Massive source stars: nucleosynthesis and models with rotation
3.3 Massive stellar models including rotation
3.3.1 Transport induced by rotation
∇ad,∇radfT
fP
(3.4) with
fP = GM4πr4P
PSP
1
hgeff−1iandfT = 4πr2
P
SP
2 1 hgeffihg−1effi
and MP the mass enclosed by the sphere of radiusrP, geff the effective gravity (resulting from both gravitational and centrifugal acceleration), ∇rad and ∇ad the radiative and adiabatic tem-perature gradients,ρ¯the average density between two isobars, nucl,ν, grav the rates of nuclear energy production, neutrino losses, and gravitational energy (energy released/absorbed by con-traction/expansion of the star). The quantities in brackets are averaged over an isobar according to
hXi= 1 SP
I
isobar
X(r, θ)dσ (3.5)
whereSP is the surface of the isobar, defined bySP = 4πrP2 withrP the radius of the equivalent sphere, having the same volume than the oblate isobar. dσis a small surface element on the isobar.
In this work, the Geneva stellar evolution codeGENEC(Eggenberger et al. 2008) is used. It was mainly developed and used for the computation of massive stellar models (e.g. Maeder 1987, 1992) with a sophisticated treatment of rotation (Meynet & Maeder 1997). The physical and modeling ingredients relevant for this work are discussed in the following sections.
3.3.1 Transport induced by rotation
In a differentially rotating star, the equipotentials are differently spaced as a function of the colatitudeθbecause the star is distorted by the centrifugal effect. The centrifugal force makes the equatorial radius larger than the polar radius so that the equipotentials are closer to each other at the poles and more spaced at the equator. Consequently, the effective gravity is larger at the poles. The radiative flux is proportional to the local effective gravity (von Zeipel 1924) meaning that there is an excess of flux along the polar axis and a deficiency close to the equatorial plane.
This imbalance triggers global circulation motions called meridional circulation. Such currents, which are advective, contribute to the transport of angular momentum and chemical elements in rotating stars. In differentially rotating stars, because the stellar layers have different angular ve-locities, shear instabilities arise and also contribute (in a diffusive way) to the transport of angular momentum and chemical elements.
Transport of angular momentum
In the radial direction, the transport of angular momentum obeys the advective-diffusive equa-tion (Chaboyer & Zahn 1992)
ρd
dt(r2Ω)¯ Mr = 1 5r2
∂
∂r(ρr2U(r) ¯Ω)
| {z }
advection
+ 1 r2
∂
∂r
ρr4D∂Ω¯
∂r
| {z }
diffusion
(3.6)
whereΩ¯is the angular velocity of a shell,ρthe density,Dthe total diffusion coefficient, taking into account the various instabilities transporting angular momentum (especially convection and shear turbulence) andU(r)the amplitude of the radial component of the meridional velocity (Maeder
& Zahn 1998). A large3 |U(r)|implies an efficient transport of angular momentum in the radial
3U(r)can be positive or negative. Here the absolute value is considered.
3.3. Massive stellar models including rotation
Figure 3.4: Internal profile of the amplitude of the radial component of the meridional velocity U(r) in rotating 20 M models with various metallicities. The radius is normalized to the total radius Rmax. For all the models, the central H mass fraction is about 0.40 (figure from Ekström et al. 2008).
direction. U(r)scales withEΩ, a term of particular importance in the expression ofU(r)and that can be approximated by
EΩ' 8 3
1− Ω2 2πG¯ρ
Ω2r3 GMr
. (3.7)
The second term in brackets is the Gratton-Öpik term. In the outer layers of stars, the densityρ¯is small so thatEΩ, henceU(r), can become largely negative (see solid line in Fig. 3.4). It contributes to redistribute efficiently the angular momentum and therefore smooth theΩ-profile.
Low metallicity stars contain less metals so they are less opaque and therefore more compact.
Consequently, ρ¯is larger so that the Gratton-Öpik term is generally smaller. This reduces U(r) (see Fig. 3.4), meaning that the angular momentum is less redistributed inside the star. It leads to steeper internalΩ-profiles. This tends to produce largerΩ-gradients which lead to stronger shear instabilities and then to a more efficient shear mixing.
Transport of chemical elements
Chaboyer & Zahn (1992) have shown that for chemical elements, the combination of meridional circulation and horizontal turbulence can be described by a pure diffusive process. The associated diffusion coefficient is
Deff = 1 30
|rU(r)|2
Dh (3.8)
with Dh the horizontal (i.e. on an isobaric surface) shear diffusion coefficient from Zahn (1992).
