streamer waves, final version

Aberystwyth University

Streamer Waves Dissertation Project

Presented for BSc Astrophysics

Mr Nicholas David Mellor Institute of Mathematics, Physics and Computer Science

Nicholas David Mellor Streamer Waves 130053235

1

Abstract

A program was devised and written using IDL, to calculate the properties of the oscillation of

a coronal streamer after a shock front from a coronal mass ejection (CME) collided with it. Two CME/

streamer collision events were examined, on the dates 23/06/2012 from 15:48 to 16:24 and

21/04/2013 from 07:12 to 11:00. After the limit of the electron density data was reached, the event

on 23/06/2012 from 15:48 to 16:24, had to be discarded. However, from the remaining event, results

were gathered. These results displayed a positive relationship for the wave speed and Alfvén speed

with increasing heliocentric height, and a negative relationship for ion plasma density, magnetic field

strength and magnetic tension force with increasing heliocentric height.


2

Table of Contents Abstract 1

Table of Figures 3

1. Introduction 4

1.1. Experimental Purpose 4

1.2. Introduction to the Sun, the Solar Corona, Coronal Streamers and Coronal Mass Ejections 4

1.3. Introduction to SoHO, LASCO-C2, IDL and SolarSoftWare 6

2. Literature Review 7

2.1. Introduction 7

2.2. Magneto-Hydrodynamics (MHD), Coronal and Alfvén Waves 7

2.3. CMEs and Magnetic Reconnection 11

2.4. Streamers, the Slow and Fast Solar Wind and LASCO 12

2.5. Conclusions 13

2.6. Bibliography 13

3. Assumptions and Equations 15

3.1. Assumptions made about the Streamer Model 15

3.2. Assumptions made about the Calculations 15

3.3. Equations used in the Experiment 15

4. Experimental Method 17

4.1. Data Acquisition 17

4.2. Data Manipulation and Calculations 18

4.2.1. Data Manipulation 18

4.2.2. Calculations 19

4.3. Data Analysis 22

5. Results, Discussion and Error Analysis 23

5.1. Results 23

5.2. Discussion 25

5.3. Error Analysis 27

6. Conclusion and Acknowledgements 30

6.1. Conclusion 30

6.2. Acknowledgements 30

7. Additional Literature 31

7.1. Bibliography 31

8. Appendices 33

8.1. Appendix 1 33

8.2. Appendix 2 34


3

8.3. Appendix 3 41

Table of Figures Figure 1: Four different MHD oscillation modes of cylindrical fluxtubes: fast MHD sausage mode (a),

fast MHD kink mode (b), slow MHD (acoustic) mode (c) and MHD torsional mode (d). (Aschwanden,

2006) 10

Figure 2: Diagram of a chord of a circle. 16

Figure 3: Two images, one from 2012/06/23 showing an example of a 'thin' streamer on the lower

right of the image; another image from 2013/10/31 showing an example of a 'fat' streamer on the

lower left of the image. 17

Figure 4: An example of original file used in the experiment. 18

Figure 5: An example of a manipulated file used in the experiment. 18

Figure 6: A flowchart to show the process the IDL program follows. 19

Figure 7: An example of a running difference image (RDI) used in the experiment 20

Figure 8: A diagram to show points along the wavelength 21

Figure 9: A graph showing wave speed against height from solar centre. 23

Figure 10: A graph to show Alfven speed against height from solar centre. 23

Figure 11: A graph to show ion plasma density against height from solar centre. 24

Figure 12: A graph to show magnetic field strength against height from solar centre. 24

Figure 13: A graph showing magnetic tension force against height from the solar centre. 25

Figure 14: A diagram showing wave propagation. 25

Figure 15: An image showing the oscillation along the streamer on 21/04/2013 from 07:12 to 11:00.

26


26

Figure 17: A table comparing calculated and literature wave speeds. 27

Figure 18: A table to show further wave speed results. 27

Figure 19: A graph showing the electron density along the normalised heliocentric height, at an angle

of 578o of the data structure. 28

Figure 20: A graph to show the average ion plasma density with heliocentric height. 28

Figure 21: A table comparing the ion densities to the average ion densities. 29

Figure 22: A table displaying all the data gathered during the experiment. 33

Figure 23: A table showing some average results. 33

Figure 24: A graph showing the wave frequency against heliocentric height. 33


4

1. Introduction

1.1. Experimental Purpose

The purpose of this experiment was to determine the physical properties of the oscillation of

coronal streamers (from this point referred to simply as streamers), after a collision with a shock wave

from a coronal mass ejection (CME). The observed properties include the wavelength and wave speed

of the propagating wave. The properties from the calculations; include the Alfvén speed, the magnetic

field and the magnetic tension force along the wave, the frequency of the fast kink mode, and the time

period of the wave. These latter properties were determined using coronal electron plasma density

values, provided by Dr Huw Morgan at Aberystwyth University (Quémerais and Lamy, 2002; Morgan,

2015). Two dates were considered and measured during this experiment, 23/06/2012 from 15:48 to

16:24 and 21/04/2013 from 07:12 to 11:00. These dates and times were used as both contained CMEs

occurring close to a streamer, this gave a strong and clear oscillation of the streamer as the shock wave

collided with it. The times taken were from as the CME starts, to when the streamer stopped

oscillating.

The secondary aim for this experiment is to provide another method of measuring the

streamer waves’ physical properties. Very little literature can be found solely about calculating these

properties, as such the aim was to provide a method of calculating approximate values for a simple

fundamental streamer wave. This method can then be built upon to give a more complete and accurate

picture of the coronal streamer. As such, an IDL program (henceforth known as “the program”) was

written to carry out all of the calculations needed to return the required results.

1.2. Introduction to the Sun, the Solar Corona, Coronal Streamers and Coronal Mass

Ejections At approximately 4.5 billion years old and of type G2V (Guenther, 1989), the Sun is the centre

of our solar system. It is a population I star, or a metal-rich star; all elements above hydrogen and

helium in the periodic table are considered metals, in astronomical terms. The sun has a metallicity of

Z=0.0166, this suggests that the Sun could have formed from a supernova explosion, where the metals

found in the Sun would have formed. There is no definite boundary to the Sun, in its outer regions the

density decreases exponentially as the distance from the centre increases (Zirker, 2002). However, for

measurement purposes the Sun’s radius is taken to be the distance from its centre to the edge of the

photosphere, which is the apparent visible surface of the Sun (Phillips, 1995).The Sun goes through

solar cycles of varying solar activity, beginning approximately every 11 years. A solar minimum is the

time period of the least number of sunspots and solar flares, often neither will be spotted for days at

a time during this period. A solar maximum is the opposite. It is the time period with the greatest solar

activity (Moussas, Polygiannakis, Preka-Papadema and Exarhos, 2005).

The solar corona is an aura of plasma surrounding the sun that extends millions of kilometres

into space. During a total solar eclipse, the corona is most apparent this is due to the moon blocking

out all light from the solar disk, as the corona is far dimmer than the disk. As such, instruments used

to observe the solar corona utilize an artificial eclipse to make the corona more visible. The corona is

a region of very hot plasma, much hotter than the surface of the sun, with a maximum temperature of

approximately 1 million Kelvin (Markus, 2004). The light in the corona is produced by 3 primary

sources; The K-corona, which is created when sunlight is scattered by free electrons in the corona. The

F-corona, when sunlight is reflected and scattered by dust particles, observable because the light

contains Fraunhofer absorption lines. Fraunhofer lines are dark absorption lines in the Sun’s spectrum


5

that can be detected when the full visible spectrum is analysed. And the E-corona, which is created by

the spectral emission lines from the ions that are present in the plasma (Ridpath, 2012). For this

experiment, it has been assumed that the only light being received is light due to the K-corona, this is

due to the Doppler broadening of the absorption lines completely obscuring them, creating the

appearance of a continuous spectrum with no absorption lines (Golub and Pasachoff, 2010). The solar

corona’s structure is in three-parts, comprising of: dark coronal holes, where the fast solar wind

originates as plasma escaping outward along open magnetic fields. Bright coronal loops that connect

regions in the photosphere that are of opposite polarity, these are magnetically closed. Finally, smaller,

intense regions called X-ray bright points, that consist of tiny loops and are scattered over the solar

disk (Priest, 2014).

Coronal streamers are the most prominent feature in the solar corona. Due to the high energy

electrons trapped within their magnetic field lines, they stand out against their background and have

a very definitive boundary. Streamers often tend to overlie active regions or prominences and are an

equilibrium between the confining magnetic field and the expanding plasma. They consist of a base

(or an arcade) of closed field lines, that is topped by a blade or a fan of open field lines. Streamers are

formed when the upper reaches of a large coronal arcade extend high enough to be stretched open

by the solar wind. Helmet streamers can exist in two types of magnetic topology (Hundhausen, 2012)

(Zhao and Webb, 2003). A unipolar streamer or pseudo-streamer, lies over a quadrupolar field;

meaning two oppositely directed bipolar fields adjacent to each other (Wang, Biersteker, Sheeley Jr et

al., 2007) (Hundhausen, 2012), and separates coronal holes of the same polarity. Or bipolar streamers

that lies above a singular bipolar arcade and between coronal holes that have opposite polarities,

creating an overlying blade in the form of a current sheet with oppositely directed magnetic fields on

both sides. During the solar minimum, streamers tend to appear from the mid-latitude regions. As the

minimum starts to turn to the maximum the streamers begin migrating to the poles, and the slow solar

wind is then seen from all latitudes (Chen, 2013).

Streamers are thought to follow the magnetohydrodynamic (MHD) kink mode (Chen, Song, Li

et al., 2010), as seen in figure 1. This means that the magnetic field is arranged in a cylinder that is

almost incompressible, and can cause a displacement of the axis of the plasma structure, or more

simply the streamer will be displaced from its equilibrium position. MHD modes guided by the tube,

with phase speeds larger than the minima of the Alvfén and sound speeds ( 𝑣𝐴𝑖 , 𝑣𝐴𝑒 , 𝑐𝑠𝑖 , 𝑐𝑠𝑒) are called

fast, whereas those smaller than one or the other minimum are called slow. The sound speed is

dependent on temperature and the Alfvén speed depends on the density and magnetic field. For fast

kink waves in a tube, the phase speed is the kink speed, 𝑐𝐾 = [𝐵𝑖2 + 𝐵𝑒

2]/[𝜇(𝜌𝑖 + 𝜌𝑒)]1

2 , which is the

mean Alfvén speed (Nakariakov and Verwichte, 2005). The frequency of this mode is given by the

equation 𝜔𝐾 = √2𝑘𝑧

2𝐵2

𝜇(𝜌𝑖+𝜌𝑒) (Nakariakov, Melnikov and Reznikova, 2003; Doorsselaere, Nakariakov and

Verwichte, 2008). It is this displacement that is being investigated, as such, it is assumed that streamers

follow the fast kink mode model.

Coronal mass ejections (CMEs) are sudden and often violent releases of solar material into

interplanetary space. There are two types, fast and slow CMEs. Fast CMEs release energy in a more

explosive manner than slow CMEs, however they do not show signs of accelerating unlike slow CMEs

(Gosling, Bame, McComas et al., 1994; J. T. Gosling, 1996). Coronal mass ejections have an associated

shock wave preceding them, it is this shock wave that interacts with the streamer.


6

1.3. Introduction to SoHO, LASCO-C2, IDL and SolarSoftWare The Solar and Heliospheric Observatory (SoHO) was launched on 2nd December 1995. Its

purpose; to study the internal structure of the Sun, its outer atmosphere and the origins of the solar

wind. SoHO is a collaboration between the European Space Agency (ESA) and the National Aeronautics

and Space Administration (NASA) (Domingo, Fleck and Poland, 1995). It was originally planned only to

be a two-year mission but it is still operational over 20 years later. SoHO is in a halo orbit about the

Sun-Earth L1 point. This is the point between the Sun and the Earth, where the balance of the Earth’s

gravity and the Sun’s gravity, equals the centripetal force required for an object to have the same

orbital period as the Earth around the Sun. This results in the object staying in that relative position

(Domingo, Fleck et al., 1995). A consequence of this, is that the Earth does not impede SoHO’s ability

to capture images of the sun, resulting in it being able to monitor the Sun constantly.

