Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.atnf.csiro.au/research/pulsar/array/uploads/Projects/TheInsAndOutsOfPulsarlinFirstRun.doc Дата изменения: Tue Dec 12 04:09:28 2006 Дата индексирования: Sun Jun 27 22:30:14 2010 Кодировка: Поисковые слова: вторая космическая скорость

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.