metzler@ ophysics, e, KS 66045 enc gy Astr awr or, MI 48109 rb du nn A atory for High Ener or elt, MD 20771 ab enb emp erature Correlation from a e ay-5.gsfc.nasa.gov d@xr hard F. Mushotzky onomy, University of Kansas, L [email protected] Christina M. Bird Ric tbir de 666, Gr richar Co al Journal Limited Cluster Sample e Flight Center, L ac dSp ophysic View metadata,citationandsimilarpapersatcore.ac.uk y Disp ersion { T ddar 〉 Astr PostScript processed by the SLAC/DESY Libraries on 5 May 1995. d: ASTRO-PH-9505020 elo cit eive c e artment of Physics and Astr artment of Physics, University of Michigan, A R yp es NASA/Go Dep Christopher A. Metzler Dep The V h all p ossess y and optical Submitted to the y disp ersion and elo cit een v w h the study of X-ra en from Bird (1994), whic d: y and optical prop erties of galaxy clusters e approac epte 1 c c ailable, regardless of the morphological t v A Abstract een X-ra ; w olution, w aluate the relationship b et eev en the increasing evidence that morphology is related to a .W Giv tly e used the largest samples of data a Most studies of correlations b et v temp erature for a limited set of galaxy clusters tak correlations di eren cluster's degree of dynamical ev of clusters included. ha provided byCERNDocumentServer brought toyouby CORE
dominant central galaxies and whichhave b een explicitly corrected for the presence of
0:610:13
substructure. We nd that / T .We use a Monte Carlo computer routine to
r
0:5
estimate the signi cance of this deviation from the / T relationship predicted by
r
the virial theorem. We nd that the simulated correlation is steep er than the observed
value only 4% of the time, suggesting that the deviation is signi cant. The combination
of protogalactic winds and dynamical friction repro duces nearly exactly the observed
relationship b etween and T .
r
1 Intro duction
Galaxy clusters o ccupy a unique p osition in the dynamical evolution of the universe.
Unlikelower-mass systems such as galaxies, which for the most part retain little dynamical
information ab out their formation, clusters of galaxies are within one or two crossing times of
their formation. This suggests that they may retain valuable clues to their initial conditions
(as well as hints ab out the collapse and formation of structure in the early universe). The
e ect of the dense cluster environment on galaxy evolution, as well as other trends in the
physical prop erties of clusters (see, for instance, Dressler 1984; Giovanelli & Haynes 1985;
Edge & Stewart 1991), suggests that they are gravitationally b ound and that their galaxies
no longer participate in the Hubble ow. This distinguishes clusters from sup erclusters and
other large-scale structures. The study of galaxy clusters thus provides a unique opp ortunity
to explore gravitational interactions and dynamical evolution in the universe.
Clusters of galaxies contain two luminous comp onents, hot gas and galaxies. If a clus-
ter is suciently old and unp erturb ed, these tracer particles will have equilibrated within
the cluster gravitational p otential. This enables use of the equations of hydrostatic and
dynamical equilibrium to explore the physical prop erties of these systems. For a hot gas in
equilibrium with a spherical gravitational p otential, the equation of hydrostatic equilibrium 2
may b e written
kT r dlnn dlnT
gas
M (r )= ( + ) (1)
X
Gm dlnr dlnr
(e.g. Fabricant, Lecar & Gorenstein 1981), where M is the X-ray determined virial mass,
X
T is the temp erature of the X-ray emitting gas, n is the gas density andm is the average
gas
mass p er gas particle. Similarly, the Jeans equation relates the kinetic energy of the galaxies
to the virial mass of the cluster:
2
Gn M (r ) d(n ) 2n
gal opt gal gal
r
2 2
= + (1 = ) (2)
r t
2 2
r dr r
r
2
d(n ) 2n
gal gal
r
= + A
2
dr r
r
(Merritt 1987), where M is the optically-determined virial mass, r is the clustercentric
opt
radius, n is the galaxy density, and are the radial and tangential velo city disp ersions
gal r t
resp ectively, and A is the anisotropy parameter describing the distribution of galaxy orbits.
For an isothermal cluster in dynamical equilibrium, with no source of energy other than
gravity, the masses as determined by the galaxies and by the gas are exp ected to b e equal.
As shown by Bahcall & Lubin (1994) among others, the ratio of the kinetic energies of the
galaxies and gas is then equal to the ratio of the logarithmic slop e of the gas density pro le
to that of the galaxies:
2
dlnn =d l n r
gas
r
= : (3)
kT
dlnn =d l n r +2A
gal
m
dlnn =d l n r
gas
=
dlnn =d l n r
gal
(where A = 0 for an isotropic distribution of galaxy orbits). Therefore, using the assumptions
that the gas and galaxies are b oth in equilibrium with the cluster gravitational p otential,
and that gravity is the only source of energy, allows us to predict that the velo city disp ersion
(as measured from galaxy velo cities) and the temp erature of the intracluster medium (as
0:5
determined from X-ray sp ectra) should b e correlated, with / T . The ratio of the
r 3
kinetic energies is called . The ratio of the logarithmic slop es of the density pro les is
spec
.
fit
Despite the many diculties in accurately measuring cluster temp eratures and velo c-
ity disp ersions, studies of X-ray and optical cluster samples reveal a well-b ehaved corre-
lation b etween these quantities (Mushotzky 1984; Edge & Stewart 1991, hereafter ES91;
Lubin & Bahcall 1993, hereafter LB93). The relationship b etween and T exp ected from
r
virial considerations is consistent with the data, although there is a large scatter ab out the
0:5
/ T line. This scatter has b een attributed to incomplete gas thermalization, co oling
r
ows, velo city anisotropies in the galaxy orbits, foreground/background contamination, and
substructure in the clusters (cf. ES91; LB93 and references therein).
