Which physics determines the location of the mean molecular weight minimum in red giants?

Transcription

1 doi: /mnras/stu1195 Which physics determines the location of the mean molecular weight minimum in red giants? Ross P. Church, 1,2 John Lattanzio, 2 George Angelou, 2 Christopher A. Tout 2,3 and Richard J. Stancliffe 4 1 Department of Astronomy and Theoretical Physics, Lund Observatory, Box 43, SE Lund, Sweden 2 Monash Centre for Astrophysics, School of Mathematical Sciences, Monash University, VIC 3800, Australia 3 Institute of Astronomy, The Observatories, Madingley Road, Cambridge CB3 0HA, UK 4 Argelander Institute for Astronomy, University of Bonn, Auf dem Huegel 71, D Bonn, Germany Accepted 2014 June 16. Received 2014 June 16; in original form 2014 April 24 ABSTRACT Stars ascending the red giant branch develop an inversion in mean molecular weight (μ)owing to the burning of 3 He in the region immediately above their hydrogen-burning shells. This inversion may drive thermohaline mixing and thereby be responsible for the extra mixing which is observationally indicated on the red giant branch. In this paper, we investigate the physical influences that determine the mass and temperature at which the inversion in μ develops. We find that it depends most strongly on the thermal structure of the envelope the profiles of density and temperature in the region of the star immediately above the shell and is otherwise relatively insensitive to abundances and nuclear reaction rates. The changes in the effects of thermohaline mixing as stars proceed up the giant branch can mostly be understood in terms of their changing thermal structure, driven by their increasing core mass. Key words: methods: numerical stars: abundances stars: evolution. 1 INTRODUCTION There is now a substantial body of evidence to support the occurrence of some form of non-convective mixing in red giant stars. The clearest indications of such mixing are the changes in abundances seen in giants above the position of the bump in the luminosity function. Below the bump the stellar compositions largely agree with the predictions from standard stellar evolution theory, in which the only change is due to the first dredge-up (Gilroy & Brown 1991; Charbonnel 1994). Beyond this luminosity, however, the surface abundances of both lithium and carbon are observed to decrease further. The situation for Li is beautifully shown in fig. 5 of Lind et al. (2009). For carbon the most pronounced effects are found in the 12 C/ 13 C isotopic ratio, which decreases with increasing luminosity along the giant branch. In some globular clusters there is now enough data to show unambiguously that [C/Fe] decreases as stars ascend the giant branch. All of these phenomena can be explained by some extra form of deep mixing that connects the hot region at the top of the hydrogen-burning shell with the bottom of the convective envelope. This allows material that has been exposed to some hydrogen burning to be mixed into the convective envelope and thereby be transported to the stellar photosphere. ross@astro.lu.se Initial attempts to elucidate the physical origin of this additional mixing were focused on rotationally induced mixing (Sweigart & Mengel 1979). However, recent models have shown that it is insufficient to reproduce the observed abundance variations (Palacios et al. 2003, 2006). Eggleton, Dearborn & Lattanzio (2006) showed that an inversion in the mean molecular mass μ should be produced during the normal evolution of red giants and that this occurs exactly at the luminosity function bump. This inversion causes a double-diffusive instability which is generally known as thermohaline mixing by analogy with oceanic science, where it was first discovered. Further exploration of the consequences of this mechanism (Charbonnel & Zahn 2007; Eggleton, Dearborn & Lattanzio 2008) within the formalism of Ulrich (1972) and Kippenhahn, Ruschenplatt & Thomas (1980) has shown that, provided that the mixing that it generates is sufficiently strong, it has many of the properties required to solve the abundance problems, for example in globular clusters (Angelou et al. 2011, 2012), metal-poor stars (Stancliffe et al. 2009) and the long-standing 3 He problem (Eggleton et al. 2006; Lagarde et al. 2011, 2012). What is missing at the moment is a study of the physical processes by which the mechanism proceeds, their relative importance and the observational consequences. For example, why do low-metallicity stars produce a lower 12 C/ 13 C ratio than solar metallicity stars? How sensitive is the position of the μ minimum, where the mixing starts, to composition and reaction rates? What is the main determinant of the position of the μ minimum? Is it the composition, the structure C 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2 978 R. P. Church et al. of the star or something else? How does it depend on the speed of mixing? It is the