Geo- and Astro-physical Fluid Dynamics Seminar

April 2013 talk at UC Santa Cruz

Transcript

1. ￿ermal Instability and Gravity Waves in Galaxy Clusters Mike McCourt, Eliot Quataert, Prateek Sharma, & Ian Parrish April, ￿￿￿￿
2. “￿ermal Instability and Gravity Waves in Clusters” ￿ Galaxy Cluster Primer ￿ Linear and Nonlinear ￿ermal Instability ￿ Astrophysical Applications ￿ Future Directions
3. Abell 370 (optical) Representative Numbers ￿ R ∼ 3 − 7 × 106 ly ￿ M ∼ 1014 − 1015 M⊙ ￿ T ∼ 5 × 107 K ￿ n ∼ 10−2 cm−3 ￿ Lx ∼ 1045 erg/s
4. Introduction ￿ermal Instability Applications Including Conduction Conclusion Cluster Mass Budget stars (optical) hot gas (x-ray)
5. Introduction ￿ermal Instability Applications Including Conduction Conclusion Cluster Mass Budget stars (optical) hot gas (x-ray) dark matter (lensing)
6. Introduction ￿ermal Instability Applications Including Conduction Conclusion Cluster Mass Budget stars (optical) hot gas (x-ray) Cluster Galaxy Kravtsov 2009 dark matter halos are largely self-similar…
7. Introduction ￿ermal Instability Applications Including Conduction Conclusion Cluster Mass Budget stars (optical) hot gas (x-ray) Andreon 2010 …but clusters have more gas and fewer stars than galaxies.
8. Introduction ￿ermal Instability Applications Including Conduction Conclusion Non-Self-Similarity If the gas properties scale with the dark matter: ￿ ρ ∼ M0 ￿ T ∼ M/r ∼ M2/3 ￿ L ∼ ρ2T1/2r3 ∼ T2 T2 T3 Gas in the centers of clusters has lower density and higher entropy than gravitational self-similar models predict. (more so for lower masses.)
9. Introduction ￿ermal Instability Applications Including Conduction Conclusion Multi-Phase Gas in Clusters xray optical Fabian et al. 2011 Some clusters have neutral, ~￿￿￿ K gas colocated with the ionized, ~￿￿￿ K plasma!
10. How does cool gas form in clusters?
11. How does cool gas form in clusters? (And why doesn’t it always form?)
12. Introduction ￿ermal Instability Applications Including Conduction Conclusion ￿ermal Instability in Clusters Conselice et al. 2005 Θ ≡ Net cooling rate ￿ ∂Θ ∂T ￿ P < ￿ ￿ermally unstable (Field ￿￿￿￿) Assume local thermal instability
13. Introduction ￿ermal Instability Applications Including Conduction Conclusion Linear ￿ermal Instability Standard WKB analysis: ￿ initial equilibrium with H = L everywhere. ￿ L ∝ n2T1/2 for clusters. ￿ assume H ∝ nα Subtlety: gas is globally stabilized against cooling (see Balbus & Soker ￿￿￿￿). cooling timescale: pTI = ￿ 3 2 − α ￿ (γtcool) −1 (tcool ≡ E/L) dispersion relation: ω2 + iωpTI − N2 k2 ⊥ k2 = 0 ω →    −ipTI , tcool short ±N − ipTI 2 , tcool long
14. Introduction ￿ermal Instability Applications Including Conduction Conclusion How Does ￿ermal Instability Saturate? (Assuming it exists…) HQ; 10 (ρ/ρ0 ) xf> t+QQH/t77 = 1/10 tf> −R y R y R k j −k −R y R HQ; 10 (ρ/ρ0 ) xf> t+QQH/t77 = 10 tf> −R y R y R k j −R.y −y.8 y.y ￿ermal Instability does not necessarily imply Multi-phase gas.
