1.2 Which parameters are used to characterize a plasma? Briefly discuss their physical meaning.

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1 Solutions to Exercises 3 rd edition Space Physics 1 Chapter Define a plasma. Discuss the importance of the density. Is there a limit for the relative or the absolute size of the neutral component? Traditionally, a plasma is defined as an at least partly ionized gas. Physically, the important aspect of a plasma are mobile charges, thus the idea of a plasma can be extended to metals and semiconductors. Since mobility of the charges is a basic concept, density is important if the plasma is not fully ionized: in a tenuous medium, charges can move freely according to the outer fields while in a very dense medium the motion of the charges is determined by the interactions with the neutrals but not by the fields. The relative size of the neutral component is unimportant as long as the interactions between charged particles and neutrals are infrequent and the neutrals do not determine the motion of the charges. 1.2 Which parameters are used to characterize a plasma? Briefly discuss their physical meaning. The most important parameters temperature and density which both give a measure for the mobility of the charges: temperature for the particle speed and density for the collision rates. Additional parameters include the Debye length as the characteristic spatial scale over which particles in a plasma exert electrostatic forces on each other. The Debye length increases with decreasing density and increasing temperature. 1.3 Which energies/temperatures can be used to classify a plasma? For space plasmas, the most important one is the binding energy which separates neutral gases from fully ionized gases. If we look into the interior of stars, the Fermi energy also becomes important which separates degenerate (non-maxwellian) and nondegenerate (Maxwellian distribution function) plasmas. 1.4 What do you need to explain a magnetosphere? What determines its spatial extent? Ingredients are a magnetic field and the solar wind. If the solar wind were missing, we just would have a dipole field without structures and dynamics. If the planetary magnetic field were missing, the solar wind would be deflected around the planet by the ionosphere. 1.5 Describe the basic features of a magnetosphere. How do these properties change if the axis of the magnetic field changes with respect to the solar wind direction; if the terrestrial magnetic field decreases; if solar wind pressure and speed increase? The basic features of the magnetosphere are (a) the magnetopause as equilibrium between solar wind and terrestrial magnetic field; (b) the cusps, where the magnetosphere is open and solar wind can penetrate into it; (c) the tail which extends in the anti-sunward direction. For the change of the axis of the geomagnetic field, we consider only the extreme case of a the axis being shifted into the plane of ecliptic. If rotation axis of the planet and axis of the dipole are parallel, one pole would be directly hit by the solar wind all the time (pole-on magnetosphere). If only the field axis is shifted but not the rotation axis, magnetospheric configuration would be highly variable (oscillation between pole-on and 1

2 Earth-like). Examples for illustration can be taken from Chapter 9. A decreasing planetary magnetic field would lead to a smaller magnetosphere: with the unchanged solar wind, the point of equilibrium between magnetic field and solar wind moves towards the planet while the structure would remain unchanged. For increasing solar wind speed/pressure, the magnetosphere also would shrink because the magnetopause is shifted towards the planet. 1.6 Why is space dominated by plasmas? The dominant source of space plasmas are the solar and stellar wind, which already are ionized because of the star s high temperatures and the hard electromagnetic radiation emitted by them. 1.7 Where in the near-earth environment do plasmas exist? Would we miss them if they were neutral matter instead? Natural plasma populations in the near-earth environment are (a) lightning bolts, which are part of the global electric circuit, their loss would have consequences for atmospheric chemistry and dynamics; (b) the ionosphere, its loss would affect radio communication; (c) the molten core of Earth, the loss of its plasma properties would imply a stop of the MHD dynamo and thus a decay of the geomagnetic field. 1.1 Chapter 2: Charged Particles in Electromagnetic Fields 2.1 Describe the concept of the guiding center. What is the reason for drifts? The concept of the guiding center allows a simple description of the particle motion separating it into two parts: the gyro-motion and the motion of a fictitious guiding center which is the momentary center of the gyro-orbit. The general motion of the particle is described by the guiding center, its momentary location always is within one gyro-radius of the guiding center. The motion of the guiding center can result from direct forces on the particle, for instance an electric field, or from a drift. Drifts occur when the gyro-radius varies during the gyration. This can happen if the magnetic field changes (such as in gradient drift) or if the particle is accelerated during the gyro-period (such as in E B-drift or g B-drift). In all cases the deviation from the circular orbit does not allow for a closed gyro-orbit the resulting offset after one gyration period corresponds to the drift. 2.2 What is an adiabatic invariant? Describe examples. Adiabatic invariants are constants of motion. They exist in periodic motions in slowly varying outer circumstances. Formally, the are related to the constancy of the action integral over one period of motion. A mechanical example is the period of motion of a pendulum with varying length of the string: if the latter changes only weakly during one swing, the ration between the pendulum s energy and frequency is a constant of the motion, that is an adiabatic invariant. In space physics, adiabatic invariants are related to the gyration period: if a magnetic field varies only weakly on the spatial scale of the gyro-radius, the first adiabatic invariant leads to the constancy of the magnetic moment. The main application of this invariant is the magnetic mirror. The second adiabatic invariant is related to the particle motion parallel to the magnetic field. Here the field properties vary only weakly on the length scale parallel to the field traversed by a particle during one gyration. If the motion is oscillatory along the field, the 2

