Analysis of Spherical Particle Distributions Observed on the WIND Spacecraft

2007:087 MASTER'S THESIS

Katharina Nowak

Luleå University of Technology Master Thesis, Continuation Courses Space Science and Technology Department of Space Science, Kiruna

2007:087 - ISSN: 1653-0187 - ISRN: LTU-PB-EX--07/087--SE

Analysis of spherical particle distributions observed on the WIND spacecraft

Katharina Nowak Wurzburg,¨ September 2007

Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Germany and Institutet för rymdfysik, Lule˚atechniska universitet, Sweden

Analysis of spherical particle distributions observed on the WIND spacecraft

Katharina Nowak Wurzburg,¨ September 2007

Examiner: Prof. Dr. Wolfgang Droge¨ Institut fur¨ Theoretische Physik und Astrophysik, Universitat¨ Wurzburg,¨ Germany

Second examiner (Sweden): Dr. Johnny Ejemalm Institutet for¨ rymdfysik, Lule˚a techniska universitet, Sweden Abstract

WIND is a NASA spacecraft built to observe the solar wind. Since many phenomena like auroras are related to the solar wind and it has also an influence on all the spectra measured on Earth, therefore it is important to explore its behaviour. On board the spacecraft the SST, EESA and PESA instruments measure the electron and ion flux of the solar wind. These data have been further investigated. Access to existing data sets from the database of the WIND mission provided by University of Berkeley, USA, has been gained through some IDL library functions that have been specifically programmed for the purpose of preprocessing and presenting data. A statistical analysis of the present data is conducted using mostly Mathematica. After examining the data for false measurements and general characteristic attributes, like minima, maxima, mean value and variance, more sophisticated statistical derivations have been computed for gaining the knowledge about the spherical statistical behaviour which is thought to be describable by the von Mises-Fisher distribution or similar functional distributions. Therefore the data were examined first for the mean direction which defines one major part of the simulating, theoretical distribution expec- tation. It was found out that in the main solar events can be modelled by distributions defined on a sphere and therefore missing or wrong data can be reconstructed.

i Contents

1 Introduction 1 1.1 Basicprinciples...... 1 1.1.1 Thesunandthesolarwind ...... 1 1.1.2 Pitchangle ...... 3 1.1.3 Thetransportequation ...... 4 1.2 WINDspacecraft...... 8 1.2.1 Instruments...... 9 1.2.2 Datainformation...... 11 1.3 Stereomission...... 18

2 Analysis and Methods 21 2.1 Model ...... 21 2.1.1 von Mises-Fisher distribution ...... 21 2.1.2 Kentdistribution...... 23 2.2 Simulationofsphericaldata ...... 25 2.2.1 Calculating the mean, the major and the minor axis of the data separately 27 2.2.2 NMinimize ...... 27 2.2.3 FindFit ...... 28 2.2.4 FindFitwithstartvalues ...... 28 2.2.5 Comparison of the results of the diﬀerent methods ...... 29 2.2.6 Reconstructionofdata...... 35 2.3 Physicalmeaning...... 37

3 Outcome 45 3.1 Issuesthatoccurred ...... 45 3.2 Result ...... 45 3.3 Furtherwork ...... 46 Listofﬁgures...... 47 Listoftables ...... 51 References...... 52

ii CONTENTS CONTENTS

A Appendix 58 A.1 Notations ...... 58 A.2 Mathematicanotebooks ...... 59

iii Chapter 1

Introduction

1.1 Basic principles

1.1.1 The sun and the solar wind

The sun in the system of the Earth is the most interesting star in the whole universe even if it is an average star of the spectral class two. Due to the fact that is is just one astronomical unit (AU) away from the Earth, it is possible ”to study not only its electromagnetic radiation but also solar emissions of a different kinds - plasmas and energetic particles” ([1]). This is important to understand on the one hand other suns in the universe and on the other hand the effects of the Sun on the Earth, which can be seen in the atmosphere, in the weather on the Earth as well as for example in the field of biology. The chemical composition of this star is not different to other stars. About 92 % of the volume of the matter inside is hydrogen needed for the energy generation in the sun and nearly 8 % is helium, consisting of a leftover from the energy production due to nuclear fusion as well as originally present matter. In terms of mass these are 75 % hydrogen, 23 % helium and about 2 % are heavy elements, which is in total more than 99.8 % of the mass of the whole solar system ([2]). The temperature on the surface of the sun is approximately 5500 K. Further basic properties are summarised in table 1.1. The sun consists of different zones. The nuclear burning zone, which is also known as the

Table 1.1: Properties of the sun [1], [3]

Radius 696000 km Mass 1.99 1030 kg · Mean distance from the Earth 1 AU=150 106 km · Average density 1.91 g/cm3 Eﬀective blackbody temperature 5785 K Radiation emitted (luminosity) 3.86 1026 W · Mass loss rate 109 kg/s core of the sun, is the innermost part, where about 50 % of its mass is concentrated. Since the energy is transported by radiation the second one is named radiation zone. From about 0.74

1 1.1. Basic principles 1.1.1 solar radii to the surface lies the hydrogen convection zone while the visible part itself is known as the photosphere. The solar atmosphere consists of three different parts and is located above the photosphere. These parts are the chromosphere, the transition region and the corona. The outer boundary of the corona is not well-defined but more hackly structured. The atmosphere itself is not stable. It emits a particle flow called solar wind. This was first demonstrated by E. N. Parker in 1958 ([4]). One year after Parker’s predictions it was measured by Lunik II, one of the first Soviet satellites. The US Venus probe Mariner 2 analysed the particles afterwards more precise and studied the number density, the different velocities of the particles and the magnetic field strength between the Earth and Venus, which is about 5 10 9 T · − in average. This weak magnetic field is directed almost parallel to the ecliptic plane. The 2 B 11 3 related energy density in the solar wind yields to Umag = 2µ0 = 10− Jm− . The direction of the interplanetary magnetic field lines frozen-in in the solar wind is not radial due to the rotation of the sun. The sidereal rotation period of the sun is 27 days. Directly at the outer boundary of the sun the solar wind flows radially away from its origin. But then the plasma circles westward and forms an Archimedian spiral. The deformation of the field lines is known as the garden-hose effect ([1]). This shape of the field lines and therefore also of the solar wind is not as steady-going in reality as it was described. There are several reasons, which would cause this deviation. One factor might be the different velocities of the streams of the solar wind. But also a change in the polarity of the interplanetary magnetic field seen from a position, which is not rotating with the sun, caused by the sector structure of the magnetic field of the sun. Another reason could be that coronal mass ejections (CMEs) or co-rotating interaction regions (CIRs) produce shocks, which generate discontinuities themselves. There is also a forth influence that should be named. Alfvén waves originated by the sun could be a reason for the discontinuities, plasma waves and magneto-hydrodynamic waves with low frequencies, which are causing fluctuations. Those have an influence on the magnetic field described above due to superposition ([5]). The solar wind itself consists roughly of an equal number of protons and electrons, also of some heavier ions. These particles travel from the Sun with supersonic velocities and carry away a mass of 1.6 1012 g/s. There are two distinct kind of plasma flow. One is called the fast solar · wind and the other one is called the slow solar wind. The first type has a speed of 400 km/s to 800 km/s and is generated in the so called coronal holes, the dark parts of the corona, which are dominated by open field lines. From these holes open field lines extend, which cannot restrict the speed of the solar wind. In contrast the speed of the second type is only 250 km/s to 400 km/s whereas the density of the slow solar wind is with 8 ions/cm3 higher than the one of the fast solar wind, which is only about 3 ions/cm3 at 1 AU. If there is a solar minimum the slow solar wind comes from regions at the magnetic equator of the sun but during a solar maximum it originates in the streamer belt as it is described in [1]. The coronal streamer belt is limited to a small region of about 20 degrees around the equator. The speed is limited by the closed ± magnetic field lines to which this type of the solar wind is associated to ([6], [7]). One effect on the interplanetary magnetic field emerging from the occurrence of the two different types of the solar wind can be depicted with picture 1.1. As it can be seen in the illustration the sun rotates counterclockwise. Therefore the Archmedian spiral arises. The arrows specify the velocity of

2 1.1. Basic principles 1.1.2 the solar wind. Due to the diﬀerent velocities of the two types at the transition from the slow solar wind to the fast one a rarefaction of the plasma occurs while a compression results from the approach of the fast one with the slower type. When the faster wind catches up with the slow solar wind a shock front will be created because of the supersonic speed. Actually this happens after reaching the Earth ([8]).

Figure 1.1: The interplanetary magnetic ﬁeld frozen in the solar wind [8]. The arrows, which are pointing away from the sun describe the velocity of the solar wind. The other arrow speciﬁes the rotation of the sun.

1.1.2 Pitch angle

The movement of single particles perpendicular to a static homogeneous magnetic field is a circular path. The radius of gyration is named rG and the velocity of one particle in the circle is ~vG. ~v , which is perpendicular to the field, is always tangential to the circle and its absolute ⊥ value ~v stays constant. If there is also a velocity component parallel to the magnetic field, | G| called ~v , the resultant movement is a helical gyration around the magnetic field line as it can k be seen in picture 1.2. Both components of the velocity of a particle define the pitch angle ϑ with v tan ϑ = ⊥ v k or expressed with the total velocity v

v = v sin (ϑ) v = v cos (ϑ) ⊥ k

3 1.1. Basic principles 1.1.3

Figure 1.2: Movement of a particle in a homogeneous magnetic field with an illustration of the gyration radius and the pitch angle ([9]). The axes x1, x2 and x3 define the coordinate system, rgyro is the gyroradius, ϑ is the pitch angle and B is the magnetic field. as introduced in [1]. In a more general definition for ϑ it is defined as the angle between the direction of the magnetic field B~ and the one of the particles ~v which can be expresses as their scalar product:

~v B~ cos (ϑ)= · (1.1) ~v B~ | || |

([9], [10]). For the case ϑ = 0◦ or ϑ = 180◦ the particles follow the ﬁeld lines without gyrating.

That means rG = 0. The maximal gyration radius with respect to a certain ﬁxed kinetic energy will be obtained if ϑ = 90◦. Expressed in dependence of the relativistic energy Ekin rG is deﬁned as 2 2Ekin Ekin m c 2 + 2 0 r m0c m0c r = sin (ϑ). (1.2) G qB 2 q is the charge of the observed particle, m0 is known as the rest mass and therefore m0c is the rest energy.

1.1.3 The transport equation

Diﬀusion is a process, which balances the distinctions of the diﬀerences in density. Interactions of particles in the interplanetary space will only change the direction of the movement but not the absolute value of their velocities in the ideal case. If the density of the particles is bigger in some parts of the plasma the probability of collisions will be higher. Therefore more particles from the high density region will be carried out than vice versa. The result will be a reduced gradient in the density.

4 1.1. Basic principles 1.1.3

Figure 1.3: Illustration of the deﬁnition of the pitch angle ϑ ([9]). The vector ~v is the direction of a particle splitted up in the perpendicular component ~v and the parallel component ~v . ⊥ k

Diffusion equation The so called diffusion equation describes this situation and is defined as

∂U = (D U) (1.3) ∂t ∇ ∇

U is the particle density, t is the time and D is the diffusion tensor for the anisotropic diffusion. Hereby D U If this is an isotropic diffusion, which means that it is independent of the particles − ∇ direction, the equation changes to ∂U = (D U) (1.4) ∂t ∇ ∇ with the diffusion coefficient D instead of the diffusion tensor D. Equation 1.4 can be simplified to ∂U = D U (1.5) ∂t 4 if D is also independent of the spatial coordinate. To generalise the equation above a source has to be introduced. The source function Q(r0,t) describes a movement from the source point to a position r0. It leads to a diffusion equation in the form

∂U D U = Q(r ,t). (1.6) ∂t − 4 0

If it will be assumed that the propagation of all particles is spherically symmetric, so that they are moving from one shell to another, at r +∆r the equation becomes

∂U 1 ∂ ∂U r2D = Q(r ,t), (1.7) ∂t − r2 ∂r r ∂r 0 with the radial diffusion coefficient Dr. During a solar flare the source function Q(r0,t) reduces to an injection, where approximately r0 = 0 at the time t0 = 0. In this case the solution of the equation 1.7 is N r2 U(r, t)= 0 exp , (1.8) 3 (4πDrt) −4Drt p 5 1.1. Basic principles 1.1.3

where N0 is the number of particles. By calculating the ﬁrst derivative in time and setting it to zero, the resultant time at the maximal intensity tm is

r2 r2 tm(r)= = , (1.9) 6Dr 2rλr where λr = 3Dr/v is the radial mean free path of the particles. Since the magnetic field, where the particles are propagating, is not radial but an Archimedian spiral, as it was described in section 1.1.1, the solution stays the same but λr is defined differently as

2 λr = λ cos (τ) . (1.10) k

In this case τ is the angle between the radial direction and the field line direction. λ is the k mean free path parallel to the magnetic field. The diffusion perpendicular to the magnetic field lines is assumed to be negligible.

Diffusion-Convection Equation In the interplanetary space the particles are interacting while they are propagating with the magnetic field frozen into the solar wind, a medium which is moving. This can be described by the diffusion-convection equation

∂U + (U~u)= (D U). (1.11) ∂t ∇ ∇ ∇

Here U~u is the convective streaming with the bulk velocity ~u of the medium. In the case that D is independent of the spatial coordinate and as ~u is approximately constant the equation | | above changes analogue to the diﬀusion equation to

∂U + ~u U = D∆U. (1.12) ∂t ∇

If a spherical symmetric geometry is assumed it can be written as

∂U 1 ∂ ∂U + (U~u)= r2D (1.13) ∂t ∇ r2 ∂r r ∂r with r and Dr deﬁned as for equation 1.7. The solution for a pulse-like injection is calculated as

N u(r ut)2 U(r, t)= 0 exp − . (1.14) 3 (4πDrt) 4Drt p Again N0 is the number of injected particles.

Pitch angle diffusion As it was mentioned before spatial scattering was assumed to be inexistent. The pitch angle cosine α is defined as α = cos ϑ with the pitch angle ϑ as it was introduced in section 1.1.2. At interactions the pitch angle cosine will change for a small value, which is known as a diffusion in pitch angle space. Analogue to the diffusion term before, a

6 1.1. Basic principles 1.1.3 scattering term can be calculated as

∂ ∂f K(α) , (1.15) ∂α ∂α where K(α) is known as the pitch angle diffusion coefficient and f is the phase space density. K(α) can be expressed as a function of the particle mean free path λ acting parallel to the k magnetic field, which can be described as the distance the particles are moving before the direction of motion has been reversed:

1 3 (1 α2)2 λ = v − dα (1.16) k 8 Z K(α) 1 − with the velocity of a particle v. To describe the propagation of particles by encountering scattering processes the transport equation can be derived as ∂f ∂f ∂ ∂f + αv = K(α) , (1.17) ∂t ∂s ∂α ∂α

∂f where ∂s is the spatial gradient acting along the magnetic field line ([1]). Parker’s transport equation does not describe the situation if the scattering caused by magnetic turbulence is not frequent enough to make the distribution of the particles isotropic. To overcome the problem the quasi-linear theory was developed in 1966. It is based on the perturbation theory and only weakly turbulent interactions between particles and waves can be described ([11]). Therefore a transport equation for anisotropic distribution was developed. Roelof published the equation ∂f ∂f 1 α2 ∂f ∂ ∂f + αv + − v K(α) = Q(s,α,t) (1.18) ∂t ∂s 2L ∂α − ∂α ∂α in 1969. In this equation L is the focussing length in the diverging magnetic field B~ and can be described by B(z) L(z)= . (1.19) − ∂B ∂z Q is again the injection of particles which are close to the sun ([12]). This can be a delta-peak injection as mentioned in the previous parts. Alternatively a so called Reid-Axford profile

C tC t Q(s0,t)= exp ( ), (1.20) t − t − tL which defines the injection at a position s0 onto the magnetic field line, can be used. Here tC is the rise time, tL is the time of decay and C is a constant ([13]). The profile can be seen as a description of a generic injection process with a big positive gradient at the the beginning and a slow decay afterwards, which can describe coronal diffusion as well as an injection of particles in the outer part of the corona ([14]).

7 1.2. WIND spacecraft

1.2 WIND spacecraft

The WIND spacecraft, as is can be seen in figure 1.4, which was build by NASA, was launched by a Delta rocket from Cape Canaveral on 1 November 1994 ([15]). The objective target of the mission was “to measure the properties of the solar wind before it reaches the Earth“ ([16]). More specific the aim of the satellite is the exploration of the interplanetary particle population as well as the acceleration of the particles at the Sun and in the interplanetary medium. Another main task of this mission was the measurement of the upstream from the Earth. WIND studies also the transport of particles, basic plasma processes in the interplanetary medium and the plasma input and output related to the magnetosphere of the Earth. This mission is part of the Global Geospace Science (GGS) program [17], which belongs to the International Solar-Terrestrial Physics (ISTP) Science Initiative [18]. GGS was mainly founded to “measure the mass, momentum and energy flow and their time variability throughout the solar wind-magnetosphere-ionosphere system that comprises the geospace environment” [17] and therefore to understand the processes of the space plasma in a better way and to evaluate its correlation to the terrestrial environment. At first it was thought about observing the solar wind from the Lagrangian point L1 but this

Figure 1.4: The WIND satellite and its instruments [17]. plan was changed and the Solar and Heliospheric Observatory SOHO was placed in the vicinity of this Lagrangian point [19]. L1 is one of ﬁve points, that are called L1 to L5, where objects with a comparatively negligible mass can be stationary since it will be dominated by gravity of two larger objects. These points can also be seen as stationary solutions of the circular restricted three-body problem. In the case of WIND the objects, which have a gravitational inﬂuence, are

8 1.2. WIND spacecraft 1.2.1 the Earth and the Sun. The orbit was changed as can be seen in figure 1.5. The advantage of the new path is that the satellite can measure at many more points of differing characteristics than it would do in the Halo orbit around L1 ([8]) but it visited this Lagrange Point in May 1997. The new orbit leaves the ecliptic plane, which is not a common orbit for a spacecraft as leaving the ecliptic plane involves a big energetic effort and therefore the measured data are even more interesting for the analyse of the solar wind. Another aspect in the orbit of WIND is that it also passes through the Earth’s magnetosphere. For the interplanetary ions and electrons entering this sphere the physical conditions and where the entry takes place should be analysed since it was not much known in this field ([20]). The satellite itself has a total weight of 1200 kg while 300 kg of it are hydrazine fuel for orbit changes and corrections.

