Maurizio Salaris
Astrophysics Research Institute – Liverpool John Moores University
EaHS12 - Oxford
The determination of
mass, radius, Teff and
spectral energy
distribution of planet host
stars are necessary to
characterize the properties
of the planets. Stellar ages
are also critical to provide
evolutionary scenarios for
these planets (and
‘biological’ scenarios in
case signs of life will one
day be detected)
Theoretical stellar
evolution models are
the essential tool to
determine the
properties of
exoplanet host stars
Equation of stellar evolution, inputs
Boundary conditions
Convection
Mass loss
Atomic diffusion + radiative levitation
Additional mixings
Equations of ‘classical’ stellar
evolution models
Mechanical |
evolution
|
|
Chemical
evolution
For each element s we have chemical reactions of the type
w production reactions nh h+nk knp s
l destruction reactions
nd s + nj j  nz z
Timescale for mechanical evolution  τKH≡|Ω|/L
Timescale for chemical evolution  τn ≡En/L
In general, but for the most advanced evolutionary phases of massive
stars, τKH« τn
The inequality between τKH and τn means that the equations of chemical
evolution can be decoupled from the other 4 equations.
One can solve the mechanical part of the star at a given instant t with a
given set of chemical abundances, then apply a timestep Δt, solve the
equations for the chemical evolution to determine the new abundances
using the values of the physical variables determined at the previous
instant t. The equations of the mechanical structure are then integrated
again at time t+Δt using these chemical abundances, and so on.
The independent variables are the mass coordinate m and time t
Unknowns are r, P, T, L, (Xs s=1, …I)
At each stellar layer we have 4+I equations for 4+I unknowns
Energy generation (and neutrino energy loss) coefficients,
equation of state, opacities and nuclear cross sections have to be
known
The total stellar mass M and initial chemical composition
(assumed to be uniform at the start) must be specified
Additional element
transport mechanisms
Diffusion
equation
diffusion velocity
ws
A stellar astronomer once said:
“We provide evolutionary tracks, yields
and physics from the pre-main sequence
through to the thermally pulsing AGB. It’s
certainly wrong, but it’s freely available.”
Isochrones
Ingredients
• Surface boundary conditions
• Equation of state
• Opacity
• Nuclear cross sections
• Convection
• Mass loss
• Atomic diffusion
• Radiative levitation
How do input physics/macro- and microscopic processes
affect the theoretical stellar models of fixed initial mass and
chemical composition ?
•
Equation of State  luminosity – effective temperature - age
•
Opacity  luminosity – effective temperature - age
•
Nuclear reaction rates  luminosity – age – surface abundances (in some cases)
•
Efficiency/extension of the convective energy transport 
luminosity - effective temperature – surface abundances
•
Microscopic diffusion/radiative levitation  luminosity – effective
temperature – surface abundances
•
Mass loss  luminosity - effective temperature (not always) – surface
abundances (not always)
•
Boundary conditions  effective temperature
Treatment of surface boundary conditions
The surface boundary of stellar model
calculations is located at a layer where the
diffusion approximation for the radiative energy
transport starts to be applicable. This
corresponds to a layer where the optical depth
τ~10
Calculations of model atmospheres provide the
pressure at the photosphere Ps as a function of
the effective temperature Teff and surface gravity
g (that depends on M and R).
The surface boundary condition is
‘critical’ for stellar models with
convective envelopes
•The RGB based on model atmospheres shows a slightly different slope, crossing
over the RGB of the models computed using the Krishna-Swamy solar T()
•Differences are within ~50K
The treatment of convection in stellar modelling has to be able to determine:
i)
when a layer becomes convectively unstable
Chemical composition gradient
Uniform chemical composition
ii) whether convection can extend beyond its formal border;
iii) the timescale of mixing;
iv) the temperature gradient in the convective region;
Desirable (and/or necessary) features of the chosen treatment are:
Coverage of as many evolutionary phases as possible
Economy of computer time + simplicity
The problem with convection
Turbulence is a flow regime characterized by chaotic and
stochastic property changes.
Richard Feynman described turbulence as
"the most important unsolved problem of classical physics."
Superadiabatic convection:
The mixing length theory (Böhm-Vitense 1958)
Almost universally used in stellar evolution codes.