3 product of field parallel momentum and distance between the reflection points is an invariant of the motion. Application is the acceleration of particles between converging magnetic mirrors (Fermi acceleration). The third adiabatic invariant is related to the temporal variation of the magnetic field: if this is only small during one gyration, the magnetic flux through the area enclosed by the drift orbit of the particle is constant. An application in the terrestrial atmosphere is the flux invariant of the radiation belt particles drifting around the earth. 2.3 Derive the general drift equation v F = F B qb 2 = 1 ω c ( F m B ) F = B ω c m t B and discuss the conditions under which it can be applied. a simple way to derive the drift formula is the transformation into a frame of reference moving with the drift speed given in the equation. A more formal derivation starts from the solution of the equation of motion m v = q v B + F v = q m v B + F m (1) with the external force F being independent of time. This is an inhomogeneous ordinary differential equation of first order with the inhomogeneity F /m. With a = q/m and b = F /m its components read v x = a(v y B z v z B y ) + b x, v y = a(v z B x v x B z ) + b y, v z = a(v x B y v y B x ) + b z. (2) Since the components are coupled, we do not get three independent differential equations but a system of three coupled equations. To solve the homogenous part of (1) we can rewrite the equation into a form v = A v: v = 0 ab z ab y ab z 0 ab x v = A v. (3) ab y ab x 0 We make an exponential ansatz v = v 0 e λt. This implies v = λ v 0 e λt. Inserting into (3) yields λ v 0 = A v 0 Such an equation gives an eigenvalue problem, that is an equation defining the eigenvalues and eigenvectors of a matrix. The Eigenvalues can be obtained from A λe = λ ab z ab y ab z λ ab x =! 0 (4) ab y ab x λ or λ ab z ab y ab z λ ab x ab y ab x λ = λ 3 + a 3 B x B y B z a 3 B x B y B z λa 2 B 2 y λa 2 B 2 z λa 2 B 2 x = λ 3 a 2 λb 2 = 0 (5) and thus λ 1 = 0 and λ 2,3 = ±ia B = ±iω c. (6) 3