One of the instruments aboard SoHO is the Large Angle and Spectrometric Coronagraph

(LASCO) this consists of three coronagraphs:

C1- a Fabry-Pérot interferometer coronagraph, which images from 1.1-3 solar radii.

C2- a white light coronagraph imaging from 2-6 solar radii, which uses an orange filter.

C3- a white light coronagraph imaging from 3.7-32 solar radii, that uses a blue filter.

The solar corona is monitored by these coronagraphs by implementing an optical system to create an

artificial eclipse, to block out the light from the solar disk. Both the C2 and C3 coronagraphs produce

images over most of the visible spectrum, whereas the C1 interferometer produces images of the

corona in very narrow visible wavelength bands (G. E. Brueckner, 1995). For the purposes of this

experiment, only the C2 coronagraph was used, as the other two were either of the wrong distance

range, or wrong spectrum or both.

Interactive Data Language (or IDL) is a programming language used for data analysis and image

manipulation. It is popular in various areas of science such as medical imaging, atmospheric physics

and astronomy. IDL has been used in many space projects and missions, for example the calibration of

high-energy X-ray telescopes and the rescue of the Hubble Space Telescope. IDL’s use in Earth sciences

include being used in improving the accuracy of long-range weather predictions, and the tracking of

sea ice to keep ships safe (Stern, 2000). There are many libraries of IDL procedures, all of them contain

procedures that are specific to a certain area of science. There are however, a few astronomical

libraries containing similar procedures but maintained by different organisations. Examples include;

the IDL astronomy user’s library by the Goddard Space Flight Center (Center), the Coyote library by

Coyote’s guide to IDL programming (Consulting) and SolarSoftWare by S.L. Freeland and R.D. Bentley

(Freeland). IDL originated from an early version of Fortran, this can be seen through the similar use of

syntax and the base integer being 0 and not 1. IDL is owned and maintained by Exelis Visual Information

Solutions, Inc., a subsidiary of Harris Corporation (Corporation).

SolarSoftWare (SSW) is a set of integrated software libraries, system utilities and databases

which provide a common programming and data analysis environment for solar physics. SSW is

primarily an IDL based system, it is a collection of data management and analysis routines obtained

from the SoHO and the Yohkoh missions, the Solar Data Analysis Center and the Astronomy libraries,

along with other packages (Freeland and Handy, 1998). All of the SSW routines can then be called into

the IDL program once the environment has been loaded. In this experiment, IDL was used with the

SSW environment installed, this allowed for the required data manipulation to be carried out.


7

2. Literature Review

2.1. Introduction Streamers are a very prominent structure in the Solar Corona, and understanding the physical

properties of these streamers can be vital to the understanding of the functions and dynamics of the

corona and sun itself. The effect of the magnetic field of the streamers is an interesting phenomena in

which the streamers appear to oscillate during coronal mass ejection events. The goal of this project

is to calculate various physical properties of steamer waves, such as; phase speed, the magnetic

restoring force and the period of oscillation. This literature review was carried out in preparation for

the project in order to better understand the various properties and phenomena of the corona and

the sun.

2.2. Magneto-Hydrodynamics (MHD), Coronal and Alfvén Waves Magneto-hydrodynamics (MHD) is the study of the magnetic properties of electrically conducting

fluids, such as plasmas. The basic concept of MHD is that; magnetics fields in a moving conductive fluid

can induce currents within that fluid. These currents polarizes the fluid, which in turn, changes the

magnetic field itself.

Energy is carried upward through the solar interior by sound waves generated in the hydrogen

convection zone, this is below the photosphere. These waves propagate into the “fast” mode,

becoming increasingly magneto-hydrodynamic in nature as they pass through the photosphere, out

into the chromosphere. This is due to the negative density gradient. Very little energy is emitted by

the hydrogen convection zone in the “slow” or “Alfvén” modes as these modes are strongly absorbed

by the photosphere. Cross-sections for collisions between ions and neutral atoms in the chromosphere

is large, meaning that the dissipation of the fast-mode waves by the frictional damping mechanism is

very small. These waves build into shock waves, the dissipation of which provides the main energy

source for the chromosphere (Osterbrock, 1961). Energy, in the form of acoustic energy, reaches the

corona preferably along almost vertical magnetic fields, these vertical fields are inside the vertical flux-

tubes. Most of the magnetic energy stays concentrated beneath the transition region due to the

refraction of the magnetic waves, as well as the continuous conversion of acoustic-like waves into fast

magnetic waves; at the equipartition layer, located in the photosphere where plasma beta equals 1.

However, some of the magnetic energy (that propagates in the region where the arcades are located)

does reach the low corona (Withbroe and Noyes, 1977) (Santamaria, Khomenko and Collados, 2015).

Plasma beta, β, is the ratio of the plasma pressure to the magnetic pressure, this can be expressed by

the following equation (K. L. DeRose, 2008):

𝛽 =𝑝

𝑝𝑚𝑎𝑔=

𝑛𝑘𝐵𝑇

𝐵2(2𝜇0)⁄

The coronal magnetic field has multiple effects on the hydrodynamics of the coronal plasma. It can

play a passive role, meaning the magnetic geometry does not change. For example, the magnetic field

can maintain thermal insulation between the plasmas of adjacent loops or flux-tubes, or it can channel

plasma and heat flows or particles and waves along the magnetic field lines. However, the magnetic

field can also play an active role, such that the magnetic geometry does change, for example; the build-

up and storage of non-potential energy, the triggering of an instability, the changing of topology by

the means of various types of magnetic reconnection, or by exerting a Lorentz force on the plasma.

The acceleration of plasma structures such as, filaments, prominences and coronal mass ejections

(CMEs) are also a result of the magnetic field playing an active role.


8

MDH is expressed in terms of macroscopic parameters, such as pressure, density, temperature, and

flow speed of the plasma. In the macroscopic scale the plasma reacts to electric and magnetic forces

as described by the Maxwell equations. The particle motion of the plasma, however, can also be

described by microscopic physics, called Kinetic theory, in terms of the Boltzmann equation or the

Vlasov equation(Aschwanden, 2005).

Many astrophysical plasmas are characterized by the ideal MHD equations, which includes the MHD

continuity equation (1), the momentum equation (2), Maxell’s equations (3-5), Ohm’s law (6) and a

specialized equation of state for energy conservation (7). For example, incompressible, isothermal, or

adiabatic. The full set of ideal MHD equations are shown below, for an adiabatic equation of

state(Aschwanden, 2005):

1. dρ

dt= −ρ∇. 𝐯

2. ρd𝐯

dt= −∇p − ρg + (𝐣 x 𝐁)

3. d

dt(pρ−γ) = 0

4. ∇ x 𝐁 = 4π𝐣

5. ∇ x 𝐄 = −1

c

∂𝐁

∂t

6. ∇. 𝐁 = 0

7. 𝐄 = −1

c(𝐯 x 𝐁)

A magnetic flux tube is defined as a surface, generated when a set of field lines intersect a simple

closed curve. The magnetic flux that crosses a section S of the flux tube is given as:

𝐹 = ∫𝑩.𝑑𝑺

𝑆

EQ.1

From the divergent free nature of the magnetic field and using Gauss’s law, the following can be

obtained:

∫∇.𝐁

V

dV = −

−F1

∫ 𝐁d𝐬

S1

⏞ +

F2

∫ 𝐁d𝐬

S2

⏞ + ∫0

𝐁. �̂�⏞ dσ

tubesurface

= 0 EQ.2

Where the flux tube surface is parallel to the magnetic field lines, so, 𝐁. �̂� = 0. As such,∇. 𝐁 = 0.

Therefore, it can be can be stated that F1=F2, this means that the magnetic flux tubes conserves the

magnetic flux. However, at the flux tube surface, when embedded in a field-free or weak-field medium

a discontinuity occurs that can be described as the following step function:

𝜃(𝑥) = {1 𝑤ℎ𝑒𝑛 𝑥 > 00 𝑤ℎ𝑒𝑛 𝑥 < 0

It can be calculated that 𝐣∗ =1

μ[B], from this it can be shown that a current sheet occurs at the flux-

tube surface, where the current flows perpendicular to the magnetic force lines. Then using the

momentum equation and Ampère’s law;

ρd𝐯

dt= −∇p + (𝐣x𝐁) + ρ𝐠

∇x𝐁 = μ𝐣


9

It is possible to obtain:

Where “i” and “e” refer to the interior and exterior of the flux-tube, but in the immediate vicinity of

the surface. This result shows a discontinuity in pressure occurs at the flux-tube surface, such that the

Lorentz force is balanced by another pressure force (Roberts and Webb, 1978). This equation also

shows the total pressure, consisting of gas pressure plus magnetic pressure, is constant across the flux-

tube’s surface. (Steiner, 2006)

The following is the procedure for calculating the magnetic structure of a vertical, axisymmetric and

untwisted magnetic flux-tube. It is worth noting however, that this is calculated for a perfectly

symmetric object, which probably does not occur in reality. These steps can still be followed for more

complex situations.

i. An initial magnetic configuration must be specified. This flux-tube will be embedded within a

plane-parallel atmosphere which is thought to remain unchanged.

ii. Calculate gas pressure in all space using hydrostatics along magnetic force lines, 𝑝 =

𝑝0(Ψ) exp [−∫𝑑𝑧′

𝐻(𝑇(Ψ,𝑧′))

Ψ,𝑧

Ψ,0]. Since this does not solve an energy equation, the pressure scale

height as a function of field line and height, 𝐻(Ψ, 𝑧), must also be specified. It is acceptable to

guess that H is independent of Ψ and identical to scale height of the surrounding field-free

atmosphere, this is approximately filled if the temperature is constant in the planes parallel to

the solar surface. From the initial, embedding atmosphere the pressure and pressure scale

height in the surrounding field-free atmosphere can be found.

iii. Evaluate the current sheet and volume using 𝑗𝜙∗ =

2(𝑝𝑒−𝑝𝑖)

𝐵𝑖+𝐵𝑒 and 𝑗𝜙 = 𝑟

𝜕𝑝

𝜕Ψ|𝑧.

The first equation applies to the locations of discontinuous gas pressure and field strength,

such as the surface of the flux-tube. The second applies to regions of continuous gas pressure.

iv. Solve the Grad-Shafranov equation, 𝜕2Ψ

𝜕𝑟2−1

𝑟

𝜕Ψ

𝜕𝑟+𝜕2Ψ

𝜕𝑧2= −𝜇𝑟𝑗𝜙, using appropriate boundary

conditions. This will give a new field configuration, Ψ(r, z), that cannot necessarily be equal to

the initial magnetic configuration. Therefore, steps (ii) to (iv) must be reiterated until

convergence has been achieved (Steiner, 2006).

There are four distinct kinds of MHD modes, all of which have different propagation, dispersion and

polarisation properties (Nakariakov, Melnikov et al., 2003) (Doorsselaere, Nakariakov et al., 2008).

These different modes are listed below:

Sausage modes, these are oblique fast magnetoacoustic waves guided by a plasma structure.