aim of this paper to provide answers to these fundamental questions. In this work, we study thermohaline mixing in isolation, without considering other forms of additional mixing such as rotation or internal gravity waves. This is because we wish to understand how thermohaline mixing works on its own, without the additional complication of considering the interaction with other mechanisms. Maeder et al. (2013) show that the presence of significant horizontal turbulence in red giants is expected to suppress the thermohaline instability; even if there is only weak rotational mixing, the location of the μ minimum may be modified. We do not take these complexities into account here, but they may be significant; our results are only strictly valid in the absence of rotational shear. In Section 2, we develop a simplified treatment to obtain the mass location and temperature of the μ minimum in the absence of mixing, and use it to investigate the effects of varying the chemical abundances, nuclear reaction rates and thermal structure of the star. In Section 3, we investigate the difference that thermohaline mixing of various efficiencies makes to the location of the μ minimum. Finally, we conclude in Section 4. 2 THE μ MINIMUM WITHOUT MIXING It is initially convenient to consider the location of the μ minimum in the absence of thermohaline mixing: that is, either the case that prevails initially, before the instability has developed significantly, or if the mixing is for some reason much less rapid than we currently believe it to be. First, this assumption allows us to ignore the feedback from the mixing on the structure of the star. Secondly, it allows us to develop a simple, yet remarkably accurate, model of the location of the μ minimum to use for parameter sensitivity tests. In order to validate our simplified treatment, and to obtain some of the quantities that it requires, we utilize models made with the Monash version of the Mt Stromlo stellar evolution code, MONSTAR (see Campbell & Lattanzio 2008). A detailed description of the implementation of thermohaline mixing in MONSTAR is given by Angelou et al. (2011). 2.1 Simplified treatment We make two simplifying assumptions about the evolution after the bump in the absence of mixing, which we go on to test by comparison with a detailed stellar model. Our assumptions are that (i) the hydrogen-burning shell moves outwards in mass coordinate m through the star at a rate ṁ shell = dm shell /dt which is constant on time-scales small compared to the giant-branch lifetime, and (ii) the abundance profile belonging to the 3 He-burning region remains fixed with respect to the burning shell. This means that we can transform into a mass coordinate m = m ṁ shell t m 0 in which we choose the constant m 0 so that the top of the burning shell, defined as the point where X = 0.66, is stationary at the origin. In this frame we can express the rate of change of the abundances X i via X i t = X i m t + X i m m m t t. (1) m We identify term on the left-hand side with the nuclear reaction rates, R i, and the final term on the right-hand side with the advection of material owing to the motion of the shell. From assumption (ii) the other term on the right-hand side is zero, and hence, we obtain X i R i = ṁ shell m, (2) where R i is the total rate of change of abundance X i owing to nuclear reactions. Hence, we can obtain the chemical structure of the burning shell from a static stellar model. The mean molecular mass 1 μ is defined as 1 μ = X i, (3) μ i i where μ i is the mean mass of particles making up a pure sample of species i. We assume realistically for the temperatures and densities that prevail near the shell that the material is fully ionized and hence μ i includes contributions from the electrons. Taking the derivative of equation (3) with respect to m we obtain the condition for the location of the μ minimum, 1 dμ = 1 X i = 1 R i = 0. (4) μ 2 dm μ i i m ṁ shell μ i i This approach does not yield an analytic solution to the location of the μ minimum because, in order to calculate R i, we must still integrate equations (2) to obtain X i (m ). We integrate explicitly inwards from the point where T = 10 6 K, chosen to be sufficiently far above the top of the hydrogen-burning shell that no nuclear reactions operate. In order to evaluate the reaction rates R i we require the temperature T and density ρ of material in the star, which we parametrize according to T (m ) = T 0 exp(γ T m ); ρ(m ) = ρ 0 exp(γ ρ m ), (5) where T 0 = T(m = 0), ρ 0 = ρ(m = 0) and γ T = ln T m ; t γ ρ = ln ρ m (6) t both evaluated at the top of the H-burning shell (i.e. m = 0). Empirically this parametrization is sufficient for two reasons: first, the density and temperature really do behave exponentially immediately outside the burning shell and, secondly, at the point where the temperature and density start to diverge from the parametrized values the nuclear