Free Access
Volume 508, Number 3, December IV 2009
Page(s) 1359 - 1374
Section Stellar structure and evolution
Published online 08 October 2009

Online Material

Appendix A: Dredge up prescriptions

A.1 First dredge up

The change in surface abundance of isotopes j at first dredge up, $\Delta X_{j}$, is interpolated from a table of detailed models with Z=10-4 in the mass range $0.5\leq M/M_{\odot}\leq12$. A correction factor $f_{{\rm CNO}}=X_{{\rm CNO}}({\rm TMS})/X_{{\rm CNO}}({\rm ZAMS})$, the ratio of CNO mass fraction at the terminal-age main sequence (TMS) and zero-age main sequence (ZAMS), is then applied to CNO elements to take into account accretion during the main sequence.

In the Izzard et al. (2006) model first dredge up is considered as an instantaneous event. In terms of time evolution this is a reasonable assumption because giant-branch evolution is fast, but in terms of luminosity or gravity this approximation is not good and it proves difficult to compare to e.g. the $[{\rm C}/{\rm Fe}]$ vs. $\log\left(L/{\rm L_{\odot}}\right)$ data of Lucatello et al. (2006). To resolve this problem the changes in abundances are modulated by a factor $f_{{\rm p}}=\min\left[\left(M_{{\rm c}}-M_{{\rm c,BAGB}}\right)/\left(M_{{\rm c,1DUPMAX}}-M_{{\rm c,BAGB}}\right),1\right]$ where $M_{{\rm c}}$ is the core mass, $M_{{\rm C,1DUPMAX}}$ is the core mass at which first dredge up reaches its maximum depth and $M_{{\rm c,BAGB}}$ is the core mass at the base of the giant branch, before first dredge up starts. $M_{{\rm c,BAGB}}$ is known from the stellar evolution prescription and  $M_{{\rm C,1DUPMAX},}$ is interpolated from a grid of models constructed with the TWIN stellar evolution code (Eggleton & Kiseleva-Eggleton 2002).

In summary, the surface abundances changes at first dredge up are given by $f_{{\rm CNO}}f_{{\rm p}}\Delta X_{j}$. They agree well with the detailed models, as a function of  $M_{{\rm c}}$, $\log L$ and time.

A.2 Third dredge up

Abundance changes at third dredge up are treated in a similar way to the prescription of Izzard et al. (2004) and Izzard et al. (2006). Intershell abundances are interpolated from tables based on the Karakas et al. (2002) detailed models the metallicities of which extend down to Z=10-4.

In low-metallicity TPAGB stars dredge up of the hydrogen-burning shell enhances the surface abundance of $^{13}{\rm C}$ and $^{14}{\rm N}$ (at higher metallicity the effect is negligible because the initial abundance of $^{13}{\rm C}$ and $^{14}{\rm N}$ is relatively large). This is modelled by dredging up $\delta M$ of hydrogen-burnt material during each third dredge up, where the abundance mixture in this material is enhanced in $^{13}{\rm C}$ and $^{14}{\rm N}$ according to

  $\displaystyle X_{{\rm C}13} = 0.006\times X_{{\rm C}12}~{\rm and}$ (A.1)
  $\displaystyle X_{{\rm N14}} = 0.28\times X_{{\rm C}12},$ (A.2)

$\displaystyle %
\delta M = \left(\frac{0.01}{1+0.1^{2.2-M(t)}}\right)\times\min...
~\left(\frac{1}{1+\epsilon^{M_{{\rm env}}(t)-0.5}}\right)$   (A.3)

and M(t) is the instantaneous stellar mass, $M_{{\rm env}}(t)$ is the instantaneous envelope mass, $N_{{\rm TP}}$ is the thermal pulse number, X12 is the envelope abundance of $^{12}{\rm C}$and $\epsilon=10^{-20}$. The first term gives the amount of H-burnt material dredged up, the second term is a turn-on effect as the star reaches the asymptotic regime and the third term is a turn-off effect for small envelopes.

Appendix B: Mass-loss prescriptions

We consider three mass-loss prescriptions for TPAGB stars.

