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1 MOVEMENTS OF THE EARTH (Apparent movement of the sun, time determination and geographic coordinates) Teaching Team: Prof. Alfonso Calera Belmonte (Dpt.

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Presentation on theme: "1 MOVEMENTS OF THE EARTH (Apparent movement of the sun, time determination and geographic coordinates) Teaching Team: Prof. Alfonso Calera Belmonte (Dpt."— Presentation transcript:

1 1 MOVEMENTS OF THE EARTH (Apparent movement of the sun, time determination and geographic coordinates) Teaching Team: Prof. Alfonso Calera Belmonte (Dpt. Applied Physics, UCLM) Prof. Antonio J. Barbero (Dpt. Applied Physics, UCLM) Consultant: Prof. Kathy Walsh (Dpt. Modern Languages, UCLM) PhysicsPhysics EnvironmentalEnvironmental LESSON 1

2 2 PhysicsPhysics EnvironmentalEnvironmental CELESTIAL SPHERE Celestial sphere: fictitious sphere of arbitrary radius, whose center is the observer’s eye. The positions of the planets and the stars are projected onto it. So, we can measure planet and star positions independently of their distance, using angle units over maximum circles defined over the sphere. 1 2 Real positions Minor circle: It is a circle defined by the intersection of the sphere with a plane dividing it into two non-equal parts. Maximum circle: It is a circle defined by the intersection of the sphere with a plane dividing it into two equal parts.

3 3 REFERENCES IN THE CELESTIAL SPHERE N S Pole Line (axis of the world) Celestial Equator Maximum circle perpendicular to the axis of the world Meridians Maximum circles perpendicular to equator Spin of the Earth Spin of the celestial sphere Right hand rule PhysicsPhysics EnvironmentalEnvironmental

4 4 N S GEOGRAPHIC COORDINATES: LATITUDE Shape of the Earth: very similar to a sphere squashed at the poles and convex at the equator. This shape is called ‘geoide’. Equatorial diameter: 12,756 Km. Equatorial length: 40,075 Km. PARALLEL: Minor circle determined by cutting the sphere with a plane parallel to the equator. LATITUDE of a point: It is the angle subtended from the center of the Earth by a radius directed to the point and another radius directed to that point on the equator located on the same meridian (  in the figure). Latitude  is measured in degrees: 0º (equator) to  90º (north/south pole) All points on the same parallel have the same latitude.   is the latitude of this point PhysicsPhysics EnvironmentalEnvironmental

5 5 N S GEOGRAPHIC COORDINATES: LONGITUDE L MERIDIAN: Any maximum circle passing throught the poles. LONGITUDE of a point: It is the angle between the plane of a particular meridian and the plane of another meridian taken as reference. Longitude L is measured in degrees, from 0º until 180º, either to the East (E) or to the West (W) from the meridian of reference. Meridian lenght: 40,008 Km. Equatorial diameter: 12,756 Km. Equatorial length: 40,075 Km. Meridian of reference L is the longitude of all those points... really L is the longitude of any point lying on that meridian! PhysicsPhysics EnvironmentalEnvironmental

6 6 ORBIT OF THE EARTH: CHARACTERISTICS 1º) The orbit of the Earth around the Sun is a slightly eccentic ellipse. The Sun lies on one of its focal points. Because of this, the apparent movement of the Sun around the Earth varies throughout the year: the Sun seems to move faster whenever the Earth is closer to it. The mean distance Earth-Sun is called astronomical unit (1 A.U.  149.5 Mkm) 2º) The time the Earth takes to complete one orbit around the Sun is 365.25 days (one year). The Earth spins once around its own axis every 24 hours (one day). 3º) The Earth’s equator plane is not the same as the Earth’s orbit plane around the Sun: both planes are tilted at an angle of 23º 27’ (ecliptic obliquity). 23º 27’ PhysicsPhysics EnvironmentalEnvironmental

7 7 September 21/22 Fall equinox  = 0 December 21/22 Winter solstice  = -23º 27’ March 20/21 Spring equinox  = 0 June 21/22 Summer solstice  = 23º 27’ 0.983 U.A. 1.017 U.A. 1 U.A. April 4 January 3 PERIHELION July 4 APHELI ON October 5 Ecliptic plane 1 A.U.. = (149597890±500) km  1.496  10 8 km 23º 27’ THE ORBIT OF THE EARTH: THE SEASONS PhysicsPhysics EnvironmentalEnvironmental