15. Introduction ￿ermal Instability Applications Including Conduction Conclusion How Does ￿ermal Instability Saturate? (Assuming it exists…) HQ; 10 (ρ/ρ0 ) xf> t+QQH/t77 = 1/10 tf> −R y R y R k j −k −R y R HQ; 10 (ρ/ρ0 ) xf> t+QQH/t77 = 10 tf> −R y R y R k j −R.y −y.8 y.y ￿ermal Instability does not necessarily imply Multi-phase gas. See cold gas when tcool /tff ￿ 10
16. Introduction ￿ermal Instability Applications Including Conduction Conclusion Quantifying the Saturation t/tTI ￿δρ/ρ￿rms 0 5 10 15 10 −3 10 −2 10 −1 1 10 Linear Theory tTI/tff = 1/10 tTI/tff = 1 tTI/tff = 10 ￿ Perturbations initially grow exponentially… ￿ …but saturate at an amplitude ∝ (tcool /tff) −1 ￿ ￿is is a non-linear effect
17. Introduction ￿ermal Instability Applications Including Conduction Conclusion Quantifying the Saturation t/tTI ￿δρ/ρ￿rms 0 5 10 15 10 −3 10 −2 10 −1 1 10 Linear Theory tTI/tff = 1/10 tTI/tff = 1 tTI/tff = 10 ★ ★ ★ ★ ★ tti/tff ￿δρ/ρ￿rms 10 −1 1 10 10 −2 10 −1 1 10
18. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation “Sinking Blobs” model (i. e. no buoyancy) Long cooling-time limit: δρ/ρ ￿ 1 What stops the instability? One possibility: sinking blobs mix into surroundings g t Joung et al. 2011
19. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation “Sinking Blobs” model (i. e. no buoyancy) Assume blobs survive for ∼ one scale-height: tsink ∼ H vsink + Archimedes’ principle: dv dt ∼ δρ ρ g ∼ vsink tsink ∼ v2 sink H + steady-state: t−1 cool ∼ t−1 sink g t Joung et al. 2011
20. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation “Sinking Blobs” model (i. e. no buoyancy) ⇒ δρ ρ ∼ ￿ tcool tff ￿−2 too steep! g t Joung et al. 2011
21. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation “Sinking Blobs” model (i. e. no buoyancy) thA/t77 = 10 xf> −k −R y R k j −R.y −y.8 y.y thA/t77 = 3 −R.yy −y.d8 −y.8y −y.k8 thA/t77 = 1 −R.yy −y.d8 −y.8y −y.k8 thA/t77 = 1/10 HQ; 10 (ρ/ρ0 ) −k −R y R xf> tf> −j −k −R y R k j @j @k @R y R k j −y.k y.y y.k tf> −j −k −R y R k j −y.8 y.y y.8 tf> −j −k −R y R k j @RXy yXy RXy δρ/ρ tf> −j −k −R y R k j @RXy yXy RXy Perturbations don’t look like sinking blobs. Buoyancy is probably important.
22. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation “Sinking Blobs” model (i. e. no buoyancy) Assume blobs survive for ∼ one scale-height: tsink ∼ H vsink + Archimedes’ principle: dv dt ∼ δρ ρ g ∼ vsink ✞ ✝ ☎ ✆ tsink ∼ v2 sink H + steady-state: t−1 cool ∼ t−1 sink g t Joung et al. 2011
23. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation Including buoyancy… “strong” turbulence: tblob ∼ (kvk) −1 momentum equation: dv dt ∼ δρ ρ g ∼ v tbuoy steady state: t−1 cool ∼ t−1 blob thA/t77 = 10 xf> −k −R y R k j −R.y −y.8 y.y thA/t77 = 3 −R.yy −y.d8 −y.8y −y.k8 thA/t77 = 1 −R.yy −y.d8 −y.8y −y.k8 thA/t77 = 1/10 HQ; 10 (ρ/ρ0 ) −k −R y R xf> tf> −j −k −R y R k j @j @k @R y R k j −y.k y.y y.k tf> −j −k −R y R k j −y.8 y.y y.8 tf> −j −k −R y R k j @RXy yXy RXy δρ/ρ tf> −j −k −R y R k j @RXy yXy RXy ⇒ δρ ρ ∼ ￿ tcool tff ￿−1 ￿ tff tbuoy ￿
24. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation ★ ★ ★ ★ ★ tti/tff ￿δρ/ρ￿rms 10 −1 1 10 10 −2 10 −1 1 10
25. Applications to Cluster and Galaxy Formation
26. Introduction ￿ermal Instability Applications Including Conduction Conclusion Hα Filaments Conselice et al. 2005 Expect multiphase gas when tcool /tff ￿ 10.
27. Introduction ￿ermal Instability Applications Including Conduction Conclusion Hα Filaments ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ✽ ▲ r (Mpc) tcool /tff 10 −3 10 −2 10 −1 1 10 10 2 10 3 Extended Hα No Extended Hα min(tcool ) (Gyr) min(tcool /tff ) Abell 133 Abell 478 Abell 496 Sersic 159-03 Abell 1991 Abell 2597 Abell 1795 Hydra A Centaurus Abell 85 Abell 644 Abell 4059 Abell 1650 Abell 2029 Abell 2142 Abell 744 10 −2 10 −1 1 10 1 10 10 2 Extended Hα No Extended Hα x-ray data from the ￿￿￿￿￿￿ catalog (Cavagnolo et al. ￿￿￿￿) Hα data from McDonald et al. (￿￿￿￿, ￿￿￿￿)
28. Introduction ￿ermal Instability Applications Including Conduction Conclusion Triggering AGN Feedback ￿ermal Instability Develops Cold Gas Accretes Residual Cold Gas No ￿ermal Instability 10 kpc Sharma, McCourt et al. 2012a
29. Introduction ￿ermal Instability Applications Including Conduction Conclusion Triggering AGN Feedback Feedback and cooling self-regulate to the critical threshold for non-linear thermal stability: min(tcool /tff ) ∼ 10
30. Introduction ￿ermal Instability Applications Including Conduction Conclusion Non-Self-Similarity ￿e condition min(tcool /tff) ∼ 10 implies a maximum gas density as a function of halo mass. Sharma, McCourt et al. 2012b
