1. Estimate* the Total Mechanical Feedback Energy in Massive Clusters Bill Mathews & Fulai Guo University of California, Santa Cruz *~ ±15-20% version 2

2. estimate feedback energy from potential energy of gas after each feedback heating event cluster gas expands and feedback energy becomes PE compare PE of gas between: (1) observed gas profiles in clusters (2) idealized gas profiles in adiabatic clusters evolved to zero redshift without: radiative cooling non-gravitational feedback energy star formation compare  PE of (1) and (2) at same M gas ( r) this determines feedback energy < r independent of the time when feedback occurred

3. why this works: (1) NFW dark halo and adiabatic gas grow from inside out (2) PE is integrated from inside out Diemand+07 cluster potential  ( r) remains constant within r v (t) total cluster potential cluster gas density constant-mass radii during halo formation

4. observed gas profiles in relaxed clusters Vikhlinin+06 observed gas fraction f g =  g /  tot  tot gg consider pairs of similar galaxies:

5. compare NFW and adiabatic cluster gas profiles density  dispersion  entropy     Faltenbacher+07 using GADGET2  dm f b /(1-f b ) gg beyond small core, gas density is NFW:  g = f b  t gas entropy: S g =  g   g   g = [3kT/  m p ] 1/2 (thermal dispersion) dark matter entropy: S dm =  dm   dm   dm = 3D velocity dispersion S dm ~ r 1.2 S g ~ r 1.2 S g = (0.70 +/- 0.25) S dm (Faltenbacher+07) S g ≈ S dm => gas and dm experience identical gravitational dissipation NFW dm gas

6. adiabatic cluster gas profiles gg  dm grid-based adiabatic cosmogical simulations mix more and have larger density cores in cluster gas Vazza 11 adopt two limiting assumptions for adiabatic  g ( r): universal baryon (1) no core: fraction  g ( r) = f b  t,nfw ( r) f b = 0.17 (2) with core:  g ( r) = c( r)f b  t,nfw ( r) NFW total cluster density

7. adiabatic cluster atmosphere (without density core) total NFW cluster profile  t ( r) for observed M v and c(M v )

8. adiabatic cluster atmosphere ignoring density core, adiabatic gas profile is scaled NFW  ( r) = f b  t ( r) = 0.17  t ( r)  ( r) contains all information about dissipative entropy-increasing events in filaments, accretion shock, and mergers

9. adiabatic cluster atmosphere using  ( r) = f b  t ( r), integrate hydrostatic equation for temperature  and entropy S: entropy S ad ( r) ~ r 1.2 (a point-slope boundary value problem) a uniform slope near r vir is the boundary condition, but its value is not imposed in advance.

10. observed cluster atmosphere  obs ( r) = f g (r )  t,nsf ( r) (Vikhlinin+06) gas fraction for composite cluster 2 (A478 & A1413)

11. observed cluster atmosphere fit to observed gas density profile:

12. observed cluster atmosphere using  obs ( r) integrate again for observed gas temperature  ( r) and entropy S obs ( r) which resembles observations: Pratt+10 S ad ( r)

13. how to recover universal adiabatic S ad ( r) ~ r 1.2 from S obs ( r) Pratt+10 (assume no significant heating by recent feedback) S obs ( r) is more sensitive to low  (from old feedback) than high T (from recent feedback heating)

14. total feedback energy is similar, with or without core small effect of core in adiabatic density  ad ( r)

15. total feedback energy  |PE| ≈ 1-3 x 10 63 ergs M v = 4x10 14 r v = 1.9 Mpc M v = 1x10 15 r v = 2.7 Mpc 10 63 ergs = 5 x 10 8 M sun c 2 is huge! L mech ≈ 10 46 erg/s over t cl = 7 Gyrs from central black hole? is spin energy needed? (McNamara+09)  obs or  ad gas outflow due to feedback (spreads metals)

16. review some assumptions for clusters (1,2): 1. ignore stellar baryon fraction f * : for massive clusters (1,2) f * = 0.01 is small (Andreon10) total stellar mass r < r 500 = (0.25, 0.65)x10 13 total gas mass flowing out beyond r 500 = (1.9, 3.8)x10 13 2. feedback energy ~10 63 ergs is from central black hole (a) total supernova energy is small: E SNII = (0.03, 0.1)x10 63 ergs in r < r v E SNIa = (0.03, 0.1)x10 63 ergs in r < r v (b) energy lost by radiation E rad is small: at cooling radius r cool = (98, 120) kpc cooling time equals age of cluster t cl ~ 7 Gyrs E rad = L X (r cool )t cl = (0.03, 0.1)x10 63 ergs (c ) most energetic known single AGN event is < 10 63 ergs E ~ 10 62 ergs (McNamara+05)

17. estimated feedback stops cooling flows! rate that unheated gas cools and flows in at r cool : cluster (1,2) 2 1 (unrelated to feedback estimate)

18. estimated feedback stops cooling flows! rate that unheated gas cools and flows in at r cool : rate that gas flows out at r due to feedback: t cl = 7 Gyrs  M( r) t cl an excellent independent check of feedback estimate < 1% of feedback energy is deposited inside r cool cluster (1,2) 2 1 2 1

19. other recent Guo-Mathews feedback results: dynamical jet models of  -ray emitting Fermi bubbles in Milky Way theory for expanding radio lobes in Virgo -- explains bright radio rims Galactic coords. VLA 90 cm b (degrees) l (degrees) projected image of (electron) cosmic ray energy density -- with viscosity in co-mixed plasma and CR diffusion 10 kpc

20. other recent Guo-Mathews feedback results: Six images of (unprojected) CR energy density with increasing viscosity in co-mixed plasma: viscosity suppresses instabilities and makes IC image uniform

21. other recent Guo-Mathews feedback results: Smooting effect of CR diffusion, increasing from left to right top 3 images: unprojected CR energy density in kpc bottom 3 images: projected CR energy density in Galactic coords. (viscosity held constant) kpc b (degrees)