8 8 EARTH-SUN DISTANCE Daily angle (radians) Spencer Formula Duffie y Beckman Formula Eccentricity factor J = day of the year (J = 1.. 365) r 0 = 1 U.A. Inverse relative distance PhysicsPhysics EnvironmentalEnvironmental

9 9 Earth-Sun distance PhysicsPhysics EnvironmentalEnvironmental

10 10 EARTH-SUN DISTANCE: XY representation Calculus from Duffie-Beckman formula PhysicsPhysics EnvironmentalEnvironmental

11 11 Declination angle 23º27’ Spring equinox Celestial south pole Celestial north pole Fall equinox Apparent path of the Sun on the ecliptic plane Winter solstice Summer solstice Celestial equator plane Declination angle  YEARLY APPARENT MOVEMENT OF THE SUN PhysicsPhysics EnvironmentalEnvironmental

12 12 23º 27’ SOLSTICES SUMMER WINTER PhysicsPhysics EnvironmentalEnvironmental

13 13 Daily angle (radians)  (degrees) On equinoxes  = 0 On the summer solstice  = +23º27’ On the winter solstice  = -23º27’ Spencer formula for declination PhysicsPhysics EnvironmentalEnvironmental

14 14  (degrees) Day of the year Spencer Crop Evapotranspiration Declination formula (Crop Evapotranspiration/FAO) http://www.fao.org/docrep/X0490E/x0490e00.htm PhysicsPhysics EnvironmentalEnvironmental

15 15 Spencer Crop Evap. COMPARING RESULTS FOR DECLINATION FROM SPENCER FORMULA AND CROP FORMULA Summer solstice Winter solstice Fall equinox Spring equinox Number of day of the year  (degrees) PhysicsPhysics EnvironmentalEnvironmental

16 16   horizon Celestial north pole 90-  Celestial equator Observer in Northern hemisphere CELESTIAL EQUATOR AND CELESTIAL NORTH POLE  latitude PhysicsPhysics EnvironmentalEnvironmental

17 17 S N E W Celestial north pole Zenit  90-   CELESTIAL EQUATOR AND CELESTIAL NORTH POLE (II) Observer in northern hemisphere PhysicsPhysics EnvironmentalEnvironmental

18 18 Celestial NP Polar CIRCUMPOLAR STARS PhysicsPhysics EnvironmentalEnvironmental

19 19 N S E W Celestial south pole Zenit 90-    CELESTIAL EQUATOR AND CELESTIAL SOUTH POLE Observer in southern hemisphere PhysicsPhysics EnvironmentalEnvironmental

20 20 S N E W Celestial north pole Zenit APPARENT PATH OF THE SUN IN THE NORTHERN HEMISPHERE SKY Summer solstice Tropic of Capricorn Tropic of Cancer Celestial equator  Winter solstice 23º 27’ -23º 27’ Equinoxes PhysicsPhysics EnvironmentalEnvironmental

21 21 S N E W Celestial north pole Zenit Season: spring/summer  Observer in northern hemisphere   Any day  declination  latitude APPARENT PATH OF THE SUN PhysicsPhysics EnvironmentalEnvironmental

22 22 Celestial north pole Season: spring/summer  Observer in northern hemisphere  POSITION OF THE SUN RELATED TO A HORIZONTAL SURFACE Zenit S N E W    zz  z zenith angle  solar altitude  solar azimut  declination  latitude   Hour angle 15º/hour COORDINATES measured from the center of sun disc PhysicsPhysics EnvironmentalEnvironmental

23 23 Celestial north pole Season Spring / Summer  Observer in northern hemisphere  Zenit S N E W   max  declination  latitude  = 0 MAXIMUM SOLAR ALTITUDE PhysicsPhysics EnvironmentalEnvironmental

24 24 Celestial north pole Season Spring / Summer  Observer in northern hemisphere  HOUR ANGLE AT THE SUNRISE Zenit S N E W   = 0  z zenith angle  solar altitude  solar azimut  declination  latitude ss  z = 90º   s Hour angle at the sunrise PhysicsPhysics EnvironmentalEnvironmental

