep function of temperature especially at low pressure. In the temperature range where dissociation occurs, the fraction of molecular hydrogen is also a strong function of pressure. Even at relatively high pressures, essentially all hydrogen is dissociated by 3500 K. According to the nebula models we studied earlier, such a high temperature is reached very close to the star only, at least after the accretion rate of gas is no more very high. For T<1000 K, all gas is molecular. From this we conclude that in most of the disk, we have H2, not H. In our calculation, we have made a number of assumptions which may not always be justified: ideal gas law, constant specific heat, etc. For more accurate calculations, it is important to include all these effects.

18 Example II: Condensation of iron I At equilibrium, for Fe (g) Fe (s), we can write from equation (1) for the reaction constant (activity for pure solids is set to unity) To compute the reference free enthalpy as a function of the temperature, we proceed as in the example before. Looking up appropriate tables, we obtain: T T S Fe (s)(t ) = ln ; S Fe (g)(t ) = ln T S(T ) = S Fe (s)(t ) S Fe (g)(t ) = ln 298 T G 0 (T ) = (T 298) 0.58 T ln 298 (T 298) The first term on the RHS is the enthalpy of vaporization at standard conditions. With these equation, we can calculate the partial pressure of Fe vapor as a function of temperature. We note that for T T0=298 K, we have so that the vapor pressure is This corresponds to the classical form of the vapor pressure law p(t)=p0 e -C/T.

19 Condensation of iron II To actually compute the condensation temperature of iron in the solar nebula we must have a suitable model of the solar nebula. In this simple example, let us assume that we have a constant total pressure (the increase of the total pressure due to vaporized Fe is neglected). In a good approximation this total pressure is equal to the hydrogen and helium partial pressures and the iron partial pressure follows from abundance considerations: p tot p(h 2 ) + p(he) For an element i we can write (Xi =mole fraction) On the cosmochemical scale, atomic abundances are normalized to the number of silicon atoms of log(ε(si))=6. Therefore, p(f e) = p tot (F e) 0.5(H) + (He) = p tot Here we have assumed hydrogen to be in molecular form. The standard abundances for the solar nebula are: log(ε(fe))=5.95; log(ε(h))=10.45; log(ε(he))=9.45. This partial pressure plots as a horizontal line in the diagram. The intersection between the two curves yields the condensation temperature of iron as condensation occurs when the vapor pressure is equal the partial pressure. We find about 1350 K for ptot=10-4 atm i.e. p(fe)=5.31 x 10-9 atm.

20 1.4 The full sequence

21 The full sequence The computation of the full condensation sequence is a complicated task (Grossman & Larimer 1974, Rev. Geophys., 12, 71). We present here some of their results. Condensation of two refractory solids: 1) Corundum: 2 Al (g) + 3O (g) Al2O3 (s) 2) Spinel: Mg (g) + 2 Al (g) + 4 O (g) MgAl2O4 (s) Note that the track for the vapor phase is not a horizontal line as in the previous example. Here Grossman & Larimer assumed a solar nebula model, including relative abundance, which means that pressure, temperature have to be considered. For corundum, the condensation temperature is found as before at the intersection of the two lines and gives T=1758 K. For spinel, the situation is somewhat more complicated. If corundum would not condense first, Spinel would condense at T=1685 K. However, the condensation of corundum removes aluminum and oxygen and thus changes the slope of the partial pressure curve below T=1758 K (arrow). According to Grossman & Larimer, spinel only condenses at about T=1500 K.

22 The full sequence II Grossman & Larimer (1974) computed the full sequence of condensation for a number of elements. The abundance of the different elements were taken to be solar and the total pressure was set to 10-4 atm. In order for the equilibrium condensation model to be correct, the various timescales for the chemical reactions (gas-gas, gassolids, solids-solids) must all be significantly shorter than the cooling time of the nebula.this was not the case at all times and non-equilibrium models should be considered. Individual bodies in the solar system do not match exactly this condensation sequence. For example, Mercury's bulk uncompressed density from the condensation model is 4.3 g/cm 3 as opposed to the 5.5 g/cm 3 observed. Mercury contains about 70% iron, Venus 30%. This large difference is in contrast to the close proximity of the condensation curves of Fe and Mg2SiO4 ( rock ). Finally, simulations of the last stages of planetary accumulation have shown that planets are not made from materials originating from narrow feeding zones but rather are collected over sizable areas of the solar nebula implying considerable mixing.

23 The full sequence III The condensation calculations have been further improved by e.g. Lodders 2003 or Ebel Equilibrium stability relations of vapor, minerals and silicate liquid as a function of temperature (T) and total pressure (P) in a system with solar bulk composition.