This mode causes the contraction and the expansion of the plasma structure, but the axis of

which is not displaced. These modes can be compressed and can cause significant variation of

the modulus of the oscillating structures magnetic field. For sausage modes, the azimuthal

wave number, m, equals 0. This would be shown as a tube with fixed end points, with the

space in between expanding and contracting. The frequency of this mode is given as:

𝜔𝑆 = √𝑘𝑧2𝐵2

𝜇𝜌𝑒

Kink (or transverse) modes, these are also oblique, fast magnetoacoustic waves guided by a

plasma structure. Unlike sausage modes, kink modes are weakly compressible and do cause

the displacement of the axis of the plasma structure. They could be imaged by instruments as

𝑝𝑒 +

𝐵𝑒2

2𝜇= 𝑝𝑖 +

𝐵𝑖2

2𝜇 EQ.3


10

periodic propagating or standing displacements of coronal structures. The frequency of this

mode is given as: 𝜔𝐾 = √2𝑘𝑧

2𝐵2

𝜇(𝜌𝑖+𝜌𝑒)

Torsional (Alfvén or twist) modes are transverse, incompressible perturbations of the magnetic

field along certain individual magnetic surfaces. Unlike kink modes, torsional modes cannot be

observed with imaging instruments, this is because they do not affect the displacement of

either the plasma structure axis or its boundary. The frequency of torsional modes is given by:

𝜔𝐴 = √𝑘𝑧2𝐵2

𝜇𝜌𝑖

Longitudinal (slow or acoustic) modes, are slow magnetoacoustic waves which propagate

mainly along the magnetic field lines of the plasma structure. These modes are compressible

and the magnetic field perturbation in these modes is negligible. The frequency of this is given

below, where CS is the sound speed, and CA is the Alfvén velocity: 𝜔𝐿 = √𝑘𝑧2 (

𝐶𝑆2𝐶𝐴

2

𝐶𝑆2+𝐶𝐴

2)

Below is figure 1; it shows each of the above mentioned modes as wire cylinders. It shows in detail

how the plasma structure of each mode can change.

Figure 1: Four different MHD oscillation modes of cylindrical fluxtubes: fast MHD sausage mode (a), fast MHD kink mode (b), slow MHD (acoustic) mode (c) and MHD torsional mode (d). (Aschwanden, 2006)

An Alfvén wave in a plasma is defined as a low frequency travelling oscillation of the ions and the

magnetic field. The wave propagates with the magnetic field lines; waves do exist at oblique incidence,

the interface between dielectric media, and smoothly change into a magnetosonic wave when

propagation is perpendicular to the magnetic field. The ion mass density provides the inertia, and the

tension of the magnetic field lines provides the restoring force. The wave is also dispersionless, and

the motion of the ions and the perturbation of the magnetic field, are in both the same direction, and

transverse to the direction of propagation. Alfvén waves are also considered as suitable transporters

of non-thermal energy, which is required to heat the Sun’s inactive atmosphere (Hasegawa and Uberoi,

1982) (Mathioudakis, Jess and Erdélyi, 2013).

The Alfvén velocity can be found, by knowing the low-frequency relative permittivity, ε, of a

magnetised plasma. This can be calculated by:

𝜖 = 1 +1

𝐵2𝑐2𝜇0𝜌


11

Where, c is the speed of light, µ0 is the permeability of free space, B is the magnetic field strength

and 𝜌 = ∑𝑛𝑠𝑚𝑠 is the total mass density of all the charged plasma particles; s denotes all plasma

species, including electrons and most ions. From this, the phase velocity of an electromagnetic wave

in this medium can be calculated:

𝑣 =𝑐

√𝜖=

𝑐

√1 +1𝐵2𝑐2𝜇0𝜌

Or, 𝑣 =

𝑣𝐴

√1 +1𝑐2𝑣𝐴2

[EQ. 4]

The Alfvén velocity is given by;

𝑣𝐴 =𝐵

√𝜇0𝜌 [EQ. 5]

2.3. CMEs and Magnetic Reconnection Coronal Mass Ejections (or CMEs) are the sudden releases of solar material into interplanetary space.

There are two classified types of CME, fast and slow (Gosling, Bame et al., 1994; J. T. Gosling, 1996).

Fast CMEs show no sign of acceleration, with the energy released in a more explosive manner. Slow

CMEs which accelerate in the direction of the chronograph suggests a continuous ejection of matter

and energy. The average energy released in a CME is between 1022 and1024𝐽(Howard, Sheeley,

Koomen and Michels, 1985). Based on SoHO and LASCO observations between 1996 and 2003 the

speed of the CMEs have been measured to vary greatly, from 20 to 3000km/s, and their transit time

to Earth’s radius can range from a few hours to a few days (V. Yurchyshyn) (V. Bothmer, 1996). CMEs

directed towards Earth are known as Halo CMEs due to their circular ‘halo’ appearance on the

chronograph. Full halo events extend 360O around the sun. A partial halo event on the other hand, is

one with an apparent width that is greater than 120O, and appears in projection above at least one

pole (Hudson, Lemen, St. Cyr et al., 1998; Burlaga, Plunkett and St. Cyr, 2002). The frequency of CMEs

are very dependent on the phase of the solar cycle, during solar minimum about one CME every 5 days

is expected, this increased to between 3 and 4 every day.

CMEs appear to originate predominantly from the mid latitude regions and from active regions such

as sunspots. CMEs were originally thought to be driven by the heat of an explosive flare, however it is

now suggested that they are unrelated. This is because the energy released in a CME must logically be

built up over time and released quickly, the mechanism of this release is still up for debate, but one

idea is magnetic reconnection. CMEs produce evident shock waves that disrupt local and global solar

structures, this occurs when the speed of the CME exceeds the local Alfvén speed (Raymond,

Thompson, St. Cyr et al., 2000).

The majority of CMEs, 60-70%, appear to emerge from flux ropes embedded in coronal streamers and

seem to occur when large amounts of plasma collect in the streamer tips (Subramanian, Dere, Rich

and Howard, 1999) (Chen, 2013). Many CMEs appear to display a flux rope topology characteristic, an

explanations for this is, on occasion, some field lines embedded deep within the CME seem to only

have one end connected to the sun (Gosling, Birn and Hesse, 1995).

Magnetic reconnection seems to play a vital role in the explosive events in the corona, especially in

solar flares. A reconnection is driven by two oppositely directed inflows which collide in an area known


12

as the diffusion region. In this diffusion region the plasma β is high but the magnetic field is low. (Priest,

Forbes and Forbes, 2000) It is still unclear however how the energies produced play into CMEs. On the

reverse of this, CMEs may drive reconnections along streamer current sheets.

2.4. Streamers, the Slow and Fast Solar Wind and LASCO Streamers are the most prominent feature in the global coronal structure, they are clear against their

background with a definitive brightness boundary. They are much brighter than the background

corona; this is due to the high energy electrons trapped in magnetic loops. Other properties of

streamers include closed magnetic field arcades, streamer cusps, and high density plasma sheets(Chen,

Song et al., 2010). Streamers are only visible during a solar eclipse, or by the use of a chronograph.

They tend to overlie active regions and are a balance between the confining of the magnetic field and

the expanding plasma.

Streamers are also thought to be a source for the slow solar wind and during solar minimum (period

of lowest solar activity) tend to appear from the mid latitude regions. As the minimum wanes,

streamers migrate towards the poles and the slow solar wind is then seen from all latitudes. (Chen,

2013) The slow solar wind averages 400km/s at 1AU and as mentioned appears to come from

streamers, some evidence for this source of solar wind is through magnetic reconnection in streamers,

giving to a release of matter and energy. Specifically this is outside the streamer-coronal hole

boundaries where the velocity of the wind depends greatly on the magnetic topology of flux tubes. In

contrast however, the fast solar wind is thought to originate in coronal hole regions and averages

speeds between 700 and 800km/s at 1AU. (U. Feldman, 2005).

Streamers have been thoroughly observed through the use of LASCO-C2 observations. Past

observations indicated that the magnetic structure of streamers consist of closed magnetic arcades,

however more recent data from LASCO-C1 seem to suggest a series of closed loops

The first part of a streamer is considered as a bulge or a helmet, and they typically stretch to around

1.5 solar radii. The width and radial extension of streamers is also dependent on solar cycle, being

smaller and shorter around solar maximum, the converse is true for solar minimum (Koutchmy and

Livshits, 1992). The second noticeable structure in a streamer, is the stalk. The stalk can stretch over

many solar radii, and is an effect of both the solar wind and the magnetic properties within the

streamer.

Large Angle Spectrometric Chronograph (or LASCO), is the primary instrument used to observe the

solar corona. It is based on The Solar and Heliospheric Observatory (or SoHO), and is often used in

tandem with another instrument on-board SoHO, the Extreme ultraviolet Imaging Telescope (or EIT).

LASCO consists of 3 different instruments, known as C1, C2 and C3. The LASCO-C1 is a Febry-Pérot

interferometer which images from 1.1 to 3 solar radii. C2 and C3 are both white light chronographs

which image from 2 to 6, and 3.7 to 32 solar radii respectively (G. E. Brueckner, 1995). The Ultraviolet

Coronagraph Spectrometer on the SoHO satellite has detected coronal emission lines of H, N, O, Mg,

Al, Si, S, Ar, Ca, Fe and Ni, especially in coronal streamers. The instrument covers the 940-1350Å range

and the 470-630Å range in second order(Raymond, Kohl, Noci et al., 1997).

CMEs and streamers have an intrinsic relation to one another, whilst streamers are thought to be a

potential source for a CME, CMEs give rise to large-scale phenomena on nearby streamers. During a

CME nearby streamers can be deflected and give rise to a streamer wave; a long-period, large scale

wave, which carries energy and propagates outwards along the plasma sheet. The damping of this

wave has nothing to do with dissipative processes(Spruit, 1982), it is instead due to the effect of the


13

magnetic field lines. The well-known Lorentz equation can provide a link between fluidic equations and

electromagnetic;

𝑗̅ × �̅� =

1

𝜇(𝐵. ∇)𝐵 − ∇(

𝐵2

2𝜇), EQ.3

Where μ is the magnetic permeability This can give rise to a magnetic restoring force, the effect of which means the amplitude of the wave

decreases with time. This can also be known as the magnetic tension;

Tension Force =

1

𝜇

𝐵02

𝑎2(𝑥𝑖 + 𝑖) EQ.4

Some measurements of the magnetic field of streamers show that it decreases from 0.3G at 3 solar

radii, to 0.01G at 10 solar radii(Chen, 2013). The magnetic properties of the streamer can be deduced

if the phase speed of the wave is known, given various parameters such as the period. EQ. 2 can be

used to calculate the phase speed of an electromagnetic wave.

2.5. Conclusions Over the course of this literature review, the processes of various wave properties within the Sun’s

interior and atmosphere have been looked at, as well the origins of coronal streamers and the

mechanisms which cause wave oscillations within the streamers. It has been determined that CMEs

drive streamer waves, as well as streamers being a potential source for CMEs. Also the damping of

these waves is due to the magnetic field lines within streamer itself, rather than dissipative forces.

Furthermore, it was discovered that using the LASCO C2 and C3 instruments on board SoHO are the

best way to view streamers.

2.6. Bibliography 1. Osterbrock, D.E., The Heating of the Solar Chromosphere, Plages, and Corona by

Magnetohydrodynamic Waves. The Astrophysical Journal, 1961. 134: p. 347. 2. Withbroe, G.L. and R.W. Noyes, Mass and Energy Flow in the Solar Chromosphere and Corona.

Annual Review of Astronomy and Astrophysics, 1977. 15(1): p. 363-387. 3. Santamaria, I.C., E. Khomenko, and M. Collados, Magnetohydrodynamic wave propagation

from the subphotosphere to the corona in an arcade-shaped magnetic field with a null point. Astronomy and Astrophysics, 2015. 577: p. 70.

4. K. L. DeRose, C.C., P. Pribyl, W. Gekelman. Measurement of the Plasma Beta in the Enormous Toroidal Plasma Device (ETPD). 2008; Available from: http://www.pa.ucla.edu/sites/default/files/files/REU/Papers%202008/derose_kim.pdf.

5. Aschwanden, M.J., Physics of the Solar Corona. An Introduction with Problems and Solutions (2nd edition). Pour la Science, ed. M.J. Aschwanden. 2005. 853.

6. Roberts, B. and A.R. Webb, Vertical motions in an intense magnetic flux tube. Solar Physics, 1978. 56(1): p. 5-35.

7. Steiner, O., Photospheric Processes and Magnetic Flux Tubes. 2006. 8. Nakariakov, V.M., V.F. Melnikov, and V.E. Reznikova, Global sausage modes of coronal loops.

