Friday, May 20, 2011

Local Noon and the Hour Angle

On Thursday I made an attempt at determining local noon using the setup below. The aperature of the lens was increased to about 6mm using a hole punch for better contrast. NIO captured projected images of the Sun at approximately 5 minute intervals over roughly a two hour period.

The motion of the camera complicated comparing the data somewhat. Fortunately I had drawn a reference line of the plywood board used as a platform. The two ends of the lines and one corner of the board served as reference points to realign the data from the images. The first step was to translate all the x,y pixel coordinates so that the west side of the lines were at the same coordinates. Then the image data was resized so the distance along the line was the same for all the images. The data was then rotated to give the east ends of the lines the same direction. The result of this realignment was that the greatest amount of error was in the location of the corner and the average value was 1.37 pixels. This proved to be equivalent to an angular resolution of 2.63 minutes of arc.

Determing the time of local noon proved to be more challenging. At local noon the Sun is highest in the sky and as a result distance of its projected image from the point directly beneath the lens aperature will be a minimum. Although the images only determine directions relative to the camera the angular distances contained enough information to make an estimate of local noon. A rough polynomial fit was used to determine the approximate time of local noon. The coefficients of the smaller powers of this fit turned out to be zero. So a second polynomial fit using the difference in time between the time of the observation and the first estimate of local noon resulted in a simpler polynomial. The final fit turned out to be surprisingly accurate. The rms error in pixel position was 0.262 which corresponds to 0.505 minutes of arc. The result was a reasonably good estimate of the local time. The difference between mean solar time and local solar time was 5.226 minutes. MICA indicated that value for the Equation of Time was 3m 32.2s or 3.537 minutes. The difference, 1.689 minutes, is due to the difference in longitude from the prime meridian for the time zone. Each minute of time corresponds to 0.25 degrees so the time difference corresponds to a difference in longitude from the time zone's meridian of 0.422 degrees. This difference allows one to determine the hour angle or the number of degrees from the prime meridian of Greenwich.

I compared the estimated longitude with the longitude found in Google Earth and their ruler tool indicated that the difference was about 21 km*.

* Edit: The original estimate was too small. The distance is approximately
   R cos(lat) Δlong where R is the equatorial radius of the Earth.

   I started taking pictures shortly before local noon so my data was rather one-sided. A two hour interval centered near local noon would probably give better results.

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