Electron heating in collisionless shocks observed by the MMS spacecraft

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observed by the MMS spacecraft

Martin Svensson

Space Engineering, master's level

2018

Luleå University of Technology

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Electron heating in collisionless shocks observed

by the MMS spacecraft

Author: Martin Svensson Supervisor: Andreas Johlander,

Swedish Institute of Space Physics Examiner:

Lars-Göran Westerberg,

Luleå University of Technology Submitted for the degree

Master’s degree in Space Engineering with specialization in Space and Atmospheric Physics at Luleå Unversity of Technology

Luleå University of Technology Swedish Institute of Space Physics

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Acknowledgements

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Abstract

Shock waves are ubiquitous in space and astrophysics. Shocks transform directed particle flow energy into thermal energy. As the major part of space is a collisionless medium, shocks in space physics arises through wave-particle interactions with the magnetic field as the main contributor. The heating processes are scale dependent. The large scale processes governs the ion heating and is well described by magnetohydrodynamics. The small scale processes governs the electron heating, it lies within the field of kinetic plasma theory and is still today remained disputed. A step towards the answer for the small scale heating would be to measure the scale, in order to relate it to a known instability or other small scale processes.

The multi-spacecraft NASA MMS spacecraft carries several high resolute particle and field instruments enabling almost instantaneous 3D particle measurements and accurate measurements of the magnetic field. Also the separation between the four MMS spacecraft is as small as ≤ 8km for a certain mission phase. This allows for new approaches when determining the scale which for shocks has not been possible before when using data from previous multi-spacecraft missions with spacecraft separation much larger. The velocity of the shock is large compared to the spacecraft, thus the shock width cannot be directly measured by each spacecraft. Either a constant velocity has to be estimated or we could use gradients of a certain parameter between the spacecraft as the shock flows over them. The usage of gradients is only possible with MMS as all the spacecraft could for MMS be within the shock simultaneously. The change for a parameter within the shock is assumed to be linear between the spacecraft and measurements. It is also assumed that the gradient of the parameter maximizes in the shock normal direction. Using these assumptions two methods have been developed. They have the same working principles but are using two or four spacecraft for linear estimation at each measurement point. From the gradient and parametric data the shock ramp width could then be found. The parameter used in this thesis is the electron temperature.

The methods using one, two and four spacecraft were tested using electron temperature data from different shock crossings. Two problems with the gradient methods were found from the results, giving false data for certain time spans. To avoid these problems, the scale of the electron temperature gradient was determined for roughly half the shock ramp.

It was found using the two and four spacecraft methods that an assumption of constant velocity for the shock speed is an uncertain assumption. The shock speed varies over short time scales and in the shock crossings analysed the constant velocity estimations were generally overestimated. From the two and four spacecraft methods roughly half of the temperature rise in the shock ramp occurred over 10.8 km or 12.4 lpe. This is almost a factor of two greater than previous scale

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1 Introduction

1.1 Shocks, fluid dynamics

Shock waves are dramatic accelerations caused by an impact of a supersonic flow. The first interests in shocks arose in the late 19th century when high speed flows in the field of fluid mechanics were given a lot of attention. In 1885 (Blackmore 1972) Ernst Mach defined his famous criteria for shock development when he realized that for a shock to evolve the relative speed between a flow and an obstacle has to be greater than the local speed of sound, cs. Mach was able to visualize his idea by

drawing a sound generating projectile with speed greater than the speed of sound, creating a cone known as the Mach Cone, see Figure 1. From Mach’s criteria were also the shock characterization parameter, the Mach number M defined as

M = u cs

, (1)

where u is the speed of the flow.

Figure 1: Visualization of a Mach Cone. A projectile travelling to the right in a flow with a relative speed greater than the speed of sound. Image credit: Gibson, University of Connecticut.

1.2 Collisionless Shocks

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above in fluid mechanics, a plasma shock is formed when an obstacle is placed in a supersonic flow. Astrophysical plasmas are often carrying a magnetic field which, first according to De Hoffmann & Teller (1950) is the primary factor for energy exchange between different particles and waves in collisionless shocks. Collisionless means that the collision rate in the medium is small enough to be negligible. For the plasma in the interplanetary medium the typical scale for the mean free path is much greater than other associated scales for a shock wave (Balogh & Treumann 2013). These scales highly depend on the specific type of plasma, for example the scale in the interplanetary medium is much larger than the scale in the solar atmosphere. Thus comparing shocks in fluid mechanics where bi-collisions between molecules are the dominant interaction with collisionless shocks in low dense plasmas where the scales are to large for collisional interaction to be dominant. The dominant interaction in collisionless shocks has to be wave-particle interaction mainly caused by the magnetic field. An important property for a shock is its ability to reflect incoming upstream ions at high energy flows for high Mach numbers. The shock will then not be able to heat all the plasma fast enough through wave-particle interactions. The shock then has to reflect partitions of the inflowing ions in the upstream direction, see Edmiston & Kennel (1984).

