The effect of a magnetic field on the radiative excitation and - PDF document

The effect of a magnetic field on the radiative excitation and damping of p-modes Hideyuki Saio Abstract A rapidly oscillating Ap star pulsates in high-oder p-modes under the influence of a strong magnetic field. The strong field distorts spatially the angluar and radial the pulsa- tion amplitude (eigenfunction). To study the effect of the magnetic field on the radiative excitation and damping of p-modes, we performed a fully nonadiabatic analysis including the effect of a dipole magnetic field. A mag- netic field always tends to stabilze low oder p-modes. For high-order p-modes, on the other hand, the magnetic field enhances kappa-mechanism excitation in some range of the field strength, depending on the pulsation frequency. 1

❘ ✂ ✮ ✽ ❆ ✏ ✱ ✰ ✱ ✿ ✽ ✑ ✏ ✮ ❅ ✓ ✄ ✆ ❄ ❯ ◗ ❲ ✿ P ◗ P ❅ ✏ ■ P ❅ ✏ ❍ ❘ ✝ ✱ ✆ ✂ P ✂ ✬ P P ❆ ✂ ✰ ❱ ✏ ❏ P ✝ ✽ ✂ ✁ ❲ P ✿ ✁ � ✍ ✎ ✏ ✣ P ✽ ✓ ❘ ✝ ✂ ✏ ✱ ✏ P ❯ ✑ ✝ ✑ ✏ ❄ ✑ ✱ Rapidly Oscillating Ap (roAp) stars: ✂☎✄ min Periods; ✂ mHz) ✆✞✝✠✟ (Freq.; Magnetic fields: a few kG High-order p-modes under a strong magnetic field Balmforth et al. (2001) ✝☛✡✌☞ Unperturbed model: main-sequence star ✝✠✒✔✓✖✕ ✆✘✗ Convection is suppressed. ✙✛✚✢✜ ✂✤�✦✥ ✙✛✚✢✜ ✧✩★✫✪ ✙✛✚✢✜ ✝✭✡ ✝✠✆✯✟ Dipole magnetic field: ✵✾✽❀✿❂❁ ✽✤❆ ✲✴✳ ✚✌❃✞❄ ❃❈❇❊❉❋❄●✺ ✵✷✶✹✸ ✺✼✻ ✑ ) modes Nonadaibatic analysis for axisymmetric ( in terms of a series expansion. ❚✤❯ ❏▲❑◆▼✹❖ ✿❙❘ ❚❳❯ ❑◆▼✹❖ 2

✏ ✁ � The nonmagnetic situation 2 stable 0 unstable -2 0.5 1 1.5 2 2.5 3 Pulsation frequency versus damping rate for the 2nd to ✂ p-modes in the absence of magnetic the 40th order field. The kappa-mechanism in the He ionization zone excites low order (3rd to 7th) p-modes, while the kappa- mechanism in the H-izonization zone excites three (28th– 30th) high order-modes below the critical frequency. 3

� ✁ ❚ ✲ Magnetic damping on low-order modes 2 1 stable 0 unstable -1 0 2 4 6 8 Damping rate versus the strength of magnetic fields for the low-order modes which are excited in the absence of magnetic field. Due to the magnetic damping caused by slow waves, all the Scuti type pulsations are sup- ✂ kG. pressed if is larger than 4

✺ ✂ � ❚ ✲ ✂ ✄ ✝ ✬ ✲ ❚ ✺ ✂ ✏ ❚ ✲ Comparison adivabatic vs nonadiabatic high-order modes I 2.01 2 1.99 1.98 1.97 1.96 0.04 0.02 stable 0 unstable -0.02 -0.04 0 2 4 6 8 Pulsation frequency (upper panel) and damping rate (lower panel) as functions of for the 29th order p-mode of ✂ . Filled and open circles show data from nona- � ✁� diabatic and adiabatic analyses, respectively. Compared with the adiabatic situation, the frequency jumps (damping- rate peaks) lie at different field strengths in the nonadia- ✵ ☎✄✝✆ batic case. This mode is unstable for ✵ ☎✄✞✆ and ✆ . 5

❚ ✺ ✲ ✂ ✄ ✂ � ✏ ✳ ✲ ✝ Comparison adivabatic vs nonadiabatic high-order modes II 1.81 1.8 1.79 1.78 1.77 0.04 0.02 stable 0 unstable -0.02 -0.04 0 2 4 6 8 The same as the previous figure but for the 26th p-mode. ✑ , but becomes marginally This mode is stable at ✄ . ✵ ☎✄✝✆ unstable in a range of 6

✂ ✥ ✑ � ✝ ✂ ✑ ✑ ✝ ✏ ✞ ❚ ✲ ❄ ✂ ✏ � � ✂ ✏ Latitudinal amplitude distribution The modulus of the latitudinal amplitude distribution of ✄ . This the 29th order p-mode of at ✑ ✂✁☎✄ s ✁✝✆ . ✝✠✒ mode is excited with a growth rate of Thick and thin lines refer to the photosphere and the outer boundary (at ✆ ), respectively. The amplitude is ✥✩✄ ✠✟ . Compared to essentially confined in a range of the adiabatic case at minimum damping, the confinment to the magnetic axis is less pronounced. 7