# Modified Newtonian Dynamics and the Bullet Cluster

### A comparison with the dark matter model

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## Abstract

The Bullet Cluster is a galaxy cluster, which is often used as evidence for dark matter [10]. In this thesis, Modified Newtonian Dynamics (MOND) is studied for various interpolation functions on a model of the Bullet

Cluster. MOND is a theory proposed by Milgrom which modifies the Newtonian gravity law, such that it explains the flat rotation curves observed in all the galaxies and galaxy clusters [13]. It is an alternative theory

to the dark matter model and in this thesis, the results of MOND are compared to results from the paper of

Paraficz et al which have studied the Bullet Cluster with the dark matter model [15]. In MOND, the Newtonian

gravity law is changed at low accelerations, for a around and below the value of 1.2 · 10^{-10} m/s^{2}

, which is introduced as a constant a_{0} by Milgrom. Accelerations much smaller than a_{0} are in the deep MOND regime and

should satisfy certain conditions. Combining the low acceleration regime (a ≪ a_{0}) and the Newton regime

(a ≫ a_{0}), an interpolation function is needed. The following interpolation functions are studied: the standard

interpolation function, the Verlinde interpolation function and the Angus interpolation function.

First the Newtonian gravitational potential Φ_{N} is introduced which can be calculated for a certain mass

distribution ρ using the Poisson equation. Also the acceleration field can be obtained from ΦN . It could

be seen that in the model used in this thesis for the Bullet Cluster, the strength of the acceleration field a

is mostly below a_{0}, but not much. Similarly the MOND potential Φ_{M} can also be calculated for ρ with the

MOND equations, which are different equations for each interpolation function. The MOND equations are

non-linear and can not be solved analytically in most cases. Thus a numerical iterative method is introduced

to solve the MOND equations and to obtain Φ_{M} . After obtaining Φ_{M} for each interpolation function, the acceleration field f can also be calculated. We found that Φ_{M} is steeper than Φ_{N} for all interpolation functions.

Also the acceleration field f is larger than a, and f is mostly above a_{0} in the model of the Bullet Cluster used

in this thesis.

By substituting Φ_{M} in the Poisson equation, another mass distribution could be obtained: apparent matter. This would be the matter distribution needed to give the acceleration field f using the Newtonian gravity

law. From this matter distribution, the apparent dark matter could be obtained. We can compare this apparent dark matter distribution with the dark matter model found in the paper in Paraficz et al. Also the apparent

matter distribution can be compared to the image of the Bullet Cluster. For both apparent dark matter and

apparent matter, the MOND model is not in agreement with the dark matter model and the observation respectively. In general, the dark matter did not spatially coincide with the galaxies. By increasing the number

of galaxies or increasing the mass of the galaxies, dark matter distributions that spatially coincide with the

galaxies can be obtained