On the stability of protoplanetary disks Niklas Ehlert Supervisors: - - PowerPoint PPT Presentation

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May 19, 2023 273 likes •530 views

When the world was still flat - On the stability of protoplanetary disks Niklas Ehlert Supervisors: Hubert Klahr (MPIA) and Jrgen Schaffner-Bielich Astrocoffee FIAS, Frankfurt, January 23, 2018 Credit: ALMA (ESO/NAOJ/NRAO) Source:

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“When the world was still flat…“ - On the stability of protoplanetary disks

Niklas Ehlert

Supervisors: Hubert Klahr (MPIA) and Jürgen Schaffner-Bielich Astrocoffee FIAS, Frankfurt, January 23, 2018

Credit:ALMA (ESO/NAOJ/NRAO)

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Source: https://www.youtube.com/watch?v=E4yirtvUurA (January 5, 2018, 22:20 h)

January 23, 2018 Niklas Ehlert 2

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January 23, 2018 Niklas Ehlert 3 Credit: NASA Credit: ALMA (ESO/NAOJ/NRAO) Credit: NASA Johnson Space Center Credit: NASA/JPL/University of Arizona

size: ~ ⋅ 1013 mass: ~ ⋅ 1040

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Topics

Content-wise: Protoplanetary disks

properties
evolution

Method-wise: Linear Stability Analysis

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Assumptions

𝑁𝑒𝑗𝑡𝑙 ≪ 𝑁∗
scale height ℎ satisfies ℎ

𝑠 ≪ 1

neglect self-gravity

𝑁𝑒𝑗𝑡𝑙 𝑁∗

<

1 2 ℎ 𝑠 (Toomre criterion)

“passive disks” → heated by central star

→ slow accretion rates (𝑁 ≤ 2 ⋅ 10−8 𝑁𝑡𝑣𝑜

𝑧𝑠 )

→ late times in disk evolution

January 23, 2018 Niklas Ehlert 5

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Disk evolution

angular momentum must be transported:

→ winds → viscosity 𝜉

Navier-Stokes equation yields:
Shakura & Sunyaev (1973)
Lynden-Bell & Pringle (1974)

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𝜖Σ 𝜖𝑢 = 3 𝑠 𝜖 𝜖𝑠 𝑠

1 2 𝜖

𝜖𝑠 (𝜉Σ𝑠1/2)

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Disk evolution equation

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𝜖Σ 𝜖𝑢 = 3 𝑠 𝜖 𝜖𝑠 𝑠

1 2 𝜖

𝜖𝑠 (𝜉Σ𝑠1/2)

Diffusive form: 𝜖𝑔 𝜖𝑢 = 𝐸 𝜖2𝑔 𝜖𝑌2

𝑌 = 2𝑠1/2 𝑔 =

3 2 Σ𝑌 = 3Σ𝑠1/2

𝐸 = 12𝜉 𝑌2 = 3𝜉 𝑠

Diffusion time scale: Δ𝑌 2

𝐸

⇒ viscous time scale: 𝜐𝜉 ≈ 𝑠2

𝜉

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Viscosity?

𝛽-prescription:
Molecular collisions: 𝜉𝑛~𝜇𝑑𝑡 =

𝑑𝑡 𝑜𝜏

⇒ 𝜐𝜉 = 𝑠2 𝜉𝑛 ≈ 3 ⋅ 1013𝑧𝑠

Reynolds number: 𝑆𝑓 =

𝑉𝑀 𝜉𝑛 = 𝑑𝑡ℎ 𝜉𝑛 ~1010 ≫ 1

⇒ highly turbulent IF there is an instability

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𝜉 = 𝛽𝑑𝑡ℎ

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Rayleigh criterion

A non-self-gravitating, non-magnetized disk flow is linearly stable to axisymmetric perturbations if 𝑒𝑚

𝑒𝑠 = 𝑒 𝑒𝑠 𝑠2Ω > 0

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Ω ∝ 𝑠−3/2

⇒ no turbulence?!

