Plasma ProcessesInitial questions: We see all objects through a medium, which could be interplanetary,interstellar, or intergalactic. How does this medium affect photons? What information canwe obtain?In the preceding few lectures, we’ve focused on specific microphysical processes. In doingso, we have ignored the effect of other matter. In fact, we’ve implicitly or explicitly assumedpropagation through a vacuum for most applications. It’s time to take on matter!When we introduced Maxwell’s equations, we very solemnly defined D = ǫE and H =B/µ to include the effects of matter, where ǫ is the dielectric constant and µ is the magneticpermeability. Is this necessary? No! Remember that Maxwell’s equations explicitly includesources, in the form of ρ and j. If we do this consistently, for all charges and currents(whether or not they are in a medium), then Maxwell’s equations for E and B alone workjust fine. Thus, Maxwell’s equations for a “vacuum” work fine in that case, as long as bothfree and bound charges are included.In that case, we can again think about the propagation of radiation, this time moregenerally. Again let’s assume a space and time variation of the form exp i(k · r − ωt). Thenwe getik · E = 4πρ , ik · B = 0 ,ik × E = i(ω/c)B , ik × B = (4π/c)j − i(ω/c)E .(1)Remember, by the way, that our justification for using a ei(k·r−ωt)variation is that Maxwell’sequations are linear (there are no terms of the form E2or EB, for example). Thus everyFourier mode propagates independently. There are other physical theories (e.g., strong-fieldgeneral relativity) that are not linear, which means that these modes would mix and couldthus not be considered independently in this way. The linearity of Maxwell’s equations is alsowhy we can get away with using complex numbers; the real and imaginary parts never mix,so they can be considered to yield independent solutions based on the original equations.Now, in general this could be pretty tough, if the medium is something arbitrary (air,water, glass). In our case, though, we’re specifically interested in a plasma, which can beloosely defined as an ionized gas that is globally neutral. That means that all charges aremobile in principle. However, as we’ve done before, we’ll assume that the ions are basicallystationary for our purposes, and mainly serve to keep the plasma neutral. Another importantsimplifying assumption is that the plasma is isotropic. Ask class: what does that implyabout the magnetic field? It implies that there is no significant external magnetic field,because that would break isotropy.Let’s consider nonrelativistic electrons. A given electron follows Newton’s lawm˙v = −eE . (2)There can be an internal magnetic field, just not an ordered one. Ask class: why, then,have we neglected the magnetic component of the force? It’s because that term is of orderv/c, which is small if the motion is nonrelativistic. Given our assumption about the spaceand time variations of quantities, this meansv = eE/(iωm) . (3)The current density is j = −nev, meaning that we getj = σE , (4)where the conductivity is σ = ine2/(ωm). This is Ohm’s law; the current responds directlyto the electric field. Note, however, that this statement requires isotropy. Ask class: canthey think of a specific example in which anisotropy will lead to a different response to anapplied electric field? If there is a strong magnetic field, charges and currents move alongthe field a lot better than across, so no longer is there this linear relation. In general, infact, the conductivity is a tensor. However, for our situation we can treat it as a scalar dueto isotropy.From charge conservation, our exp i(k · r − ωt) assumption gives−iωρ + ik · j = 0 (5)so thatρ = ω−1k · j = σω−1k · E . (6)If we define the dielectric constant byǫ ≡ 1 − 4πσ/(iω) (7)(note that this is real, since σ has an i in it), we getik · ǫE = 0 , ik · B = 0 ,ik × E = i(ω/c)B , ik × B = −i(ω/c)ǫE .(8)Well, lookee here. This looks just like the “source-free” vacuum equations we had before,except for ǫ. Indeed, arguing as before, we find that k, E, and B are mutually perpendicular.However, we find thatc2k2= ǫω2. (9)Since ǫ depends on ω, we no longer have the simple vacuum situation in which all frequenciestravel at the same rate, the phase velocity is the same as the group velocity, and so on.Thus, the presence of a plasma introduces dispersion, where wave packets spread and thereis effectively an index of refraction. If we substitute in expressions, we can rewrite thedielectric constant asǫ = 1 − (ωp/ω)2, (10)where ωp=p4πne2/m is called the plasma frequency. Numerically, ωp= 5.63 × 104n1/2s−1if n is in cm−3.Ask class: from these definitions and the dispersion relation, what does this tell usabout propagation when ω < ωp? It means that k is imaginary: k = (i/c)pω2p− ω2.Ask class: what does that mean about the propagation of radiation below ωp? It meansthat there is an exponential cutoff in the amplitude, with a distance scale of order 2πc/ωp.Therefore, effectively, frequencies below ωpcan’t propagate in a plasma.Side note 1: one way to get quick intuition in a number of astrophysical situations isto have a number of characteristic quantities in mind. The plasma frequency is an example:if you have a plasma of a given number density and are considering propagation of electro-magnetic waves, you should compare the frequency of the wave with the plasma frequency.If a magnetic field is involved, think of the cyclotron frequency. If a high density plasma isrelevant, think of the Fermi energy. Stuff like that. It helps you decide quickly what regimeyou’re in and what processes are likely to be relevant.Side note 2: since σ is purely imaginary, Ohm’s law means that there is a 90◦phase shiftbetween j and E. Therefore, in a time-averaged sense, j · E = 0 and there is no net workdone by the field in an isotropic plasma. That also means there is no dissipation, so belowthe plasma frequency you have a pure reflection. Thus, you can probe the ionosphere of theEarth by finding out when a wave of a given frequency is completely reflected. You can alsocommunicate intercontinentally by bouncing low-frequency waves off of the ionosphere.Now back to our regular program. Electromagnetic radiation travels at a velocity dif-ferent from c, due to the presence of matter. The phase velocity,