The Ins and Outs of Pulsarlin
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| The Ins and Outs of Pulsarlin
This document will highlight the workings of the matlab code pulsarlin, written by Bill Coles and George Hobbs. I aim to explain in detail the graphs that are outputted by the program, using figures and references to the code as examples. The purpose of this document is to make sure I have a thorough understanding of what the code achieves and that I am able to describe what it does and why it does it. In the following examples I will use pulsar 1842-04 to which I have added a model planet in order to illustrate what the data tells us. The parameters for the model planet are: PB 500 ECC 0
OM 200 T0 50000
A1 0.001 First run When running the program for the first time it asks for the following parameters; alpha high, alpha low and ns, for which I enter -2, -6 and 301 days respectively. Alpha high is for the white noise power law and alpha low is for the red noise power law. ns is effectively the bin size. Currently I am not sure why the bin size has to be an odd number. The program works by taking a Fourier transform of the residuals for the pulsar. By doing this it is able to see if there are any sinusoidal waves in the residual data, which show up as a peak once Fourier transformed. The peak could correspond to a planet orbiting the pulsar. The axes of the transformed data are spectral density versus frequency. The reason that the alpha low and alpha high are entered separately is so that the red noise and white noise can be split up; this is done using a high pass filter. The data starts off being irregularly sampled, or ‘gappy’. The first part of the code fits a spline fit to the data so that it can now be regularly sampled. The points are then smoothed by a low pass filter. Using the bin of 301 days all the points are averaged and shifted correctly, hence smoothed. The result gives you regularly sampled red noise data, as shown in figure 1. 
Figure 1 The top graph on Figure 1 shows the residuals of the pulsar against time. The blue crosses are the actual data. The red line is the spline fit which basically ‘joins the dots’. The green line is the smoothed data. The bottom graph on figure 1 is the difference between the real and smoothed data, which in other words, is the white noise. 
Figure 2 Figure 2 shows two versions of the red power law spectral density. The top graph uses a rectangular window weighting and the bottom graph uses a Hann window weighting. 
Figure 3 – graphical representation of a rect window and a Hann window. Pictures courtesy of Wikipedia. These power spectra are ‘pre whitened’ before plotting. This is because the red noise component is -6, which is too high to use in the program. So what happens is the exponent of the red noise is multiplied by 4 in order to create an exponent of -2, which is not too steep to use. Eventually the exponent is ‘post-darkened’; i.e. returned to its natural value. In the graphs on figure 2 the outer lines are the model, so you can see where and when the data fits the model and when it doesn’t. A peak in the data corresponds to a peak in the top graph of figure 3, i.e. a planet. The period of the model planet is 500 days and the bin size is 301 days so there is currently a leak into the red noise, hence the peak. I’m not sure why the graph takes its shape, in particular the bumps at the end of the line. To get from the top figure on figure 1 to the top figure on figure 4 a Lomb-Scargle periodogram (LSP) is carried out. The LSP is the equivalent of a fast Fourier transform, but it is used for more unevenly spaced data. The bottom figure is the same plot but on a log y axis. 
Figure 4 Because figure 4 is the Fourier transform of figure 1 the peak that appears at 0.002 is due to a sinusoidal wave in the residual data. This could be a planet. The chance of the peak happening at 0.002 (1/500) is 0.000, which is pretty good! I’m not sure how the red line is calculated, but anything above the red line is considered as a possible planet. Directory: research research -> Charter School Enrollment Data Annual Report research -> Lyra Viol Composers research -> Geographies of Race and Food research -> Author (s) title research -> The Media and Intercollegiate Sports research -> Project Title: Research and Development for goes-r risk Reduction Principal Investigator research -> Why Everyday Life Matters: Class, Community and Making Life Livable Les Back research -> Ecocritique and the Materialities of Animation Download 12.43 Kb. Share with your friends:
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