Simple, local, time independent model, that assumes convective elements with mean size l, of
the order of their mean free path.
l=α Hp
mixing length
Superadiabatic convection:
The mixing length theory
l=α Hp
mixing length
a
b
c
BV58
⅛
½
24
ML1
ML2
⅛
½
1
2
24
16
ML3
1
2
16
α
calibrated
1.0
0.6 - 1.0
2.0
The value of α affects strongly the effective
temperature of stars with convective envelopes
The’canonical’ calibration is
based on reproducing the
solar radius with a theoretical
solar models (Gough & Weiss
1976)
We should always keep in
mind that there is a priori no
reason why α should stay
constant within a stellar
envelope, and when
considering stars of different
masses and/or at different
evolutionary stages
BV58 formalism
Run of the superadiabaticity
and the ratio of the convective to the total energy flux
as a function of the radial location in the outer layers of the solar convection zone. Solid lines represent
the ML2 model, dashed lines the BV58 model.
The dashed dotted line displays the ratio between the local pressure scale height and the geometrical
distance from the top of the convective region.
BV58 α=2.01
ML2 α=0.63
Salaris & Cassisi
(2008)
surface
Sound speed relative differences: ML2 vs BV58
Bottom
envelope
convection
surface
Dashed – BV58 α=2.01
Solid – ML2 α=0.63
Superadiabatic convection:
The Full Spectrum of Turbulence model
The Full Spectrum of Turbulence (FST) theory (Canuto& Mazzitelli 1991, Canuto Goldman &
Mazzitelli 1996) is a local theory based on an expression for the convective flux derived
from a model of turbulence.
Turbulent
pressure
(Pturb ≈ ρ vc2)
is also
accounted for
Superadiabatic convection:
The Full Spectrum of Turbulence model
The turbulent scale length Λ is assumed to
be equal to the harmonic mean of the
distances from the top and bottom of the
convective boundaries
CM models and
solar calibrated
MLT models are
not equivalent
Interplay between
boundary condition and
mixing length calibration
Boundary conditions play an important role.
Solar calibrated models with different boundary
conditions predict different RGB temperatures
From Salaris et al. (2002)
From Montalban et al. (2004)
2D hydro-simulations
Hydro-calibration
Previous attempts by Deupree &
Varner (1980) Lydon et al (1992,
1993)
Extended grid of 2D hydromodels by Ludwig, Steffen
& Freytag
Static envelope models
based on the mixing length
theory calibrate α by
reproducing the entropy of
the adiabatic layers below
the superadiabatic region
from the hydro-models.
A relationship α=f(Teff,g) is
produced, to be employed
in stellar evolution
modelling
(Ludwig et al.1999)
From Freytag & Salaris (1999)
How do models from different authors compare ?
~200K
From Salaris et al. (2002)
Models from different
authors, all with solar
calibrated mixing
length, predict
different RGB
temperatures
Calibration on binary systems
Contradicting results so far
Yildiz et al (2006) find the following dependence by
modelling Hyades binaries:
Other authors obtain opposite results
The α Centauri system is a good example for showing the uncertainties in
this type of calibrations:
Miglio & Montalban (2005) found a mixing length 5-10 % smaller for the more massive
component, taking into account both seismic and non-seismic constraints.
Yildiz (2007) finds a larger mixing length for the more massive component if the seismic
constraint is neglected, whereas the opposite trend is found when the seismic constraint
is taken into account
RADII OF LOW MASS STARS
Boyajian et
al (2010)
Analyses of low-mass eclipsing binary stars
have unveiled a disagreement between the
observations and predictions of stellar
structure models. Results show that
theoretical models (with solar calibrated
mixing length) underestimate the radii (by up
to 10-20%) and overestimate the effective
temperatures of low-mass stars but yield
luminosities that accord with observations.
Chabrier et al. (2007) suggest that reduced
convective efficiency, due to fast rotation and
large magnetic field strengths, and/or to
magnetic spot coverage of the radiating
surface significantly affect their evolution,
leading to a reduced heat flux and thus larger
radii and cooler effective temperatures than
for regular objects.
But problems also for
single stars (?)
Mechanical overshooting
a<0
v>0
a<0
v=0
a>0
v>0
RCC + a Hp
Convective Core overshooting
The H central burning
lifetime is longer;
The size of the He core at the end of
the central H-burning phase is
larger; The mean luminosity during
the central He-burning phase is
larger.
The central He-burning
lifetime is shorter;
How to decrease the overshooting efficiency
as the star mass decreases?
