Monday, July 11, 2011

Tackling Simple Astronomy Problems

One of the most common astronomy problems we dealt with when teaching CX astronomy was timekeeping, including by the Sun and Stars. In this and the next instalment I will show some simple ways we addressed those issues, for students and teachers.

A “time zone” was defined by taking the 360 degrees through which Earth rotates in one day, and dividing it by 24, since it requires 24 hours to make one revolution. Thus, one standard time zone would be generated via (360 deg/ 24 hr) = 15 deg/h or 15 degrees of longitude per hour - so be 15 degrees of longitude in expanse. Thus, time zones (calibrated per HOUR) were marked out by LONGITUDE differences.

Time zones don’t mean anything until referenced or calibrated to a fixed position-location, and that is the Greenwich Meridian, defined as 0 degrees longitude. All longitudes west of Greenwich mark time earlier – and all longitudes east of Greenwich mark times later. Thus, Berlin will always have a time later than London, and London will have a time later than New Orleans, just as Barbados will always have a time earlier than London and later than Miami.

The time difference is referenced to longitude difference for the central meridians. For example, if London is at approximately 0 degrees longitude, and New Orleans is at 90 degrees west longitude, then New Orleans is earlier than London by (90 deg/ 15 deg/h) = 6 hours. If the time in London is noon local mean time, then it is 6 a.m. in New Orleans

In order to solve the problem of different local mean times, Greenwich Mean Time or GMT was developed, so people could compare the same clock times around the world. GMT is based on a 24 hour clock defined at the Greenwich Meridian. So, for example, if one is listening to the BBC from New Orleans and the time given is 13h 30 m GMT, then that means it’s 1.30 p.m. LMT in London. Since New Orleans is 6 hours earlier, than that means it’s 7.30 a.m. local mean time in New Orleans.

Thus, knowing GMT, one can always work out the time at one’s location if one knows the longitude difference relative to Greenwich. (Note for the purposes here, I ‘m taking London as having the same longitude as Greenwich. It's actually off by a few thousand feet but negligible in terms of computations.)

Apparent solar time, meanwhile, is erratic because it’s based literally on sundial time, and what’s called the equation of time (E.T.) (See figure 1)

More useful was the construction of a simple shadow stick such as shown in the diagram below, and simply using it to make apparent solar time measurements. (See Fig. 2). This simple device was designed by a fifth form student at the Garrison Secondary school, and he actually used it to carry out a detailed analysis which was then published in The Journal of the Barbados Astronomical Society. His measurements of the lengths of the shadow stick are shown for March 21, 1979.

We know that the height (H) of an object placed in direct sunlight is related to its minimum shadow length (L_s) by:

tan (a) = H/ L_s

where (a) is the altitude of the Sun. So if H = 100 cm (the length of a meter stick) and L_s = 21 cm, then:

tan (a) = 100 cm / 21 cm = 4.76

And a = arc tan (4.76) = 78. 1 deg

In fact, the actual value for Barbados on this date should have equaled 77.0 degrees or the zenith distance of the Sun equal to the latitude. This is the precise measurement that would denote local noon apparent solar time.

Exact local mean time for any given longitude actually is computed via a slight adjustment to standard time. For example, if Barbados actual longitude is 59 deg 30’ W then the local mean time requires a slight adjustment equal to the time difference corresponding to 30’ of angular difference, since the meridian referencing Atlantic Standard time (A.S.T.) is 60 deg W and (60 deg W - 59 deg 30’ W) = 30’. This is half of a degree, and students in CXC astronomy were shown how to work out that every degree of rotation made by the Earth is equivalent to 4 minutes of time.

Since 15 deg = 60 minutes (1 hour), then 1 deg = 60 mins/15 = 4 minutes. Then 30’ corresponds to 2 minutes of time. So if A.S.T. (Atlantic Standard Time) at 60 deg W is 2 p.m. then the local mean time for Barbados’ specific longitude is: 2 p.m. – 2 mins. = 1: 58 p.m. L.M.T. It was only after much practice that both Caribbean students and teachers became comfortable with these sort of time conversions!

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