Figure 1.5: The orbit of the WIND spacecraft from the 16th of November over two years [18]. The unit is in Earth radii for the X and Y axis. The center of coordinate axes is deﬁned by the position of the Earth.

1.2.1 Instruments

There are nine instruments on board the WIND spacecraft. They are called WAVES, EPACT, SWE, SMS, MFI, 3-D-Plasma, TGRS, KONUS and SWIM. In the following part the instruments that are of special interest will be explained more detailed.

WAVES

On board the spacecraft there is a radio and plasma waves experiment called WAVES. The instrument is very sensitive and measures the properties of the waves which are partly radio waves emitted by the sun and the Earth carrying some of the energy ﬂow but also waves of the plasma. This will give an insight in the processes in the solar wind and of the interplanetary plasma. The radio waves that can be measured have frequencies from 20 kHz to 14 MHz. The

9 1.2. WIND spacecraft 1.2.1 instrument can also analyse electron thermal noise at a range from 4 kHz to 256 kHz as well as electric and magnetic fields with low frequencies from DC to 10 kHz. The experimental configuration consists of a magnetic search coil to measure bi-frequency magnetic fields, three antenna systems and five main receiver systems. The coil is triaxial and is mounted at the end of a 12-m radial boom so that the magnetic fields generated by the on board electronic do not have a big influence on the measurements. In the spin plane there are two coplanar wire antennas which are orthogonal to each other and one rigid spin-axis dipole. The first ones are dipoles of a size of 5.0 m and 7.5 m for each wire while the other one extends 5.28 m from opposite surfaces of the satellite. For measuring the above-named waves there are one bi-frequency receiver which works with Fast Fourier Transformation, one broadband electron thermal noise receiver, two swept-frequency radio receivers and one time domain waveform sampler. With the help of a data processing unit (DPU) the data from all operations can be controlled. This experiment has also interconnections to the 3-D PLASMA experiment and to SWE.

SWE

The WIND Solar Wind Experiment (SWE) includes three individual instruments: the Vector Electron Ion Spectrometers, a pair of Faraday Cups and the Strahl Detector. With the spectrometers, which show in three diﬀerent axes, measurements of 3D velocity distribution of ions and electrons with a Mach number below one can be made. For a chosen increasing sequence of modulated voltage values, the E/Q spectrum can be measured with the Faraday Cup. With the third instrument the narrow beam of electrons travelling from the sun in direction of the interplanetary magnetic ﬁeld, called solar wind strahl, can be explored with a high angular resolution within an energy range from 5 eV to 5 keV.

MFI

To study the structure and the fluctuation characteristics of the interplanetary magnetic field, the influence of the transport of energy, the acceleration of the particles in the solar wind and also the dynamic processes in the magnetosphere of the Earth a Magnetic Fields investigation (MFI) instrument is mounted on the WIND spacecraft. Its data are not only important for studying the solar wind but also for the interpretation of other measurements which have been done with WIND. The dual triaxial fluxgate magnetometers are mounted on top of a boom so that it is 12 meters away from the on board electronics, which can disturb the measurements due to their magnetic field. These sensors are analogue and send signals, which are proportional to the magnetic field strength. An A/D converter with a microprocessor converts the signals afterwards into digital form. The MFI provides very accurate measurements with a high resolution almost continuously and in real time. It also has a wide measurement range, which goes from 0.004 nT ± to 65.536 nT. ±

10 1.2. WIND spacecraft 1.2.2

3D PLASMA

The only one that measures three dimensional distribution of plasma, energetic ions and electrons is the 3D PLASMA instrument. It also provides information about the perturbations of the electron distribution function at wave-particle interactions and the relative energy resolution of 0.20 and of 0.30 (unitless) with an angular resolution of 5.6 5.6 for particles with an energy ◦× ◦ from 3 eV to 30 keV and with an angular resolution of 22.5 36 for an energy range from ◦× ◦ 20 keV to 11 MeV respectively as it was described in [21]. Further goals of this instrument are to measure the acceleration and the transport as well as to study the input and the output of particles from the magnetosphere of the Earth ([24]). It will cover the gap between the energy ranges which can be measured with the experiments SWE and EPACT. The instrument consists of three detector systems, one command and data handling system and a Fast Particle Correlator (FPC). These detector systems are semi-conductor telescopes (SST), the electron electrostatic analysers (EESA) and the ion electrostatic analysers (PESA) (see figures 1.6 and 1.7). The SST is a set of three arrays of semi-conductor detector telescopes where one of them is shown in figure 1.6. Each of them has two double-ended telescopes to measure the electron fluxes as well as the ion fluxes above about 20 keV. The telescopes itself consist of either two or three closely-sandwiched detectors. The detector is covered with a thin lexan foil on one end so that ions below 400 keV, which is the maximum energy of electrons, will be absorbed and therefore it leaves the electron spectrum nearly unchanged. On the other end a magnetic field generated by a common broom magnet sweeps away electrons with the energy below 400 keV. With this configuration the ions and the electrons are cleanly separated. The output of the detectors is taken in anti-coincidence each with the detector behind it to obtain a low background because most of the electrons, which have energies greater than 400 keV and penetrate the front detectors, are rejected due to the anticoincidence. The field of view of each of the telescopes is 36 20 ◦× ◦ and therefore all five telescopes cover an area of 180 20 for each of the two different kind of ◦× ◦ detectors. Since the spacecraft rotates through 360◦ the full four steradians of 4π will be covered ([20], [26]). The EESAs and PESAs measure the electron and ion population in an energy range of 3 eV to 30 keV. The structure is in such a way that the analysers study a symmetrical spherical section with a field of view of 360 180 . The pairs of the analysers on EESA and PESA are called ◦× ◦ EESA-L and -H and analogue PESA-L and -H. L stands for ’low’, while H is the sign for ’high’ which is related to energies. EESA-L measures in a range of 10 eV to 1.1 keV while EESA-H is responsible for electrons with energies between 135 eV and 27 keV. PESA-L analyses ions from 1.15 keV to 11.6 keV and suprathermal ions with an energy from 80 eV to 27 keV will be registered by PESA-H ([27], [28]).

1.2.2 Data information

All data measured with the 3DP instrument are stored in a database on a server in Berkeley, for example, diﬀerent countrates depending on the phi and theta value. Figure 1.8 illustrates

11 1.2. WIND spacecraft 1.2.2

Figure 1.6: An array of two double-ended telescopes on the solid state telescope shown from the front and from the side [20].

Figure 1.7: EESA and PESA of the 3-D PLASMA instrument [25] data from the 18th of February in 2000 over three hours. These data are the number of particles hitting different sectors on a sphere are counted from one of the different instruments, which is in this case the SST. The electron flux is plotted over the time for different energies. These energy values mark the mean values of each measurement band. It can be seen that from 09:40 UT on the flux increases significantly for all energies. The reason for that event is incoming particles originated from the sun. The point in time, which will be examined in chapter 2, is determined at about 09:45 UT, where there is a considerable rise in the electron flux. As it will be explained later on there are some sectors, which are subject to errors. These perturbations have their seeds for example in X-ray radiation, either direct or scattered, which has its source in a solar flare. Figure 1.8 was generated while these few bin data were removed before. For the further work only these countrates depending on the time and on the energy band as well as on the position will be important. The position is provided as the set of the values for the longitude and the latitude. The number of sectors varies with the instrument which is used. There are 48 for SST but 88 for EESA and PESA. The classification of the numbers of the different sectors of the SST to the dedicated angle values is shown in table 1.2. These 48 bins are composed of

12 1.2. WIND spacecraft 1.2.2

Figure 1.8: The electron flux plotted over the time range from 09:00 UT to 12:00 UT for the 18th of February in 2000 measured with the SST instrument. The different colours mark different energies from about 27.0 keV to 516.5 keV. These energy values mark the mean values of each measurement band.

40 bigger sectors and 8 smaller ones. The sectors, which have the numbers 20-23 and 44-47, are the smaller ones and contain only redundant information. In figure 1.9 there are three parts, which describe the behaviour of the electrons with an energy of 107 keV on the 18th of February in 2000 from 09:00 UT am to 13:00 UT. The behaviour of the particles looks similar for all the other energy channels so that it is not necessary to show all of the profiles. The different curves have different colours. The blue ones show the result of a numerical approximation with the assumption that the injection took an infinitely short moment of time so that it will expressed as a Dirac delta function. The measurements are demonstrated in black, while the curves that are coloured in red are theoretical graphs predicting the measurement values gained from the numerical solution of the equation of focussed transport. In reality the injection can not be seen as a short impulse since also the diffusion has to be considered for example and therefore the Reid-Axford profile as it was defined in equation 1.20 would be a better description for the release of the particles from the sun. The red graphs were not constructed according to this profile but by considering focussing and diffusion. As shown in the uppermost part of the figure several points in the injection curve have been defined following this method. By calculating the intensity and the anisotropy profile out of these points, again the numerical solution of the focussed transport equation was used, it can be seen that they are fit curves for the measurements. The injection took place at the time ti = 09:17 UT, which can also be seen in the first of the three parts of the figure 1.9, where the injections, which are normalised to

13 1.2. WIND spacecraft 1.2.2

2 1 1, is demonstrated. In the middle the intensity in (cm sr s MeV)− is shown. Below that the anisotropies can be seen. The deﬁnition of the anisotropy can be found in section 2.3. Since the decay of the intensity is very slow for this event, the parallel mean free path λ k is chosen to be constant at 1.2 AU. The velocity of the solar wind vSW is 380 km/s. For

0.5 Injection 0 2000 Feb 18 Wind 3DP

−1 107 keV Electrons 4 10 sr s MeV) 2

3 10 λ = 1.2 AU const

Intensity (cm || ti = 09:17, V = 380 km/s sw 3 2 1

Anisotropy 0 9 10 11 12 13 Time (hours)

Figure 1.9: The particle injection profile, the intensity and the anisotropy (see section 2.3) plotted over the a time range from 9 o’clock am to 12 noon for the 18th of February in 2000 for an energy of the electrons of about 107 keV. The blue lines result by an assumption that the injection of the particles is a delta function, the black curves describe the measured profile and the red lines illustrate the fit to the observations assuming an extended injection as shown in the upper panel. ([29]) one certain energy one example with the data from SST can be seen in the graphics 1.10 and 1.12. Figure 1.10 was created with Mathematica and the other one was done with the plot3d function provided by widl, an extension to the Interactive Data Language IDL. IDL is a software produced by the Research Systems, Inc. ”for data analysis, visualisation, and cross-platform application development“ ([30]). The function plot3d needs three kind of input - a collection of data, the latitude and the longitude of the zero point of the resulting illustration. The data in this example were measured on the 18th of February in 2000 for the energy range at around 66280.4 eV and the origin of the coordinate system, which was used for the illustration, was (0, 0). In the first line of the IDL plot there is information about the origin of the data. Below that the time range as well as the energy range of the measurements is provided. Some solar events happened since the first measurement of the instruments on WIND and one occurred on the date mentioned above. They can be distinguished from the background measurements due to their higher countrates and due to the fact that the data are not randomly distributed as it

14 1.2. WIND spacecraft 1.2.2

Table 1.2: Assignment of the sector numbers to the angles φ and θ for the SST instrument.

Sector number φ/◦ θ/◦ Sector number φ/◦ θ/◦ 0 0 72 24 180 72 1 270 72 25 90 72 2 22.5 36 26 202.5 36 3 337.5 36 27 157.5 36 4 292.5 36 28 112.5 36 5 247.5 36 29 67.5 36 6 225 0 30 45 0 7 202.5 0 31 22.5 0 8 180 0 32 0 0 9 157.5 0 33 337.5 0 10 135 0 34 315 0 11 112.5 0 35 292.5 0 12 90 0 36 270 0 13 67.5 0 37 247.5 0 14 213.75 -36 38 33.75 -36 15 168.75 -36 39 348.75 -36 16 123.75 -36 40 303.75 -36 17 78.75 -36 41 258.75 -36 18 180 -72 42 0 -72 19 90 -72 43 270 -72 20 202.5 -36 44 22.5 -36 21 157.5 -36 45 337.5 -36 22 112.5 -36 46 292.5 -36 23 67.5 -36 47 247.5 -36 can be seen in figure 1.14. In direct comparison of the data sets, which are illustrated in the figures 1.12 and 1.14 it can be seen that the flux values measured on the 18th of February are higher by one order of magnitude than for example on the 24th of December. Therefore it can be seen that there was an solar event in February. A direct incidence of the solar X-ray emission is seen at the position of the sectors 7, 8 and 9 and distorts the measurements there in creating much higher values for the countrates. Therefore a maximum which is some dimensions higher than the other data values would occur there and the real distribution would not be apparent. By setting the values to zero this problem can be solved. There are some more sectors which give wrong data like the sectors numbered with 31, 32 and 33, which are located in the anti- sunward direction. As those which were mentioned cause the highest perturbance only these were removed. In principle both figures (1.10 and 1.12) show the same, but the one, which was made with Mathematica is more exact. The example in figure 1.10 shows a much more detailed structure. From the data (see Table 1.3) it can be recognised that they are at the coordinates

θ = 0◦, φ = 180◦ and at the coordinates θ = 36◦, φ = 202.5◦. φ is the longitude and θ is the latitude coordinate. But in the other illustration only the lower peak with a countrate of 352 can be seen. The reason for the fact that the higher peak can not be seen is the binning of the data done by plot3d, by shifting the measurement values towards the centre of each sector. If the

15 1.2. WIND spacecraft 1.2.2

1.0 1.0

0.5 0.5 2304 17

Θ 0.0 Θ 0.0

-0.5 -0.5 0 0

-1.0 -1.0

0 1 2 3 4 5 6 0 1 2 3 4 5 6 Φ Φ

Figure 1.10: A contour plot of the countrates Figure 1.11: A contour plot of the coun- measured with SST from the 18th of February trates measured with EESA-H from the 18th in 2000 at about 09:45 UT as a function of of February in 2000 at about 09:45 UT as a φ and θ in radian. The colours are deﬁning function of φ and θ in radian. diﬀerent values according to the legend.

Figure 1.12: A ﬁgure of the data measured Figure 1.13: A ﬁgure of the data measured with SST from the 18th of February in 2000 with EESA-H from the 18th of February in at about 09:45 UT created with plot3dp. 2000 at about 09:45 UT created with plot3dp.

16 1.2. WIND spacecraft 1.2.2 maximum of the number of points that will be explicitly included in the output in Mathematica is reduced to 4 instead of 20, as it was set for drawing the figure, the higher peak disappears. The two peaks, which showed up in the illustration mentioned before, show the characteristics of a bimodal distribution. This can not be fitted with the von Mises-Fisher distribution, which will be introduced in section 2.1.1. A much better approximation of the data distribution can be given by the bimodal Kent distribution, which will be presented in section 2.1.2. The Mathematica figure is not completely quadratic since two edges are missing which can be explained due to the fact that there are no data for a theta value of 72 or 72 with a φ − bigger than 270. On Berkeley there are not only data available from the SST but also from the instruments EESA-L, EESA-H, PESA-L and PESA-H. For SST the available data for a period of time are given in an array with 7 columns and 48 rows. That means that there are data for 48 combinations of φ and θ values all for seven energy ranges. From EESA there are 1320 values, 88 data for different combinations of φ and θ for 15 energy ranges. The energies are discrete and the same for all periods of time. A list with all energy channels for the two different instruments can be seen in table 1.4. An error in the IDL program caused the error in the output of the energies so that one value is not available for the PESA-H instrument. This is noted in the table as NaN (not a number).

Figure 1.14: A ﬁgure of the background information measured with the SST instrument on WIND on the 24th of December at 8 o’clock pm. To eliminate disturbances the data of sectors with the numbers 7, 8 and 9 were set to zero.

17 1.3. Stereo mission

Table 1.3: The countrates of the event on the 18th of February 2000, dependent on the angles φ and Θ, measured at an approximate time of 09:45 UT for an energy of 66280.4 eV with the SST instrument.

