_{1}

This study was an effort to calculate geographic coordinates, solar horizontal and equatorial cooridnates, direction of Qibla, sizes of Earth and Sun through gnomon (a vertical stick casting shadow). Gnomon has been used in history for remote navigations, astronomical and geodetic measurements. Geographic coordinates were found out with accuracy of less than 0.19 decimal degree from gnomon. Sun’s horizontal and equatorial coordinates were calculated and analyzed with US Navy Observatory and NOAA Earth System Research Laboratory. These coordinates were less than 0.01 decimal degree different from coordinates provided by US Navy and following same daily trend with time. Through spherical triangle, direction of Qibla was found out. And it came out less than 0.18 decimal degree different from its value, used in Lahore. Radius of the earth measured from gnomon using concepts of Al Buruni was 1.45 km different from its true value. Using geometrical technique of similar and concurrent triangles, size of the sun was calculated and it was 1,651,088.776 km. Investigation of systematic and random errors in each calculation was the part of this research.

In ancient times, gnomon was used as sun dial and for other observations. With the passage of time it started to use as tool for navigations. Still it is a reliable instrument for the basic astronomical as well as geographical calculations. In this paper, efforts for geographic position on earth in survival conditions, Sun’s horizontal and equatorial coordinates have been done with gnomon (a vertical stick casting its shadow). In order to emphasize the importance of gnomon in present time, other astronomical and geodetic calculations have also been investigated in this paper.

Lahore (74.27985278E, 31.4421667N) is the capital city of Punjab province, Pakistan. It is the second most populated city of Pakistan. An estimated in the 2014, it has population of 7,566,000. It has flat topography. Hottest month is June, when average high temperature routinely exceeds 40˚C. The monsoon season starts in late June and the wettest month is July, with heavy rainfalls. Coolest month is January with dense fog. Elevation of Lahore from mean sea level is 208 to 213 m (see

The primary purpose of this research was to build an instrument, low cost, which could help us in finding position of observer in survivor conditions. This instrument could calculate Geographic coordinates as well as Horizontal and Equatorial coordinates of the Sun. Able us also to calculate direction of Qibla, Apparent solar noon time, Time of Sun set and Sun rise for Lahore as study area. Using techniques of Al Buruni, it may help us to calculate radius of the earth. Through geometrical techniques of similarity and concurrency it leads us to calculate diameter of the celestial bodies like sun, moon.

Solar azimuth and altitude angles have been calculated for January 2, 2015. Time of observations was different

in solar azimuth and altitude measurements. Observations were taken from 11:30 to 16:00 (GMT + 5 h). Through this data set objectives of research was achieved and for validation, results have been compared with the US Navy Observatory, NOAA Earth System Research Laboratory and Google Earth.

In order to calculate latitude of observer through gnomon we have to calculate day angle through Equation (1).

Equation (2) estimates

Declination of the sun was calculated from equations (1) and (2). It came out to be negative, indicating that sun was in southern hemisphere of the celestial sphere (see

There for using spherical triangles rules, latitude of the testing site could be calculated with Equation (3).

Time (GMT + 5 h) | Shadow of gnomon length (mm) | Gnomon height (mm) | Elevation Angle (Taken from gnomon in decimal) | Elevation angle (Taken from US Navy in decimal) |
---|---|---|---|---|

11:28 | 281 | 199 | 35.305 | 34.9 |

11:45 | 280 | 199 | 35.401 | 35.4 |

12:00 | 277 | 199 | 35.69 | 35.6 |

12:24 | 282 | 199 | 35.2 | 35.5 |

13:08 | 307 | 199 | 32.95 | 33.7 |

13:47 | 348 | 199 | 29.76 | 30.6 |

14:11 | 361 | 199 | 28.86 | 28.1 |

14:40 | 459 | 199 | 23.43 | 24.5 |

15:26 | 679 | 200 | 16.41 | 17.8 |

15:59 | 898 | 200 | 12.54 | 12.4 |

Day | Day angle (radian) | Declinations (decimal degree) |
---|---|---|

January 2, 2015 | 0.0172142 | −22.9793417 |

Latitude (Gnomon) | Latitude (Google earth) | Difference in Latitude (decimal degree) |

31.33066583 | 31.4421667 | 0.111508397 (6 min 41.43 sec) |

Difference of latitude calculated from gnomon with latitude taken from Google Earth was 6 min 41.43 seconds (see

Uncertainty of ±1 mm in height, length and shadow of the gnomon could produce error of 8 min 24.36 sec in calculation of latitude (see

Apparent Solar Noon = 12.00

Noon at Greenwich (ESRL Global Monitoring Division-GRAD Group [

Systematic Error (arc min) | |
---|---|

Declination | <3 min |

Length and height of the stick | 8 min 24.36 sec |

Coordinates (taken from gnomon) | Systematic error | Random error (±) | Ratio | Nautical miles |
---|---|---|---|---|