Astronomy and Astrophysics, 2003. 412: p. L7-L10. 9. Doorsselaere, T.V., V.M. Nakariakov, and E. Verwichte, Detection of Waves in the Solar Corona:

Kink or Alfvén? The Astrophysical Journal Letters, 2008. 676(1): p. L73. 10. Aschwanden, M.J., Coronal magnetohydrodynamic waves and oscillations: observations and

quests. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 2006. 364(1839): p. 417-432.

http://www.pa.ucla.edu/sites/default/files/files/REU/Papers%202008/derose_kim.pdf


14

11. Hasegawa, A. and C. Uberoi, Alfven wave. DOE Critical Review Series. 1982. p. Medium: ED; Size: Pages: 137.

12. Mathioudakis, M., D.B. Jess, and R. Erdélyi, Alfvén Waves in the Solar Atmosphere. From Theory to Observations. Space Science Reviews, 2013. 175: p. 1-27.

13. J. T. Gosling, P.R., The acceleration of slow coronal mass ejections in the high-speed solar wind. Geophysical Research Letters, 1996. 23(21).

14. Gosling, J.T., et al., The speeds of coronal mass ejections in the solar wind at mid heliographic latitudes: Ulysses. Geophysical Research Letters, 1994. 21(12): p. 1109-1112.

15. Howard, R.A., et al., Coronal mass ejections: 1979–1981. Journal of Geophysical Research: Space Physics, 1985. 90(A9): p. 8173-8191.

16. V. Yurchyshyn, S.Y., H. Wang, N. Gopalswamy, Statistical Distributions of Speeds of Coronal Mass Ejections. Big Bear Observatory, Centre for Solar Physics ans Space Weather, Crimean Astrophysical Observatory, Centre for Solar Research of New Jersey Institute of Technology, Laboratory for Experimental Physics: Califonia, USA. Washington, USA. Crimea, Ukraine. New Jersey, USA. Maryland, USA.

17. V. Bothmer, R.S., Signatures of Fast CMEs in Interplanetary Space. Advances in Space Research, 1996. 17(4).

18. Burlaga, L.F., S.P. Plunkett, and O.C. St. Cyr, Successive CMEs and complex ejecta. Journal of Geophysical Research: Space Physics, 2002. 107(A10): p. SSH 1-1-SSH 1-12.

19. Hudson, H.S., et al., X-ray coronal changes during Halo CMEs. Geophysical Research Letters, 1998. 25(14): p. 2481-2484.

20. Raymond, J.C., et al., SOHO and radio observations of a CME shock wave. Geophysical Research Letters, 2000. 27(10): p. 1439-1442.

21. Subramanian, P., et al., The relationship of coronal mass ejections to streamers. Journal of Geophysical Research: Space Physics, 1999. 104(A10): p. 22321-22330.

22. Chen, Y., A review of recent studies on coronal dynamics: Streamers, coronal mass ejections, and their interactions. Chinese Science Bulletin, 2013. 58(14): p. 1599-1624.

23. Gosling, J.T., J. Birn, and M. Hesse, Three-dimensional magnetic reconnection and the magnetic topology of coronal mass ejection events. Geophysical Research Letters, 1995. 22(8): p. 869-872.

24. Priest, E., T. Forbes, and T. Forbes, Magnetic Reconnection. Magnetic Reconnection, by Eric Priest and Terry Forbes, pp. 612. ISBN 0521481791. Cambridge, UK: Cambridge University Press, June 2000., ed. E. Priest. 2000.

25. Chen, Y., et al., Streamer Waves Driven by Coronal Mass Ejections. The Astrophysical Journal, 2010. 714(1): p. 644.

26. U. Feldman, E.L., N. A. Schwadron, On the sources of the fast and slow solar wind. Journal of Geophysical Research: Space Physics, 2005. 110.

27. Koutchmy, S. and M. Livshits, Coronal Streamers. Space Science Reviews, 1992. 61: p. 393-417. 28. G. E. Brueckner, e.a., The Large Angle Spectroscopic Chronagraph (LASCO). Solar Physics, 1995.

162(1). 29. Raymond, J.C., et al., Composition of Coronal Streamers from the SOHO Ultraviolet

Coronagraph Spectrometer. Solar Physics, 1997. 175(2): p. 645-665. 30. Spruit, H.C., Propagation speeds and acoustic damping of waves in magnetic flux tubes. Solar

Physics, 1982. 75(1-2): p. 3-17.


15

3. Assumptions and Equations Throughout the project assumptions had to be made, either to make the equations or the

model simpler, and are listed below.

3.1. Assumptions made about the Streamer Model These assumptions were made about the model of the streamer, how it was being visualised,

and used.

The oscillation of the streamer is a fundamental fast kink mode, this reduces the complexity

of the model and allows for simpler calculations.

The streamer is cylindrical with a radius equal to the radius visible in the image.

Within the streamer the electron density is equal to the ion density, as the streamer is not

electrically charged.

The ion plasma only consists of hydrogen ions.

3.2. Assumptions made about the Calculations These assumptions were made during the calculations that were used throughout the

experiment.

At the distance SoHO is from the sun, it can be assumed that 1 arc-second is equal to 730km.

This allows a distance to be calculated from an image, given that the number of arc-seconds

per pixel is known.

The plasma density outside of the streamer is much less than the plasma density inside the

streamer, 𝜌𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 ≪ 𝜌𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙.

The magnetic field inside the streamer and the magnetic field outside the streamer are equal,

𝐵𝑖 = 𝐵𝑒.

The wave shape follows that of a sine wave, as such the wave is symmetrical and hence the

total wavelength is double that of a half wavelength.

3.3. Equations used in the Experiment Throughout the project multiple equations are used. This section contains a list of equations

and the variables used in them, as well as some derivations.

Alfvén velocity, 𝑣𝐴 =𝐵0

√𝜇0𝜌, where, 𝐵0 is the magnetic field, 𝜇0 is the permeability of free space,

𝜌 is the plasma density. For this experiment, this equation was used in the form 𝐵0 = 𝑣𝐴√𝜇0𝜌,

to calculate magnetic field.

Kink speed, 𝑐𝐾 = (𝐵𝑖2+𝐵𝑒

2

𝜇0(𝜌𝑖+𝜌𝑒))

1

2, where the magnetic field inside the flux tube is 𝐵𝑖, and outside

the tube is 𝐵𝑒. Using the above assumptions, results in:

𝑐𝐾 = (𝐵𝑖2 + 𝐵𝑒

2

𝜇0(𝜌𝑖 + 𝜌𝑒))

12

𝑐𝐾 = (2𝐵2

𝜇0𝜌)

12

𝑐𝐾 =√2𝐵

√𝜇0𝜌= √2𝑣𝐴

So,


16

𝑣𝐴 =𝑐𝐾

√2

Wavenumber, 𝑘 =2𝜋

𝜆, where λ is the wavelength in meters.

The frequency of the fast kink mode (Doorsselaere, Nakariakov et al., 2008), 𝜔2 = 𝜔𝑘2 = 𝑣𝑘

2𝑘2,

where the kink speed, 𝑣𝑘 = 𝑣𝐴 [2

1+𝜌𝑒𝜌𝑖

]

1

2

, 𝜌𝑒 is the electron density, 𝜌𝑖 is the ion density and k is

the wave number. Substituting known values gives:

𝜔2 = 𝑘2𝑣𝐴2 [

2

1 +𝜌𝑒𝜌𝑖

]

𝜔 = √2𝑘2𝑣𝐴

2

1 +𝜌𝑒𝜌𝑖

𝜔 = √2𝑘2𝐵0

2

𝜌𝑖𝜇0 (1 +𝜌𝑒𝜌𝑖)

And hence

𝜔 = √2𝑘2𝐵0

2

𝜇0(𝜌𝑖 + 𝜌𝑒)

Time period, 𝑇 =2𝜋

𝜔.

Lorentz force, 𝒋𝑥𝑩 =(𝑩.𝛁)𝑩

𝜇− 𝛁(

B2

2𝜇), the first term on the right- hand side is the magnetic

tension force, the second term on the right-hand side is the magnetic pressure force. Only the

magnetic tension force is needed in this experiment.

Magnetic tension force, (𝑩.𝛁)𝑩

𝜇, this is non-zero if B varies along the direction of B, and can be

regarded as being produced by the effect of a tension along B of magnitude 𝐵2𝜇⁄ per unit

area. It can also be said that 𝑩 = 𝐵�̂�, in terms of the unit vector along the field, as such:

𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑓𝑜𝑟𝑐𝑒 (𝑇𝑚) =𝐵

𝜇

𝑑

𝑑𝑠(𝐵�̂�)

𝑇𝑚 =𝐵

𝜇

𝑑𝐵

𝑑𝑠�̂� +

𝐵2

𝜇

𝑑�̂�

𝑑𝑠

𝑇𝑚 =𝑑

𝑑𝑠(𝐵2

2𝜇) �̂� +

𝐵2

𝜇

�̂�

𝑅𝑐

Where, �̂�, is the unit vector principal normal to the magnetic field line, and Rc, is its radius of

curvature (Priest, 2014).

Radius of curvature, 𝑅𝑐 =ℎ

2+𝑤2

8ℎ, where, h, is the height of the

circle segment, and, w, is the width of the circle segment, shown

in figure 2.

Figure 2: Diagram of a chord of a circle.


17

4. Experimental Method 4.1. Data Acquisition

Collecting the data for this experiment proved to be challenging. Due to a solar minimum in

2008-2009, no useful data could be collected for a couple years either side of the solar minimum

peak, from approximately mid-2006 through to mid-2010. Also, because of the nature of the

experiment, and certain parameters that must be met; such as the need to see a clear oscillation and

perturbation along the streamer, and the difficulties encountered when there is no single

predominate streamer (as shown in figure 3); the number of events that were useable were reduced

drastically. This number again was reduced when the events that were considered had no associated

images, or there was a server error and the images could not be returned.

Firstly, a NASA database of CME events and LASCO daily movies was scoured to find suitable

events (NASA). This website contains daily movies from the LASCO instrument aboard the SoHO

spacecraft, the database has information on nearly all of the CME events since 1996, information such

as acceleration, mass and kinetic energy of the coronal mass ejections. When looking for events to use,

it was the events whose mass was greater than 1x1016g. The logic behind this was that the CMEs that

contained the most mass, had the most kinetic energy and therefore had the greatest chance to

displace the streamer. Once events had been found, the associated data files then had to be obtained.

These files were downloaded in the form of FITS files from the LASCO images query form, found on the

Large Angle and Spectrometric Coronagraph Experiment website ((NRL); (NRL)). A Flexible Image

Transport System (or FITS) is an open standard that defines a digital file format useful for transmission,

storage and processing of scientific images. FITS is the most common type of digital file format used in

astronomy; as the file type is designed specifically for scientific data and each file contains a header.

This is unlike many image formats as this header includes many provisions for describing photometric

and spatial calibration information (Wells, Greisen and Harten, 1981). The term image is loosely

applied, when taken in context to FITS files; as the format supports data arrays of arbitrary dimensions,

whereas, normal images are usually 2- or 3-Dimensional. However, for simplicity in this paper, FITS

files will be split into two parts, the header information and the ‘image’ (Pence, Chiappetti, Page et al.,

2010).

Figure 3: Two images, one from 2012/06/23 showing an example of a 'thin' streamer on the lower right of the image; another image from 2013/10/31 showing an example of a 'fat' streamer on the lower left of the image.


18

As the brightness of the images was not used during the experiment, uncalibrated FITS images

could be used; therefore, level 0.5 C2 files were downloaded. During the download process, it had to

be made sure that all the files were of the same size, that is 1024x1024 pixels. This is due to these files

being the largest files available, and as such the resolution of these images is the greatest. Images that

have the starting resolution of 512x512 pixels, have different properties to the 1024x1024 images,

such as a different number of arc seconds per pixel and different resolutions. Because of this, the

512x512 pixel images had to be discounted, and only the 1024x1024 images could be used, due to this

the time differences between each image could be inconsistent.

