ASTRONOMICAL AZIMUTH DETERMINATION BY THE HOUR ANGLE OF POLARIS USING ORDINARY TOTAL STATIONS

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1 Survey Review, 40, 308 pp (April 2008) ASTRONOMICAL AZIMUTH DETERMINATION BY THE HOUR ANGLE OF POLARIS USING ORDINARY TOTAL STATIONS Ε. Lambrou and G. Pantazis National Tecnical University of Atens ABSTRACT Te determination of te astronomical azimut of a line is not a difficult task for surveyors any more. Te aim of tis paper is to analyze te teoretical details and errors in order to propose te use of ordinary total stations, for an easy, efficient and accurate determination of te astronomical azimut of a line by te our angle metod via Polaris sigtings. As many modern total stations ave a built in quartz clock tey can register automatically te UTC time as well as te angle measurements (orizontal and zenit) of eac observation. Te total fieldwork time needed is about 10 minutes and te accuracy tat may be acieved is about ± 2. Tis procedure will be proven to be easier tan te determination of te geodetic azimut of te same line. Te calculation is independent and te result is free of te errors tat te coordinates of a survey mark may contain, because tey are not used. However good positional data is required from oter sources. Astronomical azimuts are an alternative solution for te surveyors in order to ceck or orient teir field surveys and arbitrary networks independent of te GPS system. KEYWORDS: Astronomical azimut. Hour angle. Total station. Polaris. Quartz clock. INTRODUCTION Te determination of astronomical azimuts is not very popular. Tis is due to te rumour about te difficulty of te calculations and te need for nigt observations. Supposedly, special skills are needed in order to operate te appropriate instruments for te observations. Fortunately, tis is not te case. Te former myt, about te level of difficulty, is no longer true. Today te revolution in geodetic instrumentation plays a major role in making te total process easier and quicker. Additionally, it would be very convenient for te surveyors to ave quick, easy and accurate astronomical azimut determination to orient or ceck a field survey or network. However, clear skies are required for astronomical observations. Tis determination is also independent of a national geodetic reference system. So it is not necessary to know any points or pillar coordinates and terefore it is free of te uncertainties tat tese coordinates may contain. Surveyors in te Nortern emispere ave a great advantage in determining te astronomical azimut of a line, especially between latitude 15 and 60 due to Polaris. Polaris is te star tat elps anyone find is orientation towards te nort. It makes a small circle, about 1 radius, around te Nort Astronomical Pole and its apparent motion is very slow. It is a brigt star wit a magnitude of 2.1, wic makes it visible at nigt and easily recognised, as it is te last star at te tail of te Ursa Minoris (Little Bear) constellation. Besides tat, te star constellations of Ursa Majoris (Great Bear) and Cassiopeia elp to find tis valuable star in te celestial spere. At latitudes less tan 15, it is not so convenient to observe Polaris as it is located very low on te celestial spere near te local orizon. At tose latitudes te errors caused by astronomical refraction are magnified. At latitudes larger tan 60 te metod of Polaris observations is affected by te error of te value of te astronomical Contact: G Pantazis gpanta@survey.ntua.gr 2008 Survey Review Ltd. 164 DOI / X290951