φ/◦ θ/◦ countrate φ/◦ θ/◦ countrate 0 72 72.00 180 72 640.0 270 72 352.0 90 72 64.00 22.5 36 34.00 202.5 36 2304 337.5 36 25.00 157.5 36 336.0 292.5 36 52.00 112.5 36 50.00 247.5 36 1088 67.5 36 30.00 225 0 704.0 45 0 19.00 202.5 0 640.0 22.5 0 18.00 180 0 112.0 0 0 20.00 157.5 0 16.00 337.5 0 10.00 135 0 15.00 315 0 19.00 112.5 0 21.00 292.5 0 10.00 90 0 22.00 270 0 42.00 67.5 0 12.00 247.5 0 208.0 213.75 -36 152.0 33.75 -36 34.00 168.75 -36 56.00 348.75 -36 34.00 123.75 -36 25.00 303.75 -36 28.00 78.75 -36 36.00 258.75 -36 64.00 180 -72 92.00 0 -72 72.00 90 -72 72.00 270 -72 64.00 202.5 -36 4.000 22.5 -36 1.000 157.5 -36 2.000 337.5 -36 4.000 112.5 -36 2.000 292.5 -36 0.000 67.5 -36 3.000 247.5 -36 3.000

An example can be seen in the ﬁgures 1.11 and in 1.13. The measurements were taken on the 18th of February in 2000 at around 09:45:00 UT in the morning. It is quite striking to note that there are some gaps in the illustration 1.13. The reason for that is the missing data for these areas. The models presented in section 2.1 can be used for a reconstruction of such missing data. In the other illustration the same data were used in Mathematica. As it was explained before this was done since this type of plot is more precise. The advance for the other plot method is the better reconstruction of the spherical conﬁguration.

1.3 Stereo mission

The STEREO (Solar TErrestrial RElations Observatory) mission has the objective to analyse coronal mass ejections (CMEs) and to investigate the structure of the solar wind in a better way than it was done before. CMEs are ejections of plasma from the solar corona, which consists mainly of electrons, protons and parts of heavier elements ([31]). It is expected that the space weather forecast increases in accuracy due to a better understanding of the physical processes related to the CMEs ([32]). The NASA mission took place in Cape Canaveral, Florida ([33]),

18 1.3. Stereo mission

Table 1.4: The energies of the EESA and PESA instruments in eV. These energy values mark the mean values of each measurement band. EESA-H EESA-L PESA-H PESA-L 27662.7 1112.99 28433.7 9521.37 18944.4 689.161 21151.4 8131.28 12965.8 426.769 15730.1 6941.45 8874.88 264.837 11699.1 5926.36 6076.48 164.957 8701.90 5060.50 4161.27 103.320 6474.69 4322.26 2849.21 65.2431 4820.53 3690.04 1952.32 41.7663 NaN 3150.10 1339.37 27.2489 28433.7 2688.70 920.296 18.2895 21151.4 2294.05 634.385 12.8144 15730.1 1958.31 432.729 9.41314 11699.1 1671.65 292.064 7.25626 8701.90 1428.19 200.056 5.92896 6474.69 1218.10 136.845 5.18234 4820.53 in October 2006. It consists of two spacecrafts, which are nearly identical. Since one is moving ahead of the Earth and one behind they are called Ahead and Behind or STEREO-A and STEREO-B ([32], [34]). They depart from the Earth so that the angle between the Earth, the

Sun and the spacecraft increases by 22.5◦ each year as it is described in [33]. On each of the two spacecrafts there are four different instrument packages. The first is SECCHI, which is the abbreviation for Sun-Earth Connection Coronal and Heliospheric Investigation. It is a group of telescopes which consists of two white-light coronographs, one UV-camera and one heliospheric imager. PLASTIC (PLAsma and SupraThermal Ion and Composition) determines the velocity, the density and the temperature of the solar wind as well as the composition of suprathermal ions. SWAVES or STEREO/WAVES is the third experiment and used for measuring the solar radio bursts. The In-situ Measurements of PArticles and CME Transients instrument package IMPACT is a combination of seven different devices and analyses the three- dimensional distribution of electrons in the solar wind plasma, the vector of the magnetic field and the solar energetic particle (SEP) ions and electrons. One of the the instruments from IMPACT is the Solar Electron and Proton Telescope (SEPT). It provides “the three-dimensional distribution of energetic electrons and protons with good energy and time resolution.” ([34]) Two dual double-ended telescopes count electrons in an energy range from 30 keV to 400 keV and protons with energies from 60 keV to 7000 keV. Electrons and protons are separated by the same principle with magnets and an absorption foil as it was used in 3D-PLASMA and described in section 1.2.1. The telescope SEPT-E is located to look in the ecliptic plane along the Parker spiral magnetic field and SEPT-NS is looking vertical to that plane ([34]). 16 detectors are measuring the energy, the direction of incidence of the electrons and protons ([32]). With this information the release of energy and the acceleration of particles in the outer part of the sun and in the interplanetary space as well as the correlation with

19 1.3. Stereo mission measurements with wavelength in the optical and in the radio wave regions can be found out. Since the SEPT instrument registers electrons in four diﬀerent directions, each with a conical

ﬁeld of view of 52.8◦, there are no measurements of the whole sphere available ([34]). For a better analysis and a better comparison to other measurements a reconstruction of the missing values could be advantageous. The problem of not complete data will be examined in section 2.2.6.

20 Chapter 2

Analysis and Methods

2.1 Model

There are several distributions for modelling spherical data as for example the Kent distribution, the von-Mises-Fisher distribution or the Watson distribution. These are all functions depending on θ and φ, which also have other inputs like for example the mean direction or a concentration parameter. A distribution is deﬁned by all of its moments if they exist. This is the case with all of the functions which will be described in this section. The kth central moment is deﬁned by

k mk(r)= E (X r) , (2.1) − where E is the expected value in probability theory. This is the probability density function times the arguments integrated over the whole area over which it is defined. The first four moments are the most interesting ones because their influences can be seen easily in the shape of the distribution. The first central moment is the expected value itself, the second central moment is known as the variance. For k = 3 the standard deviation can be calculated by taking the normalisation of the third moment. Analogue the kurtosis can be derived for k = 4 ([35]). The standard deviation and the kurtosis can be used to measure the variation from the normal distribution in the one dimensional case as well as from its generalisation for higher dimensions, the multivariate normal distribution or multivariate Gaussian distribution, especially in the two dimensional case ([36]).

2.1.1 von Mises-Fisher distribution

”The von-Mises-Fisher distribution is an unimodal and symmetric distribution and can be used as an all-purpose model for directions in the plane” ([37]). It is a probability distribution on the (p 1)-dimensional sphere in Rp. If for example p = 2, the von Mises-Fisher distribution − is described on the circle. Because of these properties it is well suited for modelling the data provided by the WIND spacecraft. The distribution corresponds to the multivariate Gaussian

21 2.1. Model 2.1.1 distribution ([38]). The probability density form of this distribution is

T fmis(~x, ~µ, κ)= Cp(κ) exp(κ~µ ~x) (2.2)

[39], where the vector ~x is a p-dimensional random unit vector, ~µ is called the mean direction and κ is the concentration parameter. It has to be satisfied that ~µ = 1 and κ 0. The || || ≥ meaning of κ is illustrated in the figure 2.1. There are three graphs, the blue one for a value of κ 1 of 3 , the red one for κ = 1 and the yellow curve was drawn for κ = 4. For a better differentiation the mean vector of these three curves is chosen differently. The normalisation constant Cp is defined as κp/2 1 C (κ)= − . (2.3) p p/2 (2π) Ip/2 1(κ) − The function Ip/2 1(κ) is known as the modified Bessel function of the first kind and of the − order p. Here it is a distribution in R3 so p = 3. In this case the von Mises-Fisher distribution is also known as the Fisher distribution or as the von Mises-Arnold-Fisher distribution as it is also described in [37]. The first trigonometric moment of this distribution is dependent on the parameter κ and is given by 1 + exp ( 2κ) 1 1 ρ = − = coth κ (2.4) 1 exp ( 2κ) − κ − κ − − where the second trigonometric moment is

4 4 4ρ α = 1 coth κ + = 1 . (2.5) − κ κ2 − κ

One example of the von Mises-Fisher distribution can be seen in the figures 2.2 and 2.3. The first one is an illustration dependent on φ and θ. κ is set to 5, φ goes from 0 to 2π and the θ π π value is in the range from 2 to 2 . The values in z-direction represent the density. The mean −T 1 1 direction ~µ is 0, √ , √ . The next depiction (figure 2.3) shows the so called Mollweide 2 2 projection of the same function as used in figure 2.2. This graphical view was introduced by the German mathematician Karl Brandan Mollweide in 1805 ([40]). It is a non-geometric map projection mostly used for geographic maps of the world and it is an area accurate mapping. Other names for it are Babinet projection, homolographic projection or elliptical projection ([41]). The two dimensional function for the projection is

2√2 η η f (λ, ζ) = (( )λ cos ( ), √2 sin ( )). (2.6) Mollw. π 2 2

η is an auxiliary angle and is deﬁned by the equation η+sin (η)= π sin (ζ) while λ is the longitude from the central meridian and ζ is the latitude ([42]). Another alternative for an elliptical projection is the so called Hammer-Aitov projection, which has a lower angular distortion, but it requires much more mathematical eﬀort.

The colours in ﬁgure 2.3 represent the density. The density will be as higher as the rainbow colours change to purple. Red denotes 0 and purple 0.7. The mean direction is the same as in

22 2.1. Model 2.1.2

Κ = 13

f Κ = 1

Κ = 4 0.6

0.4

0.2

Θ 1 2 3 4 5 6 rad

Figure 2.1: Two dimensional plots of the von Mises-Fisher distribution plotted with Mathe- matica for diﬀerent values of κ over the angle θ. The three graphs are shifted in relation to each other for a better distinction.

ﬁgure 2.2 but κ is chosen as 5.

2.1.2 Kent distribution

The Kent distribution or 5-parameter Fisher-Bingham distribution is also defined as a two dimensional function on the unit sphere ([43],[37]). It was first described in 1982 by Kent, constructed as a generalisation of the Fisher distribution and of another one, called Bingham distribution. The density function is dependent on five parameters, which are denoted as κ, β,

~γ1, ~γ2 and ~γ3. The ﬁrst two variables are shape parameters, which are non-negative. κ is, as in section 2.1.1, the parameter to describe the degree of concentration and β is known as the ovalness parameter. The higher the value of β the more ovaloid is the shape of the data. The relation between these two parameters is

κ 2 (2.7) β ≥ for an unimodal distribution and κ < 2 (2.8) β if it is a bimodal distribution. The other three are location parameters. ~γ1 is the mean direction,

~γ2 is the major axis of the distribution and ~γ3 is the minor axis. The whole function is given by

f (~x)= C exp (κ ~γ ~x + β[(~γ ~x)2 (~γ ~x)2]) (2.9) kent κ · · 1 · 2 · − 3 · while the Cκ is again the normalisation factor. It is deﬁned by

1 Cκ = 2r . (2.10) 3 1 (2r)! β (2π) 2 κ− 2 ∞ I 1 r=0 r!r! κ 2r+ 2 (κ) P 23 2.1. Model 2.1.2

10 Density 5 1

0 0 0Θ

2 Φ 4 -1

Figure 2.2: A three dimensional illustration of the von Mises-Fisher distribution plotted with Mathematica. The density depending on the angles φ and θ is shown here. The value of κ was T 1 1 set to 5, while the mean vector µ is equal to 0, √ , √ . 2 2

In this relation I2r is like in section 2.1.1 the modiﬁed Bessel function. Cκ can be written in an approximate formula if κ is large. This more simple equation is

κ2 4β2 C exp ( κ) − . (2.11) κ ≈ − p 2π

Two illustrations of the Kent distribution can be seen in the figures 2.4 and 2.5. The first graphic was made for ~γ = (0, 1, 0)T , ~γ = ( 1, 0, 0)T and a mean direction 2 3 − ~γ = (cos (0.8) cos (2) 0.5, sin (0.8) cos (2) 0.5, sin (2) 0.5)T . κ was chosen as 10 and the 1 − − − second shape parameter β is equal to 1, which fulfils equation 2.7. This example is therefore a unimodal distribution with mode at ~γ1 and with a maximum density of Cκ exp (κ). In the figure

Figure 2.3: A three dimensional illustration of the von Mises-Fisher distribution plotted with Mathematica as a Mollweide projection. The parameters of the von Mises-Fisher distribution are the same as in ﬁgure 2.2. The red colour is related to 0, while purple remarks the highest value which is 0.7.

24 2.2. Simulation of spherical data the density was plotted with respect to θ and φ. The second example was done for the bimodal

10 Density 6 5

4 0 Φ -1 2 Θ0

1 0

Figure 2.4: A three dimensional plot of the Kent distribution in the monomodal case plotted with Mathematica. The density depending on the angles φ and θ is shown here. Parameters used: ~γ = (0, 1, 0)T ,~γ = ( 1, 0, 0)T ,~γ = (cos (0.8) cos (2) 0.5, sin (0.8) cos (2) 0.5, sin (2) 0.5)T , 2 3 − 1 − − − κ = 10 and β is equal to 1. case (figure 2.5). The parameter κ was set to 1 as well as β so that equation 2.8 is fulfilled. Here the parameters are the same as in the monomodal example before but the mean direction T was chosen as ~γ1 = (cos (0.8) cos (2), sin (0.8) cos (2), sin (2)) and κ = 1. This distribution has one big advantage to the von Mises-Fisher distribution, which was described above. Since there are more parameters undefined, the distribution can also deal with asymmetric data ([44]). It can be constructed as a bimodal distribution in spite of the von Mises-Fisher function which is just unimodal and rotational symmetrical.

2.2 Simulation of spherical data

For analysing the data from the WIND spacecraft in a better way it is necessary to study if the functions introduced in section 2.1 match the measured data distribution. Since every distribution is dependent on some free parameters the ﬁrst step that has to be made is to ﬁnd the values of these parameters according to the measurements. The second step would be the analysis itself.

Since the distributions are normalised through the deﬁnition of the factors Cp and Cκ respectively, all function values are limited to 1. As the countrates are not normalised and are for example for the 18th of February in 2000 at about 09:45 UT for an energy of 66280.4 eV in the range up to 24, the distribution functions will be changed so that they also make use of the full value range. For this purpose a factor F , which is independent of any other parameter, is introduced. The values of F will be elements of the positive real numbers R+. The von Mises-Fisher

25 2.2. Simulation of spherical data

4 3 Density 6 2 1 4 Φ -1 2 0 Θ 1 0

Figure 2.5: A three dimensional plot of the Kent distribution in the bimodal case plotted with Mathematica. The density depending on the angles φ and θ is shown here. Parameters used: β,~γ2 and ~γ3 are the same as in figure 2.4, but κ was changed to 1 and ~γ1 was chosen as (cos (0.8) cos (2), sin (0.8) cos (2), sin (2))T . function for fitting the data is defined for the following as

f (~x, ~µ, κ, F )= F C (κ) exp(κ~µT ~x). (2.12) mis · p

The Kent function changes analogue to the deﬁnition above to

f (~x, κ,~γ ,~γ ,~γ ,β,F )= F C exp κ γ x + β[(γ x)2 (γ x)2]. (2.13) kent 1 2 3 · κ · · 1 · 2 · − 3 ·

As the only purpose of the factors Cp and Cκ is the normalisation, they are unnecessary for the modelling and the reconstruction of data. Therefore they can be omitted. In function 2.12 there are three parameters which can be changed. The ﬁrst one is the mean direction and this is equal to the mean direction of the WIND data. The second one is κ, which can be changed in such a way that the modelled data and those from the measurements are as similar as possible, and the last one is the factor F as described above. In the other function

(2.13) there are also the free parameters κ and F and also the mean direction ~γ1. As described above there are three parameters more to find. To get all of the parameters or at least one of them of a set of data there are some possibilities. If all of them are found it can be proved with the help of Pearson’s chi-square test if the parameters are chosen in a way so that its result is as small as possible. The chi-square Goodness-of-Fit test is defined as n (S M )2 χ2 = i − i , (2.14) M Xi=1 i where Si is the fit value for every combination of phi and theta and Mi is the related measured datum. This test has the advantage that it is a simple and fast technique to get an idea about

26 2.2. Simulation of spherical data 2.2.1 the order of magnitude of the diﬀerences between the measured data and the modelled data. The assumption for the success of the chi-square test is that the measurements are all very good. If for example just one value of the measured data, which is completely diﬀerent from the expected value than the test, will give a very high value. This can occur if there is one big failure in the measurements.

2.2.1 Calculating the mean, the major and the minor axis of the data separately

There are several methods to calculate a parameter of a distribution separately depending on the kind of parameter. As the minor and the major axis of the Kent distribution the direction of the maximum and minimum of the data respectively can be found. For calculating the mean direction, which is needed for the von Mises-Fisher distribution as well as for the Kent distribution, two possibilities will be given. The ﬁrst one is a computation of the mean values for every diﬀerent value of theta. Taking out the highest mean, where the related theta value is called θmean, is the second step of this procedure. Getting the phi value of the mean direction φmean can be done in the same way as calculating θmean. θmean and φmean specify the mean direction with T (cos (φmean) cos (θmean), sin (φmean) cos (θmean), sin (θmean)) . The assumption that the data are symmetrically distributed around the maximum leads to the second possibility. In this case the maximum would be equal to the mean and therefore the same values can be used. The problem with this idea is that even it can be used for a distribution with more parameters than the mean direction, the major and the minor axis, another method has to be used to calculate the remaining parameters.

