Journal of Gravity
Volume 2017 (2017), Article ID 9151485, 6 pages
https://doi.org/10.1155/2017/9151485

Corrigendum

Corrigendum to “Clusters of Galaxies in a Weyl Geometric Approach to Gravity”

Erhard Scholz

Faculty of Mathematics & Natural Sciences and Interdisciplinary Centre for History and Philosophy of Science, University of Wuppertal, 42119 Wuppertal, Germany

Correspondence should be addressed to Erhard Scholz; scholz@math.uni-wuppertal.de

Received 24 November 2016; Accepted 17 July 2017; Published 15 October 2017

Copyright © 2017 Erhard Scholz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the article titled “Clusters of Galaxies in a Weyl Geometric Approach to Gravity” [1], there was an error in equation () that led to a wrong relation between the acceleration due to the scale connection () and the acceleration arising from the scalar field energy density (). The correction of () has consequences for the model. It makes a new run of the data evaluation based on the corrected dynamical equations necessary. The new results are given in updated Tables 3 and 4 and Figures 1–6. The overall picture of the empirical test does not change, although now three rather than two galaxy clusters agree with the model only in the range. To facilitate controlling the correction, a detailed derivation of the corrected equation is given in Appendix.

Table 3: Empirical values () and model values () for total mass at ,  .

Cluster

Coma
1278

2300

A85
1216

1900

A400
712

1093

IIIZw54
731

1350

A1367
893

1529

MKW4
580

857

ZwCl215
1098

2093

A1650
1087

2150

A1795
1118

2136

MKW8
715

1279

A2029
1275

2286

A2052
875

1250

MKW3S
905

1450

A2065
1008

2450

A2142
1449

2364

A2147
1064

1450

A2199
957

1621

A2255
1072

2271

A2589
848

1471

Model values and empirical values in , (empirical) in ().

Table 4: Model values for halo and baryonic masses at .

Cluster

Coma
11.15
6.44
1.73
4.71
1.77
0.276
0.16
6.3

A85
8.03
4.52
1.01
3.51
1.54
0.140
0.09
5.2

A400
2.27
1.36
0.45
0.91
0.26
0.084
0.32
8.7

IIIZw54
2.77
1.65
0.54
1.11
0.25
0.080
0.32
10.9

A1367
3.77
2.21
0.65
1.56
0.42
0.089
0.21
8.9

MKW4
0.99
0.58
0.18
0.40
0.09
0.022
0.25
10.9

ZwCl215
8.00
4.55
1.11
3.45
1.19
0.137
0.12
6.7

A1650
8.15
4.69
1.24
3.45
1.10
0.162
0.15
7.4

A1795
8.04
4.59
1.14
3.45
1.15
0.14
0.12
7.0

MKW8
2.12
1.24
0.36
0.88
0.20
0.039
0.20
10.8

A2029
12.83
7.15
1.47
5.68
2.83
0.203
0.07
4.5

A2052
2.58
1.50
0.43
1.07
0.31
0.059
0.19
8.3

MKW3S
3.35
1.95
0.55
1.40
0.39
0.072
0.18
8.5

A2065
10.28
5.80
1.32
4.48
1.48
0.142
0.10
6.9

A2142
12.55
6.95
1.35
5.60
2.59
0.159
0.06
4.8

A2147
4.83
2.77
0.71
2.06
0.87
0.118
0.14
5.5

A2199
4.36
2.52
0.68
1.84
0.55
0.088
0.16
8.0

A2255
10.35
5.84
1.33
4.51
1.76
0.166
0.09
5.9

A2589
3.99
2.33
0.68
1.65
0.52
0.105
0.20
7.7

Mass values in ,  ,  ; for see Table .

Figure 1: Halo components of Coma cluster: transparent matter halo , total scalar field (SF) halo , halo of freely falling galaxies , and net phantom energy (in barycentric rest system). Empirical data (violet dot and bar): baryonic masses (dot) and (with error intervals) at  .

