The rotational velocities of stars at the edges of galaxies are higher than predicted by Newtonian dynamics. One theory that is capable of explaining this anomaly is called MOdified Newtonian Dynamics
(MOND). To further test this MOND theory, the wide binary test has been suggested. In the wide binary test, the stability of wide binary systems is tested. The MOND theory deviates significantly from the
Newtonian theory when accelerations are smaller than a0 = 1.2·10−10 m/s2 , which is present in wide binary systems. The MOND theory is non-linear, which introduces a non-trivial effect of the external field
(EF) of the Milky Way on the wide binary orbit. To perform the wide binary test, an iterative particle mesh N-body code was used with initial conditions taken from the Gaia mission database. Additionally, the EF was modeled in three ways, where the Newton potential adjustment model was found to be the most realistic. The results include nine simulations, split up into three models and three different regimes. The models include the Newton model, the MOND without EF model, and the MOND with EF model. The regimes consist of the Newtonian regime (ainternal > a0), the transition regime (ainternal ≈ a0), and the deep MOND regime (ainternal < a0). In the Newton model, only the Newtonian regime gave stable orbits. The MOND without EF model gave stable orbits for all regimes, and the MOND with EF model gave stable orbits only for the Newtonian regime. The MOND without EF gave stable wide binary orbits, whereas the other models did not. If additional measurements detect these wide binary systems, then the MOND without EF model gains credibility. The orbits of the Newton model and the MOND with EF model were similar, with only a small difference. This is because the strength of the external field is 1.6 · a0, which makes the external field dominant over the internal accelerations, making the simulation Newtonian-like. To determine which model is correct, additional long-term measurements of Gaia are required. Further research recommendations include improving the performance and accuracy of the code. This is done by better implementation of the fast Fourier transform, translating the code into a faster programming language than Python, using adaptive time steps, and reducing self-interaction. The most important extension to the model is to introduce the galactic tidal effect.

The next generation of telescopes will be able to directly measure the light curves of exoplanets. This thesis demonstrates the potential of using light curve analysis to study exoplanetary ring systems, with the goal of retrieving their albedo map and optical depth. This can improve our understanding of the composition, formation, and evolution of such systems. We construct a mathematical model that describes how light interacts with ring particles through absorption, transmission, and scattering. Applying this model to data of Saturn’s rings from high-resolution images by NASA's Cassini space probe, albedo maps for the rings are constructed and the optical depth is found to be mostly consistent with existing literature. We then derive an analytic formula for the light curve of an exoplanetary ring system. This formula is converted into a numerical linear transformation that links the albedo map of a ring system to its light curve. Using the Moore-Penrose pseudoinverse and singular value decomposition this transformation is inverted, enabling the retrieval of the albedo map from the light curve. The method is tested using synthetic data of model ring systems while varying the axial tilt, observer direction, and signal-to-noise ratio from artificial Gaussian white noise. The results show that the method is generally effective at recovering the albedo map and optical depth for signal-to-noise ratios down to 10³.

We have defined the infinity boxes in general dimensions, which are mathematical objects that allow rays of light to continuously reflect. When the collision points of these rays are are positioned where they would appear to the viewer (along the inititial ray), then the mirror pattern will be revealed. These mirror patterns can be used to create mirror-tilings of two- and three-dimensional Euclidian space. We have answered the question whether a curved infinity box can create patterns that are nearly identical to exact tilings of Euclidian space. To obtain the answer, we set up the fundamental theory of rays and infinity boxes, and computed the radii of the curved mirror faces that exhibit certain properties. These properties are the alignmnent of internal angles, and the alignment of image points in an infinity box. We have simulated infinity boxes in two- and three-dimensions to visualise the patterns from these infinity boxes curved with these radii. Furthermore, we connected the theory of infinity boxes the theory of dynamical billiards by stating that infinity boxes visualise solutions of dynamical billiards. Lastly, we analysed the connection between the dodecahedron infinity box, and concluded that curving the mirrors of the dodecahedron infinity box results in a slight approximation of the interior of a stereographically projected 120-Cell. All the numerical methods used to visualise infinity boxes have been thoroughly explained, and the computer program that we created called MiRai, is available over the internet.

