- (a) Use the fundamental potential E(V , S, N) for the nonrelativistic, classical, perfect gas [Eq. (5.9c)] to derive Eqs. (5.11) for the gas pressure, temperature, and chemical potential.(b) Show
- The form of the potential energy functional derived in the text [Eq. (19.47)] is optimal for demonstrating that the operator F̂ is self-adjoint. However, there are several simpler, equivalent forms
- Using the non-Fokker-Planck arguments outlined in the text, compute an estimate of the electron-electron equilibration rate, and show that it agrees with the Fokker- Planck result, Eq. (20.23a), to
- Collections of stars have many similarities to a plasma of electrons and ions. These similarities arise from the fact that in both cases the interaction between the individual particles (stars, or
- Another type of wave mode that can be found from a fluid description of a plasma (but requires a kinetic treatment to understand completely) is a drift wave. Just as the two-stream instability
- Show that the quasilinear evolution equations guarantee conservation of total energy —that of the resonant electrons plus that of the waves. Pattern your analysis after that for momentum, Eqs.
- (a) Show that the parameter τ along the world line (24.69) is proper time and that the 4-acceleration has magnitude |a(vector)]= 1/κ.(b) Show that the unit vectors e(vector)ĵ introduced in Sec.
- Show that the Weyl curvature tensor (25.48) has vanishing contraction on all its slots and has the same symmetries as Riemann: Eqs. (25.45). From these properties, show that Weyl has just 10
- As discussed in the text, spinning black holes contain a considerable amount of rotational energy [Eq. (26.86)]. This exercise sketches how this energy may be extracted by an electromagnetic field:
- A quite different type of accretion disk forms when the gas is unable to cool. This can occur when its gas supply rate is either very large or very small. In the former case, the photons are trapped
- (a) Consider the Penrose process, described in the text, in which a particle flying inward toward a spinning hole’s horizon splits in two inside the ergosphere, and one piece plunges into the hole
- Consider the surface of a zero-pressure star, which implodes along a time like geodesic r = R(t) in the Schwarzschild spacetime of its exterior. Analyze that implosion using Schwarzschild coordinates
- Around a Schwarzschild black hole, spherical symmetry dictates that every geodesic orbit lies in a plane that bifurcates the t = const 3-volume. We are free to orient our coordinate system, for any
- Our study of the Schwarzschild solution of Einstein’s equations in this chapter has been confined to situations where, at small radii, the Schwarzschild geometry joins onto that of a star—either
- From the Kerr metric (26.70) derive Eqs. (26.84) for the surface area of a spinning black hole’s horizon—that is, the surface area of the 2-dimensional surface {r = rH, t = constant}.Equations.
- Consider the light propagating outward from the beam splitter, along the x arm of an interferometric gravitational-wave detector, as analyzed in TT gauge, so (suppressing the subscript “x arm”
- Explore how pulsar timing can be used to detect a plane gravitational wave.(a) Derive Eq. (27.97).that underlies the rays’ super-hamiltonian; Ex. 25.7. The numerical value of the action is zero,
- Consider a small satellite in a noncircular orbit about a spherical body with much larger mass M, for which the external gravitational field is Schwarzschild. The satellite will follow a timelike
- The cosmic dawn that preceded the epoch of reionization can be probed by low frequency CMB observations using a special radio hyperfine line emitted and absorbed by hydrogen atoms. This line is
- The most likely signal to be detected using a pulsar timing array is a stochastic background formed by perhaps billions of binary black holes in the nuclei of galaxies.(a) For simplicity, suppose
- In our discussion of recombination, we related the emission of Lyman α photons to their absorption. This involves some important ideas in the theories of radiation and thermodynamics.(a) Consider a
- Suppose, as was once thought to be the case, that the universe today is flat and dominated by cold (pressure-free) matter.(a) Show that a ∝ t 2/3 and evaluate the age of the universe assuming the
- (a) Consider a plasma in which the magnetic field is so weak that it presents little impediment to the flow of heat and electric current. Suppose that the plasma has a gradient ∇Te of its electron
- Consider a plasma in which, in the local mean rest frame of the electrons, the electron stress tensor has the form (20.35) with ez the direction of the magnetic field. The following analysis for the
- In this exercise we explore nonlinear effects in ion-acoustic waves (Ex. 21.5), and show that they give rise to solitons that obey the same KdV equation as governs solitonic water waves. This version
- Derive Eqs. (21.27) for the phase and group velocities of electromagnetic modes in a plasma. Vp₁ = 7 k=c(1-2) ²1 k, (21.27a)