2.2.2 NMinimize

In Mathematica for example exists a function implemented called NMinimze which can be used to minimise a function trough several parameters and due to some restrictions concerning the parameters. It was first introduced in version 5 and is able to use several methods, that do not calculate derivatives so that the objective function does not have to be differentiable or continuous. In general, NMinimize finds global minima and not just local ones. If the function itself and also the constraints are nonlinear it could be the case that only a local minimum can be found. For the linear case this Mathematica function uses simplex and revised simplex methods but in the nonlinear case the Nelder-Mead method combined with differential evolution is taken ([45]). The Nelder-Mead algorithm was found in 1965 from the mathematicians J. A. Nelder and R. Mead and is an effective and computational compact method. It can find a solution for a multidimensional function dependent on several parameters which is the case for the spherical distributions introduced in the sections 2.1.1 and 2.1.2 ([46], [47]). The constrains that have been made for the von Mises-Fisher distribution are

F 0 • ≥

27 2.2. Simulation of spherical data 2.2.3

κ> 0 and κ< 10000 (as recommendation) •

for all θ in radian π <θ< π • 2 − 2

for all φ in radian 0 <φ< 2π. •

2.2.3 FindFit

There is also a third method to get the parameters for the best fit of the available data with a known distribution that can be used with Mathematica. There is a function defined in the program, which is called FindFit. It gives the best fit parameters for a set of data and a given fit function. This function can be used for linear fits as well as for nonlinear ones. In the first case it will find a globally optimal fit but in general it will find only a locally optimal fit in nonlinear case ([45]). The method that is used internally in FindFit is for linear cases a singular value decomposition and for nonlinear cases the Levenberg-Marquardt method, which is a relatively easy and also very popular algorithm using the gradient descent and the inverse Hessian method as it is described in [48]. It can be even chosen which of the following procedures should be used, the conjugate gradient, gradient, Levenberg-Marquardt, Newton and Quasi-Newton. If nothing is selected the choice it is done automatically according to the previous named rules. A second problem is that the parameters have to be limited to a certain maximum so that the program will stop after a while. This is only important if the accuracy goal can not be reached. So it was chosen that κ shall be below 10,000 like it has been done for the second method. Also for φ, θ and F the same constraints as before were taken. The algorithm for the function FindFit starts the search for a best fit with the parameter which is the first one that is written down in the list of parameters in the input. If the function would calculate the optimal fit in an analytical way, this order would make no difference in the output. But since it has to be determined numerically the result can depend on parameter with which the algorithm will start. In the case of the used data taken from SST this problem did not occur.

2.2.4 FindFit with start values

One problem mentioned in section 2.2.3 which can occur by using FindFit is that the algorithm finds not the global but the local minimum. To avoid this, FindFit has the function that some starting values can be specified manually. The better these values match the best fit parameters, the better the chance to find the global minimum. These values are chosen in that way that they can be varied within a relatively big interval without changing the result. An indication for a rough guess can be a calculation of the parameters with other methods like for example with NMinimize or a figure as for example 1.10 of the measured data. Also the methods as they were described in section 2.2.1 can be used. Out of this for example the approximate major axis for a fit with the Kent distribution can be found out. Finding parameters like κ or β it is much more difficult.

28 2.2. Simulation of spherical data 2.2.5

2.2.5 Comparison of the results of the diﬀerent methods

The different methods as they were described in the sections 2.2.1, 2.2.2, 2.2.3 and 2.2.4 have been tested for the data measured on the 18th of February in 2000 on the SST instrument as well as those from EESA-H. For all of the modellings the von Mises-Fisher distribution, introduced in section 2.1.1, was used. At the end of this chapter some fits have also been done with the Kent distribution, which was described in section 2.1.2. The data measured on the 18th of February in 2000 at a time of 09:45 UT measured with the SST instrument varies from 0 to 2304, which is shown in table 1.3, and the countrates from the EESA-H instrument, which will be analysed, also are varying between 0 and 17. A comparison of the range of the original data with the one of the modelled data can give a first estimation of the goodness of modelling. The results of modelling based on SST data should be compared with the figures 1.10 and 1.12, the ones based on EESA-H should be compared with the figures 1.10 and 1.13.

Data from SST Table 2.1 shows the resultant parameters found with the functions NMinimize and FindFit. It can be seen that the results for the parameters F and κ from NMinimize and FindFit are not distinguishable if start values were used. For symmetric radially distributed data the maximum direction should be the same as the mean direction. Therefore the start values for φm and θm are chosen to be the φ and the θ value of the maximum direction as it was described in section 2.2.1. The start values for κ and C were found by analysing the spreading of the data and the maximum value respectively. With the chosen start values the resultant parameters do also not vary in a wide range. The start values were found similar for all the further examples. Hence the start values for the FindFit method applied to the data measured on the 18th of February at 09:45 UT with the SST instrument are

C = 2000 • κ = 2 • φ = 0.2π • m θ = 0. • m

Table 2.1: The parameters for the von Mises-Fisher function found with diﬀerent methods. The ﬁts were based on the data measured on the 18th of February in 2000 with SST on WIND.

Method F κ φm/rad θm/rad NMinimize 2328 6.674 3.716 0.6397 FindFit with start values 2328 6.674 3.716 0.6397 FindFit without start values 2011 5.863 10 13 5.845 -0.5328 · −

An illustration of the results with the methods NMinimize and FindFit can be seen in ﬁg- ures 2.6 and 2.7. The illustrations look very similar, which leads to the assumption that the

29 2.2. Simulation of spherical data 2.2.5

1.5 1.5

1.0 1.0

0.5 2302 0.5 2302

Θ 0.0 Θ 0.0

-0.5 -0.5

0 0 -1.0 -1.0

-1.5 -1.5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Φ Φ

Figure 2.6: A contour plot of the data mod- Figure 2.7: A contour plot of the data modelled with the von Mises-Fisher function over elled with the von Mises-Fisher function over the angles φ and θ, where the parameters were the angles φ and θ, where the parameters were calculated with NMinimize. The fit is based calculated with FindFit. The fit is based on on the data measured on the 18th of Febru- the data measured on the 18th of February in ary in 2000 at about 09:45 UT with SST on 2000 at about 09:45 UT with SST on WIND. WIND. The figure was done with Mathemat- The figure was done with Mathematica. ica.

Figure 2.8: Modelling of the flux with the Figure 2.9: Modelling of the flux with the von Mises-Fisher function, where the param- von Mises-Fisher function, where the parameters were calculated with NMinimize. The eters were calculated with FindFit. The fit is fit is based on the data measured on the 18th based on the data measured on the 18th of of February in 2000 at about 09:45 UT with February in 2000 at about 09:45 UT with SST SST on WIND. The figure was done with the on WIND. The figure was done with the func- function plot3d of IDL. tion plot3d of IDL.

30 2.2. Simulation of spherical data 2.2.5

procedures come to a similar result and therefore also to similar values of φm and θm. This can be investigated by analysing the differences. The maximal difference between these is about 0.06. A plot, which is a demonstration of the differences, is figure 2.10. The figures 2.8 and 2.9 show as well the modelling of the measured data as it was demonstrated in the figures 2.6 and 2.7. In the examples constructed with Mathematica the symmetric and especially the uniform characteristics of the calculated distributions are easier to see whereas the three dimensional formation can be identified much better in the other two illustrations. The original data are depicted in the figure 1.10 as well as in the figure 1.12. The first one is a plot, which was done with Mathematica, while the second one was generated on the basis of the same data set with an IDL function. By comparing the four pictures with those of the measured data, it can be seen that they have the same structure and they resemble each other. Also the maximum countrate, which was measured as 2304, is about the same for both modellings as it can be seen in the legends of the illustrations. Therefore it becomes apparent that a good modelling within the realms of possibility can be found with the use of both methods. The result found with FindFit by using no start values is not shown. The reason for that is that this did not provide a reasonable modelling. The data, which have been calculated with this kind of 9 method lie in an interval, which has a range smaller than 10− . The modelling can also be done

1.0

0.5 0.02

Θ 0.0

-0.5 -0.04

-1.0

0 1 2 3 4 5 6 Φ

Figure 2.10: A contour plot of the diﬀerences between the data modelled with FindFit and NMinimize on the basis of the countrates measured on the 18th of February in 2000 at 09:45 UT with the SST instrument as they were illustrated for example in the ﬁgures 2.6 and 2.7. The colours, which specify the values of the countrates, are mapped over the angles φ and θ. with other data. Later even measurements from another instrument will be used.

Data from EESA The same procedure as before was also applied to the data measured with the EESA-H instrument, which has more values as it was explained in section 1.2.2. For the same reason as in the case, which is the same as here but with the SST data, only the results found with NMinimize and FindFit are presented here but not the outcome for the third method. In that case the diﬀerence between the maximum and the minimum of the data 18 calculated with FindFit without using any start values is even smaller with 10− . Since the

31 2.2. Simulation of spherical data 2.2.5

1.5 1.5

1.0 1.0

0.5 12 0.5 12

Θ 0.0 Θ 0.0

-0.5 -0.5

0 0 -1.0 -1.0

-1.5 -1.5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Φ Φ

Figure 2.11: A contour plot of the data Figure 2.12: A contour plot of the data modelled with the von Mises-Fisher function, modelled with the von Mises-Fisher function, where the parameters were calculated with where the parameters were calculated with FindFit with the start values mentioned in NMinimize. The modelling is based on the section 2.2.5. The modelling is based on the data measured on the 18th of February in 2000 data measured on the 18th of February in 2000 at about 09:45 UT with EESA-H on WIND. at about 09:45 UT with EESA-H on WIND. measurement data are not constant as it can be seen for example in table 1.3, the output does not fulfil the expection for a modelling. The resultant modellings can be seen in figures 2.11 and 2.12. The symmetric behaviour is easy to see, also that the maxima have about the same coordinates as the original data, which can be seen in figure 1.11. One deviation besides of the exact shape is that both of the modelled distributions have a maximum value of 12 and the maximum countrate of the measured data is 17. A comparison with the Mollweide projection of the data as seen in figure 1.13 can be also done with the pictures 2.13 and 2.14. Here the maxima of the data can be compared as well as the distribution of the data. Only the sectors 4 and 9 have higher values than it is expected. The reason for that is an error in the data transfer with IDL. Just as well as before it can be recognised that both of the fit methods do not differ visibly from each other. This can be explained by the fact that the two algorithms returned the same parameters listed in table 2.2. The start values that were used are again chosen as it was described in 2.2.1. For the constant C the maximum value of the original data was taken, κ is an estimation of the degree of variance and the parameters φm and θm are set to define the position of the maximum direction. This yields to the following values:

C = 17 • κ = 3 • φ = 0 • m θ = π . • m 16 The resultant parameters of these start values, which deﬁne the von Mises-Fisher distribution, can be seen in table 2.2.

32 2.2. Simulation of spherical data 2.2.5

Figure 2.13: Modelling of flux with the von Figure 2.14: Modelling of the flux with the Mises-Fisher function, where the parameters von Mises-Fisher function, where the param- were calculated with FindFit with the start eters were calculated with NMinimize. The values mentioned in section 2.2.5. The fit fit is based on the data measured on the 18th is based on the data measured on the 18th of February in 2000 at about 09:45 UT with of February in 2000 at about 09:45 UT with EESA-H on WIND. The illustration was done EESA-H on WIND. The illustration was done with the function plot3d of IDL. with the function plot3d of IDL.

Table 2.2: The parameters for the von Mises-Fisher function found with diﬀerent methods. The ﬁts were based on the data measured on the 18th of February in 2000 with EESA-H on WIND.

Method F κ φm/rad θm/rad NMinimize 10.17 13.17 3.155 0.05613 FindFit with start values 10.17 13.17 3.155 0.05612 FindFit without start values 15.99 5.863 10 19 0.03126 0.09009 · −

Simulations with the Kent distribution A modelling of the data was not only done with the von Mises-Fisher distribution but also with the Kent function introduced in section 2.1.2. Only the procedure NMinimize was successful. For the other two methods the algorithm did not converge to a tolerance of about 4.8 10 6 no matter how many iterations have been done. This · − tolerance value is defined in Mathematica internally. Anyway Mathematica returned the best fit parameters found with the FindFit methods. These nine parameters each that have been found with the three algorithms are presented in table 2.3. For the FindFit method without specifically denounced start values the procedure did not work at all. Therefore there are no results presented in this section. Instead the FindFit was used on the one hand for the case that a monomodal function is a best fit for the measurements and for the bimodal case on the other hand. Although only the condition as it was written in the equations 2.8 and 2.7 respectively differs, the resultant values disagree. The outcomes of the methods can be seen in the figures 2.15 and 2.16, which can be compared with the illustrations of the measured data as demonstrated in 1.12. The FindFit algorithm applied to the monomodal Kent distribution could not model the behaviour of the particles

33 2.2. Simulation of spherical data 2.2.5

Table 2.3: The parameters for the Kent function found with diﬀerent methods. The ﬁts were based on the data measured on the 18th of February in 2000 with SST on WIND.

Method C κ β φm/rad θm/rad NMinimize 1.289 5535 2.245 4.314 0.6290 FindFit; monomodal 151.7 2.237 1.119 0.09040 1.571 FindFit; bimodal 881.1 283.1 282.1 3.292 0.00005697

Method φ2/rad θ2/rad φ3/rad θ3/rad NMinimize 5.568 -1.571 5.030 -1.557 FindFit; monomodal 0.5455 -0.3726 0.5082 1.198 FindFit; bimodal 3.284 -0.2021 3.263 -0.001487

Figure 2.15: Modelling of the flux with the Figure 2.16: Modelling of the flux with the monomodal Kent function, where the param- bimodal Kent function, where the parame- eters were calculated with FindFit with the ters were calculated with FindFit without any start values mentioned in section 2.2.5. The start values. The fit is based on the data mea- fit is based on the data measured on the 18th sured on the 18th of February in 2000 at about of February in 2000 at about 09:45 UT with 09:45 UT with SST on WIND. The illustration SST on WIND. The illustration was done with was done with the function plot3d of IDL. the function plot3d of IDL. as good as the one applied to the bimodal function. Also the full range of the data was only reproduced by the FindFit method for the bimodal distribution. An illustration of the data calculated with the help of NMinimize is not shown because the output values reveal that the result are not reasonable. The maximum of the data is in an order of an unreasonable high number whereas the minimum was calculated to be around zero. The figures 2.15 and 2.16 look similar to each other even if the parameters are not the same as it was mentioned before. Some cases like the modelling, where all parameters of the von Mises-Fisher distribution were calculated with the FindFit method without specified start values for the algorithm, yield no realistic results. That means that the output does not describe the appearance of the data that have to be fitted. This happened for the data found with SST as well as for these measured with EESA. Out of this reason it will not be used for the reconstruction of the data, which will be described in the next section. Both, FindFit with indicated start values and NMinimize, give

34 2.2. Simulation of spherical data 2.2.6 useful results in all test cases.

2.2.6 Reconstruction of data

Almost no instrument, which measures spherical data, provides complete and correct data for every position. There are different reasons for that. Sometimes the instrument is not built for complete 360◦ measurements and in other cases some part are not working as good as others. In all cases it could be helpful to reconstruct the missing of false data. As it can be seen in figure 1.12 the 48 different data values define 48 bins, numbered from 0 to 47. Some bins do not provide information, which matches the other data. One example to demonstrate this is shown in figure 2.17. The data from the bins which are totally different from

Figure 2.17: A figure of the flux with respect to the pitch angle cosine with the data of all sectors from 0 to 47 measured on the 18th of February in 2000 from SST on WIND. the others are from the bins 8 and 9 as well as the sectors 22, 23 and 31, 32 and in some cases also the one with the number 7. One example for incomplete measurements is the STEREO mission described in section 1.3. The instrument measures only in cones with an aperture angle of 60◦. Since the data for STEREO are not available it has to be modelled with the data from 3D-PLASMA. For a similar reconstruction of the situation all bins except for those which are on both of the poles and some opposed bins on the sides (see figure 1.12) have to be taken away. As it was explained before the bins 7, 8, 9, 31, 32 and 33 will not be chosen. One example of bins that can be taken is the combination of the bins 24 and 25, which are on the North pole of the arrangement showed in figure 1.12, bins 42 and 43 on the opposite side and 11, 12, 13 as well as 35, 36 and 37, where θ is equal to zero. This is demonstrated in figure 2.18. To reconstruct the missing data the von Mises-Fisher distribution will be used. A reconstruction of missing data was done for the data measured on the 18th of February in 2000.

35 2.2. Simulation of spherical data 2.2.6

Figure 2.18: A Mollweide projection for the measured data from 18th of February 2000 at about 09:45 UT from SST on WIND as a function of phi and theta where only the bins with the numbers 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43 are visible. The illustration was done with the function plot3d of IDL.

Reconstruction for the event of the 18th of February in 2000 Two examples for the reconstructed data can be seen in the ﬁgures 2.19 and 2.20. For their generation the data as demonstrated in illustration 2.18 were used. The parameters of the distribution shown in the pictures 2.19 and 2.20 are presented in table 2.4. The result, which was calculated with the FindFit method without stating any start values, was like in some cases before, the best one that could be found in spite of that the algorithm did not converge according to Mathematica. As start values

C = 640 • κ = 2 • φ = π • m θ = 0.4π • m were chosen according to one of the rules mentioned in 2.2.1.

Table 2.4: The parameters for the von Mises-Fisher function found with diﬀerent methods. The ﬁts were based on the data measured on the 18th of February in 2000 with SST on WIND taken only from the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43.

Method F κ φm/rad θm/rad NMinimize 4586 6.509 0.2284 2.682 FindFit with start values 4604 6.510 3.369 0.4589 FindFit without start values 0.001000 1.000 0.001000 0.001000

36 2.3. Physical meaning

Figure 2.19: A reconstruction of the data Figure 2.20: A reconstruction of the data from 18th of February 2000 at about 09:45 UT from 18th of February 2000 at about 09:45 UT measured at SST on WIND was taken only for measured at SST on WIND was taken only for the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43. The remaining data were modelled and 43. The remaining data were modelled with help of the function NMinimize. The il- with help of the function FindFit with the lustration was done with the function plot3d start values mentioned in section 2.2.5. The of IDL. illustration was done with the function plot3d of IDL.

