1. Entropy in the ICM Institute for Computational Cosmology University of Durham Michael Balogh

2. Collaborators Mark Voit (STScI -> Michigan) –Richard Bower, Cedric Lacey (Durham) Greg Bryan (Oxford) Ian McCarthy, Arif Babul (Victoria)

3. Outline Review of ICM scaling properties, and the role of entropy Cooling and heating The origin of entropy Lumpy vs. smooth accretion and the implications for groups

4. ICM Scaling properties

5. Luminosity-Temperature Relation If cluster structure were self-similar, then we would expect L  T 2 Preheating by supernovae & AGNs?

6. Mass-Temperature Relation Cluster masses derived from resolved X-ray observations are inconsistent with simulations Another indication of preheating? M  T 1.5

7. Definition of S:  S =  (heat) / T Equation of state:P = K  5/3 Relationship to S:S = N ln K 3/2 + const. Useful Observable:Tn e -2/3  K Characteristic Scale: Convective stability: dS/dr > 0 Only radiative cooling can reduce Tn e -2/3 Only heat input can raise Tn e -2/3 Entropy: A Review K 200 = T 200  m p (200f b  cr ) 2/3

8. Dimensionless Entropy From Simulations Simulations without cooling or feedback show nearly linear relationship for K(M gas ) with K max ~ K 200 Independent of halo mass (Voit et al. 2003) Simulations from Bryan & Voit (2001) Halos: 2.5 x 10 13 - 3.4 x 10 14 h -1 M Sun

9. Entropy profiles Entropy profiles of Abell 1963 (2.1 keV) and Abell 1413 (6.9 keV) coincide if scaled by T 0.65 Scaled entropy: (1+z) 2 T -1 S Scaled entropy: (1+z) 2 T -0.66 S Radius (r 200 ) Pratt & Arnaud (2003)

10. Heating and Cooling

11. Preheating? Preheated gas has a minimum entropy that is preserved in clusters Kaiser (1991) Balogh et al. (1999) Babul et al. (2002) Isothermal model M=10 15 M 0 K o =400 keV cm 2 300 200 100

12. Balogh, Babul & Patton 1999 Babul, Balogh et al. 2002 log 10 L X [ergs s -1 ] kT [keV] 10 1 0.1 40424446 Isothermal model Preheated model K o =400 keV cm 2

13. Does supernova feedback work? Local SN rate ~0.002/yr (Hardin et al. 2000; Cappellaro et al. 1999) An average supernova event releases ~10 44 J Assuming 10% is available for heating the gas over 12.7 Gyr, total energy available is 2.5x10 50 J This corresponds to a temperature increase of 5x10 4 K To achieve a minimum entropy K 0  T/  2/3 :  /  avg = 0.28 (K 0 /100 keV cm 2 ) -3/2 Consider the energetics for 10 11 M sun of gas: SN energy too low by at least a factor ~50

14. Core Entropy of Clusters & Groups Core entropy of clusters is  100 keV cm 2 at r/r vir = 0.1 Self-similar scaling Entropy “Floor” Ponman et al. 1999

15. Entropy Threshold for Cooling Each point in T-Tn e -2/3 plane corresponds to a unique cooling time

16. Entropy Threshold for Cooling Entropy at which t cool = t Hubble for 1/3 solar metallicity is identical to observed core entropy! Voit & Bryan (2001)

17. Entropy History of a Gas Blob no cooling, no feedback cooling & feedback Gas that remains above threshold does not cool and condense. Gas that falls below threshold is subject to cooling and feedback. Voit et al. 2001

18. Entropy Threshold for Cooling Updated measurements show that entropy at 0.1r 200 scales as K 0.1  T 2/3 in agreement with cooling threshold models Voit & Ponman (2003)

19. L-T and the Cooling Threshold Gas below the cooling threshold cannot persist Voit & Bryan (2001) Balogh, Babul & Patton (1999) Babul, Balogh et al. (2002) log 10 L X [ergs s -1 ] kT [keV] 10 1 0.1 40424446 Also matched by preheated, isentropic cores

20. L-T and the Cooling Threshold Gas below the cooling threshold cannot persist Voit & Bryan (2001) Balogh, Babul & Patton (1999) Babul, Balogh et al. (2002) log 10 L X [ergs s -1 ] kT [keV] 10 1 0.1 40424446 Also matched by preheated, isentropic cores

21. Mass-Temperature relation Both pre-heating and cooling models adequately reproduce observed M-T relation ● Reiprich et al. (2002) Babul et al. (2002) Voit et al. (2002)

22. The overcooling problem Balogh et al. (2001) Observations imply  * /  b  0.05 f cool 0.1 0.6 0.5 0.4 0.3 0.2 Fraction of condensed gas in simulations is much larger, depending on numerical resolution Pearce et al. (2000) Lewis et al. (2000) Katz & White (1993) kT (keV) 110 Observed fraction

23. Heating-Cooling Tradeoff Many mixtures of heating and cooling can explain L-T relation If only 10% of the baryons are condensed, then ~0.7 keV of excess energy implied in groups Voit et al. (2002)