It is imp ortant to rememb er, however, that the predicted T correlation derives
r
from the virial theorem, and that in order to test it one must consider the dynamical state
of the clusters in the dataset (cf. Gerbal et al. 1994). The high frequency of substructure
in clusters of all morphologies, as determined by b oth X-ray and optical studies (see, e.g.
Davis & Mushotzky 1993; Mohr, Fabricant & Geller 1993; Beers et al. 1991; Bird 1993,
1994), is generally b elieved to indicate that clusters are dynamically-young. If clusters are
only within a few crossing times of formation, then in many cases virial equilibrium has
not b een established. This certainly in uences the broad distribution of clusters ab out the
0:5
canonical / T relation.
r
In this pap er we will quantify the e ects of morphology and substructure on the velo city
disp ersion-temp erature correlation for clusters. In Section 2 we present the limited cluster
sample, in which the morphological typ e of the cluster sample has b een restricted and the
e ects of substructure have b een minimized. Wehave supplemented the available published
X-ray temp erature data with new, more accurate temp eratures from ASCA and Ginga.In
Section 3 we present the regressions b etween the velo city disp ersion and temp erature. Section 4
4 summarizes prop osed mechanisms for mo difying the slop e of the T correlation. In
r
Section 5 we present a summary. 5
2 The Limited Cluster Sample
The morphology of a cluster may b e describ ed by its gas and/or galaxy distribution.
As our observations of clusters have improved, it has b ecome clear that morphology is re-
lated to the dynamical age of a cluster. Irregular clusters are dynamically young, and tend
to b e spiral-rich and gas-p o or. They tend to have non-Gaussian velo city distributions and
kinematically-distinct sub concentrations of galaxies. Regular clusters are dominated by el-
lipticals, have Gaussian velo city distributions and tend to b e luminous X-ray emitters (cf.
Sarazin 1988 and references therein; Bird 1993,1994).
Bird (1994) presents a detailed analysis of the dynamics of nearby clusters (z<0:1)
with central galaxies. These clusters tend to have smo oth morphologies and X-ray co oling
ows, and in the past it has b een assumed that they represent the most relaxed, dynamically-
evolved clusters in the universe. However, Bird (1994) shows that these clusters also p ossess
signi cant substructure. An ob jective partitioning algorithm called KMM (McLachlan &
Basford 1988; Ashman, Bird & Zepf 1994) is used to remove galaxies b elonging to subsystems
in the clusters, and the dynamical prop erties of the \cleaned" (i.e., substructure corrected)
cluster datasets are presented. It is the 25 clusters in this \cD database" which form the
optical sample of the present analysis.
Of the 25 clusters used in Bird (1994), 21 have accurate X-ray temp erature measure-
ments. These clusters, which will b e referred to as the limited cluster sample, are listed in
Table 1. Table 1 includes the following information: column (1), the cluster name; (2), the
1-D velo city disp ersion of the cluster (estimated using the robust biweight estimator S ,
BI
Beers, Flynn & Gebhardt 1991) without substructure correction; (3), the velo city disp ersion
corrected for substructure; (4), the X-ray temp erature, (5) the source co de for the X-ray
measurement. The optical redshifts are taken from the literature, with sources given in Bird
(1994). In addition wehave added the Centaurus Cluster (A3526), whichwas excluded 6
from the cD study b ecause of its proximity. The X-ray temp eratures are taken from single-
temp erature mo dels to ASCA or Ginga sp ectra where available, and then from EXOSAT and
the Einstein MPC. For the clusters A1736 and A3558, the GINGA observations are b est- t
bya two-temp erature mo del (Day et al. 1991), in contradiction to b oth the Einstein and
ROSAT sp ectra. Because the data are inconclusive, wehave included b oth temp eratures in
Table 1 for these two clusters, and we will consider them b oth in the statistical analysis.
Note that the velo city disp ersion presented here is measured only along our line of sight
to the cluster. We assume for the moment that anyvelo city anisotropy in these clusters is
small and therefore is comparable to (we will explore this assumption in more detail
LO S r
b elow).
In Table 2 we present the individual values of for the limited cluster sample, b oth
spec
with and without substructure correction. With no substructure correction, the mean value
+0:30 +0:24
of is 1.20 , with an rms scatter of 0.66 (GINGA: 0.99 , rms 0.43). The high mean
0:18 0:17
value and large scatter are due to the inclusion of A2052 in the dataset. The uncorrected
1
velo city disp ersion of this cluster is extremely high, 1404 km s , with corresp onding =
spec
+0:15
3:51. If this datap oint is excluded from the list, the mean drops to 1.09 with rms scatter
0:15
+0:24
0.43 (GINGA: 0.97 , rms 0.42). Including the substructure correction to the velo city
0:17
+0:10
disp ersion (and retaining A2052, which is no longer anomolous), h i =0:90 with
spec
0:15
an rms scatter of 0.37 (where the con dence intervals are the 90% b o otstrapp ed estimates)
+0:12
(GINGA: 0.87 , rms 0.38) .
