Stellar evolution with a variable mixing-length parameter

1. The convection simulations
Using the convection code of Stein & Nordlund (1989), we have performed radiation coupled hydrodynamical simulations of convection in a small box at the surface of seven stars, ranging from the sub giants Boo and Procyon and down to the 0.9 M dwarf, Cen B. About 20% of the simulation domain is above hT i = Te , and the simulations cover between 11 and 13 pressure scale heights or 10{11 decades in optical depth. Horizontally they each contain 5{7 granules. At optical depths below = 300, the simulation treats radiative transfer in detail using the Feautrier technique, in three dimensions and with monochromatic opacities. For larger the di usion approximation is used. For the atomic physics we use an updated version (Trampedach 1997) of the continuous opacities of Gustafsson et al. (1975) the opacity distribution functions of Kurucz (1992), and the so-called MHD equation of state (EOS) (Hummer & Mihalas 1988; Dappen et al. 1988).
an1
ቤተ መጻሕፍቲ ባይዱ
0.44601 0.25638 0.18447 -0.00742 -0.46139 -0.04661 -0.11952 -0.33586 0.07988 0.07867
an2
0.005005 0.005547 0.000813 0.003232 0.008598 0.008061 0.002082 0.092365 0.083088 0.016930
n a1 a2 a3 a4 a5 a6 a7 a8 a9
1.33381 0.44224 0.34174 0.06660 -2.25315 -1.12180 -1.05538 -0.62807 0.84445 1.79939
an;
0.15229 1.64329 1.56252 0.10977 14.37900 12.06989 6.60583 10.61300 -6.75028 -1.26586
1
2.
T-
relations
The T - relations from the simulations have been derived by averaging temperature over iso- surfaces and time. These T - relations have then been tted to expressions of the form
Stellar evolution with a variable mixing-length parameter
Regner Trampedach and Robert F. Stein Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA J rgen Christensen{Dalsgaard Teoretisk Astrofysik Center, Danmarks Grundforskningsfond, and Institut for Fysik og Astronomi, Aarhus Universitet, DK-8000 Aarhus C, Denmark Ake Nordlund Teoretisk Astrofysik Center, Danmarks Grundforskningsfond, and Astronomisk Observatorium, Niels Bohr Institutet for Astronomi, Fysik og Geofysik, K benhavns Universitet, DK{2100 K benhavn , Denmark
The T ( ) relation shows a distinct dependence on stellar mass on the zeroage main sequence (ZAMS), illustrated in Fig. 1a. The results indicate that the line blanketing by metal lines in the low-mass stars is more e cient than the cooling by the hydrogen lines in the higher-mass F0{G3 stars.
Abstract. By simulating the convection in the upper layers of seven
di erent stars and matching these simulations to 1D mixing-length models using the same input physics, we have been able to infer the behaviour of the T - relation and the mixing-length parameter, , as the stellar parameters change. We have then used this knowledge in stellar evolution calculations, in order to investigate the consequences of changing and the T - relation as the stars evolve.
3. Matching envelope models to the simulations
The 1D structure models do not include the core of the star and they assume a homogeneous composition; hence they are functions of Te , gsurf , mass and composition alone. We use the MHD EOS and OPAL opacities smoothly joined with the opacities used in the simulations at low temperatures. In the atmosphere we use a T - relation derived from the simulations, as described in Sect. 2. Convection is treated with the mixing-length formulation. An unambigious match to the 2 simulations necessitated the inclusion of a turbulent pressure, Pturb;1D = vconv%; in the hydrostatic equilibrium. In the envelope model the mixing-length parameter and the parameter in the expression for the 1D turbulent pressure were adjusted to obtain continuity of temperature and density at a common pressure point. These 's were then tted to an expression analogous to Eq. (2), giving a standard deviation of 2
Figure 1. a) Changes with stellar mass in the T - relation, for ZAMS models, relative to the solar-mass star. b) Derived behaviour of with stellar parameters. We have overplotted the location of the seven simulations, as well as the evolutionary tracks also displayed in Fig. 2. 0.016|more than an order of magnitude smaller than the range of variation of 's for the seven stars, indicating a signi cant variation. The result is shown in Fig. 1b, and the coe cients are given in Table 1.
4. Stellar evolution
Fig. 2 shows the location of the seven simulations, as well as evolutionary tracks for the three cases; 1) xed solar and T - relation (solid), 2) varying , but xed solar T - relation (dashed) and 3) varying and T - relation (dotted), each for six di erent masses. From the tracks we see that varying has a noticeable e ect, increasing Te during the main sequence life of stars lighter than the Sun, and decreasing Te during turn-o for more massive stars. At late evolutionary stages, when they start to move up the Hayashi track and most of the star becomes convective, they all get more sensitive to changes in . Letting the T - relations change works in the opposite direction and almost cancels the e ect of the varying . The approach to the Hayashi track and the hydrogen burning phase of stars more massive than the Sun still di er somewhat from the conventional models, though. A larger means more e cient convection, gives a smaller entropy jump in the atmosphere and leads to a deeper convection zone. Increased atmospheric opacity and increased line blocking will both lead to more e cient convection, but it also changes the outer boundary conditions and we have found that the envelope models respond to these changes by decreasing the depth of the convection zone, i.e., counteracting the e ect they have on . 3
4 3
Tt Te
4
a + = a1 + 2
1 + Fconv a8 Ftot
a9
?1
? a a4 + a 3 + + a5 e 2
log 10
a7
?a6
2
; (1)
to better than 0.25%. Each of the nine parameters for the seven simulations have then been tted to linear expressions an = an; + an1 log10 (Te =Te ; ) + an2 log10 (gsurf =gsurf ; ) ; (2) to be used in the subsequent stellar evolution calculations. The coe cients are listed in Table 1. Table 1. Coe cients for Eq. (2).