burning is already small. We obtain the rate at which the shell burns out through the star by noting that the energy generation in the star is almost entirely by CNO cycling in the H-burning shell. Then, when the luminosity is L, ṁ shell = 4Lm H, (7) Q CNO X 0 where m H is the proton mass, Q CNO is the energy liberated in burning four protons to one helium nucleus via the CNO cycle and X 0 is the hydrogen mass fraction well above the burning region. This allows us to completely specify our model with nine parameters, γ T, T 0, γ ρ, ρ 0, ṁ shell, and the abundances of H, 3 He, 4 He and 12 C well above the burning region. To treat the nuclear burning we use a reduced network where only the slowest reactions are followed explicitly and those that proceed more rapidly are assumed to be instantaneous. 1 The expression mean molecular mass is a misnomer, since in all the parts of stars with which we are concerned the material is fully ionized and hence there are no molecules present. A more accurate term would be mean particular mass since it is the mean mass of all massive particles, including free electrons, that is relevant, but we reluctantly adopt the prevailing nomenclature.

3 Physics of the μ minimum 979 Table 1. The reaction network implemented in our simplified treatment. References: (a) Caughlan & Fowler (1988); (b) Li et al. (2010). Effective reaction Rate-determining step Ref. for rate 3p 3 He p(p, β + )d (a) 2 3 He α + 2p 3 He( 3 He, 2p)α (a) 3 He + α + p 2 α 3 He(α, γ ) 7 Be (a) 12 C + 2p 14 N 12 C(p, γ ) 13 N (b) For example, the rate of the reaction chain p(p, β + )d(p, γ ) 3 He is determined by the p(p, β + )d reaction. Empirically, we find that the nuclear reaction chains associated with only four reaction rates need to be considered in order to locate the μ minimum. The network that we use is specified in Table 1. The results of an example integration are presented in Figs 1 and 2. Fig. 1 shows the abundances as functions of mass around the region where the μ minimum develops. Points show the abundances derived with a full stellar model, lines those by integration. The dashed pink line shows the location of the μ minimum in the stellar model, the dotted purple line its location in our simplified treatment. Fig. 2 shows the relative magnitudes of the contributions from the various nuclear reaction rates to the terms on the RHS of equation (4), and hence to the rate of change of μ. Although our simplified treatment matches the detailed model quite well there are small differences visible in Fig. 1, which are caused by four effects. (i) As the star ascends the giant branch it becomes more luminous and hence ṁ shell increases, contrary to our assumption (i). This contributes the majority of the difference between the two locations of the μ minimum. (ii) Our parametrization of T and ρ underestimates them at locations away from the shell. This causes the μ minimum to move towards the shell, but the effect is small because the parametrization is a sufficient fit in the regions where 3 He is burning. (iii) The calculation of the reaction rates in the simplified model omits the effects of weak screening. This changes the reaction rates by at most a few per cent and has an insignificant effect on the location of the μ minimum. (iv) The simplified treatment does not include the 14 N(p, γ )or 16 O(p, γ ) reactions. These reactions are insignificant in the region of the μ minimum but cause the abundance profiles to be incorrect closer to the shell. Subject to the caveats listed above, inspection of the plot shows that the location of the μ minimum in the simplified model agrees with that in the detailed structure model within δm 10 4 M, which is approximately equal to the mesh spacing at that location in the MONSTAR model. Thus, we conclude that, despite neither of our assumptions (i) and (ii) being entirely correct, they are both sufficiently valid that we can use the simplified treatment to study the effects of the various parameters on the location of the μ minimum. 2.2 Sensitivity to physical processes Using our simplified model, we test the sensitivity of the location of the μ minimum to the various physical processes taking place above the shell in the absence of any mixing. Figure 1. Variations of chemical abundances X i as functions of mass m across the region where the μ minimum develops. Symbols are a full MONSTAR model, lines our simplified treatment. In the top panel, black (crosses, solid line) are hydrogen and red (squares, dashed line) 3 He. In the lower panel, blue (open circles, solid line) 4 He, brown (triangles, dashed line) 12 C, pink (diamonds, dotted line) 14 N and grey (filled circles) 16 O. The vertical dashed pink line shows the location of the μ minimum in the detailed model, the vertical dotted purple line its location in the simplified treatment. The model has a mass of 1 M, Z = 0.02 initially and is taken immediately after the luminosity function bump when the μ minimum has just developed Surface abundances Fig. 3 shows the effects of varying the abundances above the burning shell. The black line shows the variation of the helium abundance (where the difference from standard is taken from or put into hydrogen). The top panel shows the temperature at the μ minimum and the bottom panel its offset in mass from the top of the H-burning shell. The μ minimum is located at the point where the rate of increase of μ caused by proton burning is balanced by its rate of decrease caused by 3 He burning. As the helium abundance is increased, the hydrogen abundance is decreased. Lowering the abundance of H reduces the p+p and 12 C(p,γ) rates and hence shifts this point