0:17
To demonstrate the e ect of morphology on , these numb ers should b e compared
spec
to the values from the LB93 study. Lubin & Bahcall use 41 clusters of widely varying
+0:08
morphology. Their mean value of is 1.14 with an rms scatter of 0.57. The ES91
spec
0:08
sample, b eing based on an X-ray ux-limited catalog of clusters, is biased toward X-ray
luminous systems, which are less likely to b e a ected by ma jor substructure. This sample 7
+0:11
yields h i =0:91 with an rms scatter of 0.38. It is clear that when examining
spec
0:13
correlations b etween temp erature and velo city disp ersion, uncertaintymaybeintro duced by
neglecting the e ects of morphology and substructure in the dataset.
3 The Velo city Disp ersion { Temp erature Correlation
In Figure 1, we present the velo city disp ersion and temp erature data for the 22 clusters
in the limited sample. The velo city disp ersions are corrected for substructure. The dashed
lines are the correlations predicted by the virial theorem, for = 1 and for =0:67.
spec spec
Recall that for these data h i =0:90. The solid line is the b est t to the data using the
spec
lower temp eratures for A1736 and A3558:
2:500:09 0:610:13
=10 T (4)
r
Similarly,we nd that
3:150:60 1:310:21
T =10 (5)
r
For the higher GINGA temp eratures for these two clusters, we nd that
2:390:09 0:760:11
=10 T (6)
r
and
3:210:61 1:340:21
T =10 (7)
r
In b oth equations the uncertainties quoted are the b o otstrapp ed 1- values. This t includes
the errors in the measurements, using a linear tting technique develop ed by Akritas, Ber-
shady & Bird (1995, in preparation). This algorithm, based on the ordinary least-squares
bisector rst de ned by Isob e et al. (1990), explicitly includes b oth intrinsic scatter in the
relation and uncorrelated measurement errors. The bisector metho d assumes that neither
variable is dep endent on the other, which is probably appropriate for the currentphysi-
cal situation. The velo city disp ersion and X-ray temp erature are b oth determined by the 8
depth of the gravitational p otential (and p erhaps other physical e ects), and are therefore
independent of each other.
This subtlety in the application of linear regression algorithms has b een previously noted
by astrophysicists for other applications, such as the Tully-Fisher e ect (see Isob e et al. 1990
for a detailed discussion), but not yet applied to the problem of X-ray and optical correlations.
The use of an inappropriate or biased regression technique can have a signi cant e ect on
the co ecients of the linear t, as we demonstrate in Table 3. To simplify this discussion,
in Table 3 we present the following:
the published linear regressions given in ES91 and LB93
the linear regressions determined from an ordinary least squares t, without measure-
ment errors
the linear regressions from the bisector lines, with and without measurement errors
for the ES91 and LB93 datasets, as well as similar regressions for our limited cluster dataset.
The uncertainties in the linear co ecients are the 1- values, determined using a b o otstrap
metho d which is the preferred estimator for small datasets.
First of all, we see that the published linear regressions are recovered for b oth the ES91
and the LB93 datasets using the ordinary least squares (OLS) regressions, without errors.
For these ts, the velo city disp ersion is assumed to b e dependent on the temp erature, which
as discussed ab ove do es not seem likea physically well-motivated assumption. In addition,
simulations suggest that the OLS regressions are severely biased for such small sample sizes.
The bisector slop es for all three datasets are much steep er than the OLS slop es, varying
from 0.61 for our limited cluster dataset and the Einstein data to 0.87 for the LB93 dataset.
The regression for our limited cluster dataset is marginally consistent with the slop e of 0.5 9
predicted by the virial theorem. For the ES91 and LB93 datasets, the tted slop es are at
least 3 away from the canonical value of 0.5.
Given the large disp ersions b etween the individual linear regressions, as well as the
co ecients of the regressions for the three datasets, how signi cant is this di erence? To
estimate the signi cance of the observed deviation, we utilize a Monte Carlo computer rou-
tine. This co de simulates 22 cluster temp eratures b etween 2.0 and 10.0 keV and generates
velo city disp ersions using the virial relation and a value of 1. It then includes a velo city
term for the intrinsic scatter in the relationship (which is generated bycho osing a velo city
1
p erturbation from a uniform distribution of width 150 km s )aswell as measurement er-
rors in b oth velo city and temp erature (these are mo delled as Gaussians; the disp ersion in
1
velo cities is 150 km s and in temp erature is 0.5 keV). For 1000 simulations, only 40 of the
random datasets had measured bisector slop es greater than 0.61, the lowest value obtained
for the limited cluster dataset. The average value for the 1000 runs was 0.55 0:03. The
highest value of the slop e obtained for any of the simulated datasets is 0.64, which is com-
parable to the value obtained for the ES91 dataset but still strongly inconsistent with the
LB93 regression and the limited cluster dataset (with the high temp eratures for A1736 and
A3558).
These simulations suggest that while the deviation b etween the observed correlation b e-
tween velo city disp ersion and temp erature and that predicted by the virial theorem is small,
it is signi cant. Clearly larger individual cluster datasets, higher-quality X-ray sp ectra, and
a larger dataset of clusters will b e vital to improving our understanding of this fundamental
correlation.