4.2. Data Manipulation and Calculations

4.2.1. Data Manipulation Data manipulation is the process by

which data is changed in an effort to make it

easier to read and use. The manipulation of the

data in this experiment was relativity simple.

Starting with an original FITS file, an example of

which is given in figure 4, a black mask is added

to the image so it covers the bright ring at the

centre of the image. This was done to remove

the brightest part of the image, so that the

second part of the data manipulation can be

completed. After the mask was added, the top

and bottom 1 percent of the intensity

distribution were removed. This can be achieved

by thinking of all the intensity values, from light

to dark, being put into a histogram; the quantity

of each intensity value was then calculated and

plotted on the graph. If for arguments sake,

that the darkest intensity value has a value of

1, and the most intense value is equal to 100,

then all the values of intensity 1 and 100 are

ignored and removed. An example of the

resulting file is shown figure 5. This did not

affect the ultimate outcome of the

experiment, as stated above the brightness

was not used, it simply made the coronal

features such as streamers and coronal mass

ejections more visible.

All the data manipulation, calculations

and analysis was carried out in an IDL program,

which is provided in the appendices under

appendix 2. This code was particularly required

to carry out the calculations that this

experiment demanded; these calculations are

outlined below.

Figure 4: An example of original file used in the experiment.

Figure 5: An example of a manipulated file used in the experiment.


19

4.2.2. Calculations Due to the data being used in the experiment, existing in FITS file format, the basis for most of

the calculations is somewhat subjective. To tell the program which pixel coordinates to take for various

calculations, such as; the streamer length, the wavelength and radius of curvature; the user is required

to use the cursor to click the image, on the points that the program should use. These points are also

the basis for the speed calculations and as such the calculations of the magnetic field, frequency and

time period. Although the program informs the user where they should click, the exact location will

vary depending on the image, the user, and what the user perceives is the correct position of the point.

However, despite the subjective nature of the experiment in that sense, the program does

return suitable and somewhat accurate results, but this shall be discussed later. The calculations

themselves are well tested and are laid out in a logical manner. The following figure displays a flow

chart that summerises the process taken by the program, the actual code of which is given in appendix

2.

Figure 6: A flowchart to show the process the IDL program follows.

As shown in the flowchart, the first calculation required is to calculate the length of the

streamer. This was achieved by the user selecting the ‘base’ of the streamer, the region of the streamer

closest to the mask. The user then needs to select the last visible part of the streamer, the IDL program

then calculated the difference in these points in pixels. From the header of the FITS file, the program

extracts information about the number of arc-seconds per pixel, in this case 11.9, and so the number

of pixels is multiplied by this. Using the assumption that at the distance the SoHO spacecraft is from

the Sun, there is approximately 730km per arc-second, the streamer length in arc-seconds is then

multiplied by 730km. Finally, 1 solar radius is added to this value, this then gives the approximate

length of the streamer. This is the same technique that is used in future distance measurements.

The program creates a series of running difference images (RDIs) so that the kink speed can be

calculated. An RDI is the resulting image where one image is subtracted from another, for example,

RDI=image2-image1, this is shown in figure 7 as an example. The bright regions representing where

the wave is in the second image, the dark regions showing where the wave was initially. Using RDIs it

is possible to see where the streamer was and is, in two consecutive images. This means that by

Data manipulation.

Streamer length calculation.

Running difference

images created.

Kink speed calculation.

Mid-distance calculation.

Alfvén speed calculation.

Approximate wavelength calculation.

Ion plasma density

calculation.

Magnetic field strength

calculation.

Radius of curvature

calculation.

Magnetic tension force calculation.

Frequency calculation.

Time period.

Plotting the results in graphs.

Printing results to console.

Printing results to a file.


20

selecting two points on an RDI, e.g. the two

crests of the wave at different times, and

dividing by the time period between the

images (speed =distance

time), the speed can be

calculated. Each time the kink speed is

calculated; the average distance between the

two points from the centre of the Sun

(henceforth called mid-distance), is also

calculated and stored in an array. At each

mid-distance point, the wave/kink speed,

Alfvén speed, magnetic field strength,

magnetic tension force, ion plasma density,

time period and frequency are calculated.

Using the relation between kink speed, cK, and

Alfvén speed, 𝑣𝐴 =𝑐𝐾

√2, the Alfvén speed is

calculated; this coupled with the ion plasma

density and the permeability of free space

gives the magnetic field strength, by 𝐵0 =

𝑣𝐴√𝜇0𝜌𝑖.

The ion plasma density is calculated from the data provided by Dr Huw Morgan at Aberystwyth

University (Quémerais and Lamy, 2002; Morgan, 2015). By using the assumption that ion and electron

plasma densities are equal, the electron plasma density needs to be converted from cm-3 to m-3

(multiplied by 1x106) and then multiplied by the mass of a proton. The data structure in which the

electron density is stored is set out in a radial system, where there are 720 radial components, so data

for every 0.5 degree. Then, for each radial component, from 2 solar radii (Rʘ) to 5 Rʘ, there are 100

data points at intervals of 1 and starting at 0. As such, to select a data point, both a radial component

and a height above 2 Rʘ, is required. It is here that the mid-distance is also used, for each mid-distance

height, the electron density is selected and stored in an array to be used in the magnetic field strength

calculation. However, to comply with the layout of the data structure, the mid-distances had to be

normalised, such that the mid-distance is given in terms of integer values of 100. This was

accomplished by the following method:

x =middistance

(5Rʘ − 2Rʘ

100 )=middistance

26112.825km

Due to the maximum height constraint implemented during the creation of the electron density data

structure, any height above 5 Rʘ will not return a density value and will cause the program to crash. As

such a limit had to be put into place, so any mid-distance value above 99 would be given a value of 0,

hence there being some 0 values at higher heights. Since the ion plasma density has been calculated

at various heights above the 2 Rʘ, the magnetic field strength is also calculated for the same mid-

distance heights. This shows how the magnetic field strength within the streamer changes along the

length of the streamer.

It was also necessary to calculate the radius of curvature; this was needed in the magnetic

tension force equation. The radius of curvature calculates the radius of the circle that is contained

within the curve of the displaced streamer. The equation, 𝑅𝑐 =ℎ

2+𝑤2

8ℎ was used to calculate this, as

shown in figure 2, w, is the width of the chord, and, h, is the height of the chord. This was calculated

Figure 7: An example of a running difference image (RDI) used in the experiment


21

for the images where the streamer is oscillating below 5 Rʘ. Again these values were calculated using

user selected clicks, and distances calculated as described above. The calculated radius of curvature

and the magnetic field strength, B, can then be combined to calculate the magnetic tension force, Tm,

the SI units of which are Pam-1, or Pascal per meter. Using the equation mentioned in section 3.3. for

magnetic tension force, as given below, where μ is the permeability of free space:

𝑇𝑚 =𝑑

𝑑𝑠(𝐵2

2𝜇) �̂� +

𝐵2

𝜇

�̂�

𝑅𝑐

When both of the unit vectors, �̂� and �̂�, are taken to be equal 1, the equation reduces to:

𝑇𝑚 =𝐵2

2𝜇+𝐵2

𝜇

1

𝑅𝑐

Since the magnetic field is calculated at various points along the streamer, and the radius of curvature

is also calculated as the wave propagates along the streamer. It is also possible to calculate the

magnetic tension force, along the streamer.

The wavelength of the streamer is also calculated and

again the user click input method is implemented. Figure 8 is a

diagram that is showing a simplified version of the displaced

streamer, the black circles and diamonds show two sets of points

along the streamer. If the user selects one set of points the

program multiplies that calculated distance by 2, which gives the total approximate wavelength of the

streamer. From this wavelength, in meters, the wavenumber can also be calculated. In the case of this

experiment the angular wavenumber is found, using the equation 𝑘 =2𝜋

𝜆. The wave number is

required as it is needed to calculate the frequency, and subsequently the time period of the wave.

The equation for the frequency of the fast kink mode was derived in section 3.3., in the

simplest form it is given as 𝜔 = √2𝑘2𝐵0

2

𝜇0(𝜌𝑖+𝜌𝑒). Where, k is the wavenumber, ρi is the ion plasma density,

ρe is the electron density. Again, due to the magnetic field strength and the wavenumber having been

calculated at various points along the streamer, it is possible for the frequency to be calculated at the

same points along the streamer. This allows for the user to see how the frequency varies with

heliocentric height along the streamer. From the frequency of the kink mode, it is then possible to

calculate the time period of the wave, using 𝑇 =2𝜋

𝜔. This equation will return the time period in

seconds, for the purpose of the experiment it was decided to change the time period to hours. This

was due to the time intervals between each image taken by LASCO on-board SoHO, being 10. 6̇ minutes

and 12.8 minutes, as such it was assumed that the time period will be an hour or more.

Figure 8: A diagram to show points along the wavelength


22

4.3. Data Analysis After the program carries out all of the required calculations, it then plots the results and saves

all of the data calculated into a file. The program does both of these procedures, so that not only is the

data presented so that it is easier to see trends, it is also possible to have the exact numbers for

comparison purposes. There is more data that the program does not print to file, but instead, will print

to the console. This is due to the data either not being relevant for comparison, or because the data is

in the form of a number, that is needed for a subsequent calculation. However, it is interesting to see

the data nonetheless, hence it is printed to the console.

Unfortunately, due to the limitations of the electron density data only extending out to 5Rʘ,

and the streamer on the 23/06/2012 from 15:48 to 16:24, starting to oscillate from approximately 3Rʘ;

it was not possible to calculate results from this date. As such only the streamer on 21/04/2013 from

07:12 to 11:00 was actually used in the calculations, as this data was within the 5Rʘ limit and so would

return results.


23

5. Results, Discussion and Error Analysis

5.1. Results In this section the important results from the experiment are presented in graphical form. All

other results from this experiment are given in appendix 1, in the form of tables of data from all of the

graphs, as well as some data that is not in graphs.

Figure 9, shows the relationship

between the wave speed, or the kink speed,

of the propagating wave along the streamer,

and the radial heliocentric distance. As can

be seen from the figure there is, on average,

a positive relationship between these two

parameters. The points at around 3.5 and 4.5

solar radii (Rʘ), seem to be erroneously high

and low, respectively. This is mostly caused

by human error due to the subjective nature

of this program. As such these will slightly

affect the results later on.

As in the previous figure, figure 10

shows the relationship between the Alfvén

speed, against the radial heliocentric

distance. Due to the relation between the

kink speed and the Alfvén, as mentioned in

section 4.2.2. figure 9 has a similar shape to

figure 10. As such, the points that were

erroneous in figure 9 are also erroneous in

figure 10.

Figure 9: A graph showing wave speed against height from solar centre.

Figure 10: A graph to show Alfven speed against height from solar centre.


24

Figure 11, displays the ion plasma

density at the same radial heliocentric

distances as in previous figures. As can be

seen from the figure, there is a negative

relationship between the two parameters,

this is to be expected as the particle density

decreases with distance from the sun.

However, it can also be seen that the graph

goes to zero, this is due to the density data

structure limitations, so any height value

above 5Rʘ will return a zero-value.

In figure 12, the magnetic field

strength is shown against the radial

heliocentric height; it can be seen from the

graph, that there is a negative relationship

between the two. Because the magnetic field

strength relies upon the plasma density, the

magnetic field strength above 5Rʘ cannot be

calculated and so a zero value is given. The

first point on the graph at approximately

3.1Rʘ, is erroneously low, however, this could

be due to an exceptionally low Alfvén speed

or ion plasma density. Aside from these two

points, the graph shows a negative trend

which is as expected as the magnetic field will

decrease with increasing height.

Figure 11: A graph to show ion plasma density against height from solar centre.

Figure 12: A graph to show magnetic field strength against height from solar centre.


25

Figure 13 displays the magnetic

tension force against the radial heliocentric

distance. As before with figure 12, the graph

goes to zero, and the first point at

approximately 3.1Rʘ is lower than expected.

Both of these points are due to the vaules of

the magnetic field strength, because of the

equation for magnetic tension force:

𝑇𝑚 =𝐵2

2𝜇+𝐵2

𝜇

1

𝑅𝑐

This has caused the first point that was

already low, to become even more

pronounced, and the last point to be zero.