4 980 R. P. Church et al. Figure 2. The contributions of various nuclear reactions to the gradient in μ with mass, plotted as functions of mass across the region where the μ minimum develops. Symbols are calculated from a full MONSTAR model, lines from our simplified treatment. Black (crosses, solid line) are the p+p reaction, red (squares, short dashed line) 3 He + 3 He, blue (open circles, dotted line) 3 He(α, γ ), brown (triangles, dash dotted line) 12 C(p,γ), pink (diamonds) 14 N(p,γ) and grey (filled circles, long-dashed line) the sum of all the rates. The vertical dashed pink line shows the location of the μ minimum in the detailed model, the vertical dotted purple line its location in the simplified treatment. The model has a mass of 1 M, Z = 0.02 initially and is taken just after the luminosity function bump when the μ minimum has just developed. The disagreements between the models have two causes: the different magnitudes of the 3 He + 3 He curve are caused by ṁ shell not being constant. The disagreement in the 12 C abundance visible to the left of the figure is caused by the omission of the 14 N(p,γ) reaction in the simplified treatment. inwards to higher temperatures, where those rates are faster and the 3 He abundance is lower. Hence, the μ minimum moves towards the shell, and to a higher temperature. The effect is shown in Fig. 4. The effects of varying the other surface abundances can be understood in the same way. Increasing the 3 He abundance increases the rate of the reaction that decreases μ and hence moves the μ minimum inwards. Conversely, increasing the 12 C abundance increases the rate of proton burning and thus moves the μ minimum outwards, as found by Stancliffe (2010) in detailed models of thermohaline mixing in asymptotic giant branch stars that dredge up primary carbon from their cores Reaction rates The effects of varying the four reaction rates are shown in Fig. 5. Perhaps counterintuitively, increases in any of the rates cause the μ minimum to move outwards. For the p+p, 12 C(p,γ)and 3 He(α, γ ) reactions the explanation is the same as that given for increasing H and 12 C abundances above. Increasing the rate of a reaction which increases μ causes the point at which it balances the reduction in μ caused by 3 He burning to move outwards. Increasing the rate of the 3 He burning also moves the μ minimum outwards, in this case because the 3 He burns at lower temperatures, and hence further from the shell. Figure 3. The variation of the location of the μ minimum in mass (lower panel) and temperature (upper panel) as the abundances of material above the shell are varied. On the abscissa, we plot the abundance divided by the default value; the defaults are given in Table 2. The black line with crosses shows the effect of varying the helium abundance, with the difference from the standard value being put into hydrogen. The red line (squares) shows variations in 3 He and the blue line (circles) 12 C; in both these cases the difference is put into or taken from the abundance of unreactive species. The model has mass of 1 M, Z = 0.02 initially and is taken immediately after the luminosity function bump when the μ minimum has just developed Thermal structure As the star evolves up the giant branch the growth of the core causes the shell to become hotter and denser. This changes the state of the material just above the shell and hence the location of the μ minimum moves. As the shell becomes hotter its luminosity increases and hence it consumes mass more rapidly, increasing the rate at which mass is advected into the 3 He-burning region. In Fig. 6, we compare the magnitudes of these various contributions. The black dots show the variation of the temperature and mass of the μ minimum as the star evolves up the giant branch, taken from