The deviation of the T relationship from that predicted by the equilibrium mo del
r
describ ed in Section 1 implies that is a function of the depth of the gravitational p otential,
as estimated by either the temp erature or the velo city disp ersion. In this case, de ning an 10
average (unweighted) value of for a cluster sample whichcovers a wide range of physical
spec
parameters yields a quantity which is p o orly de ned. The dep endence of on temp erature
and/or velo city disp ersion is no doubt partially resp onsible for the high scatter ab out the
T relation, which remains even after elimination of the e ects of substructure from the
r
optical dataset.
Wehave seen in Section 2 that consideration of morphology and substructure signi -
cantly reduces the scatter in the values of for the individual clusters. Examination of
spec
Table 3 reveals that the same e ect do es not hold true for the determination of the T
r
correlation. Inclusion of the substructure correction actually raises the scatter in the param-
eters of the t slightly, although it remains comparable to the values obtained by b oth ES91
and LB93. It is clear that although substructure in uences the scatter in the relationship,
other physical e ects must also b e signi cant (see also Gerbal et al. 1994).
Previous authors have claimed that their data was consistent with the canonical virial
0:5
theorem dep endence of velo city disp ersion on temp erature, / T (ES91, LB93). We
r
have seen that this \consistency" is due to the inaccurate use of the least squares linear
regression, and that none of the three datasets are consistent with the canonical prediction.
Correction for substructure has very little e ect on the slop e of the T correlation. The
r
scatter to high velo city disp ersions implied by the \steep er than virial" relation has b een
noted by all previous studies and generally attributed to velo city substructure. However, we
demonstrate that correction for substructure has little e ect on the correlation.
4 Mechanisms for Explaining the Discrepancy
The virial theorem prediction of the relationship b etween galaxy velo city disp ersion and
gas temp erature is based on three assumptions: that the galaxy orbits are isotropic, that the
gas and the galaxies o ccupy the same p otential well, and that gravity is the only source of 11
energy for either the gas or the galaxies. Any pro cess whichmay contribute to the deviation
of the slop e from the virial value must op erate to a di erent degree in hot, high- clusters
v
than in co oler, low- systems, to skew the relationship in the observed fashion (although
v
the e ect need not b e large). Mechanisms whichhave b een prop osed include anisotropyin
the distribution of galaxy orbits, incomplete thermalization of the gas, pressure supp ort of
the ICM from magnetic elds, biasing and protogalactic winds.
4.1 Anisotropy and Magnetic Pressure Supp ort
The anisotropy parameter A is not well-determined for more than one or two clusters.
2 2
Recall that A =1 = .For radial orbits, with > ,A<0 and is increased (relative
r t
r t
to the value determined by pro le tting; see eqn. 3). For circularized orbits, <,
r t
A>0 and is decreased. To repro duce the observed trend in the T relation, we
r
estimate that hot clusters require A 0:1 (slightly radial orbits), and co ol clusters require
A 0:6 (mo derately circular orbits). Such an extreme variation in galaxy anisotropy is not
predicted byany current theory of cluster formation. Kau mann & White (1993) do nd
some evidence for a dep endence of formation history on mass, but this variation is negligible
13 15
over the range of masses included in the limited cluster sample (5 10 1 10 M ;S.
White, 1994, private communication).
In most observations, the temp erature pro le of the ICM is at out to the radius where
the background dominates the cluster sp ectrum (Mushotzky 1994). Nonetheless, simulations
by Evrard (1990) suggest that the cluster gas will not b e completely thermalized after only
one crossing time. This e ect is evident in more detailed calculations by Metzler & Evrard
(1995, in preparation), who nd that the degree of thermalization is not systematically
dep endent on temp erature. Incomplete thermalization clearly a ects the distribution of
temp eratures measured for the limited cluster sample, but do es not a ect the slop e of the
T relationship in the required direction.
r 12
In an attempt to resolve the discrepancy b etween cluster masses determined by grav-
itational lensing and those determined from X-rays (Miralda-Escude & Babul 1994), Lo eb
& Mao (1994) prop ose magnetic pressure supp ort of the intracluster medium, at least in
the cores of co oling ows. To b e dynamically signi cant, tangled magnetic elds must con-
tribute a similar amount of p otential energy to the ICM as the gravitational p otential. The
required eld strength (on the order of 50 G) is large, but Lo eb & Mao argue that such
elds may b e generated within co oling ows, where gas and magnetic eld lines are con ned
and compressed.
Comparison of the limited cluster sample with Table 1 of Edge, Stewart&Fabian
1992 reveals that the ma jority of the limited cluster sample p ossesses co oling ows (as
determined from depro jection analysis) and therefore may b ene t from magnetic pressure
supp ort. Rememb er, however, that the Lo eb & Mao (1994) analysis is restricted to the inner
1
120h kp c of A2218 (inside the radius of the co oling ow), whereas our temp eratures and
velo city disp ersions are determined for the entire cluster (again assuming that the cluster
ICM temp erature pro les are at outside the co oling radius, as ASCA data suggest). It
is unclear whether the variation in deriving from magnetic pressure supp ort would b e
detected in our analysis of the X-ray and optical data.
4.2 Protogalactic Winds
Protogalactic winds provide an additional source of heating of the ICM. Yahil & Os-
triker (1973), Larson & Dinerstein (1975) and White (1991) discuss ram pressure stripping
and protogalactic winds as mechanisms for the metal enrichment of the ICM. In the winds
scenario, the sp eci c energy of the ICM is a ected by the initial collapse of the cluster, the
relative motions of galaxies in the cluster, and winds from sup ernova explosions during the
formation of elliptical galaxies at early times. Of these three physical pro cesses, White (1991)
demonstrates that only protogalactic winds can b o ost the energy of the gas ab ove the value 13
determined through the virial theorem. In addition he shows that the energy contribution
due to winds will b e larger in co ol clusters than in hot ones.
