You can skip to the following subsections, then scroll from there. Subsection 5.3 appears to offer the most favorable observational tests for distinguishing between pulsar models.
5.1 The low frequency cut-off
5.2 The second resonance
5.3 Bi-drifting subpulses and rapid subpulse phase swings
5.4 Subharmonic phase lock
5.5 Pulse microstructure
5.6 The rotational signature
5.7 Phase jumps
5.8 Glitches and more
5.9 Total oscillatory power output as a function of "spin down" rate
5.1 The low frequency cut-off
[Chandra]
X-ray image of pulsar J1811-1925 [49].
It shed its cocoon over 1600 years ago.
Do you see the "lighthouse beacon"? Is it pointing at us?
Click on the image to zoom in.
The pulsar is still very young, but after hundreds of thousands
of years it will reach a low frequency limit for solitary pulsars.
The oldest radio pulsars drop out at about 0.25 Hz. Baring & Harding [22]. have proposed that the pulsars become radio-quiet because of photon splitting in the intense magnetic field. The cutoff point is the same as the equation for Bm with b=1. There are several isolated X-ray pulsars in the 0.1 to 0.2 Hz range, suggesting that the pulsars continue to oscillate after they become radio-quiet. The gamma ray bursts sometimes exhibit a light curve that resembles the exponential decay of a resonant system with a frequency of 0.1 to 0.2 Hz. In the case of the 1998 August 27 burst from SGR 1900+14 [23] the period of the decaying light curve was the same as the period of the pulsar, tending to establish a connection between the two phenomena. The lowest frequency isolated pulsar known has a frequency of 0.085 Hz [1]. In view of the inaccuracies inherent in a Newtonian solution near the Schwarzschild radius, and also considering the neglect of the gravitational redshift, these observed limiting relationships are in reasonable agreement with the predicted cutoff at the Schwarzschild radius. The pulse frequency could be at the second harmonic of the oscillation frequency, in which case the discrepancy is greater.
The accretion powered X-ray pulsars, which are found in binary systems, have periods up to 1400 seconds [2], and are known to be routinely spun up by accretion. Pulsars necessarily do rotate, and the oscillatory model does not apply to the long period accretion powered pulsars, except that detecting sustained high frequency oscillation in one of these systems would constitute an excellent observational test.
5.2 The second resonance
There are an infinity of spherical resonance modes, and it may be possible for more than one mode to oscillate simultaneously. Even if the other modes are not oscillatory, the main pulse is narrow and the harmonic content high, so it can excite the high frequency resonances, and the effect is about the same.
The drifting subpulses of of the 1.0605 Hz radio pulsar B0031-07,
from reference [37].
The pulsar frequently switches modes, and the pattern shown is just
one of the three. Time progress from left to right for each pulse, and
from bottom to top in progressing from one pulse to the next.
Under these conditions the voltage in the polar regions will be the sum of two oscillating voltages, and the pulses will be emitted when the sum of the voltages exceeds some charged particle production threshold. The second resonance is not harmonically related to the pulsar frequency, so the phase of the pulses will shift systematiclly from one pulsar cycle to the next.
It would be possible for more than one high frequency resonance to be excited, and resonances that are close to a harmonic of the pulsar frequency are particularly interesting. Due to the nonlinear coupling between the oscillating voltages, the higher frequency will tend to phase lock to a pulsar harmonic, resulting in narrow fixed drift bands. The pulses would only be emitted when all three oscillations are near a peak. Many pulsars do have two or more drift bands. When there are no significant drifting subpulses the system degenerates to a mean profile with multiple fixed peaks, and this configuration is very common.
The drifting subpulses are almost universally believed to be caused by a ring of sparks drifting about the magnetic pole of a rotationally powered pulsar. That is the Ruderman and Sutherland model [51]. They will be analyzed here in terms of the oscillatory model. Future observations will be required to determine which model is the correct one, and it will be very difficult to tell which is which.
The following two figures have not previously been published.
I computed the spectra from a data record supplied by S. Ord.
