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200 (Evrard et al. 2008),
the expected scaling is YSZ∝σ4.76(we assume that
M100∝M500). The right panels of Figure 2 shows this
predicted slope (dashed lines).
The bisector of the least-squares ﬁts to the data has
a slope of 2 .94±0.74, signiﬁcantly shallower than the
predicted slope of 4.8.
We recompute the velocity dispersions σp,Afor all
galaxies within one Abell radius (2.14 Mpc) and in-
side the caustics. Surprisingly, the correlation is slightly
stronger (99.4% conﬁdence level). This result supports
the idea that velocity dispersions computed within a
ﬁxedphysicalradiusretainstrongcorrelationswith other
cluster observables, even though we measure the velocity
dispersion inside diﬀerent fractions of the virial radius
for clusters of diﬀerent masses. Because cluster veloc-
ity dispersions decline with radius (e.g. Rines et al. 2003;
Rines & Diaferio 2006), σp,Amay be smaller than σp,100
(measured within r100,c) for low-mass clusters, perhaps
exaggerating the diﬀerence in measured velocity disper-
sionsrelativeto the diﬀerences in virialmass(i.e., σp,Aof
a low-mass cluster may be measured within 2 r100while
σp,Aof a high-mass cluster may be measured within r100;
the ratio σp,Aof these clusters would be exaggerated rel-Hectospec Virial Masses and SZE 3
Fig. 1.— Redshift versus projected clustrocentric radius for the 15 HeCS clusters studied here. Clusters are ordered left-to-r ight and
top-to-bottom by decreasing values of YSZD2
A(r2500). The solid lines show the locations of the caustics, which w e use to identify cluster
members. The Hectospec data extend out to ∼8 Mpc; the ﬁgure shows only the inner 4 Mpc to focus on the viria l regions.
ative to the ratio σp,100). Future cluster surveys with
enough redshifts to estimate velocity dispersions but too
few to perform a caustic analysis should still be suﬃcient
for analyzing scaling relations.
Because of random errors in the mass estimation, the
virial mass and the caustic mass within a given radius
do not necessarily coincide. Therefore, the radius r100
depends on the mass estimator used. Figure 2 shows
the scaling relationsfor two estimated masses M100,cand
M100,v;M100,cis the mass estimated within r100,c(where
bothquantitiesaredeﬁnedfromthecausticmassproﬁle),
andM100,vis the mass estimated within r100,v(both
quantities are estimated with the virial theorem, e.g.,
Rines & Diaferio 2006). including galaxies projected in-
sider100,v. Similar to σp, there is a clear correlation
between M100,vandYSZD2
A(99.0% conﬁdence with a
Spearman test). The strong correlation of dynamical
mass with SZE also holds for M100,cestimated directlyfrom the caustic technique (99.8% conﬁdence).
The bisector of the least-squares ﬁts has a slope of
1.11±0.16, again signiﬁcantly shallower than the pre-
dicted slope of 1.6. This discrepancy has two distinct
origins. By looking at the distribution of the SZE sig-
nals in Figure 2, we see that, at a given velocity disper-
sion or mass, the SZE signals have a scatter which is a
factor of ∼2. Alternatively, at ﬁxed SZE signal, there
is a scatter of a factor of ∼2 in estimated virial mass.
Unless the observational uncertainties are signiﬁcantly
underestimated, the data show substantial intrinsic scat-
ter. Moreover, this scatter is comparable to the range of
our sample and, therefore, the error on the slope derived
from our least-squares ﬁt to the data is likely to be un-
derestimated (see Andreon & Hurn 2010, for a detailed
discussionofaBayesianapproachtoﬁttingrelationswith
measurement uncertainties and intrinsic scatter in both
quantities).4 Rines, Geller, & Diaferio
TABLE 1
HeCS Dynamical Masses and SZE Signals
Cluster z σ p M100,vM100,c YSZD2
AYSZD2
ASZE
(350 kpc) ( r2500)
km s−11014M⊙1014M⊙10−5Mpc−210−4Mpc2Ref.
A267 0.2288 743+81
−616.86±0.82 4.26 ±0.14 3.08 ±0.34 0.42 ±0.06 1
A697 0.2812 784+77
−596.11±0.69 5.96 ±3.51 – 1.29 ±0.15 1
A773 0.2174 1066+77
−6318.4±1.7 16.3 ±0.7 5.40 ±0.57 0.90 ±0.10 1
Zw2701 0.2160 564+63
−473.47±0.42 2.69 ±0.30 1.46 ±0.016 0.17 ±0.02a2
Zw3146 0.2895 752+92
−676.87±0.89 4.96 ±0.91 – 0.71 ±0.09 1
A1413 0.1419 674+81
−606.60±0.85 3.49 ±0.15 3.47 ±0.24 0.81 ±0.12 1
A1689 0.1844 886+63
−5215.3±1.4 9.44 ±5.66 7.51 ±0.60 1.50 ±0.14 1
A1763 0.2315 1042+79
−6416.9±1.6 12.6 ±1.5 3.10 ±0.32 0.46 ±0.05a2
A1835 0.2507 1046+66
−5519.6±1.6 20.6 ±0.3 6.82 ±0.48 1.37 ±0.11 1
A1914 0.1659 698+46
−386.70±0.57 6.21 ±0.21 – 1.08 ±0.09 1
A2111 0.2290 661+57
−454.01±0.41 4.77 ±1.23 – 0.55 ±0.12 1
A2219 0.2256 915+53
−4512.8±1.0 12.0 ±4.7 6.27 ±0.26 1.19 ±0.05a2
A2259 0.1606 735+67
−535.59±0.60 4.90 ±1.69 – 0.27 ±0.10 1
A2261 0.2249 725+75
−577.13±0.83 5.10 ±2.07 – 0.71 ±0.09 1
RXJ2129 0.2338 684+88