Since the pressure scale height steadily increases when moving deeper and deeper
inside a star (HP when r0), OV has to decrease to zero for stars with small
convective cores (M1.5M, Roxburgh 92, Woo & Demarque 01)
Different prescription
for the “ramping” are adopted
The trend of OV with mass introduces
an additional degree
of freedom in stellar evolution model
calculations
Calibration of the overshooting extension with:
CMD morphology
Eclipsing binaries
NG2420
2 Gyr (canonical –
solid)
3 Gyr
oversh dashed
solid
5.37 Gyr
oversh
(0.08 Hp)
dashed
[Fe/H]=-0.44
From Pietrinferni et al. (2004)
Calibration of the overshooting
extension with star counts
An example:
The case of NGC1978 in the LMC
TO mass ~1.5Mo Age 2-3 Gyr
Preferred value Λ~0.10
From Mucciarelli et al. (2007)
Calibrations with eclipsing
binaries: an example
Schroeder et al. (1997)
Pols et al. (1997)
Ribas et al. (2000)
= Prad/Pgas
= 0.12
From Schroeder et al. (1997)
Approximate trend of λov with mass
(from eclipsing binaries)
M~1.2-1.3 Mo
M~1.5 Mo
M ~2.0-2.5 Mo
M ~3.0-4.0 Mo
M ~6.0 Mo
M ~10.0 Mo
λov~0.0 Hp
λov~0.10-0.15 Hp
λov~0.20 Hp
λov~0.25 Hp
λov~0.30 Hp
λov~0.40-0.50 Hp
Miglio et al. (2007) found, for 12Boötis,
M=1.42 λov~0.37 Hp
M=1.37 λov~0.15 Hp
The mixing scheme during core He-burning:
connection between nuclear burning & convection
 rad
 d ln T 

  κLr
 d ln P rad
He (low k) -> C/O (high k)
convective
rad  ad
radiative
He
C/O
induced overshooting
(Castellani, Giannone, Renzini 1971)
II phase: Development of a
‘partial mixing zone’
When Yc decreases below ~0.7, a ‘partial
mixing’ (semiconvective) zone develops
beyond the boundary of the convective core.
Semiconvective
region
Semiconvection and
HRD evolution
Semiconvection increases central
He-burning lifetime by a factor ~1.5
-2
Breathing Pulses
(Start when Yc~0.10)
Numerical artifact ??
Parameter R2=Nagb/Nhb
Observations
R2=0.14±0.02
Semiconv +BPs R2~0.08
Semiconv no BPs R2~0.12
Mimicking semiconvection with overshooting
Caloi & Mazzitelli (1990)
Sweigart (1990)
Extension of mixing (by ~0.1Hp) in regions
beyond the boundary of all convective regions
forming within the He-rich core
The edge of the convective core is let
propagate with velocity
High overshooting ~ 1 Hp
Seismic data of pulsating White Dwarfs (Metcalfe et al. 01 – 02, GD358) can provide
constraints on the internal chemical stratification:
central O
abundance
q
Model
M core
M tot
XO
0.840.03
0.490.01
q
The C/O profile
O
C
No semiconvection
No overshooting
0.56
Semiconvection
+ no Breathing Pulses
0.79
Semiconvection
+ no Breathing Pulses
(by Dorman & Rood)
0.58
“Low” Overshooting
0.60
0.43 – 0.47
“High” Overshooting
0.56
0.49 – 0.53
O
0.48 - 0.52
C
041 – 0.45
O
C
Time dependent mixing: An example
See also Eggleton (1972), Pols & Tout
(2001), Langer et al. (1983), Ventura &
Castellani (2005)
Herwig et al. (1997)
Radiative region
Convective region
f=0.02
vc , vo from MLT
Overshooting
Following Freytag et al.
(1996) hydro-simulations
Time dependent mixing in convective cores
Ventura & Castellani (2005 – but
see also, e.g., Eggleton 1972,
Deng et al. 1996, Salasnich et al.
1999, Woosley et al. 2002)
Approach similar to Herwig
et al. (1997)
Mass loss on the RGB
The luminosity and
effective temperature
of HB stars depends
on the amount of mass
lost along the RGB
Mass loss on the RGB
Different parametrizations
different results
dM RGB
L
 4 1013 R
dt
gR
high mass loss
efficiency
Origlia et al. 2007
Uncertain dependence
the metallicity
on
intermediate
mass loss
efficiency
Reimers
low mass loss
efficiency
Atomic diffusion + radiative levitation
Radiative levitation alters the surface abundances when <grad> is larger than
the local acceleration of gravity g below the base of the convective envelope
g
grad
0.8M - Z=0.00017
Richard et al. 2002
[Fe/H]=-2
Predicted surface abundances at the TO
Are atomic diffusion + radiative levitation really efficient ?