It is the same as the one analyzed in
[50], and the subpulse drift pattern
is shown there.
The J1231-47 power spectrum. The off-scale lines on the left are harmonics of
the pulsar frequency. The first harmonic is the largest, and it has an
amplitude of 1.0 on the scale shown. The cluster of spectral
lines near 18 F1 is associated with
the drifting subpulses.
The spectral line at 7.315 F1 is artificial
interference, and there are a few other weak interfering signals.
An expanded portion of the above figure showing the
14th harmonic of the pulsar frequency
and one of the sidebands associated with the drifting subpulses.
The linewidth of both spectral lines was determined by the duration
of the data record (283 pulses). The linewidths would be much
narrower with a longer data record.
The largest sideband in the above J1231-47 power spectrum is at 17.948 F1, where F1 is the basic pulsar frequency. This frequency is the most probable value of the second resonance. Two or three cycles of the resonance are visible in each pulsar cycle, resulting in subpulses spaced by P1/17.948 = 0.0557 P1, where P1 = 1/F1 is the basic pulsar period. The subpulse spacing from the two dimensional autocorrelation function was 0.0557 ±0.0009 P1 [50], so the time and frequency domain solutions are commensurate, which is to be expected for any physical model.
The subpulses are only visible for about 50 degrees out of each pulsar cycle. The visibility window effectively amplitude modulates the 17.948 F1 signal at the pulsar frequency, generating an extensive sideband system at (n+0.948) F1. In each pulsar cycle there are (n+0.948) cycles of each sideband. There is slightly less than an integral number of cycles of each sideband frequency in each pulsar cycle, so the nearest peak occurs slightly later in the next pulsar cycle. That causes the subpulses to drift toward later times.
The pattern repeats when each sideband drifts through one full cycle of itself. The number of pulsar cycles required is 1/(1-0.948) = 19.2. That is known as the P3 interval, and the result obtained from the two dimensional autocorrelation function was 19.3 ±0.2 [50]. As before, the time and frequency domain solutions are commensurate.
The drift pattern is more elaborate when the sidebands are near (n+½) F1. Pulsar B0943+10 exemplifies these systems, and its sideband system is shown in reference [38]. The sidebands are at (n+0.5355 ±.0003) F1, peaking at about 35 F1. Then in each pulsar cycle there are n+0.5355 cycles of each sideband. The nearest peak of each sideband frequency occurs 1-0.5355 cycle of itself later in the next pulsar cycle. The P3 interval is P1/(1-0.5355) = 2.15 P1. This drift pattern is shown in figure 4 of [52]. The signal is periodic, so there is another peak 0.5355 cycle early in the next pulsar cycle. This peak defines another set of drift lines with a repeat interval of P1/0.5355 = 1.87 P1. This drift pattern is shown in figure 8 of [38].
Actually, there are two ways of drawing drift lines for all pulsars with drifting subpulses, and they can be revealed by correlation techniques, but for most signal processing methods only the longer of the two intevals is noticeable if the repeat interval is long. When the sidebands are at (n+e) F1, with e < ½, the dominant drift is toward earlier times with a repeat interval of P1/e. When the sidebands are at (n-e) F1, still with e < ½, the dominant drift is toward later times, and also with a repeat interval of P1/e. One case is about as likely as the other. When e is near ½ the difference between lower and upper sidebands is not well defined, and the subpulses appear to be drifting in both directions simultaneously. There is only one pattern, but there are two ways of drawing drift lines through it.
The method of analysis utilized here is unconventional,
but it would apply equally well to the popular drifting spark ring model.
More refined methods will be required for distinguishing between the two models.
5.3 Bi-drifting subpulses and rapid subpulse phase swings
The bi-drifting subpulse drift pattern of pulsar
J0815+0939, from [54].
I have added the blue drift lines.
The bi-drifting pulsars are only recently discovered, and there is currently (as of August 2005) not enough published observational data to determine if the pattern is permanent. Additional data will presumably be forthcoming, and if the subpulse drift pattern is found to be permanent, or at least recurrent, then the possible explanations for the pattern will be greatly restricted.
