A new concept in stellar astrophysics A new concept in stellar astrophysics based on internal rotationbased on internal rotation::

Effective mass and its place Effective mass and its place in the A- and B-star puzzlein 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 : -0.7Λs (Stothers, 1974)-0.9Λs (Claret & Gimenez,1993)-0.7Λs (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 129929 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 Teff = 0.025

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 Teff = 0.025

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 Msun:

From the rotating and NR models:

Combination of these equations

The effective mass of PV Cas A

For the metal rich chemical composition: Meff = 2.58 Msun

For the solar composition: Meff = 2.60 Msun

The MS life-time and the effective mass

NR models

M=2.82 Msun, t(MS) =280 My

M=2.57 Msun t(MS) =370 My

DR model of PV Cas A

M=2.82 Msun t(MS) = 370 My (Meff = 2.58 Msun )

The mass-luminosity relation

Observational M- L: dlogL / dlogM= 2.3

The minimum value obtained from the models (ZAMS):

dlogL / dlogM= 3.9 - 4.0

For DR models, but, Meff in place of M:

dlogL / dlogMetkin = 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.