288 REVIEWS OF MODERN PHYSICS JULY 1971 size.2 Realistically,though,the observed larger currents conductivity of the plasma,so that e is infinite in this and the huge fluctuations within the same hemisphere limit.Nevertheless,one is enticed by the form of these certainly meant that one would have had to weed out equations to introduce finite photon mass simply by the "real"currents to obtain as low a mass limit as that subtracting u2 from the right-hand side.In fact this found via the "external field." procedure can be justified by introducing the Proca However,there is a more elementary reason for giving Eqs.(2.19)in place of Maxwell's,and following care- up this approach.The effect produced by a massive fully the modifications in the standard derivations of the photon via its "eccentric dipole"compared to that dispersion equations (Jackson,1962,Sec.10.8).We produced via its"external field"is intrinsically smaller are left with the new relations by a number of order (b/R),which for the earth is 1/19.Thus,the "eccentric dipole effect'will not k2=02/VA2-u2 (magnetosonic), (4.25) yield as good a mass limit as that which could be and obtained by a similar experimental effort using the k2 cos20=02/V2-u2 (Alfven). (4.26) "external field effect,"even if the real currents could be separated out. For 440,k2 would become negative for low enough w,so that the waves would be damped.In principle,if C.Long-Period Magnetic Waves the critical frequency at which the waves did not propagate could be found,the photon mass could be Gintsburg (1963)proposed a method to limit the measured directly.Of course,the lowest mass that photon mass that deals with the propagation of long could be measured for any given frequency or period period magnetic waves.The idea focuses on the form of (T)would be the index of refraction (N=kc/w)of electromagnetic radiation in a cold,nondissipative,magnetized plasma 4=w/c=(2.09X10-10cm-)/T(sec),(4.27) (cf.Thompson,1962,Chap.7;and Tackson,1962 Sec.10.8).There are two types of magnetic waves in no matter what the plasma density or field strength. such a plasma.In the first,or acoustic type,the wave Patel (1965)proposed a mass limit from this idea by propagates perpendicular to the direction of the static using the data of Patel and Cahill (1964).These magnetic field Ho,and the oscillating field points in the authors had combined ground magnetometer and same direction as Ho.The dispersion equation for these Explorer XII results to surmise that hydromagnetic magnetosonic waves is waves having a period of 200 sec were generated at about 50 000 km from the center of the earth and then k2=02/VA2 (4.23) traveled along the magnetic field lines to the earth's where Va is the Alfven velocity V=Ho/(4mp)12,and surface in about 1.5 min,during which time their p is the mass density.If the plasma were not cold,so amplitude was attenuated by one-third.From this Patel conjectured that it was reasonable to consider that the speed of sound s was no longer negligible compared to Va,then Eq.(4.23)would be modified by these waves as being at or above the critical frequency. the replacement V(V2+s2)1/2. By taking values for p of 50 Mproton/cm3 and for The second,and purely magnetic,wave is the Alfven Ho of 10-G,and using these values in Eqs.(4.25)- wave,in which the magnetic field oscillates perpendicu- (4.26)for k=0,Patel obtained the mass value (re- storing an omission of lar to Ho,and the wave travels in the plane perpen- dicular to the oscillating field,at an angle 0 to Ho.Now u=10-9cm-1. (4.28) the dispersion equation is He then took this value as the upper limit to the photon k2 cos20=@2/VA2 (4.24) mass,meaning The factor cos20 arises from the powerful constraints 4≤10-9cm-1 relating electric and magnetic fields in a highly con- =4X10-7g. (4.29 ductive plasma.It is natural to describe the quantity Va-2 in Eqs.(4.23)and (4.24)as ec-2,where e is the However,there are numerous reasons for questioning effective dielectric constant.However,this is a highly this value.Patel himself pointed out that it could easily sophisticated use of the term,as one may see from the be off by an order or two of magnitude because of fact that,as long as the plasma is motionless,the uncertainties in the values assumed for the plasma electric field must vanish because of the nearly infinite density and the magnetic field strength.Along the path of the waves there could have been very large fluc- #Schrodinger (1943a)[and a note added in proof in his later tuations in these parameters,which would have red)had developednry Field produced a marked effect on the wave propagation characteristics. 4-l=君T()A(6r)-a2(e/mc)2(hc/k)2=29715km, To add more confusion,there obviously is huge which agreed well with his "Schrodinger Method"mass limit uncertainty in the assumption that the damping came 1=36300km. from a massive photon,and not from other dissipative