This paper aims to improve the Particle Mesh (PM) method for Newtonian Dynamics (ND) and MOdified Newtonian Dynamics (MOND). The PM method involves discretizing the mass density on a grid using Gaussian smoothing, which is used to calculate the gravitational potential using Fast Fourier Transforms (FFTs). A major downside of Gaussian smoothing is that the mass density of two particles overlaps whenever they are in close proximity, which leads to incorrect orbital dynamics. Hence, the linearity of ND is used to correct the accelerations of particles whenever this overlap in mass density occurs. These short-range corrections involve the use of analytical expressions to eliminate the inaccurate influence of the PM method, correcting the interactions between particle pairs to direct particle-particle interactions. These corrections can introduce instabilities whenever two particles are too close in proximity and the time steps are too large. This issue is mitigated by using a softening parameter in the corrections. In MOND, the linearity in the accelerations is removed, which means that these short-range corrections cannot always be used. They may only be used in the limit of high accelerations, for which MOND reduces to ND. The short-range corrections are tested for circular and elliptical two-body systems, showing a significant reduction in error of their expected orbital trajectories. To test the limitations of the PM method under non-linear accelerations in MOND where no corrections may be applied, a wide binary system is used to determine the minimum distance between particles that still yields somewhat accurate. This also demonstrates that without the use of short-range corrections large grids are required, highlighting a major inefficiency of PM methods. Lastly, the corrections are tested in both ND and MOND using the Plummer and Miyamoto-Nagai models, which represent a sphere and a disk, respectively. The Plummer model showed significant improvements in stability in both ND and MOND. However, the Miyamoto-Nagai model shows less optimistic results in MOND due to the difficulty in placing the particles at a minimum separating particles whenever the accelerations are non-lineair. Overall, careful use of these short-range corrections allows for significant improvements in the PM method for both ND and MOND. Further research into optimizing the computational performance of FFTs and short-range corrections using tree codes is still available. Additionally, a time-adaptive PM can be implemented to prevent possible instabilities due to large time steps when using the corrections.

Maps, which have been in used for centuries, are constructed by map projections. A map projection is a transformation of the latitudes and longitudes positions on the globe to the x and y coordinates on a flat map. There are probably over a hundred of different map projections, some more practical than others. A perfect map is impossible to create, so it is essential to determine the next best alternative. The different maps can be classified in five different categories: Conformal, Equal-area, compromise, equidistant and true-direction. To evaluate this, various distortions are considered. The six distortions defined in this thesis are: area (A), isotropy (I), flexion (F), skewness (S), distance (D), and boundary cut (B). These distortions together form a comprehensive set of all distortions that occur due to map projections. Since these distortions are quite difficult to compute directly, they are numerically approximated. To achieve this, the data points of the coordinates need to be sampled, which can be done through various methods. This thesis examines systematic generation and random sampling of points. Additionally, it investigates how many points are needed to achieve a sufficiently homogeneous covering of the globe and a stable solution for the distortion, this is achieved from around 20.000 point on the globe. These methods are then applied to different map projections, including the Mercator, Equirectangular, Winkel Tripel, Gott-Wagner, and Azimuthal Equidistant and the Azimuthal Equidistant split into two-hemispheres.. Among these map projections, the Gott-Wagner projection has the lowest total distortion value and is therefore considered the best overall map projection. However, each distortion has its own advantages and disadvantages, so it is not possible to definitively declare one map projection as the best in all scenarios. Selecting the best map projection depends on minimising the distorting specific to its needs.

This bachelor's thesis explores the realm of map projections, aiming to establish a systematic approach for evaluating and ranking them based on angular and area distortions. Beginning with a definition of map projections, the study progresses to quantify the distortions that occur when projecting a spherical surface onto a flat one. Using these distortions, various maps are assessed; for each, figures of the map are displayed and a short summary of the map's features and the angular and area distortion are given. These distortions are also visualised in figures. To rank the map projections, distortion numbers are defined and refined throughout the study. The findings reveal a ranked list of map projections, providing a structured basis for selecting appropriate projections in cartographic endeavours. Through mathematical analysis, this thesis contributes to the understanding and evaluation of map projections, offering valuable insights for cartographers and researchers in the field.