An important parameter characterizing a shock is the Mach number. The Mach number is described in section 1.1 as the speed of the flow divided by the speed of sound. For a collisionless plasma the physical circumstances is different and hence the Mach number has to be described in another way. A common way in space physics is to use the Alfvén Mach Number M_A, which is the magnitude of the upstream flow speed, Vu divided by the Alfvén speed, vA. The Alfvén speed is

the phase speed of an Alfvén wave and is given by vA=

B0

√ µ0ρ

, (2)

where B₀ is the solar wind magnetic field, µ₀ is the vacuum permeability and ρ is the mass density of the plasma. Thus MA reads

MA= Vu vA = √VuB0 µ0ρ . (3)

An Alfvén wave is propagating with its wave group velocity parallel to the magnetic field. In magnetohydrodynamics, there also exist a wave type which has a group velocity that could be perpendicular to the magnetic field: the magnetosonic wave, see Balogh & Treumann (2013). For a magnetosonic wave the background magnetic field and the oscillating electric field E₁, generated by the oscillating ions, will compress the plasma by the E1× B0 drift which lies in the same direction

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where θBV is the angle between the magnetic field and the plasma bulk velocity. cms is the

perpendicular magnetosonic speed given by

c2_ms= v2_A+ c2_s (5)

and c_s is the speed of sound as

c2_s = γekBTe+ γikBTi me+ mi

. (6)

The specific heat ratios for electrons and ions, γe,i, is typically three (3) for electrons and one (1)

for ions (Chen 1984), k_B is the Boltzmann constant, m_e,i are the electron and ion masses and T_e,i are the electron and ion temperature. The two solutions in equation 4 refers to the fast, v+_ms and slow magnetosonic wave, v_ms− , where v−_ms < vA < v+ms. The particle bulk speed relative to the

speed of the obstacle is in general larger than v+_ms. In these common cases the shock is considered a fast-mode shock i.e a steepened fast-mode wave. Thus v_ms+ is used for the magnetosonic Mach number as

Mms=

Vu

vms(θBV)

. (7)

Also, as Mms indicates the shock wave steepening for perpendicular flows it is the ‘real’ Mach

number when considering shocks. The magnetosonic Mach number can be calculated by combining equations 4-6.

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Figure 2: Illustration of the Shock normal angle θ_Bn and particle reflection for different θ_Bn at the shock surface. Image credit: Balogh & Treumann (2013)

The third characteristic shock parameter important to the large scale structure of the shock is the plasma beta β. The plasma β is the ratio of plasma thermal pressure to magnetic field pressure, given by

β = nkT B2_/2µ

0

. (8)

β simply describes the ratio of pressure issued by thermic movements to pressure issued by the magnetic field. Thus it is a measure of the relative importance between kinetics and magnetic fields for a flow, see Chen (1984).

1.2.1 Structure of supercritical quasi-perpendicular shocks

Shocks of all kinds are steepened waves between an obstacle and a flow. Depending on the characteristics of the flow and obstacle they might, however, have a different structure in detail. For supercritical shocks parts of the ion flow is reflected. For quasi-perpendicular shocks the reflected ions are convected back to the shock as the magnetic field lines are tangential to the shock surface. The shock foot is the upstream region that is dominated by the incoming plasma and the shock reflected ions. The size of the shock foot is proportional to the ion gyroradius, rciof the reflected ions.

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the shocked downstream region in the magnetosheath there is a large scale rise in the magnetic field called the overshoot which forms above some critical Mach number for quasi-perpendicular shocks. The overshoot region is directly followed by an undershoot region. The changing from overshoot to undershoot will follow the structure of a damped oscillator and is caused by electron currents in the shock ramp (Balogh & Treumann 2013). The spikes are then thought to represent energetic ions trapped in the shock front (Saxena et al. 2005). See Figure 3 for an overview picture of the structure using MMS ion density data from a shock crossing. The overall shock structure varies with the characteristic parameters for shocks, a quasi-perpendicular shock is for instance typically steeper and has less fluctuations than a quasi-parallel shock.

Figure 3: Illustration of the different physical regions for a shock. Arrows mark these regions. Data from a shock crossing measured by MMS.

1.2.2 The Earth’s bow shock

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and the magnetopause is called the magnetosheath. The Earth’s bow shock has a curved geometry and is often considered symmetric around the Earth-Sun line with a close to parabolic shape. An illustration of the Earth’s bow shock can be found in Figure 4. Here one also can see the dependence of shock normal angle as discussed in section 1.2, with the escaping ion and electrons into their respective foreshock region. The angle between the Earth-Sun line and the interplanetary magnetic field also represent the realistic case in the figure as it generally is approximately 45 degrees, see Parker (1958).