Latitude | 0.15517^{ } | 0.1901 | 0.816255 | 59.70518514 (North-South difference) |

Longitude | 0.179853 | 0.01667 | 10.78902 | 0.1534 (East-West difference) |

Since the 1h of time corresponds to 15 degrees of longitude, so our derived longitude was T × 15 = 4.94 × 15 = 74.1E decimal degree. From Google Earth longitude of the observational site (Lahore) was 74.27985278E. Therefore systematic error in longitudes was = +10 min 47.47 sec. Total uncertainty in time measurements and equation of the time was ± 1 minute. In order to check robustness in calculation, ratio of these errors was taken. This ratio was closed to 1.0, it would imply that we have no significant systematic errors and observations could be improved using the same apparatus (see

Since lines of longitude converge at the earth’s geographic poles, in order to determine linear error in longitude, we must multiply the longitude error in angular units with cosine of latitude. Therefore, our east west error from the true position was 0.17985 × cosine (31.4421667) = 0.15344 Nautical miles.

Reference [

Islam which is the second largest religious in world has center significance for calculation of direction of Qibla. This figure has main direction for five time prayers per day in Islam. Holy Kaaba was cubical in geometry and located in Makah, which is a city in Saudi Arabia.

It was not reliable method to determine direction of Holy Kaaba, through magnetic compass. The following method which used the sun was more reliable. It has been observed for centuries and reported in many books by Muslims around the world that, two times in a year sun came overhead above Kaaba. This was observational fact for centuries, and used to set correct prayers direction in places far from Makah by Muslims for last so many centuries.

Those two dates and times were:

May 28 at 9:18 GMT,

July 16 at 9:27 GMT.

In order to find out direction of Qibla from Lahore, we used rule of spherical trigonometry [

Equation (5) was just final form of four part formula which given by

Cosine (inner side) × Cosine (inner angle) = Sine (inner angle) × Cot (other side) − Sine (inner angle) × Cot (other angel)

In Equation (5),

Inner angle =

Inner side =

Other side =

Other angle =

Cosine (

Sine (

Rearranging the terms

Sine (

From

We used gnomon, having horizontal plane surface (remember wood changes its shapes with seasons). We cut this sheet of wood to a circle of convenient diameter. At the center of the sheet make two small holes and adjust a meter rod and movable antenna (gnomon) in these holes. Put two bubble levelers on the wood sheet which tell us about level of gnomon in a plane. To achieve the horizontal level, gnomon was mounted with net bolt

Coordinates | ||||
---|---|---|---|---|

Taken from gnomon | 74.27985E | 31.33066583N | NaN | NaN |

Taken from Google Earth | 74.27985278E | 31.4421667N | 39.82624E | 21.42249444N |

Uncertainty in magnetic declination | ±0.29^{˚} |
---|---|

Difference from true value | 0.1783641^{˚} |

screws of 3 inches length. Magnetic compass on gnomon was used to measure solar azimuth angle from North Pole. Before a day or two the experiment, we made our quartz clock as accurate as we can do with local time, also we took a look on the weather condition from the National Weather Forecasting Center. We select a site where gnomon’s shadow throughout the day was easily measureable and there was not obstacle between the sun and gnomon. We used a plum line to ensure the verticality of the gnomon (see

We prepared a plot to investigate solar altitude variation with time (see

Time of observations for solar azimuth was different than solar altitude angle. Solar azimuth angle was measured, from direction of gnomon’s shadow with respect to true geographic pole of earth, in anticlockwise direction. We used magnetic compass to find the direction of magnetic north pole. NOAA gave us magnetic field calculator based on two Earth’s magnetic field models, IGRF (1590-2019) & WMM (2014-2019) [

Time of observations | Azimuth from gnomon (with magnetic north) | Magnetic declination | Azimuth (with geographic north) | Solar Azimuth (US Navy) |
---|---|---|---|---|

(GMT + 5 h) | Azimuth (E of N ) | (+E/−W) | True Azimuth (E of N) | Azimuth (E of N ) |

11:32 am | 162 | 1.8283 | 163.8283 | 170.2 |

11:48 am | 169 | 1.8283 | 170.8283 | 174.7 |

12:01 pm | 172 | 1.8283 | 173.8283 | 178.4 |

12:24 pm | 178 | 1.8283 | 179.8283 | 184.9 |

13:09 pm | 191 | 1.8283 | 192.8283 | 197.2 |

13:47 pm | 197 | 1.8283 | 198.8283 | 206.9 |

14:12 pm | 204.9 | 1.8283 | 206.7283 | 212.8 |

14:38 pm | 212 | 1.8283 | 213.8283 | 218.4 |

15:23 pm | 228 | 1.8283 | 229.8283 | 227.1 |

15:54 pm | 231.9 | 1.8283 | 233.7283 | 232.3 |

showed, plot of solar azimuth angle measured from gnomon. This graph indicated that solar azimuth angle was going to increase to 270˚ till sunset.