4 The eigenvector for λ 1 can be determined from 0 ab z ab y ab z 0 ab x u 1 u 2 =! 0. (7) ab y ab x 0 u 3 It is thus defined by a linear system of equations: Solving this system yields 0 = B z u 2 B y u 3 0 = B z u 1 + B x u 3 0 = B y u 1 B x u 2. (8) u λ1 = B x B y B z = B. (9) The first eigenvalue and -vector thus describe the field parallel motion. For λ 2 = iω c the eigenvector can be determined from the relation iω c ab z ab y ab z iω c ab x u 1 u 2 =! 0. (10) ab y ab x iωc u 3 The resulting linear system of equations 0 = ibu 1 + B z u 2 B y u 3, 0 = B z u 1 ibu 2 + B x u 3, 0 = B y u 1 B x u 2 ibu 3 (11) gives the eigenvector u λ2 = (12) Derive an expression for the gyro-radius and frequency of a relativistic particle. The relativistic momentum is p = γm o v, where γ = 1 v 2 /c 2 and m o is the rest mass. What is the expression for the magnetic moment of a relativistic particle? Show that the relativistic magnetic moment is conserved. The expression for the relativistic equation of motion is F = d dt (γm 0 v) = p. With the Lorentz force the equation of motion becomes γm 0 v = q v B (13) which is the same as in the non-relativistic case except for the factor γm 0 instead of m. Thus the solution for gyro-radius and cyclotron frequency also are the same except for γm 0 instead of m: ω c = q B and r L = p γm 0 q B = γm 0v q B. (14) The expression for the magnetic moment can be derived from the first adiabatic invariant J 1 = p 1 dq using p = γm 0 and q = r L Ψ as generalized coordinates: µ = γm 0v 2 2B = W kin, B. (15) 4

5 2.5 How is the magnetic moment defined. Give examples for magnetic moments. Why is the magnetic moment an important physical quantity? A moving charge is a current and thus according to Ampere s law it creates a magnetic field. A gyrating particles corresponds to a ring current. Similar to the current in a long straight coil it gives rise to a magnetic field similar to that of a dipole. It s magnetic moment is current times area encircled by the current: µ = I A. The magnetic moment is an important quantity in space physics because it is one of the adiabatic invariants: it stays constant, even if the magnetic field changes weakly during the gyration. This constancy of the magnetic moment is used, for instance, in the description of magnetic mirrors. 2.6 Show that j = j in the derivation of Ohm s generalized law ( ) j = σ E + v B. Ohm s law in the observer s rest frame reads j = σ E. In the rest frame of a plasma moving with speed v relative to the observer it reads j = σ E. If conductivity is high, i.e. E/B 1, the field s are transformed according to E = E + v B and B = B (16) because in the latter equation the term v E/c 2 is much smaller than B. In addition, high conductivity combined with the requirement of changes occurring at speed slow compared to the speed of light, reduces Ampere s law to B = µ 0 j. The current in the plasma s rest frame therefore is µ 0 j = B = B = µ 0 j. (17) 2.7 Derive Coulomb s law from Gauss law of the electric field E ds ϱ c = d 3 r or E = ϱ c. (18) ε 0 ε 0 O(V ) V We integrate over a sphere centered at charge Q with radius r and and obtain for the electric field on the sphere s surface E = Q 4πε 0 r 2 e r. (19) Now assume a second charge q on the surface of the sphere. The electric field exerts a force F = qe and thus q is attracted (or repulsed, depending on the sign of the charges) with the Coulomb force F coul = 1 qq 4πε 0 r 2 e r. (20) 2.8 Why does give a maximum Larmor radius? r L = p eb = W kin ceb = 1021 m (21) The correct expression for the Larmor radius considers only the field parallel energy. Thus if the total kinetic energy is considered, the above equation gives a 5