2 E LAMBROU AND G PANTAZIS latitude Φ of te station point, wic is needed in te calculations. In addition, te geometry of te sperical triangle is very weak and special equipment, for example, an astronomical diagonal eyepiece is indispensable for te observation. In suc a case oter metods, suc as te observation of star pairs near elongation or Black s metod, are used [7],[8]. In te Soutern emispere a corresponding star is Octantis. Tis is located about 45 from te Sout Celestial Pole and as a magnitude of 5.5. It is difficult to recognize on te celestial spere and for tis reason it is necessary to know te approximate azimut of te reference meridian. In general, te precision of te astronomical azimut determination depends on te quality of te instrument used, te metod of determination, te number of measurements, te successful removal of systematic errors, te experience of te observer and te latitude of te instrument station. [8] A basic parameter in most observations in geodetic astronomy, as in te astronomical azimut determination, is te registration of te time of eac observation. Many of te modern total stations, even tose manufactured for common surveying fieldwork, seem to solve tis problem, as tey ave built in crystal quartz clocks. Some uses of te astronomical azimut are: - Orientation of airport radar and satellite or telecommunication antennas. - Navigation - Orientation of army systems. - Control of underground surveying work in caves and mines from above ground to te start or te end of te traverses. In addition, a common use of tis determination is for te correct orientation of a land survey or an arbitrary network. In most surveying fieldwork a precision of ±2 to ±5 is sufficient. ASTRONOMICAL AZIMUTH DETERMINATION Te astronomical azimut Α ΑΒ of a line AB, defined between two points on te eart s surface, may be determined by measuring te orizontal angle between te given direction and te vertical plane of a celestial body and applying tis angle to te astronomical azimut of te celestial body. Te following metod is te most convenient for determining astronomical azimuts of stars for geodetic purposes, as te best precision will be acieved. Observing a close circumpolar star at any our angle (our angle metod). By tis metod te UTC time of eac observation to te star must be measured. Ten te our angle of te star is computed by te equation [8]: =θ + Λ a ( 1) were a = Rigt ascension of te star Λ = Astronomical longitude of te instrument station and te Greenwic sidereal time θ can be calculated as follows [8]: 0UT θ = θ + ( UTC + DUT ) f ( 2) were UTC = Coordinated Universal Time 0UT θ = Greenwic sidereal time at 0 UT DUT = Correction at te Coordinated Universal Time (UTC ) 165

3 ASTRONOMICAL AZIMUTH BY HOUR ANGLE OF POLARIS USING TOTAL STATIONS f = Ratio of te duration of te mean solar day to te mean sidereal day = Te astronomical azimut Α s of te star is calculated by te formula from te basic rules of sperical trigonometry applied to te astronomical triangle [8]: sin tanα s = cos Φ tan δ sin Φ cos ( 3) were δ = Declination of te star Φ = Astronomical latitude of te instrument station Polaris is te most suitable star for te application of tis metod. Te precision acieved will be analyzed in te following paragraps. PROPOSED METHODOLOGY An ordinary total station tat is to be used for astronomical observations must be equipped wit a built - in crystal quartz clock, of one second time resolution, a dual axis level sensor, illumination of te cross airs and, for convenience, a diagonal eyepiece. In tese total stations te levelling errors are largely eliminated as tey ave digital levelling sensors. Also for instruments wit dual axis level sensors, small levelling deviations are corrected instantly and automatically, so tat te instruments display te correct angle value in eac measurement. It must be cecked tat te level sensors and te angle correction system are switced on during te observations. Te reading and estimation errors of te observer, wic are te booking errors, are eliminated since te measurements are registered automatically. Consequently te only remaining error is te observer s sigting error tat may be reduced by repeated pointings to te target. Tus a ig angle measuring precision may be acieved. Tese instruments are easy to operate and te surveyors use tem for simple construction, land surveys and setting out tasks in general. For te determination of te astronomical azimut of a line by te our angle metod mentioned above, te time of eac observation must be recorded. Tis can be acieved by using a total station equipped wit a built in crystal quartz clock, wic permits te measurement and te registration of te time of eac observation, to one second resolution. Te main criterion, wic distinguises a good timekeeper from a poor one, is te stability of its oscillator. Modern tecnology provides frequency standards wit standard deviation of no more tan 1µs [11]. However, te point is to sycronize te instrument clock wit te UTC time wit a sufficient accuracy. Te UTC time results by adding or subtracting te integral number of ours of te zone time and te Dayligt Saving Time to or from te civil time at every place. Te sycronization may be done by: - Te conventional metod were te observer ears te broadcast radio time signals and sets te correct time in te instrument clock, as it was done wit te old portable cronometer used in geodetic astronomy. Tis syncronization creates an error of te order of ± 0.5 seconds, wic mainly depends on te skill and te experience of te observer. - Te connection of a GPS receiver wit te total station [1], [3], [4]. Some GPS receivers can provide accurate time information in te form of 1pps (pulse per second) output, wic is syncronised wit te UTC time to an accuracy of few microseconds [9]. Tis accurate procedure is not easy to apply, as it needs te appropriate software and instrumentation [3]. 166

