A new concept in stellar astrophysics based on internal rotation: Effective mass and its place in the A- and B-star puzzle Mutlu Yıldız Ege University, Dept. of Astronomy and Space Sciences, Turkey

The basic effect of rotation: One dimensional hydrostatic equilibrium:

For the case of solid-body rotation: Correction in log k2 : -0.7Λ s (Stothers, 1974) -0.9Λ s (Claret & Gimenez,1993) -0.7Λ s (Yıldız 2004) From model properties

For the case of differential rotation: Similar equations for L and R as a function of ’rotation parameter’?

Profile of the rotation rate:  =  (r) Steep rotation rate gradient near the surface of PV Cas’ components :

Is such a DR reasonable? Motivations If angular momentum transportation is not a sudden process, such a DR can be anticipated, at least for some time. Time-scale of decay of differential rotation in radiative envelopes of Cp stars ~ life-time of A-type stars (Arlt et al. 2003) Aerts et al. (2003) ruled out rigid rotation of β Cep type star HD as a result of its seismic analysis. The temperature difference between the blue sides of magnetic Ap stars and normal stars (Hubrig et al. 2000).

Hubrig et al. (2000) The temperature difference between the blue sides of magnetic Ap stars and normal stars dlog T eff = For T=10000 K, dT=590 K For T= 9000 K, dT=530 K Compare NR models with DR models with steep rotation rate gradient near the surface!

Hubrig et al. (2000) The temperature difference between the blue sides of magnetic Ap stars and normal stars dlog T eff = For T=10000 K, dT=590 K For T= 9000 K, dT=530 K Compare NR models with DR models with steep rotation rate gradient near the surface!

NR and DR models Temperature difference between the ZAMS lines = 550 K

Luminosity level as a function of rotation parameter Consider the simplest case: Homogeneous mass distribution Integrate the equation of Hydrostatic equilibrium :  = constant

The effective mass and rotation parameter Ideal gas pressure Average rotation parameter Effective mass for homogeneous mass distribution: Luminosity primarily depends on it rather than the real mass

Luminosity vs. average rotation parameter NR, SBR and DR models (t is constant).

For more realistic mass distribution From the models of 2.55 and 2.82 M sun : From the rotating and NR models: Combination of these equations

The effective mass of PV Cas A For the metal rich chemical composition: M eff = 2.58 M sun For the solar composition: M eff = 2.60 M sun

The MS life-time and the effective mass NR models M=2.82 M sun, t(MS) =280 My M=2.57 M sun t(MS) =370 My DR model of PV Cas A M=2.82 M sun t(MS) = 370 My (M eff = 2.58 M sun )

The mass-luminosity relation Observational M- L: dlogL / dlogM= 2.3 The minimum value obtained from the models (ZAMS): dlogL / dlogM= For DR models, but, M eff in place of M: dlogL / dlogM etkin = 3.6

Results Rapid rotation of the inner regions of stars can solve some fundamental problems in the stellar astrophysics. Therefore, we introduce effective mass as a novel approach. It may have also cosmological implications! If we find the mass of an early type star from its brightness, the mass we find is primarily its effective mass The effective mass of PV Cas A is about 10% less than its real mass Internal rotation can be the dominant discriminator for the chemically peculiar stars Irregular mass-luminosity relations may be due to internal rotation

Conclusion The nature is always much more complicated than we anticipate, but, the effective mass may help us to find some more pieces of the puzzle.