APPENDIX B - Physics & Astronomy



APPENDIX B

INTEGRATING GLOBAL POSITIONING SYSTEMS

IN THE COSMIC RAY PROJECT

The purpose for integrating the global positioning system (GPS) into the cosmic ray detector was to provide a time stamp for recorded events, thus pinpointing exactly where and when an event took place.

Although the GPS system would give us the exact location and time, we wanted to also see how accurately we could determine our exact location by using a US Geological Survey topographic map. A topographic map is a line-and-symbol representation of natural and selected man-made features of a part of the Earth’s surface plotted to a definite scale (Topographic Map Symbols).

Study began with the USGS topographic map for the Edwardsville Quadrangle available in the Lovejoy Library map room on the SIU campus. The first rough estimate of our location in the Science Building was:

Lat. N38( 47’ 35”

Long. W89( 59’ 50”

Elevation 552 ft. above MSL

Revised maps were ordered from Travelden in Kirkwood, Mo.

In the mean time, David Kraus installed the first GPS. We placed the antenna on the ledge of the patio on the north side of the building. We set it to take 10,000 readings, then averaged them, giving us statistically good numbers. There are 24 GPS satellites orbiting the earth at any given time, 12 of which may be visible. You need to get readings from at least 4, you hope for 8. We were receiving approximately 8 at this point.

Resulting GPS data:

Lat. 38( 47’ 40” .047

Long. 89( 59’ 57” .818

Height 140.82m

The next step was to determine our location using the USGS map.

To find the exact location you need to determine:

Longitude, which is measured in degrees east (+) or degrees west (-) of the Greenwich Meridian.

The earth is divided into 360( (degrees)

Each degree is then divided into 60’ (minutes)

Each minute is then divided into 60” (seconds)

Latitude which is measured in degrees north (+) or degrees south (-) of the equator.

The Equator is 0(.

The northern hemisphere is divided into 90( as is the southern hemisphere.

Each degree is then divided into 60’ (minutes)

Each minute is then divided into 60” (seconds)

The third component is height, which is normally measured with respect to your position above or below mean sea level (MSL).

USGS maps are marked with tic marks indicating latitude and longitude. Once you have determined your location on the map, you then need to measure the distance on the map to the nearest latitude tic and the nearest longitude mark. (there are four tic marks in the interior of the map at 2.5’ intervals, in addition to tic marks on the borders)

The measuring tool used in this experiment was a sliding microscope, with precision to 0.001 mm.

USGS maps have a scale of 1:24,000, so 1 cm on the map equals 24,000 cm (240 m).

The following is the procedure used to determine location on the map:

LATITUDE

Once you have determined the distance from your location to the nearest tic mark, you need to convert this to degrees.

Conversion factor for cm to degrees latitude

1 degree latitude = 30.922 meters per second

This is determined by

2 ( R( = length

360

R( = radius of the earth (6.378 140 x 106 m)

2 ( (6.378 140 x 106 = 111.3195 km/degree

360

111.3195 km/degree = 30.922 m / second

You then add or subtract the converted value to the value of the tic mark you measured from.

If your location is above the tic mark you measured from, you will add the distance measured to the value of the tic mark. If your location is below the tic mark, you will subtract the value.

LONGITUDE

To calculate the conversion factor for cm on your map to degrees in longitude you must first know you latitude. This is because the length of a degree of longitude is longest at the equator and decreases as it nears the poles.

The conversion factor is calculated as follows:

r = R cos (

r ~ conversion factor

R~ radius of earth in km/degree

(~ latitude

Example:

r = 111.3195 km/degree (cos 38(47”39.5776’ latitude)

r = 86.7624 km/degree

Tip: Check your calculator for the correct way to enter value of degrees.

86.7624 km/degree = 24.1006 meters/second

The first three trials resulted in the following:

| |Latitude |Longitude |

|Trial 1 |38:47:39.53 |89:59:56.52 |

|Trial 2 |38:47:39.53 |89:59:56.42 |

|Trial 3 |38:47:39.53 |89:59:56.22 |

|GPS |38:47:40.047 |89:59:57.818 |

|Mean difference |1.29% ( 25 ft.) |2.47% (81 ft.) |

It was determined after these trials that a more exact conversion factor needed to be calculated. Also, more exact measurements needed to be taken. After this was done Trials 4 and 5 were done.

| |Latitude |Longitude |

|Trial 4 |38:47:40.019 |89:59:58.387 |

|Trial 5 |38:47:39.929 |89:59:58.486 |

|GPS |38:47:40.047 |89:59:57.818 |

|Mean difference |0.18% (2.8 ft.) |1.06% (45 ft.) |

A second GPS system was installed and the antenna placed on the roof of the building. Again the system was allowed to record 10,000 readings and an average taken. Signals were being received from 12 satellites. Again, the location was determined using the USGS map.

| |Latitude |Longitude |

|Trial 7 |38:47:38.860 |89:59:55.710 |

|GPS |38:47:39.237 |89:59:55.792 |

|Difference |0.960% (38 ft.) |0.146% (6.48 ft.) |

Next the two GPS readings were compared:

Latitude

Antenna on Roof 38:47:39.237

Antenna on Ledge 38:47:40.047

Difference .810 x 30.922 m/sec = 82.153 ft.

Map difference 85.58 ft.

Longitude

Antenna on Roof 89:59:55.792

Antenna on Ledge 89:59:57.818

Difference 2.026 x 24.1007 m/sec = 160.15 ft.

Map difference 163 ft.

For comparison, we took the measurements on the maps using an inexpensive ruler, similar to those available to high school students.

| |Latitude |Longitude |

|Trial 6 |38:47:39.903 |89:59:56.923 |

|GPS |38:47:40.047 |89:59:57.818 |

|Difference |0.359% (14.5 ft.) |1.54% (71.0 ft.) |

The following steps were used to determine elevation.

The USGS map has contour marks that indicate height above mean sea level. These are marked in feet or meters above mean sea level. Determine the location of the Science Building on the map, measure the distance from the location to the nearest contour line and using the 1:24,000 scale, determine the height.

The GPS height position is given in meters and some adjustment is necessary here.

Because the Mean Sea Level fluctuates with the topography of the earth’s surface, GPS systems measure height using the geoid, an arbitrary smooth line drawn around the earth, as a reference point. The height we determined using the map used MSL as the reference point. To compare these two valueswe needed to adjust for the difference between the geoid reference point and MSL.

First, find we found our location in reference to the Geoid using the NIMA Geoid Calculator at

The Geoid calculator asked for latitude and longitude in degrees, minutes and seconds.

Tip: Eastern Hemisphere (+)

Western Hemisphere (-)

A negative geoid number means that the MSL at your location is below the Geoid so the geoid value needs to be added to the GPS number. A positive number would be subtracted. Our geoid value was –31.78m. So we added this number to the value for height given by the GPS.

| |Elevation (ft.) | |Elevation (ft.) |

|Trial One |560.91 |Trial 2 |608.5 |

|GPS on ledge |566.06 |GPS on roof |610.7 |

|Difference |5.15 . |Difference |2.2 |

In conclusion, exact location can be determined using a USGS map with a fair amount of accuracy, but errors due to map printing fluctuations and human measurement error are evident. GPS systems are more consistent.

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