6 maximum Larmor radius. If the filed parallel component of the motion does not vanish, the kinetic energy perpendicular to the field is reduced and consequently also the Larmor radius. 2.9 Solve the equation of motion F = q v B. (22) This is the homogenous part of the equation of motion in exercise 2.3 and has been solved there Develop a simple (numerical) model for the depletion of the radiation belt. Start with the information from examples 5 and 8, assume losses to occur at a height of 1.05 Earth radii from the center of Earth (about 300 km height in the atmosphere) and that during each bounce period 2% of the remaining particles are scattered into the loss cone Determine the gyro-radii and frequencies for electrons and protons moving with thermal speeds in the following fields: (a) the Earth s magnetosphere with n e = n p = 10 4 cm 3, T e = T p = 10 3 K, B = 10 2 G; (b) the core of the Sun with n e = n p = cm 3, T e = T p = K, B = 10 6 G; (c) the solar corona with n e = n p = 10 8 cm 3, T e = T p = 10 6 K, B = 1 G; (d) the solar wind with n e = n p = 10 cm 3, T e = T p = 10 5 K, B = 10 5 G. The relavant equations for thermal speed, Larmor radius and cyclotron frequency are 2kB T v th = m, r L = mv q B and ω c = q B m. (23) The numbers then are electrons protons v th [m/s] r L [m] ω c [1/s] v th [m/s] r L [m] ω c [1/s] (a) (b) (c) (d) A particle gyrates in a homogeneous magnetic field. (a) Determine the size of a volume V which contains an amount of magnetic energy equal to the particle s kinetic energy. (b) Determine the height of a cylinder with this volume and a base given by the Larmor orbit. (c) Discuss this result. (a) The magnetic energy density is ε B = B 2 /2µ 0 or in cgs-units ε B B = B 2 /8π. With the kinetic energy of the particle W kin = mv 2 /2 the volume containing the same amount of magnetic energy must be V same = W kin ε B = 4π mv2 B 2. (24) (b) With the Larmor radius r L = mv/( q B) (all energy is in the motion perpendicular to the field) the volume of a cylinder with base Larmor orbit is V = πrlh 2 = πh m2 v 2 q 2 B 2 = πhm mv 2 q 2 B 2 = hm 8q 2 V same. (25) To caontain the same energy, the cylinder therefore must have a height h = 8q 2 /m.... 6

7 2.13 In the equatorial plane, the Earth s magnetic field can be described as B = B o (R E /r) 3 with B o = 0.3 G, R E being the Earth s radius, and r being the geocentric distance. Determine the time a particle with pitch angle 90 needs to drift around the Earth in the equatorial plane. What is the meaning of this time? Determine the period for electrons and protons with an energy of 1 kev drifting in a height of 5r E above the center of the Earth. Compare with the drift due to the gravitational field and the period of an uncharged particle (e.g. a satellite) in the same orbit A proton of cosmic radiation is trapped between two magnetic mirrors with R m = 5. Initially, it has an energy of 1 kev and v = v in the meridional plane between the two mirrors. Each mirror moves with v m = 10 km/s towards the other. Draw a sketch of the configuration. Determine the acceleration of the proton. (a) Does the acceleration continue until the mirrors are in contact with each other or does the particle escape? Determine the maximum energy acquired by the particle. Determine the maximum energy for other pitch angles, too. (b) How long does the particle need to acquire maximum energy? (Hint: assume the mirrors to be planes moving with speed v m and show that the energy gain in each interaction is 2v m. How many interactions are required for the particle to acquire maximum speed?) 2.15 The magnetic field of a magnetic mirror varies as B z = B o (1 + αz 2 ) along the axis. (a) At z = 0 an electron has a speed of v 2 = 3v 2 = 1.5v2. Where does reflection occur? (b) Determine the motion of the guiding center. (c) Show that the motion is sinusoidal. Determine the frequency. (d) Determine the longitudinal invariant belonging to this motion. (a) The mirror point is determined from the constancy of the magnetic moment: mv,0 2 = mv2 B mp = v2 2B o 2B mp v 2 B o = 1.5B o. (26) Thus at the reflection point it is (b) 1 + α 2 z mp = 1.5 or z mp = α 2 /2. (27) 2.16 A particle of mass m and charge q is at rest in a uniform magnetic field B. At time t = 0, a uniform electric field perpendicular to B is switched on. Show that the maximum energy gain is 2m(E/B) 2. The geometrical situation is quite similar to the Wien-Filter discussed in example 3, the only difference is the electron s initial speed. In this case, the electron is initally at rest. Thus only a force q E acts on it, accelerating it. As the electron s speed increases, the Lorentz force q v B becomes increasingly important forcing the electron into a curved orbit. The resulting motion is a special form of the E B drift with the electron speed being maximal at one turning point of its orbit (as usual) and being zero at the other: the path is a cycloid. At the turning point with non-vanishing speed the electrons kinetic energy is maximal. Here its motion is perpendicular to both the magnetic and the electric field. In the gyrocenter s rest frame the force exerted by the electric field cancels that of the magnetic field, qe = qv R B and the electron s speed thus is v R = E/B. The gyrocenter, on the other hand, moves with the drift speed v D = E/B into the same direction. Thus in the 7