24 The full sequence IV The result of such calculations are the condensation temperatures of important minerals as listed in the table. Jones, total pressure 10-3 bars Lodders 2003, total pressure 10-4 bars We note that hydrogen and helium do not condense for temperatures expected in the nebula. Methane only condenses at large distances.

25 1.5 Water ice condensation

26 Water ice condensation The condensation of water ice sets the representative temperature for the appearance of volatile ices (e.g. methane ice). Note that some oxygen is removed from the gas by the formation of silicates and oxides. Namely ~23% of all oxygen is incorporated into rocky elements (Al, Ca, Mg, Si, and Ti) before water ice condenses. Lodders 2003 As we can understand from the table, the clearly most important condensation temperature is the one of water ice at about 180 K at a pressure of 10 4 bar (some other calculations indicate lower temperatures of T 150 K). The reason is that for a solar composition, the surface density of condensible materials rises dramatically once water ice forms (by about a factor 4). Thus, Σ(ices + rock) 4 Σ(rock). The exact ratio is uncertain. Classical calculations (Weidenschilling 1977) found 4.2. Recent calculations (Min et al. 2011) indicate a smaller jump factor of about 2.8. It is a long standing, classical (and plausible) assumption to associate the large change of the surface density due to ice condensation with the global structure of the Solar System, in particular the division between terrestrial planets inside, and giant planets outside. We will see later how the increase of the surface density affects planetary growth (specifically the so called isolation mass, and the growth timescale).

27 Initial solid surface density profile Under the simplistic assumption that the fraction of material that condenses out of the gas is constant except for the increase at the iceline, we can write for the initial solid surface density profile Here, Σ(r,t=0) is the gas surface density at t=0 (which is obviously ill defined) and fd/g is the dust to gas ratio. For the later, it is assumed that it is the same in the disk as in the star. Then, we can relate it to the so-called stellar metallicity [Fe/H]. f D/G f D/G, = 10 [Fe/H] This formula implies that we assume that iron is a good tracer of the relevant overall amount of solids available for planet formation i.e. a scaled solar composition. The metallicity is defined as This means that a star with the same Fe content as the sun has [Fe/H]=0. [Fe/H] can be determined spectroscopically. For solar like FGK stars in the solar neighborhood, one finds a Gaussian distribution for [Fe/H] with a mean µ ~0.0, and a dispersion σ~ 0.2 (e.g. Santos et al. 2003). Mordasini et al The spread in [Fe/H] by about 1 dex shows that initial dust-to-gas ratios in disks vary by about one order or magnitude.

28 Iceline position The factor fr/i represents the effect of the iceline: Fit for α=0.007 (Alibert et al. 2011) r ice AU = where Σ5.2AU 10g/cm Mstar M is the initial gas sur For a disk similar as the MMSN, we find an iceline position of about 3.7 AU. Mordasini et al For an actively accreting disk, the iceline position depends on disk mass (viscous heating). The plot shows the position of the iceline aice as a function of the initial gas surface density Σ0 at 5.2 AU (upper three lines). It corresponds to an initial disk midplane temperature Tmid of 170 K. The iceline is plotted for three values of the viscosity parameter α: 0.01 (dashed line), (solid line) and (dotted line). The lower three lines correspond to an initial Tmid of 1600 K, roughly the evaporation temperature of rock. The figure on the right shows the initial solid surface density (black line). The initial gas surface density is derived from the similarity solutions of the viscous accretion disk problem.

29 Iceline position II Passively irradiated disks are found to have the ice line at a smaller radius, sometimes closer than 1 AU (Garaud & Lin, 2007; Lecar et al., 2006). :7; &' <=/.1)#>/--#?)/807,@# '#!&#!%#!$#!"# Saas Fee Course 2011 J9./0K1#L/)1,.#673*#<LA6A@#M78/07,-# /,3#I/.1)#87,.1,.#/.#.C1#0>1#7?#.C1# +/).CH-#;)7I.C# 2)34,/)*# ()*# +,-./0.1# 5/)67,/8179-# +/).CH-#I/.1)#87,.1,.# #####'A$#################&AB####%A'####%AB#####$A'############"A'## (4-./,81#?)7>#.C1#D9,E#FG# In the Solar System, the composition of the putative parent bodies of different classes of meteorites indicates that water-rich asteroids exist in the outer asteroid belt (Morbidelli et al., 2000). This suggests that the ice line in the Solar Nebula was located at about 3 AU. One should however understand that the iceline position was dynamic and evolved in time. It is an active subject of research (e.g. Min et al. 2011).