5 Physics of the μ minimum 981 Figure 4. The contributions of various nuclear reactions to the gradient in μ for a model with the abundance of H decreased and an enhanced 4 He abundance. The contributions are plotted as functions of mass across the region where the μ minimum develops. Symbols are abundances from a full MONSTAR model with the standard abundances, i.e. without the enhanced helium abundance but instead with the abundances from Table 2. Lines are our simplified treatment applied with surface mass fractions of X = andy = Black (crosses, solid line) are the p+p reaction, red (squares, dashed line) 3 He + 3 He, blue (open circles, dotted line) 3 He(α, γ ), brown (triangles, dash dotted line) 12 C(p,γ), pink (diamonds) 14 N(p,γ) and grey (filled circles, long dashed line) the sum of all the rates. The vertical dashed pink line shows the location of the μ minimum in the standard model, the vertical dotted purple line its location in the model with modified abundances. The model has a mass of 1 M, Z = 0.02 initially and is taken immediately after the luminosity function bump when the μ minimum has just developed. an evolutionary sequence calculated with MONSTAR and which hence includes all of the physical changes. The blue dot-dashed lines show the location of the μ minimum if the temperature and density in the shell (T 0 and ρ 0 ) are held constant, but their gradients (γ T and γ ρ ) and ṁ shell are allowed to vary as in the MONSTAR model. As can be seen, this makes very little difference to the location in mass of the shell, which depends principally on how steep the changes in temperature and pressure are. The temperature in the shell, however, varies considerably along the giant branch and this effect dominates the temperature of the μ minimum as can be seen from the top panel. The red dashed lines show the effect of allowing only ṁ shell to vary; it can be seen that this has less effect on the mass location of the μ minimum than the other thermal properties Discussion Comparison of Fig. 6 with Figs 3 and 5 reveals that the magnitude of the changes caused by the evolving thermal structure are much greater than those that could be attributed to changes in the composition and rates. It is worth stressing that the variations of the abundances and reaction rates that we have considered are for the most part unreasonably large, whereas the temperatures and densities are those that actually arise during the evolution of the star. Thus, we can draw the following conclusions from our analysis of the unmixed case: Figure 5. The variation of the location of the μ minimum in mass (lower panel) and temperature (upper panel) as various reaction rates are varied. On the abscissa, we plot the constant by which we multiply the default reaction rate; references for the defaults are given in Table 1. The black line (crosses) shows the effect of varying the p+p rate, the red line (squares) the 3 He + 3 He rate, the blue line (circles) the 3 He + α rate and the brown line (triangles) the 12 C(p, γ ) rate. The model has mass of 1 M, Z = 0.02 initially and is taken immediately after the luminosity function bump when the μ minimum has just developed. (i) in the absence of structurally significant mixing, changes in the position and temperature of the μ minimum as a star evolves up the giant branch are primarily due to the changes in the temperature and density profiles caused by the increasing core mass, and (ii) the feedback of the mixing on the location of the μ minimum owing to changes in abundance alone is unlikely to be significant unless almost all the 3 He has been burnt. The typical extent of these differences can be seen by considering a star of the same mass but different metallicity. Fig. 7 shows the evolution of the μ minimum location along the giant branch of a star of globular cluster metallicity (Z = 10 4 ; red dots) compared

6 982 R. P. Church et al. Figure 6. Evolution of the location of the μ minimum as a star evolves up the giant branch. The variation of the location of the μ minimum in mass (lower panel) and temperature (upper panel) are plotted as functions of the luminosity. Black dots show the full evolution from MONSTAR models. The blue dot dashed lines show the effect of holding the temperature and density at the top of the shell constant, but allowing the steepness of the shell (γ T and γ ρ )andṁ shell to vary as in the detailed model. The red dashed lines show the situation if only ṁ shell is allowed to vary. The model has a mass of 1M, Z = 0.02 initially and the plot starts immediately after the luminosity function bump when the μ minimum has just developed. to our reference star (Z = 0.02; black dots). The first thing to note is that the luminosity bump, and hence first occurrence of a global minimum in μ, occurs at higher luminosities in the lower metallicity star, owing to the reduced depth of first dredge-up. The μ minimum is at a somewhat greater mass from the H-burning shell at a given luminosity but at a similar temperature; this is because the thermal structure in the low-metallicity star is less steep as a function of mass. The brown points show the effect of changing the abundances to those of higher metallicity but ignoring the feedback Figure 7. Effects of metallicity on the mass m μ (bottom panel) and temperature T μ (top