8 laboratory frame, the electron speed is v R + v D = 2E/B, corresponding to a kinetic energy of E kin = 2mE 2 /B A solar proton with energy 1 MeV starts with an initial pitch angle of 85 at 2.5 solar radii. The interplanetary magentic field decreases as r 2. Determine the proton s pitch angle at Earth s orbit (213 solar radii) from the conservation of the magnetic moment. The magnetic moment is conserved: µ = mv2 2B = const or sin 2 85 B sun = sin2 α Earth ( B 1 ) 2 (28) sun 213 which gives α Earth = which is essentially field parallel. Thus without any interfering processes such as scattering all particles injected from the Sun would arrive almost fieldparallel. Note that the change in pitch angle is independent of particle energy A 10 kev α particle is trapped inside the radiation belt at a height of 10 6 km above the surface of the Earth in a magnetic field of about 10 6 T. Determine the drift speeds for the curvature and the gradient drift. Compare to the drift caused by the gravitational field One model for particle acceleration in solar flares uses the second adiabatic invariant. A shock propagates outward through a magnetic field loop of sinusoidal form. Particles gyrate on this loop and bounce back and forth from the shock front. Develop a model for particle acceleration. The acceleration can be described by the second adiabatic invariant, v s = const. If we neglect all effects such as focussing in diverging and converging magnetic field or scattering,. The geometry is s h H x with H as the height of the loop and h(t) as the height of the shock in the corona. If we describe the loop by a cosine, we have a symmetric situation. In a cartesian system, the path can be described by x = W t with t [ π, π] and W the width of the loop and y = H cos t. The path length then is s = +x rp x rp r dt = +x rp x rp 2 Chapter 3: Magnetohydrodynamics W 1 + H 2 sin 2 t/w 2 dt = (29) 3.1 Explain the difference between convective and partial derivatives. Find examples to illustrate the differences. 8

9 3.2 Recall simple hydrodynamics and give other examples of the momentum balance. Discuss the different forms and compare with the Navier Stokes equation. 3.3 Derive the hydrostatic equation from the Navier Stokes equation. Which terms do you need? 3.4 Give a quantitative discussion of the stability of a sunspot (all important numbers are given in Tab.6.1). 3.5 Is the filament sketched in Fig. 3.5 realistic? Why does it not dissolve towards the sides (remember, it is a plasma, not a solid body)? 3.6 What is the meaning of viscosity and Reynolds number? What are the formal differences between hydrodynamics and magnetohydrodynamics? What are the differences in substance? 3.7 Explain the consequences of stationary flows parallel and oblique to the magnetic field. 3.8 Why has pressure the unit of an energy density? 3.9 Show that is a solution of B z = B z tanh ξ, where ξ = B z 2H p B x x B z x + α 2 B2 z = const Show that in an ideal, non-relativistic magnetohydrodynamic plasma the ratio between the electric and the magnetic energies is (v /c) Determine the dissipation times for a copper block (side length 10 cm, conductivity 260 A/Vm) and the interstellar medium (linear dimension m, conductivity 2.6 µv/am). Compare with the age of the universe (about s) Determine the Debye length and the number of particles inside a Debye sphere for electrons and protons moving with thermal speeds in the following fields: (a) the Earth s magnetosphere with n = 10 4 cm 3, T = 10 3 K, B = 10 2 G; (b) the core of the Sun with n = cm 3, T = K, B = 10 6 G; (c) the solar corona with n = 10 8 cm 3, T = 10 6 K, B = 1 G; and (d) the solar wind with n = 10 cm 3, T = 10 5 K, B = 10 5 G. 9

