Radiative Rayleigh-Taylor instabilities

Emmanuel Jacquet (ISIMA 2010)

Mentor: Mark Krumholz (UCSC)

Outline

I. Introduction and motivation

II. Fundamentals and generalities

III. The (very) optically thin limit

IV. The (very) optically thick limit

V. Conclusion

I. Introduction and motivation

Classical Rayleigh-Taylor instability

• Two immiscible liquids in a gravity field

• If denser fluid above unstable (fingers).

Motivation 1: massive star formation

• Radiation force/gravity ~ Luminosity/Mass of star.

• >1 for M>~20-30 solar masses.

• But accretion goes on… (Krumholz et al. 2009) : radiation flows around dense fingers.

Motivation 2: HII regions

• Neutral H swept by ionized H

• Radiative flux in the ionized region RT instabilities?

And more!

II. Fundamentals and generalities

The general setting

Width Δz of interfaceignored.

z=0+- - - -z=0-

Equations of non-relativistic RHD

gas

Radiation

Rate of 4-momentum transfer from radiation to matter

Energy

Momentum

Linear analysis: the program (1/2)• Dynamical equations:

• Perturbation:

• Search for eigenmodes:

• Eulerian perturbation of a quantity Q:

• If Im(ω) > 0: instability!

• Lagrangian perturbation:

Linear analysis: the program (2/2)

• Perturbation equations still contain z derivatives:

• Everything determined at z=0 so should dispersion relation.

• Importance of boundary conditions.

Boundary conditions

• Normal flux continuity at interface in its rest frame:

• From momentum flux continuity:

• Perturbations vanish at infinity.

z>0

z<0

≈ 0

III. The (very) optically thin limit

Absorption and reradiation in an optically thin medium

• Higher opacity for UV photons dominate force

Radiative equilibrium

Hard photon attenuation

visible near infrared

So we should solve:

Let us simplify…

with:

?

Isothermal media with a chemical discontinuity

• Discontinuity in sound speed.• Assume ρ-independent opacity and constant

F in each region

constant T and effective gravity field:

• Constant 2x2 matrix A:

eff

Instability criterion

• (Pure) instability condition:

• Dispersion relation:

• Growth rates:

Ex. ofunstableconfiguration

with:

1

2

IV. The (very) optically thick limit

Optically thick limit

• Radiation Planckian at gas T (LTE)

• Radiation conduction approximation.

• Total (non-mechanical) energy equation:

• Conditions:

Meet A again:

with:

Adiabatic approximation

• Rewrite energy equation as:

• If we neglect Δs=0.

• …under some condition:

with

« Reduced » set of equations

with:

Perturbations evanescent on a scale height

• A traceless must be eigenmode of A:

• Pressure continuity:

Rarefied lower medium

• Dispersion relation: in full:

• In essence:

• Really a bona fide Rayleigh-Taylor instability!

Unstable if g>0

Domain of validity

Not local

Not adiabatic

No temperature locking

Not optically

thick

E=x=1

Window if:

Convective instability?

So what about massive star formation?• Flux may be too high for

« adiabatic RTI »

• But if acoustic waves unstable : « (RHD) photon bubbles » (Blaes & Socrates 2003)

• In dense flux-poor regions, « adiabatic RTI » takes over.

growth time a/g (i.e. 1-10 ka).

• Tentative only…

Summary: role of radiation in Rayleigh-Taylor instabilities & Co.

Characteristic length/photon mean free path

1

OPTICALLY THICKOPTICALLY THIN adiabaticisothermal

<< 1 >> 1

Radiation modifies EOS, with radiation force lumped in pressure gradient

Radiation as effective gravity(« equivalence principle violating »)

Flux sips in rarefied regions: buoyant photon bubbles (e.g. Blaes & Socrates 2003)