You can skip to the following subsections, or scroll down to them.     Home

5.1   The low frequency cut-off
5.2   The second resonance
5.3   Bi-drifting subpulses
5.4   Subpulse phase lock
5.5   Gamma ray pulsars
5.6   The rotational signature
5.7   Phase jumps
5.8   Glitches
5.9   Total oscillatory power output as a function of "spin down" rate

5.1   The low frequency cut-off

• Scatter plot of 643 pulsar frequencies

[Chandra] X-ray image of the 15 Hz pulsar J1811-1925 [49]. It shed its cocoon over 1600 years ago. Click on the image to zoom in. The pulsar is still very young. As it ages, it will eventually 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 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 systematically 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 14^th 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 intervals 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

The bi-drifting subpulse drift pattern of pulsar J0815+0939, from [54]. I have added the blue drift lines.

Bi-drifting cannot be explained by reflection from something in the vicinity of the pulsar. The direct signal is beamed in the rotational model, so the reflection would also have to be beamed to achieve a commensurate signal strength. A diffuse reflector would not work, and a specular surface aimed at the earth is not believable. There are other possibilities, but mostly the models do not focus on the earth.

The figure below shows a simulation of the Ruderman-Sutherland drifting subpulse model for two concentric and contra-rotating spark rings of 9 sparks each. The parameters of the simulation were selected to illustrate the characteristics of the model rather than to reproduce the J0815+0939 drift pattern. That would not have been possible anyway, because the two inner bands drift in the same direction, while for J0815+0939 they drift in opposite directions. Qiao et al. [54]. proposed a model in which the emission process interchanges the beam order, causing the two inner bands to drift in opposite directions. Refer to the reference for a simulation that does reproduce the J0815+0939 drift pattern.

The second panel in this figure shows the effect of shifting the differential phase of the contra-rotating spark rings by half of the spark spacing, 20 degrees in this case. The differential phase of the two drift patterns is shifted by 180 degrees. The pattern is very sensitive to a differential phase drift, and it becomes more sensitive if there are more sparks in the rings. The angular velocities of the rings have to be in inverse proportion to the spark count if it is not the same for both. The differential phase of the two patterns will drift steadily if the angular velocities are not exactly right.

With the oscillatory model, both drift patterns are phase locked to the main pulse, causing them to be synchronous with each other. The drift patterns will also interact with each other, resulting in a preferred phase relationship. This three way phase lock will be rare, but then bi-drifting pulsars are also rare. B1839-04 is another one [57]. Bi-drifting pulsars where two sets of subpulses march in lockstep in the same general direction should also exist.

5.4   Subpulse phase lock

In pulsar B1702-19, the drifting subpulses in the interpulse region are phase locked with those of the main pulse [59]. If the interpulse represents emission from the other pole, which is mostly though to be the case, then some mechanism would have to exist to synchronize the circulating spark rings at the two poles. In the oscillatory model the oscillation is global, so the synchronization just represents two perspectives of the same waveform.

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. Even when they are not harmonically related, the subpulses can phase lock to the pulsar frequency when there are an integral number of cycles of the oscillation in an integral number of pulsar cycles.

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   Gamma ray pulsars

The top panel of this figure shows the pulsed gamma ray radiation from pulsar J0631+1036, as observed with the Fermi satellite. The bottom panel is the radio pulse, properly phase aligned. The figure shows two full pulsar cycles. Adapted from [60]. Refer to the reference for the complete figure. The solid black regions represent photon energies above 3 GeV.

In one half of the cycle, an oscillating pulsar will preferentially eject electrons from a given pole, while simultaneously ejecting particles of the other sign from the other pole. 180 degrees later the roles will reverse, with positrons, protons, or other positive ions being preferentially ejected from the first pole. Except for positrons, the difference in particle mass between the two polarities can result in different emission characteristics. Particles of the wrong sign will collide with the surface of the pulsar, spallating more charged particles. Some radio pulsars have deep and abrupt nulls, where they become nearly undetectable for prolonged intervals, which supports the interpretation that the emission process is a periodic charged particle avalanche.