In the first figure all sectors were filled with data fitted with NMinimize, for the second figure FindFit was used. Only for the data of the ten sectors that were used for the reconstruction the measured countrates were taken. By comparing both of the pictures with was done for the measured values (figure 1.12) it can be seen they look very similar. The maximum stays at the same sector and the shape of the models looks also equal to the original data plotted with the IDL function plot3d. Only the range of the measurement values is double as wide for the modelled data as for the measured data. In spite of the different parameters for φm and

θm as it was shown in table 2.4, which occured due to the fact that the fit methods that were used are not stable for only ten data values, the outputs are similar to each other and also to the measured data. The reason for that is the symmetry of the sine and the cosine. These trigonometric functions were used to calculate for example the mean direction in the karthesian coordinates with the angles φm and θm. In this case the mean directions are about the same for both of the algorithms NMinimize and FindFit with manually denounced start values. Only the figure 2.21 looks different. It was generated with the help of the function FindFit without an manual indication of start values. The the data vary from 0 to 2304, just as well as the original values but they are not distributed in the same way.

2.3 Physical meaning

The diﬀerential intensity or the energy spectrum I is a measured variable which can be achieved by in-situ measurements in the interplanetary space and characterises the components of the solar wind like electrons or protons completely. I(~x, E, n,tˆ )dEdtdΩdσ can be explained as the

37 2.3. Physical meaning

Figure 2.21: A reconstruction of the data from 18th of February 2000 at about measured at SST on WIND was taken only for the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43. The remaining data were modelled with help of the function FindFit without any start values. The illustration was done with the function plot3d of IDL. number of particles within an energy range from E to E + dE that are falling in the solid angle Ω on the area dσ, which is itself deﬁned by the position vector ~x, in a time interval t. Some of the parameters are also shown in picture 2.22. The diﬀerential intensity is mainly dependent on the directionn ˆ and therefore it can be seen

Figure 2.22: The diﬀerential intensity ([3]). as an angular distribution. This yields that the omnidirectional intensity is deﬁned as

I(~x, E, n,tˆ )dΩ I¯(~x, E, t) := . (2.15) R R dΩ R R 38 2.3. Physical meaning

Two diﬀerent possibilities can occur. The angular distribution is called isotropic if the diﬀerential intensity is not dependent onn ˆ. In this case the intensity I can be written as

I(ˆn)= I¯ (2.16) for everyn ˆ. I¯ is the averaged intensity, which can be calculated as the sum of all data of one event for one energy range divided by the number of data. In the second case the calculation is more diﬃcult. The anisotropy can be deduced from the particle current density S~, which is deﬁned by

S~(~x, E, t)= I(~x, E, n,tˆ )dΩ. (2.17) ZZ

If the distribution is symmetrical the equation for S~ can be written as

2π π

S~(~x, E, t)= I(~x, E, ϑ, t) cos (ϑ)dΩ eˆB (2.18) Z Z · φ=0 ϑ=0 1

= 2π I(~x, E, α, t)αdα eˆB, (2.19) Z · α= 1 − wheree ˆB is the unit vector in the direction of the magnetic field and ϑ is known as the pitch angle, which was defined in section 1.1.2. α is defined as the cosine of ϑ as already defined in section 1.1.3 and so particles with an pitch angle of 90◦ do not contribute to the particle current density because these particles do not flow in the direction of the magnetic field. For ϑ = 0 the particles have the highest velocity in the direction of the magnetic field and therefore they have the most influence on S~. The connection of the anisotropy denoted as A~ and the particle current density S~ is as following:

3 S~ A~(~x, E, t)= | | sˆ . (2.20) vU · 0 v is the velocity of the particles and U is the diﬀerential particle density deﬁned by

1 U(~x, E, t)= I(~x, E, n,tˆ )dΩ. (2.21) v ZZ

Equation 2.20 means that A~ is the directed flow S~ compared with the flow for all particles 1 1 moving in a volume with a differential particle density U with a velocity of 3 v. The value 3 comes from the three orthogonal directions in space. Under the assumption that the particles gyrate around the same axis equation 2.20 changes to

1 3 α= 1 I(~x, E, α, t)αdα A~(~x, E, t)= − eˆ . (2.22) R 1 · B α= 1 I(~x, E, α, t)dα R −

39 2.3. Physical meaning

The probability to be in every interval [φ, φ+dφ] is the same for every particle, which means that they are equally distributed for the angle φ, while φ is the azimuth angle. Here the anisotropy has the direction of the vector of the magnetic field B eˆ ([9], [49]). · B Since the assumptions match with the von Mises-Fisher distribution this can be taken as pitch angle distribution. As described in section 1.1.2 the pitch angle is the angle between the direction of the magnetic field B~ and the direction of the particles ~s. In the ideal case the vector of the magnetic field is the symmetric axis of the distribution. Since the event of the 18th of February in 2000 is quite normal and equally distributed the data measured on this day, the data can be taken for calculating the anisotropy with formula 2.22. In this case the maximum direction can be used as the direction of the magnetic field B~ . For that the pitch angle ϑ can be calculated | | as B~ ~s cos ϑ = · . (2.23) B~ ~s | || | Since the distribution is on a sphere the length of the vectors of the particle direction and of the magnetic field direction are both 1 and the equation 2.23 reduces to

Bx sx α = cos ϑ = B~ ~s =  B   s  = B s + B s + B s . (2.24) · y · y x x y y z z  B   s   z   z 

With I(~x, E, α, t)= fmises(κ, ~µ, ~x, α) the anisotropy can be calculated as

1 3 α= 1 fmises(κ, ~µ, ~x, α)αdα A~ (~x, E, t)= − eˆ ([50]) (2.25) mises R 1 · B α= 1 fmises(κ, ~µ, ~x, α)dα R − Insertion of the used deﬁnition of the von Mises-Fisher distribution from formula 2.12 in equation 2.25 yields to

1 T 3 α= 1 F Cp(κ) exp(κ~µ ~x)αdα A~mises(~x, E, t)= − · eˆB R 1 T · α= 1 F Cp(κ) exp(κ~µ ~x)dα − · R 1 T 3 α= 1 exp(κ~µ ~x)αdα = − eˆB R 1 T · α= 1 exp(κ~µ ~x)dα − R 1 ~ 3 α= 1 exp(κB ~x)αdα = − · eˆ R 1 ~ · B α= 1 exp(κB ~x)dα − · R 1 3 α= 1 exp(κα)αdα = − eˆ R 1 · B α= 1 exp(κα)dα − R κα 1 1 exp (κα) κ−2 1 = − eˆB 1 1 κ exp (κα) 1 − 3 [exp (κ)(κ 1) exp ( κ)( κ 1)] = − − − − − eˆ κ [exp (κ) exp ( κ)] B − − 1 =3(coth (κ) )ˆe − κ B

40 2.3. Physical meaning

For the Kent distribution the anisotropy is deﬁned as

1 3 α= 1 fkent(κ, ~µ, ~x, α)αdα A~ (~x, E, t)= − eˆ kent R 1 · B α= 1 fkent(κ, ~µ, ~x, α)dα R − analogue to the equation 2.25. Inserting the deﬁnition from section 2.1.2 this can be calculated as

1 2 2 3 α= 1 Cκ exp (κ ~γ1 ~x + β[(~γ2 ~x) (~γ3 ~x) ])αdα A~kent(~x, E, t)= − · · · · − · eˆB R 1 2 2 α= 1 Cκ exp (κ ~γ1 ~x + β[(~γ2 ~x) (~γ3 ~x) ])dα − · · · · − · R 1 ~ ~ 2 ~ 2 3 α= 1 exp (κ B ~x + β[(B ~x) (B ~x) ])αdα = − · · · − · eˆB R 1 ~ ~ 2 ~ 2 α= 1 exp (κ B ~x + β[(B ~x) (B ~x) ])dα − · · · − · R 1 2 2 3 α= 1 exp (κ α + β[(α) ( α) ])αdα = − · − − eˆB R 1 2 2 α= 1 exp (κ α + β[(α) ( α) ])dα − · − − R 1 3 α= 1 exp (κ α + β 0)αdα = − · · eˆ R 1 B α= 1 exp (κ α + β 0)dα − · · R κα 1 1 exp (κα) κ−2 1 = − eˆB 1 1 κ exp (κα) 1 − 3 [exp (κ)(κ 1) exp ( κ)( κ 1)] = − − − − − eˆ κ [exp (κ) exp ( κ)] B − − 1 =3(coth (κ) )ˆe − κ B with the assumption that ~γ3 is independent of B~ . As it was said beforee ˆB can be taken as the maximum direction, in the case of the Kent distribution as well as in the case of the von Mises-Fisher distribution.

In the calculation of A~kent(~x, E, t) it can be seen that the solution is only dependent on the shape parameters of the distribution, like for the derivation of A~mises(~x, E, t). In figure 2.23 the dependence of the anisotropy on the parameter κ is shown. If A~(~x, E, t) as the relation of the direct flow will increase the part of the directed flow and the not directed flow will increase, the part of the directed flow will also increase. This means that the distribution of the particle flow will be more directed. These characteristics are directly dependent on κ, which was already shown in figure 2.1. For a higher concentration parameter κ the distribution will be more directed. The function as is was seen in the plot 2.23 converges to 3 with an increase of κ. This behaviour can be verified by introducing an delta function as the intensity in the definition of the anisotropy as it was defined in equation 2.22, which give the same result. This matches the relation to the anisotropy as it was calculated above. It can be observed that A~ (~x, E, t) is the first moment of the von Mises-Fisher distribution | mises | as it was mentioned in section 2.1.1. Since the meaning of the first trigonometric moment in the used case can be compared with the variance on a sphere, which defines the degree of broadening,

41 2.3. Physical meaning

Anisotropy 3.0

2.5

2.0

1.5

1.0

0.5

Κ 10 20 30 40 50

Figure 2.23: The dependence of the anisotropy, calculated with the von Mises-Fisher distribution and the Kent distribution, on κ. the calculated moment and the anisotropy signify the same. For every time as for example for the 18th of February in 2000 at 09:45 UT for a specific energy one magnetic field vector can be measured and therefore one anisotropy value can be calculated. The derived magnitude A~(~x, E, t) is demonstrated in table 2.5 for all the different | | situations, which were discussed before in section 2.2.5. First the anisotropy was calculated for the measurements modelled with the von Mises-Fisher distribution for the events of the 18th of February and afterwards it was done for the modelling with the Kent distribution. By analysing

Table 2.5: The Anisotropy calculated for different fits of the events, which occurred on the 18th of February in 2000. It is also distinguished between the instrument, which was measuring the specific data on WIND, if it was modelled for the reconstruction as well as the algorithms for simulating the data. Since the anisotropy is only dependent on the parameter κ, it is also included in the table. The function, which was used for all the modellings, is the von Mises-Fisher distribution. Instrument Simulation method Reconstr. κ Anisotropy SST NMinimize No 6.674 0.8502 SST FindFit with start values No 6.674 0.8502 SST FindFit without start values No 5.863 10 13 0.0 · − EESA-H NMinimize No 13.17 2.772 EESA-H FindFit with start values No 13.17 2.772 EESA-H FindFit without start values No 5.863 10 19 0.0 · − SST NMinimize Yes 6.509 0.8464 SST FindFit with start values Yes 6.510 0.8464 SST FindFit without start values Yes 1.000 0.3130 the table 2.5 it can be seen that the values calculated for the measurements taken on the 18th of February at 09:45 UT with the SST instrument and modelled with the NMinimize and the FindFit algorithm with manually set start values have about the same value. Therefore it can be assumed that the anisotropy is about 0.85 in this case. The data measured with EESA-H are

42 2.3. Physical meaning modelled only three times whereof two procedures returned reasonable results. For these two algorithms the anisotropy was calculated as 2.772.

Table 2.6: The Anisotropy calculated for diﬀerent ﬁts of the events, which occurred on the 18th of February in 2000 and was measured with the SST instrument on WIND. It is distinguished between the algorithms for simulating the data. Since the anisotropy is only dependent on the parameter κ, they are also included in the table. The function, which was used for all the modellings, is the Kent distribution.

Simulation method κ Anisotropy NMinimize 5535 2.999 FindFit; monomodal 2.237 2.679 FindFit; bimodal 283.1 2.205

The analysis for the case that the angular distribution is anisotropic are completed. In the isotropic case the differential intensity can be calculated as described in equation 2.16. This is the average intensity for a special time range. For the data from the 18th of February in 2000 at 09:45 UT measured with the SST it is 47.5833 /s and for the data measured with the EESA-H it is 1.27273 /s. The flux as defined in equation 2.18 was also calculated and compared with the flux from the measured data, which can be seen in figure 2.24. On this picture only the data from the sectors, which give error-free data, are shown. The problems with the wrong data was mentioned in section 1.2.2.

Figure 2.24: A plot of the ﬂux with respect to the pitch angle cosine for the measured data from 18th of February 2000 from SST on WIND only for the sectors with the numbers 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43.

43 2.3. Physical meaning

Flux cm2 eV s sr

0.1

0.01

0.001 Pitch angle cosine Α -0.5 0.0 0.5 1.0 rad

Figure 2.25: A plot of the ﬂux with respect to the pitch angle cosine for the data modelled with the von Mises-Fisher distribution with respect to the data measured on the 18th of February 2000 from SST on WIND.

44 Chapter 3

Outcome

3.1 Issues that occurred

There were several issues which occurred while working on this task. The major ones will be shortly mentioned and a solution is given if appropriate:

As it was denoted before the FindFit method and the NMinimize method do not always • ﬁnd the global minima but local onew. Also the increase in the number of iterations did not solve this problem.

Since it is not possible to insert matrices with a structure of 88 15 because the dimensions • × are to big, a way around had to be found. The solution was to process the whole list of data element-wise and compose smaller structures that can be joined afterwards when writing to a ﬁle.

Some of the measurement bins provide perturbed or erroneous values. It should be avoided • to rely on this sectors for reconstruction of data points and also these should be disregarded in calculations.

3.2 Result

The von Mises-Fisher distribution and the Kent distribution were used as a model for the data from the instruments SST and EESA on 3D PLASMA. In section 2.2 on the one hand the measurement data were fitted and on the other hand a reconstruction of some data was done. As it was shown the model works relatively good. In the modelling of the reconstruction of the situation, as it nearly occurred for the mission STEREO, the model operates well. Three different kind of methods have been used to model and reconstruct the data. The FindFit method did not work out in most cases as it was explained before. In principle both of the remaining procedures, NMinimize and FindFit with indicated start values, worked out well and returned similar results. Therefore the way of choosing the start values as it was used and described in section 2.2 can be determined to be useful. Only in one case different results were produced for these two types. The model of the Kent distribution did not work fine for the

45 3.3. Further work

NMinimize algorithm as it could be seen in section 2.2.5 but for the FindFit algorithm. The reason for that might be that the algorithm cannot cope with so many parameters since the Kent distribution is dependent on more parameters than the von Mises-Fisher distribution. Fitting a function in nine parameters is much more difficult than to fit it only for four and therefore it is more probable that it leads not to the best result. But the model of the Kent distribution worked only out for the bimodal case as it was said also in section 2.2.5. But by comparing not only the shape and the range of the models that were found but also the anisotropy values it can be seen that these values differ from each other. The anisotropies calculated with the Kent function are not only different from each other but also from the approximate value of 0.85 found out with the von Mises-Fisher distribution. For this reason it can be said that the Kent distribution is not such a suitable model for the used purpose. It is obvious that it is more easy to model data, which have a wider range in the values and those, that are more distributed over the sphere. For the reconstruction it is even harder to handle not wide spreaded data as only four sections with measurement values can be used and therefore sometimes the main part of the data is not considered. Altogether it can be said that it is possible on the one hand to model this kind of data with the von Mises-Fisher distribution and on the other hand to reconstruct it if only data from four different directions is available.

3.3 Further work

Since the von Mises-Fisher distribution is not ideal for every set of data as it is measured on spacecrafts like WIND because it is not flexible enough for using it for example on asymmetric data, other distribution functions have to be tested. The conditions for such a function, which had to be thought to model spherical data, are that they can describe for as many shapes as possible. Some other well-known distributions on a sphere are the Point distribution, the Spherical Uni- form distribution, the Watson distribution and the Wood distribution. For the first two, which were mentioned, it is not reasonable to analyse them for the purpose of simulating data mentioned before, because normally the data are not concentrated on one point or equally distributed in all directions, which are the assumptions for these two functions. Even if these cases would occur, a reconstruction on the basis of such a distribution would be dispensable. A more flexible function is needed to model all kind of data sets, which could be thought of in connection to spacecraft measurements. Since the Watson distribution is rotationally symmetric, like the von Mises-Fisher distribution, this would not be a solution in the present case. The last remaining distribution would be the Wood distribution. It is only dependent on five parameters, which are the same as the four of the von Mises-Fisher distribution and the location parameter β. A further task would be to test the modelling and reconstruction with this function.