panel) of the μ minimum as a function of surface luminosity L as a star evolves up the giant branch. Black dots show our reference 1M Z = 0.02 star. Red dots show the full evolution of a model of the same mass but with Z = Blue squares (superimposed on red dots) show the same calculation, but using our simplified treatment rather than a full evolutionary model. Brown triangles show the effect of changing the abundances to those of the Z = 0.02 model whilst retaining the stellar structure and ṁ shell of the Z = 10 4 model. on the stellar structure through different opacities and μ.see Table2 for the exact values used. As one might expect from the previous discussion, the increases in abundances cause the μ minimum to move outwards and become cooler, but the changes are again small compared to those caused as the star evolves to higher luminosities. 3 THE EFFECTS OF MIXING Once the μ minimum has developed we expect mixing to occur by the thermohaline double-diffusive instability. Following Ulrich

7 Physics of the μ minimum 983 Table 2. The initial chemical mass fractions of our models. Species Abundance (Z = 0.02) Abundance (Z = 10 4 ) H He He C N O (1972), we treat the process diffusively within MONSTAR, adopting a diffusion coefficient 2 μ D thm = C t K (8) ad in which K is the thermal diffusivity given in terms of the temperature T, density ρ, specific heat at constant pressure c P and opacity κ by K = 4acT 3, (9) 3κρ 2 c P the radiative and adiabatic temperature gradients and ad are defined in their usual way and μ = ln μ ln P. (10) t The constant C t is a dimensionless coefficient related to the aspect ratio α of the fingers of material that descend into the star to cause thermohaline mixing by C t = 8 3 π2 α 2. (11) Thermohaline mixing with C t 1000 matches well various abundance observations (see references in the Introduction); however, multidimensional hydrodynamic simulations suggest that the appropriate α may be much smaller (see e.g. Denissenkov 2010;Traxler, Garaud & Stellmach 2011). Here, we consider a range between C t = 1, which is unreasonably small, to C t = 10 4 which is larger than suggested by either theory or observations. In Fig. 8,weshow the effect of thermohaline mixing with various values of C t on the location of the μ minimum. The initial model is our standard 1 M, Z = 0.02 model with mixing applied between the appearance of the μ minimum and the tip of the red giant branch. There are three mechanisms by which mixing could have an effect on the location of the μ minimum. First, the opacity in the envelope changes owing to the different abundances, and hence changes the thermal structure. Empirically, this effect is found to be very small. Secondly, the change in abundances well above the burning shell will change the location of the μ minimum as described in Section Finally, the mixing process brings fresh 3 He into the region in which it is burnt, moving the μ minimum inwards and to hotter temperatures. In order to differentiate between these last two effects, in Fig. 8 we plot the location of the μ minimum in both our full evolutionary models that include mixing (points in the figure), and in our simplified approach without mixing but taking the same thermal structure and surface abundances (lines in the figure). Inspection of the figure shows that, at the onset of thermo- 2 We note that Maeder et al. (2013) show that in an inhomogeneous region the denominator should be ad μ ; however, for thermohaline mixing in red giants this represents a fractional correction to D thm of the order of 10 5 and hence can be safely neglected. Figure 8. Effects of mixing on the mass m μ (bottom panel) and temperature T μ (top panel) of the μ minimum as a function of surface luminosity L as a star evolves up the giant branch. Filled black squares (partially hidden beneath the various lines) show our reference 1 M Z = 0.02 star with no thermohaline mixing. Sequences of full models with mixing are plotted for C t = 1 (open red squares), C t = 10 (filled blue circles), C t = 100 (open brown triangles), C t = 1000 (open pink diamonds) and C t = 10 4 (filled grey triangles). The points have been averaged across sets of 100 adjacent timesteps to reduce the scatter in the plot and hence make it clearer. Lines show unmixed models from our simplified treatment made with surface abundances taken from the mixed model of the same surface luminosity. The red solid lines show calculations with abundances from the models mixed with C t = 1, blue dashed lines have abundances from the C t = 10 models, brown dotted lines have abundances from the C t = 100 models, pink dot dashed lines have abundances from the C t = 1000 models and grey long dashed lines have abundances from the C t = 10 4 models. In the top panel, the lines for C t values of 1, 10 and 100 lie on top of one another; in the bottom panel all the lines other than that for C t = 10 4 lie on top of one another.