Pulsars are not known to eject ion beams. The jets show that something is being ejected, but recombination spectral lines should be present if the jet begins as an ion beam. On the other hand, the spectral lines of a relativistic beam are likely to be broad and difficult to recognize. A pulsed ion beam is fully capable of producing a radio pulse, although that is not the association that would be expected.

The gamma ray interpulse will be visible when the spin axis (and the magnetic axis) is nearly perpendicular to the line of sight. For pulsars with only one large gamma ray peak per cycle, the interpulse will be approximately coincident with the radio pulse, but from the other pole. The interpulse will be visible over a wider viewing angle when the emission occurs at a higher altitude, and plasma in the equatorial regions will block the radio view of the other polar region to a much greater extent than it does the gamma ray pulse.

This figure shows the gamma ray profile for pulsar J1952+3252, adapted from [61]. For this particular figure, the gamma ray pulse closest to the radio pulse may be interpretable as a phase shifted interpulse, but the other figures in the reference suggest the need for second and higher order harmonics.

Many radio pulsars have multiple peaks in their mean profile, which are interpretable as representing high order spherical resonances, with their amplitudes being exaggerated by threshold effects in the emission process. Resonance at the second harmonic could explain the this figure. The energy flow at the second harmonic could be due to ringing, or it could be that the second resonance is oscillating and phase locked to the fundamental. Either way, the harmonic content of the fundamental frequency is increased. More generally, in the presence of strong time-dependent nonlinearities, there are no compelling reasons for expecting large amplitude oscillation to be sinusoidal.

Even harmonics would affect the radio pulse from the other pole. When the harmonics are combined with threshold effects, the radio pulse might be missing altogether. Even harmonics would also result in asymmetrical polar jets. Conversely, a high velocity through the interstellar medium parallel to the spin axis would result in asymmetrical jets, which in turn would be manifested as even harmonics.

The magnetosphere of the binary eclipsing pulsar is about 0.05 light second across [62]. It is influenced by the wind from the other pulsar, so it is probably not representative of isolated pulsars, but if we were to receive signals from both the nearside and farside of the magnetosphere of a 0.1 second pulsar with a similar magnetosphere then the differential delays would be significant. A delayed interpulse would look much like one cycle of a second harmonic. On the other hand, the bridge between the gamma ray peaks of pulsars such as J1709-4429 is less ambiguous in favoring a second harmonic interpretation, combined with an emission threshold. It may well be that both effects play a role. It is probably not possible to reliably tell the difference yet. The primary complication is that the pulsar's orientation relative to the line of sight is usually not known.

With the rotational model, the radio pulse is caused by a narrow beam that sweeps past the earth. Since the gamma ray pulses are usually much broader, most of the radio beams should miss the earth. This evaluation is subject to a strong observational bias, as gamma ray pulsars are frequently discovered first at radio frequencies. Another complication is that the radio signal should be less predictable when there are two peaks in the gamma ray profile. The equations derived in Section 4 do not work for millisecond pulsars, so this model should not be applied to them without further considerations.

5.6   The rotational signature

Estimates of the spin rate after a core collapse vary greatly. Magnetic braking during collapse is capable of substantially lowering the final spin rate [63]. Analysts tend to assume that a model is wrong if it does not predict the correct spin rate for young pulsars, encouraging a model selection process that may be premature.

If the pulsars do oscillate then any low frequency and highly periodic change in the pulsar signature might represent the rotational period. 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.

This image is a 1.5 hour recording of pulsar B1702-19. The data record used in producing the image is from reference 59. The image spans 360 degrees of the pulsar cycle. The narrow band on the right is the interpulse. Most pulsars do not have an interpulse. With either model, the presence of an interpulse probably means that the pulsar's spin axis is nearly perpendicular to the line of sight, with the emission from the other magnetic pole being hidden from view in other cases. The slow amplitude variations are caused by interstellar scintillation.