The formalism of Vassiliadis & Wood (1993, VW93) relates the mass-loss rate to the Mira pulsation period of the star, given by
$\displaystyle %
P_{0} = 10^{-2.07-0.90L/L_{\odot}+1.94\log_{10}\left(R/R_{\odot}\right)}~{\rm days}.$   (B.1)

The mass loss rate is then given by, as in Karakas et al. (2002), i.e. without the $M/M_{\odot}-2.5$ term of the original VW93 prescription,
$\displaystyle %
\dot{M}=\dot{M}_{{\rm VW0}} = f_{{\rm VW}}\left(-11.4+0.0125P_{0}\right)M_{\odot}~{\rm yr}^{-1}$   (B.2)

unless $P_{0}>\left(500-\Delta P_{{\rm VW}}\right)~{\rm days}$ in which case a superwind is applied
$\displaystyle %
\dot{M} = \max\left(\dot{M}_{{\rm VW0}},f_{{\rm VW}}\frac{Lc}{v_{{\rm w}}}\right)$   (B.3)

$\displaystyle %
v_{{\rm w}} = 10^{5}\left(-13.5+0.056P_{0}\right)~{\rm cm}~{\rm s}^{-1}.$   (B.4)

The free parameters $f_{{\rm VW}}$ and $\Delta P_{{\rm VW}}$ subtly affect the mass-loss rate. The factor  $f_{{\rm VW}}$ is a simple multiplier, which is 1 by default (see model set 27). The period shift  $\Delta P_{{\rm VW}}$ allows the onset of the superwind to be delayed, e.g.  $\Delta P_{{\rm VW}}=-100~{\rm days}$ in model set 33 - it is zero by default.

The Reimers mass-loss rate is given by
$\displaystyle %
\dot{M} = 4\times10^{-13}\eta\frac{RL}{M}{M_{\odot}}~{\rm yr}^{-1},$     (B.5)

where $\eta$ is a parameter of order unity (Reimers 1975) which we vary in model sets 10, 11 and 12.

van Loon.
In model set 13 we use the split form of van Loon et al. (2005) appropriate to oxygen-rich red giants,

$\displaystyle %
\log_{10}\left[\dot{M}/({M_{\odot}}~{\rm yr}^{-1})\right] =
...<0.9~{\rm and~}\\
-5.3+0.82l_{4}-10.8t_{35} & l_{4}\geq0.9,
\end{array}\right.$   (B.6)

where $l_{4}=\log_{10}\left(L/10^{4}~{L_{\odot}}\right)$ and $t_{{\rm 35}}=\log_{10}\left(T_{{\rm eff}}/3500~{\rm K}\right)$. Note, if $T_{{\rm eff}}>4000~{\rm K}$ we enforce a minimum mass-loss rate of $10^{-4}~{M_{\odot}}~{\rm year}^{-1}$ because the above formula can approach zero as the temperature rises (and the envelope mass becomes small) as a star approaches the white-dwarf cooling track.

Appendix C: Binary distributions

Our default binary-star distribution is the combination of

The initial mass function (IMF) of Kroupa et al. (1993, KTG93) for the initial primary mass M1

\psi(M_{1})=\left\{ \begin{array}{ll}
0 & M_{1}/{M_{\odot}}...
...t}}\leq m_{{\rm max}}\\
0 & m>m_{{\rm max}}\end{array}\right.
\end{displaymath} (C.1)

where p1=-1.3, p2=-2.2, p3=-2.7, m0=0.1, m1=0.5, m2=1.0 and $m_{{\rm max}}=80.0$. Continuity requirements and $\int\psi(M){\rm d}M=1$ give the constants a1, a2 and a3.

A distribution flat in q=M2/M1 for the initial secondary mass M2, where $M_{2}\leq M_{1}$

A distribution flat in $\ln a$ (i.e. probability ${\sim}1/a$) for the separation a where $3\leq a\leq10^{5}$.

Initially circular binaries (except for model set Ae5).

Appendix D: All model sets and results

Table D.1 shows the full set of models we considered, of which Table 1 is a subset.

Table D.2 shows the full set of CEMP, CNEMP and NEMP to EMP ratios for all our model sets of which Table 3 is a subset.

Table D.1:   The full list of our binary population models (a subset is shown in Table 1). The meanings of the symbols are given in Sect. 2.

Table D.2:   Percentage of CEMP, CNEMP and NEMP (sub-)giants relative to total EMP giants in all our model binary populations (see Sect. 2.6 for selection criteria).

Appendix E: Observation database

Our observational selection is taken from the SAGA database as compiled by Suda et al. (2008) combined with data Lucatello et al. (2006).

When data exists for the same star from more than one source, we take the arithmetic mean of the values and add errors in quadrature. In the case of log-values, e.g.  $[{\rm Fe}/{\rm H}]$ or $\log g$, we simply average the log-values rather than attempt a more sophisticated approach. This makes little difference to our final results. In the case of data limits (e.g. x<4) we ignore the data - few data are of this type and the general result is not affected.

We ignore error bars in the sense that, e.g. a star with $[{\rm Fe}/{\rm H}]=-2.9$ $\pm $ 0.2 is not included in our selection, even though it may well have - in reality - $[{\rm Fe}/{\rm H}]=-2.7$ and hence qualify. This is the price we pay for a simple selection procedure and in the large number limit (the database has about 1300 stars) it is not a problem.

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