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

Authors: Erhard Scholz
Journal: Journal of Gravity
 (2017)

---

## Body

In the article titled “Clusters of Galaxies in a Weyl Geometric Approach to Gravity” [1], there was an error in equation (47) that led to a wrong relation between the acceleration due to the scale connection (aφ) and the acceleration arising from the scalar field energy density (asf). The correction of (47) 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 2σ range. To facilitate controlling the correction, a detailed derivation of the corrected equation 47 is given in Appendix.Table 3
Empirical values (M500,M200) and model values (Mtot) for total mass at r500,  r200.
Cluster r 500 M t o t ( r 500 ) M 500 r 200 M t o t ( r 200 ) M 200 Coma 1278 5.32 + 0.83 - 0.58 6.55  + 0.79 - 0.79 2300 13.19  + 2.36 - 1.65 13.84  + 1.49 - 1.41 A85 1216 4.67 + 0.51 - 0.36 6.37  + 1.00 - 1.00 1900 9.70  + 1.09 - 0.77 7.71  + 0.8 - 0.74 A400 712 1.26 + 0.25 - 0.16 1.83  + 0.39 - 0.39 1093 2.62  + 0.55 - 0.35 1.48  + 0.21 - 0.18 IIIZw54 731 1.33 + 0.34 - 0.27 1.91  + 0.58 - 0.58 1350 3.09  + 1.30 - 1.02 2.81  + 2.74 - 1.10 A1367 893 1.82 + 0.29 - 0.19 1.76  + 0.27 - 0.54 1529 4.28  + 0.81 - 0.62 4.06  + 0.45 - 0.40 MKW4 580 0.54 + 0.09 - 0.06 0.50  + 0.14 - 0.14 857 1.10  + 0.19 - 0.13 0.71  + 0.07 - 0.06 ZwCl215 1098 3.74 + 0.47 - 0.32 4.93  + 0.98 - 0.98 2093 9.32  + 1.49 - 1.07 10.37  + 3.51 - 2.62 A1650 1087 3.42 + 0.63 - 0.50 3.44  + 0.66 - 0.66 2150 9.40  + 2.87 - 2.31 11.14  + 5.77 - 3.46 A1795 1118 3.40  + 0.40 - 0.26 3.41  + 0.63 - 0.63 2136 9.32  + 1.11 - 0.74 10.99  + 2.26 - 2.09 MKW8 715 0.86 + 0.18 - 0.14 0.62  + 0.12 - 0.12 1279 2.35  + 0.67 - 0.54 2.38  + 1.04 - 0.59 A2029 1275 6.39 + 0.61 - 0.43 14.7  + 2.61 - 2.61 2286 15.83  + 1.57 - 1.12 13.42  + 2.43 - 2.26 A2052 875 1.63 + 0.26 - 0.17 1.39  + 0.28 - 0.28 1250 2.94  + 0.47 - 0.32 2.21  + 0.06 - 0.08 MKW3S 905 1.90 + 0.27 - 0.18 1.45  + 0.34 - 0.34 1450 3.81  + 0.60 - 0.40 3.46  + 0.36 - 0.34 A2065 1008 3.92 + 0.72 - 0.66 11.18  + 1.78 - 1.78 2450 11.90  + 8.20 - 4.61 16.69  + 21.34 - 6.73 A2142 1449 7.15 + 0.68 - 0.50 7.36  + 1.25 - 1.25 2364 15.29  + 1.53 - 1.14 15.03  + 3.9 - 2.64 A2147 1064 3.32 + 0.53 - 0.40 4.44  + 0.67 - 0.67 1450 5.81  + 1.14 - 0.90 3.46  + 1.17 - 0.74 A2199 957 2.26 + 0.38 - 0.28 2.69  + 0.42 - 0.42 1621 4.98  + 0.93 - 0.67 4.81  + 0.37 - 0.36 A2255 1072 3.94 + 0.44 - 0.31 7.13  + 1.38 - 1.37 2271 12.27  + 1.87 - 1.37 13.32  + 1.44 - 1.19 A2589 848 1.92 + 0.33 - 0.22 3.03  + 0.75 - 0.75 1471 4.60  + 0.84 - 0.58 3.58  + 3.86 - 1.54 Model valuesMtot(rN00) and empirical values MN00 in 1014M⊙, rN00 (empirical) in kpc (N=1,2).Table 4
Model values for halo and baryonic masses atr200.
Cluster M t M s f M s f 2 M p h 1 M g a s M ∗ f ∗ f t 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 in1014M⊙,  f∗=M∗/Mgas,  ft=Mt/Mgasr200; for r200 see Table 2.Figure 1