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Common coordinates used when working with Sun-Earth related interactions like the bow shock is the Geocentric Solar Ecliptic, GSE coordinates. For GSE coordinates the origin is placed at the Earth’s center and the x-axis points towards the Sun. The positive z-axis is the Ecliptic north pole of earth. The y-axis completes the right-handed system and points in the dusk-ward direction. The GSE coordinate system is used throughout this thesis.

1.3 Reversible and irreversible heating within shocks

The energy distribution for an ideal gas where collisions exists is, typically, a Maxwellian to minimize the entropy. For a collisionless medium the distribution does typically not follow a Maxwellian. For a particle flow in a collisionless medium, the distribution is obtained by measuring the energy for every particle in all directions. From the distribution the temperature can be considered of as the width. A wider distribution means that there are more particles with higher speed in the gas. Heating can then be thought as the broadening of the energy distribution. Thus if one observe the distribution at different times, with the same number of particle counts but with a broader distribution for the later, the gas has been heated. In plasma physics it is often interesting to differ between the distributions for different particle velocity directions like parallel, antiparallel and perpendicular directions relative to the magnetic field lines. In Figure 5 one can see the electron energy distributions for different times from a shock crossing of the Cluster spacecraft. Here the distribution broadening and thus the heating of the electrons can be observed for different directions.

Figure 5: Broadening of particle distributions for different direction and time. Data from a Cluster shock crossing. Image credit: Schwartz et al. (2011).

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particles by the magnetic field is related to the magnetic moment invariance µ = mv

2 ⊥

2B (9)

where m is the particle mass, v⊥ is the particle velocity perpendicular to the magnetic field and B

is the magnitude of the magnetic field. Heating of particles caused by the electric field is related to the induced electric field in the shock ramp. The electric field is induced as the ion penetrates deeper into the ramp before being reflected compared to the electrons. The electrons will then also be accelerated by the Lorentz force

F = q(E + v × B) (10) where q is the particle charge and E is the electric field. This acceleration will for quasi-perpendicular shocks be directed perpendicular to the magnetic field. Here the magnetic field lines are quasi-tangential to the shock surface and E is directed in the shock normal direction. Thus from equation 9 and 10 this represents the reversible heating because the equations allows for both increasing and decreasing B and E over large scales. One can then for shocks use the distribution for perpendicular particle velocities to observe the the reversible heating.

In Figure 5 one can observe heating in both the perpendicular and parallel direction. As described above the reversible processes are related to the perpendicular temperature. There also has to be irreversible heating as the distribution for parallel velocity follows the perpendicular closely for different time cuts. The origin of this irreversible heating has not been determined. One theory is that it may be caused by scattering of electrons from a plasma instability within the shock region as Debye-scale electric fields has been observed, see Bale et al. (1998). Clues to the answer of the origin of the irreversible heating should be found by knowing how the spatial width of the shock ramp scales.

1.4 Magnetospheric Multiscale

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The MMS mission is divided into several phases in which the orbits and the spacecraft separation are changed. Figure 6 shows how the mean spacecraft separation at apogee has varied with time, the times for the different orbit phases 1a, 1b and 2a are marked as the shaded green areas.

Figure 6: MMS spacecraft separation as a function of time. The black line shows the mean separation distance, the shaded yellow area shows the variance in separation and the green shadowing marks the specific orbital phases. In the lower part the local time at apogee is plotted as a function of time. Image credit: Johlander, IRF Uppsala.

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1.4.1 Fast Plasma Investigation

The Fast Plasma Investigation (FPI) instruments of MMS is described by Pollock et al. (2016). The FPI instruments are measuring the differential particle flux for MMS with an unprecedented high time resolution. At each spacecraft the FPI consists of four dual 180 × 90◦ ion spectrometers and four dual 180 × 90◦ electron spectrometers separated around the periphery of the MMS spacecraft. This provides MMS with 3D distributions within the energy range of 10 eV/q to 30 keV/q at a time resolution of 150 ms for ions and 30 ms for electrons. The FPI uses the electrostatic type of spectrometers with a hemispheric geometry for the electron instrument and a toroidal geometry for the ion instrument. The general working principle for an electrostatic analyzer is that over two parallel curved conducting plates with adjustable voltages an electric field is created. The incoming particle then needs to have a specific velocity (Energy) in order to pass between the curved plates as the centripetal and electrostatic forces need to match. If the particle passes through the curved plates it will be detected and counted by a particle detector. This process can be described by balancing the equation

W q =

1

2ER, (11)

where W is the particle energy, q is the charge, E is the applied electric field and R is the radius of the curved plates. Thus by adjusting the electric field the particle distribution for different energies can be measured.