25 25  z zenith angle  solar altitude  solar azimut  s hour angle at the sunrise  hour angle It varies from 0º (horizon) to 90º (zenit) It varies from 0º (cénit) to 90º (horizonte) It varies from 0º (south) to 180º (north). Sign: + towards E, - towards W It varies 0º (Sun on the meridian) to a value dependent on the day of the year and on the latitude. Sign: + before noon, - after noon Value dependent on the day of the year and on the latitude. SIGN CRITERION PhysicsPhysics EnvironmentalEnvironmental

26 26 RELATIONSHIPS BETWEEN THE POSITION ANGLES Zenital angle / solar elevation vs declination, latitude and hour angle Azimut angle vs solar elevation, declination and latitude Hour angle at the sunrise vs declination and latitude What rate does the hour angle vary? PhysicsPhysics EnvironmentalEnvironmental

27 27 SOLAR DAY Solar day is the time interval the Sun takes to verify a complete revolution around a stationary observer lying on the Earth. THIS INTERVAL IS NOT NECESSARY A 24-HOURS INTERVAL! An observer located on the northern hemisphere is looking at the south and turns on a clock which goes on uniformly when the Sun lies directly on the local meridian (then it’s noon!). The duration of the solar day varies throughout the year for the two main following reasons: When moving in the ecliptic plane, the Earth sweeps different areas at different dates because its velocity varies depending on the distance Earth-Sun. The axis of the Earth is tilted a constant angle onto the ecliptic plane. That observer may find 24 hours later that... the Sun does not lie on the meridian. Maybe the Sun has already passed the meridian, maybe the Sun has not yet reached the meridian. It depends on the day of the year! PhysicsPhysics EnvironmentalEnvironmental

28 28 MEAN SOLAR DAY Mean Solar Day is the average of the solar days and corresponds to a fictitious sun moving on the equatorial plane, whose apparent movement around the Earth have a constant orbital velocity. ALL MEAN SOLAR DAYS HAVE THE SAME DURATION Cenit S N E W Celestial equator PhysicsPhysics EnvironmentalEnvironmental

29 29 TIME EQUATION The disagreement between the movement of the mean sun (fully homogeneous, with 24-hours intervals for every two next passes across the local meridian) and the apparent movement of the real Sun, is taken into account for calculus by defining the TIME EQUATION. The time equation reaches its maximum value (about 16 minutes) in november and its minimum in february (about 14 minutes). SPENCER FORMULA FOR TIME EQUATION Daily angle J number of the day of the year We can obtain data for each day of the year from this one or some similar formula TIME EQUATION = SOLAR MEAN TIME - SOLAR APPARENT TIME 0 to 365, or 0 to 366 for leap year PhysicsPhysics EnvironmentalEnvironmental

30 30 http://averroes.cec.junta-andalucia.es/ies_gaviota/ fisiqui/relojsol/horas.htm TIME EQUATION: GRAPHICS TIME EQUATION = SOLAR MEAN TIME - SOLAR APPARENT TIME PhysicsPhysics EnvironmentalEnvironmental

31 31 DETERMINATION OF TIME: GMT GMT = Greenwich Mean Time www.greenwichmeantime.com 0º GMT is the Greenwich time according to the fictitious movement of the mean sun. It counts from midnight, when the mean sun passes across the lower Greenwich meridian. When the mean sun passes across the upper Greenwich meridian, it is noon: GMT = 12:00:00 PhysicsPhysics EnvironmentalEnvironmental

32 32 DETERMINATION OF TIME: UNIVERSAL TIME UT = Universal Time UTC = Universal Time Coordinated Universal time coordinated (UTC) is GMT updated by adding additional seconds (“leap seconds”) to having in mind the lack of uniformity in the rotation of the Earth. UTC means the average value from a certain number of measurements made using different atomic clocks around the world. In aviation UTC is called ZULU time. http://www.its.bldrdoc.gov/fs-1037/dir-009/_1277.htm Definition UTC http://www.hyperdictionary.com/search.aspx?Dict=&define=UTC&search.x=32&search.y=10 UT measurements are based on the standard second. Actual definition for a second (1967): a second equals to 9 192 631770 periods of the radiation from a particular transition between two hiperfine levels of the ground state of cesium 133. PhysicsPhysics EnvironmentalEnvironmental

33 33 DETERMINACIÓN OF TIME: GREENWICH OBSERVATORY PhysicsPhysics EnvironmentalEnvironmental

34 34 DETERMINATION OF TIME: GMT y UTC Zenit S N E W UTC = 00:00:00 GMT = 00:00:00 UTC = 12:00:00 GMT = 12:00:00 PhysicsPhysics EnvironmentalEnvironmental

