5. What form does a gravitational-wave pulsar signal take at an Earth-based detector?

If a gravitational wave detector were floating in empty space, far from any massive bodies such as the Sun, Jupiter, or the Earth, gravitational-wave pulsar detection would be very simple. As seen from the detector, the gravitational wave signal from such a source would be a sinusoidal 'pure tone' with a fixed frequency^5.1. The signal could be detected using a simple method, known as the Fourier Transform [25]. There is a well-known efficient computer implementation of this method, known as the Fast Fourier Transform, which would make searching for a pulsar signal an easily soluble problem with even a small computer cluster [26].

However a great deal of complication is introduced because our gravitational-wave detectors (LIGO [5] and GEO [6]) are Earth-based. The Earth's motion as it spins about its axis and orbits the Sun changes the gravitational wave as measured by the detector. To put it another way, the relative motion of the star and detector "modulates the waveform". This modulation is very similar to the way that an FM radio transmitter encodes information onto a radio-frequency carrier signal.

Some graphs help to explain this. Shown below as Fig. 5.1 is the gravitational wave frequency of a fictional pulsar whose intrinsic spin frequency is 100 cycles per second (100 Hz). The frequency of the gravitational wave received by a hypothetical 'space based' detector far from our solar system is 200 Hz as shown in the figure. Even over a span of one year, this frequency does not change: it remains exactly 200 Hz ^5.2.

Figure 5.1: The gravitational wave frequency received by a hypothetical 'space based' detector far from our solar system or any other mass is 200 Hz. [TOP] Annual (no) modulation of the (gravitational wave) signal frequency. [BOTTOM] Daily (no) modulation of the signal frequency for the first two days of the year.

Shown in Fig. 5.2 is another curve, illustrating the frequency of the same pulsar as seen from an Earth-based detector. Notice there is a slow frequency drift with a period of 12 months, caused by the Earth's motion about the Sun. Superimposed on this is a faster frequency shift, moving up and down once every 24 hours as the Earth rotates on its axis. This faster frequency drift (once per day) shown in the second graph is barely visible on the first graph, because its amplitude is very small.

Figure 5.2: The frequency received by a detector on the Earth. [TOP] Annual modulation of the signal frequency over one year. The overall modulation is due to the Earth's orbital motion. The modulation due to the Earth's rotation is barely visible. [BOTTOM] Daily modulation of the signal frequency for the first two days of the year. The modulation due to the Earth's rotation is now clearly visible.

These frequency shifts are due to the Doppler Effect [27]. This is the same effect which makes a train whistle appear to change in pitch as the train passes by a fixed spectator. As the Earth's motion carries the detector toward the pulsar, the frequency of the signal at the detector is shifted to a larger value (this effect is also known as ``blue-shift'' of the frequency). Later, when the Earth's motion carries the detector away from the pulsar, the frequency is shifted to lower values (``red-shift'' of the frequency).

These frequency variations are small but important. The magnitude of the frequency shift $\Delta f$ away from the nominal frequency is $\Delta f = (\vert v\vert/c)f$ where $v$ is the detector velocity along the direction pointing toward the source and c is the speed of light (c$\simeq$300000 km/sec). The velocity of the LIGO and GEO detectors due to the Earth's rotational motion is $\vert v\vert/c=10^{-6}$ and the part due to motion around the Sun is $\vert v\vert/c=10^{-4}$. Hence the frequency modulation introduced by the Earth's orbital is at most about one part in ten thousand. This in turn is as much as one hundred times larger than the frequency modulation introduced by the Earth's rotational motion.

Since the detector velocity along the direction pointing toward a source is different for different source positions, the pattern of frequency modulation that would be observed in the detector's output depends on the exact location of the source in the sky. Searching for pulsars is a challenging computational problem because every sky location has a different pattern and we have to search in the data for all the different patterns.

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Einstein@Home S3 Analysis Summary	Last Revised: 2005.09.11 16:22:17 UTC
Copyright © 2005 Bruce Allen for the LIGO Scientific Collaboration	Document version: 1.97