OS
O.A. Schokker
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On Aspects of Magnetic Signature Source Modelling using Near-Field Measurements
Using Multipolar Bases and Bayesian Inference on Complex Ship Geometries
his thesis investigates the modelling of a ship’s magnetic field, by means of an equivalent source model defined on an arbitrarily shaped surface enclosing the source. This setup allows for the use of measurements close to the enclosing surface, such as on the seabed in shallow waters. A multipolar basis is introduced to reduce the dimensionality of the problem, which allows for a convenient mapping to the Decreasing Spherical Harmonic Expansion to describe the far field with low computational costs. It is found that the method performs well in determining a source model and predicting the magnetic field in new locations, provided that enough well-distributed measurements are available. The method remains accurate in the presence of noise, although measurements very close to the enclosing surface reduce performance, a reduction attributed to an approximation that may be improved. Bayesian inference is used to stabilise the method when a low number, badly distributed and/or noisy measurements are available. We find that this regularisation indeed enables the use of the method in such situations. In addition, it is investigated whether orthogonality of the basis is required. It is found that it is not, and that a non-orthogonal basis can yield slightly better results. This suggests that the discretisation of the surface and the choice of solution space of the equivalent source are the main limiting factors in improving the solutions.
...
his thesis investigates the modelling of a ship’s magnetic field, by means of an equivalent source model defined on an arbitrarily shaped surface enclosing the source. This setup allows for the use of measurements close to the enclosing surface, such as on the seabed in shallow waters. A multipolar basis is introduced to reduce the dimensionality of the problem, which allows for a convenient mapping to the Decreasing Spherical Harmonic Expansion to describe the far field with low computational costs. It is found that the method performs well in determining a source model and predicting the magnetic field in new locations, provided that enough well-distributed measurements are available. The method remains accurate in the presence of noise, although measurements very close to the enclosing surface reduce performance, a reduction attributed to an approximation that may be improved. Bayesian inference is used to stabilise the method when a low number, badly distributed and/or noisy measurements are available. We find that this regularisation indeed enables the use of the method in such situations. In addition, it is investigated whether orthogonality of the basis is required. It is found that it is not, and that a non-orthogonal basis can yield slightly better results. This suggests that the discretisation of the surface and the choice of solution space of the equivalent source are the main limiting factors in improving the solutions.
Relating Modified Newtonian Dynamics to Dark matter
Application to a Virgo-like Galaxy cluster
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. ...
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. ...
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.
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.