4 E LAMBROU AND G PANTAZIS After Polaris as been observed, its astronomical azimut at te time of eac observation is calculated by equation (3). Te celestial coordinates of Polaris a,δ, 0UT wic are required as well as te θ and DUT values are available for eac day of te year on an appropriate website wit te digital astronomical epemeris [10]. Instead of te astronomical coordinates Φ, Λ of te instrument station, te geodetic coordinates φ, λ may be used in te equations (1) and (3). A map or a small and eld GPS receiver may provide tese coordinates. Terefore, one sets up te total station at point A and sigts to te desired point B. Repeatedly sigt to Polaris and register te measured orizontal angle b (Fig.1) and te UTC time. After tese measurements in face left (FL) of te instrument, repeated measurements to Polaris and one measurement to point B follow in face rigt (FR). Te astronomical azimut Α ΑΒ of te line AB will be derived, by te equation (4), by adding or subtracting (depending on te relative position of te points) te orizontal angle b between te direction AB and Polaris to or from te calculated astronomical azimut of Polaris A P via equation (3). = 360 b + Α ( 4) Α ΑΒ P Fig. 1. Te astronomical azimut A AB of te direction AB From eac sigting to Polaris a different azimut of te star A P will be calculated according to te observation time and a different orizontal angle b will be registered between te star and te eart s direction AB as te star is moving. So te sum A AB of te two above values will be always te same. Ten to fifteen sigtings are sufficient to eliminate te observer s sigting error. ERROR ANALYSIS Te total error of te determination of te astronomical azimut Α ΑΒ of a line AB contains: - Error of te measured angle b, wic includes: - Error of te sigting to te point B - Error of te sigtings to te star - Error of te declination δ of Polaris - Error of te used value of te astronomical latitude Φ of te instrument station - Error of te determination of te our angle Eac one of te above errors will be analyzed in te following paragraps. Sigting error to point B. An experimental investigation of several types of targets, using several total stations tat ave nominal precision of ± 0.5, ± 1, ± 3 and ± 7, was carried out to determine tis error. 167

5 ASTRONOMICAL AZIMUTH BY HOUR ANGLE OF POLARIS USING TOTAL STATIONS A luminous target, a cone target and a single prism were used as targets. Te single prism is te most commonly used target by te surveyors. Te targets were put at distances of 300 m and 100 m and repeated sigtings were carried out in four different measurement series. Te results proved tat te mean value of te sigting error is of te order of alf of te nominal direction measuring precision for eac total station. Te smallest sigting error was found in te luminous target at distance of 300 m. Error of te sigtings to te star. Tis can be reduced by repeated measurements. Ten to fifteen sigtings to te star may be carried out witin 5 to 10 minutes. Te standard deviation of te single azimut value determination is calculated by formula (5) [ υυ ] = ± ( 5) x ( n 1) and correspondingly te standard deviation of te mean azimut value is calculated by te following formula [8]. [ υυ] = ± ( 6) x n ( n 1) Te value of te declinationδ of Polaris is given by te digital astronomical epemeris wit a precision of ±0.01 [10]. Te derivation of equation (3) gives te following equation (7), wic presents te influence of te error inδ, to te error in te azimut determination. cos Φ sin Α δ = ± 2 2 δ (sin z cos Α ) (1 + sin ) s ( 7) Tis error takes a maximum value of alf te given precision of te coordinateδ, about ± Error of te used value of astronomical latitude Φ. In equation (3) te astronomical latitude Φ of te instrument station must be used. Since it is quite difficult to determine tis te geodetic coordinate φ may be used instead. Te difference of te two types of coordinates depends on te fitting of te used ellipsoid to te geoid at te station point. Te value of te difference fluctuates between 5-20 in most areas. Te differentiation of equation (3) leads to equation ( 8) tat calculates te error in te azimut caused by te uncertainty of te Φ value. = ± sin A cot z ( 8) AΦ Figure 2 depicts te error relative to te error Α Φ s Φ in te calculated Polaris astronomical azimut Φ in te used value of Φ, and to te astronomical latitude Φ of te instrument station. According to tis diagram, an error of 15 in te Φ value causes error of ±0. 2 in te determination of te astronomical azimut if te observation point is located at te latitude of