31. What about thermal conduction?
32. Introduction ￿ermal Instability Applications Including Conduction Conclusion Including Conduction λF ∝ (ˆ b ⋅ ˆ k) × (χet￿ɪ )￿￿￿ e 5), to the point where the magnetic pressure n the filaments. The regions over which the ed are coincident with, but significantly longer on of the cold filaments. The field enhancement freezing as the cooling plasma is compressed o the initial field direction in the nonlinear state by contrast, in one at 1.43 Gyr (Figur state in two dimen regions to becom (because of the s 0 x(kpc) t=0.95 Gyr 0 20 40 −1 −0.5 0 x(kpc) z(kpc) t=0.475 Gyr 0 20 40 0 20 40 −0.13 −0.12 −0.11 −0.1 plots of log10 temperature (in keV) for the fiducial run (MWC) at linear (0.475 Gyr; left) he arrows show the magnetic field direction. this figure is available in the online journal.) Sharma et al. 2010 ✽ ✽ ✽ ➙ ➙ ➙ ✽ ➙ ￿ anisotropic isotropic ￿ tχ /tff tTI /tff Cold Fraction (by Mass) 10 −1 1 10 −2 10 −1 1 10 √ 2 √ 2 √ 2/10 10 √ 2 √ 2
33. Introduction ￿ermal Instability Applications Including Conduction Conclusion Including Conduction, cont. tχ /t77 = ∞ xf> @k @R y R k j tχ /t77 = 10 √ 2 tχ /t77 = √ 2 tχ /t77 = √ 2/4 tχ /t77 = √ 2/10 thA/t77 = 10−0.75 xf> @k @R y R k j thA/t77 = 10−0.25 tf> xf> −j −k −R y R k @j @k @R y R k j tf> −j −k −R y R k tf> −j −k −R y R k tf> −j −k −R y R k tf> thA/t77 = 1 j −j −k −R y R k
34. Introduction ￿ermal Instability Applications Including Conduction Conclusion Summary ￿ermal Instability ￿ Assuming that the ICM is thermally unstable, multi-phase gas forms only when tcool /tff ￿ 10. ￿ We can understand this threshold in terms of the non-linear saturation of the instability. ￿ Cooling and feedback self-regulate to the threshold for non-linear stability. Applications ￿ermal instability enables us to understand: ￿ Why some clusters show filaments of cool gas while others don’t, ￿ How AGN feedback is triggered/regulated, and ￿ Why the gas profiles in clusters are not self-similar
35. Introduction ￿ermal Instability Applications Including Conduction Conclusion Future Directions? Clusters ￿ Gravity waves sourced by cooling? Other Objects ￿ Coronae (e. g. in the sun or accretion disks) may also have local thermal instability.
36. Backup Slides
37. Introduction ￿ermal Instability Applications Including Conduction Conclusion Atomic Gas in Clusters Hydra A Abell ￿￿￿ Abell ￿￿￿￿ McDonald et al. 2010
38. Introduction ￿ermal Instability Applications Including Conduction Conclusion Particle Mean Free Path Spitzer-Härm collisionless q0 ￿ Bale et al. ￿￿￿￿ ￿ Derived from the ￿￿ electron distribution function measured with the NASA Wind spacecraft.
39. Introduction ￿ermal Instability Applications Including Conduction Conclusion Heating Fluctuations thA/t77 = 10 xf> R k j thA/t77 = 1 thA/t77 = 1/10 6B/m+BH xf> R k j RyyW 6Hm+iX tf> xf> −R y R y R k j tf> −R y R tf> jyyW 6Hm+iX −R y R
40. Introduction ￿ermal Instability Applications Including Conduction Conclusion Feedback Regulation Feedback and cooling self-regulate to the critical threshold for non-linear thermal stability: min(tcool /tff ) ∼ 10
41. Introduction ￿ermal Instability Applications Including Conduction Conclusion Physics of the Saturation Including buoyancy… “weak” turbulence: twave ∼ (kvg) −1 tshear ∼ (kvk) −1 velocity perturbation: ￿ δv v ￿ int ∼ (shear rate) × (int. time) ∼ twave tshear ∼ vk vg ￿ 1 add incoherently: tblob ∼ ￿ vg vk ￿2 twave ￿ twave steady state: t−1 blob ∼ t−1 cool momentum equation: dv dt ∼ δρ ρ g ∼ v tbuoy ⇒ δρ ρ ∼ ￿ tcool tff ￿−1/2 ￿ tff tbuoy ￿3/2
42. Introduction ￿ermal Instability Applications Including Conduction Conclusion MTI Temperature (t = 0) Temperature (t = 5 tbuoy) HBI Temperature (t = 5 tbuoy) ∆T
43. Introduction ￿ermal Instability Applications Including Conduction Conclusion t = 0 t = 4 tbuoy t = 6 tbuoy t = 30 tbuoy t = 0 t = 7.5 tbuoy t = 20 tbuoy t = 40 tbuoy
44. Introduction ￿ermal Instability Applications Including Conduction Conclusion t = 0 t = 3 tbuoy t = 12 tbuoy t = 17 tbuoy t = 50 tbuoy Temperature 1.0 1.05