Inference for the internal rotation profile of stars based on - PowerPoint PPT Presentation

Inference for the internal rotation profile of stars based on dipolar modes of oscillations —2 general theoretical comments and 2 stars— Masao Takata Department of Astronomy, School of Science, University of Tokyo 13 September 2016

Introduction: inference for the internal rotation of stars ◮ one of the hottest topics in asteroseismology ◮ subgiants and red giants ◮ main-sequence stars (early-type and solar-type) ◮ dipolar modes ◮ most easily observed nonradial modes of stellar oscillation ◮ play a major role in constraining the internal rotation of stars (other than the Sun) m = − 1 m = 0 m = +1

Introduction: rotational splitting m = − 1 m = 0 m = 1

Theory: effects of slow rotation ✓ ✏ rotational splitting ¯ Ω n , 1 δν n , 1 = β n , 1 2 π ✒ ✑ ◮ average rotation rate (2-d problem) � R � π ¯ Ω n , 1 = K n , 1 ( r , θ ) Ω ( r , θ ) d θ d r 0 0 ◮ (global) sensitivity of each mode � R ( ξ r − ξ h ) 2 ρ r 2 d r 0 β n , 1 = 1 − C n , 1 = � R ρ r 2 d r ξ 2 r + 2 ξ 2 � � h 0

Theory: special characteristics of the kernel of dipolar modes ✓ ✏ The rotation kernel is multiplicatively separable. K n , 1 ( r , θ ) = 3 K n , 1 ( r ) sin 3 θ ˆ 4 ✒ ✑ ◮ reduction of the 2-d problem to the 1-d problem, � R ¯ K n , 1 ( r ) ˆ ˆ Ω n , 1 = Ω ( r ) d r , 0 with the model-independent angular average � π Ω ( r ) = 3 Ω ( r , θ ) sin 3 θ d θ . ˆ 4 0 ◮ straightforward reinterpretation of the 1-d problem that assumes Ω depends on only the radius. Question: Is there any physical reason for the separability?

Theory: dipolar modes in the absence of rotation ✓ ✏ reduced displacement The essential part of the dipolar oscillation can be described by the reduced displacement � ζ . ◮ formulation in terms of the (normal) displacement � ξ leads to the 4th-order ordinary differential equation, ◮ from which a 2nd-order equation of � ζ can be separated ◮ impacts on ◮ mode classification � ζ = � ξ − � r g ◮ asymptotic analysis ✒ ✑

Theory: slow-rotation effect on dipolar modes All quantities about the slow-rotation analysis can be expressed by the reduced displacement in place of the normal displacement. ✓ ✏ ξ r → ζ r , ξ h → ζ h ✒ ✑ ( ζ r − ζ h ) 2 ρ r 2 ˆ K n , 1 ( r ) = � R ( ζ r − ζ h ) 2 ρ r 2 d r 0 � R ( ζ r − ζ h ) 2 ρ r 2 d r 0 β n , 1 = � R ζ r2 + 2 ζ h2 � ρ r 2 d r � 0

KIC 11145123: a δ Sct– γ Dor hybrid with clear rotational splittings Kurtz et al. (2014) ◮ A type (terminal-age) main-sequence star [ T eff = 8050 ± 200 K ; log g = 4 . 0 ± 0 . 2 (cgs) ] ◮ nearly rigid rotation with a period of ∼ 100 day, but with the slightly faster rotating envelope than the core “I have never seen such a simple spectrum.” by Don in Tokyo 2013

KIC 11145123: clear splittings �� � � �� �

KIC 11145123: questions ◮ Why such unusually slow rotation? ◮ The physical reason for the faster rotating envelope? ✓ ✏ ◮ How reliable is the argument for the faster rotating envelope? ✒ ✑

KIC 11145123: the envelope rotates faster than the core! model-independent argument ☛ ✟ ◮ observation: 2 δν g < δν p (according to Don) ✡ ✠ ¯ Ω δν = (1 − C n , 1 ) 2 π ☛ ✟ C n , 1 < 1 2 for high-order g modes ◮ ✡ ✠ ⇒ 2 δν g is the upper limit of the core rotation rate ☛ ✟ C n , 1 > 0 for p modes ◮ ✡ ✠ ⇒ δν p is the lower limit of the envelope rotation rate ✓ ✏ [core rotation rate] < 2 δν g < δν p < [envelope rotation rate] ✒ ✑

KIC 11145123: C n , 1 < 1 2 for high-order g modes Asymptotic analysis tells that C n , 1 approaches 1 2 from below in the low frequency limit. 0.55 ���������������� C n,1 0.50 0.45 1 10 Frequency [d -1 ]

KIC 11145123: C n , 1 > 0 for low-order p modes? Most evolutionary calculations support this. � Exceptions in the polytropic models with large indices

KIC 9244992: another δ Sct– γ Dor hybrid with clear splittings Saio et al. (2015) ◮ F type main-sequence star T eff = 6900 ± 300 K, log g = 3 . 5 ± 0 . 4 (cgs) ◮ clear series of high-order g-mode triplets ◮ nearly uniform rotation with a period of ∼ 65 days, with the core rotating 4 % faster than the envelope

KIC 9244992: a problem of the non-equally split triplets ☛ ✟ s rot = ν 2 s = ν m =1 + ν m = − 1 rot − ν m =0 40 ν (e.g. Dziembowski & Goode 1992) 2 ✡ ✠ 1.0 0.5 s [10 − 4 d − 1 ] 0.0 -0.5 -1.0 s rot -1.5 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Frequency [d − 1 ] The second-order rotation effect cannot explain the observation.

Summary Inference of the stellar internal rotation based on dipolar modes ◮ theory ◮ multiplicative separability of the rotation kernel ◮ complete description of the effects of slow rotation on the frequencies based on the reduced displacement ◮ two particular stars ◮ KIC 11145123: model-independent argument for the faster rotating envelope than the core ◮ KIC 9244992: an unresolved problem about the non-equally split triplets of the high-order g modes

Asymptotic formula for the second-order rotation effects s = ν m =1 + ν m = − 1 − ν m =0 2 ✓ ✏ s rot = ν 2 rot 40 ν ✒ ✑ (e.g. Brassard et al. 1989; Dziembowski & Goode 1992) ◮ high-order dipolar g modes ◮ uniform rotation ◮ negligible centrifugal distortion

Asymptotic rotation inversion of red giants based on dipolar mixed modes with no reference model Goupil et al. 2013; Deheuvels et al. 2015 ✓ ✏ δν = ζ Ω core + (1 − ζ ) ¯ ¯ Ω envelope 2 ✒ ✑ � − 1 1 + ν 2 ∆Π 1 � sin (2 θ g ) ζ = ∆ ν sin (2 θ p ) � − 1 √ � √ � N � N ¯ ˆ Ω core = 2 r d r 2 Ω d r r G G � − 1 � ˆ �� d r Ω ¯ Ω envelope = c d r c P P confirmed by the revised asymptotic analysis of dipolar modes (Takata 2016), which takes account of the strong interaction between the core and envelope oscillations