One model that might may be able to explain a permanent bi-drifting pattern is a dual path model in which reflection or refraction effects make two different regions of the spark ring simultaneously visible [57]. However, B1839-04 is also a bi-drifting pulsar [57], and the preliminary data show that the P2 intervals of the two bands are appreciably different. Different P2 intervals would be difficult to explain with the model. More complete signal processing and detailed modeling attempts will be required to determine if the dual path model is plausible.
Qiao et al. [54]. proposed that the bi-drifting pattern is caused by two counter-rotating spark rings. If the magnetic inclination angle were very small the drift bands would be wider, causing both sets of drifting subpulses to be visible in the vicinity of the intersections of the two sets of drift lines in the figure.
Now, a sturdy observer standing on the the pulsar at the longitude of the intersecting drift lines could watch the two rings, and observe when two of the drifting columns are immediately adjacent. At that instant, and assuming that the pattern is at least recurrent, the observer would know that Earth was at the local zenith -- not by viewing the earth, but by watching the drifting sparks.
There are phase ambiguities in this analysis, but they can be neglected for now. And actually, since the P3 interval is generally not a multiple of the pulsar period, the localization would not be exact, but the approximation becomes better as the P3 interval becomes longer.
One way of seeing the implausibility of this kind of interaction is to apply the equivalence principle to the problem. The only direct way that the observer could even tell if the pulsar is rotating is by viewing the distant stars. Indirect methods, such as measurements of the coriolis force and other effects could be used, but the important consideration is that, for a truly isolated pulsar, there can be no detectable local interactions synchronous with the basic pulsar period.
Referring to subsection 5.2 above, in a bi-drifting pulsar one set of sidebands is at (n+k) F1, and the other set is at (n-k) F1. There is no reason to expect that both sideband systems peak for the same value of the integer n, meaning that the P2 intervals in the two drift bands are unrelated, even though the P3 interval is the same for both. (The above drift lines are drawn for the same P2 interval in both bands, but more refined signal processing methods show that they are different [58], and in pulsar B1839-04 [57] they are greatly different).
In order for the two patterns to synchronize with each other they must interact, leading to the expectation that both patterns will be present in one of the drift bands. The appearance of the first and fourth bands in the above figure suggests that it is in one of them that the patterns interact, but specialized signal processing of the original data will be required to determine if both are in fact present in one of them. (The result would be an "X" shaped drift band).
The rapid subpulse phase swings in some pulsars, such as B0320+39 [56], do not appear to represent a differnt phenomenon than that of the bi-drifting pulsars, with the only essential difference being that both P2 intervals are present in the same band. The subpulses of the two interleaved patterns could drift in either the same or in opposite directions, depending on which spherical resonances are excited. In practice, because pulsar emission tends to be erratic, the two patterns could probably not be visually separated when both drift directions are the same, so the composite would look like a single pattern with a strongly curved drift line. The erratic individual pulses can be averaged, but the averaging would also effectively blend the two patterns into one. Pattern separation would be more favorable when the two drift directions are opposite.
In the oscillatory model the multiple drift bands in the above figure are caused by still other spherical resonances that are excited at harmonics of the basic pulsar frequency. These resonances are phase locked to the harmonics, even though their precise resonance frequency would not be exactly at a harmonic. The preferred set of phase locked frequencies evidently changes from time to time in some pulsars, with that representing mode switching. The reason that the preferred set should change is unclear. (Non-linear interactions between the drifting subpulses and the fixed bands could also result n rapid phase swings. See subsection 5.4.)
Depending on the outcome of future observations, analysis of
the bi-drifting pulsars
could prove to be a powerful tool for distinguishing between pulsar
models. For the purpose of distinguishing between models, bi-drifting
subpulses in one drift band would be just as effective as when in two
bands -- if they can be convincingly separated.
