GOLDHABER AND NIETO Limits on the Photon Mass 293 one could increase the frequency to the magic value Jupiter is about 11 times larger in radius than the k aaheu/4,reducing Johnson voltage in proportion. Earth,and has a magnetic field ten to 1,000 times as This would give a factor 15 improvement,in AV/V. strong at the planetary surface (Michaux,1967). for Williams,or a factor 4 in u.One could imagine Further,the Jovian magnetosphere appears to be other improvements:Increasing the applied voltage, 40-50 Jovian radii from the planet,while the Earth's increasing the input resistance (this reduces Johnson magnetosphere is at 8-10 Earth radii (Frank,1967; voltage as the square root of the increased resistance), Parker,1967,1970).As a consequence,one may expect and increasing the dimensions of the apparatus.The a low charge and current density out to a much larger first and third of these cause large increases in power distance,in Jovian radii,than holds for the Earth, requirements.The third (increasing asheu)would measuring in Earth radii.The higher field,the larger appear most promising because the limit on u goes radius,and the enormous magnetosphere should make inversely as the diameter of the spheres.While the an improvement of a factor ten on the geomagnetic optimum frequency goes inversely with dabeu,the limit quite easy with orbiting satellite magnetometer Johnson noise V o(@C)-1 remains constant since C data,and a factor 100 improvement possible.The latter increases.Hence,the attainable limit really does go would imply greater than 10%corrections to the Jovian inversely with the diameter.An optimistic guess would field at 30R,where it is between 10-4 and 10-2G.The seem to be a further improvement by a factor 100 in main potential difficulty would be rapid fluctuations in AV/V,leading to a final limit 40 times smaller than the field during satellite orbits.The large magnetosphere present geomagnetic value. would make the altitude-dependent method more useful than it is for the geomagnetic limits. 3.Pulsar Signals Feinberg (1969)has mentioned the possibility of 5.Magnetic fields in the Galaxy observing pulsar signals from another galaxy.For two Yamaguchi (1959)proposed that a photon mass reasons,however,this is not very promising.The first limit could be obtained from magnetic fields detected is because the distance to the nearest galaxies (the in astronomical observations.His idea was that if fields Magellenic Clouds)is of the order of 50X103 kpc (about extend over a distance D then the photon mass must 50 times the distance to the Crab nebula pulsars) be usD-1.Using the dimensions of the Crab nebula, (Allen,1963).This means that signals will be down by he deduced us10-17 cm-1.In a recent private com- the factor of 2500.Thus,a successful pulsar search will munication to the present authors,he pointed out that probably have to depend on a statistical analysis of the same technique applied to the field in one of radio signals rather than on the direct observations that the spiral arms of our galaxy could yield a limit are needed for a mass limit. 4≤10-21cm-1 Further,even if such a pulsar could be observed There is an immediate naive objection to this reason- directly,on the average the signal would have to travel ing.Even for u=0,magnetic fields extend over a great through about 1,000 kpc in each galaxy.Thus,even if distance only in the presence of a conducting medium, intergalactic space were empty of plasma,the plasma e.g.,plasma.Therefore,the relevance of free space so- dispersion in the galaxies would limit the improvement lutions of the Proca equations,with their Yukawa in the mass limit from pulsars to a factor of only falloff factor,is not at all manifest.A closer look is (50/2)1/2=5,so that the ultimate mass limit from required. pulsars is about 500 times worse than the present If a magnetic field has average axial component (H) geomagnetic value. in a cylinder of radius R,then the average azimuthal It would be interesting to see if the low frequency vector potential satisfies the relation pulse broadening and/or intensity oscillations with w became more noticeable for such a distant pulsar. (u2A2)之(u2R2/8)(H2), (6.20) 4.Other Magnelostatic Tests assuming H varies slowly.Thus,if uR is large,the A energy is much greater than the ordinary magnetic From the discussion of the Schrodinger method energy.If more could be learned about the source of applied to the Earth's magnetic field,it seems unlikely galactic fields,and if it could be shown that the ordinary that an improvement of more than a factor 2 in the H energy saturates the capacity of the source,then it limit on u could be achieved,and a factor 4 would be would be clear that the Yamaguchi proposal uRs1 the most optimistic estimate.The two most con- is correct. spicuous alternative objects for magnetic measurement However,at present the source of the fields is un- are the dipole fields of the Sun and of Jupiter.The Sun known.It may have been an event in the early develop- has the advantage of large size (Ro=109Re),but the ment of the galaxy,or there may be a continuous disadvantage of enormous and rapidly varying plasma pumping of energy into the fields.Further,even the currents which make accurate magnetic surveys quite characteristic survival time for the fields is not well difficult.Also,the relevant region is very hot,adding determined.Old ideas that cosmic ray flux gives a lower substantially to the challenge! bound to the field variation time are less convincing