1 movements of the earth (apparent movement of the sun, time determination and geographic...

1

MOVEMENTS OF THE EARTH(Apparent movement of the sun,

time determination and geographic coordinates)

PROBLEM SOLUTIONS

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)

Physics

Environmental

LESSON 1

2

Physics

Environmental

Eratóstenes of Cirene (284-192 B.C), a greek astronomer, geographer, mathematician and philosopher, was the first wise man who got an accurate measure of the circumference of the Earth. His procedure to do that based on the following: he knew that at noon of the summer solstice day the rays of the sun dropped vertically on the bottom of a well in the ancient city of Siena (near the actual Assuan, Egipt), located almost exactly on the Cancer tropic. Using a gnomon* he could measure the inclination angle of the solar rays in the city of Alexandria, the same day at the same time. Alexandría was located about 800 km to the North from Siena (Eratostenes was director of the famous Library of Alexandría). The angle measured by Eratostenes was 7º14’.From those data, determine the Earth circumference (or its radious).

(At the Eratostenes time, the most difficult data to obtain was an accurate value for the distance between the two cities).

PROB. 0101 / MEASURING THE EARTH RADIUS

http://www.astromia.com/biografias/eratostenes.htm

* Gnomon: a vertical stick over a flat platform, by measuring the length of the shadow it is possible calculating the incident angle of the rays of the sun.

3

Physics

Environmental

Siena

Alexandria

TROPIC OF CANCER

0

0

d

R

36º

24º

28º

32º

http://www.lib.utexas.edu/maps/africa/egypt_pol97.jpg

dR

= 7º14’ = 7.23º d 800 km

d

R km6340

18023.7

800

km398342 R

PROB. 0101 / MEASURING THE EARTH RADIUS (CONTINUED)

4

The geographic coordinates of Palma de Mallorca are 39º34’ N, 2º39’ E and those of Edinburgh are 55º57’, N 3º10’ WCompare the following quantities for the days December 4th and June 4th in both cities:A. The official times of sunrise and sunset.B. The lenght of the day.C. Solar elevation and azimut at noon (official time). D. What time (official) is the sun crossing the local meridian? You have to have in mind the fact that in Spain the official winter time and the official summer time are forwarded respect GMT by 1 hour and 2 hours, respectively. In Britain, the official summer time is forwarded by 1 hour, but the official winter time has no forward respect GMT.

PROB. 0102 / TIME AND DAY LENGTH

Physics

Environmental

5

ΦΦcoscos

Φcos tantan

sinsins

Sunrise hour angle:

33611.0)º57.39()(-22.14ºtΦcos tanantantans º36.70s

Declinations: º14.22

June 4th (J=155) º34.22

December 4th (J=338)

Palma Mallorca º57.39'34º39

Edinburgh

º95.55'57º55

(Dec 4th)

33972.0)º57.39()(22.34ºtΦcos tanantantans º86.109s (Jun 4th)

60209.0)º95.55()(22.34ºtΦcos tanantantans º98.52s (Dec 4th)

60821.0)º95.55()(22.34ºtΦcos tanantantans º46.127s (Jun 4th)

PROB. 0102 / TIME AND DAY LENGTH (CONTINUED) PART A)

Physics

Environmental

6

horas

/º15)(º

00:00:12 (LAT) time Sunrise

Local apparent time at sunrise:


Physics

Environmental

7

Sunrise time (LST) LST = LAT - 4(Ls-Le) - Et

Palma M.4-dic, Calculus. Exemple

LST = 7:18:35 - 4(0-(-2.65º)) - 9.59

Converting longitude degrees to minutes

Time equation of the day

Local longitude (decimal fraction of a degree)

LAT in minutes

Standard meridian longitude (in this case, Greenwich)

= 438.56 min -10.60 min -9.59 min = 418.39 min

= 6.973 h = 6:58:24


Physics

Environmental

8

Sunrise time (LST)

LST = LAT - 4(Ls-Le) - Et

Winter (Dec 4th): Official time = LST + 1 = 7h 58m 23s

Summer (Jun 4th): Official time = LST + 2 = 6h 27m 53s

Palma de Mallorca:

Official time:

Edinburgh:

Winter (Dec 4th): Official time = LST = 8h 31m 09s

Summer (Jun 4th): Official time = LST + 1 = 4h 40m 43s


Physics

Environmental

9

horas

/º15

)(º2day theofLength

52.98

127.46

4-dic

4-jun

º sEdinburgh

4-dic

4-jun

º sPalma M.