200 (Evrard et al. 2008),

the expected scaling is YSZ∝σ4.76(we assume that

M100∝M500). The right panels of Figure 2 shows this

predicted slope (dashed lines).

The bisector of the least-squares ﬁts to the data has

a slope of 2 .94±0.74, signiﬁcantly shallower than the

predicted slope of 4.8.

We recompute the velocity dispersions σp,Afor all

galaxies within one Abell radius (2.14 Mpc) and in-

side the caustics. Surprisingly, the correlation is slightly

stronger (99.4% conﬁdence level). This result supports

the idea that velocity dispersions computed within a

ﬁxedphysicalradiusretainstrongcorrelationswith other

cluster observables, even though we measure the velocity

dispersion inside diﬀerent fractions of the virial radius

for clusters of diﬀerent masses. Because cluster veloc-

ity dispersions decline with radius (e.g. Rines et al. 2003;

Rines & Diaferio 2006), σp,Amay be smaller than σp,100

(measured within r100,c) for low-mass clusters, perhaps

exaggerating the diﬀerence in measured velocity disper-

sionsrelativeto the diﬀerences in virialmass(i.e., σp,Aof

a low-mass cluster may be measured within 2 r100while

σp,Aof a high-mass cluster may be measured within r100;

the ratio σp,Aof these clusters would be exaggerated rel-Hectospec Virial Masses and SZE 3

Fig. 1.— Redshift versus projected clustrocentric radius for the 15 HeCS clusters studied here. Clusters are ordered left-to-r ight and

top-to-bottom by decreasing values of YSZD2

A(r2500). The solid lines show the locations of the caustics, which w e use to identify cluster

members. The Hectospec data extend out to ∼8 Mpc; the ﬁgure shows only the inner 4 Mpc to focus on the viria l regions.

ative to the ratio σp,100). Future cluster surveys with

enough redshifts to estimate velocity dispersions but too

few to perform a caustic analysis should still be suﬃcient

for analyzing scaling relations.

