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Issues Relating to Observables of Rapidly Rotating Stars
Robert Deupree, DirectorInstitute for Computational AstrophysicsSaint Mary’s UniversityHalifax, NS Canada
What Does Rapid Rotation Do?
Rotation changes the force balance, which alters the structure. This affects the intrinsic properties (e.g., luminosity, oscillation frequencies) of the model (or star)
Rotation changes the shape of the surface Rotation introduces a variation in the flux
flowing through the surface as a function of latitude (von Zeipel’s law – more flux flows out at higher latitudes)
Rapid Rotation Affects what the Observer Sees
Apparent location in the HR diagram– Both deduced L and Teff depend on inclination
Observed Spectral Energy Distributions (SEDs) depend on inclination
– SED is weighted integral over the nonspherical (and nonuniform in Teff and geff) surface
Individual line profiles are affected– Doppler shift– Same integration as SED
The Goal
The goal is to develop a self-consistent picture of rapidly rotating stars which can be compared to all the observational evidence:– Apparent location in the HR diagram– Spectral Energy Distribution (SED)– Individual line profiles– Oscillation frequencies– Interferometry
How can I Define Rapid Rotation?
Before going on to discuss the tools needed to deal with rapid rotation and its effects, it is reasonable to define what rapid rotation means in this context
Suggested Rotation Division(for 10 Mo ZAMS Model)
Slow Rotation – Veq ≤ 100 km/s– Inclination effects very small– Oscillation frequencies by standard methods– Departure from sphericity small
Moderate Rotation – 100 km/s ≤ Veq ≤ 300 km/s– Inclination effects noticeable (range in “observed” log L ≈ 0.2,
corresponds to 1 Mo uncertainty at 10 M0)– Some oscillation frequencies require more complex treatment– Relatively spherical [R(polar) ≈ 0.95 R(eq) at Veq = 300km/s]
Rapid Rotation - 300 km/s ≤ Veq ≤ Vcrit (zero effective gravity: Vcrit ≈ 600 km/s)
– All effects are large
Rotation Rate and Surface Equatorial Velocity
For slow rotation, Veq (surface equatorial velocity) is proportional to Ω (the rotation rate)
For very rapid rotation, Ω is approximately constant (Veq grows because Req is growing)
Plan
With this definition of rapid rotation, one can see that much is required of the tools to be utilized
I will first present an introduction to the modelling tools used
Then I will provide results on the effects of rotation using this collection of tools
Modelling Tools
Rapidly Rotating Stars Toolkit
2.5 D Stellar Structure and Evolution Code– Evolution on thermal and nuclear time scales– Non-Lagrangian => need velocities
3D Hydrodynamics Code Linear Nonradial pulsation code Model stellar atmospheres code (plane
parallel mostly good enough) Integration code to obtain flux = f(λ,i)
2.5 D Stellar Models with Rotation
Conservation Laws– Mass– 3 components of momentum (includes azimuthal symmetry)– Energy– Composition
Poisson’s equation Need to solve composition equations implicitly and simultaneously with
the other equations Subsidiary relations: Equation of state, opacity,… Inertial frame Independent variables: fractional surface equatorial radius, colatitude Dependent variables: density, temperature, three velocity components,
composition abundances, gravitational potential
Why not a 3D Evolution Code?
Lagrangian evolution code is not practical (knots)– => need to compute velocities to determine where material
goes with respect to your coordinate system
Implicit code accuracy limitation has Δt < Δx / v– 3D evolution is useful only if have non-uniform rotation– Then, v above replaced by Δv = (Vrot - <Vrot>)– This can give a large value of Δv and thus a small value
(essentially hydrodynamic) of Δt
3D Hydrodynamics Code
Hydrodynamic instabilities Magnetic fields required Must be able to determine long time scale
effect of calculations which can be carried only over a short (hydrodynamic) time scale– Mixing– Angular momentum redistribution
Linear, Nonradial Pulsation Code
Must be able to handle significant latitudinal variation
Apply to multi-dimensional stellar models Oscillation frequencies to match with
observations
Modelling Basic Stellar Properties
Model stellar atmospheres code– Plane parallel adequate in most cases– NLTE
Integration code to compute observed flux as function of inclination
– SEDs: needed to determine deduced luminosity (integral over all wavelengths of the observed flux corrected for distance) and effective temperature (shape of SED)
– Line profiles
Our Specific Tools Used
ROTORC – 2.5 D Stellar structure and evolution code
3D hydro code (under construction) NRO – linear, adiabatic nonradial pulsation
code PHOENIX – NLTE model atmospheres code CLIC – Integration code
Structural Results
Stellar Models with Rotation
a
1
0
Uniform Rotation– 12 Msun, 0 ≤ Veq ≤ 575
km/s
Differential Rotation
Why this Rotation Law?
Jackson, MacGregor, and Skumanich (2005) used this rotation law to model Achernar shape to compare with observed interferometry (Domiciano de Souza, et al. 2004)– No longer believed that interferometry shows the
surface How would we know if this rotation law was
correct?