1.4.2 Fluxgate Magnetometers

The FluxGate Magnetometers (FGM) of MMS is described by Russell et al. (2016). Precise measurement of the magnetic field is of utmost importance for MMS to obtain its scientific objectives. MMS uses two different kinds of triaxial fluxgate magnetometers provided by two different manufacturers. These are mounted on separated booms and are called the Digital FluxGate magnetometer (DFG) and the Analogue FluxGate magnetometer (AFG). Their maximum sampling frequency is 128 Hz. Data obtained from the two sensors are then continually inter-compared. A fluxgate magnetometer consists of drive windings and sense windings around a magnetic core. The drive windings are fed with an alternating current which is strong enough to saturate the coil for each half-period. This periodic variation will become asymmetrical if an external magnetic field is added to the system by which the degree of asymmetry is proportional to the added magnetic field. This is measured by means of the sense winding (Hoymork 2002).

2 Scope and motivation

2.1 Scales hints the physics

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and increase our understanding of the shock ramp and what processes within the ramp that are causing this irreversibility. An important parameter which should give clues about the physics within the ramp is the scale or thickness. This can then be attributed to candidate processes like kinetic instabilities, dispersion or unstable shock reformation. One should also relate the measured thickness to the corresponding Mach number, shock normal angle and plasma beta value as these are believed to affect the ramp width. In this thesis the main focus is to test and compare novel techniques for determination of the temperature gradient scale. The obtained results will be compared to previously reported results.

2.2 Previous work and the problem with scaling

A similar study as this has been made using data from the ESA Cluster mission which also was a multi-spacecraft system, exploring the interaction between the solar wind and the magnetosphere of the Earth. By statistical studies Hobara et al. (2010) argues that partitions of the electron temperature gradient in a quasi-perpendicular shock scale as the electron inertial length, λ_pe. Schwartz et al. (2011) show data from one Cluster measurement of a shock, which supports the theory of electron inertial length scaling. Also the multi-spacecraft ISEE performed measurements of shock crossings.

The main issues for both Cluster and ISEE with scaling of small scale boundary parameters (like the temperature in a shock) is the large spacecraft separation and instrumental time resolution. For Cluster the spacecraft separation varied but was never small enough for the possibility to have all the spacecraft situated within the ramp at once. Thus they could not directly measure spatial differences in the ramp. Schwartz et al. (2011) assumed a constant shock velocity which was estimated using a timing method. In this timing method they used the time when the spacecraft of Cluster measured a specific magnitude of the magnetic field in the shock ramp. The time difference, together with the spacecraft positions gave a constant velocity estimate (Schwartz 1998), see Figure 7.

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As the particle instrument of Cluster requires a complete rotation about the spacecraft spin axis for a complete 3D particle distribution the time resolution for a 3D particle distribution is ≈4 s, (Johnstone et al. 1997). This limits the particle measurements at thin boundary layers like the steep ramp of a shock. Complete 3D distribution for shocks with high propagation speeds could therefore not be obtained. However approximations of gyrotropic particle distributions were made for some special cases when the spacecraft spin axis were aligned with the magnetic field lines, see Schwartz et al. (2011).

2.3 Improvements with MMS

As described in section 1.4 MMS is primary a mission for further understanding of the magnetic reconnection process. Thus, as also described in section 1.4, MMS is highly suitable for in-situ measurements at small scales giving a new tool for deeper understanding of the physics inside the plasma boundary layers. This means that MMS will provide measurements from within the steep ramp of the bow shock which gives the opportunity to directly measure gradients within the shock ramp. The gradients could then be used to estimate the spatial span for the steepening parameters. The FPI instrument allows MMS to make instantaneous 3D measurements of particle distributions with a time resolution of 30 ms for electrons and 150 ms for ions see Pollock et al. (2016). This is a major improvement compared to previous similar missions where the full 3D distributions only could be obtained after the spacecraft had rotated 360◦.

3 Methods

MATLAB has been the program used for analysis of the data. It was used since it typically has short development time and there is a large collection of routines for spacecraft data analysis available at https://github.com/irfu/irfu-matlab from which several routines has been highly applicable for the scope of this thesis. All the used functions and scripts needed to generate the results in section 4 can be found at https://www.dropbox.com/sh/9cdwtr0zvv8tbr1/AACp1jwi7Uodz_YXPwWaVkBna? dl=0.

3.1 Shock crossing selection

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than otherwise. Thus more data is available at these events of interest for MMS. The SITL then reports the time spans chosen with descriptions of these events through a text document.

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Figure 8: Example of an image obtained by using ’Quicklook’ for fast data investigation. The black arrow marks the time for an example of a quasi-perpendicular shock. The blue arrow marks a

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3.2 Parameters

The large scale structure of a shock is mainly characterized by the three shock parameters mentioned in section 1.2. The applicability for the shock scaling methods described below are partly determined by the large scale structure as a clear distinction between the shock and the upstream region is required. Also the shock ramp should be steep, with as few plasma fluctuations in the data as possible.

