286 REVIEWS OF MODERN PHYSICs.JULY 1971 the average of any component of H over the sphere However,the more recent work on the geomagnetic must vanish.Therefore,if true external currents be limit on u does exploit satellite data as well as earth- estimated,detection of any anomalous uniform "ex- based measurements.Furthermore,the Schrodinger ternal"field on the surface of the Earth would demon- method might be applied to other planets in the not too strate a violation of Maxwell's theory of just the form distant future.These facts seem a sufficient defense for produced by a finite photon mass. our classification. We turn to a quantitative derivation of the external Recently the present authors (1968)improved on field effect.Recalling the definition of a magnetic Schrodinger's results by using Cain's fitib(Cain,1966 dipole moment Cain et al.1965,1968;Hendricks and Cain,1965)to De=D=(-)∫drJ×r/c, (4.13) geomagnetic data from earthbound and satellite meas- urements.For epoch 1960.0 Cain obtains values16 of and the static limit of Proca's Eq.(2.18) D=31044yR (-72+)A=J(4π/c), (4.14)） Hxt…=(21士5)Y, we have the dipole vector potential Hext~0=Hxt·(sX2)/八X2|=(14土5)Y A=V x D eHrr (4.15) Hxt0X2=(8±5)Y, (4.19)】 Since H is V x A,the dipole contribution to the earth's where s points towards the south geographic pole field.measured from coordinates centered at the To obtain our mass limit from this number,we had to dipole,12 is take into account the "true external"sources,which H=(Der/r3)[(1+ur+u22)(32-7-2)-u22] were unknown when Schrodinger wrote his paper.As discussed in Goldhaber and Nieto (1968)these include (4.16) ~9y from the quiet-time proton belt,perhaps 15-30 This is to be compared to the ordinary dipole field. due to currents in the geomagnetic tail,and~15y from the hot component of the plasma in the magnetosphere HD=（D/r)(32r-). (4.17) which are all parallel to the dipole moment.In addition, If observations are made near the surface of a sphere there is a true external field antiparallel to the dipole of centered at the dipole,(here it will be the surface of the ~20y at the equator,which is due to the compression earth rR=const),the factors in the first term of of the geomagnetic field by the solar wind.Finally,the Eq.(4.16)would just make D appear to have a slightly interplanetary field of ~5y points in an unknown different value when compared to Eq.(4.17).However. direction at the earth's surface.17 the last term in Eg.(4.16)is new.It will be observed Thus,the total external field parallel to D due to as an apparent external magnetic field antiparallel known sources is <40y.Subtracting this from 2.Hext to the direction of the dipole.The ratio of the"external in Eq.(4.18)gives an upper limit on the antiparallel field"(Hxt)to the dipole field at the equator (HpE)is external field which could be due to a finite photon mass,Hoxt(antiparallel)<20y. Hex/HDE=(uR)2/[1+4R+3(uR)2](4.18) The significance of this limit depends crucially on the Using the 1922 magnetic surveys discussed by Schmidt reliability of the fit of Cain to the geomagnetic field. (1924),13 Schrodinger obtained a ratio of (Hex/HDE)= In Goldhaber and Nieto (1968)we made a number of (539)/(31089y),where 1y=10-5 G.14 Putting this into observations concerning this reliability:(1)The Eq.(4.18)yields u=2.76X10-10cm-1.Later,Bass and existence in Eq.(4.19)of external components perpen- Schrodinger (1955)argued that multiplying this dicular to D is hard to explain in any known physical estimate by a factor of 2 would give a reliable upper model;(2)There appears to be an irreducible "noise" limit4≤5.5×10-10cm-1≡1.9X10-7g. in data from earthbound observatories of about 100 y, The critical reader will have noticed that Schrodinger's in large part due to magnetic anomalies in the earth's results did not use extraterrestrial measurements at all. crust;(3)Earthbound data are very sparse in Asia and In fact the ingenuity of the external field method is much of the southern hemisphere.Despite this,the fit precisely that it requires only ground observations is made by an expansion in spherical harmonics of a potential whose gradient gives H,even though these 1s The orientation of the earth's dipole moment is given by Finch and Leaton (1957).A more recent value is to be found 1sOther reports on this study are contained in Hendricks and in Cain and Hendricks (1968).In the usual physics convention, Cain (1966),and Cain et al.(1965). this dipole points to the southern hemisphere. 16 In consulting the literature,care should be taken to dif. See also Chapman and Bartels (1940). ferentiate between the sign conventions used for the spherical 1 The equations we are using are in cgs electrostatic units, harmonics,and between north-seeking and north poles.See, so that magnetic fields are not measured in the Gauss of cgs for example,Table 3 in Fougere (1965),as well as Cain et al. electromagnetic units.In Sec.IV.B.1,this does not matter sin we are calculating ratios of fields.However,in Sec.IV.B.2, (1967),Appendix.This is a later fit than that of Cain (1966), but it does not calculate possible external terms. one must be sure to insert the correct unit conversion factor 17A more detailed discussion on these points and pertinent (which is essentially c). references can be found in Goldhaber and Nieto (1968)