Using White's Equation 2, we generated a distribution of temp eratures for velo city
1
disp ersions ranging from 350-1200 km sec (taking his values for the fraction of intracluster
gas coming from winds (w =0:5) and the typical wind velo city in terms of the galactic
velo city disp ersion (f = 3)). Fitting these simulated data, we nd that the protogalactic
w
winds mo del predicts a correlation b etween the velo city disp ersion and the temp erature of
a cluster:
0:68
/ T (8)
r
0:62
This dep ends slightly on the choice of w and f ; for f =2 we nd that / T . The
w w r
protogalactic wind mo del repro duces nearly exactly the dep endence of velo city disp ersion on
ICM temp erature that we nd in the limited cluster sample (and which is consistent with
the slop es found by earlier studies).
4.3 Winds and Biasing
Another e ect whichmay pro duce the steepness of the T relationship is a velo city
r
bias b etween cluster galaxies and the background dark matter, which is driven by dynamical
0:5
friction (Carlb erg 1994; Carlb erg & Dubinski 1991). Simple virial analysis predicts / T
r
if the collisionless comp onent has exp erienced no co oling or heating. If and refer
DM gal
to the background dark matter and galaxy velo city disp ersions resp ectively, and assuming
the virial equilibrium holds for the dark matter, then we can write
gal gal
0:5
= / T (9)
gal DM
DM DM
If the ratio of velo city disp ersions is temp erature{dep endent, then this will mo dify the ob-
served T relation. 14
For the purp oses of illustration, we take the distribution of background dark matter
velo cities to b e Maxwellian,
n(r)
2 2
f (v )= exp v =2 ; (10)
3=2
2
(2 )
in which case the Chandrasekhar dynamical friction formula for a galaxy of mass M in a
dark matter p otential well with density can b e written as
" #
2
dv 4 ln G M 2X
M
2
p
= er f (X ) exp X v ; (11)
M
3
dt v
M
p
2 (Binney and Tremaine 1987). This can b e rearranged for a characteristic with X = v =
M
timescale, and writing the bias for the individual galaxy of mass M, b = v = ,wehave
M
2 3
s
1
3 3
p
b 2
2
4 5
t = er f b= 2 b exp b =2 : (12)
fric
2
4 ln G M
Again, for the purp oses of illustration, we assume a p ower law density pro le for the back-
GM ( 2 2 . Substituting ground dark matter, = Ar ; then ' implies G ' 2 R 4R in, we nd that the dynamical friction timescale for galaxies at a radius R roughly scales 2 as t / (R) R .At a xed radius R, more massive (thus typically higher temp erature) fric clusters will have a higher velo city disp ersion, and thus a longer characteristic timescale for dynamical friction to b e signi cant. This translates into a temp erature{dep endentvelo city bias. Simulations provide an ideal mechanism to test these ideas. Metzler & Evrard (1995) have conducted an ensemble of N{b o dy + hydro dynamic simulations of the formation and evolution of individual clusters, explicitly including galaxies and galactic winds. These sim- ulated clusters are compared to a ensemble drawn from the same initial conditions | but without galaxies and winds | to isolate the e ects of winds on clusters. The metho d is explained in Metzler & Evrard (1994). Figure 2 shows velo city disp ersion { temp erature data drawn from their mo dels. A \virial radius" is identi ed for each simulated cluster as the radius with a mean interior 15 overdensity of 170. The temp eratures used are mass{averaged over all gas within the virial radius; the velo city disp ersions are averages drawn from the full 3D velo city information for all dark matter or galaxies within r . A solid line corresp onding to = 1 has also b een vir spec placed on the plots. Comparing the dark matter velo city disp ersion to the average interior temp erature shows that in the simple two{ uid mo dels, the simulated clusters are well{ t by the virial relation 0:5 / T . This is sensible; there is no physics in these mo dels b eyond that used to derive the exp ected relation. Note that the values of are consistently larger than one; this is spec a signature of the incomplete gas thermalization previously seen in other studies. It is not clear whether this is physical or numerical in origin; a series of runs with di erent resolution would clarify this. The mo dels including galaxies and winds show di erent b ehavior. Here, the inclusion of energetic winds, plus dynamical friction of the galaxy comp onent, provide the necessary physics to deviate from the virial T relation. For the dark matter, the temp erature dep en- dence is steep er than 0.5, a result of the inclusion of energetic winds. When galaxies are used 0:65 to calculate the velo city disp ersion, however, the relation steep ens to / T , comparable to our observed result. The simulations