在星系尺度或更大尺度的天體中，發現由觀測到的光度所推得的質量比由牛頓動力學量測到的質量要小。科學家們提出了不同的假設來解決這個問題。最為人熟知的是暗物質理論，暗物質理論描述在這些天體中存在一些物質，他們具有質量但我們無法經由光度觀測看到。另一個重要的理論為修正牛頓力學理論。理論上，如果我們可以得到天體的密度分布及重力位能，我們就可以來判定這兩個假說的正確性。
Wojtak 在 2011 年的文章中利用史隆數位巡天資料庫，綜合了 7800 個星系團的速度資料為一集合星系，且第一個利用這集合星系測量出星系團中的重力紅移。因為重力紅移可以直接得知星系團的位能，所以我們可以用此資訊來驗證這兩個假說。實際上，Wojtak 宣稱修正牛頓力學無法模擬他們的資料。然而很快的 Bekenstein 和 Sanders 在 2012 的文章中提出修正牛頓力學理論仍然可以模擬這些重力紅移的資料。
在這個研究中我們重新檢視這個問題。我們延長速度彌散分布的資料從1.2 到 3.5 百萬角秒差距並重新分析重力紅移的資料。我們假設了三個質量密度的分布(King, Hernquist 及 Eta3 模型)。也假設了三個修正牛頓力學的形式(Bekenstein, simple 及 standard 形式)。我們的結果與 Bekenstein 和 Sanders的提出的修正牛頓力學理論仍然可以模擬這些重力紅移的資料的結果相符。
然而在速度彌散分布的模擬結果中顯示，修正牛頓力學理論在加速度參數比在星系尺度下所量測到的參數小十倍下可以模擬這個集合星系團的速度彌散分布。我們認為修正牛頓力學理論可以模擬出這個集合星系的速度彌散分布及重力紅移，但加速度參數在星系團中與在星系中的不一致仍然無法解決。

In the framework of Newtonian dynamics, for astrophysical objects at scale of galaxy or larger the mass estimated from luminous matter is smaller than the mass estimated from motion. Scientists proposed different solutions to this so called "missing mass" problem. The most famous one is the dark matter paradigm, in which some matter exist in a form that it has mass but is invisible. A serious contender is Modified Newtonian Dynamics (MOND) which belongs to the modified gravity paradigm. Ideally, if both the density distribution and the gravitational potential can be probed separately, it is possible to distinguish the two paradigms definitively.

Using data from Sloan Digital Sky Survey (SDSS), \cite{Wojtak201109} combined velocity measurements of galaxies of 7800 clusters of galaxies into an assembled cluster, and produced the first serious gravitational redshift measurement from a cluster of galaxies. Since gravitational redshift probes the potential directly, it helps to distinguish the two paradigms. In fact, \cite{Wojtak201109} claimed that MOND could be ruled out by their data. This claim was quickly rebutted by \cite{Bekenstein201203}, who maintained that MOND was in harmony with the gravitational redshift data.

We revisit this problem. We extend the velocity dispersion distribution of \cite{Wojtak201109} from 1.2 Mpc to 3.5 Mpc, and also re-analysis the gravitational redshift data. We examine three one-component mass density profiles (King, Hernquist and Eta3 models) and three MOND interpolation functions (Bekenstein, simple and standard forms). Our result on gravitational redshift confirms the conclusion of \cite{Bekenstein201203} that MOND is in harmony with the gravitational redshift data. In fact, there exists some degeneracy among model parameters when we fit the gravitational redshift data. However, our result on velocity dispersion distribution shows that although MOND can fit the data, the MOND acceleration parameter $a_0$ is about 10 times less than the one observed at the scale of galaxy. We conclude that MOND satisfies both velocity dispersion and gravitational redshift data of the assembled cluster, but the difference between the values of $a_0$ at cluster of galaxies scale and galactic scale is difficult to reconcile.

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