by Dave Robinson
11th October 2011
For information only
Note: The data used for this analysis was setiQuest Data, available at http://setiquest.org/wiki/index.php/SetiQuest_Data_Links
The Crab Pulsar is the remnants of a relatively large star that destroyed itself less than 1000 years ago, relative to the Earth (i.e. ignoring the time of flight from the Crab to the Earth). We know a considerable amount regarding the remnant neutron star; specifically its rotation speed, as indicated by the characteristic 'lighthouse' flashes from the inclined magnetic axis and rotation axis. However this document raises some interesting questions regarding information that is apparent from the low frequency data, that is normally thrown away when normalizing the data to produce the waterfall diagram.
The model that the Author was considering is that the neutron star is so fresh, in astronomical terms, that there is a good chance that it has yet to reach hydrostatic equilibrium; in other words, the size of the remnant is still oscillating, as illustrated below:
The remnant is assumed to be an oblate spheroid due to its rapid rotation rate. I think that the physics of such an oscillating system will have a different oscillation rate in the polar direction from the equatorial direction, such that the polar oscillation will be faster than the equatorial direction – due to the fact that the mean oscillation axis is shorter in this direction. For those not used to mental gymnastics with frequency domain processing, we can simply illustrate what the time domain waveform would look like, if we have a frequency domain signal consisting of two fairly closely spaced but unrelated frequencies. This is shown below:
The important thing to notice is that although the time domain signal only contains the two close frequencies we put in them, the resulting signal appears to have an envelop which contains a difference component. Note that this effect is an illusion. There is no signal containing this apparent beat frequency (it would be different if the two signals were multiplied and not simply added as we have done).
In the case of the Crab Pulsar, if the oscillating model is correct, we would expect to see a somewhat similar situation, the signal would consist of the sum of several relatively closely spaced frequency components; however these would tend to be smeared, simply because at any intermediate line of latitude, the oscillation period would be faster than the equatorial oscillation, but slower than the polar oscillation. But in principal the result should show analogous trends.
There is of course a basic assumption here, and that is the radial oscillation period is sufficiently fast as to show an appreciable number of oscillations in the overall period of time that we have a continuous observation of the Crab, which for the 26th March 2010 observations amounts to 609 SETISeconds (circa 9minutes). I personally have no idea what the oscillation period for a 26 mile diameter of solid neutronium actually is, however what I have done is to split the 5 files provided by SetiQuest into 609 files of 2^23 samples (1 SETISecond). I have then averaged each of these shorter files to get the mean value of the intensity of the Crab over a period of 1 SETISecond, then plotted the resulting time domain image. The result is fascinating.
The results are remarkably similar to the grossly simplified 2 frequency model we showed earlier. The next illustration shows the Fourier transform of this waveform.
Here we see that we are not getting a clear two frequency spectrum consisting of the polar and equatorial oscillation frequency, but rather a continuous distribution, as we would have expected from the previous argument. Measurements from this graph show that the mean value of the oscillation period is approximately 3.5 seconds.
If this interpretation of the results is correct, then I am sure it aught to be possible to extract a lot more information from such scans, such as the ratio of the polar to equatorial diameter, and knowing the oscillation period, it aught to be possible to make some estimate of the physical attributes of the neutronium forming the remnant, in order to support such an oscillation period; thus creating a new science of Pulsar seismology.