Ever since their discovery, the Trojans have raised many questions among astrophysicists. In 1989 it was found that there were more L4 than L5 Trojans and the current L4:L5 ratio value is estimated to be 1.6. However, this asymmetry could not be explained by Jupiter’s current orbit and must therefore have arisen during the early development of the Solar System. It was suggested that an outward migration as described by the Nice model could cause the asymmetry. To investigate this, we extended a recent research by modelling an outward migration of Jupiter on the actual Trojans and their symmetric copies. In this manner, we had a broad initial Trojan distribution, which allowed us to investigate several effects of initial values like the Trojan inclinations and maximum angular deviations from the Lagrange point. The simulations used a recently found midpoint Yoshida integrator, which allowed fourth order symplectic simulations for our non-separable Hamiltonian problem. From the simulations it followed that the ratio could only be explained if the Trojans initially had large angular deviations from the Lagrange point, which is possible if they later lost energy due to for example mutual collisions. Also the small initial inclinations as predicted for the early Trojans would in general have risen due to the migration, but it could not explain the the inclinations up to 40 degrees for today Trojans, implying that the increase in inclination must have been caused by other reasons. We found that not only the total migration distance and duration are important, but also the function that describes the evolution of Jupiter’s semi-major axis and the eccentricity of Jupiter’s orbit. For further research we suggest to use data sets on those quantities from simulations with the Nice model, to investigate these related problems together. We believe that our model and code is of great advantage for such a research, because of its optimisations and it allows to implement the data on the distance and eccentricity directly into the model without having to alter the equations of motion.

There has been interesting research on the superconducting diode recently. A possible structure for this is the Superconductor/Ferroelectric/Superconductor (S/FE/S) Josephson junction. Our goal is to simulate a S/FE/S junction with a magnetic field and see if this leads to a superconducting diode effect. First this junction and its properties are studied through a numerical simulation. Then a magnetic field is applied and we look at how this affects the junction. For our simulation we use the Velocity Verlet method. We start by simulating a normal RCSJ. The results agree with what we expect from the theory, however there is small but noticeable error which we attribute to the numerical method. Next a S/FE/S junction is simulated. The polarisation of the ferroelectric only affects the RCSJ when it oscillates at a frequency which is close to the plasma frequency. If we apply a magnetic field we see small gaps for the retrapping current however these are close to the numerical error thus they are to small too be detected by our simulation.

Humans are efficient at moving due to their exceptional mastery of bipedal locomotion. Several models have been made that attempt to model the motion of the centre of mass with a spring-mass system with various degree of success. For example, a two dimensional model tracks the height of the centre of mass well and a three dimensional model explains the normal force exerted on the ground.

In this report a three dimensional model is constructed to describe the motion of the centre of mass, with the purpose to determine the spring constant and the rest length in a simple prosthetic leg. The research question is: can a three dimensional spring-mass model accurately track the centre of mass and can this be used to determine the optimal properties of the spring of a simple prosthetic leg?

The spring mass model consists of two springs connected to the ground and the centre of mass. The springs do not exert a force on the centre of mass if they are extended. The model incorporates steps by moving the ground connection points instantaneously over fixed points on a rail. The model can be used in two distinct ways. The first generates a periodic trajectory by finding the optimal initial positions y₀ and z₀ (PPFA) and the second finds the optimal spring parameters k and u to fit a data set (DFA). The optimal conditions are found by discretizing the parameter space and using an iterative process. The space around the best parameters becomes the parameter space for the next iteration.

The PPFA shows that the initial lateral displacement is nearly constant with respect to the spring constant when the rest length is chosen such that the centre of mass is at constant height if the model is stationary. The PPFA also shows that there are two distinct y-trajectories possible for walking. Running, however, has only one trajectory. Several versions of the model are fit to the data: only discretised step widths, discretised step widths and a x₀-offset, and distinct legs. There is not much difference between these models. The x₀-offset is only Δs = -0.02m confirming that the position of the feet does align with the extreme values of the y- and z-position. The model fits the general characteristics well but fine details in the motion are lost. Future research should investigate how cumbersome these deviations of the data are perceived by people using a simple prosthetic.