8 984 R. P. Church et al. haline mixing, the rate of mixing has a very significant effect on the position and temperature of the μ minimum, yet the surface abundances are all unchanged and hence the unmixed calculations all coincide. As the evolution progresses two things take effect. The first is that the shell becomes hotter and denser, and the gradients steeper. This means that the effect of the extra 3 He provided by the mixing is lessened, and all the models converge as they evolve to large luminosities. The second is that, in the cases where C t = 1000 and C t = 10 4 the burning reduces the surface 3 He abundance significantly. As a result the μ minimum becomes shallower, μ falls and the mixing becomes weaker. By the tip of the giant branch, the model with C t = 10 4 has burnt sufficient 3 He that the mixing has reduced to the point where it no longer has a significant effect on the location of the μ minimum, as seen by the coincidence of the grey points and line at large luminosities in Fig. 8:ithasburntitself out. 4 SUMMARY AND DISCUSSION As a star ascends the first giant branch and the burning shell passes through the composition discontinuity produced by first dredge-up, an inversion in the mean molecular weight is produced which can cause thermohaline mixing. Under the simplifying assumption that thermohaline mixing is the only extra mixing process taking place, we have investigated the physical processes and conditions in the star that determine where this μ minimum is located in mass and temperature and hence the physical properties of the material that lies at the bottom of the mixed region. We find that the majority of the changes in the location of the μ minimum between models of the same star at different phases on the giant branch, and between stars of different metallicities, are caused by the differences in the temperature and density profiles of the material just above the hydrogen-burning shell. These are set by the mass of the core and thermal conductivity of the envelope and are essentially independent of the thermohaline mixing. The abundance of 3 He is also significant but only in models where it has almost entirely been consumed does its abundance contribute significantly to the evolution of the location of the μ minimum. Mixing, on the other hand, does have a significant effect; by bringing fresh 3 He all the way down to the region where it is being burnt it moves the burning closer to the hydrogen-burning shell and thus increases the temperature at the μ minimum. The temperature at the μ minimum is probably the most significant quantity in terms of consequences for the star. Prior to the discovery of the thermohaline instability in red giants, phenomenological models of extra mixing were developed which typically placed the maximum extent of the mixing at a fixed temperature relative to the base of the hydrogen-burning shell, equivalent to log T = log 10 (T shell /T μ ) 0.2 (Wasserburg, Boothroyd & Sackmann 1995; Boothroyd & Sackmann 1999; Sackmann & Boothroyd 1999; Denissenkov & Pinsonneault 2008). We find that log T varies over the giant-branch evolution with a range typically between 0.2 and 0.3. This relative constancy of the temperature difference makes thermohaline mixing an attractive candidate mechanism for the additional mixing seen in red giant stars; however, as discussed in Section 1, there are some theoretical indications that it should operate weakly if at all. The relatively smooth variation of the temperature at the base of the thermohaline