35 35 PhysicsPhysics EnvironmentalEnvironmental DETERMINATION OF TIME: LOCAL APPARENT TIME (LAT) HORA SOLAR LOCAL (HSL) / LOCAL APPARENT TIME (LAT) It refers to the position of the Sun from the local meridian. S N E W Cenit  HSL = 12:00:00  = 30º Example: HSL = 10:00:00  = 0º Sun apparent movement:

36 36 DETERMINATION OF TIME: LOCAL STANDARD TIME (LST) HORA SOLAR ESTÁNDAR (HSE) / LOCAL STANDARD TIME (LST) It refers to the standard meridian time (taken as a reference) on each point of a particular zone. All standard meridians are multiple of 15º either E or W from Greenwich. http://stj.chihuahua.gob.mx/asamblea/horarios.htm PhysicsPhysics EnvironmentalEnvironmental

37 37 Relationship between local apparent time (LAT, HSL) and local standard time (LST, HSE) LST = LAT - 4·(L s -L e ) - E t Longitude correction Sun apparent movement 15 degrees / hour 4 min / degree L s, L e >0 towards W <0 towards E Degrees Longitude correction in minutes Time equation (minutes) L s Standard meridian longitude L e Local meridian longitude LAT = LST + 4·(L s -L e ) + E t DETERMINATION OF TIME: LOCAL STANDARD TIME (LST) (II) PhysicsPhysics EnvironmentalEnvironmental

38 38 DETERMINATION OF TIME: LOCAL STANDARD TIME (EXEMPLE) The local standard time is the same for all points in the same time zone.... but the local apparent time is NOT the same! Each point has yours. www.greenwichmeantime.com 0º 1º52’ LeLe LsLs Find LST in Albacete when it is 12:00:00 LAT on January 1 st. (Longitude of Albacete 1º52’ W) Jan 1 st E t = -2.90 min Longitud correction -7.47 min LST = LAT - 4·(L s -L e ) - E t L s, L e >0 towards W <0 towards E 4·(-1.867) =- 7.47 min = -7 min 28 s 1º52’ = 1.87º LST = 12:00:00 -(-7.47) -(-2.90) LST = 12:00:00 +10.37 min = = 12:10:23 GreenwichGreenwich AlbaceteAlbacete PhysicsPhysics EnvironmentalEnvironmental

39 39 Find the local apparent time at 10:00:00 h LST on the 16th October in a city where the longitude is 58º 29’ W. LAT = LST + 4·(L s -L e ) + E t 4·(60.00-58.48) = 6.08 min 16th Oct E t = +14.62 min = 10:00:00+ 6.08 + 14.62 = 10 h + 20.70 min 10 h + 20.70 min = 10:20:42 Reference meridian DETERMINATION OF TIME: LOCAL APPARENT TIME (EXEMPLE 2) PhysicsPhysics EnvironmentalEnvironmental

40 40 Legal time is the time corresponding to a reference meridian on each time zone (on a general sense, it is the time corresponding to a certain time zone). DETERMINATION OF TIME: OFFICIAL TIME Official time is the time established by the government. It can differ from the legal time by an enter number having in mind criteria of energetic sparing (it is usual having different times on winter and on summer). Spain belongs to the center european time zone.. Winter time: OFFICIAL TIME = LEGAL TIME = GMT + 1 Summer time: OFFICIAL TIME = LEGAL TIME + 1 = GMT + 2 DETERMINATION OF TIME: LEGAL TIME http://nist.time.gov/ PhysicsPhysics EnvironmentalEnvironmental

41 41 GEOGRAPHIC POSITION: LATITUDE DETERMINATION For determining the latitude we must know the altitude over the horizon of some fixed reference. We will see two of them: 1º) SOLAR ALTITUDE WHEN THE SUN IS CROSSING THE LOCAL MERIDIAN Zenit S N E W PhysicsPhysics EnvironmentalEnvironmental

42 42 2º) POLAR STAR ALTITUDE: THIS IS A DIRECT MEASUREMENT OF LATITUDE Application: at night, only on the northern hemisphere Zenit S N E W Celestial North Pole GEOGRAPHIC POSITION: LATITUDE DETERMINATION (II) PhysicsPhysics EnvironmentalEnvironmental