On January 2, 2015 the sun had declination in southern hemisphere, therefore using concepts of spherical triangles

Actual value taken from the US Navy was δ = −22.89585 decimal degrees, it was −0.01752 decimal degree more than calculated from gnomon.

Hour angle defined as the number of hours between the solar noon and time of interest multiplied by constant 15˚/hour. Value of this constant was known with rate at which the sun appeared to move around the earth namely 360˚ in 24 h or 15˚ per hour. At solar noon, the hour angle would be 0˚ and westward direction of the solar noon would take it positive [

Hour set, was expressed in degrees and divided by 15. The result was the number of hours after local solar noon at which the sun sets. To find the hour of sunrise, this same number of hours was subtracted from the noon [

Hourset = cos^{−1} [−tan (−22.867833) tan (31.33067)] = 75.12326106˚ = 5.008217404 h (see

Solar apparent noon occurred at 12 h (see

Hour rise = 12 − 5.008217404 = 6.99765066˚

Eratosthenes and a Muslim scientist Al Buruni were prominent names in calculation of the earth circumference. Eratosthenes used concept of solar declination while the Al Buruni apply mathematical techniques of trigonometry. Al Buruni was the first who know that, earth, not flat rather spherical in shape. Using the same concept of Al Buruni, through gnomon we calculated radius of the earth for Lahore. In order to do this, we find height of the gnomon from the ground (including the height of the instrument from the ground). It showed a vertical plane through the center of the Earth and the gnomon (see

R represent radius of the earth, and h showed height of gnomon above earth’s surface. Geometric horizon (also horizon distance) from top of gnomon is shown in tangent to radius of the earth in

Coordinates | Hour Angles | US Navy Data (GMT + 5 h) | NOAA Data (GMT + 5 h) | |||
---|---|---|---|---|---|---|

Latitudes | Hour set | Hour rise | Hour rise | Hour set | Hour rise | Hour set |

31.33067N (from gnomon) | 5.008217 | 6.99765066 | 7.05 | 17.2 | 6.9333333 | 17.0833333 |

31.44217N (from Google Earth) | 5.002349 | 6.997651 | 7.0333333 | 17.183333 | 6.95 | 17.1 |

Coordinates | Difference from US Navy (decimal degrees) | Difference from NOAA (decimal degrees) | ||
---|---|---|---|---|

Hour set | Hour rise | Hour set | Hour rise | |

Google earth | 0.180984 | 0.0356823 | 0.09765066 | −0.047651 |

Gnomon | 0.191783 | 0.05234934 | 0.075115896 | −0.06431736 |

As

Height of the gnomon from ground = h = 0.2 m

Distance to horizon = R tan (d_{g}), by taking the average radius of the earth 6371 km [_{g} = 0.01435814155˚^{ }

In literature (13) given by great scientist Al Buruni [

From equation (13) radius of the earth came out to be 6369.55 km which was different from the true value about 1.45 km. With this value circumference of the earth came to 40021.063 km.

Aristarchus was first to estimate distance of the sun. After that, he calculated radius of the sun on lunar eclipses. We used property of two similar triangles, side angle side (S.A.S), formed when button on gnomon just full cover disk of sun. To measure size of celestial bodies, we put a button of diameter 1.1 cm (taken from the venial caliper) on gnomon. We recommend using solar glasses to avoid harmful effect from the sun radiations (see ^{6} km) and Aphelion when it is farthest from the sun (152.10 × 10^{6} km) [

In order to calculate solar diameter, we used Equation (14)

Diameter of the sun was come out to 1651088.776 km which is 259404.8 km different from the actual diameter of the sun (see

Fewer errors in each calculation suggest us that gnomon can be used in the basic astronomical, astrophysical

Coordinates | Hour Angles | US Navy Data (GMT + 5 h) | NOAA Data (GMT + 5 h) | |||
---|---|---|---|---|---|---|

Latitudes | Hour set | Hour rise | Hour rise | Hour set | Hour rise | Hour set |

31.33067N (from gnomon) | 5.008217 | 6.99765066 | 7.05 | 17.2 | 6.9333333 | 17.0833333 |

31.44217N (from Google Earth) | 5.002349 | 6.997651 | 7.0333333 | 17.183333 | 6.95 | 17.1 |

remote navigations and geodetic calculations.

I would like to acknowledge assistance of U.S. Naval Observatory, NOAA Earth Science laboratory Global Monitoring Division-GRAD Group, for providing me data for this study. Special thanks to open journal of applied sciences, who give a place to publish this work. I would like to mentioned assistance of my student Muhammad Mohid for collecting data in this research.

Muhammad UsmanSaleem, (2016) Gnomon Assessment for Geographic Coordinate, Solar Horizontal & Equatorial Coordinates, Time of Local Sunrise, Noon, Sunset, Direction of Qibla, Size of Earth & Sun for Lahore Pakistan. Open Journal of Applied Sciences,06,100-111. doi: 10.4236/ojapps.2016.62011