There is however, a negative relationship

that can be seen; this as expected, if the

magnetic field strength is weakening, then

less force will be required to straighten the

magnetic field lines.

5.2. Discussion As a starting point for the method used in this experiment, a paper in the Astrophysical Journal

(Chen, Song et al., 2010), was heavily used for inspiration. From an attempted replication of this earlier

study, the initial method was created. However, this replicated method was eventually abandoned,

and a new method adopted. This newer method, the one described in this experiment, arose due to

the software that was available during the experiment.

The results presented in the paper are an example of the results that are produced by the

program. These will obviously change with regards to the streamer/ CME shock collision event, and

the electron plasma density. However, these results all

show roughly the correct changes, based on the

calculations, as they move from higher to higher distances.

There are some obvious anomalies, these were mentioned

in the results section as they were presented. These

anomalies can be avoided, but for the sake of the experiment, they were included to give some insight

as to why they would arise, and how they can be avoided.

For example, when selecting the points to use for the speed calculations, it is important to

select the peak of the wave, as indicated in figure 14 by the black circles, as the peak propagates along

the streamer. As mentioned in the calculations section, this selection was carried out using running

difference images; as such, the wave can be in what is effectively two places at two different times in

one image, as seen in figure 7. The difficulty with this method is due to the image subtraction. If the

same pixel in the two images contains the same information, then that information is lost when the

subtraction takes place. This means that it can be hard sometimes to judge where the peak is, and so

the user has to decide. This is where the subjective nature of the experiment arises. This is also why

the wave speed varies so much and why anomalies occur, if the speed equation is examined, speed =distance

time, if the user judges the two peaks to be further apart than they actually are, then the speed will

Figure 13: A graph showing magnetic tension force against height from the solar centre.

Figure 14: A diagram showing wave propagation.


26

increase and vice versa. However, another reason that the wave speed varies, is a problem that cannot

be avoided easily. This is the amount of time between each image from LASCO. In the case of the date

focussed upon in this experiment, there were two different time periods between images. This has

already been touched on, these are 10. 6̇ minutes and 12.8 minutes. Depending over which of these

time periods the two images where taken, will also affect the overall wave speed. These two reasons

would explain the anomalous points at approximately 3.5Rʘ and 4.5Rʘ in figure 9, and subsequently in

figure 10.

As previously mentioned, the basis of

gathering data points from the FITS files in this

experiment, is purely subjective. It is up to the

user to judge where they should click on the

image, when prompted to do so. Whilst this does

return results, the results are not always

consistent; this could be due to the program not

registering a click, or the program accepting one

click for two sets of points, for example. In this

sense the program is not very robust. An

improvement to the program would be

implementing a piece of edge detection

software. Exelis Visual, the company behind IDL,

have some imaging software called ENVI

(Corporation), that can carry out edge detection

procedures, unfortunately this software was not

available to use during this experiment.

For this experiment to be even more

successful, a greater range of electron density

data would be required. If the electron density

data was available out to 10Rʘ for example, it

would then be possible to study the propagation

of waves along the streamer, as in figure 16,

using both the LASCO-C2 and LASCO-C3

instruments. This would provide a greater and a

more reliable understanding of streamers and

magnetohydrodynamics. Having a greater range

of electron density data would also stop the ion

density data going to zero, at distances greater

than 5Rʘ; hence also stopping the magnetic field

strength and magnetic tension force going to

zero.

It must also be noted that between 2Rʘ and

5Rʘ, a full wavelenth was only witnessed on a

small number of the images during the event,

such as figure 15. In figures 15 and 16, the red line indicates the streamer that was analysed, and the

red diamond indicates where the CME originated. Once the oscillation was set into motion and started

to travel along the streamer, the oscillation very quickly passed beyond 5Rʘ, as so could no longer be




27

measured. For the earlier images, where there was no complete wavelengh, it has to be assumed that

the half wavelength that can be seen, would be symmetrical to the half that has not yet formed. This

assumption is essential to the calculations, as without it, the wavenumber, the frequency of the wave,

and hence the time period cannot be calculated reliably. Another factor that was found to determine

the wavelength of the streamer oscillation, was how far radially the streamer was from the source of

the CME. For example, the streamer shown in figure 16 was further away from where the CME

originated, than the streamer in figure 15 was. It was seen that the streamer in figure 16 started to

oscillate noticably at 3Rʘ, and the oscillations would extend out beyond 5Rʘ. However, it was also

noted that the wavelength of this oscillation, was shorter than the wavelength of the streamer

oscillation in figure 15, that was closer to the origin of the CME.

Comparing the results from this experiment with the results presented in the Astrophysical

Journal (Chen, Song et al., 2010); it can be seen in figure 17, that the wave speeds calculated in this

experiment are substantially higher than the literature values, and increases with distance rather than

decreases. This discrepancy is most likely due to the way the wavelength is calculated in this

experiment, the problems of which have already been discussed. However, since this paper was

published, the temporal resolution of LASCO-C2 observations have decreased from approximately 30

minutes, to 10. 6̇ and 12.8 minutes. This decrease in the time interval between observations could also

cause the discrepancy in the two sets of results.

Calculated wave speed (kms-1) 643 888 801 813 1273

Literature wave speed (kms-1) (Chen, Song et al., 2010)

429 391 344 369 325

Figure 17: A table comparing calculated and literature wave speeds.

After the initial results were gathered and it was seen that the wave speeds were much higher than

the literature values, further results were gathered to be compared. Figure 18 shows these results.

Attempt 1 in the table below is the set of results presented in this paper, and then 4 further attempts

were made. It can be seen from this table that the results from this experiment are consistently higher

than the literature values. Despite a few anomalous values such as the last value of the first attempt,

and the middle three values of attempt two.

Average mid-distance 53.27 66.09 77.49 91.30 103.08

Attempt 1 Wave speed (km/s) 643 888 801 813 1273




Attempt 5 Wave speed (km/s) 561 673 719 716 799 Figure 18: A table to show further wave speed results.

5.3. Error Analysis After considering figure 18, it can be seen that the wave speeds cannot be the main source of

error throughout this experiment. While some errors do occur from these values, it is not enough to

cause the inconsistencies seen in the magnetic field calculations. As such, other sources had to be

considered. When looking at the equation for the magnetic field strength, 𝐵0 = 𝑣𝐴√(𝜇0𝜌), it can be

seen that the only variables in the equation are the Alfvén speed, and the plasma density. It has already

been stated that the Alfvén speed is related to the wave speed by, 𝑣𝐴 =𝑐𝐾

√2, so the Alfvén speed will

also not be the major cause of the errors. This leaves the plasma density, when the plasma density is

plotted for the angle required against the heliocentric height as in figure 19, it can be seen that there

is no definite correlation.


28

Aside from the lack of a consistent correlation with heliocentric height, it can be seen that there are

also negative values of electron density. However, this is not logical, as there cannot be negative

density values, and as the heliocentric height increases the density value should decrease. It is these

factors that cause the discrepancy in the magnetic field. Without this consistent correlation, the

magnetic field strength values calculated from these values will not have a reliable correlation.

However, the magnetic field strength values are correct for the electron density values that have been

returned. It must also be noted that from figure 18, the mid-distance points start from approximately

50 on the x-axis, so any data before an x-axis value of 50 can be dismissed.

Discrepancies in the electron density

data could arise from the model that is used

to generate the data (Quémerais and Lamy,

2002). Whilst this model was being

implemented it could have been possible

that there were large variations within the

coronal plasma density itself. These

variations would then be shown in the

electron density data. Therefore, it will be

variations such as these that cause the errors

within the calculations.

After these initial results found that

the electron density model is the main

source of error throughout this experiment,

a new method for gathering the electron

density was devised, this is shown in

appendix 3. The radial line that has been

used so far is 578, along this line at each mid-

distance point the electron density value was

taken. In the revised method, two additional radial lines where taken either side of the main radial

Figure 19: A graph showing the electron density along the normalised heliocentric height, at an angle of 578o of the data structure.

Figure 20: A graph to show the average ion plasma density with heliocentric height.


29

line; e.g. 576, 577, 579 and 580. Along each of these lines, the electron density values were taken for

each mid-distance point; these were then averaged and manipulated to find an average ion density.

The difference can be seen in figure 20 when compared to figure 11. It can be seen from figure 21,

which compares the numerical data for figures 11 and 20, that the values of the averaged ion density

are mostly lower than the ion plasma density, except the values at 4.47Rʘ. Hence, it is thought that

using an averaged ion density set would increase the accuracy of the results.

Heliocentric height (R_sun)

Ion plasma density (m-3)

Average ion plasma density (m-3)

3.09 1.86E-15 1.61E-15

3.49 1.49E-15 1.47E-15

3.94 1.32E-15 1.29E-15

4.47 7.50E-16 7.75E-16

5.17 0 0 Figure 21: A table comparing the ion densities to the average ion densities.


30

6. Conclusion and Acknowledgements

6.1. Conclusion The main aim of this experiment was to calculate the properties, of an oscillating streamer

after a collision with the shock front from a coronal mass ejection. This was achieved successfully with

IDL and Solarsoft, in the form of a program. Two streamer/ shockwave collision events were examined,

on the 23/06/2012 from 15:48 to 16:24 and the 21/04/2013 from 07:12 to 11:00. However, due to

limitations with electron plasma density, the event on 23/06/2012 from 15:48 to 16:24, had to be

dismissed. All the calculations were therefore only carried out on the event on 21/04/2013 from 07:12

to 11:00. After analysing the results that were gathered from this event, it can be seen that the wave

speed and Alfvén speed increases with heliocentric height; ion plasma density, magnetic field strength

and the magnetic tension force, all decrease with heliocentric height. It was also found that the

frequency and time period of the wave decreases with heliocentric height, this would suggest a

damping effect. However, it is the magnetic tension force that acts as the damping force. The magnetic

tension force is the force that straightens bend magnetic field lines.

This method proves to be fairly successful, despite the subjective nature of it, it relies on the

user’s own perception of where boundaries are within the images. The IDL program written, uses the

Solarsoftware (ssw) environment, which provides a library of procedures written specifically for LASCO,

among other instruments. It is believed that this method is a good starting point to be built upon to

produce a more robust, and accurate method for calculating these properties.

The results of this experiment were compared to an earlier study in the Astrophysical Journal

(Chen, Song et al., 2010). After the comparison had been made it was found that the wave speed

results were vastly different and had different trends. Possible explanations for this were discussed in

the above section, however, as different methods and different streamer/ shockwave collision events

were used, the results would be different regardless.

6.2. Acknowledgements I would like to thank Dr. Xing Li from the Institute of Maths, Physics and Computer Science

(IMPACS) at Aberystwyth University, without his help and guidance this project would not have been

possible. I would also like to thank Dr. Huw Morgan also from the Institute of Maths, Physics and

Computer Science (IMPACS) at Aberystwyth University. For without the electron density data that he

supplied the project would have become much, much harder. Finally, I would like to thank the authors

who contributed to the IDL libraries used, such as Solarsoft, again without their contribution, the

project would not have come to fruition.


31

7. Additional Literature This section contains any literature that was used after the completion of the literature

search.