10 3 Chapter 4: Plasma Waves 4.1 Explain, in your own words, the important quantities characterizing a wave. What are group and phase speeds? 4.2 Show that in an electron plasma wave the energy contained in the electron oscillation exceeds the energy in the ions by the mass ratio m i /(Z i m e ). 4.3 On re-entry into the Earth s atmosphere, a spacecraft experiences a radio blackout due to the shock developing in front of the spacecraft. Determine the electron density inside the shock if the transmitter works at 300 MHz. 4.4 Show that the maximum phase speed of a Whistler wave is at a frequency ω = ω ce /2. Prove that this is below the speed of light. 4.5 Show that if a packet of Whistler waves with a spread in frequency is generated at a given instant, a distant observer will receive the higher frequencies earlier than the lower ones. 4.6 Show that if the finite mass of the ions is included, the frequency of Langmuir waves in a cold plasma is given by ω 2 = ω 2 pe + ω 2 pi. 4.7 How would you use pulse delay as a function of frequency to measure the average plasma density between the Earth and a distant pulsar? 4.8 Determine the Alfvén speeds and the electron plasma frequencies for the situations described in Problem Use Fig. 4.4 to describe the properties of magnetohydrodynamic waves propagating parallel to B o for v A > v s and v A < v s Show that in an Alfvén wave the average kinetic energy equals the average magnetic energy Discuss an Alfvén wave with k B o. (a) Determine the dispersion relation under the assumption of a high but finite conductivity (the displacement current nevertheless can be ignored). (b) Determine the real and the imaginary parts of the wave vector for a real frequency. 10

11 4 Chapter 5: Kinetic Theory 5.1 Describe the meaning of the mean free path. What are the physical and formal differences in a neutral gas and in a plasma? 5.2 The solar wind is a proton gas with a temperature of about 1 million K. Plot the distribution function and determine the most probable speed and energy. Compare with the flow speed of 400 km/s and the kinetic energy contained in the flow. 5.3 A spacecraft measures the proton distribution in the solar wind. Above an energy of about 20 kev, the distribution can be described as a power law in E with E 4. Plot the distribution and compare with the results of Problem 5.2. What kind of distribution is this? 5 Chapter 6: Sun and Solar Wind 6.1 Explain the basic conservation laws across a shock front. What are the differences between a gas-dynamic and a hydrodynamic shock? 6.2 Explain the differences between a fast and a slow shock. 6.3 Derive for the length of the archimedian spiral. s = 1 u ( sowi ψ { ψ 2 ω ln ψ + }) ψ Consider a 10 MeV proton in interplanetary space. Determine its gyro-radius, gyration period, and the wave numbers of Alfvén waves in resonance with the proton (assume different pitch angles: 10, 30, and 90 ). Compare with the same values for a 1 MeV electron. 6.5 For an observer on Earth, calculate the length of the magnetic field line to the Sun and its longitude of origin (connection longitude). Do the same for an observer at 5 AU. (Assume a plane geometry with the field line confined to the plane of ecliptic.) 6.6 Imagine a slow solar wind speed starting on the Sun. 30 east of this stream, a fast stream with twice the speed of the slow stream originates. Where would they meet? (Simple assumption of an Archimedian magnetic field spiral.) 6.7 An electron beam with c/3 propagates through the interplanetary plasma and excites a radio burst (cf. example 15). Assume a decrease in plasma density 1/r 2. Calculate the 11