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Magnetorotational Instability

Chandrasekhar 1961, Balbus & Hawley 1991
Modified stability condition: 𝑙𝑤𝐵 2 +

𝑒Ω2 𝑒ln 𝑠 > 0

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[1]

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Dead zones

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[1]

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Hydrodynamical Instabilities

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Gravitational Instability Vertical Shear Instability Subcritical Baroclinic Instability

Goldreich- Schubert-Fricke Instability Critical Layer Instability Convective Overstability Streaming Instability?

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Convective Overstability

neglect vertical structure:

𝜍0 = 𝜍 0 ⋅ 𝑆 𝑆0

𝛾𝑟

, 𝑑𝑡

2 = 𝑑 𝑡 2 ⋅

𝑆 𝑆0

𝛾𝑞

, 𝑞 = 𝑑𝑡

2𝜍

Consider thermal relaxation: 𝑇 = 𝑞

𝑈 𝑈−𝑈 𝜐

𝜐 → 0: locally isothermal limit 𝜐 → ∞: adiabatic limit

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Worst slide in this talk…

𝜖𝑢 (𝜍0 + 𝜍′) + 𝜍0 + 𝜍′ 𝜖𝑦 𝑣′ + 𝑣′ 𝜖𝑦 (𝜍0 + 𝜍′) = 0 𝜖𝑢 𝑣′ + 𝑣′ 𝜖𝑦 𝑣′ + 1 𝜍0 + 𝜍′ 𝜖𝑦 (𝑞0 + 𝑞′) + 𝜖𝑦 Φ = 0 𝜖𝑢 𝑞0 + 𝑞′ + 𝑣′ 𝜖𝑦 (𝑞0 + 𝑞′) + 𝛿 𝑞0 + 𝑞′ 𝜖𝑦 𝑣′ = 0

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𝜖𝑢 𝜍′ + 𝜍0 𝜖𝑦 𝑣′ + 𝑣′ 𝜖𝑦 𝜍0 = 0 𝜖𝑢 𝑣′ + 𝜖𝑦𝑞′ 𝜍0 − 𝜍′ 𝜍0

2 𝜖𝑦 𝑞0 = 0

𝜖𝑢 𝑞′ + 𝑣′ 𝜖𝑦 𝑞0 + 𝛿 𝑞0 𝜖𝑦 𝑣′ = 0

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Dispersion relation

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Eigenvectors

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𝒘 = 𝜍′, 𝑣𝑠

′ , 𝑣𝜚 ′ , 𝑣𝑨 ′ , 𝑞′ 𝑈 𝑁 𝜕 ⋅ 𝒘 = 0

[3]

⇒

verstable epicycle

growth rate

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Numerical Simulations

test non-linear instabilities
PLUTO code (Mignone et al.):

– discretization, static or adaptive mesh – finite volume or finite difference approach – 1D, 2D or 3D simulations

shows vortex formation and growth

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Credit: ALMA (ESO/NAOJ/NRAO)

Summary

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…hydrodynamic instabilities operate ... Pp-disks are turbulent. MRI creates turbulence, but… … there are dead zones, where… … under baroclinic conditions and finite cooling times to create vortices.

Credit: N. Raettig

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Next talk in about 6 months…

Thank you and stay tuned!

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Credit: ALMA (ESO/NAOJ/NRAO)

Summary

January 23, 2018 Niklas Ehlert 21

…hydrodynamic instabilities operate ... Pp-disks are turbulent. MRI creates turbulence, but… … there are dead zones, where… … under baroclinic conditions and finite cooling times to create vortices.

Credit: N. Raettig

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Sources

[1] Philip J. Armitage: „Astrophysics of Planet Formation“, Cambridge University Press, New York, USA, 2013 [2] https://upload.wikimedia.org/wikipedia/ commons/d/d4/Johannes_Kepler_1610.jpg (January 6, 2018, 19:05 h) [3] Lyra W., 2014, ApJ, 789, 77

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Backup

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