Estimate* the Total Mechanical Feedback Energy
in Massive Clusters
Bill Mathews & Fulai Guo
University of California, Santa Cruz
*~ ±15-20%
version 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 Mgas ( r)
this determines feedback energy < r
independent of the time when feedback occurred
why this works:
(1) NFW dark halo and adiabatic gas grow
from inside out
constant-mass radii
during halo formation
cluster potential ( r) remains
constant within rv(t)
Diemand+07
(2) PE is integrated from inside out
cluster gas density
total cluster potential
observed gas profiles in relaxed clusters
tot
g
observed gas fraction fg = g/tot
consider pairs of similar galaxies:
Vikhlinin+06
compare NFW and adiabatic cluster gas profiles
dispersion 
density 
dm
gas
entropy 
dm
dm
Sdm ~ r1.2
gas
gas
Sg ~ r1.2
dmfb/(1-fb)
g
NFW
beyond small core, gas density is NFW: g = fbt
gas entropy:
Sg = gg
Faltenbacher+07 using GADGET2
g = [3kT/mp]1/2 (thermal dispersion)
dark matter entropy: Sdm = dmdm dm = 3D velocity dispersion
Sg = (0.70 +/- 0.25) Sdm (Faltenbacher+07)
Sg ≈ Sdm => gas and dm experience identical gravitational
dissipation
adiabatic cluster gas profiles
grid-based adiabatic cosmogical simulations mix more and
have larger density cores in cluster gas
dm
adopt two limiting assumptions
for adiabatic g( r):
universal baryon
(1) no core:
fraction
g
NFW
g( r) = fbt,nfw( r)
fb = 0.17
(2) with core:
Vazza 11
g( r) = c( r)fbt,nfw( r)
total cluster density
adiabatic cluster atmosphere (without density core)
total NFW cluster profile t( r)
for observed Mv and c(Mv)
adiabatic cluster atmosphere
ignoring density core,
adiabatic gas profile is scaled NFW
( r) = fbt( r) = 0.17t( r)
( r) contains all information about
dissipative entropy-increasing events
in filaments, accretion shock, and mergers
adiabatic cluster atmosphere
using ( r) = fbt( r),
integrate hydrostatic equation
for temperature  and entropy S:
(a point-slope boundary value problem)
entropy Sad( r) ~ r1.2
a uniform slope near rvir
is the boundary condition,
but its value is not imposed
in advance.
observed cluster atmosphere
gas fraction for
composite cluster 2
(A478 & A1413)
obs( r) = fg(r )t,nsf( r)
(Vikhlinin+06)
observed cluster atmosphere
fit to observed gas density profile:
observed cluster atmosphere
using obs( r) integrate again
for observed gas temperature
( r) and entropy
Sobs( r) which resembles observations:
Sad( r)
Pratt+10
how to recover universal adiabatic Sad( r) ~ r1.2 from Sobs( r)
(assume no significant heating
by recent feedback)
Pratt+10
Sobs( r) is
more sensitive to low  (from old feedback)
than
high T (from recent feedback heating)
small effect of core in adiabatic density ad( r)
total feedback energy is similar, with or without core
total feedback energy |PE| ≈ 1-3 x 1063 ergs
obs or ad
Mv = 4x1014
rv = 1.9 Mpc
1063 ergs = 5 x 108 Msun c2 is huge!
Lmech ≈ 1046 erg/s over tcl = 7 Gyrs
from central black hole?
is spin energy needed? (McNamara+09)
Mv = 1x1015
rv = 2.7 Mpc
gas outflow
due to feedback
(spreads metals)
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 < r500 = (0.25, 0.65)x1013
total gas mass flowing out beyond r500 = (1.9, 3.8)x1013
2. feedback energy ~1063 ergs is from central black hole
(a) total supernova energy is small:
ESNII = (0.03, 0.1)x1063 ergs in r < rv
ESNIa = (0.03, 0.1)x1063 ergs in r < rv
(b) energy lost by radiation Erad is small:
at cooling radius rcool = ( 98, 120) kpc
cooling time equals age of cluster tcl ~ 7 Gyrs
Erad = LX(rcool)tcl = (0.03, 0.1)x1063 ergs
(c ) most energetic known single AGN event is < 1063 ergs
E ~ 1062 ergs (McNamara+05)
estimated feedback stops cooling flows!
rate that unheated gas cools and flows in at rcool:
(unrelated to feedback estimate)
cluster (1,2)
2
1
estimated feedback stops cooling flows!
rate that unheated gas cools and flows in at rcool:
rate that gas flows out at r due to feedback:
tcl = 7 Gyrs
cluster (1,2)
2
1
M( r)
tcl
< 1% of feedback energy
is deposited inside rcool
2
1
an excellent
independent check
of feedback estimate
other recent Guo-Mathews feedback results:
dynamical jet models of -ray emitting
Fermi bubbles in Milky Way
b (degrees)
10 kpc
theory for expanding radio lobes
in Virgo -explains bright radio rims
l (degrees)
VLA 90 cm
Galactic coords.
projected image of (electron) cosmic ray
energy density -- with viscosity
in co-mixed plasma and CR diffusion
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
other recent Guo-Mathews feedback results:
b (degrees)
kpc
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)