6 E LAMBROU AND G PANTAZIS Α (arcseconds) Φ '' Φ 25'' 20'' 15'' 10'' 0.2 5'' Astronomical Latitude Φ ( o ) Fig. 2. Diagram of error Α Φ Error in te determination of te our angle. Anoter differentiation of equation (3) gives equation (9) tat determines te error in te azimut caused by te total uncertainty in te value. A ± cos Φ (tan Φ cos As cot z) = ( 9) Te our angle is determined by equation (1). Terefore, te total error in te our angle value, is calculated by equation (10): = Λ ( 10) 2 2 ± θ + + θ depends on te error of te measured UTC time. Te measurement of te UTC time includes: - Te syncronization error of te instrument s clock to te UTC time, wic is about ±0.5 s. If te UTC time information is provided by a GPS receiver connected to a ig accuracy total station ten te accuracy of te time information is of te order of ±1 ms. - Te observer delay. Te personal delay of te observer is te time interval between te moment tat te observer sigts te star in te centre of te reticule s cross-airs to te moment tat e puses te key of te instrument display to record te measurement. Tis time delay is different for eac observer and depends mainly on is pysical caracteristics and is reflexes. Te magnitude of tis delay must be subtracted from te UTC time registered at eac measurement. It can be determined only experimentally as follows. A GPS receiver, wic can provide accurate UTC time information in te form of 1pps output (e.g. Trimble 4000 DL), is connected to a total station (e.g. Leica TDM 5000) by te appropriate ardware and software (Fig.3) [3]. Eac time te observer takes a measurement, te UTC time can be registered by te instrument wit 1ms precision. Simultaneously te GPS receiver is connected by te appropriate cable wit a ligt emitting diode (LED) tat flases every 2 α 169

7 ASTRONOMICAL AZIMUTH BY HOUR ANGLE OF POLARIS USING TOTAL STATIONS integral second as te GPS receiver emits a pulse per second. Te observer sees te LED on te instrument s reticule and takes measurements wen te LED is flasing. As te LED flases exactly every UTC integral second and te instrument registers te time tat te observer puses te key to take te measurement, te difference between te two times is te personal delay of te observer. Tis time interval was calculated for different observers, according to te above experiment and was found to be about 0.3 s. Tis time error causes an almost insignificant error in te azimut determination as illustrated in te following diagram (Fig.4) and may be ignored in Polaris observations. It may be mentioned tat most instruments record te values of angles and time simultaneously. If an electronic delay of some µs occurs, it is similarly insignificant. LED TOTAL STATION CIRCUIT TRIGGER PPS OUTPUT GPS RECEIVER ANTENNA GPS RS 232 PPS INPUT 12 V BATERY 12 V Fig. 3. Te instrument s connection for te determination of te observer s delay [3] Te error Λ in te astronomical longitude Λ is about ±5 to ±20. Tis is because te geodetic value of λ, is used in lieu of te astronomical value of Λ. α is te error in te rigt ascension a of Polaris tat is provided wit a precision of ±0.01 [10]. Α (arcseconds) '' '' '' '' '' '' Astronomical Latitude Φ ( o ) Fig. 4. Diagram of error Α 170