It is written as Dh = c−1h r|2V −αU(r)|with α = 12dln(rdln2rΩ)¯ and ch = 1. As a consequence, in contrast with the angular momentum transport equation (Eq. 3.6), the equation for the change of abundance of a given chemical element iin a given shell at coordinate r is a pure diffusive equation:
CHAPTER 3. MASSIVE SOURCE STARS: NUCLEOSYNTHESIS AND MODELS WITH ROTATION
ρdXi
dt = 1 r2
∂
∂r
ρr2(D+Deff)∂Xi
∂r
+ dXi
dt
nucl
. (3.9)
The last term accounts for the changes due to the nuclear reactions (cf. Sect. 3.3.3). In radiative zones,D= Dshear withDshear the diffusion coefficient accounting for the shear instabilities. Sev-eral prescriptions exist in the literature (Maeder 1997; Talon & Zahn 1997; Maeder et al. 2013). The one used in this work is from Talon & Zahn (1997). It can be expressed as
DshearTZ97=fenergHp gδ
K+Dh (∇ad− ∇) +ϕδ∇µ(DK
h + 1) 9π
32Ωd ln Ω d lnr
2
(3.10) where K is the thermal diffusivity, and fenerg = 2Ric with Ric the critical Richardson number below which the medium becomes dynamically unstable and turbulent. Although the canonical value for Ric is 1/4 some works have shown that turbulence may happen for larger Richard-son number. Canuto (2002) has shown that turbulence may arise for Ri ≤Ric ∼ 1. Brüggen &
Hillebrandt (2001) have done numerical simulations suggesting that shear mixing occurs already at Ri ∼ 1.5. The physical range of values may be around 0.25 < Ric < 2, corresponding to 0.5< fenerg<4.
I computed two 15Mmodels at solar metallicity withυini = 300km s−1 and withfenerg= 1 and 4. At core H depletion, the surface N/H ratios are enhanced by a factor of 2 (fenerg = 1) and 3 (fenerg = 4) compared to the initial surface N/H ratio. Such surface enrichments qualitatively agree with observation of10−20Mrotating stars (e.g. Gies & Lambert 1992; Villamariz & Herrero 2005; Hunter et al. 2009). In this work, except stated explicitly,fenerg= 4. The effect offenergon the internal mixing of low metallicity massive stars is investigated in Sect. 4.3.3.
A comparison of 3 different prescriptions for the shear mixing
Although not used in the present work, I have included inGENEC the global diffusion coeffi-cient proposed by Maeder et al. (2013), taking into account different instabilities together and their possible interaction (see Appendix A.2 for calculation details).
Fig. 3.5 shows the tracks of 20M models withυini/υcrit = 0.4,Z = 10−5 and computed with various Dshear coefficients: from Maeder (1997), from Talon & Zahn (1997) and the global coeffi-cient from Maeder et al. (2013). The new coefficoeffi-cient DglobalM13 combining the different instabilities leads to a bluer and more luminous evolution, characteristic of stars experiencing a high degree of mixing. The surface 4He mass fraction at the end of the main sequence is the highest in the DM13globalmodel (0.56 against 0.25 and 0.31 for theDshearTZ97andDM97shearmodels, respectively), which is also a signature of an efficient rotational mixing. An estimation of the mixing efficiency in the deep stellar interior can be obtained with the amount of central22Ne after the main sequence (more de-tails in Sect. 4.1). 22Ne in the center reaches a much higher mass fraction in theDTZ97shear andDM13global models. It suggests that, after the main sequence, a more efficient mixing in the deep interior is at work in these two models compared to theDM97shearmodel. The efficiency of the s-process in massive stars is impacted by the amount of 22Ne (main neutron source, cf. Sect. 3.1.2). From Fig. 3.5, we see that the s-process should operate similarly in theDshearTZ97andDM13globalbecause of the similar22Ne content. On the opposite, theDshearM97 model predicts a less efficient operation of the s-process.
The dashed track in Fig. 3.5 shows the main sequence of theDTZ97shear model but withfenerg=40 instead of 4. In this case, the track is rather close to the track of the model with the global diffusion coefficient. A higherfenergvalue for theDshearTZ97model may mimic the more complex physics con-tained in theDglobalM13 coefficient. It shows that finally, even iffenerg= 4was proposed as a possible upper limit for the Dshear of Talon & Zahn (1997), a higher fenerg (i.e. higher mixing efficiency) probably cannot be excluded.
3.3. Massive stellar models including rotation