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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 |
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