3.2.1 Shock normal vector

The shock normal vector ˆn is a unit vector with direction normal to the shock surface. In spacecraft observations ˆn can be determined in various ways using the field and plasma measurements. Often used is the assumption of 1D-planar shock geometry. The method used in this report uses a mix of the co-planarity assumption for both the magnetic field and the bulk velocity fpr ions (Abraham-Shrauner 1972). In Paschmann & Daly (1998) at page 255 the mixed method number three

ˆ

n = ± (∆B × ∆V) × ∆B

|(∆B × ∆V) × ∆B| (12) where ∆B is the difference of the magnetic field vectors upstream and downstream, B_u and B_d. The same goes for the velocity difference ∆V. The ± symbol refers to if the normal should point upstream or downstream. Throughout this report ˆn is pointing upstream. Equation 12 was then implemented in MATLAB and by taking averages upstream and downstream for the magnetic field and ion drift velocity, ˆn was found.

3.2.2 Shock normal angle

The shock normal angle, θ_Bn, is the angle between the upstream magnetic field and the shock normal vector. After determining ˆn, θBn can easily be calculated. Simply use the scalar product

Bu· ˆn = cos θBn||Bu|| (13)

to get

θBn= cos−1

Bu· ˆn

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3.2.3 Perpendicular and parallel temperature

The temperature data was for each measurement obtained as a second order tensor Tij, which can

be represented as a 3x3 matrix. The scalar mean temperature is given by T = Tr(Tij) 3 = 1 3 3 X i=1 Tii (15)

where the trace, Tr means a summation of the diagonal terms. See Paschmann et al. (1998) for further description of extracting parameters from velocity distributions. In collisionless plasmas, particle distribution functions are typically non-isotropic and it is for this thesis interesting to show the rise of parallel Tk and perpendicular temperature T⊥ separately. As described in section

1.3, Tk refers to the small scale fast irreversible heating while T⊥ refers to the large scale slow

reversible heating. Parallel and perpendicular refers to the direction relative to the magnetic field. To extract these from the the temperature tensor Tij, one has to perform a tensor rotation

using transformational matrices to align T_ij to the magnetic field. The magnitudes of the different directional temperatures are then found as the pivot elements in the matrix representation of the rotated Tij.

3.2.4 Mach number, beta and shock scale values

Besides the above mentioned shock normal angle, from section 1.2 it is mentioned that the Mach number and beta are also characterizing parameters for a shock. These values were calculated based on equation 3 for the Mach number and equation 8 for the beta-value.

The ion and electron plasma inertial length, l_piand l_pe, used for scaling the obtained results are given by

lpi,e =

c ωpi,e

(16) where c is the speed of light and ω_pi,e is the electron and ion plasma frequency given by

ωpi,e=

s ni,ee2

0mi,e

. (17)

Here e is the elementary charge, n_i,e is the number density, ₀ is the vacuum permittivity and m_i,e is the particle mass, see Chen (1984).

3.3 Temperature gradient scale

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be thought of as the shock moving forward and backward over the spacecraft. It is thus not possible to directly from data products measure the shock ramp width. To determine the width of the shock ramp we then have to use other approaches. For this thesis three different methods are used for determination of the spatial scaling. They have been categorized below by how many spacecraft required for the specific method.

3.3.1 Single Spacecraft

For determination of the spatial scale when using only one spacecraft, one way is to first assume that the shock is moving relative to the spacecraft with a constant velocity, Vsh. The spatial scale

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Figure 9: Magnetic field data measured by all the spacecraft of MMS, plotted on the same time axis. The Figure visualize the close separation between the spacecraft of MMS. Making timing an inefficient method for velocity estimation

Thus timing the exact same data value will give inaccurate results because of possible small scale shock fluctuations. In this report another method is used developed by Gosling & Thomsen (1985), and is hereby called the Gosling-Thomsen Method (GTM). With the GTM the constant shock velocity is given by

Vsh= (Vu· ˆn)

x0

1 ± x0

(18) where ∆t is temporal width of the shock foot, Vu is the particle bulk velocity upstream and ωcithe

reflected ion cyclotron frequency. This reflection is assumed to be specular. x₀ is the turnaround distance for a reflected ion, given by

x0 =

ωcit1(2cos2θBn− 1) + 2sin2θBnsin(ωcit1)

ωci∆t

(25)

where the turnaround time for the reflected ion t1 is given by t1 = cos−1 1 − 2 cos2_θ Bn 2 sin2θBn . (20)

Thus by using equation 19-20, equation 18 could be solved and the shock velocity V_sh be found. Then for the spatial distance, the obtained Vsh was multiplied with the time span for the shock

crossing.

