GOLDHABER AND NIETO Limits on the Photon Mass 287 harmonics form a complete orthonormal set only over an that this may be too low by a factor of 2 or 3.After entire sphere;(4)Even low noise satellites cannot clear reconsidering his numbers in the light of Eg.(4.21) up the picture because there are peculiar secular (time) we [Goldhaber and Nieto (1968)felt that the same variations of the fits (Cain,private communication)to conservative error estimation that we had used would the satellite data of the order of tens of y,and also make Gintsburg's limit (8-10)X10-48 g.Thus,it is because up to now only the magnitude but not the nearly a geometric mean between the old and new direction of H has been measured with satellites. results of the Schrodinger method.The main limitation For these reasons,the possibility of systematic on the altitude-dependent method is that externa errors in the fits to a particular spherical-harmonic perturbations become quite significant beyond 3R coefficient of many tens of y cannot be excluded.To (Frank.1967). take account of this,and any errors in the estimates of true external fields,we added 100 y to our estimate, 3.Eccentric Dipole or "Vertical Current"Effect meaning that the left-hand side of Eg.(4.18)is less than 4X10-3.Simple numerical work then gives us a In his early paper,Schrodinger(1943b)pointed out an limit on the mass of the photon ofis effect due to the displacement b of the magnetic dipole origin from the geocenter.From Eq.(4.15),the lines u≤10-10cm-1 of vector potential are circles around the magnetic =3X10-15eV dipole axis,and hence would intersect the surface of the Earth at a small angle of order b/R1/19.From =4X10-48g, (4.20) Eq.(2.19),this means that there is a "feigned"vertical current from the component of A perpendicular to the which is five times better than the number that Earth's surface.That is,the integral of H along a Schrodinger suggested on the basis of much less precise closed path on the surface would fail to vanish-H and very much less reliable data.To our knowledge this could not be written as the gradient of a scalar potential. is the smallest energy or energy limit yet measured for To look for this effect,Schrodinger consulted Schmidt's anything.[The limit in Eq.(4.20)corresponds to a (1924)vertical current measurements for the independ- photon Compton wavelength20 of 80R.] ent surveys of 1885 and 1922.Schmidt had generally found ascending currents in the Eurasian-African 2.Altitude-Dependent Method hemisphere,and descending currents in the American- One might hope to use satellite measurements at Pacific one in 1885.This pleased Schrodinger because varying altitudes to detect the exponential decay of H. these signs are implied by a finite u.However,the later From Eq.(4.16)the magnitude of the dipole field for survey indicated that the currents were smaller and 4≠0is fluctuated in sign from point to point within the same hemisphere.This eventually caused Schmidt to doubt Hn(a)=(1+3(r)[(1-5cos28)/(1+3cos20)]}HD, his earlier "hemispheric current effect"( (Schmidt. =F(4,Y,0)HD (4.21) 1924.1939).21 Still,from the data,Schrodinger was able to speculate where cos=D.7,Hn is the magnitude of the orthodox at a figure of +50.8X10-3 A/km2 for the maximum and dipole field of Eq.(4.19),and cubic terms in ur are minimum vertical currents.To compare this to his neglected. theory he used his then quoted result u=2.76X10-10 Gintsburg (1963),the first to apply the altitude- cm-1,from the "Schrodinger Method"of Sec.IV.B.1. dependent method,used the assumption F=1-(ur)? He obtained to obtain a mass limit from magnetic measurements at varying altitudes by Vanguard,Explorer,and Pioneer Jmax theoretical=6.4X10-3 A/km3. (4.22) satellites.He gives a limit u<3X10-4 g,and states Although this was smaller than the "measured" iAfter our comments on numerical errors by others,we value of 50.8X10-1 A/km2,it was within an order of we imtend magnitude,and of the correct sign.Schrodinger took hope from this since at that time he had field-theoretic cm-1);this was chiefly due to using an incorrect value for the radius of the Earth! reasons for believing in a photon mass of about this mass difference.for example,Is We observe,however,that from observations on the sizes clusters of galaxies one could set a limit to the mass of the a With hindsight we can say that perhaps Schmidt was too graviton"possibly as low as 102 cm-1=4X10-mg.But this pessimistic in dismissing his early observations.As we now ignores the troublesome point that no one has ever observed know,sunspot activity plays a heavy role in the currents cir- real live single graviton (nor should we expect that anyone will culating in the radiation belts.Schmidt's 1885 survey was in a soon).Also,it is not totally clear that the gravitational field should be quantized in the first place.[See,e.g.,Lecture 1,p. a722e 12 of Feynman (1962-1963).]We shall comment further on (14.2)near the sunspot minimum of 1923.6,which could have these points elsewhere. caused the change in his results.See Waldmeier (1941),who 20 Note that in the units we are using,the photon Compton discusses sunspot observations.Waldmeier's figures are also wavelength is given by =2r/u. given by Kiepenheuer (1951)