43 43 To determine the longitude of a point we must know simultaneously LAT on that point and LST on some reference meridian, in orden to obtain L e from the equation: LAT = LST + 4(L s -L e ) + E t LAT measurement on the point LST on L s meridian Having in mind the daily correction GEOGRAPHIC POSITION: LONGITUDE DETERMINATION PhysicsPhysics EnvironmentalEnvironmental

44 44 LAT = LST + 4(L s -L e ) + E t 4 (L s -L e ) = LAT – LST – E t degrees minutes minutes/degree minutes HSL – HSE – E t = 4 (L s -L e ) =  L  L > 0 LsLs W E LeLe  L < 0 LsLs W E LeLe GEOGRAPHIC POSITION: LONGITUDE DETERMINATION. SIGNS The sun spends 4 minutes to go over a degree PhysicsPhysics EnvironmentalEnvironmental

45 45 On July 28th the sun passes across the local meridian at 12:13 LST. Find the longitude of that point respect to the reference meridian. Whenever the sun passes across the meridian it is 12:00 LAT Time equation on July 28th: E t = –6.60 m (-6 m 36 s) LAT – LST – E t = 4 (L s -L e ) =  L Difference LAT-LST = -13 m 1 4 = (-13 – (-6.60)) (-6.4) 1 4 =  L < 0 LsLs W E LeLe 1º 36’ (Ls-Le) =  L (Ls-Le) =  L = -1.6º = -1º36’ Le = Ls + 1º36’ GEOGRAPHIC POSITION: LONGITUDE DETERMINATION. EXEMPLE PhysicsPhysics EnvironmentalEnvironmental

46 46 S N E W Cenit THE LENGTH OF THE DAY (SUNRISE AND SUNSET TIME) ss LAT = 12:00:00  = 0º The time (in hours) the sun takes to reach the local meridian is Lenght of the day Lack of some corrections SUNRISE SUNSET (because the sun goes over 15º/hour in its path on the sky) The maximum of hours of sun for a day is twice PhysicsPhysics EnvironmentalEnvironmental

47 47 PhysicsPhysics EnvironmentalEnvironmental -16’ SOLAR ALTITUDE ANGLE CORRECTION ON SUNRISE AND SUNSET  = 0 -16’ -34’ -50’ Correction  3-5 minutos Sunrise ahead of time Sunset behind time Solar altitude (solar disc center) I. CORRECTION BY ATMOSPHERIC REFRACTION

48 48 II. Correction by variation of declination Throughout a day the apparent movement of the sun goes on, so its declination varies continuously. As a consequence, the declination is not the same at sunrise than at sunset. So, the lenght of a day is not exactly 2  s /15 hours. Associated variation  1 minute III. OPTICAL EFFECTS BY THERMAL INVERSIONS http://www.astrored.org/usuarios/xgarciaf/orto1.htm SOLAR ALTITUDE ANGLE CORRECTION ON SUNRISE AND SUNSET (II) PhysicsPhysics EnvironmentalEnvironmental

49 49 BIBLIOGRAFÍA y DOCUMENTACIÓN M. Iqbal, An Introduction to Solar Radiation, Academic Press (1983) Main text: Yearbooks and tables. Sunrise and sunset time. Observatorio astronómico nacional Horas de salida y puesta de Sol en capitales provincia España http://www.oan.es/servicios/agenda/2003/index.html U.S. Naval Observatory Horas de salida y puesta de Sol en coordenadas cualesquiera http://aa.usno.navy.mil/data/docs/RS_OneYear.html#formb Royal Greenwich bservatory http://greenwichmeantime.com/ PhysicsPhysics EnvironmentalEnvironmental

50 50 BIBLIOGRAFÍA y DOCUMENTACIÓN (II) http://www.astrored.org/usuarios/xgarciaf/orto1.htm Sunrise and sunset corrections http://www.infoplease.com/ce6/society/A0850108.html Glossary and definitions (English) http://www.sunlitdesign.com/infosearch/hourangle.htm?indexref=3 http://www.sundialsoc.org.uk/glossary/frameset.htm http://www.rediris.es/red/zona_horaria.es.html Spain hour zones See also quotations on the text. http://rubens.anu.edu.au/student.projects97/naval/home.htmhttp://rubens.anu.edu.au/student.projects97/naval/home.htm (no longer available) The problem of the longitude W J H Andrewes, “Crónica de la medición del tiempo”, Investigación y Ciencia, nov 2002 PhysicsPhysics EnvironmentalEnvironmental


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