8 E LAMBROU AND G PANTAZIS Figure 4 illustrates, according equation (9), te error A in te determined Polaris astronomical azimut, relative to te total error of te calculated our angle according to equation (1) and te astronomical latitude Φ of te observation place. According to equation (10) and taking = ±0.8 s = ±12, Λ = ±15 and α = ±0.01 te total error in te determination of te our angle is = ±1.3 s = ±20. At latitude of 40, a negligible error of ± is caused in te determined Polaris astronomical azimut. EXPERIMENT A practical test was carried out on a pillar of te test field of te National Tecnical University of Atens. Te astronomical coordinates Φ, Λ of te pillar are known. Two different ordinary total stations (Leica TCR303 and Leica TC307) were used for te determination of te astronomical azimut of te line connecting te pillar to an illuminated target at a distance of 300 m. Tese instruments ave all te required features tat were defined previously and are widely used in surveying fieldwork. An observation set consisted of one sigting to te target (FL), 15 sigtings to Polaris (FL), 15 sigtings to Polaris (FR) and one sigting to te target (FR). Te total stations were selected for use after aving performed all tests tat ceck teir proper function. Eac measurement set wit one total station lasted about 10 minutes. Table 1 illustrates te results. Table 1. Test results θ Total Station Date Nominal Direction measuring precision Time measuring precision Time syncronization by: Mean value of Astronomical Azimut Standard deviation of te Mean value of te Astronomical Azimut Leica TCR303 Leica TC307 25/7/ s Te observer ±2.0 28/7/ s Te observer ±3.9 DISCUSSION 1. Te determination of te astronomical azimut of a line today, is quick and easy to perform by using ordinary total stations. 2. Observing a close circumpolar star, as Polaris, at any our angle is very convenient. Polaris, is an easily recognisable star on te celestial spere of te Eart s Nortern emispere. 3. Ten-minutes of fieldwork is sufficient for tis determination. 4. Te built in quartz clock in most modern total stations elps wit automatic registration of te UTC time of eac observation. 5. Te synconization of te instrument built in quartz clock to te UTC time is not a difficult task as an accuracy of ±1 s in te syncronization is easily acieved. 6. For te astronomical azimut determination by te our angle metod using Polaris observations te need for ig time accuracy is not essential as it causes very small errors in te final calculated azimut value. Even a total error of four seconds in te 171

9 ASTRONOMICAL AZIMUTH BY HOUR ANGLE OF POLARIS USING TOTAL STATIONS our angle used causes a maximum error of ±0.01 in te finally determined astronomical azimut value. 7. Te precision tat may be acieved mainly depends on te total station tat will be used. Typically, it is alf of te nominal direction measuring precision of te instrument. CONCLUSIONS Te use of modern ordinary total stations leads to fast, easy and accurate determination of te astronomical azimut A of a given line. Tis is acieved by: - Calculating te astronomical azimut of te star by a simple formula witout any computer programming needed. - Reducing te random error of te observer by repeated sigtings to te star. - Registering directly te UTC time, as te built in total station clock may be syncronized to it. - Recording automatically and digitally all measured values (angle and time) eliminates reading errors and decreases te data acquisition time needed. Using an ordinary total station, by sigting to Polaris via te our angle metod, te astronomical azimut of a given line can be determined in ten minutes. Tis determination is independent of any national geodetic reference system and te uncertainties or biases tat publised pillar coordinates usually ave. Te previously mentioned advantages, te easy operation, te general availability of modern digital total stations and te simple calculations make te proposed metodology efficient, easy and convenient to use by all surveyors in many land survey applications. References 1. Balodimos, D. D., Korakitis R., Lambrou E. and Pantazis, G., Fast and Accurate Determination of Astronomical Coordinates Φ, Λ and Azimut, Using a Total Station and GPS Receiver, Survey Review, Vol. 37, No. 290, p Bomford G., Geodesy, 3rd edition, Oxford University Press, UK. 3. Lambrou, E., Development of a Metodology for Astrogeodetic Determinations, Using Digital Geodetic Instruments, NTUA, Scool of Rural and Surveying Engineering (In Greek), PD Tesis. 4. Lambrou, E., and Pantazis, G., Accurate Orientation of te Gyroscope s Calibration System, oral presentation at te FIG Working Week 2004, May 22-27, Atens, Greece (Digital Proceedings in CD). 5. Leica Heerbrugg AG, Users Manual for TM 5000/ TDM 5000 System, V2.2, Switzerland 6. Leica Heerbrugg AG, Users Manual for TCR 303, Switzerland. 7. Mackie, J. B., Te Elements of Astronomy for Surveyors, Sevent edition, Carles Griffin & Company Ltd, London. 8. Mueller, I., Sperical and Practical Astronomy as Applied to Geodesy, Frederick Ungar Publising Co, New York, U.S.A. 9. Trimble Navigation, Operation Manual for Model 4000DL, Revision A, Sunnyvale, California, U.S.A. 10. ttp:// viewed in ttp:// viewed in