Surface Shape – Uniform Rotation
Equatorial radius increases significantly
Polar radius decreases slightly
Curves for Veq = 150, 210, 255, 310, 350, 405, 450, 500, 550, and 575 km/s
Surface Shape – Effects of Differential Rotation
Surface Equatorial velocity = 240 km/s
Increasing β increases oblateness
For sufficiently high β and Veq, can get cusp at the pole
– Flux integration logic violated
Spectral Information
SEDs Lines
To Obtain SED’s of Rotating Stars
o Zone up surface into about 80000 zoneso Use locally plane parallel PHOENIX model atmospheres for local
intensity as function of angle from local surface normalo Surface properties (Teff, geff, Vrot, and R as f(θ) from 2.5D model)
o ξ is angle between local surface normal and the observero d is the distanceo Iλ is the intensity emitted
o W computes the rotational Doppler shift
projdA
d
WiIiF
2
0,,,0
PHOENIX Code Provides Iλ(μ=angle from local normal) as f(Teff,geff)
PHOENIX treats about 100,000 lines for 24 elements in NLTE
Fe I atom transitions computed in NLTE at right
SED’s cover extensive wavelength range to capture flux ( B stars: 300Å ≤ λ ≤ 10000Å with Δλ = 0.02Å)
SED for Rapidly Rotating Star Seen Pole on and Equator on
Where are the Models in the HR Diagram?
Inclination curves– Locus of apparent L
and Teff as functions of inclination (i = 0:10:90)
– Higher termperature and luminosity seen pole on
– Luminosity from total flux and distance
– Teff from shape of SED
Inclination Curves (12 Msun ZAMS)
Move to the right in HR diagram as rotation increases
Get longer as rotation increases (increasing β [differential rotation parameter] also makes inclination curves longer)
Pole to Equator differences– Δm ≈ 0.5 mag, ΔTeff ≈ 1200K for Veq = 310 km/s– Δm ≈ 2.1 mag, ΔTeff ≈ 6100K for Veq = 575 km/s
How do the Deduced Temperatures Compare to the Model Temperatures?
Model Temperatures as a function of colatitude
Temperatures deduced from composite SED as a function of the inclination of the observer from the rotation axis
Lines
Line profiles have the potential to provide much information
– Chemical composition– Inclination– Differential rotation
Even moderate rotation makes this much more difficult
Line Profiles and Differential Rotation
Differential rotation changes the shape of the line
– Decreases the depth of the core
– Broadens the wings– These are same sorts of
changes that people use to determine inclination
What Causes the Change?
Increasing β increases the rotation rate closer to the rotation axis
The rotational velocities are larger at all surface locations except the equator
Effects Appear to be Largest at Mid Latitudes
The differences in rotational velocity introduced by this particular rotation law are largest at mid latitudes
Are the Line and Broadband Parameters Consistent?
Have determined Teff from broadband information (inclination curves)
Do lines provide the same information?– Ignore Doppler broadening of lines– Compare equivalent widths of lines to PHOENIX
plane parallel equivalent widths as functions of Teff and log g to determine line Teff as a function of inclination
Lines Compared
He I 4471 He II 4686 C II 4267 N II 4631 O II 4642 Mg II 4481 Al III 1855 Si II 4130
Results for Pole and Equator
Temperatures obtained from He lines agree with photometric temperatures
Temperatures obtained from metals generally do not
Can We Talk about Asteroseismology Now?
Rotation affects the oscillation frequencies one would observe
Approximate methods exist for determining the effects of rotation on the frequencies if the rotation is not too large
Computation of Pulsation Frequencies
Use linear, adiabatic pulsation code developed by Clement (ApJS, 116, 57)– Write horizontal variation in terms of sum of
selected Yℓm’s
– Numerical radial integration of five, first order partial differential equations
– Updated for differential rotation
Terminology
As rotation rate increases, mixing of Yℓm’s other than the one present at zero rotation makes mode classification tricky– m is still a valid quantum number– ℓ is not
Define a parameter, ℓ0, which is the ℓ that the mode can be traced back to at zero rotation– Becomes more difficult for more rapid rotation
Focus on Lower Order p Modes
Here we shall restrict our attention to lower order p modes– 0 ≤ n ≤ 3– 0 ≤ ℓ0 ≤ 3
Six Yℓm’s M = 10 Msun, 0 ≤ Veq ≤ 360 km/s Axisymmetric modes only
Rotation Decreases the Pulsation Frequencies
What one would expect based on Period – mean density relation
– Each mode frequency is scaled to be unity at zero rotation
– Trend correct– Pulsation constant changes
if use mass divided by actual volume
Volume increases too much to keep Q constant
Large Separation
Δνℓ = νℓ,n+1 - νℓ,n
Uniform rotation generally decreases the large separation
Small Separation
Δνℓ,n = νℓ,n - νℓ+2,n-1
Moderate and rapid rotation increase the small separation appreciably
Note that large and small separation become close to same size for sufficiently large rotation
Effects of Differential Rotation
Clement updated NRO to include differential rotation– σ = ω + mΩ now varies as function of location
Does not interfere with solution algorithm
– Radial and latitudinal momenta equations have added term Also does not change solution algorithm
rrr r
22ˆcosˆsin
Remember Differential Rotation Model?
a
1
0
Differential Rotation Law
Differential Rotation Affects Frequencies
Effects are comparatively modest in magnitude
May either increase or decrease frequencies, depending on ℓ0
Small Effects on Large Separation
This particular differential rotation law does not affect the large separation greatly
Small Separation
Effects of increasing β mimic those of increasing the rotation rate for the small separation
The parameter β does affect the convective core boundary and shape
A Comment on Mode Identification for Rapidly Rotating Models
Lines– Lines get fairly washed out except when seen
nearly pole on
Photometric– When rotation becomes sufficiently rapid, the
amplitude ratios in different photometric bands begin to depend on the inclination between the observer and rotation axis
Nearing the End
Rotation affects just about everything we see when we observe a rotating star
– When the rotation is sufficiently small, some things can be ignored and some things treated with present approaches
– Once the rotation becomes moderate, most effects of rotation must be included
– For rapid rotation, all effects must be accounted for
Bear in mind that one does not have a solution unless it solves everything
Thanks to
Maurice Clement Chris Geroux Aaron Gillich Catherine Lovekin Ian Short Nathalie Toqué