thus provide evidence for a temp erature{dep endent 0:1 velo city bias, = / T . Both this bias and the increase in gas temp eratures due to gal DM energetic winds are resp onsible for the nal correlation. It should b e noted, of course, that the agreementbetween the simulated ensemble and our real clusters is to some degree fortuitous. The wind mo del used in the simulations of Metzler & Evrard is intentionally of much greater wind luminosity than exp ected for real early{typ e galaxies, and the dynamical accuracy of mo delling galaxies by heavy collisionless particles in the cluster p otential is unclear (Frenk et al. 1995). Nonetheless, this corrob orates the theoretical exp ectation that b oth energetic winds and velo city bias can result in the 16 observed T relation. 5 Discussion Although Lubin & Bahcall (1993) found that the correlation b etween cluster velo city disp ersion and temp erature was somewhat steep er than that predicted by the virial theorem, the scatter in their dataset was to o broad for them to rule out consistency with the hydro- 0:610:13 static isothermal mo del. We show that for our limited dataset, / T (GINGA: r 0:76 / T ), slightly but signi cantly (at 96% con dence) steep er than that predicted by the r virial theorem. For the ES91 and LB93 datasets, this discrepancy is signi cantatthe>99% level. It seems improbable that this is an artifact of the substructure correction algorithm. The mixture mo delling technique used to remove substructure from the cluster datasets do es not preferentially raise the velo city disp ersion of high- clusters and lower that in low- r r systems, as examination of Table 1 reveals. The protogalactic winds mo del of White (1991), in addition to p ossible velo city bias due to dynamical friction acting on the cluster galaxies, quantitatively repro duces the ob- served variation in the T relationship. Preliminary measurements of cluster emission r line diagnostics from ASCA show metal abundances typical of Typ e I I sup ernovae, also supp orting the protogalactic winds mo del (Mushotzky 1994). (Contrary to the mo del, how- ever, there is as yet no conclusive evidence that low-temp erature clusters have higher global abundances than hot systems.) It seems plausible that other physical mechanisms, suchas velo city anisotropy, incomplete thermalization of the gas and/or the galaxies, and magnetic pressure supp ort in cluster cores (which are all likely to b e present in some unknown and variable degree in clusters) are resp onsible for the large scatter ab out the b est- t T r line. This scatter is apparenteven after morphology and substructure are considered in the determination of cluster parameters. 17 Finally we can relate our revised determination of to the long-standing -discrepancy. spec Early studies of cluster X-ray sp ectroscopy and imaging revealed an imp ortant inconsistency: h i =1:2 (Mushotzky 1984) but h i =0:7 (Jones & Forman 1984). Wehave seen that spec fit the corrections for morphology and substructure bring h i down to 0.9, only marginally spec consistent with h i (but con rming the earlier results of ES91). For many individual clus- fit ters, h i and h i are completely di erent. Perseus (A426) is the most obvious example, spec fit with =1:53 and =0:57. So what is the current status of the -discrepancy? spec fit First of all, we can compare current data on the distribution of gas and galaxies in clusters. Schomb ert (1988) summarizes the data on cluster density pro les determined from avariety of tracer particles: 2:60:3 / r gal 2:10:2 / r (13) gas In the hydrostatic isothermal mo del, fit / gas gal = (14) fit spec 2:3 For our value of , / r , which is at b est only marginally consistent with the spec gas 2:1 dep endence / r determined by Jones & Forman (1984). gas As Gerbal et al. (1994) p oint out in their theoretical analysis of the -discrepancy, however, in order to test the consistency of the gas and galaxy scale lengths one must simultaneously observe their radial dep endence independently, not tting them together as Jones & Forman did. In the next stage of this pro ject (Bird & Mushotzky 1995), we present non-parametric determinations of the galaxy and gas density pro les based on the MAPEL package (Merritt & Tremblay 1994). MAPEL, a constrained maximum likeliho o d algorithm, allows us to determine the b est- t mo del to the surface density pro les without assuming 18 a King-mo del (or other isothermal) t to the data (Merritt & Tremblay 1994). This is imp ortant b ecause there is growing evidence from gravitational lensing exp eriments and computer simulations that the King mo del t is not a go o d description of the gravitational p otential of a galaxy cluster (Navarro, Frenk & White 1994; see also Beers & Tonry 1986). These pro les will allow us to test on a cluster-by-cluster basis whether the galaxy and gas pro les di er { a comparison which in the past has only b een p ossible in a statistical sense (cf. Bahcall & Lubin 1994). Note also that in the time since White (1991) app eared, ROSAT PSPC and ASCA surface density pro les of co ol clusters have b ecome publicly available. These clusters will b e included in the continuation of this pro ject (velo city data are published in Beers et al. 1994). The protogalactic winds mo del predicts that co ol clusters will have a larger scale length of gas density than hot clusters (again, b ecause the relative energy contribution of winds to the ICM is greater in co ol systems). Use of the expanded dataset for these clusters will allow us to directly test this prediction and to prob e the e ects of protogalactic winds on . fit Wewould like to thank Lori Lubin, Neta Bahcall, Ray White I I I, Bill Forman, Christine Jones and the other attendees of the Asp en Summer Workshop for their contributions to this pro ject. Claude Canizares, Keith Ashman and Alistair Edge also provided useful con- versations during the course of this work. Andy Fabian's critical reading of the manuscript greatly improved our statistical analysis. We are grateful to Simon White for clari cation of issues relating to cluster evolution and parametrization of cluster density pro les. This researchwas supp orted in part by NSF EPSCoR grant No. OSR-9255223 to the University of Kansas. 19 REFERENCES Ashman, K.M. & Carr, B.J. 1988, MNRAS, 234, 219 Ashman, K.M., Bird, C.M. & Zepf, S.E. 1994, AJ, 108, 2348 Bahcall, N.A. & Lubin, L.M. 1994, to app ear ApJ Beers, T.C., Forman, W., Huchra, J.P., Jones, C. & Gebhardt, K. 1991, AJ, 102, 1581 Beers, T.C., Kriessler, J., Bird, C.M. & Huchra, J.P. 1995, AJ, 109, to app ear March Beers, T.C. & Tonry, J.L. 1986, ApJ, 300, 557 Binney,J.& Tremaine, S. 1987, Galactic Dynamics (Princeton: Princeton University Press) Bird, C.M. 1993, Ph.D thesis, University of Minnesota and Michigan State University Bird, C.M. 1994, AJ, 107, 1637 Bird, C.M. & Mushotzky, R.F. 1995, in preparation Buote, D.A. & Canizares, C.R. 1992, ApJ, 400, 385 Carlb erg, R.G. 1994, ApJ, 433, 468 Carlb erg, R.G. & Dubinski, J. 1991, ApJ, 369, 13 Davis, D.S. & Mushotzky, R. F. 1993, AJ, 105, 491 David, L.P., Slyz, A., Jones, C., Forman, W., Vrtilek, S.D. & Arnaud, K. 1993, ApJ, 412, 479 Day, C.S.R., Fabian, A.C., Edge, A.C. & Raychaudhury, S. 1991, MNRAS, 252, 394 Dressler, A. 1984, ARA&A, 22, 185 Edge, A.C. 1991, MNRAS, 250, 103 Edge, A.C. & Stewart, G.C. 1991, MNRAS, 252, 428 Edge, A.C., Stewart, G.C. & Fabian, A.C. 1992, MNRAS, 258, 177 20 Table 1: The Cluster Sample 1 1 Cluster S (uncorr) km s S (corr) km s T (keV) Source Co de BI BI X +76 +76 +1:8 A85 810 810 6.6 E91 80 80 1:4 +165 +214 +1:0 A119 862 1036 5.1 E91 140 221 0:8 +130 +176 +1:6 A193 726 515 4.2 E91 108 153 0:9 +149 +98 +1:0 A194 530 470 2.0 JF84 107 78 1:0 +126 +131 +2:1 A399 1183 1224 6.0 E91 108 116 1:5 +132 +111 +1:4 A401 1141 785 8.6 E91 101 81 1:6 +171 +171 +0:2 A426 1262 1262 6.3 D93 132 132 0:2 +96 +86 +0:06 A496 741 533 4.0 W94 83 76 0:06 +143 +234 +1:8 A754 719 1079 8.7 E91 110 243 1:6 +66 +78 +0:2 A1060 630 710 3.3 Ikeb e 1994 ASCA 56 78 0:2 +156 +1:4 168 A1644 919 921 4.1 E91 114 0:6 141 +107 +136 +0:7 A1736y 955 528 4.6 D93 114 87 0:6 +0:7 6.2 DFER 0:7 +142 +192 +0:1 A1795 834 912 5.6 W94 119 129 0:1 +401 +143 +0:6 A2052 1404 714 3.4 E91 348 148 0:5 +148 +117 +0:35 A2063 827 706 3.4 Yamashita 1992 119 109 0:35 +126 +177 +4:4 A2107 684 577 4.2 D93 104 127 1:6 +124 +124 +0:07 A2199 829 829 4.5 W94 118 118 0:07 +212 +142 +0:2 A2634 1077 824 3.4 D93 152 133 0:2 +109 +203 +1:6 A2670 1037 786 3.9 D93 81 239 0:9 +118 +100 +0:3 A3526 1033 780 3.8 F94 79 100 0:3 +120 +111 +2:0 A3558y 923 781 3.8 D93 101 98 2:0 +0:3 DFER 6.2 0:3 +98 +138 +0:5 DC1842-63 522 565 1.4 D93 82 117 0:4 21 Table 2: with and without Substructure Correction spec Cluster (uncorr) (corr) spec spec A85 0.60 0.60 A119 0.88 1.27 A193 0.76 0.38 A194 0.85 0.67 A399 1.41 1.51 A401 0.92 0.43 A426 1.53 1.53 A496 0.83 0.43 A754 0.36 0.81 A1060 0.73 0.92 A1644 1.25 1.25 A1736 1.20 0.37 Einstein 0.89 0.27 GINGA A1795 0.75 0.90 A2052 3.51 0.91 A2063 1.22 0.89 A2107 0.67 0.48 A2199 0.92 0.92 A2634 2.07 1.21 A2670 1.67 0.96 A3526 1.70 0.97 A3558 1.36 0.97 Einstein 0.83 0.60 GINGA DC1842-63 1.18 1.38 22 Table 3: Fitting the T Correlation r Source Best Fit 2:600:08 0:460:12 Edge & Stewart 1991 =10 T r 3:220:77 1:350:27 N = 23 (pub) T =10 clus r 2:610:06 0:450:09 Ordinary least squares (no errors) =10 T r 2:460:06 0:680:10 Bisector (no errors) =10 T r 2:410:51 