3.3.2 Two Spacecrafts

As mentioned in section 2.3 one of the major benefits with MMS compared to Cluster is the small separation distance between the spacecraft. This allows for all the MMS spacecraft to be inside the shock ramp at the same time. Hence, it is possible to measure the gradient of a specific parameter inside the shock ramp. From the gradient data we then know how the parameter spatially changes which we can use to determine distance by combining with means of measured changes for the parameter. An important concept when determine the change of a parameter χ at an arbitrary point in a flow is to consider the material derivative

Dχ Dt =

∂χ

∂t + u · ∇χ, (21) where u is the flow velocity and t is time. Comparing the two right hand terms in equation 21 for χ in a space within a shock one can conclude that the spatial term u · ∇χ dominates. This as χ mainly is changed because the shock with the velocity u is flowing with a steep gradient ∇χ much larger than the temporal change in χ itself. Thus we assume that the shock profile is time-independent between two measurement points. We also assumes that the gradient for χ is largest in the shock normal direction. Thus we use the simplification that shock only moves in the shock normal direction. One can then simplify equation 21 by cancelling the temporal term and also rewrite by using the shock velocity Vsh as flow velocity with respect to an arbitrary point

Dχ

Dt = Vsh(∇χ · ˆn). (22) Shi et al. (2006) used a technique for instantaneous determination of velocity between data points, using Cluster magnetic field data. A similar method will here be applied for shocks using an arbitrary parameter χ instead of the magnetic field. We could then use equation 22 and measurements of χ from two spacecraft for determination of the spatial width. This technique has the main assumption that the χ has to linearly increase or decrease between two spacecraft and two measurement points. The material derivative in equation 22 is the total derivative for the measured parameter χ between two measurement points. It can be rewritten as

Dχ Dt =

hχi_i+1− hχi_i

(26)

where dt is the time lapsed between the measurements and hχii is the mean value between the two

spacecraft at measurement i given as

hχii =

χi,m+ χi,n

2 . (24)

The index m and n denotes the two spacecraft used. Consider ∇χi as the gradient of χ between

two spacecraft as

∇χ_i= χi,m− χi,n xi,m− xi,n

(25) where x_i,m,n denotes the position of the spacecraft m or n for each measurement i along the shock normal. From the assumption that χ increases linearly between two measurements, the ∇χ mentioned in equation 22 refers to the mean of ∇χi between two measurements which can be

rewritten as

h∇χii =

∇χi+1+ ∇χi

2 . (26)

To not confuse h∇χi with means between spacecraft, the variable name is changed as

h∇χii= βi. (27)

Vsh can be rewritten for a specific measurement i as

Vsh = Vi =

hxi_i+1− hxi_i

dt (28)

where hxii is the mean position of spacecraft m and n at measurement i given as

hxii =

xm+ xn

2 . (29)

By substituting equation 23-28 in equation 22 one obtains hχii+1− hχii

dt =

hxi+1i − hxii

dt βi. (30) Then cancel the time derivatives on each side and solve for hx_i+1i which gives

hxi+1i =

hχi_i+1− hχi_i βi

+ hxii. (31)

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3.3.3 Four Spacecrafts

Once again as a major benefit with MMS is allowing all spacecraft to be inside the shock ramp, a possible method to determine the gradient of a specific parameter inside the ramp is to use the concept of reciprocal vectors, see Vogt et al. (2011) and Chanteur (1998). By the same principles as for the two spacecraft method described above the increase or decrease of a parameter χ between spacecraft and between measurement is assumed linear. The difference is how to obtain the ∇χ_i term. We can by using the positions for the four spacecraft construct four reciprocal vectors as linear estimators between the spacecraft (Chanteur 1998). The reciprocal vectors are given by

kα=

rβγ× rβλ

rβα· rβγ× rβλ

(32) where k_αis the reciprocal vector for each spacecraft α and r is the distance vector between spacecraft α, β, γ and λ, see Vogt et al. (2011). The formula for the gradient between the four spacecraft can then be constructed as ∇χ_i = 4 X α=1 kαχi,α. (33)

We can then obtain the average position of the spacecraft using hxi_i+1= hχii+1− hχii

∇χ_i + hxii (34) where hχi is the four-spacecraft average of χ, and ∇χ_i is the gradient linearly interpolated between i and i + 1.

Before applying this method for spatial scale determination a good methodology is to plot the from the reciprocal vectors obtained gradient data. The gradient data over the shock region should then be well distinguishable from the solar wind gradient data. One should also plot the gradient in the shock normal direction for evaluation of the assumption that the gradient is largest in this direction. For this one could also perform a maximum variance analysis (Paschmann & Daly 1998) which if the assumption is correct should give approximately the normal vector direction.

4 Results

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Table 1: List of shock crossings with corresponding upstream parameters used in this thesis. Crossing 1 2 3 4 Date 2016-11-10 2016-12-06 2016-12-18 2017-01-15 MA 12.9 9.14 11.75 11.2 θBn [◦] 83.21 82.7 58.58 83 βe 1.68 1.68 2.45 2.18 lpe [km] 0.87 1.27 1.73 1.50

The obtained results for each shock using the three methods will be presented and analysed, ordered by the given crossing names in Table 1. Since the analysis methods are the same for all the crossings only the obtained final results for crossing 2, 3, and 4 will be shown for time/space efficiency.