5.4 Subharmonic phase lock
For many pulsars the drift rate varies across the window, which in the oscillatory model is interpretable as a "phase pulling" effect. Such effects are universal in free running oscillators, and they can lead to phase lock when two coupled oscillators are almost harmonically related. (See subsection 5.3 for another possible explanation of rapid phase swings.)
Two-dimensional autocorrelation function for pulsar B1918+19. The
figure is from reference 43. I have
added the grid. The vertical axis
represents 20 pulsar cycles, and the horizontal axis represents 90 degrees
of the 0.8210 second pulsar cycle.
The horizontal and diagonal lines are both spaced vertically by 4 pulsar
cycles. The slope of the lines was selected visually to be parallel to the
drift bands. The vertical lines represent the phase angle through
which the pulses drift in 4 pulsar cycles. The bands are spaced
by quite nearly 4 P1, so approximately
4 pulses occur in each repeating drift pattern.
Time progresses from left to right and bottom to top.
Click on the image to expand it.
In the above figure the repeat interval is about 4 P1, so, for an advancing phase, the second resonance must be at (n+1/4) F1. The vertical lines are spaced by 19.4 degrees, with the spacing representing one cycle of the signal at the second resonance. Based on the grid construction, which is approximate, there are 360/19.4 = 18.5 cycles in 360 degrees, and the nearest frequency satisfying the n+1/4 constraint is 18.25 F1. The construction is not accurate enough to reliably determine n, but the Fourier transform will show a sharp peak at the correct value.
Observe that each of the four pulses that make up a cycle is visible in most of the diagonal drift bands. The four locations are easier to see by clicking on the image to expand it. That does not necessarily mean that the four pulses occur at preferred phases of the pulsar cycle. When the P3 is long the two dimensional autocorrelation reveals the drift bands. When P3 is short the bands become a diagonal sequence of correlation dots. One of the dots coincides with the reference phase, and if the reference phase is changed the whole pattern will shift to one side if the subpulses are drifting asynchronously. The occurrence of correlation dots at fixed phases therefore cannot be used to infer that the subpulses tend to occur at fixed positions. Other statistical tests could be applied, and the phenomenon is impossible for a rotationally powered pulsar.
The figure also shows bands where the subpulses do not drift. They coincide with pulse number 2 of the 4-pulse sequence. Referring to the reference shows that pulse number 2 is special by being at the peak of the mean profile. The nonlinear interactions are plausibly stronger at that point, resulting a ringing of the second resonance, and with two cycles of the ringing being visible after each occurrence of pulse 2.
The pulse sequence shown (22 pulses) is much too short for any analysis to be conclusive. The pulsar frequently switches modes, making it difficult to obtain long correlations. Drift modes are usually recurrent, and by isolating many copies of this mode it may be possible to determine if the ringing bands are consistently associated with pulse number 2. For an observer riding on the surface of a rotating pulsar there is nothing special about pulse number 2, while in the presence of nonlinearly interacting oscillations such things are possible. Significant interactions would only be expected when the repeat interval is short and nearly an integral number of pulsar cycles, making satisfactory test cases rare.
In a statistical sense, an observer at a different angular location about the spin axis of a rotationally powered pulsar must see the same thing at a different time. The other locations are inaccessible to us, but observing from the same location at different times indirectly provides some of the same information. The available data are not sufficient for drawing any conclusions, but they illustrate the kind of tests that can be applied.
5.5 Pulse microstructure
The intense E field near the pulsar will sweep the local region free of charged particles, and the pulsar frequency is far too low to radiate effectively without charged particle interactions. But if material is accreted from the supernova debris or if charged particles are pulled from the pulsar surface by the E field then most of them will move parallel to the magnetic flux lines toward the other magnetic pole, where they will collide with the surface to produce more charged particles. The new charged particles with the right sign will in turn be accelerated toward the other pole by the E field, where they will again collide to produce more charged particles. This back and forth motion of charged particle clumps will result in a quasi-periodic high frequency structure in the pulse.