0:750:08 Bisector (errors) =10 T r 2:530:06 0:620:09 Lubin & Bahcall 1993 =10 T (unweighted) r 2:520:07 0:600:11 y N = 41 (pub) =10 T (weighted ) clus r 2:540:06 0:610:09 Ordinary least squares (no errors) =10 T r 2:380:05 0:840:08 Bisector (no errors) =10 T r 2:360:05 0:870:08 Bisector (errors) =10 T r 2:480:25 0:730:38 This pap er, no substructure correction =10 T r 2:791:54 1:160:52 N = 22 (bisector with errors) T =10 clus r 2:750:08 0:310:13 Ordinary least squares (no errors) =10 T r 2:510:07 0:690:12 Bisector (no errors) =10 T r 2:500:09 0:610:13 This pap er, substructure correctionyy =10 T r 3:150:60 1:310:21 N = 22 (bisector with errors) T =10 clus r 2:620:07 0:420:11 Ordinary least squares (no errors) =10 T r 2:450:09 0:690:13 Bisector (no errors) =10 T r 2:390:09 0:760:11 This pap er, substructure correctionyy =10 T r 3:210:61 1:320:21 N = 22 (bisector with errors) T =10 clus r 23 Evrard, A.E. 1990, in Clusters of Galaxies, eds. Oegerle, W.R., Fitchett, M.J. & Danly, L., (New York: Cambridge University Press), 287 Fabricant, D.M., Lecar, M. & Gorenstein, P. 1980, ApJ, 241, 552 Frenk, C.S., Evrard, A.E., White, S.D.M., & Summers, F.J. 1995, ApJ, submitted Fukuzawa, Y., Ohashi, T., Fabian, A.C., Canizares, C.R., Ikeb e, Y., Makishima, K., Mushotzky, R.F. & Yamashita, K. 1994, PASJ, 46, L55 Gerbal, D., Durret, F. & Lachieze-Rey, M. 1994, a, 288, 746 Giovanelli, R. & Haynes, M. 1985, ApJ, 292, 404 Jones, C. & Forman, W. 1984, ApJ, 276, 38 Kau mann, G. & White, S.D.M. 1993, MNRAS, 261, 921 Kent, S.M. & Gunn, J.E. 1982, 87, 945 Kent, S.M. & Sargent, W.L.W. 1983, 88, 697 Larson, R.B. & Dinerstein, H.L. 1975, PASP, 87, 911 Lo eb, A. & Mao, S. 1994, ApJ, 435, L109 Lubin, L.M. & Bahcall, N.A. 1993, ApJ, 415, L17 McLachlan, G.J. & Basford, K.E. 1988, MixtureModels, (New York: Marcel Dekker) Merritt, D. 1987, ApJ, 313, 121 Merritt, D. & Tremblay, B. 1994, AJ, 108, 514 Metzler, C.A. & Evrard, A.E. 1994, ApJ, 437, 564 Metzler, C.A. & Evrard, A.E. 1995, ApJ, in preparation Miralda-Escude, J. & Babul, A. 1994, preprint Mohr, J., Fabricant, D. & Geller, M. 1993, CfA preprint 24 Mushotzky, R.F. 1984, Phys. Scripta, T7, L157 Mushotzky, R.F. 1994, New Horizons,(Tokyo: publisher unknown), in press Navarro, J., Frenk, C. & White, S.D.M. 1994, MNRAS, in press Sarazin, C.L. 1988, X-ray emissions from clusters of galaxies, (Cambridge: Cambridge Uni- versity Press) Schomb ert, J.M. 1988, ApJ, 328, 475 Sharples, R.M., Ellis, R.S. & Gray,P.M. 1988, MNRAS, 231, 479 White, R.E. I I I. 1991, ApJ, 367, 69 White, R.E. I I I, Day, C.S.R., Hatsukade, I. & Hughes, J.P. 1994, ApJ, 433, 583 Yahil, A. & Ostriker, J.P. 1973, ApJ, 185, 787 Yamashita, K. 1992, Frontiers of X-ray Astronomy,(Tokyo: publisher unknown), 473 A This manuscript was prepared with the AAS WGAS L T X macros v2.2 E 25 NOTES TO TABLES Table 1. Source co de: E91 = Edge 1991, JF84 = Jones & Forman 1984, DFER = Day et al. 1991, D93 = David et al. 1993, W94 = White et al. 1994, F94 = Fukuzawa et al. 1994; y Two-temp erature sp ectral mo dels based on GINGA observations (Day et al. 1991) suggest that these clusters mayhave higher temp eratures than the Einstein data suggest. Wehave p erformed our statistical analysis for b oth sets of temp eratures. Table 2. y: LB93 did not published a regression for temp erature on velo city disp ersion. The rst regression of velo city disp ersion on temp erature do es not include weighting by the 2 measurement errors; the second regression is weighted following a algorithm. yy: The rst set of regressions uses the lower temp eratures for A1736 and A3558; the second set uses the higher temp eratures. 26 Figure 1: The T correlation for the limited cluster sample. Errors in the velo city dis- r p ersions (the vertical axis) are taken from Bird (1994). Errors in the temp eratures are taken from the literature, identi ed in Table 1. The dashed lines are the predicted correlations for 0:5 the isothermal -mo del, with =1 or =0:67 and / T . The solid line is the b est t r to the data. 27 Figure 2: The true T correlation for the simulated clusters of Metzler & Evrard (1995). Velo city disp ersions are the average for all galaxies or dark{matter particles within an over- density of 170. Temp eratures are the mass{average temp erature for all gas within an over- density of 170. The upp er panel shows the results for the ensemble of two{ uid simulations 0:50 (without galaxies or energetic winds); here / T . The lower panel shows the results DM from the ensemble including galaxies and winds; the crosses show the T relation for the 0:55 dark matter in these runs, while the b oxes use cluster galaxies. The results, / T DM 28 0:65 and / T , are steep er than the simple virial relation. gal