4.1 Crossing 1

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(a) Zoomed out view of a shock crossing. (b) Zoomed in view of a shock crossing.

Figure 10: Shock crossing which occured 2016-11-10. Grey section in Figure a) marks time span for Figure b)

4.1.1 Single Spacecraft

Using the method for a single spacecraft described in section 3.3.1. Using ∆t = 2.3 s and f_cp = ωci/(2π) = 0.17 Hz, the constant shock velocity, Vsh was estimated as -70 km/s. A negative value

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from the spacecraft. Figure 11 was then obtained. The left figure shows the electron temperature data and the distance travelled by the shock. The right figure displays the electron temperature as a function of distance travelled by the shock. The shock ramp is marked as a yellow area.

(a) Upper panel shows the electron temperature data over time. Bottom panel shows the distance travelled by the shock over time.

(b) Electron temperature vs distance.

Figure 11: Obtained distance by using the single spacecraft method. The yellow shading marks the shock ramp

The single spacecraft method is using a constant velocity thus the bottom panel is expected to show a linear result, which it does. Using this method the obtained spatial width for the complete shock ramp is 68 km which scales to 78lpe. The temperature rose from 23.4 eV to 40.8 eV, see the

borders of the yellow sections in Figure 11. Thus using this method over the complete temperature rise one can conclude that the temperature rose 17.4 eV over 78lpe. By looking at the temperature

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Figure 12: Electron temperature and magnetic field data from a shock crossing by Cluster. Image credit: Schwartz et al. (2011)

2016-11-10 UTC 0 5 10 15 20 25 30 35 40 MMS1 16:58:47.0 .5 2016-11-10 UTC 5 10 15 20

-250 -200 -150 -100 -50 0 50 100 150 200 250 x[km] 0 10 20 30 40 50 60 T[eV]

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Schwartz et al. (2011) as seen in Figure 12 estimated that half of the temperature rise in the shock ramp occured over a 6.4l_pe. Here the temperature rose from 23.4 eV to 34.1 eV over 23.2 km which corresponds to a 11.3 eV rise over 26.7l_pe.

4.1.2 Two Spacecraft

When using the two spacecraft method, described in section 3.3.2 first the spacecraft pair which were closest separated tangential to the shock normal had to be found. Using the position data for all the spacecraft and the obtained shock normal, Figure 14 was generated.

-10 -8 -6 -4 -2 0 2 4 6 8 10 0 2 4 6 8 10 12 13 ₁₄ 23 24 34 12 13 ₁₄ 23 24 34 12 13 ₁₄ 23 24 34 12 13 ₁₄ 23 24 34

Figure 14: Separation distance between two spacecraft along the tangential (y-axis) and normal (x-axis) direction relative the shock. Every mark consists of two numbers representing the specific spacecraft pair.

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Figure 15: Obtained distance by using the two spacecraft method. The yellow shading marks the shock ramp

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is a restriction when using this method. Figure 15a was then adjusted with a cut-off for times later than 16:48:57.28. Figure 16 was then obtained.

Figure 16: Obtained distance by using the two spacecraft method. The yellow shading marks the shock ramp.

The unavoidable error is thereby removed. Figure 16 then shows a temperature rose of 11.24 eV (≈ 67% of the total rise) over 18.51 km or 21.23lpe

4.1.3 Four Spacecraft

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Figure 17: Temperature gradient obtained from the four spacecraft method. The upper panel is showing the gradient in x,y, z direction. The middle shows the total gradient compared to the gradient in the normal direction. The third panel shows the electron temperature for all MMS spacecraft.

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Figure 18: Distance obtained by the four spacecraft method over the whole shock ramp. Upper panel shows temperature, middle panel shows the gradient data in the shock normal direction and the bottom panel shows the distance profile.

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Figure 19: Distance obtained with the four spacecraft method with a cutoff at 16:48:57.28 were the method breaks down.

From Figure 19 one can then observe that for the 11.24 eV rise(≈ 67% of the total rise) in temperature the spatial width was measured as 16.28 km or 18.67l_pe.

4.1.4 Comparison and discussion

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(a) Comparison of the different methods for the complete shock ramp. Upper panel shows the electron temperature, bottom panel shows the profiles obtained from the three methods.

(b) Distribution at two different time cuts. The dashed lines representing the initial shock ramp close to the solar wind. The solid lines representing a time the ending of the shock ramp close to the downstream region.

Figure 20: Comparison between the scaling methods.

