Issues Relating to Observables of Rapidly Rotating Stars

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