Figure 2: Comparison of the contribution of the scalar field halo of the galaxies with the baryonic mass for the Coma cluster (empirical data for violet dot).

Figure 3: Contribution of the baryonic mass , of the scalar field, and the phantom energies ,   to the transparent mass and to the total mass of the Coma cluster in the WST model. Model errors indicated at , (black). Empirical data for (violet dot) and for the empirically determined total mass with error bars (violet) from [2]. Additional empirical data at (yellowish) from [3].

Figure 4: Halo models for clusters 2–9 in Table : total mass (black line) with model error bars at ,  , transparent matter halo constituted by scalar field halo and net phantom halo (in barycentric rest system of cluster) , and baryonic mass (gas and stars) . Empirical data for the total mass with error intervals at (violet) from [2]. Additional empirical data at (yellow) from [3].

Figure 5: Halo models for clusters 10–19 in Table . For description, see Figure 4.

Figure 6: Comparison of dark/transparent/phantom mass halos for Coma in NFW, WST, and TeVeS models and free parameters of halos for NFW and TeVeS ( with neutrino core) adapted to mass data (black error bars) at and  kpc. Red error bar at expresses variability of the NFW halo at this distance due to adapting it to the given error interval of mass data at .

Minor Corrections. “15” should be changed to “14” in the following two sections. In the abstract, the number should be corrected in the phrase “the total mass for 15 of the outlier reduced ensemble of 17 clusters seems to be predicted correctly (in the sense of overlapping 1σ error intervals).” In paragraph 6 of Introduction, it should be corrected in the sentence “For 15 of the 17 main reference clusters the empirical and the theoretical values for the total mass agree in the sense of overlapping 1σ error intervals.” Equations (), (), and () should read as follows.

Major Corrections, Section  2. Equation () should read as follows:

The paragraph following equation (), including the old equation (), has to be cancelled. The correction of () entails the following modifications in the equations of Section :

In the comment below equation (59) “three quarters” and “one quarter” should be replaced with “one-half” and the inline formula below equation (62) should be replaced with .

Major Corrections, Sections  3–5. The modifications of the equations in Section lead to the following changes in Section :

This makes a new run of the data evaluation necessary, now based on the corrected dynamical equations. The new results are given in updated Tables 3 and 4 and Figures 2–6. In the evaluation on p. 16, right column, first two sentences of the first paragraph should be replaced with the following:

 For 5 clusters A85, A2255, and A2589 and the outliers A2029 and A2065, the error intervals of empirical data and model data do not overlap. For the first three of them (A85, A2255, and A2589) the model predictions are consistent with the empirical data within doubled error intervals ( range).

In page 20, right column (Section ), 20, at the middle of the second paragraph “15 clusters” should be replaced with “14 clusters,” and “Two clusters  …” should be substituted by “Three clusters  …” at the beginning of the last phrase.

Appendix

Energy Component of the Scalar Field

In scalar field (Einstein) gauge , and . Similarly

Moreover,

leads to

Thus

Following () and () of the main article and using (, the Hubble parameter at present), the energy momentum of the scalar field,

becomes

In the static case the first two terms of vanish and the total energy component of the scalar field turns into

Assuming conditions under which the terms of cosmological orders of magnitude and those of order can be neglected, we arrive at

like in the corrected equation . In the central symmetric case () implies . In spherical coordinates and

Because of the approximation is justified.

References

E. Scholz, “Clusters of Galaxies in a Weyl Geometric Approach to Gravity,” Journal of Gravity, vol. 2016, Article ID 9706704, 22 pages, 2016.

Y.-Y. Zhang, “Corrigendum to star-formation efficiency and metal enrichment of the intracluster medium in local massive clusters of galaxies,” Astronomy & Astrophysics, vol. 544, no. C3, 1 page, 2012.

Reiprich and Thomas, Cosmological Implications and Physical Properties of an X-Ray Flux-Limited Sample of Galaxy Clusters [Dissertation, thesis], Dissertation University Munich, Munich, Germany, 2001.