12 frequency drift under the assumption that the electron beam propagates radially. How do these results change if the curvature of the Archimedian field line, along which the electrons propagate, is considered? How would the curvature of the field line influence the frequency drift of a type II burst in front of an interplanetary shock? 6.8 A magnetic loop on the Sun has a parabolic shape with B = B o (1 + s 2 /H 2 ) with H = m being the height of the loop and s the distance from the top of the loop. Calculate the bounce period of particles with a speed of 2c/3. As the particles interact with the atmosphere at the mirror points, they create hard X-rays. What is the time interval between two subsequent elementary bursts? 6.9 The plasma instrument on an interplanetary spacecraft detects a sudden increase in plasma density. No other changes in plasma or field are observed. Is this a shock? 6.10 The plasma instrument on an interplanetary spacecraft detects a discontinuity with a jump in plasma density from 4 cm 3 to 8 cm 3 and a jump in plasma flow speed from 400 km/s to 700 km/s (all quantities in the spacecraft frame). Determine the shock speed. What is the meaning of this speed? 6.11 Assume average solar wind properties at Earth s orbit: proton density 7 cm 3, electron density 7.5 cm 3, He 2+ density 0.25 cm 3, flow speed 400 km/s almost radial, proton temperature K, electron temperature K, magnetic field 7 nt. Calculate the flux densities and the flux through a sphere with radius 1 AU for the following quantities: protons, mass, radial momentum, kinetic energy, thermal energy, magnetic energy, and radial magnetic flux. 6 Chapter 7: Energetic Particles 7.1 Determine the Larmor radius, gyro-period, and speed of galactic cosmic rays with an energy of 10 GeV in a 5 nt magnetic field. Compare with the same values for a solar proton with an energy of 10 MeV. Determine the travel time between the Sun and the heliopause at 100 AU for a straight path and a path following an Archimedian magnetic field line. 7.2 Assume a Galton board with n rows of pins. For each pin, the possibility of a deflection to the left or right is 0.5. (a) Give the probability distribution in the nth layer. (b) Show that for large n this distribution converges toward the bell curve. Give the standard deviation. (c) Write a small computer program to simulate a Galton board. Compare the runs of your simulation with the expected result for a different number of rows. Alternatively, simulate the results for a Galton board with 5 rows and 100 balls by tossing a coin. Compare with the expected results. 7.3 Get an idea about changes of time scales in diffusion. Imagine a horde of ants released at time t o = 0 onto a track in the woods. The speed of the ants is 1 m/min, their mean free 12

13 path 10 cm. How long do you have to wait until the number of ants passing your observation point at 1 m (10 m, 100 m) is largest? How do your results change if you get faster ants (10 m/min, 50 m/min) or ants moving more erratically (mean free paths reduced to 5 cm, 1 cm). Can you imagine different populations of ants characterized by different speeds and different mean free paths reaching their maximum at the same time at the same place? (More realistic numbers for interplanetary space: particle speeds of 0.1 AU/h, 1 AU/h, and 6 AU/h, distances of 0.3 AU, 1 AU, 5 AU, and mean free paths of 0.01 AU and 0.1 AU). 7.4 In interplanetary space, propagation should be described by the diffusion convection equation instead of a simple diffusion equation. The flow speed of the solar wind is about 400 km/s. Calculate profiles with the diffusion convection equation with the numbers given in the parentheses for Problem 7.3. Compare with solutions of the simple diffusion model. Discuss the differences: how do they change with particle speed and mean free paths and why? (Note: Solving this problem you should get an idea about the influence of convection. And this influence is quite similar when additional processes in the transport equation are considered too.) 7.5 Explain the shape of κ(µ) in Fig for isotropic scattering. Why is it not a straight line? 7.6 Shock acceleration is important for many of the particle populations discussed in this chapter. Describe them and find arguments for the differences, in particular the maximum energy gained by the different populations. 7.7 In Fig. 7.4 the composition slowly evolves from one characteristic of flare acceleration to another one characteristic of shock acceleration. Can you explain this slow evolution in terms of a δ-like solar acceleration, a continuous acceleration of particles at the shock, and interplanetary propagation? 7.8 An interplanetary shock propagates with a speed of 800 km/s in the space craft frame into a solar wind with a speed of 400 km/s. The ratio of upstream to downstream flow speed in the shock rest frame is 3, and the upstream diffusion coefficient is cm 2 /s. Determine the characteristic acceleration time. Determine the power-law spectral index for times longer than the acceleration time. 7.9 A shock propagates with a speed of 1000 km/s through interplanetary space. The solar wind speed is 400 km/s. The particle instrument on a spacecraft observes an exponential intensity increase by two orders of magnitude starting 3 h prior to shock arrival. Determine the diffusion coefficient in the upstream medium (losses from the shock can be ignored, the shock is assumed to be quasi-parallel) Perpendicular transport in modulation: compare the travel path of a particle at r = 80 AU if the particle has to follow the Archimedian spiral around the Sun for one winding 13