Because of random errors in the mass estimation, the

virial mass and the caustic mass within a given radius

do not necessarily coincide. Therefore, the radius r100

depends on the mass estimator used. Figure 2 shows

the scaling relationsfor two estimated masses M100,cand

M100,v;M100,cis the mass estimated within r100,c(where

bothquantitiesaredeﬁnedfromthecausticmassproﬁle),

andM100,vis the mass estimated within r100,v(both

quantities are estimated with the virial theorem, e.g.,

Rines & Diaferio 2006). including galaxies projected in-

sider100,v. Similar to σp, there is a clear correlation

between M100,vandYSZD2

A(99.0% conﬁdence with a

Spearman test). The strong correlation of dynamical

mass with SZE also holds for M100,cestimated directlyfrom the caustic technique (99.8% conﬁdence).

The bisector of the least-squares ﬁts has a slope of

1.11±0.16, again signiﬁcantly shallower than the pre-

dicted slope of 1.6. This discrepancy has two distinct

origins. By looking at the distribution of the SZE sig-

nals in Figure 2, we see that, at a given velocity disper-

sion or mass, the SZE signals have a scatter which is a

factor of ∼2. Alternatively, at ﬁxed SZE signal, there

is a scatter of a factor of ∼2 in estimated virial mass.

Unless the observational uncertainties are signiﬁcantly

underestimated, the data show substantial intrinsic scat-

ter. Moreover, this scatter is comparable to the range of

our sample and, therefore, the error on the slope derived

from our least-squares ﬁt to the data is likely to be un-

derestimated (see Andreon & Hurn 2010, for a detailed

discussionofaBayesianapproachtoﬁttingrelationswith

measurement uncertainties and intrinsic scatter in both

quantities).4 Rines, Geller, & Diaferio

TABLE 1

HeCS Dynamical Masses and SZE Signals

Cluster z σ p M100,vM100,c YSZD2

AYSZD2

ASZE

(350 kpc) ( r2500)

km s−11014M⊙1014M⊙10−5Mpc−210−4Mpc2Ref.

A267 0.2288 743+81

−616.86±0.82 4.26 ±0.14 3.08 ±0.34 0.42 ±0.06 1

A697 0.2812 784+77

−596.11±0.69 5.96 ±3.51 – 1.29 ±0.15 1

A773 0.2174 1066+77

−6318.4±1.7 16.3 ±0.7 5.40 ±0.57 0.90 ±0.10 1

Zw2701 0.2160 564+63

−473.47±0.42 2.69 ±0.30 1.46 ±0.016 0.17 ±0.02a2

Zw3146 0.2895 752+92

−676.87±0.89 4.96 ±0.91 – 0.71 ±0.09 1

A1413 0.1419 674+81

−606.60±0.85 3.49 ±0.15 3.47 ±0.24 0.81 ±0.12 1

A1689 0.1844 886+63

−5215.3±1.4 9.44 ±5.66 7.51 ±0.60 1.50 ±0.14 1

A1763 0.2315 1042+79

−6416.9±1.6 12.6 ±1.5 3.10 ±0.32 0.46 ±0.05a2

A1835 0.2507 1046+66

−5519.6±1.6 20.6 ±0.3 6.82 ±0.48 1.37 ±0.11 1

A1914 0.1659 698+46

−386.70±0.57 6.21 ±0.21 – 1.08 ±0.09 1

A2111 0.2290 661+57

−454.01±0.41 4.77 ±1.23 – 0.55 ±0.12 1

A2219 0.2256 915+53

−4512.8±1.0 12.0 ±4.7 6.27 ±0.26 1.19 ±0.05a2

A2259 0.1606 735+67

−535.59±0.60 4.90 ±1.69 – 0.27 ±0.10 1

A2261 0.2249 725+75

−577.13±0.83 5.10 ±2.07 – 0.71 ±0.09 1

RXJ2129 0.2338 684+88