46 List of Figures

1.1 The interplanetary magnetic field frozen in the solar wind [8]. The arrows, which are pointing away from the sun describe the velocity of the solar wind. The other arrowspecifiestherotationofthesun...... 3 1.2 Movement of a particle in a homogeneous magnetic field with an illustration of

the gyration radius and the pitch angle ([9]). The axes x1, x2 and x3 deﬁne

the coordinate system, rgyro is the gyroradius, ϑ is the pitch angle and B is the magneticfield...... 4 1.3 Illustration of the definition of the pitch angle ϑ ([9]). The vector ~v is the direction of a particle splitted up in the perpendicular component ~v and the parallel ⊥ component ~v ...... 5 k 1.4 The WIND satellite and its instruments [17]...... 8 1.5 The orbit of the WIND spacecraft from the 16th of November over two years [18]. The unit is in Earth radii for the X and Y axis. The center of coordinate axes is definedbythepositionoftheEarth...... 9 1.6 An array of two double-ended telescopes on the solid state telescope shown from thefrontandfromtheside[20]...... 12 1.7 EESA and PESA of the 3-D PLASMA instrument [25] ...... 12 1.8 The electron flux plotted over the time range from 09:00 UT to 12:00 UT for the 18th of February in 2000 measured with the SST instrument. The different colours mark different energies from about 27.0 keV to 516.5 keV. These energy values mark the mean values of each measurement band...... 13 1.9 The particle injection profile, the intensity and the anisotropy (see section 2.3) plotted over the a time range from 9 o’clock am to 12 noon for the 18th of February in 2000 for an energy of the electrons of about 107 keV. The blue lines result by an assumption that the injection of the particles is a delta function, the black curves describe the measured profile and the red lines illustrate the fit to the observations assuming an extended injection as shown in the upper panel. ([29]) 14 1.10 A contour plot of the countrates measured with SST from the 18th of February in 2000 at about 09:45 UT as a function of φ and θ in radian. The colours are defining different values according to the legend...... 16 1.11 A contour plot of the countrates measured with EESA-H from the 18th of Febru- ary in 2000 at about 09:45 UT as a function of φ and θ inradian...... 16

47 LIST OF FIGURES LIST OF FIGURES

1.12 A figure of the data measured with SST from the 18th of February in 2000 at about 09:45 UT created with plot3dp...... 16 1.13 A figure of the data measured with EESA-H from the 18th of February in 2000 at about 09:45 UT created with plot3dp...... 16 1.14 A figure of the background information measured with the SST instrument on WIND on the 24th of December at 8 o’clock pm. To eliminate disturbances the data of sectors with the numbers 7, 8 and 9 were set to zero...... 17

2.1 Two dimensional plots of the von Mises-Fisher distribution plotted with Mathe- matica for diﬀerent values of κ over the angle θ. The three graphs are shifted in relation to each other for a better distinction...... 23 2.2 A three dimensional illustration of the von Mises-Fisher distribution plotted with Mathematica. The density depending on the angles φ and θ is shown here. The T 1 1 value of κ was set to 5, while the mean vector µ is equal to 0, √ , √ . . . . 24 2 2 2.3 A three dimensional illustration of the von Mises-Fisher distribution plotted with Mathematica as a Mollweide projection. The parameters of the von Mises-Fisher distribution are the same as in ﬁgure 2.2. The red colour is related to 0, while purpleremarks the highest value which is 0.7...... 24 2.4 A three dimensional plot of the Kent distribution in the monomodal case plotted with Mathematica. The density depending on the angles φ and θ is shown here. Parameters used: ~γ = (0, 1, 0)T ,~γ = ( 1, 0, 0)T ,~γ = (cos (0.8) cos (2) 0.5, 2 3 − 1 − sin (0.8) cos (2) 0.5, sin (2) 0.5)T , κ = 10 and β isequalto1...... 25 − − 2.5 A three dimensional plot of the Kent distribution in the bimodal case plotted with Mathematica. The density depending on the angles φ and θ is shown here.

Parameters used: β,~γ2 and ~γ3 are the same as in figure 2.4, but κ was changed T to 1 and ~γ1 was chosen as (cos (0.8) cos (2), sin (0.8) cos (2), sin (2)) ...... 26 2.6 A contour plot of the data modelled with the von Mises-Fisher function over the angles φ and θ, where the parameters were calculated with NMinimize. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with SST on WIND. The figure was done with Mathematica...... 30 2.7 A contour plot of the data modelled with the von Mises-Fisher function over the angles φ and θ, where the parameters were calculated with FindFit. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with SST on WIND. The figure was done with Mathematica...... 30 2.8 Modelling of the flux with the von Mises-Fisher function, where the parameters were calculated with NMinimize. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with SST on WIND. The figure was donewiththefunctionplot3dofIDL...... 30 2.9 Modelling of the flux with the von Mises-Fisher function, where the parameters were calculated with FindFit. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with SST on WIND. The figure was done withthefunctionplot3dofIDL...... 30

48 LIST OF FIGURES LIST OF FIGURES

2.10 A contour plot of the differences between the data modelled with FindFit and NMinimize on the basis of the countrates measured on the 18th of February in 2000 at 09:45 UT with the SST instrument as they were illustrated for example in the figures 2.6 and 2.7. The colours, which specify the values of the countrates, are mapped over the angles φ and θ...... 31 2.11 A contour plot of the data modelled with the von Mises-Fisher function, where the parameters were calculated with FindFit with the start values mentioned in section 2.2.5. The modelling is based on the data measured on the 18th of February in 2000 at about 09:45 UT with EESA-H on WIND...... 32 2.12 A contour plot of the data modelled with the von Mises-Fisher function, where the parameters were calculated with NMinimize. The modelling is based on the data measured on the 18th of February in 2000 at about 09:45 UT with EESA-H onWIND...... 32 2.13 Modelling of flux with the von Mises-Fisher function, where the parameters were calculated with FindFit with the start values mentioned in section 2.2.5. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with EESA-H on WIND. The illustration was done with the function plot3d ofIDL...... 33 2.14 Modelling of the flux with the von Mises-Fisher function, where the parameters were calculated with NMinimize. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with EESA-H on WIND. The illustration was done with the function plot3d of IDL...... 33 2.15 Modelling of the flux with the monomodal Kent function, where the parameters were calculated with FindFit with the start values mentioned in section 2.2.5. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with SST on WIND. The illustration was done with the function plot3d ofIDL...... 34 2.16 Modelling of the flux with the bimodal Kent function, where the parameters were calculated with FindFit without any start values. The fit is based on the data measured on the 18th of February in 2000 at about 09:45 UT with SST on WIND. The illustration was done with the function plot3d of IDL...... 34 2.17 A figure of the flux with respect to the pitch angle cosine with the data of all sectors from 0 to 47 measured on the 18th of February in 2000 from SST on WIND. 35 2.18 A Mollweide projection for the measured data from 18th of February 2000 at about 09:45 UT from SST on WIND as a function of phi and theta where only the bins with the numbers 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43 are visible. The illustration was done with the function plot3d of IDL...... 36 2.19 A reconstruction of the data from 18th of February 2000 at about 09:45 UT measured at SST on WIND was taken only for the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43. The remaining data were modelled with help of the function NMinimize. The illustration was done with the function plot3d of IDL...... 37

49 LIST OF FIGURES LIST OF FIGURES

2.20 A reconstruction of the data from 18th of February 2000 at about 09:45 UT measured at SST on WIND was taken only for the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43. The remaining data were modelled with help of the function FindFit with the start values mentioned in section 2.2.5. The illustration was donewiththefunctionplot3dofIDL...... 37 2.21 A reconstruction of the data from 18th of February 2000 at about measured at SST on WIND was taken only for the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43. The remaining data were modelled with help of the function FindFit without any start values. The illustration was done with the function plot3d of IDL. . . . 38 2.22 The differential intensity ([3])...... 38 2.23 The dependence of the anisotropy, calculated with the von Mises-Fisher distribution and the Kent distribution, on κ...... 42 2.24 A plot of the flux with respect to the pitch angle cosine for the measured data from 18th of February 2000 from SST on WIND only for the sectors with the numbers 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43...... 43 2.25 A plot of the flux with respect to the pitch angle cosine for the data modelled with the von Mises-Fisher distribution with respect to the data measured on the 18thofFebruary2000fromSSTonWIND...... 44

50 List of Tables

1.1 Propertiesofthesun[1],[3] ...... 1 1.2 Assignment of the sector numbers to the angles φ and θ for the SST instrument. 15 1.3 The countrates of the event on the 18th of February 2000, dependent on the angles φ and Θ, measured at an approximate time of 09:45 UT for an energy of 66280.4eVwiththeSSTinstrument...... 18 1.4 The energies of the EESA and PESA instruments in eV. These energy values mark the mean values of each measurement band...... 19

2.1 The parameters for the von Mises-Fisher function found with different methods. The fits were based on the data measured on the 18th of February in 2000 with SSTonWIND...... 29 2.2 The parameters for the von Mises-Fisher function found with different methods. The fits were based on the data measured on the 18th of February in 2000 with EESA-HonWIND...... 33 2.3 The parameters for the Kent function found with different methods. The fits were based on the data measured on the 18th of February in 2000 with SST on WIND. 34 2.4 The parameters for the von Mises-Fisher function found with different methods. The fits were based on the data measured on the 18th of February in 2000 with SST on WIND taken only from the sectors 11, 12, 13, 24, 25, 35, 36, 37, 42 and 43. 36 2.5 The Anisotropy calculated for different fits of the events, which occurred on the 18th of February in 2000. It is also distinguished between the instrument, which was measuring the specific data on WIND, if it was modelled for the reconstruction as well as the algorithms for simulating the data. Since the anisotropy is only dependent on the parameter κ, it is also included in the table. The function, which was used for all the modellings, is the von Mises-Fisher distribution. . . . 42 2.6 The Anisotropy calculated for different fits of the events, which occurred on the 18th of February in 2000 and was measured with the SST instrument on WIND. It is distinguished between the algorithms for simulating the data. Since the anisotropy is only dependent on the parameter κ, they are also included in the table. The function, which was used for all the modellings, is the Kent distribution. 43

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55 Acknowledgement

This master thesis was written during my studies at the Institut f¨ur Theoretische Physik und Astrophysik in W¨urzburg as well as in the Institut of Space Science in Lule˚a. It is my pleasure to express my gratitude to all the people who contributed, in whatever manner, to the success of this work. Particularly

Professor Dr. Wolfgang Dr¨oge for providing me an opportunity to conduct my master • thesis under him as well as for his support during this period,

all the students and doctorands working at the Institut f¨ur Theoretische Physik und Astro- • physik for the very good working atmosphere and for helping me, especially with computer related issues.

I would especially like to thank Oliver Kurz, not only for helping me by reviewing the whole work but, more importantly, for encouraging me at every step of a sometimes diﬃcult journey. I am also very thankful for the strong support of my parents, which occured not only in a ﬁnancial way, to ensure my studies, which I greatly appreciate. Declaration of Authorship

I, Katharina Nowak, conﬁrm that this work submitted for assessment is my own and is expressed in my own words. Any uses made within it of the works of any other author, in any form (e.g. ideas, equations, ﬁgures, text, tables, programs), are properly acknowledged at their point of use. A list of the references used is included.

Signed: Date: Appendix A

Appendix

A.1 Notations

It is important to distinguish the different notations of distributions, in particular of the von Mises-Fisher distribution. The most common notations are the probability density function (pdf ) and the probability density element (pde) ([37]). The pdf is defined as the derivative of the distribution function as described in [51] whereas the connection between pdf, denoted as g(θ, φ), and the pde h(θ, φ) can be described by g(θ, φ) = h(θ, φ) sin (Θ). Sometimes even the standardised notation is referred to be the definition of the distribution without mention that it is a special case. In this form the axis of symmetry is chosen as (0, 0 , 1)) . In most definitions it is not said which form they use. One example is written down on [52]. This is the pdf of the standardised form of the von Mises-Fisher distribution. Sometimes there are also different denotations of the same form of a distribution. In these cases it has to be shown that they are the same. This was found with the two definitions of the von Mises-Fisher distributions, which were used for this work. The one found in 2.2 looked unequal to the one mentioned in the book [37]. It will be demonstrated that even if they look different, they do not describe different functions. In [37] the function of the distribution was written as

f(θ, φ)= C exp (κ(sin (θ) sin (α) cos (φ β) + cos (θ) cos (α))) (A.1) · − and the other one was

κ f(θ, φ)= exp (κ cos (θ)) sin (θ). (A.2) 4π sinh (κ)

Now the function deﬁned in equation A.1 without the normalisation factor will be transformed so that it can be seen that it corresponds to the function 2.2, also without the factor C:

exp (κ(sin (θ) sin (α) cos (φ β) + cos (θ) cos (α))) − cos (θ) sin (φ) = exp (κ  cos α sin (β) sin (α) sin (β) cos (α)  sin (θ) sin (φ)    cos (θ)    

58 A.2. Mathematica notebooks

cos (θ) sin (φ) With ~µT = cos (α) sin (β) sin (α) sin (β) cos (α) and ~x =  sin (θ) sin (φ)  it can be  cos (θ)    seen that it is equal to equation 2.2 without the factor C. Now both of the factors have to be compared.

p/2 1 equation 2.3 κ C (κ) = − p p/2 (2π) Ip/2 1(κ) 3 − 1 p=3 κ 2 − = 3 2 (2π) Ip/2 1(κ) 1− κπ 2 = 3 3 1 2 2 π 2 2 2 sinh (κ) κ = , 4π sinh (κ) which is the same as deﬁned in [37]. Now it is shown that both deﬁnitions are the same.

A.2 Mathematica notebooks

Attached are two Mathematica notebooks, Mises-distribution.nb and Kent-distribution.nb, which have been used for the computation of the modelling functions.

59 Clear@"Global‘*"D Needs@"PlotLegends‘"D

The Fisher-distribution as a special case of the Von Mises-Fisher Distribution for dimension 2, i.e. on a circle

1 Κ Cos Θ-Μ misfunc1@Θ_, Μ_, Κ_D := ã @ D 2 Π BesselI@0, ΚD normMF[p_,Κ _] defines the normalisation factor for p-dimensional Von Mises-Fisher distribution

Κp2-1 normMF@p_, Κ_D := p2 H2 ΠL BesselI@p 2 - 1, ΚD misfunc2 defines the 3-dimensional von Mises-Fisher distribution using the above normalisation factor. misfunc3 defines the 3-dimensional von Mises-Fisher distribution using the above normalisation factor and an additional factor F.

ΚΜ.x misfunc2@x_, Μ_, Κ_D := normMF@3, ΚD ã

ΚΜ.x misfunc3@x_, Μ_, Κ_,C_D := C normMF@3, ΚD ã

PlotKVerschieden2D = Plot@8misfunc1@Θ, 1, 0.3‘D, misfunc1@Θ, 2, 1D, misfunc1@Θ, 3, 4D<, 8Θ, 0, 2 Π<, AxesLabel ® 8Θ rad, f<, PlotLegend ® 8"Κ = 13", "Κ = 1", "Κ = 4"<, LegendPosition ® 80.3, 0.2<, LegendShadow ® 80, 0

Export@"PlotKVerschieden2D.eps", PlotKVerschieden2DD;

T Μ x = Μ1 x1 + Μ2 x2 + Μ3 x3 Μ = 1 È È as well as x = 1 È È Π Π so that we can write for - £ Θ < ; 0 £ Φ < 2 Π K 2 2 O x = r Cos Θ Cos Φ ; @ @ xD @ xD r Cos Θ Sin Φ ; @ xD @ xD r Sin Θ @ xD withD r = 1 and Μ = r Cos Θ Cos Φ ; A A ΜE A ΜE r Cos Θ Sin Φ ; A ΜE A ΜE r Sin Θ A ΜE E with r = 1 ΜT x = Cos Θ Cos Φ Cos Θ Cos Φ + Cos Θ Sin Φ Cos Θ Sin Φ + Sin Θ Sin Θ @ xD @ xD A ΜE A ΜE @ xD @ xD A ΜE A ΜE @ xD A ΜE

r = 1; x@Φ_, Θ_D := 8r Cos@ΘD Cos@ΦD , r Cos@ΘD Sin@ΦD, r Sin@ΘD<; 2 Mises-distribution.nb

Plot of an example of the von Mises - Fisher distribution.

