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Chapter 1: Astrophysics

Local Mean Sidereal Time

Dear Dr. SETI:
I am attempting to understand the calculation of Local Sidereal time and am running into trouble. I have managed to get to Greenwich Mean Sidereal Time in radians but cannot figure out how to go from that to LST. Any help out there?
Jim (Project Argus participant)

The Doctor Responds:
Local Mean Sidereal Time (LMST -- sometimes called simply LST) is a time scale which tells your instantaneous position relative to the stars. It is useful for radio astronomy because it takes into account not just the Earth's rotation on its axis (as conventional time does), but also the Earth's revolution around the Sun. The two motions combine such that one day on the sidereal scale equals 23 hours 56 minutes on the terrestrial clock. And when your antenna is pointed along a North-South meridian line, at any instant the Right Ascension (RA) you are scanning exactly equals your LMST.

Since LMST is a measure of time, it is normally expressed in hours, minutes and seconds (hh:mm:ss). However, it is most readily calculated in angular measure (degrees, or radians), and then converted to clock time. There are programs which enable your computer to display LMST on a real-time clock display. A good one is available on the Naval Observatory Time Service website. And, we have a spreadsheet online (titled ra_dec.xls) to facilitate these and related calculations. But it's worthwhile to go through the computational exercise yourself, just to make sure you understand the principles involved.

Jim is halfway there, because he has already calculated LMST for the Prime Meridian (conveniently defined for us by the location of the Greenwich Observatory, England). Thus, what he has figured out is called GMST (Greenwich Mean Standard Time). But he has expressed that number in radians, and needs next to convert it to degrees.

You remember how to convert between radians and degrees, don't you, Jim? Multiply radians by 180 and then divide by pi, to get degrees (that's because 1 cycle equals 2*pi radians equals 360 degrees). Now you have GMST in degrees.

The next step is to convert GMST to LMST, which involves introducing a correction for your local longitude. (Of course, if you happen to live on the Prime Meridian, this step is unnecessary.) If you know your station's longitude in degrees with respect to Greenwich, simply add it to GMST to get LMST, in degrees. The only caveat is that all latitudes east of the Prime Meridian are defined as having a positive sign, and those west of Greenwich are expressed as negative numbers.

Of course, every location on Earth is both east and west of the prime meridian! We tend to think in terms of the eastern and western hemispheres, but even that distinction is really not necessary. This is one of those cases where you can't go wrong. Let's say you're at longitude 90 West. That is of course the same as being at 270 East. So adding negative 90 degrees to GMST will give you one solution for LMST (in degrees). Adding positive 270 degrees to GMST will give you an entirely different (but totally equivalent) solution. If one of the answers is a negative number of degrees, adding 360 will make it positive. If one of the solutions yields something greater than 360 degrees, just subtract 360 to end up in the proper range.

Finally, if you want LMST in hh:mm:ss instead of degrees, all you have to remember is that there are 24 time zones in 360 degrees of longitude, thus each hour equals (360/24)=15 degrees.

Let's work a simple example: Say it is now 2 radians GMST. That equals (2 rad * 180 / pi) = 114.6 degrees for GMST. I live 77 degrees West of Greenwich, so my LMST is [114.6 + (- 77)] = 37.6 degrees. Since each hour equals 15 degrees, my LMST is (37.6 / 15) = 2.5 hours, or 02:30:00. (That means if I have my antenna on the meridian transit, right now it is looking at RA 02:30:00.)

But what if I had chosen to go the long way around the world, and measure my latitude in degrees east of Greenwich? 77 degrees west is the same as 283 degrees east, so instead of adding (-77) to GMST, I could just as easily have added (+283). The sum of 114.6 + 283 is 397.6 degrees, which is obviously too large a number. Subtracting 360 gives us 37.6 degrees for LMST, which is the same answer we arrived at above (and which, when converted to hh:mm:ss, still equates to 02:30:00).

A good way to check the accuracy of your work is to compute LMST going both ways around the world, and compare your results. Try it, Jim, and let me know how it works out!

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