1. Issues Relating to Observables of Rapidly Rotating Stars Robert Deupree, Director Institute for Computational Astrophysics Saint Mary’s University Halifax, NS Canada

2. 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)

3. 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

4. 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

5. 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

6. Suggested Rotation Division (for 10 M o 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 M o uncertainty at 10 M 0 ) – 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

7. 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)

8. 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

9. Modelling Tools

10. 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)

11. 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

12. 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 - ) – This can give a large value of Δv and thus a small value (essentially hydrodynamic) of Δt

13. 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

14. Linear, Nonradial Pulsation Code Must be able to handle significant latitudinal variation Apply to multi-dimensional stellar models Oscillation frequencies to match with observations

15. 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

16. 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

17. Structural Results

18. Stellar Models with Rotation Uniform Rotation – 12 Msun, 0 ≤ Veq ≤ 575 km/s Differential Rotation

19. 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?

20. 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

21. 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

22. Spectral Information SEDs Lines

23. To Obtain SED’s of Rotating Stars o Zone up surface into about 80000 zones o Use locally plane parallel PHOENIX model atmospheres for local intensity as function of angle from local surface normal o Surface properties (Teff, geff, Vrot, and R as f(θ) from 2.5D model) o ξ is angle between local surface normal and the observer o d is the distance o I λ is the intensity emitted o W computes the rotational Doppler shift

24. PHOENIX Code Provides I λ (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)

25. SED for Rapidly Rotating Star Seen Pole on and Equator on

26. 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

27. 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

28. 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

29. Lines Line profiles have the potential to provide much information – Chemical composition – Inclination – Differential rotation Even moderate rotation makes this much more difficult

30. 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

31. 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

32. Effects Appear to be Largest at Mid Latitudes The differences in rotational velocity introduced by this particular rotation law are largest at mid latitudes

33. 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

34. 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

35. Results for Pole and Equator Temperatures obtained from He lines agree with photometric temperatures Temperatures obtained from metals generally do not

36. 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

37. Computation of Pulsation Frequencies Use linear, adiabatic pulsation code developed by Clement (ApJS, 116, 57) – Write horizontal variation in terms of sum of selected Y ℓm ’s – Numerical radial integration of five, first order partial differential equations – Updated for differential rotation

38. Terminology As rotation rate increases, mixing of Y ℓm ’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, ℓ 0, which is the ℓ that the mode can be traced back to at zero rotation – Becomes more difficult for more rapid rotation

39. Focus on Lower Order p Modes Here we shall restrict our attention to lower order p modes – 0 ≤ n ≤ 3 – 0 ≤ ℓ 0 ≤ 3 Six Y ℓm ’s M = 10 Msun, 0 ≤ Veq ≤ 360 km/s Axisymmetric modes only

40. 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

41. Large Separation Δν ℓ = ν ℓ,n+1 - ν ℓ,n Uniform rotation generally decreases the large separation

42. Small Separation Δν ℓ,n = ν ℓ,n - ν ℓ+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

43. 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

44. Remember Differential Rotation Model? Differential Rotation Law

45. Differential Rotation Affects Frequencies Effects are comparatively modest in magnitude May either increase or decrease frequencies, depending on ℓ 0

46. Small Effects on Large Separation This particular differential rotation law does not affect the large separation greatly

47. 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

48. 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

49. 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

50. Thanks to Maurice Clement Chris Geroux Aaron Gillich Catherine Lovekin Ian Short Nathalie Toqué