70.36

109.86

Length of the day.

PROB. 0102 / TIME AND DAY LENGTH (CONTINUED) PART B)

Physics

Environmental

10

zsinsinsin coscosΦcoscosΦ

Φcoscos

ΦΨcos

sinsinsin

Solar elevation

Azimut

Official time LST

Palma M., 4-dic 12:00:00 11:00:00

Palma M., 4-jun 12:00:00 10:00:00

Edimburgo, 4-jun 12:00:00 11:00:00

Edimburgo, 4-dic 12:00:00 12:00:00

LAT

11 56 56

10 49 26

11 20 12

10 12 42

h m s

PROB. 0102 / TIME AND DAY LENGTH (CONTINUED) PART C)

LAT = LST + 4·(Ls-Le) + Et

Physics

Environmental

11

)12(15 HSL

Hour angle determination (degrees) from LAT

Palma M., Dec 4th

Palma M., Jun 4th

Edinburgh, Jun 4th

Edinburgh, Dec 4th 0.8

17.6

)(º

10.0

26.8

11.91 0.73

53.95 28.44

27.60 10.41

61.46 60.86

)(º )(º

zsinsinsin coscosΦcoscosΦ

Φcoscos

ΦΨcos

sinsinsin

Solar elevation

Azimut

PROB. 0102 / TIME AND DAY LENGTH (CONTINUED) PART C)

Physics

Environmental

12

Determination of the official time when the sun passes across the local meridian. We add 1 or 2 hours to LST corresponding to LAT = 12:00

Palma M., Dec 4th

Palma M., Dec 4th

Edinburgh, Jun 4th

Edinburgh, Dec 4th 11 50 24

11 57 54

11 50 24

11 57 54

h m s

11:50:24

12:57:54

12:50:24

13:57:54

LST Official time

The sun is passing through the local meridian

PROB. 0102 / TIME AND DAY LENGTH (CONTINUED) PART D)

Physics

Environmental

13

Physics

Environmental

On the evening of the 15th April Her Majesty’ agent 007, James Bond, is kidnapped in London by enemy agents from a secret organization attempting to swap him with one of their own main ringleaders, who is getting a holiday paid by the government as a convict in a britain prison.Bond is inmediately carried by plane out of the country and that night he is locked up in an secret hiding-place abroad. However before the sunrise, a few hours later, our clever agent 007 escapes and finds a safe refuge in the tower of a church from which he can watch over an extensive area around him.Once he’s convinced to have eluded his kidnappers, Bond waits for the sunrise and when the sun is just rising in the horizon he uses the Polar star as the north point reference and, using two straight sticks and his splendid analogical watch (the stupid kidnappers did not take it away, they didn’t notice that these watch was also a pocket computer), he gets a measure of the angle between the rising sun and the north direction. The result of that measure is 70º towards the East. After that, he catches a carrier pigeon from the pigeon loft of the tower, and uses some old wooden boards he finds there to built a cage to keeping the animal. Furthermore, he uses a flat board and another stick to build a simple ‘gnomon’ (see next page picture).

PROB. 0103 / DETERMINATION OF LATITUDE AND LONGITUDE

70º

Polar star

14

Physics

Environmental

When the sun is about to reach its maximum elevation, Bond watches carefully the shadow of the stick and writes down the time of the tower clock at the moment when the shadow reaches its minimum length. This clock indicates 11:44 h, whereas on his own watch he can see 9:44 h. Using these measurements along with a table he gets from the memory of his wonderful watch, and after using also the calculator that of course it has, Bond pulls out a sheet from his notebook, writes on it the geographical coordinates of the place where he actually stands and a brief note addressed to the local government of the country where he actually is, asking for a rescue helicopter. Straight away, he ropes the piece of paper to the leg of the carrier pigeon and releases it. Then, he sits calmly down and waits for the rescue.

Which are the geographic coordinates? The government of which country has he asked for rescue?