The scintillation is caused by inhomogeneities in the free electron density along the line of sight. Multipath effects occur as the pulsar moves transversely through the medium, sometimes resulting in constructive interference between different paths, and sometimes resulting in destructive interference. The variations make the rotational signature difficult to detect if it is on the same time scale. Discrimination is easier in this case, because as the magnetic poles move alternately over the pulsar's horizon the signals from the poles should be out of phase with each other. They are in phase. A periodic variation in the pulse arrival time would also occur, but when the spin and magnetic axes are nearly aligned and the emission occurs at low altitude it will be very small. Scintillation effects are less pronounced for nearby pulsars.

Pulsars J0737-3039 A and B are a rare system with a millisecond pulsar and a mainstream pulsar in a tight orbit. General relativity predicts that the pulsars' spin axes will precess with a period of 71 years, and secular changes in the system have been observed for several years now. They are being modeled with the rotational model [62].

The precession equation does not explicitly contain the pulsar's rotation rate, so an oscillating pulsar will precess by the same equation, even though it spins at a much lower rate. That makes it easy to confuse the two models in short term observations. As the system evolves, the difference between modeling the magnetosphere in eclipse as a wobbly torus and that of an aligned rotor will become apparent.

This figure was adapted from reference 64. It shows the signal from the millisecond pulsar as the line of sight to it moves through the magnetosphere of the other pulsar during the eclipse.

Using the model sketched in Section 5.5, and assuming the P₁ oscillation mode with no significant harmonics, the sine wave represents the oscillating E field (with an undetermined polarity). The radio pulse from the nearer pulsar is not detectable during the eclipse, but the grid lines show where it would be, based on observations in other regions of the orbit. The radio pulse from the visible pole occurs at the peak of the sine wave.

When the line of sight first enters the magnetosphere, just after an orbital phase of 89 degrees, both poles of the oscillator are at a comparable distance, so they have an approximately equal influence on the signal. The E field passes through zero twice per cycle, and at those times no new ions are created, resulting in a maximum radio transparency for the plasma. There are two transparent intervals per pulsar cycle, one at each zero crossing. They have about the same amplitude, showing that the spin angular momentum vector is approximately parallel to the orbital angular momentum vector. (This conclusion is model-dependent.)

As the line of sight approaches one of the poles, the plasma near that pole becomes dominant. The transparent interval corresponding to the second zero crossing after the radio pulse persists well into the eclipse. However, the plasma becomes increasing radio-opaque at the first zero crossing. The plasma will be denser in that region, so a plausible interpretation is that the charged particles generated by the radio pulse have not had time to recombine sufficiently at the first crossing. A different eclipse is shown in [62]. The half-period peaks at eclipse ingress are better defined in the other figure.

The negative peaks of the sine wave indicate the location of the radio pulse from the other pole. Plasma in the equatorial regions will block the radio view of the other polar region over a wider angle than the body of the pulsar would, but precession should nevertheless eventually bring the other polar region into view, where the signal from it will appear as the interpulse.

5.7   Phase jumps

Another way of distinguishing between the oscillatory and rotational models is to look for phase jumps. 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. Similar shifts are possible for a rotating pulsar, but there are angular momentum constraints.

5.8   Glitches

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 -- if the assumption is made that the pulsars do have a neutron crust.

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 undergo similar mode changes. They presumably represent a change in the preferred scheme of 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. It would be interesting to determine if major glitches are usually accompanied by profile changes.

5.9  Total oscillatory power output as a function of "spin down" rate

The rotational pulsar 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. If the frequency, mass, and spin-down rate are known, then the total energy flow is computable from the requirement that the system conserve energy.

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 1/P dP/dt, where P is the period. m is the mass. The equation includes the internal gravitational energy of a solid sphere, which is 1/6 of the total.

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 π² 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 perturbations. Do the perturbations decay with time, or do they grow? If they grow then the gravitational potential energy becomes available for radiation, and it approaches m c².