# Relating Modified Newtonian Dynamics to Dark matter

### Application to a Virgo-like Galaxy cluster

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## Abstract

In this thesis Modified Newtonian Dynamics (MOND) is explored in galaxy clusters similar to

the Virgo cluster. MOND is a theory proposed to explain the flat rotation curves of galaxies

and the velocities of galaxies within galaxy clusters, as an alternative to the Dark Matter (DM)

model. MOND states that Newton’s law of gravitation is incorrect at accelerations of the order

of and smaller than Milgrom’s constant a_0 = 1.2 · 10^−10m/s^2

[1].

The MOND potential φ_M created by a certain mass distribution ρ satisfies the MOND equation,

a non-linear partial differential equation. For accelerations much smaller than a0 this equation

gives a quadratic relation between the gradient of the potential (∇φ_M) and the mass distribution

ρ, this is called deep MOND. This is much different from the Poisson equation, that infers a

linear relation between ∇φ_M and the mass sources, and which still holds for accelerations

much larger than a_0 [1], referred to as Newtonian Dynamics (ND). For accelerations around a0

an interpolation of deep MOND and ND is used. It appears that the potential and acceleration

in Virgo-like clusters is according to ND at the center, and approaches deep MOND at the edge.

Therefore an interpolation function µ is necessary to model such clusters accurately.

When the MOND potential φ_M is substituted into the Poisson equation, a new mass distribution

is found, the apparent mass distribution ρ_AM, which would need to match the actual mass

distribution in DM models, which use the Poisson equation. This apparent mass distribution

ρ_AM is the sum of the actual mass distribution ρ extracted from optical observations and the

apparent dark mass distribution ρ_ADM, a distribution that is interpreted as a theoretical DM halo.

This allows us to compare MOND and DM. With our method, realistic mass configurations

of galaxy clusters that are Virgo-like, generate apparent mass distributions ρ_AM with regions

containing negative mass. The existence, shapes and locations of these regions are in agreement

with what Milgrom found [2]. The total mass of the actual mass distribution is M = 10^15M_sun,

while the sum of the negative mass is M_negative ≈ −0.09 · 10^15M_sun = −0.09M is approximately 9%

of the total mass. Since negative mass is not acceptable, this gives us the opportunity to create

conditions to falsify either the MOND model or the DM model.