- Ion-acoustic waves can propagate in an unmagnetized plasma when the electron temperature Te greatly exceeds the ion temperature Tp. In this limit, the electron density ne can be approximated by ne =
- Verify Eq. (21.28). (AB) = AB* + A*B 4 (21.28)
- Consider a transverse electromagnetic wave mode propagating in an unmagnetized, partially ionized gas in which the electron-neutral collision frequency is νe. Include the effects of collisions in
- A narrow bundle of magnetic field lines with cross sectional area A, together with the plasma attached to them, can be thought of as like a stretched string. When such a string is plucked, waves
- Derive Eq. (21.52) for Faraday rotation. dx dz @newce 2w²c (21.52)
- A radio pulsar emits regular pulses at 1-s intervals, which propagate to Earth through the ionized interstellar plasma with electron density ne ≈ 3 × 104 m−3. The pulses observed at f = 100MHz
- Derive Eq. (21.65). n² = 1 - 1- y² sin² 0 2(1-X) ± X Y4 sin¹0 4(1-X)² 1/2 + Y² cos² €]¹/²* (21.65)
- The free electron density in the night-time ionosphere increases exponentially from 109 m−3 to 1011m−3 as the altitude increases from 100 to 200 km, and the density diminishes above this height.
- Consider a wave mode propagating through a plasma—for example, the ionosphere— in which the direction of the background magnetic field is slowly changing. We have just demonstrated that so long
- Verify that the group velocity of a wave mode is perpendicular to the refractive-index surface (Fig. 21.6b).Fig. 21.6b. ck/w B (b) const @= const
- Verify Eq. (21.76). Wi 31/2al/3wp 24/3 (21.76)
- Derive the two-fluid equation of motion (22.12) by multiplying the Vlasov equation (22.6) by v and integrating over velocity space. afs at + (v.V) fs + (a · V₁) fs = . af, Ət + dx; əfs dx;
- In a very strong magnetic field, we can consider electrons as constrained to move in 1 dimension along the direction of the magnetic field. Consider a beam of relativistic protons propagating with
- For each of the following modes studied earlier in this chapter, identify in the CMA diagram the phase speed, as a function of frequency ω, and verify that the turning on and cutting off of the
- Jeans’ theorem is of great use in studying the motion of stars in a galaxy. The stars are also almost collisionless and can be described by a distribution function f(v, x, t). However, there is
- The particle distribution function f(v, x, t) ought not to become negative if it is to remain physical. Show that this is guaranteed if it initially is everywhere nonnegative and it evolves by the
- Derive expression (22.23) for the zz component of the dielectric tensor in a plasma excited by a weak electrostatic wave, and show that the wave’s dispersion relation is ∈3 = 0. €3(w,k) =1+
- Consider a plasma in which the electrons have a Maxwellian velocity distribution with temperature Te, the protons are Maxwellian with temperature Tp, and Tp ≪ Te. Consider an ion acoustic mode in
- Use Laplace-transform techniques to derive Eqs. (22.29)–(22.31) for the time evolving electric field of electrostatic waves with fixed wave number k and initial velocity perturbations Fs1(v, 0). A
- Consider the three-wave processes shown in Fig. 23.5, with A and C Langmuir plasmons and B an ion-acoustic plasmon. The fundamental rate is given by Eqs. (23.56a) and (23.56b).(a) By summing the
- Consider a plasma with cold protons and hot electrons with a 1-dimensional distribution function proportional to 1/(v02 + v2), so the full 1-dimensional distribution function is(a) Show that the
- (a) Show that the cruise-control feedback system described at the beginning of Box 22.2 has G̃(z) = 1/(1− iz) and H̃ = −κ/(iz), with z = ωτ and κ = Kτ, as claimed.(b) Show that the Nyquist
- Consider an unmagnetized electron plasma with a 1-dimensional distribution function:where v0 and u are constants. Show that this distribution function possesses a minimum if v0 > 3−1/2u, but the
- Consider a plasma with a distribution function F(v) that has precisely two peaks, at v = v1 and v = v2 [with F(v2) ≥ F(v1)], and a minimum between them at v = vmin, and assume that the minimum is
- A clever technique for studying the behavior of individual electrons or ions is to entrap them using a combination of electric and magnetic fields. One of the simplest and most useful devices is the
- One method for confining hot plasma is to arrange electric coils so as to make a mirror machine in which the magnetic field has the geometry sketched in Fig. 20.6a. Suppose that the magnetic field in
- Use the relativistic equation of motion to show that the relativistic electron cyclotron frequency is ωc = eB/(γme), whereis the electron Lorentz factor. What is the relativistic electron Larmor
- The most energetic (ultra-high-energy) cosmic rays are probably created with energies up to ∼1ZeV = 1021 eV in sources roughly 100 million light-years away.(a) Show that they start with the
- A 100-MeV α-particle is incident on a plastic object. Estimate the distance that it will travel before coming to rest. This is known as the particle’s range.