The rotational velocities of stars in galaxies indicate that either Newtonian gravity breaks down on this scale, or that there is extra mass in the universe that has not been observed. The former is known as Modified Newtonian Dynamics (MOND), whereas the latter option is known as the dark matter paradigm. In this thesis a version of MOND is considered where the field equation for gravity is modified and becomes nonlinear. To simulate MOND, new N-body codes are needed due to the field equation not being linear anymore and in this thesis a new particle mesh method is developed. This new particle mesh method follows the same steps as the normal particle mesh method, except at the step where the acceleration field on the grid is calculated. This step is replaced by an iterative method to find the MONDian acceleration field on the grid. The code was then tested in the deep MOND regime using four cases of which analytical solutions of the MOND Poisson equation are known. These cases are: linear motion, the two-body problem, a ring system and an isothermal sphere. Using these test cases, numerous conclusions were drawn. From the linear motion test it was found that in the code bodies can interact with themselves, resulting in non-physical accelerations. The two-body test showed that the magnitude of the relative error in the force as calculated by the particle mesh code is roughly on the same order as the same error when calculating the Newtonian force using a particle mesh code. Furthermore, all tests showed that energy is conserved quite well, even though momentum need not be conserved due to the self interactions, and that generally the conservation of angular momentum is violated by the code. The tests also showed that the analytical formulae that were derived for these systems gave largely the same predictions as the algorithm, hence verifying both the formulae as the algorithm. The code could be sped up by writing it in C, the accuracy can be improved by reducing self-interactions and it can be extended by adding a method to handle close encounters. Some applications of the code are given, including: simulating the solar system, wide binary stars, tidal dwarf galaxies and galaxy clusters.

This research investigates the impact of gravitational scatterings caused by close encounters between particles in an N-body Kepler system, addressing three main questions: (1) the influence of scatterings on system evolution, (2) the correspondence between simulated and expected average times between scatterings, and (3) the effect of increasing different parameters individually on the average scattering time. Simulations demonstrate an average scattering angle of 15.2 degrees for particles involved in the top 10 percent of scatterings. This would indicate a non-negligible impact of gravitational scatterings, especially for systems with heavier bodies. The results indicate that the simulated average time between scatterings is higher than the expected average, necessitating further research for accurate estimation. Moreover, the time between scatterings decreases over time, before reaching a stationary state after roughly 300 scatterings. On this domain, the correlation coefficient between the scattering time and the scattering counter was found to be  -0.08. By varying the test domains for different parameters, a new expression for the expected time between two scatterings is proposed based on simulation data. A clear connection was found between the scattering time and the number of particles, the maximum orbital radius and the maximum inclination angle. The study acknowledges limitations, including the non-stationary initialization state and linear approximations to most computations, suggesting avenues for future improvement. Overall, this research aims to find the role of gravitational scatterings in Kepler systems and underscores the need to consider these interactions, which are now often considered to be negligible.

This report has been made to gain more insight and knowledge of the Keplerian method for collision detection simulations by simplifying N-body systems to a list or particles with orbital parameters. Collision detection is an essential part in modeling the evolution of protoplanetary disk becoming planetary systems by the merging of planetesimal objects. Presumably Kep- lerian systems allow for a computationally efficient algorithm, having its computational time influenced by orbital parameters, unlike the efficient octree method which only scales with the number of particles.
The aim of this research was to create a collision detection model for a simplified proto- planetary disk system using Kepler systems and analyse the results of the model to check the reliability of this simulation method, i.e. if the quality of the results are trustworthy, perform- ing consistently well, and are accurate. In addition, the method was tested on its computational efficiency by investigating the run time of separate parts of the algorithm.
In the initialisation of the algorithm a particle list is created and sorted by increasing apoap- sis. Then by a sweep and prune filter the list is checked for possible collision pairs. For pairs that have a small enough minimal orbital intersection distance that they can collide, the colli- sion time is calculated and added to a time-ranked collision list. The algorithm keeps merging the pairs on top op the list, creating a new particle with new possible collision pairs and new collision times, and updates the time-ranked collision list after each merge. An empty collision list implies the end of the simulation. Various parameters as well as length of the lists are saved within the algorithm for subsequent examination.
The results of the Keplerian model found showed similarities to other simulation studies, and include the perceived critical eccentricity of 0.02, preservation of shape of body radius distribution, and the formation of large bodies. Simulations showed three stages of collisions, characterised by collisions occurring homogeneously throughout the disk, large body colli- sions, a final stage of small body collisions. The performance of the algorithm could be sum- marised by the number of checks by the sweep and prune filter #checks ∝ N 2 and #checks ∝ ε for eccentricity ε < 0.1, the number of pairs in the collision pair list was found to be propor- tional to the number of starting particles N, the body radius s, the inverse of the inclination 1/I, and the eccentricity ε, so #pairs ∝ N2, s, 1/I, ε. For N < 1000 the computation time of the algorithm appeared to be proportional to N2, but for N < 10.000 proportional to to a higher power. For ε < 10−9 and N < 10.000 the computation time was found to be runtime ∝ s.
The conclusion of this research is that the Keplerian model could be used to model N-body systems for collision detection because of its behavioural resemblance to observable astrophys- ical systems. The computation time of the Keplerian system scales with a higher order per N than for instance the three code algorithm, but has showed to be influenced by orbital parame- ters in which this method differentiates itself from other simulation methods, with a remarkable influence on the total computation time by the body radius s of the system. More research could be done for higher orders of N to better model reality as well as providing more insight on the proportionality of the computation time. Recommendations to the algorithm consist of adding planetary spin as a particle parameter, and methods to incorporate apsidal precession in a com- petent manner.