M4 Mucciarelli, Salaris
et al. (2011)
In the Sun YES
(helioseismology)
In Pop II stars …. maybe not
….. But why ?
diffusion
‘ Ad hoc’ solution proposed
by Richard et al. (2005)
T0 is a reference temperature, and is
a free parameter
This additional term adds to the
counterbalancing effect of
chemical element gradients
Same type of turbulent
transport as rotational
induced mixing, when the
combined effect of
meridional circulation and
horizontal turbulence
leads to vertical diffusion
(Chaboyer & Zahn 1992)
0.8M - Z=0.00017
Richard et al. 2002
ADDITIONAL TRANSPORT MECHANISMS
“The H-burning front moves outward into the stable region, but preceding the Hburning region proper is a narrow region, usually thought unimportant, in which
3He burns.
The main reaction is 3He (3He, 2p)4He: two nuclei become three nuclei, and the
mean mass per nucleus decreases from 3 to 2. Because the molecular weight (µ)
is the mean mass per nucleus, but including also the much larger abundances of
H and 4He that are already there and not taking part in this reaction, this leads
to a small inversion in the µ gradient. “
Eggleton et al. (2006)
1Mo solar
composition
Gratton et al. (2000)
Free parameter
Thermoaline mixing
Lagarde &
Charbonnel
(2010)
Superadiabatic convection
MLT (3 + 1 free parameters) - FST (2 + 1 free parameters)
Non-local Time-independent
›››› Affects depth of convective envelopes, Teff , P/rho stratification in
the convective envelopes, surface abundances/nucleosynthesis,
asteroseismic properties.
i) Calibration of the MLT parameters (and the FST free parameter) depends
on input physics, MLT formalism and boundary conditions chosen.
ii) Uncertainties associated to the calibrators.
iii) Different solar calibrations may provide different results in earlier or
later evolutionary phases (uncertainties in boundary conditions play a role
here !)
“Semiconvection” + “Overshooting”
Free parameters (or assumptions) related to the
extension and efficiency of mixing.
Instantaneous or time-dependent mixing
››››››› Affect evolutionary times (star counts), luminosities, Teff , loops
in the Colour-Magnitude-Diagrams, predicted populations of variable stars
in stellar populations, chemical profiles, asteroseismic properties.
Empirical calibrations are needed.
The calibrated ‘extended’ size of the mixed regions might be due not just to convective
overshooting but also to other neglected physical processes and/or inadequacies of the
input physics.
How do we move forward ?
Need to keep the number of equations and free
parameters to a minimum (ideally no free parameters)
i) Introduce additional ingredients in the MLT (or FST) formalisms (this
usually increases the number of free parameters ) ? (e.g. Gough 1977,
Balmforth 1992)
ii) Calibrate the MLT parameter(s) (and overshooting) on 2D or 3D hydro
simulations ? (e.g. Ludwig et al. 1999, Freytag & Salaris 1999)
iii) Employ Reynolds stress models ?
(e.g. Xiong 1978, Canuto 1992, 1993, Yang & Li 1997)
… again free parameters/assumptions (Yang & Li model has 6 free
parameters
Efficiency of atomic diffusion + radiative levitation
Affects effective temperatures luminosities, evolutionary timescales
Not clear why it is not fully efficient in Pop II stars. Current modelling of a
moderating ‘turbulent mixing’ are completely ‘ad hoc’ and calibrated on
observations
Empirical calibrations are (again) necessary
Additional mixing on the RGB
Affects surface chemical abundances of important elements like CNO
Thermoaline mixing is the current most accepted candidate. Free parameter to
be calibrated
Hydro-simulations suggest that thermoaline mixing is insufficient to explain
the observations
RGB mass loss
Affects mainly the mass and CMD location during the He-burning
phase.
No theory about RGB mass loss (but see recent paper by Cranmer &
Saar 2011)
Empirical calibrations are needed
WILL WE EVER BE ABLE TO MAKE REALISTIC
FULL HYDRO-SIMULATIONS OF A WHOLE STAR ?
WILL FULL HYDRO-SIMULATIONS OF THE
‘IMPORTANT’ STELLAR REGIONS BE SUFFICIENT
TO SOLVE THESE PROBLEMS ?
WILL ASTEROSEISMOLOGY BE ABLE TO
LEAD US TOWARDS A SOLUTION FOR
THESE (OR AT LEAST SOME) PROBLEMS ?