Halo components of Coma cluster: transparent matter haloMt=Msf+Mph1, total scalar field (SF) halo Msf, halo of freely falling galaxies Msf2, and net phantom energy Mph1 (in barycentric rest system). Empirical data (violet dot and bar): baryonic masses M∗500+Mgas500 (dot) and M500 (with error intervals) at r500=1280 kpc.Figure 2
Comparison of the contribution of the scalar field halo of the galaxiesMsf2 with the baryonic mass for the Coma cluster (empirical data for Mbar500 violet dot).Figure 3
Contribution of the baryonic massMbar, of the scalar field, and the phantom energies Msf,  Mph1 to the transparent mass Mt and to the total mass Mtot=Mt+Mbar of the Coma cluster in the WST model. Model errors indicated at r500, r200 (black). Empirical data for Mbar (violet dot) and for the empirically determined total mass M500 with error bars (violet) from [2]. Additional empirical data at r200 (yellowish) from [3].Figure 4
Halo models for clusters 2–9 in Table1: total mass Mtot (black line) with model error bars at r500,  r200, transparent matter halo Mt constituted by scalar field halo Msf2 and net phantom halo (in barycentric rest system of cluster) Mph1, and baryonic mass (gas and stars) Mbar. Empirical data for the total mass with error intervals at r500 (violet) from [2]. Additional empirical data at r200 (yellow) from [3].Figure 5
Halo models for clusters 10–19 in Table1. 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 (μ2 with neutrino core) adapted to mass data (black error bars) at r500=1280kpc and r200=2300 kpc. Red error bar at r200 expresses variability of the NFW halo at this distance due to adapting it to the given error interval of mass data at r500.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 (28), (57), and (58) should read as follows.28 L ϕ 3 = 2 3 ξ 2 η ϕ - 2 D ν ϕ D ν ϕ 3 / 2 = ξ ϕ 2 η - 1 ϕ - 1 ∇ ω 3 , 57 ρ p h = ν ~ a N a o ρ m - 4 π G a o - 1 ν ~ ′ a N a o ∇ a N · a N . 58 ρ t = a o a N 1 / 2 ρ m + 8 π G - 1 a o a N 1 / 2 ∇ a N · a N a N .Major Corrections, Section  2. Equation (47) should read as follows: 47ρsf≈˙4πG-1∇2ω. The paragraph following equation (47), including the old equation (48), has to be cancelled. The correction of (47) entails the following modifications in the equations of Section 2: 49aadd=aφ+asf=2aφ.50aadd=2GMra~or.51a~o=ao4≈124Hc≈2.4·10-9cms-2.59ρsf=ρph=12ρt63∇·∇ω∇ω=πGaoρm65ρsf=12aoam1/2ρm+8πG-1∇am·amam66ρph=ρsf.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 aadd=aφ+asf=2aφ.Major Corrections, Sections  3–5. The modifications of the equations in Section 2 lead to the following changes in Section 3: 69ρsf1=12aoabar1/2ρm+8πG-1∇abar·abarabar,70ρph1=ρsf1.71ρsf2=12aoastar1/2ρstar+8πG-1∇astar·astarastar,74ρph1=ρsf1.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 datado 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 (2σ range).In page 20, right column (Section5), 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.



---

*Source: 9151485-2017-10-15.xml*