The aim of this thesis was to test the efficiency in practice of an analytical propagator with collision detection for N-body Keplerian systems. This can be used to simulate the evolution of a protoplanetary disk, which gives insight into how planetary systems form. The analytic propagator calculates collisions one by one, while a numerical propagator would compute each time step. The idea of using the analytic propagator is that collisions are rare in astronomical scales, such that jumping from collision to collision and calculating it, is more efficient than calculating all the time steps that are between collisions. Simplifying the orbits of the planetesimals into perfect Keplerian orbits, analytical solutions exist which are used by the analytic propagator.

In this thesis, the runtimes of simulations were measured as well as other properties directly related to the runtime. The overall efficiency of the algorithm with respect to N seemed to be O(N³), which is one power less than previously predicted. The prediction was that the runtime of the full simulation is O(N²ε + N⁴s³/Ia³). Here ε is the maximum eccentricity, s/a is the ratio of a planetesimal's radius to the semi-major axis of its orbit, and I is the maximum inclination. This was calculated by estimating the total number of collisions to be O(N²s²/Ia²) and the runtime for each collision to be O(N²s/a). But the number of collisions turns from quadratic to linear in N, implying that above a certain N almost all planetesimals collide, which reduces the power of N by one. For comparison, the octree code has an algorithmic efficiency of O(N logN) per time step, and the number of steps for a fixed integration time grows as O(N^4/3 log N).

Chiral induced spin selectivity (CISS) is an unexplained phenomenon in physics, in which the interaction within chiral molecules brings about a selectivity in the spin of the electrons.
The Breit interaction has been suggested as a possible explanation, but has not yet been researched in the context of CISS.
In this thesis, first a derivation of the Breit equation is given, which in the second part is being used to construct a simulation of electron transport through a simple chiral molecule.
The spin-polarization for the transmission including the Breit interaction was compared to the case excluding the Breit interaction, and the two were found to be substantially different. From that it followed that the influence of the Breit interaction on the transport of an electron through a chiral molecule is significant and it should therefore not be disregarded in calculations on CISS.

This thesis develops a method to map the surface of an exoplanet. The problem of exoplanet mapping sounds easy to solve. Take a few photographs of an exoplanet and sew them together to create a map of the surface. The telescope required to do this is far beyond our technological capabilities. However, our current telescopes could allow us to measure the light reflected from the star on the surface of the star as a point source. This thesis constructs the surface map of an exoplanet using only a point source of reflected light as information. This method of planet mapping is called spin-orbit tomography, introduced by Cowan and Agol [2008] and more in depth by Fujii and Kawahara [2012]. Spin Orbit tomography is a method to construct a surface map from the reflected light curve, the total intensity of the light from the star, reflected on the surface of the planet, directed towards an observer.

