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Chapter 1: Astrophysics

Sign of the Doppler Shift

Dear Dr. SETI:
I've run into a problem where math and intuition seem to be giving different answers. When that has happened in the past, either I was looking at the problem wrong, or I had done the math wrong.

Here is the intuition part: if I live right on the equator and point my antenna to the Eastern horizon (90 degrees azimuth at zero degrees of elevation), that point should give me maximum positive Doppler, right? Then, if I keep the antenna locked onto that position, the point being monitored will rise in the sky until it is at its zenith. My antenna should now be pointed up at 90 degrees elevation, and Doppler should be at a minimum. Then, as the point starts to set in the West, and I track it with my antenna, the Doppler shift should go negative, and reach its maximum negative just as the radiant sets - right?

So, the Doppler should go through a one-half wave of a sine from Max positive to Max negative, and the slope of the Doppler curve should always be negative.

Here is the math part: using the Excel spreadsheet for Geo-Rotational Doppler, it never seems to go negative. I can set the hour angle at -6 hrs (point at the Eastern horizon) or +6 hrs (point at the Western horizon) and the Doppler sign does not change.

What have I missed?
Jim, California

The Doctor Responds:
I don't think you've missed anything, Jim, although your spreadsheet may have. Your intuitive assessment is roughly correct (though at latitudes other than on the equator, the curve is not exactly sinusoidal). What you have described is the classic Doppler 'S' curve for orbital motion: maximum positive frequency excursion at acquisition of signal (AOS), zero Doppler directly overhead, and maximum negative frequency change at loss of signal (LOS).

I believe what the spreadsheet may be calculating is not absolute received frequency change (Doppler shift), but rather the first time derivative of Doppler shift (the rate of change of frequency over time). In other words, you are seeing the slope of the Doppler curve, which is nearly zero at AOS, maximum at closest approach (zenith), and nearly zero at LOS.

By the way, as you suspected, the sign of this slope is always negative (frequency continuously decreasing), and can never go positive. I suspect the spreadsheet is merely displaying the magnitude of the rate of change of Doppler shift, since the sign is constant (therefore irrelevant). That is to say, if the Doppler rate (first time derivative of frequency) is "100 Hz per minute", it goes without saying that this is a change in frequency which is negative over time.

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