zone implies that the nucleosynthetic products of extra mixing should vary in a characteristic way with luminosity, which may make it possible to test whether the mixing process that operates is actually thermohaline mixing. In particular, changes in the abundance of lithium are very sensitive to the temperatures that the material mixed up to the surface of the star has experienced. We show in a companion paper (Lattanzio et al. submitted) that the evolution of the surface Li abundance can only be calculated accurately by taking great care to resolve the mixing at the μ minimum both in space and time. Read in conjunction with that work this paper should offer the reader some consolation that, provided that the degree of extra mixing is known, the location of the μ minimum is well defined, relatively robust to solution method and input physics and largely determined by the structure of the star rather than our solution of the thermohaline mixing. ACKNOWLEDGEMENTS The authors would like to thank Alexander Heger, Simon Campbell and Carolyn Doherty for useful discussions, and the referee, Robert Smith, for his positive and constructive comments and suggestions. This research was supported under the Australian Research Council s Discovery Projects funding scheme (project numbers DP , DP and DP ). RPC is supported by the Swedish Research Council (grants and ). CAT thanks Churchill College for his fellowship and Monash University for support as a Kevin Westfold distinguished visitor. RJS is the recipient of a Sofja Kovalevskaja Award from the Alexander von Humboldt Foundation. Calculations presented in this paper were carried out at LUNARC using computer hardware funded in part by the Royal Fysiographic Society of Lund. REFERENCES Angelou G. C., Church R. P., Stancliffe R. J., Lattanzio J. C., Smith G. H., 2011, ApJ, 728, 79 Angelou G. C., Stancliffe R. J., Church R. P., Lattanzio J. C., Smith G. H., 2012, ApJ, 749, 128 Boothroyd A. I., Sackmann I.-J., 1999, ApJ, 510, 232 Campbell S. W., Lattanzio J. C., 2008, A&A, 490, 769 Caughlan G. R., Fowler W. A., 1988, At. Data Nucl. Data Tables, 40, 283 Charbonnel C., 1994, A&A, 282, 811 Charbonnel C., Zahn J.-P., 2007, A&A, 467, L15 Denissenkov P. A., 2010, ApJ, 723, 563 Denissenkov P. A., Pinsonneault M., 2008, ApJ, 684, 626 Eggleton P. P., Dearborn D. S. P., Lattanzio J. C., 2006, Science, 314, 1580 Eggleton P. P., Dearborn D. S. P., Lattanzio J. C., 2008, ApJ, 677, 581 Gilroy K. K., Brown J. A., 1991, ApJ, 371, 578 Kippenhahn R., Ruschenplatt G., Thomas H.-C., 1980, A&A, 91, 175 Lagarde N., Charbonnel C., Decressin T., Hagelberg J., 2011, A&A, 536, A28 Lagarde N., Romano D., Charbonnel C., Tosi M., Chiappini C., Matteucci F., 2012, A&A, 542, A62 Li Z. et al., 2010, Sci. China Phys. Mech. Astron., 53, 658 Lind K., Primas F., Charbonnel C., Grundahl F., Asplund M., 2009, A&A, 503, 545 Maeder A., Meynet G., Lagarde N., Charbonnel C., 2013, A&A, 553, A1 Palacios A., Talon S., Charbonnel C., Forestini M., 2003, A&A, 399, 603 Palacios A., Charbonnel C., Talon S., Siess L., 2006, A&A, 453, 261 Sackmann I.-J., Boothroyd A. I., 1999, ApJ, 510, 217 Stancliffe R. J., 2010, MNRAS, 403, 505 Stancliffe R. J., Church R. P., Angelou G. C., Lattanzio J. C., 2009, MNRAS, 396, 2313 Sweigart A. V., Mengel J. G., 1979, ApJ, 229, 624 Traxler A., Garaud P., Stellmach S., 2011, ApJ, 728, L29 Ulrich R. K., 1972, ApJ, 172, 165 Wasserburg G. J., Boothroyd A. I., Sackmann I.-J., 1995, ApJ, 447, L37 This paper has been typeset from a TEX/LATEX file prepared by the author.