20-23 November 2006 Joint Helas & CoRoT/ESTA Workshop 1

Effects to consider on oscillation frequencies

The effect of rotation (perturbative approach)

J. C. Suárez et al.

Instituto de Astrofísica de Andalucía (CSIC), Granada, Spain

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

2

Plan

Introduction Slow, Moderate & Fast rotators

Main effects (perturbative approach)Differential rotationÉchèlle DiagramsAnalysis of splitting asymmetries &

rotation profile variations (RPV).

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

3

Introduction

Type of rotators: Slow (ε,μ <<1) Moderate (ε,μ ≤ 0.5) Fast (ε,μ > 0.5)

FastSlow-Moderate

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

4

mnss

Lsnmn TDmDJCm 2

3

12

00

2

0,,0 1

Perturbative approach

Effect of rotation separated in orders O(μ) O(μ): slow rotation O(μ2,μ3): moderate rotation

Dziembowski & Goode (1992), Soufi et al. 1998

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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120

0

2

2 DmD

Main effects

First order (Coriolis, geometric) Symmetry in multiplets

Second order (Coriolis,Centrifugal) Asymmetries in multiplets

01 1 JCm l

m=-1m=-1 m=0m=0 m=+1m=+1splittingglobal shift asymmetry

1

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Near degeneracy

Close – frequency modes Avoided crossing, mixed modes Selection rules: Δm=0, Δℓ=0,2

...),(

),(),( 20

3,32

210

j j

jddd

ωω0 0 ~ ~ ωωjj

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Effects on adiabatic frequencies

)(ˆ ,,0 Onmn

Goupil al. 1999

mnss

Lsnmn TDmDJCm 2

3

12

00

2

0,,0 1

Near degeneracy (2nd order)

geff & Coriolis (1st order)

Centrifugal (2nd order)

No rotation: ω=ω0(0)

Cubic effects (3rd order)

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

8

Differential rotation

Shellular rotation gives rise to additional terms Eigenfunctions (CL, J0, D0, D1)

Structure terms (J0, D0, D1)

2

1

12

00

2

00 1

DmDJCm s

ls

)](1[)( 0 rr s

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Differential rotation

Intermediate-mass, moderately rotating stars

DR vs. UR modelsDR as shellular (local

conservation of angular momentum)

)](1[)( 0 rr s Suárez et al. 2006 A&A 449, 673

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Differential rotation. The effects on frequencies

Mainly affect ω1 & ω2

Significant effects for g and mixed modes

Up to 3μHz for g-mixed and 1μHz for p modes

Suárez et al. 2006 A&A 449, 673

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Differential rotation. The effects on eigenfunctions

Suárez et al. 2006 A&A 449, 673

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Rotation & Échèlle diagrams

Solar-like pulsators (high-frequency modes excited)

Significant effects for large frequencies

Near degeneracy naturally affects small separation. Largest effects for ℓ=(0,2) modes.

m components may complicate diagnostics but amplitudes may alleviate the problem

50 km/s

30 km/s

20 km/sℓ=2 ℓ=0 ℓ=1ℓ=3

Lochard et al. 2006, in prep1.5 M●

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Rotation & Échèlle diagrams

Seismic modelling based on δω fitting diagnostics generally neglect rotation.

For a 1.5 M● (MS, 1Gyr) model δω(Ω)- δω(0) ~1μHz

But rotation effects can be “depolluted”

Lochard et al. 2006, in prep

~1.2 Hz

Ω

Ω=0

1.5 M●, 30km/s

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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obsma

obsmamaaaa sssV ,,,3,2, 2

1,3

Rotation & Échèlle diagrams

Main idea: to obtain the “true” (non-rotating) small separations from observed δω

Rotation depullution is performed for close modes a & b from the structure quantity Va~Vb

aabat VV 02.064.0

2/122

Goupil et al. 2003

Δν(0)

Δν(t)

Δν(Ω)

skmvM sba /30,3,1,,50.1

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Analysis of splitting asymmetries & RPV

Asymmetries of rotation splittings (Sj) are observed in pulsating stars (δ Scuti, β Cephei, etc.)

Rotation profile variations (RPV) affects the rotation splittings: asymmetries (Aj)

Study of RPV through Aj is a promising tool to search for the true RP.

2mm

jS

Suárez et al. 2006, in prep.

02 mmmjA

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Analysis of splitting asymmetries & RPV

Aj cannot be explained by UR models DR models in the approximation of shellular

rotation.

Suárez et al. 2006, in prep.

S3: Smoothed Prof.S2: Shellular-typeS1: Linear-type.

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Analysis of splitting asymmetries & RPV

Aj(g) behaves differently to Aj(p) Al of low-order g & p modes are the most affected

RPV near the core (μ-gradient zone).

Suárez et al. 2006, in prep

S1

S3

S2

S1

S2S3

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Analysis of splitting asymmetries & RPV

Asymmetries are directly proportional to the second –order term D0

Analysis of D0 Kernels (work in progress) will help to understand the behaviour of Aj:

• Mode dependency (n,ℓ)• RPV effects

Suárez et al. 2006, in prep.

dxxKxA Dj )()(0

1

0

2

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Bibliography

Zahn et al. 1992 A&A 265, 115 Dziembowski & Goode 1992 A&A Goupil et al. 2003 ESA Eddington Conference, SP Soufi et al. 1998 A&A 334, 911 Suárez et al. 2006 A&A 449, 673

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Analysis of splitting asymmetries & RPV

Low order, low-degree modes (g & mixed) modes are the most affected by variations near the core & the μ-gradient zone

Suárez et al. 2006, in prep

20-23 November 2006

Joint Helas & CoRoT/ESTA Workshop

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Analysis of splitting asymmetries & RPV

RPV modify the asymmetry of rotationally split multiplets

Suárez et al. 2006