The Period and Period Derivative

Figure 8 &mdash These two panels show the Period and Period-dot. The y-axis is the Reduced $\chi^2$. Ideally, you should see a sharp peak in Reduced $\chi^2$, which indicates that the Period or Period derivative have been well measured.

Now we turn our attention to the last three plots. Take a look at the two panels in Figure 7. They look similar to the plot of dispersion measure, once again having "Reduced $\chi^2$" on the y-axis. However, in the panel on the left the x-axis is labeled as "Period (ms)" and in the panel on the right the x-axis is labeled as "P-dot (s/s)". You will also notice that Period and P-dot have a number being subtracted from them. This number is simply the center of the plot, and we subtract it so that we can easily tell how far the peak of the curve lies from this central value. Once again, we want to see a nice, sharp peak, telling us that we have measured the period well.

Figure 9 &mdash In this panel, the Period and Period-dot are on the x- and y-axes, and the Reduced $\chi^2$ is represented by the different colors. A high value of the Reduced $\chi^2$ is shown in red and a lower value in white or purple. This is similar to a topographic map that shows elevation using closely spaced lines. You will notice that the x- and y-axis are also labeled with Frequency and Frequency-dot. Remember that frequency is essentially one divided by the period, so it is fairly easy to go between using Period and Period-dot and Frequency and Frequency-dot.

The P-dot, or Period Derivative, is a the first derivative of the period, and a measure of how much the period changes with time. It has units of "seconds per second" 1. These strange units simply tell us how much the period appears to slow down or speed up. These numbers are typically very small, since pulsars are quite stable. The Period Derivative is an important quantity to measure when really trying to understand an individual pulsar. Generally speaking, pulsars ought to be slowing down, but the motion of the pulsar relative to our solar system can cause the rotation to appear to speed up. This motion could be induced by something like a companion star or planets orbiting the pulsar. When determining whether a candidate is a true pulsar or RFI you should look for a sharp peak in the Reduced $\chi^2$ vs. P-dot curve. This indicates that the candidate has a well defined Period Derivative. Isolated pulsars typically have a P-dot that is zero (within errors), but if the P-dot is non-zero \emph{and} the pulsar is real, then you have found an exciting binary pulsar.

Finally, we turn our attention to Figure 8. This may seem like a very complicated plot, but it is really nothing more than the previous two plots put together. The x-axis is the Period or Spin Frequency, once again shifted so that the center of the plot corresponds to zero. The y-axis is P-dot or F-dot (F-dot being the change in spin frequency). The colors represent the Reduced $\chi^2$, with red meaning a high Reduced $\chi^2$ and purple meaning a low Reduced $\chi^2$. This is very similar to an elevation map. Instead of lines of constant elevation, we have colors of constant Reduced $\chi^2$. To make this plot, we take the two curves from the previous two plots, and combine them. If you imagined the plot coming out of the page and being three dimensional, the red areas would correspond to the peaks and the purple areas to the "lowlands" in the Reduced $\chi^2$ curves. You can also think of the two plots in Figure 7 as slices through the peak of Figure 8.

Ideally, we should see a well defined region of red, and only one well defined region of red, in this plot. If there are other well defined regions of red, then our measurements might not be very good. Usually, this is only an issue for weak pulsars or strong, variable RFI.

The Search Information

If you look at the original prepfold plot, you will also see a lot of text at the top. All of this is information about the search, the observations, and the candidate pulsar. The part that you should be most familiar with is the Dispersion M easure and Period, which is labeled as $\mathrm{P_{topo}}$ or $\mathrm{P_{bary}}$ 2. While both of these numbers can be determined by looking at the plots, they are also printed in the Search Information area for you. You will also notice that there are numbers in paranthesis at the end of each value. This paranthetical number is simply the error in the last decimal place. If you are interested in what the other terms mean, I have included them in the glossary.


1Can you think of what the P-dot would be if it were in different units, say years/year?
2$\mathrm{P_{topo}}$ is the period measured at the telescope, while $\mathrm{P_{bary}}$ is the period that would be measured if our telescope were placed at the center of mass of the solar system (also called the Barycenter). The solar system center of mass is actually actually just beyond what we typically think of as the edge of the Sun, called the photosphere, so this correction is made on computers. Why should the two be different? The answer is that the telescope is accelerating as the Earth rotates on its axis and orbits around the Sun. In effect, the telescope is "running" towards or away from the pulsar, affecting when pulses appear to arrive. The solar system center of mass isn't accelerating very much, and so doesn't suffer from this problem. It is a reference frame that all observers can agree on.
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