There is no particular path length along the flux lines from one end of the pulsar to the other that is special, but for a diameter of 20 km a typical distance of perhaps three times that would be reasonable. In most cases only one pole will be visible, resulting in a typical round trip distance in the vicinity of 120 km. If the particle energy is high then the particle velocity will be near c, resulting in a quasi-periodic signal with a period of, very roughly, 400 microseconds.
The highest frequency observed fluctuations are too short to represent the round trip transit time, but the dominant periods are in the range of interest [47]. There is unfortunately little possibility of determining if the dominant period represents the round trip transit time between the poles, since it is plausible that localized mechanisms could produce a quasi-periodic structure.
The observed microstructure period correlates positively with pulsar period [48], which is contrary to the predictions of the oscillatory model. But the pulse width of high frequency pulsars is systematically shorter, which reduces the visibility of long period fluctuations. The correlation could be due to a visibility limitation.
5.6 The rotational signature
If the pulsars do oscillate then any low frequency and highly periodic change in the pulsar signature might represent the rotational period. The rotation would cause a small variation in the pulse arrival time, but it is probably better technique is to search for periodic changes in the pulse profile. Several precessional effects are known in orbiting pulsars [10,12, 13], so only isolated pulsars can be considered in looking for the rotational signature unless the rotation rate is so high that precession can be ruled out.
If the angular velocity is high then from the equatorial bulge argumet of the introduction the magnetic and spin axis will have a greater tendency to be aligned. The restoring force is less at low spin rates, making it easier to detect a rotational signature.
5.7 Phase jumps
Another way of distinguishing between the oscillatory and rotational models is to look for phase jumps. A phase jump is possible in a rotating pulsar, but it requires that the angular velocity first change with one sign, then after a short time change again with the other sign, which is improbable. On the other hand for an oscillating pulsar a major disturbance would be expected to affect the frequency, and the frequency shift will integrate to a phase change after the event. High frequency oscillators are more susceptible to phase jumps.
5.8 Glitches and more
The radius decreases steadily as the pulsar ages, so the neutron crust must rupture and buckle at times. The buckling will increase the effective radius, increasing the frequency of resonance. The signatures of the oscillatory and rotational model glitches are very similar in this respect.
There is another factor to be considered in evaluating the glitch characteristics. The pulse profile of SGR 1900+14 changed at the time of the August 1998 outburst, with the change persisting for 1.5 years and showing no signs of recovery [40]. Some radio pulsars with drifting pulses undergo similar mode changes. They presumably represent a change in the preferred scheme of phase locked frequency ratios, and occasional readjustments will be necessary due to random disturbances, and systematically so as the Schwarzschild radius is approached. Such a change will affect both the basic pulsar frequency and the spindown rate. The possibility can be considered that a major and sudden reshuffling of the frequency ratios is the cause of the glitches. There is no drifting when the frequencies are locked in exact integer ratios. It would be interesting to determine if major glitches are usually accompanied by profile changes.
The equations cannot be used to estimate the intensity of the E field, except that fields up to 1018 V/m are possible if no mechanism limits the build up of the oscillator's intensity. It is unlikely that the field is that intense during normal oscillation, but with a major disturbance fields of that order may be possible. At 1.4 solar mass the Schwarzschild radius is 4.1 km. In the polar regions charged particles will be accelerated by the E field for a distance that is of the same order as the radius, thereby acquiring an energy of up to 1021 electron volts. The most energetic cosmic ray ever observed was at 3 x 1020 ev [6], suggesting that a connection could exist.
These relationships suggest the possibility of detecting a periodic pulsar signal in the cosmic ray data. At 1016 electron volts a 0.1 Hz proton beam has a vacuum coherence distance of about 105 light years. At 1017 electron volts a 0.1 Hz beam of iron nuclei has a vacuum coherence distance of 6,000 light years. These calculations assume a 10:1 energy spread.
[Chandra]
X-ray image of the jet from quasar 3C273.