24. Heating + Cooling McCarthy et al. in prep Start with Babul et al. (2002) cluster models, which have isentropic cores Allow to cool for time t in small timesteps, readjusting to hydrostatic equilibrium after each step Develops power-law profile with K  r 1.1

25. Entropy profiles of CF clusters McCarthy et al. in prep Observed cooling flow clusters show entropy gradients in core Well matched by dynamic cooling model from initially isentropic core Observations Model

26. Simple cooling+heating models McCarthy et al. in prep Data from Horner et al., uncorrected for cooling flows

27. Simple cooling+heating models McCarthy et al. in prep Data from Horner et al., uncorrected for cooling flows Non-CF clusters well matched by preheated model of Babul et al. (2002) CF cluster properties matched if gas is allowed to cool for up to a Hubble time

28. The origin of entropy Voit, Balogh, Bower, Lacey & Bryan ApJ, in press astro-ph/0304447

29. Important Entropy Scales K 200 = T 200  m p (200f b  cr ) 2/3 Characteristic entropy scale associated with halo mass M 200 K sm = v 2 acc  (4  in ) 2/3 Entropy generated by accretion shock  (Mt) 2/3 (d ln M / d ln t) 2/3

30. Dimensionless Entropy From Simulations How is entropy generated initially? Expect merger shocks to thermalize energy of accreting clumps But what happens to the density? (Voit et al. 2003) Simulations from Bryan & Voit (2001) Halos: 2.5 x 10 13 - 3.4 x 10 14 h -1 M Sun

31. Smooth vs. Lumpy Accretion Smooth accretion produces ~2-3 times more entropy than hierarchical accretion (but similar profile shape) SMOOTH LUMPY Voit et al. 2003

32. Preheated smooth accretion If pre-shock entropy K 1 ≈K sm, gas is no longer pressureless = (M 2 -1) 2 M 2 4 8/3 K sm 5 K 1 K 2 ≈ K sm + 0.84K 1, for K sm /K 1 » 0.25 + 0.84K 1 v in 2 3(4  1 ) 2/3 ≈ Note adiabatic heating decreases post-shock entropy

33. Lumpy accretion Assume all gas in haloes with mean density  f b  cr K(t) ≈ (  1 /  f b  cr ) 2/3 K sm (t) ≈ 0.1 K sm (t) Two solutions: K  v in 2 /  1. distribute kinetic energy through turbulence (i.e. at constant density) 2. v sh ≈ 2 v ac (i.e. if shock occurs well within R 200 )

34. Preheating and smooth accretion M(t o )=10 14 h -1 M o K mod K sm K 200 K1K1 K c (T 200 ) 2 1.5 1 0.5 0 0 0.5 1 f g =M g /f b M 200 K (10 34 erg cm 2 g- 5/3 ) 2 1.5 1 0.5 0 0 0.5 1 f g =M g /f b M 200 K (10 34 erg cm 2 g- 5/3 ) M(t o )=10 13 h -1 M o K mod K sm K 200 K1K1 K c (T 200 ) Voit et al. 2003 Early accretion is isentropic; leads to nearly-isentropic groups

35. Entropy gradients in groups

36. Entropy in groups Entropy profiles of Abell 1963 (2.1 keV) and Abell 1413 (6.9 keV) coincide if scaled by T 0.65 Cores are not isentropic Scaled entropy (1+z) 2 T -1 S Scaled entropy (1+z) 2 T -0.66 S Radius (r 200 ) Pratt & Arnaud (2003)

37. Excess entropy in groups Entropy “measured” at r 500 (~ 0.6r 200 ) exceeds the amount hierarchical accretion can generate by hundreds of keV cm 2

38. Entropy gradients in groups 1 10 0.1 1 10 T lum (keV) L x /T 3 lum (10 42 h -3 erg s -1 keV -3 M o =5×10 13 h -1 M o  eff =5/3  eff =1.2 100 1000 0.1 1 10 1000 K(0.1r 200 ) keV cm 2 Voit et al. 2003

39. Excess entropy at R 200 Entropy gradients in groups with elevated core entropy naturally leads to elevated entropy at R 200 Voit et al. 2003  eff = 1.2  eff = 1.3 ≈ 2.6 K(R 200 ) K 200 (d ln M / d ln t) -2/3 ≈ 3.5 for 10 13 h -1 M o ≈ 1.7 for 10 15 h -1 M o

40. Excess Entropy at R 500 Entropy “measured” at r 500 (~ 0.6r 200 ) exceeds the amount hierarchical accretion can generate by hundreds of keV cm 2

41. Smooth accretion on groups?  Groups are not isentropic, but do match the expectations from smooth accretion models  Relatively small amounts of preheating may eject gas from precursor haloes, effectively smoothing the distribution of accreting gas.  Self-similarity broken because groups accrete mostly smooth gas, while clusters accrete most gas in clumps

42. Conclusions Feedback and cooling both required to match cluster properties and condensed baryon fraction Smooth accretion models match group profiles Difficult to generate enough entropy through simple shocks when accretion is clumpy Similarity breaking between groups and clusters may be due to the effects of preheating on the density of accreted material