Π Π Show Plot3DK2MisesFisher = Plot3D misfunc3 x ph, th , x , , 2, 18 , B B B @ D B 2 4 F F Π Π ph, 0, 2 Π , th, - , , PlotRange ® All, AxesLabel ® "Φ", "Θ", "Density" , 8 < : 2 2 > 8

findroot@Φ_D = Module@8a = Pi<, fr = FindRoot@2 * Θ + Sin@2 *ΘD a * Sin@ΦD, 8Θ, 0

ShowBParametricPlot3DB:H2 * Sqrt@2D *Λ* Cos@findroot@ΦDDL Pi, Π Π Sqrt 2 * Sin findroot Φ , 0, FaceForm Hue misfunc2 x Λ, Φ , x , , 5 , @ D @ @ DD B B B @ D B 2 4 F FFF> 8Λ, -Pi, Pi<, 8Φ, -1 2 * Pi, 1 2 * Pi<, FormatType ® OutputForm, PlotRange ® All, PlotPoints ® 40, ViewPoint ® Above, ColorFunction ® "Rainbow"F,

DisplayFunction ® HPopupWindow@Button@"Click here"D, ðD &LF

MollweideEx = ParametricPlot3DB:H2 * Sqrt@2D *Λ* Cos@findroot@ΦDDL Pi, Π Π Sqrt 2 * Sin findroot Φ , 0, FaceForm Hue misfunc2 x Λ, Φ , x , , 5 , @ D @ @ DD B B B @ D B 2 4 F FFF> 8Λ, -Pi, Pi<, 8Φ, -1 2 * Pi, 1 2 * Pi<, FormatType ® OutputForm, PlotRange ® All, PlotPoints ® 40, Axes ® False, ViewPoint ® Above, ColorFunction ® "Rainbow", Boxed ® FalseF; Off@FindRoot::nlnumD Off@ReplaceAll::repsD Off@ParametricPlot3D::legacycolfuncD

ShowBParametricPlot3DB:H2 * Sqrt@2D *Λ* Cos@findroot@ΦDDL Pi, Sqrt@2D * Sin@findroot@ΦDD, Π Π Π Π misfunc2 x Λ, Φ , x , , 5 , FaceForm Hue misfunc2 x Λ, Φ , x , , 5 , Λ, -Pi, Pi , B @ D B 2 4 F F B B B @ D B 2 4 F FFF> 8 < 8Φ, -1 2 * Pi, 1 2 * Pi<, FormatType ® OutputForm, PlotRange ® All, ViewPoint ® 81.3, -2.4, 2

DisplayFunction ® HPopupWindow@Button@"Click here"D, ðD &LF Off@ParametricPlot3D::legacycolfuncD Export@"MollweideEx.eps", MollweideExD;

Import of other data: From originally three different arrays of the size 7x48, for phi, theta, related data the second column of each combined in phithetadata. The data comes from Berkeley loaded with idl and was get with get_sf(tr).

phithetadata1 = Import@"homekanowakDocuments18-02-00-phithetadata.dat", "Table"D; phithetadataall = Import@"homekanowakDocuments18-02-00-phithetadataall.dat", "Table"D; phithetadata = phithetadata1; For@i = 1, i £ Length@phithetadataD, i++, phithetadata@@i, 1DD = phithetadata@@i, 1DD; phithetadata@@i, 2DD = phithetadata@@i, 2DD; phithetadata@@i, 3DD = phithetadataall@@i, 3DDD phithetadata; Mises-distribution.nb 3

phithetadatarad = Table@8phithetadata@@i, 1DD Degree, phithetadata@@i, 2DD Degree, phithetadata@@i, 3DD<, 8i, 1, Length@phithetadataD

Max@Table@phithetadatarad@@i, 3DD, 8i, Length@phithetadataradD

Show@ShowLegend@ListContourPlot@phithetadatarad, PlotRange ® All, AxesLabel ® 8Φ, Θ<, ColorFunction ® "LakeColors"D, 8ColorData@"LakeColors"D@1 - ð1D &, 7, "2304", "0", LegendPosition ® 81.1, -0.4<, LegendSize ® 84, 8<

H*ListContPlPhiThetaData= ShowLegend@ListContourPlot@phithetadatarad,PlotRange®All,AxesLabel®8Φ,Θ<, FrameLabel®8Φ,Θ<,ColorFunction®"LakeColors"D,8ColorData@"LakeColors"D@1-ð1D&, 7,"2304","0",LegendPosition®81.1,-0.4<,LegendSize®84,8<

Calculation of the parameters of the von Mises-Fisher distribution so that the distribution fits as good as it gets the data from WIND/phithetadata.

Settings for following minimizing algorithm

maxiterations = 20 000; minkappa = 0; maxkappa = 10 000; seed = 0;

’squares’ defines the sum of quadratic differences of the model function and the data table. ’squares’ is minimized using ’NMinimize’ for the free parameters Κ the, concentration parameter,

and the mean direction Μ , a vector defined by {ΘΜ , ΦΜ= with r = 1 Calculation of the parameters of the von Mises-Fisher distribution with a kind of a chi-square test where its result will be minimized with the funtion NMinimize defined in Mathematica.

squares = Sum@Hmisfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm, ΘmD, Κ,CD - phithetadatarad@@i, 3DDL^2, 8i, Length@phithetadataradD

NMinimizeB8squares, Κ ³ 0<, Π Π Θm, - , , Φm, 0, 2 Π , Κ, minkappa, maxkappa ,C , MaxIterations ® maxiterations :: 2 2 > 8 < 8 < > F F;

Calculation of the parameters of the von Mises-Fisher distribution with a kind of a chi-square test where its result will be minimized with the funtion FindFit defined in Mathematica.

nfitsolohnestart = Timing@FindFit@phithetadatarad, 8misfunc3@x@Φ, ΘD, x@Φm, ΘmD, Κ,CD, 8Κ > 0<<, 8C, Θm, Φm, 8Κ, 1<<, 8Θ, Φ<, MaxIterations ® 1 000 000DD;

Calculation of the parameters of the von Mises-Fisher distribution with a kind of a chi-square test where its result will be

nfitsol = TimingBFindFitBphithetadatarad, Π Π misfunc3 x Φ, Θ , x Φm, Θm , Κ,C , 0 <= Φm <= 2 Π, - <= Θm <= , Κ > 0, C > 0 , : @ @ D @ D D : 2 2 >> 88Φm, 0.2 Π<, 8Θm, Π<, 8Κ, 2<, 8C, 2000<<, 8Φ, Θ<, MaxIterations ® 1000FF;

Φ and Θ values in a range of [- Π /2,Π /2] and [0,Π ] respectively

Mod@Θm . nminsol@@2, 2DD, 2 ΠD; Mod@Φm . nminsol@@2, 2DD, 2 ΠD; Mod@Θm . nfitsolohnestart@@2DD, 2 ΠD - 2 Π; Mod@Φm . nfitsolohnestart@@2DD, 2 ΠD; Mod@Θm . nfitsol@@2DD, 2 ΠD; Mod@Φm . nfitsol@@2DD, 2 ΠD;

Calculation of the simulated data values with nminsol, nfitsolohnestart and nfitsol.

phithetanminsol = phithetadatarad; For@i = 1, i £ Length@phithetadataradD, i++, phithetanminsol@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nminsol@@2, 2DD, Θm . nminsol@@2, 2DDD, Κ . nminsol@@2, 2DD,C . nminsol@@2, 2DDDD phithetanminsol;

phithetanfitsolohnestart = phithetadatarad; For@i = 1, i £ Length@phithetadataradD, i++, phithetanfitsolohnestart@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nfitsolohnestart@@2DD, Θm . nfitsolohnestart@@2DDD, Κ . nfitsolohnestart@@2DD,C . nfitsolohnestart@@2DDDD phithetanfitsolohnestart;

phithetanfitsol = phithetadatarad; For@i = 1, i £ Length@phithetadataradD, i++, phithetanfitsol@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nfitsol@@2DD, Θm . nfitsol@@2DDD, Κ . nfitsol@@2DD,C . nfitsol@@2DDDD phithetanfitsol;

Calculation of the simulated data values with nminsol, nfitsolohnestart and nfitsol but without " " for export.

phithetanminsoldeg = Table@phithetanminsol@@i, 3DD, 8i, 1, Length@phithetadataradD

Export@"phithetanminsoldeg.dat", phithetanminsoldeg, "Table"D; Export@"phithetanfitsolohnestartdeg.dat", phithetanfitsolohnestartdeg, "Table"D; Export@"phithetanfitsoldeg.dat", phithetanfitsoldeg, "Table"D;

Maximum and minimum values.

Max@Table@phithetanminsol@@i, 3DD, 8i, 1, Length@phithetadataradD

Max@Table@phithetanfitsolohnestart@@i, 3DD, 8i, 1, Length@phithetadataradD

Max@Table@phithetanfitsol@@i, 3DD, 8i, 1, Length@phithetadataradD

Contour plots for nminsol, nfitsolohnestart and nfitsol.

cnminsol = ShowLegendB

ContourPlotBChop@misfunc3@x@ph, thD, x@Φm . nminsol@@2, 2DD, Θm . nminsol@@2, 2DDD, Π Π Κ . nminsol 2, 2 ,C . nminsol 2, 2 , ph, 0, 2 Π , th, - , , @@ DD @@ DDDD 8 < : 2 2 > Mesh -> None, PlotRange ® All, ColorFunction ® "LakeColors", FrameLabel ® 8Φ, Θ

cnfitsolohnestart = ContourPlotB Chop@misfunc3@x@ph, thD, x@Φm . nfitsolohnestart@@2DD, Θm . nfitsolohnestart@@2DDD, Κ . nfitsolohnestart@@2DD, 1 47.5832 * C . nfitsolohnestart@@2DDDD, Π Π ph, 0, 2 Π , th, - , , Mesh -> None, PlotRange ® All, 8 < : 2 2 > ColorFunction ® "LakeColors", FrameLabel ® 8Φ, Θ

cnfitsol = ShowLegendBContourPlotBChop@misfunc3@x@ph, thD, x@Φm . nfitsol@@2DD, Θm . nfitsol@@2DDD, Π Π Κ . nfitsol 2 ,C . nfitsol 2 , ph, 0, 2 Π , th, - , , @@ DD @@ DDDD 8 < : 2 2 > Mesh -> None, PlotRange ® All, ColorFunction ® "LakeColors", FrameLabel ® 8Φ, Θ

H*Export@"cnminsol.eps",cnminsolD; Export@"cnfitsol.eps",cnfitsolD;*L diffminfit = phithetadatarad; For@i = 1, i £ Length@phithetadataD, i++, diffminfit@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nfitsol@@2DD, Θm . nfitsol@@2DDD, Κ . nfitsol@@2DD,C . nfitsol@@2DDD - misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nminsol@@2, 2DD, Θm . nminsol@@2, 2DDD, Κ . nminsol@@2, 2DD,C . nminsol@@2, 2DDDD Max@Table@diffminfit@@i, 3DD, 8i, 1, Length@phithetadataD

Differences between the measured data and the simulated data. 6 Mises-distribution.nb

phithetanminsoldiff = phithetadatarad; For@i = 1, i £ Length@phithetadataradD, i++, phithetanminsoldiff@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nminsol@@2, 2DD, Θm . nminsol@@2, 2DDD, Κ . nminsol@@2, 2DD, C . nminsol@@2, 2DDD - phithetadatarad@@i, 3DDD phithetanminsoldiff; Max@Table@phithetanminsoldiff@@i, 3DD, 8i, 1, Length@phithetadataD

Plots of the differences between the measured data and the simulated data.

ShowLegend@ListContourPlot@phithetanminsoldiff, PlotRange ® All, ColorFunction ® "LakeColors"D, 8ColorData@"LakeColors"D@1 - ð1D &, 7, "110", "-227", LegendPosition ® 81.1, -0.4<, LegendSize ® 84, 8<

Import of other data: arrays of the size 15x88 statt 7x48, wobei ich da nur die 2. Spalte genommen habe. Die Daten wurden mit eh statt wie vorher sf in widl in Berkeley geladen.

phigross = Import@ "homekanowakDocumentsDatenoutputsvielePhiTheta-dat-Dateienphigross.dat", "Table"D; thetagross = Import@ "homekanowakDocumentsDatenoutputsvielePhiTheta-dat-Dateienthetagross.dat", "Table"D; datagross = Import@ "homekanowakDocumentsDatenoutputsvielePhiTheta-dat-Dateiendatagross.dat", "Table"D ; datagrossnew = Import@"homekanowakDocumentsdataehnew.dat", "Table"D ;

Of the arrays of the size 15x88 take the [spalte]. column of thetagross, phigross and datagross and produce an array with these data (one energy range)). This new array is called phithetadatagross with values of phi in the first column, values of theta in the second column and values of data in the third column. Mises-distribution.nb 7

phithetadatagross = Table@0, 8i, 1, Length@thetagrossD<, 8i, 1, 3

phithetadatagrossrad = Table@8phithetadatagross@@i, 1DD Degree, phithetadatagross@@i, 2DD Degree, phithetadatagross@@i, 3DD<, 8i, 1, Length@phithetadatagrossD

Calculation of the parameters of the von Mises-Fisher distribution so that the distribution fits as good as it gets the data from WIND/phithetadatagross.

Calculation of the parameters of the von Mises-Fisher distribution with a kind of a chi-square test where its result will be minimized with the funtion NMinimize defined in Mathematica.

squares = Sum@ Hmisfunc3@x@phithetadatagrossrad@@i, 1DD, phithetadatagrossrad@@i, 2DDD, x@Φm, ΘmD, k, CD - phithetadatagrossrad@@i, 3DDL^2, 8i, Length@phithetadatagrossradD

NMinimizeB8squares, k ³ 0<, Π Π Θm, - , , Φm, 0, 2 Π , k, minkappa, maxkappa ,C , MaxIterations ® maxiterations :: 2 2 > 8 < 8 < > F F;

Calculation of the parameters of the von Mises-Fisher distribution with a kind of a chi-square test where its result will be minimized with the funtion FindFit defined in Mathematica.

nfitsolohnestartgross = Timing@FindFit@phithetadatagrossrad, 8misfunc3@x@Φ, ΘD, x@Φm, ΘmD, k, CD, 8k > 0<<, 8C, 8k, 1<, Θm, Φm<, 8Θ, Φ<, MaxIterations ® 1 000 000DD;

Calculation of the parameters of the von Mises-Fisher distribution with a kind of a chi-square test where its result will be

minimized with the funtion FindFit defined in Mathematica

nfitsolgross = TimingBFindFitBphithetadatagrossrad, Π Π misfunc3 x Φ, Θ , x Φm, Θm , k, C , 0 <= Φm <= 2 Π, - <= Θm <= , k > 0, C > 0 , : @ @ D @ D D : 2 2 >> 88C, 50<, 8k, 3<, 8Φm, Π 2<, 8Θm, 0<<, 8Φ, Θ<, MaxIterations ® 1000FF;

Calculation of a set of simulated data values (with more phi-theta combinations) for nminsolgross, nfitsolgross, nfitsolohnestartgross 8 Mises-distribution.nb

PhiThetaSimudatagrossMin = Table@0, 8i, 1, Length@phithetadatagrossD<, 8i, 1, 3

PhiThetaSimudataohnestartgrossFit = Table@0, 8i, 1, Length@phithetadatagrossD<, 8i, 1, 3

PhiThetaSimudatagrossFit = Table@0, 8i, 1, Length@phithetadatagrossD<, 8i, 1, 3

Max@Table@phithetadatagrossrad@@i, 3DD, 8i, 1, Length@phithetadatagrossD

Export@"SimudatagrossMin.dat", SimudatagrossMin, "Table"D; Export@"SimudataohnestartgrossFit.dat", SimudataohnestartgrossFit, "Table"D; Export@"SimudatagrossFit.dat", SimudatagrossFit, "Table"D;

Plots for the parameters found with the data of phithetadatagross (with more phi-theta combinations) and a plot of the measured data (from EESA-H)

H*CoolColor@ z_ D :=RGBColor@1,1-z,0.1 zD;*L Mises-distribution.nb 9

ShowLegendBContourPlotB Chop@misfunc3@x@ph, thD, x@Φm . nminsolgross@@2, 2DD, Θm . nminsolgross@@2, 2DDD, Π Π k . nminsolgross 2, 2 ,C . nminsolgross 2, 2 , ph, 0, 2 Π , th, - , , @@ DD @@ DDDD 8 < : 2 2 > PlotRange ® All, PlotPoints ® 12, FrameLabel ® 8Θ, Φ<, ColorFunction ® "LakeColors"F, 8ColorData@"LakeColors"D@1 - ð1D &, 7, "8", "0", LegendPosition ® 81.1, -0.4<, LegendSize ® 84, 8<

ShowLegendBContourPlotBChop@misfunc3@x@ph, thD, x@Φm . nfitsolgross@@2DD, Θm . nfitsolgross@@2DDD, k . nfitsolgross@@2DD,C . nfitsolgross@@2DDDD 1.27272, Π Π ph, 0, 2 Π , th, - , , PlotRange ® All, PlotPoints ® 12, FrameLabel ® Θ, Φ , 8 < : 2 2 > 8 < ColorFunction ® "LakeColors"F, 8ColorData@"LakeColors"D@1 - ð1D &,

7, "8", "0", LegendPosition ® 81.1, -0.4<, LegendSize ® 84, 8<

Export@"cgrossmeasure.eps", cgrossmeasureD;

Calculatation of dicrete values with the von Mises-Fisher distribution and of the differences between these values and the measured values (phithetadatagross).

phithetadatagross1dim = Table@0, 8i, 1, Length@phithetadatagrossD

For@i = 1, i £ Length@phithetadatagrossD, i++, phithetadatagross1dim@@iDD = phithetadatagross@@i, 3DDD differencesmisfun2completephithetadatagrossMin= phithetadatagross; For@i = 1, i £ Length@phithetadatagrossD, i++, differencesmisfun2completephithetadatagrossMin@@i, 3DD = HPhiThetaSimudatagrossMin@@i, 3DD - phithetadatagross@@i, 3DDLD differencesmisfun2completephithetadatagrossMin1dim= Table@0, 8i, 1, Length@differencesmisfun2completephithetadatagrossMinD

For@i = 1, i £ Length@differencesmisfun2completephithetadatagrossMinD, i++, differencesmisfun2completephithetadatagrossMin1dim@@iDD = differencesmisfun2completephithetadatagrossMin@@i, 3DDD differencesmisfun2completephithetadatagrossFit= phithetadatagross; For@i = 1, i £ Length@phithetadatagrossD, i++, differencesmisfun2completephithetadatagrossFit@@i, 3DD = HPhiThetaSimudatagrossFit@@i, 3DD - phithetadatagross@@i, 3DDLD differencesmisfun2completephithetadatagrossFit1dim= Table@0, 8i, 1, Length@differencesmisfun2completephithetadatagrossFitD

For@i = 1, i £ Length@differencesmisfun2completephithetadatagrossFitD, i++, differencesmisfun2completephithetadatagrossFit1dim@@iDD = differencesmisfun2completephithetadatagrossFit@@i, 3DDD differencesmisfun2completephithetadatagrossFitohnestart= phithetadatagross; For@i = 1, i £ Length@phithetadatagrossD, i++, differencesmisfun2completephithetadatagrossFitohnestart@@i, 3DD = HPhiThetaSimudataohnestartgrossFit@@i, 3DD - phithetadatagross@@i, 3DDLD 10 Mises-distribution.nb

differencesmisfun2completephithetadatagrossFitohnestart1dim= Table@0, 8i, 1, Length@differencesmisfun2completephithetadatagrossFitohnestartD

For@i = 1, i £ Length@differencesmisfun2completephithetadatagrossFitohnestartD, i++, differencesmisfun2completephithetadatagrossFitohnestart1dim@@iDD = differencesmisfun2completephithetadatagrossFitohnestart@@i, 3DDD

differencesmisfun2completephithetadatagrossFitohnestart1dim; phithetadatagross;

H*Export@"PhiThetaSimudataohnestartgrossFit.dat", PhiThetaSimudataohnestartgrossFit,"Table"D;*L

Plots of the differences.