- (a) By taking the trace of the time-dependent tensorial virial theorem and specializing to an MHD plasma with (or without) self-gravity, show thatwhere I is the trace of Ijk, Ekin is the system’s
- Carry out an analysis of the flute instability patterned after that for rotating Couette flow and that for convection in stars (Fig. 18.5): Imagine exchanging two plasma-filled magnetic flux tubes
- (a) Show that the potential energy (19.59) can be transformed into the following form:Here symbols without tildes represent quantities in the plasma region, and those with tildes are in the vacuum
- Consider a slender flux tube with width much less than its length which, in turn, is much less than the external pressure scale height H. Also assume that the magnetic field is directed along the
- Estimate the Debye length λD, the Debye number ND, the plasma frequency fp ≡ ωp/2π, and the electron deflection timescale tDee ∼ tDep, for the following plasmas.(a) An atomic bomb explosion in
- Show that the condition ne ≪ (2πmekBT)3/2/h3 [cf. Eq. (20.3)] that electrons be nondegenerate is equivalent to the following statements.(a) The mean separation between electrons, l ≡ ne−1/3,
- (a) Express the Coulomb logarithm in terms of the Debye number, ND, in the classical regime, where bmin ∼ b0.(b) What range of electron temperatures corresponds to the classical regime, and what
- When analyzing the stability of configurations for magnetic confinement of a plasma (Sec. 19.5), one needs boundary conditions at the plasma-vacuum interface for the special case of perfect MHD
- The solar wind is a supersonic, hydromagnetic flow of plasma originating in the solar corona. At the radius of Earth’s orbit, the wind’s density is ρ ∼ 6 × 10−21 kgm−3, its velocity is v
- In an equilibrium state of a very low-β plasma, the plasma’s pressure force density −∇P is ignorably small, and so the Lorentz force density j × B must vanish [Eq. (19.10)]. Such a plasma is
- Compute the velocity profile of a conducting fluid in a duct of thickness 2a perpendicular to externally generated, uniform electric and magnetic fields (E0ey and B0ez) as shown in Fig. 19.7. Away
- The currents that are sources for strong magnetic fields have to be held in place by solid conductors. Estimate the limiting field that can be sustained using normal construction materials.
- Another magnetic confinement device which brings out some important principles is the spheromak. Spheromaks can be made in the laboratory (Bellan, 2000) and have also been proposed as the basis of a
- (a) A physical quantity that turns out to be useful in describing the evolution of magnetic fields in confinement devices is magnetic helicity. This is defined by H = ∫dV A · B, where A is the
- Many self-gravitating cosmic bodies are both spinning and magnetized. Examples are Earth, the Sun, black holes surrounded by highly conducting accretion disks (which hold a magnetic field on the
- Consider cells that reside in a heat and particle bath of a classical, relativistic, perfect gas (Fig. 5.1). Each cell has the same volume V and imaginary walls. Assume that the bath’s temperature
- Derive Eq. (6.31). 2(ỹ(f)ỹ* (ƒ')) = S₂(f)8(ƒ - f'). (6.31)
- Consider the flow of water along a horizontal channel of constant width after a dam breaks. Sometime after the initial transients have died away6 the flow may be described by the nonlinear,
- Derive Eqs. (10.49). F₁1GW m-2, A₁ 10-³ ¹/² m-¹, K105 J-1/2, KA₁₂ ≤1cm-¹. (10.49)
- After the earthquake that triggered the catastrophic failure of the Fukushima-Daiichi nuclear power plant on March 11, 2011, reactor operation was immediately stopped. However, the subsequent tsunami
- Verify Eqs. (22.79) and (22.80). Uc=-T2 Uc = -T² (Ə(FT)) n (22.79)
- Consider the control system discussed in the last two long paragraphs of Box 22.2. It has G̃H̃ = −κ(1+ iz)[iz(1− iz)]−1, with z = ωτ a dimensionless frequency and τ some time constant.(a)
- Consider a steady, 1-dimensional, large-amplitude electrostatic wave in an unmagnetized, proton-electron plasma. Write down the Vlasov equation for each particle species in a frame moving with the
- For a tokamak plasma, compute, to order of magnitude, the two-point correlation function for two electrons separated by(a) A Debye length, and(b) The mean interparticle spacing.