7.1. Bibliography

(NRL), N.R.L., Large Angle and Spectrometric Coronagraph Experiment. (NRL), N.R.L., LASCO/EIT Images Query Form. Aschwanden, M.J., 2005, Physics of the Solar Corona. An Introduction with Problems and Solutions

(2nd edition). Aschwanden, M.J., 2006, Philosophical Transactions of the Royal Society of London A: Mathematical,

Physical and Engineering Sciences 364(1839): 417. Burlaga, L.F., Plunkett, S.P., & St. Cyr, O.C., 2002, Journal of Geophysical Research: Space Physics

107(A10): SSH 1. Center, G.S.F., The IDL Astronomy User's Library. Chen, Y., 2013, Chin. Sci. Bull. 58(14): 1599. Chen, Y., Song, H.Q., Li, B., et al., 2010, The Astrophysical Journal 714(1): 644. Consulting, F.S., Coyote's guide to IDL programming. Corporation, H., IDL. Domingo, V., Fleck, B., & Poland, A., 1995, Space Science Reviews 72(1-2): 81. Doorsselaere, T.V., Nakariakov, V.M., & Verwichte, E., 2008, The Astrophysical Journal Letters 676(1):

L73. Freeland, S., & Handy, B., 1998, Solar Physics 182(2): 497. Freeland, S., L. Bentley, R, D, Solarsoft. G. E. Brueckner, e.a., 1995, Solar Physics 162(1). Golub, L., & Pasachoff, J.M., 2010, The solar corona. (Cambridge University Press). Gosling, J.T., Bame, S.J., McComas, D.J., et al., 1994, Geophysical Research Letters 21(12): 1109. Gosling, J.T., Birn, J., & Hesse, M., 1995, Geophysical Research Letters 22(8): 869. Guenther, D., 1989, The Astrophysical Journal 339: 1156. Hasegawa, A., & Uberoi, C., 1982, Alfven wave. DOE Critical Review Series, (Medium: ED; Size: Pages:

137). Howard, R.A., Sheeley, N.R., Koomen, M.J., & Michels, D.J., 1985, Journal of Geophysical Research:

Space Physics 90(A9): 8173. Hudson, H.S., Lemen, J.R., St. Cyr, O.C., Sterling, A.C., & Webb, D.F., 1998, Geophysical Research Letters

25(14): 2481. Hundhausen, A.J., 2012, Coronal expansion and solar wind. (Springer Science & Business Media). J. T. Gosling, P.R., 1996, Geophysical Research Letters 23(21). K. L. DeRose, C.C., P. Pribyl, W. Gekelman, 2008, Measurement of the Plasma Beta in the Enormous

Toroidal Plasma Device (ETPD), (UCLA, Department of Physics and Astronomy - University of Califonia, LA, CA, 90095-1547).

Koutchmy, S., & Livshits, M., 1992, Space Science Reviews 61: 393. Markus, J., 2004, Physics of the Solar Corona: An Introduction, (Springer-Verlag, Berlin). Mathioudakis, M., Jess, D.B., & Erdélyi, R., 2013, Space Science Reviews 175: 1. Morgan, H., 2015, The Astrophysical Journal Supplement Series 219(2): 23. Moussas, X., Polygiannakis, J., Preka-Papadema, P., & Exarhos, G., 2005, Advances in Space Research

35(5): 725. Nakariakov, V.M., Melnikov, V.F., & Reznikova, V.E., 2003, Astronomy and Astrophysics 412: L7. Nakariakov, V.M., & Verwichte, E., 2005, Living reviews in solar physics 2: 3. NASA, Index of /CME_list/daily_movies. Osterbrock, D.E., 1961, The Astrophysical Journal 134: 347.


32

Pence, W.D., Chiappetti, L., Page, C.G., Shaw, R., & Stobie, E., 2010, Astronomy & Astrophysics 524: A42.

Phillips, K.J., 1995, Guide to the Sun. (Cambridge University Press). Priest, E., 2014, Magnetohydrodynamics of the Sun. (Cambridge University Press). Priest, E., Forbes, T., & Forbes, T., 2000, Magnetic Reconnection. Quémerais, E., & Lamy, P., 2002, Astronomy & Astrophysics 393(1): 295. Raymond, J.C., Kohl, J.L., Noci, G., et al., 1997, Solar Physics 175(2): 645. Raymond, J.C., Thompson, B.J., St. Cyr, O.C., et al., 2000, Geophysical Research Letters 27(10): 1439. Ridpath, I., 2012, A dictionary of astronomy. (Oxford University Press). Roberts, B., & Webb, A.R., 1978, Solar Physics 56(1): 5. Santamaria, I.C., Khomenko, E., & Collados, M., 2015, Astronomy and Astrophysics 577: 70. Spruit, H.C., 1982, Solar Physics 75(1-2): 3. Steiner, O., 2006. Stern, B., 2000, Proceedings of SPACE: 1011. Subramanian, P., Dere, K.P., Rich, N.B., & Howard, R.A., 1999, Journal of Geophysical Research: Space

Physics 104(A10): 22321. U. Feldman, E.L., N. A. Schwadron, 2005, Journal of Geophysical Research: Space Physics 110. V. Bothmer, R.S., 1996, Advances in Space Research 17(4). V. Yurchyshyn, S.Y., H. Wang, N. Gopalswamy, Statistical Distributions of Speeds of Coronal Mass

Ejections, (Big Bear Observatory, Centre for Solar Physics ans Space Weather, Crimean Astrophysical Observatory, Centre for Solar Research of New Jersey Institute of Technology, Laboratory for Experimental Physics, Califonia, USA. Washington, USA. Crimea, Ukraine. New Jersey, USA. Maryland, USA).

Wang, Y.-M., Biersteker, J., Sheeley Jr, N., et al., 2007, The Astrophysical Journal 660(1): 882. Wells, D.C., Greisen, E., & Harten, R., 1981, Astronomy and Astrophysics Supplement Series 44: 363. Withbroe, G.L., & Noyes, R.W., 1977, Annual Review of Astronomy and Astrophysics 15(1): 363. Zhao, X., & Webb, D., 2003, Journal of Geophysical Research: Space Physics 108(A6). Zirker, J.B., 2002, Journey from the Center of the Sun. (Princeton University Press).


33

8. Appendices

8.1. Appendix 1 The data within this appendix is the tabulated form of the graphs given in the results section.

The graph provided here in figure 24, shows relationship between wave frequency and heliocentric

height, but this is not directly needed for the experiment.

Height from solar

centre (R_sun)

Streamer wavelength

(m)=

Wave speed

(km/s)=

Alfven speed

(km/s)=

Ion plasma density

(m-3)

Magnetic field

strength (T)

Magnetic tension

force (Pa/m)

Angular frequency (rads/s)=

Time period

(hours)=

3.08839 1.79E+06 643 455 1.86E-15 2.20E-05 1.93E-04 2.26E-03 7.72E-01

3.49369 2.28E+06 888 628 1.49E-15 2.72E-05 2.94E-04 2.45E-03 7.12E-01

3.93747 2.72E+06 801 566 1.32E-15 2.31E-05 2.12E-04 1.85E-03 9.43E-01

4.46815 3.39E+06 813 575 7.50E-16 1.76E-05 1.24E-04 1.50E-03 1.16E+00

5.17242 3.83E+06 127 900 0 0 0 -NaN -NaN Figure 22: A table displaying all the data gathered during the experiment.

Visible streamer length (km) 2.71E+06

Average streamer Wavelength (m) 2.80E+06 Figure 23: A table showing some average results.

Figure 24: A graph showing the wave frequency against heliocentric height.


34

8.2. Appendix 2

This appendix displays the code used to calculate the required results; the code varies

depending on the data set used. However, this appendix shows the code required for the 21/04/2013

dataset.

PRO mainprogram_21042013

DEVICE, Decomposed=0

LOADCT,0

!p.multi = 0

!p.background = 0

;;;changing the directory to find the images

dir='C:\Users\Nick\Documents\Uni_stuff\Third_year\Disseration\Fits_img\2104

201305'

;;;search folder for number of files

filesearch=FILE_SEARCH(dir, '*', COUNT=n)

PRINT,'number of files=', n

;WINDOW, n, xsize=1250, ysize=725

;;;finding the files

PRINT,findfile(dir+'\*.fts')

file=findfile(dir+'\*.fts')

;;;Giving the files a name

imgs=INTARR(1024,1024,n)

obvtime=STRARR(n)

exptime=FLTARR(n)

arcsec=FLTARR(n)

datee=STRARR(n)

FOR i=0,n-1 DO BEGIN

imgs[*,*,i]=LASCO_READFITS(file(i),headerinfo)

obvtime(i)= headerinfo.TIME_OBS

exptime(i)= headerinfo.EXPTIME

arcsec(i)=headerinfo.PLATESCL

datee(i)=headerinfo.DATE_OBS

ENDFOR

;;;to calculate time

tsec=fltarr(n)

tsecbetween=fltarr(n)


tsec(i)=ANYTIM2TAI(ANYTIM2CAL(datee(i)+' '+obvtime(i),form=11))

ENDFOR


tsecbetween(i)=tsec(i+1) - tsec(i)

ENDFOR

PRINT,arcsec

PRINT,datee

PRINT,obvtime

;;;setting up the image plots, giving the orientation in the window.

;;;incase of odd number of n

WINDOW,n, xsize=1024, ysize=725

m=EVENODD(n) ;m=0 when n is even, m=1 if n is odd

IF m EQ 1 THEN BEGIN

o=((n/2.0)+0.5)

ENDIF ELSE BEGIN

o=n/2

ENDELSE

!P.multi=[0,o,2,1024] ; 3X2 multi-plots

!P.CHARSIZE=1.

!X.CHARSIZE=1.


35

!Y.CHARSIZE=1.

;;;plotting the images.


PLOT_IMAGE,imgs[*,*,i], title='Image'+STRING(i)

ENDFOR

;;;plotting each of the images in their own window and making them all

512x512

images=INTARR(1024,1024,n)

!P.multi=0

minim=MIN(imgs,dim=3)

x=FINDGEN(1024)-headerinfo.crpix1

y=FINDGEN(1024)-headerinfo.crpix2

xx=REBIN(x,1024,1024)

yy=REBIN(REFORM(y,1,1024),1024,1024)

ht=SQRT(xx^2+yy^2)

mask=ht GT 175

HELP, images


WINDOW,i,xsize=1024,ysize=1024

images[*,*,i]=HIST_EQUAL((imgs[*,*,i]-minim)*mask,per=0.01)

TVSCL,images[*,*,i];,/nosquare, xma,rgin=[0,0], ymargin=[0,0]

ENDFOR

sizee=SIZE(images)

rdi=INTARR(sizee(1),sizee(2),n)

rdisize=(SIZE(rdi[*,*,1]))

rdiresize=INTARR(rdisize(1)*3,rdisize(2)*3,n)

for i=1,n-2 DO BEGIN

rdi[*,*,i]=images[*,*,(i+1)]-images[*,*,i]

; tvscl,rdi[*,*,i];,/nosquare, ymargin=[0,0]

rdiresize[*,*,i]=CONGRID(rdi[*,*,i],rdisize(1)*3,rdisize(2)*3,/interp)

endfor

!P.multi=0

PRINT, 'click at each node'

;;;calculating the approximate length of streamer

WINDOW,n+1,xsize=1024,ysize=1024

!P.multi=[0,1,1,612]

TVSCL,images[*,*,3]

PRINT, 'Select the lower right hand point and the upper left hand point

of the wavelength, to be calculated.'

PRINT, 'Change in the code if different dimensions are necessary.'