ListPlot3D@differencesmisfun2completephithetadatagrossMin, PlotRange ® All, ColorFunction ® "Rainbow", ViewPoint ® 80, -Infinity, 0

Reconstruction of data with data from 10 different sectors.

Exclusion of sectors ("-1" since the sector numbers start with "0" in idl)

phithetadatarad10 = Table@0, 8i, 1, 10<, 8i, 1, 3

IfBi H11 + 1L ÈÈ i H12 + 1L ÈÈ i H13 + 1L ÈÈ i H24 + 1L ÈÈ i H25 + 1L ÈÈ

i H35 + 1L ÈÈ i H36 + 1L ÈÈ i H37 + 1L ÈÈ i H42 + 1L ÈÈ i H43 + 1L,

phithetadatarad10@@jDD = phithetadatarad@@iDD; j++FF phithetadatarad10; Max@Table@phithetadatarad10@@i, 3DD, 8i, 1, Length@phithetadatarad10D

Calculation of the parameters of the von Mises-Fisher distribution.

squares10 = Sum@ HChop@misfunc3@x@phithetadatarad10@@i, 1DD, phithetadatarad10@@i, 2DDD, x@Φm, ΘmD, Κ,CD - phithetadatarad10@@i, 3DDDL^2, 8i, Length@phithetadatarad10D

NMinimizeB8squares10, Κ ³ 0<, Π Π Θm, - , , Φm, 0, 2 Π , Κ, minkappa, maxkappa ,C , MaxIterations ® maxiterations :: 2 2 > 8 < 8 < > F F;

nfitsolohnestart10 = Timing@FindFit@phithetadatarad10, 8misfunc3@x@Φ, ΘD, x@Φm, ΘmD, Κ,CD, 8Κ > 0<<, 88Κ, 1<, Θm, Φm, C<, 8Θ, Φ<, MaxIterations ® 1000DD; Mises-distribution.nb 11

nfitsol10 = TimingBFindFitBphithetadatarad10, Π Π misfunc3 x Φ, Θ , x Φm, Θm , Κ,C , 0 <= Φm <= 2 Π, - <= Θm <= , Κ > 0, C > 0 , : @ @ D @ D D : 2 2 >> 88Φm, Π<, 8Θm, 0.4 Π<, 8Κ, 2<, 8C, 640<<, 8Φ, Θ<, MaxIterations ® 1000FF;

Calculation of the simulated data with the parameters of the von Mises-Fisher distribution, that were calculated above.

phithetanminsol10 = phithetadatarad; For@i = 1, i £ Length@phithetadataradD, i++, phithetanminsol10@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nminsol10@@2, 2DD, Θm . nminsol10@@2, 2DDD, Κ . nminsol10@@2, 2DD,C . nminsol10@@2, 2DDDD phithetanminsol10;

phithetanfitsolohnestart10 = phithetadatarad; For@i = 1, i £ Length@phithetadataradD, i++, phithetanfitsolohnestart10@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nfitsolohnestart10@@2DD, Θm . nfitsolohnestart10@@2DDD, Κ . nfitsolohnestart10@@2DD,C . nfitsolohnestart10@@2DDDD phithetanfitsolohnestart10;

phithetanfitsol10 = phithetadatarad; For@i = 1, i £ Length@phithetadataradD, i++, phithetanfitsol10@@i, 3DD = misfunc3@x@phithetadatarad@@i, 1DD, phithetadatarad@@i, 2DDD, x@Φm . nfitsol10@@2DD, Θm . nfitsol10@@2DDD, Κ . nfitsol10@@2DD,C . nfitsol10@@2DDDD phithetanfitsol10;

nmin10 = Table@phithetanminsol10@@i, 3DD, 8i, 1, Length@phithetadataradD

Plots of the data of the simulated data reconstructed from 10 different sectors.

ContourPlotBChop@misfunc3@x@ph, thD, x@Φm . nminsol10@@2, 2DD, Θm . nminsol10@@2, 2DDD, Κ . nminsol10@@2, 2DD,C . nminsol10@@2, 2DDDD, 8ph, 0, 2 Π<, Π Π th, - , , Mesh -> None, PlotRange ® All, ColorFunction ® "LakeColors" ; : 2 2 > F ContourPlotBChop@misfunc3@x@ph, thD, x@Φm . nfitsolohnestart10@@2DD, Θm . nfitsolohnestart10@@2DDD, Κ . nfitsolohnestart10@@2DD,C . nfitsolohnestart10@@2DDDD, 8ph, 0, 2 Π<, Π Π th, - , , Mesh -> None, PlotRange ® All, ColorFunction ® "LakeColors" ; : 2 2 > F ContourPlotBChop@misfunc3@x@ph, thD, x@Φm . nfitsol10@@2DD, Θm . nfitsol10@@2DDD, Κ . nfitsol10@@2DD,C . nfitsol10@@2DDDD, 8ph, 0, 2 Π<, Π Π th, - , , Mesh -> None, PlotRange ® All, ColorFunction ® "LakeColors" ; : 2 2 > F ListContourPlot@phithetanminsol10D; ListContourPlot@phithetanfitsol10D;

Export of the data of the simulated data reconstructed from 10 different sectors. 12 Mises-distribution.nb

Export@"nmin10.dat", nmin10, "Table"D; Export@"nfitohnestart10.dat", nfit10, "Table"D; Export@"nfit10.dat", nfit10, "Table"D;

Calculation of the simulated data with the parameters of the von Mises-Fisher distribution, that were calculated above but with the measured data for the 10 specific sectors.

nmin10andmeasured = nmin10; ForBi = 1, i £ Length@phithetadataradD, i++,

IfBi H11 + 1L ÈÈ i H12 + 1L ÈÈ i H13 + 1L ÈÈ i H24 + 1L ÈÈ i H25 + 1L ÈÈ

i H35 + 1L ÈÈ i H36 + 1L ÈÈ i H37 + 1L ÈÈ i H42 + 1L ÈÈ i H43 + 1L,

nmin10andmeasured@@iDD = phithetadatarad@@i, 3DDFF nmin10andmeasured;

nfitohnestart10andmeasured = nfitohnestart10; ForBi = 1, i £ Length@phithetadataradD, i++,

IfBi H11 + 1L ÈÈ i H12 + 1L ÈÈ i H13 + 1L ÈÈ i H24 + 1L ÈÈ i H25 + 1L ÈÈ

i H35 + 1L ÈÈ i H36 + 1L ÈÈ i H37 + 1L ÈÈ i H42 + 1L ÈÈ i H43 + 1L,

nfitohnestart10andmeasured@@iDD = phithetadatarad@@i, 3DDFF nfitohnestart10andmeasured;

nfit10andmeasured = nfit10; ForBi = 1, i £ Length@phithetadataradD, i++,

IfBi H11 + 1L ÈÈ i H12 + 1L ÈÈ i H13 + 1L ÈÈ i H24 + 1L ÈÈ i H25 + 1L ÈÈ

i H35 + 1L ÈÈ i H36 + 1L ÈÈ i H37 + 1L ÈÈ i H42 + 1L ÈÈ i H43 + 1L,

nfit10andmeasured@@iDD = phithetadatarad@@i, 3DDFF nfit10andmeasured;

Export of the data of the simulated data reconstructed from 10 different sectors but with the measured data for the 10 specific sectors.

Export@"nmin10andmeasured.dat", nmin10andmeasured, "Table"D; Export@"nfitohnestart10andmeasured.dat", nfitohnestart10andmeasured, "Table"D; Export@"nfit10andmeasured.dat", nfit10andmeasured, "Table"D;

Differences with the measured values.

data = Table@phithetadata@@i, 3DD, 8i, 1, Length@phithetadataD AllD; ListPlot@data, PlotRange ® AllD; Mises-distribution.nb 13

diffdatared = Table@phithetadata@@i, 3DD, 8i, 1, Length@phithetadataD

diffdatared@@iDD = 0, diffdatared@@iDD = data@@iDD - nmin10andmeasured@@iDDFF diffdatared; ListPlot@diffdatared, PlotRange -> AllD; phithetadiffnmin10andmeasured = phithetadatarad;

ForBi = 1, i £ Length@phithetadataD, i++,

IfBi == 7 ÈÈ i == 8 ÈÈ i == 9, phithetadiffnmin10andmeasured@@i, 3DD = 0,

phithetadiffnmin10andmeasured@@i, 3DD = data@@iDD - nmin10andmeasured@@iDDFF ListPointPlot3D@phithetadiffnmin10andmeasured, PlotRange ® All, Filling ® Bottom, ColorFunction ® "Rainbow"D; phithetadiffnmin10andmeasured; Clear "Global‘*" @ D

The Kent-distribution for dimension 2 shape parameters: K : concentration parameter Β :Β <=2K

kent x_, Κ_, Γ1_, Γ2_, Γ3_, Β_ := Exp Κ Γ1.x + Β Γ2.x 2 - Γ3.x 2 @ D A H L IH L H L ME kent4 x_, Κ_, Γ1_, Γ2_, Γ3_, Β_,C_ := C * Exp Κ Γ1.x + Β Γ2.x 2 - Γ3.x 2 @ D A H L IH L H L ME normMF[p_,Κ _] defines the normalisation factor for the Kent distribution

1 2 2 normMFKent Κ_, Β_ := iExp -Κ "######################Κ - 4 Β y @ D 2 Π j @ D z k { kent2 defines the 3-dimensional Kent distribution using the above normalisation factor

kent2 x_, Κ_, Γ1_, Γ2_, Γ3_, Β_ := normMFKent Κ, Β * kent x, Κ, Γ1, Γ2, Γ3, Β @ D @ D @ D

kent3 x_, Κ_, Γ1_, Γ2_, Γ3_, Β_,C_ := normMFKent Κ, Β * kent x, Κ, Γ1, Γ2, Γ3, Β * C @ D @ D @ D

Example for the kent funtion

x Φ_, Θ_ := r Cos Θ Cos Φ , r Cos Θ Sin Φ , r Sin Θ ; @ D 8 @ D @ D @ D @ D @ D< r = 1; Exkentmono = Plot3D kent2 x Phi, Theta , 10, x 0.8, 2 - 0.5, x -Π 2, Π , x Π, 0 , 1 , B @ @ D @ D @ D @ D D -Π Π Theta, , , Phi, 0, 2 Π , PlotRange ® All, AxesLabel ® "Φ", "Θ", "Density" ; : 2 2 > 8 < 8 8 < 8

T Γi x = Γi1 x1 + Γi2 x2 + Γi3 x3 Γ = 1 È È 2 Kent-distribution.nb

as well as x = 1 È È Π Π so that we can write for - £ Θ < ; 0 £ Φ < 2 Π K 2 2 O x = r Cos Θ Cos Φ ; @ @ xD @ xD r Cos Θ Sin Φ ; @ xD @ xD r Sin Θ @ xD withD r = 1 and Μ = r Cos Θ Cos Φ ; A A ΜE A ΜE r Cos Θ Sin Φ ; A ΜE A ΜE r Sin Θ A ΜE E with r = 1 ΜT x = Cos Θ Cos Φ Cos Θ Cos Φ + Cos Θ Sin Φ Cos Θ Sin Φ + Sin Θ Sin Θ @ xD @ xD A ΜE A ΜE @ xD @ xD A ΜE A ΜE @ xD A ΜE

r = 1;

x Φ_, Θ_ := r Cos Θ Cos Φ , r Cos Θ Sin Φ , r Sin Θ ; @ D 8 @ D @ D @ D @ D @ D< Needs "PlotLegends‘" @ D Π Π ShowLegend ContourPlot kent4 x ph, th , 0.7, x , , x -Π 2, Π , x Π, 0 , 1 3, 300 , B B B @ D B 2 4 F @ D @ D F Π Π ph, 0, 2 Π , th, - , , PlotRange ® All, AxesLabel ® "Φ", "Θ", "Density" , 8 < : 2 2 > 8

maxiterations = 20 000; minkappa = 0; maxkappa = 10 000; seed = 0;

the mean direction Γ1, a vector defined by {ΘΜ1 , ΦΜ1= with r = 1

the major direction Γ2, a vector defined by {ΘΜ2 , ΦΜ2= with r = 1

the minor direction Γ3, a vector defined by {ΘΜ3 , ΦΜ3= with r = 1 Kent-distribution.nb 3

squaresKent = Sum kent3 x phithetadatarad i, 1 , phithetadatarad i, 2 , k, x Φm, Θm , x Φ2, Θ2 , AH @ @ @@ DD @@ DDD @ D @ D x Φ3, Θ3 , Β,C - phithetadatarad i, 3 2, i, Length phithetadatarad ; @ D D @@ DDL 8 @ D 0 , Θm, - , , Φm, 0, 2 Π , k, 0, 10 000 , 2 2 > :: 2 2 > 8 < 8 < Π Π Π Π Θ2, - , , Φ2, 0, 2 Π , Θ3, - , , Φ3, 0, 2 Π , Β,C , MaxIterations ® maxiterations : 2 2 > 8 < : 2 2 > 8 < > F ; F nfitsolkent = Timing FindFit phithetadatarad, B B Π Π kent4 x Φ, Θ , k, x Φm, Θm , x Φ2, Θ2 , x Φ3, Θ3 , Β,C , 0 <= Φm <= 2 Π, - <= Θm <= , : @ @ D @ D @ D @ D D : 2 2 Π Π Π Π k > 0, - £ Θ2 <= , 0 <= Φ2 <= 2 Π, - <= Θ3 <= , 0 <= Φ3 £ 2 Π, k ³ 2 Β,C > 0, Β > 0 , 2 2 2 2 >> Φm, Θm, k, 2 , Θ2, Φ2, Θ3, Φ3, Β, 1 , C, 380 , Φ, Θ , MaxIterations ® 100 ; 8 8 < 8 < 8 << 8 < FF nfitsolkentbi = Timing FindFit phithetadatarad, B B Π Π kent4 x Φ, Θ , k, x Φm, Θm , x Φ2, Θ2 , x Φ3, Θ3 , Β,C , 0 <= Φm <= 2 Π, - <= Θm <= , : @ @ D @ D @ D @ D D : 2 2 Π Π Π Π k > 0, - £ Θ2 <= , 0 <= Φ2 <= 2 Π, - <= Θ3 <= , 0 <= Φ3 £ 2 Π,C > 0, Β > 0 , 2 2 2 2 >> Φm, Θm, k, 2 , Θ2, Φ2, Θ3, Φ3, Β, 1 , C, 380 , Φ, Θ , MaxIterations ® 100 ; 8 8 < 8 < 8 << 8 < FF cp2min = ContourPlot Chop kent4 x ph, th , B @ @ @ D k . nminsolkent 2, 2 , x Φm . nminsolkent 2, 2 , Θm . nminsolkent 2, 2 , x Φ2 . nminsolkent@@ 2,DD 2 @, Θ2 . nminsolkent@@ 2, 2DD , x Φ3 . nminsolkent@@ DDD2, 2 , @Θ3 . nminsolkent @@2, 2 DD, Β . nminsolkent @@2, 2 DDD,C .@ nminsolkent 2, 2@@ ,DD Π@@ Π DDD @@ DD @@ DDDD ph, 0, 2 Π , th, - , , PlotRange ® All, PlotPoints ® 42 ; 8 < : 2 2 > F

ContourPlot kent4 x Φ,@ Θ , k . nfitsolkent 2 , x Φm . nfitsolkent 2 , Θm . nfitsolkent 2 , x Φ2@ .@ nfitsolkentD 2 , Θ2 .@@ nfitsolkentDD @ 2 , @@ DD @@ DDD x@Φ3 . nfitsolkent@@2DD, Θ3 . nfitsolkent@@2DDD, Β . nfitsolkent 2 , C@ . nfitsolkent 2@@ ,DD Φ, 0, 2 Π , Θ, -Π @@2, DDDΠ 2 , PlotRange ® All@@;DD @@ DDD 8 < 8 < D ContourPlot kent4 x Φ,@ Θ , k . nfitsolkentbi 2 , x Φm . nfitsolkentbi 2 , Θm . nfitsolkentbi 2 , x Φ2@ .@ nfitsolkentbiD 2 , Θ2 .@@ nfitsolkentbiDD @ 2 , @@ DD @@ DDD x@Φ3 . nfitsolkentbi@@2DD, Θ3 . nfitsolkentbi@@2DDD, Β . nfitsolkentbi 2 , C@ . nfitsolkentbi 2@@ ,DD Φ, 0, 2 Π , Θ, -Π 2,@@Π DDD2 , PlotRange ® All ; @@ DD @@ DDD 8 < 8 < D ListContourPlot phithetadata, PlotRange ® All ; @ D 4 Kent-distribution.nb

dataminkent = Table 0, i, 1, Length phithetadata ; For i = 1, i £ Length@ phithetadata8 ,@ i++, D