In the near future, next-generation telescopes will be able to observe Earth-like exoplanets illuminated by their parent stars for long periods of time. As the distance between exoplanet and observer is enormous, exoplanets will only make up a pixel on our image. However, the observed intensity of this pixel will fluctuate over time as the exoplanet rotates about its axis and orbits around its parent star. These fluctuations in the reflected light contain information about the planet’s surface. Previous researchers have developed a retrieval method known as spin-orbit tomography which uses these intensity fluctuations to construct a surface map of the exoplanet assuming fully diffuse reflection as a model for the reflection of starlight. In this thesis, we aim to build on this method by introducing two new reflection models, namely Fresnel and Lommel-Seeliger reflection, and by using this composite reflection model for the starlight we attempt to reconstruct the surfaces of exoplanets based on their observed light curve. We will derive an analytical expression for the observed intensity of the light reflected off an exoplanet. Next, we will simulate intensity observations along the orbit of an exoplanet with a pseudo-randomly generated surface containing terrain types found on Earth.
As we assign a measure of reflectance for each reflection model, we will construct three distinct reflectance maps for one planet, nd simulate the observed light curve by linearly transforming these maps. Afterward, we attempt to retrieve the reflectance maps from the observed signal by inverting the transformation.
We show that the retrieval of all three maps is successful if the light curve is observed without noise, even when the number of samples is low. If artificial shot noise is imposed on our signal, we are still able to retrieve glossy reflectance maps for Lambertian and Lommel-Seeliger reflection by truncating the transformation matrix. Lastly, we show that even if the original light signal originates from an exoplanet that only reflects according to the Lambertian model, then the composite reflection model still retrieves the Lambertian map accurately. On the other hand, if the original light curve is simulated by the composite reflection model, then the Lambertian method does not retrieve an accurate surface map.

Context: In astronomy, collision detection is the computational problem of finding when planets, asteroids or satellites collide. Computer simulations make it possible to show very precisely how bodies move through space, which makes it possible to detect collisions. However, these simulations require a lot of computing power if there are many bodies, which is of course undesirable.
Aims: The aim of this report is to provide more insight into a collision detection method that does not use the computationally expensive simulations, but rather fast mathematical techniques.
Methods: It can be shown that the collision detection problem between two planets is equivalent to finding the integer point (𝑘, 𝑙) between two parallel lines, closest to the origin. The solution uses the continued fraction of the ratio of the orbital periods of the planets, as the slopes of the lines are this ratio. The convergents of the continued fraction form basis vectors with which the point (𝑘, 𝑙) is searched for. This is called lattice basis reduction.
Results: Where brute force always takes 𝑘 steps to find the integer point (𝑘, 𝑙), the method with continued fractions and lattice basis reduction often finds the point in about √𝑘 steps. The worst case however is still 𝑘 steps, the same as brute force. This only happens if there are no basis transformations, so luckily 𝑘 is then often small.
Conclusions: When there are many bodies between which collisions need to be detected, lattice basis reduction provides a fast method. For the basis vectors, the convergents of the continued fraction of the ratio of the orbital periods of the planets must be used. If the solution cannot be reached exactly, (𝑘, 𝑙) can also be estimated.

The Atmoscope is the idea to use the atmosphere of a planet as a telescopic lens. It uses the refractive properties of an atmosphere to converge light rays to a focal line. It has been hypothesized that using Earth’s atmosphere could, under favorable circumstances, yield an amplification of 55; 000 for a detector that has a surface area of 1m2. However, the precise effects of the oblateness of Earth in combination with the effects of physical effects such as turbulence inside the atmosphere have not yet been researched. Here we show that incorporating the eccentricity of Earth diminishes the amplification greatly. We found using a ray tracing model for gradient-index media that the amplification for ellipsoidal planets has the following relation with the detector or pixel size: A / D􀀀0:66. For spherical planets this relation is A / D􀀀1. Furthermore, we implemented velocity diffusion to simulate physical effects inside the atmosphere, which surprisingly did not affect the amplification significantly. We anticipate this result to be a starting point for more sophisticated ray tracing models. For example, a full map for the refraction in the atmosphere could be used to test the effect of a non-homogeneous atmosphere. Much research still needs to be done, before it can be decided definitively whether the Atmoscope could be a useful telescope.