In Figure 20a one can see that the curves for spatial widths using two and four spacecraft follow each other closely. This was expected because the methods are using the same principles but with different number of spacecraft when linearizing between measurements. By looking at the plots for two and four spacecraft in the bottom panel it is clear that the shock velocity varies at these time scales. Thus as the spatial width obtained when using the single spacecraft method was larger, the estimated constant shock velocity was overestimated. Another property observed in Figure 20a is that the shock appears to decelerate to a velocity similar to the spacecraft, as the distance travelled by the shock relative the spacecraft is zero for time spans in the bottom panel. This is a property observed in several shock crossings but for obvious reason could not been observed when using a single spacecraft method with a constant shock velocity estimate.

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estimated the width of roughly half of the temperature rise, where the temperature gradient was largest. For the single spacecraft method the steep temperature rise marked in Figure 13b was used. This part of the shock ramp contains the error described in the results for both two and four spacecraft where the temperature gradient crossed zero. Also when estimating the gradient using the four spacecraft method in Figure 17, it can be seen that the other steep part of the shock ramp has a larger gradient in the normal direction. Thus the other steep part of the shock ramp is used for comparison, see Figure 21.

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to previous scaling estimates by Schwartz et al. (2011) using Cluster data, the results obtained by using the gradient methods shows a factor of two greater shock width.

4.2 Crossing 2

The third crossing which occured in 2016-12-06 had a solar wind density of 17.4 cm−3 and a temperature of 14.3 eV. The overview map is found in Figure 22 and the resulted distance profiles over the steepest half of the shock ramp in Figure 23.

(a) Zoomed out view (b) Zoomed in view.

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Figure 23: Comparison of the different methods for roughly half the shock ramp. Upper panel shows the electron temperature, bottom panel shows the profiles obtained from the three methods. This part of the shock ramp represents a rose of 11.7 eV (≈61% of the total rise). Then as seen in Figure 23 the distance obtained from the two methods using two and four spacecraft follows each other closely while the distance obtained from the single spacecraft method appears overestimated. The resulted distance over this part of the shock was found as 15.9 km for the two spacecraft method and 13.8 km for the four spacecraft method or 12.5l_pe and 10.8l_pe.

4.3 Crossing 3

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(a) Zoomed out view (b) Zoomed in view.

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Figure 25: Comparison of the different methods for roughly half the shock ramp. Upper panel shows the electron temperature, bottom panel shows the profiles obtained from the three methods. This part of the shock ramp represents a rose of 19.4 eV (≈46% of the total rise). Then as seen in Figure 25 the distance obtained from the two methods using two and four spacecraft follows each other closely while the distance obtained from the single spacecraft method appears overestimated. The resulted distance over this part of the shock was found as 26.3 km for the two spacecraft method and 30.1 km for the four spacecraft method or 15.2l_pe and 17.4l_pe.

4.4 Crossing 4

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(a) Zoomed out view of a shock crossing which occured 20170115

(b) Zoomed in view of a shock crossing which occured 20170115.

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Figure 27: Comparison of the different methods for roughly half the shock ramp. Upper panel shows the electron temperature, bottom panel shows the profiles obtained from the three methods.

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5 Summary and Conclusion

Shocks are ubiquitous in space. Through wave-particle interactions between a magnetic field and a particle flow, the flow bulk speed is decelerated and heated. The origin of the electron heating observed in collisionless shocks is a poorly understood subject consisting of a complex interplay between the large scale reversible heating and the small scale irreversible heating. To relate the small scale irreversible heating to a known plasma property, the scaling of the steep shock ramp has to be determined. The electron heating in the nearby bow shock of the Earth has been measured by the MMS spacecraft which by their small spacecraft separation and high instrumental time resolution allowed for new approaches for the scale estimation. The new possibilities by the use of gradients within the shock obtained from MMS data has been tested with, for shocks, novel techniques using one, two and four spacecraft. From the data presented in the result section above one can conclude that an assumption of a constant shock velocity will lead to errors in spatial scaling as the shock velocity changes throughout the ramp. The methods for spatial scaling using two and four spacecraft gave similar results but the four spacecraft method showed smaller errors. In theory the gradient methods could be generalized enabling us to use even more spacecraft than four. The results from this work show that for one shock crossing roughly half of the temperature increase happens on a length scale of 10.8 km or 12.4 l_pe. Three other shock crossings by MMS show similar results. These results differ from previous results using Cluster by Schwartz et al. (2011), where they found a length scale 2-3 times smaller than what is found here. The results from MMS also highlights that the four spacecraft methods shows a clear overview of the shocks movement for each crossing. Thus with the high resolute data of MMS, the usage of gradients between measurements within a shock has been shown as a new possible method for a better scaling estimation.

6 Future work

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Electron heating in collisionless shocks observed by the MMS spacecraft

observed by the MMS spacecraft

Martin Svensson

Space Engineering, master's level

2018

Electron heating in collisionless shocks observed

by the MMS spacecraft

Acknowledgements

Abstract

Contents

1

Introduction

2

Scope and motivation

3

Methods

4

Results

5

Summary and Conclusion

6

Future work

References