30 1.6 Condensation in extrasolar systems

31 Condensation in extrasolar systems In the last section, we have assumed a scaled solar composition for other stars. Detailed spectroscopic observations however show that some other stars have a different elemental composition in their photosphere. This implies that in the nebula there was a different condensation sequence leading to planetary building blocks consisting of minerals different than in the Solar system. This could in turn lead to a different composition of the planets forming around such stars (even though we have seen that in Solar System it is not straightforward to go from the condensation sequence to the composition of the final planets). In the end, this possibly affects the ability of extrasolar terrestrial planets around other stars to be habitable. C/O The plot shows the measured photospheric Mg/ Si vs. C/O for known planetary host stars. Solar values are shown by the black star (Asplund et al. 2005). The dashed line indicates a C/O value of 0.8 and marks the transitions between a silicate-dominated composition and a carbidedominated composition at 10 4 bar Bond et al Mg/Si Here we discuss some results of Bond et al who combined condensation sequence calculations for extrasolar planet host stars with planet accretion simulations.

32 Condensation in extrasolar systems II The C/O ratio controls the occurrence of C in form of graphite and other carbide phases like SiC, TiC. At high C/O>0.8, SiC becomes the dominant form of Si instead of silicates which are Si-O compounds (as found in the solar system). Additionally, a significant amount of solid C is also present as a planet-building material. Therefore, at C/O>0.8, so-called carbon planets form (Seager et al. 2007). The sun has C/O=0.54. The exact composition of silicates that form is controlled by the Mg/Si value. The minerals vary from pyroxene (MgSiO3) and various feldspars (for Mg/Si<1), to a combination of pyroxene and olivine (Mg2SiO4) (for 1<Mg/Si<2) and finally to olivine with other MgO or MgS species (for Mg/Si>2). The solar Mg/Si value is 1.05 while the bulk Earth Mg/Si value is Planets with Mg/Si<1 i.e. lower than Solar System will form species will have melts with a felsic composition. Such magma is very viscous, so extrusive volcanism could be very explosive on such planets. Solar Normalized Mass HD 4203 HD27442 HD HD r (AU) Bond et al Figure 15. The plot shows the solid mass distribution obtained from the condensation sequence within the disk for four known extrasolar planetary systems. All distributions are normalized to the solar distribution. Mass distributions are shown for -HD4203 (solid; Mg/Si = 1.17, C/O = 1.86) -HD27442 (dash-dotted; Mg/Si = 1.17, C/O = 0.63) -HD (long dash; Mg/Si = 1.91, C/O = 0.83) -HD72659 (short dash; Mg/Si = 1.23, C/O = 0.40).

33 Condensation in extrasolar Sim.3 systems III Starting with the planetesimal surface density and composition obtained from the condensation sequence, Bond et al. run accretion simulations of the final phase of terrestrial planet formation. Semimajor Axis (AU) Final Composition - HD72659 (0.5 Myr) Final Composition - HD4203 (0.5 Myr) Sim.4 Sim.1 Sim.2 Sim.3 Sim.4 Sim.1 Sim.2 Sim.3 Sim.4 O Fe Mg Si C S Al Ca Other Semimajor Axis (AU) Bond et al Semimajor Axis (AU) Final Composition - HD (0.5 Myr) Figure 14. Schematic of the bulk elemental planetary composition for the Schematic of the bulk elemental planetary composition for the Earth-like systems high-c-enrichment systems HD19994 (top), HD (middle), and HD4203 (bottom). All values are wt% of the final simulated planet. Values are shown for the terrestrial planets produced in each of the four simulations run for the 1.86). All values are wt% of the final simulated planet. system. Values Size of bodies are is not shown to scale. Earth for values the taken terrestrial from Kargel & Lewis (1993)areshownintheupperrightofeachpanelforcomparison. found to form around HD72659 (Mg/Si = 1.23, C/O = 0.40) and HD4203 (Mg/Si = 1.17, C/O = planets produced in each of the four simulation run. (An Size extended of version bodies of this figure is not is available to scale. in the online Earth journal.) values are shown in the upper right of each panel for comparison. However, for planetesimals initially forming under disk conditions at later times the planetary composition for all simulated planets changes to more closely resemble a C- enriched Earth-like planet, with planets dominated by O, Fe, Mg, and Si and a significant amount of C. Up to 4.37 wt% Cispredictedtoexistintheplanetsforthediskconditions at yr. These planets are essentially C-enriched Earths, containing the same major elements in geochemical ratios within limits to be considered Earth like, but also an enhanced inventory Note that planets forming around HD72659 at 1 AU are roughly Earth like. Planets closer in contain more Al and Ca, which are more refractory (higher condensation temperature). The model planets forming at about 0.3 AU around HD4203 are however very different. They should have a crust made of graphite. The different composition also affects the geological evolution in terms of plate tectonics, or the existence of magnetic fields. Both are thought to be important for the emergence of life.

34 Questions?