The radiated power of the oldest pulsars is low only because the wavelength exceeds the diameter by a factor of about 105. For a constant voltage, the power radiated by a short dipole varies as ω4. Thus, if the oldest pulsars could oscillate at a wavelength comparable to their diameter then the radiated power would be far greater. That is not possible in a global spherical oscillation mode, but the system will become increasingly nonlinear as the Schwarzschild radius is approached. The nonlinearity will result in harmonics of the fundamental frequency. The harmonics would be of a lower magnitude than the fundamental, but in being at a higher frequency they radiate more effectively.
Another consideration is that, while the observational data indicate that the overtone modes are below the threshold of oscillation in normal pulsars, it might nevertheless be that at a point quite near the Schwarzschild limit they become oscillatory, further increasing the power output. In not being harmonically related to the fundamental frequency the system would exist in a state of chaos if several of the overtone modes became active.
Still another consideration is that if, due to a disturbance, granularization were to occur then each of the granules could individually radiate, and at a much higher frequency than the pulsar frequency. At the ultimate field strengths the electrical energy density is 108 kg/m3, so the disturbed fields would be highly effective in causing further disturbances. These disturbances are nonlinearly coupled to the gravitational potential energy by the equations of Section 4, so the possibility of a run-away condition may exist. If so, the radiation becomes field limited, and, at one solar mass near the Schwarzschild limit, the system radiates a sustained 1041 watts (1048 ergs/sec) of Maxwellian radiation. Field limiting would cause the signal to be flat-topped. To place the enormity of this energy flow in perspective, the sun radiates only 3.9 x 1026 watts. The 23 Jan 99 gamma ray burst was 9 billion light years away, but for a few seconds it was bright enough that it could have been seen with a pair of binoculars .
The energy flow computed here is in the range of interest for these objects. The basis of the calculation is the Dirac field limits, combined with the gravitational solution of the Einstein field equations. Carrying through these calculations leads to the equation (MKS units)
power = 8 π b2 ε0 c3 (G m me2/(hbar qe))2
G=gravitational constant, ε0 =vacuum permittivity, m=mass of source, me = mass of electron, qe = charge of electron, hbar = h/(2 π), h = Planck's constant.
b is an undetermined coefficient with a magnitude near unity.
[HST] image showing the galaxy responsible for the 14 Dec 97 burst (expandable, marked by arrow).
The energy of the 14 Dec 97 burst has been estimated as 1/6 of a solar mass . If b is 1 then the above equation would require over a day to radiate that much energy at one solar mass. The time drops to 60 seconds at 45 solar masses. The bursts last for about a minute, so the numbers indicate that the sources of the most energetic gamma ray bursts are massive. If the object is surrounded by a fireball, and with the internal energy flow being primarily non-electromagnetic, then these calculations overestimate the mass of the source. However, without the interior electrograv coupling of Section 4 the radiation would not occur in the first place, so the surface area of the fireball is probably not too important in estimating the mass of the source. Forms of radiation other than the electromagnetic are not included in the equation, and they may well dominate in such events. Since the electromagnetic output is evidently less than 0.4% of the rest mass, there is the observationally intriguing possibility that such events radiate a similar amount of gravitational radiation at a relatively high frequency.
These calculations assume that the radiation is not beamed. If it
is beamed then the minimum mass is substantially less.