CURSOR, X1,Y1, /DEVICE

WAIT, 1s

CURSOR, X2,Y2, /DEVICE

x_b=ABS(X1-X2)

y_b=ABS(Y2-Y1)

xy_b=SQRT((x_b)^2+(y_b)^2) ;streamer in

pixels

wlarcsec=xy_b*arcsec[2] ;streamer in

arcsecs

stlength=wlarcsec*730 ;streamer

length in km

stlength=stlength+(696000)

;streamer length + 1 solar radii

PRINT, 'streamer length in arcsecs', wlarcsec

PRINT, 'Streamer length (km)=', stlength


36

;;;to calculate the wavespeed of the streamer

a=n-4

b=4

wavespeed=FLTARR(5)

alfvenspeed=FLTARR(5)

middistance=FLTARR(5)

middistance1=FLTARR(5)

WINDOW,n+2, xsize=1024, ysize=1024

FOR i=4, n-6 DO BEGIN

TVSCL,rdi[*,*,i]

PRINT,'Select two nodes of waves'

WAIT,1s

CURSOR, X4,Y4,/DEVICE

WAIT, 1s


X3=headerinfo.crpix1

Y3=headerinfo.crpix2

XX=ABS(X4-X3)

YY=ABS(Y4-Y3)

XY=SQRT((XX)^2+(YY)^2) ;distance

from centre of sun to first wave node

XXX=ABS(X5-X3)

YYY=ABS(Y5-Y3)

XY2=SQRT((XXX)^2+(YYY)^2) ;distance

from centre of sun to second wave node

xydiff=ABS(XY2-XY)

;difference between the first and second wave node

xydiffarcsec=xydiff*arcsec[2]

;difference in arcsecs

difference=xydiffarcsec*730

;sdifference in km

middistance(i-4)=(((((xydiff/2)+xy)*arcsec[2]*730)-696342.0)/26112.825)

;distance from limb to midpoint

middistance1(i-4)=((((xydiff/2)+xy)*arcsec[2]*730)/696342)

wavespeed(i-4)=(difference/(tsecbetween(i)))*1000

alfvenspeed(i-4)=wavespeed(i-4)/(SQRT(2.0))

IF wavespeed(4) GT 0 THEN BREAK

ENDFOR

PRINT, 'wavespeed',wavespeed

PRINT, 'Alfven speed', alfvenspeed

PRINT, 'middistance', middistance

stop

averagewavespeed=(TOTAL(wavespeed))/(N_ELEMENTS(wavespeed))

wavelength=FLTARR(5)

frequency=FLTARR(5)

timeperiod=FLTARR(5)


;;;Calculate the wavelengths

TVSCL,rdi[*,*,i]


37

PRINT,'Select two crests of waves'

WAIT,1s


WAIT, 1s


XX=ABS(X6-X3)

YY=ABS(Y6-Y3)


from centre of sun to first wave node

XXX=ABS(X7-X3)

YYY=ABS(Y7-Y3)


from centre of sun to second wave node

xydiff=ABS(XY2-XY)

;difference between the first and second wave node

xydiffarcsec=xydiff*arcsec[2]

;difference in arcsecs

difference=xydiffarcsec*730

;sdifference in km

wavelength(i-b)=(difference)*2

frequency(i-b)=wavespeed(i-b)/wavelength(i-b)

timeperiod(i-b)=(1.0/frequency(i-b))

IF wavelength(4) GT 0 THEN BREAK

ENDFOR

k=FLTARR(5)

;wavenumber

pi=3.141592654

FOR i=0,4 DO BEGIN

k(i)=(2*pi)/(wavelength(i)*1000)

PRINT,'Angular wavenumber=', k(i)

ENDFOR

;;;radius of curvature

R_c=FLTARR(5)

h=FLTARR(5)

w=FLTARR(5)


WINDOW, n+3, xsize=1024,ysize=1024

TVSCL, images[*,*,i]

PRINT,'First click'

WAIT,1

CURSOR, X8,Y8,/device,/wait

WAIT, 1

PRINT,'second click'

CURSOR, X9,Y9,/device,/wait

PLOTS, [X8,X9], [Y8,Y9], /DEVICE

WAIT,1

PRINT,'third click'

CURSOR, X10,Y10,/DEVICE,/wait

;stop

WAIT, 1

PRINT,'fourth click'

CURSOR, X11,Y11,/DEVICE,/wait

WAIT,1

XX=ABS(X8-X3)

YY=ABS(Y8-Y3)


38


from centre of sun

XXX=ABS(X9-X3)

YYY=ABS(Y9-Y3)


from centre of sun to second

xydiff=ABS(XY2-XY) ;difference

between the first and second wave node

xydiffarcsec=xydiff*arcsec[2] ;difference

in arcsecs

w(i-4)=xydiffarcsec*730 ;sdifference in

km

;;;;;;;;;;;;;;;;;;;;;;;

XX=ABS(X10-X3)

YY=ABS(Y10-Y3)


from centre of sun

XXX=ABS(X11-X3)

YYY=ABS(Y11-Y3)


from centre of sun to second

xydiff=ABS(XY2-XY) ;difference

between the first and second wave node

xydiffarcsec=xydiff*arcsec[2] ;difference

in arcsecs

h(i-4)=xydiffarcsec*730 ;sdifference in

km

R_c(i-4)=(h(i-4)/2.0)+(w(i-4)^2)/(8.0*h(i-4))

ENDFOR

;;;Electron Density

dir2='C:\Users\Nick\Documents\Uni_stuff\Third_year\Disseration\ElectronDens

ity'

a=DIALOG_PICKFILE(path=dir2);choose file

RESTORE,a,/verbose ; read file

HELP,d ;file is restored into variable “d”, which is a data structure.

;TVSCL,d.nel ; display electron density

;TVSCL,d.nelf ; display a more smoothed (less noise) version of electron

density

;PLOT,d.nelf[*,20],/xst

;plot a latitudinal slice at a given height

;OPLOT,d.nel[*,20],thick=2

;print,'d.rra', d.rra ;print the height range of the polar-coordinate

array

hts=d.rra[0]+FINDGEN(d.nht)*(d.rra[1]-d.rra[0])/(d.nht-1) ;create vector

of height coordinates

pa=d.para[0]+FINDGEN(d.npa)*(d.para[1]-d.para[0])/d.npa ;create vector

of position angle coordinates (0 to 359)

rho=FLTARR(5)

rhoo=FLTARR(5)

middis=FLTARR(5)

FOR i=0,4 DO BEGIN

IF middistance(i) GT 99.0 THEN BEGIN

middistance(i)=0.0

ENDIF ELSE BEGIN

middis(i)=ROUND(middistance(i))


39

rho(i)=d.nel[578,middis(i)]

rhoo(i)=rho(i)*1.67E-27*1000000

ENDELSE

PRINT,'Plasma density=', rhoo(i)

ENDFOR

magneticfield=FLTARR(N_ELEMENTS(wavespeed))

mu0=1.2566E-6

FOR i=0, 4 DO BEGIN

magneticfield(i)= alfvenspeed(i)* SQRT(mu0*rhoo(i))

ENDFOR

averagemagnetic=(TOTAL(magneticfield))/(N_ELEMENTS(magneticfield))

averagewavelength=(TOTAL(wavelength))/(N_ELEMENTS(wavelength))

tensionforce=FLTARR(5)

FOR i=0,4 DO BEGIN

tensionforce(i)=(magneticfield(i)^2)/(2*mu0)+((magneticfield(i)^2)/(mu0))*(

1/(R_c(i)/1000))

ENDFOR

angfreq=FLTARR(5)

caltperiod=FLTARR(5)

FOR i=0,4 DO BEGIN

angfreq(i)=SQRT((2*(k(i)^2)*(magneticfield(i)^2))/(mu0*(rhoo(i))))

PRINT, 'angular frequency=',angfreq(i)

caltperiod(i)=((2*pi)/angfreq(i))/3600

PRINT, 'calculated time period (hours)=',caltperiod(i)

ENDFOR

!p.multi = 0

!p.background = 255


PLOT,middistance1,wavespeed,$

TITLE='Wave speed against height from solar centre',$

XTITLE='Height from solar centre (R_sun)', YTITLE='Wave speed (m/s)',$

THICK=1.0, /YNOZERO, PSYM=-7, COLOR=0, xmargin=[13,3]

WRITE_JPEG, FILEPATH('speedagainstheight.png',

ROOT_DIR='C:\Users\Nick\Documents\Uni_stuff\Third_year\Disseration\Data'),$

TVRD()


PLOT,middistance1,alfvenspeed,$

TITLE='Alfven speed against height from solar centre',$

XTITLE='Height from solar centre (R_sun)', YTITLE='Alfven speed

(m/s)',$


WRITE_JPEG, FILEPATH('alfvenagainstheight.png',


TVRD()

;stop


PLOT,middistance1,magneticfield,$

TITLE='Magnetic field against height from solar centre',$

XTITLE='Height from solar centre (R_sun)', YTITLE='Magnetic field

strength (T)',$


WRITE_JPEG, FILEPATH('magfieldagainstheight.png',


TVRD()


40


PLOT,middistance1,rhoo,$

TITLE='Ion plasma density against height from solar centre',$

XTITLE='Height from solar centre (R_sun)', YTITLE='Ion plasma density

(m^-3)',$


WRITE_PNG, FILEPATH('rhoagainstheight.png',


TVRD()


PLOT,middistance1,tensionforce,$

TITLE='Magnetic tension force against height from solar centre',$

XTITLE='Height from solar centre (R_sun)', YTITLE='Magnetic tension

force (Pa/m)',$


WRITE_PNG, FILEPATH('tensionforceagainstheight.png',


TVRD()


PLOT,middistance1,angfreq,$

TITLE='Wave frequency against height from solar centre',$

XTITLE='Height from solar centre (R_sun)', YTITLE='Frequency of the

wave (Rads/s)',$


WRITE_PNG, FILEPATH('freqagainstheight.png',


TVRD()

PRINT, 'Height from solar centre (R_sun)=',middistance1

PRINT, 'Time between images (s)', tsecbetween

PRINT, 'Average wavespeed (m/s) =',wavespeed

PRINT, 'Alfven speed (m/s)=',alfvenspeed

PRINT, 'Wavelength (km)=',wavelength

PRINT, 'Magnetic field strength (T)=', magneticfield

PRINT, 'Max Amplitude=',h

PRINT, 'w=',w

PRINT, 'Radius of curvature (km)=',R_c

PRINT, 'Magnetic Tension force (T)=', tensionforce

PRINT, 'Angular frequency (rads/s)=', angfreq

PRINT, 'Time period (hours)=', caltperiod

dir2='C:\Users\Nick\Documents\Uni_stuff\Third_year\Disseration\Data\'

OPENU,1, dir2+'STREAMER_DATA.dat'

PRINTF,1, 'Streamer length (km)=', stlength

PRINTF,1, 'Streamer wavelength (m)=', wavelength

PRINTF,1, 'Wave speed (m/s)=',wavespeed

PRINTF,1, 'Average streamer Wavelength (m)', averagewavelength

PRINTF,1, 'Height from solar centre (R_sun)=',middistance1

PRINTF,1, 'Time between images (s)', tsecbetween

PRINTF,1, 'Alfven speed (m/s)=', alfvenspeed

PRINTF,1, 'Ion plasma density=', rhoo

PRINTF,1, 'Magnetic field strength (T)',magneticfield

PRINTF,1, 'Magnetic Tension force (Pa/m)=', tensionforce

PRINTF,1, 'Angular frequency (rads/s)=', angfreq

PRINTF,1, 'Time period (hours)=', caltperiod

CLOSE,1

STOP

END


41

8.3. Appendix 3 This appendix contains the revised bit of code to calculate the average ion density. This

replaces the initial code finding the original ion density.

rho=FLTARR(5)

rho1=FLTARR(5)

rho2=FLTARR(5)

rho3=FLTARR(5)

rho4=FLTARR(5)

rhoo=FLTARR(5)

rhoo1=FLTARR(5)

rhoo2=FLTARR(5)

rhoo3=FLTARR(5)

rhoo4=FLTARR(5)

middis=FLTARR(5)

FOR i=0,4 DO BEGIN

IF middistance(i) GT 99.0 THEN BEGIN

middistance(i)=0.0

ENDIF

middis(i)=ROUND(middistance(i))

rho(i)=abs(d.nel[576,middis(i)])

rho1(i)=abs(d.nel[577,middis(i)])




rhoo(i)=rho(i)*1.67E-27*1000000

rhoo1(i)=rho1(i)*1.67E-27*1000000

rhoo2(i)=rho2(i)*1.67E-27*1000000

rhoo3(i)=rho3(i)*1.67E-27*1000000

rhoo4(i)=rho4(i)*1.67E-27*1000000

;PRINT,'Plasma density=', rhoo(i)

ENDFOR

averagerho=FLTARR(5)

averagerhoo=FLTARR(5)

FOR i=0,4 DO BEGIN

averagerho(i)=(rho(i)+rho1(i)+rho2(i)+rho3(i)+rho4(i))/5

averagerhoo(i)=(rhoo(i)+rhoo1(i)+rhoo2(i)+rhoo3(i)+rhoo4(i))/5

ENDFOR