14 of the spiral or if it travels the same distance straight along a radial, that is perpendicular to the spiral. 7 Chapter 8: Magnetosphere 8.1 Determine the magnetic flux density and the direction of the field for your home town (assume a simple dipole field). Compare with Fig To which L-shell is your home town connected magnetically? Determine also the cut-off rigidity and the Størmer unit for this rigidity. 8.2 A 10 kev proton with an equatorial pitch angle of 40 moves from L = 6 to L = 1.5. Assume a dipole field and calculate the energy gained by Fermi acceleration (second adiabatic invariant) during this motion. 8.3 Assume a sinusoidal variation with period T = 1 h in the solar wind speed with an amplitude of ±40 km/s around an average of 400 km/s. Determine the speed of the magnetopause and the maximum and minimum stand-off distances. 8.4 Størmer orbits are calculated for a terrestrial dipole field. Give a qualitative statement about the errors made in neglecting the actual shape of the magnetosphere. Try to consider the influence of the shifted dipole as well as the different topologies in the noon and midnight directions. 8.5 Fig shows the inward L-shell diffusion of energetic electrons. Try to estimate the diffusion coefficient for this process (assume that the energy of the particles does not change during the inward motion). 8.6 The magnetopause is determined as the equilibrium between the gas-dynamic pressure of the solar wind and the magnetic pressure of the geomagnetic field. The magnetic pressure of the interplanetary magnetic field is neglected as is the gas-dynamic pressure of the plasmasphere. Determine the error due to this approximation. 8.7 Give an order of magnitude estimate of the Chapman Ferraro current. 8 Chapter 9: Planetary Magnetosphares 9.1 Determine the energies of electrons and protons in resonance with the waves in the foreshock of the different planets. Use the numbers given in Fig

15 9.2 Confirm the calculated stand-off distances given in Table 9.3(assume that the solar wind density decreases as r 2 ). 9.3 Discuss the relationship between the spin period and the magnetic moment of the planet in terms of the magnetohydrodynamic dynamo. Would this relationship be in agreement with the assumption of a similar process working in all planets? The source of the magnetic field energy is the rotational energy. Thus for a given planet, the energy contained in the rotation increases as the angular speed increases and we would expect the magnetic field strength to increase with the rate of rotation, that is to decrease with the spin period. In the solar system spin periods and dipole moments are ordered as follows: Spin period ( ) Dipole moment ( ) Jupiter Jupiter Saturn Saturn Neptune Uranus Uranus Neptune Earth Earth Mars Mercury Mercury Mars Venus Venus Deviations from the rule occur at Neptune and Uranus and at Mars and Mercury. In the former pair, deviation is weak and both planets ar peculiar in such that the angle between their axis of rotation and the magnetic field s dipole axis is around 50. The deviation between Mars and Mercury is more pronounced and more significant: Mars s spin period is close to one day, thus we would expect a dipole moment close to that of Earth instead it is observed to be almost 3 orders of magnitude smaller. Thus Mars is the planet with strongest deviation from the rule in our current understanding because it is lacking the liquid core required for the magnetohydrodynamic dynamo to work. The martian magnetic field probably is a field remnant from the origin of the planet. 9.4 Discuss the possibility of aurorae and their shapes and detectability on other planets. Use your knowledge about the aurorae on Earth. Aurora requires the precipitation of energetic particles either along open field lines or from some reconnection region in the magnetosphere. To be visible in the optical, this precipitation should occur on the night side of the magnetosphere. All Earthlike magnetospheres should exhibit aurorae similar to that on Earth. The situation is different for pole-on magnetospheres. 9.5 Describe and discuss the plasma sources in the different magnetospheres. Plasmas in a magnetosphere stem from three different sources: the solar wind, the planetary atmospheres, and the planet s satellites. Solar wind plasma is convected into the magnetosphere during open magnetospheric conditions. A planetary atmosphere becomes a source of magnetospheric particles if at high altitudes neutral atmospheric particles become ionized and are transported by the magnetic field into the magnetosphere. 15