Excitation of Oscillations in the Sun and Stars

Bob Stein - MSU

Dali Georgobiani - MSU

Regner Trampedach - MSU

Martin Asplund - ANU

Hans-Gunther Ludwig - Lund

Aake Nordlund - Copenhagen

P-Mode Excitation

P-modes are excited by PdV work of turbulent and non-adiabatic gas

pressure fluctuations,= Reynolds stresses and

entropy fluctuations

P-modes are excited by PdV work of turbulent and non-adiabatic gas

Pressure fluctuations,= Reynolds stresses and

Entropy fluctuations

P-Mode Excitation

Pressure fluctuation Mode compression

Mode energy

Δ⟨Eω⟩Δt

=

ω2 drδPω* (∂ξω/∂r)

r∫

2

8ΔνEω

δP=δPturb+δPgasnad

δPturb=δ ρδVz2 , δPgas

nad=P δ lnP−Γ1δ lnρ( )

Eω =12ω2 drρξω

2∫rR

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

Eigenfunction

P-Mode Excitation Alternatives

dEωdt

=ω2 d3r∫dξdr

2

h2dh0

hmax

∫ τh ρvh2

( )2+

∂P∂s

⎛

⎝ ⎜

⎞

⎠ ⎟ ρ

δsh⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Goldreich, Murray & Kumar, 1994

dEdt

=1

8Md3x∇ i∫ ξj∇ lξm dτe∫

−iωτd3∫ r

× ρ2ui'uj

'ul"um

" +δijδlm q'q"

[ ]

q=∂P∂s

⎛

⎝ ⎜

⎞

⎠ ⎟ ρ

∇ • stu→⎛

⎝ ⎜

⎞ ⎠ ⎟

Samadi & Goupil, 2001M= d3∫ xρξ

2

Use Convection Simulation

to Evaluate Excitation

Computation

• 3D, Compressible

• EOS includes ionization

• Solve– Conservation equations

• mass, momentum & internal energy

– Induction equation– Radiative transfer equation

• Open boundaries– Fix entropy of inflowing plasma at bottom

Method

• Spatial derivatives - Finite difference– 6th order compact or 3rd order spline

• Time advance - Explicit– 3rd order predictor-corrector

• Diffusion∂f∂t

⎛

⎝ ⎜

⎞

⎠ ⎟ diffusive

=∇ •αν∇f

α =max|Δ3 f |−1,0,1( )

max|Δf |−1,0,1( )

Radiation Transfer

• LTE

• Non-gray - multi-group

• Formal Solution Calculate J - B by integrating Feautrier equations along one vertical and 4 slanted rays through each grid point on the surface.

Stratified convective flow:diverging upflows, turbulent downflows

Velocity arrows, temperature fluctuation image (red hot, blue cool)

Vorticity

Downflows are turbulent, upflows are more laminar.

Vorticity Distribution

Down

Up

Fluid Parcelsreaching

the surface Radiate

away their

Energy and

Entropy

Z

SE

Q

Entropy

Green & blue are low entropy downflows, red is high entropy upflows

Entropy Distribution

P-Mode Oscillations:Stochastic Excitation

Nordlund & Stein, ApJ, 546, 576, 2001Stein & Nordlund, ApJ, 546, 585, 2001

P-Mode Excitation

Triangles = simulation, Squares = observations (l=0-3)Excitation decreases both at low and high frequencies

P-Mode Excitation

Mode energy

Δ⟨Eω⟩Δt

=

ω2 drδPω* (∂ξω/∂r)

r∫

2

8ΔνEω

Eω =12ω2 drρξω

2∫rR

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

Mode Mass

Mode mass increases toward low frequencies, because low frequency modes penetrate deeper

P-Mode Excitation

Mode compression

Δ⟨Eω⟩Δt

=

ω2 drδPω* (∂ξω/∂r)

r∫

2

8ΔνEω Eigenfunction

Mode Compression

Mode compression decreases toward low frequencies, reduces low frequency excitation.

P-Mode Excitation

Pressure fluctuation

Δ⟨Eω⟩Δt

=

ω2 drδPω* (∂ξω/∂r)

r∫

2

8ΔνEω

δP=δPturb+δPgasnad

δPturb=δ ρδVz2 , δPgas

nad=P δ lnP−Γ1δ lnρ( )

Pressure Fluctuations

Pressure fluctuations decrease toward high frequency,Reduces high frequency excitation.

P-Mode excitation

• Decreases at low frequencies because of mode properties:– mode mass increases toward low frequencies– mode compression decreases toward low

frequencies

• Decreases at high frequencies because of convection properties:– Turbulent and non-adiabatic gas pressure

fluctuations produced by convection and convective motions are low frequency.

P-Mode Excitation

Excitation primarily by downflowsdown & up flows interfere destructively

Other Stars

Excitation Spectra

Decreasing gIncreaseing Teff

Reynolds Stress vs. Entropy Fluctuations

Star A Sun

Eta Boo

Excitation Pturbulent Pnon-ad gas

Excitation (log g, Teff)

P-Mode Excitation

• Excitation increases with decreasing gravity

• Excitation increases with increasing effective temperature

• Excitation by turbulent pressure is comparable to excitation by non-adiabatic gas pressure (entropy) fluctuations

Vz =Fconvℜgas

2ρCpμ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1/3

MLT

The End