The jets from the nuclei of quasars and active galaxies consist mostly of electrons and positrons [5]. A behemoth pulsar at the galactic center is a possible source, as gamma rays are converted to electron-positron pairs in an intense enough magnetic field, and one or the other will be ejected by the E field at a velocity near c. A static E field near Em also results in pair production, but it is unlikely that such intense fields occur during normal pulsar oscillation. A static magnetic field cannot decay into single electron-positron pairs, but decay into multiple particles might be possible [31].
The model as developed has an upper limit of around 105 solar masses at 0.1 Hz, but for a high Q resonator operating in an overtone mode there is no practical upper limit to how large it can be and still maintain phase coherence across the diameter. Evidence is accumulating that the cores of the active galaxies contain massive objects of 108 solar masses or more, and these objects might be pulsars. Arguing against that possibility is the evidence that our own galaxy has an object with a mass of three million suns at the center, and no pulsar has been detected at that location. The region is badly obscured, so the final vote is not in yet.
5.9 Total oscillatory power output as a function of "spin down" rate
The core of galaxy [NGC 7052].
What's in there?
There is one relationship that is vastly different in the two models. The rotational model is powered by the kinetic energy of the rotation, which is relatively small for an old and small pulsar. On the other hand, the oscillatory model is powered by the gravitational energy, which is of order m c2 for an object near the Schwarzschild radius. Further, if the frequency, mass, and spin-down rate are known, then the total energy flow is directly computable from the requirement that the system conserve energy. This relationship forms the basis of a good observational test.
The 0.085 Hz pulsar 1E1841-045 radiates 3.5 x 1035 ergs/sec in X-rays alone. The 0.091 Hz pulsar 1RXS J170849.0-400910 is similar, with a steady X-ray output of 1.2 x 1036 ergs/sec [27]. These power levels cannot be supplied by the rotational kinetic energy of the pulsar, and it is likely that there are other and still unmeasured energy flows to be considered.
The energy flow equation is easily derived in the Newtonian approximation
power = -3/10 m c2 s/k
k is the ratio of the actual radius to the Schwarzschild radius. s is
the fractional frequency change per second, with the units of 1/time. s
is negative, and is the same as
At 1.4 solar masses the Schwarzschild radius, 2 G m/c2, is 4.1 km. (In geometrical units the mass is half this value). Assuming 1.4 solar masses, a representative spin-down rate of one part in 1015 per second, and k=3, the equation evaluates numerically to 2.5 x 1031 watts or 2.5 x 1038 ergs/sec. For b=1 and the P1 mode the frequency is 1.2 Hz and the radius is 12 km.
This calculation includes neutrino radiation and the kinetic energy of the ejected mass, although the equation is not valid when the pulsar ejects its own mass. The equation overestimates the power output in that case.
The equation for the case where the pulsar is powered by the rotational energy is
power
= -8/5 f2 m p2
r02 s
f = rotational frequency; m = mass; r0 = radius; s = fractional frequency change per second.
For f=0.1 Hz, r0=10,000 meters, s=-10-13, and 1.4 solar masses the equation evaluates to 4.4 x 1024 watts, or 4.4 x 1031 ergs/sec. Several low frequency pulsars are known that radiate at more than this power level, and these pulsars are thought to be magnetars.
It is not necessary to rely on new theory to conclude that gravitational contraction can supply power to the pulsars. As a dense magnetized sphere contracts the radial motion induces a current flow parallel to the equator [footnote]. I2 R losses convert the current flow into heat. The motion is slow, but the numbers in these equations are astronomical in their magnitude. At the highest spin-down rates the voltage around the equator would be on the order of 104 volts if the electrical resistance were high and if the radius dropped by a factor of two over the life of the pulsar. The electrical resistance is probably very low, so the actual voltage would be nearly zero, but the current flow would be high. The electrical and gravitational fields are strongly coupled in the pulsars. Before pulsar theory can be considered complete it must be determined if the coupling is stable against small pertubations. Do the pertubations decay with time, or do they grow? If they grow then the gravitational potential energy becomes available, and it approaches m c2.