- Compute the entropy of a proton-electron plasma in thermal equilibrium at temperature T including the Coulomb correction.
- Show that the nonresonant part of the diffusion coefficient in velocity space, Eq. (23.22b), produces a rate of change of electron kinetic energy given by Eq. (23.23). Dnon-res 1 fod
- Show that the Langmuir wave power radiated by an electron moving with speed v in a plasma with plasma frequency ωp is given bywhere kmax is the largest wave number at which the waves can propagate.
- Fill in the missing details in the derivation of the electron Fokker-Planck equation (23.51a). It may be helpful to use index notation. df dt = V₁. [R(v) f + D(v). Vvf], (23.51a)
- For the bump-in-tail instability, the bump must show up in the 1-dimensional distribution function F0 (after integrating out the electron velocity components orthogonal to the wave vector v).Consider
- Consider a beam of high-energy cosmic ray protons streaming along a uniform background magnetic field in a collisionless plasma. Let the cosmic rays have an isotropic distribution function in a frame
- (a) Derive the kinetic equation for the Langmuir occupation number.(b) Using the approximations outlined in part (b) of Ex. 23.5, show that the Langmuir occupation number evolves in accord with the
- Consider the circular polar coordinate system {ω̅, ∅} and its coordinate bases and orthonormal bases as shown in Fig. 24.3 and discussed in the associated text. These coordinates are related to
- Cosmic ray particles with energies between ∼1GeV and ∼1PeV are believed to be accelerated at the strong shock fronts formed by supernova explosions in the strongly scattering, local, interstellar
- The solar wind is a quasi-spherical outflow of plasma from the Sun. At the radius of Earth’s orbit, the mean proton and electron densities are np ∼ ne ∼ 4 × 106 m−3, their temperatures are
- Verify Eq. (23.73a), and show numerically that the maximum Mach number for a laminar shock front is M = 1.58.Eq. (23.73a) nok B Te ="ogT [[1-{1 - 6/Φ12] M> – {}{24 - 1} Φ(φ) = (23.73a)
- Let A, B be second-rank tensors.(a) Show that A + B is also a second-rank tensor.(b) Show that A ⊗ B is a fourth-rank tensor.(c) Show that the contraction of A ⊗ B on its first and fourth slots
- You have measured the intervals between a number of adjacent events in spacetime and thereby have deduced the metric g. Your friend claims that the metric is some other frame-independent tensor g̃
- If two events occur at the same spatial point but not simultaneously in one inertial frame, prove that the temporal order of these events is the same in all inertial frames. Prove also that in all
- For an arbitrary basis(i) The duality relation (24.8),(ii) The definition (24.9) of components of a tensor,(iii) The relationbetween the metric and the inner product to deduce the following
- Consider the circular polar coordinates {ω̅, ∅} of Fig. 24.3 and their associated bases.(a) Evaluate the commutation coefficients cαβρ for the coordinate basisand also for the orthonormal
- (a) Consider spherical polar coordinates in 3-dimensional space, and verify that the nonzero connection coefficients, assuming an orthonormal basis, are given by Eq. (11.71).(b) Repeat the exercise
- Derive the prescription 1–4 [Eqs. (24.38)] for computing the connection coefficients in any basis. . e. [ea eß] = Capp, Cap = [ae]. (24.38a)
- (a) Derive Eq. (24.39).(b) Derive Eq. (24.40). Vg=0. (24.39)
- This exercise serves two roles: It develops the relativistic stress-energy tensor for a viscous fluid with diffusive heat conduction, and in the process it allows the reader to gain practice in index
- In 3-dimensional Euclidean space Maxwell’s equation ∇ · E = ρe/∈0 can be combined with Gauss’s theorem to show that the electric flux through the surface ∂V of a sphere is equal to the