Gnomon

Minimum lenght

PROB. 0103 / DETERMINATION OF LATITUDE AND LONGITUDE (CONTINUED)

At this moment, the tower clock indicates 11:44 and Bond’s watch indicates

9:44

15

Physics

Environmental

W

E

S N

= 110º

70º

Sun crossing over the local meridian

at 11:44 LST

Besides we know from the tower clock that when the sun reaches the meridian it is 11:44 (LST) and of course it is 12:00 (LAT)


If the angle to the north is 70º at sunrise, then the solar azimut is

= 180º-70º = 110º

Moreover, from Bond’s watch we also know that in London it is 9:44

16

Physics

Environmental

Φcoscos

ΦΨcos

sinsinsinRelationship between azimuth

and latitude, declination and solar altitude angles:

At sunrise = 0 Ψcos

Φcossin

The kidnapping occurred on April 15th, and Bond escapes the following day, which is the 106th day on the year (assuming no leap year).

The latitude is obviously 60º N for the observer can see the Polar star.

Latitude determination: 500.0º011cos

º84.9Φcos sin

º60500.0cosΦ 1

April 1st


April 16th

For day J = 106 we find = 9.84º, and Et = 0.02 min, so we’ll consider negligible the Et contribution.

17

Physics

Environmental

Longitude determination

LAT = LST + 4 (Ls-Le) + Et

Local standard time Time equation

Local apparent time

12:00 = 11:44 + 4 (Ls-Le) + Et

0:00

Longitude correction

Longitude correction 4 (Ls-Le) = 12:00 - 11:44 = +0:16

+16 minutes towards E from standard meridian = +4º from standard meridian of this place

What is the standard meridian of this place?


Because the sun takes 4 minutes to going over ONE grade

Difference of longitude between the standard meridian

Ls and the local meridian Le

(Ls-Le) = 4º

18

Physics

Environmental

Standard meridian of this place

Bond’s watch indicates London time: there it is 09:44 LST (the same in Greenwich).The tower clock indicates local standard time: it is 11:44 LST.So, we conclude from these difference (2 hours) that the standard meridian lies 30º E from Greenwich: Ls = -30º.

(Ls-Le) = 4º Le Local meridian longitudeLs Standard meridian longitude

Greenwich

0º

Ls Le

-30º

-4º

E

-34º

(-30º-Le) = 4º -Le = 30º+4º=34º

(towards W, longitude > 0; towards E, longitude < 0)

Le = -34º (34º E)


19

Physics

Environmental

34º E

60º N N

Geographic coordinates: 60º N, 34º E

Country: Rusia


20

On 31st January 2005 a loner navigator happens on some point of the North Atlantic ocean. He has a sextant aboard and also an accurate clock keeping GMT time. By using his sextant the navigator finds that when the sun is passing through the local meridian its altitude upon the horizon is 30º23’24’’, and at this moment the clock indicates that GMT is 14:00:00.

a) What is the geographic position of this navigator (latitude and longitude).

b) The navigator goes always to the bed at sunset. What time will he go to the bed this day? Give the result both in LAT and GMT.

Use table of declination and time equation for January.

PROB. 0104 / DETERMINATION OF LATITUDE AND LONGITUDE ON THE SEA

Mes de ENERODía Declinación (º) Ec. tiempo (min)1 -23.06 -2.902 -22.98 -3.353 -22.89 -3.794 -22.80 -4.235 -22.70 -4.676 -22.59 -5.097 -22.47 -5.528 -22.35 -5.939 -22.21 -6.34

10 -22.07 -6.7411 -21.93 -7.1412 -21.77 -7.5213 -21.61 -7.9014 -21.45 -8.2715 -21.27 -8.6316 -21.09 -8.9917 -20.90 -9.3318 -20.71 -9.6619 -20.51 -9.9920 -20.30 -10.3021 -20.09 -10.6022 -19.87 -10.8923 -19.64 -11.1824 -19.41 -11.4525 -19.17 -11.7026 -18.92 -11.9527 -18.67 -12.1928 -18.42 -12.4129 -18.15 -12.6230 -17.89 -12.8231 -17.61 -13.00

January

Day Declination (º) Time Equation (min)

21

Celestial equator

S

E

W

N

30º23’24’’

PROB. 0104 / DETERMINATION OF LATITUDE AND LONGITUDE ON THE SEA(CONTINUED)

Maximum solar altitude

Daily solar path on 31st Jan

22


=30º23’24’’

= 90º - (-)

= -17º36’36’’

S

W

Ecuador celeste

N

E

31st January = -17.61º = -17º36’36’’

Et = -13.00 minutes =30º23’24’’= 30.39º

Latitude:

= 90º - (-) =90º - (30º23’24’’-(-17º36’36’’)) =

= 90º - (30º23’24’’-(-17º36’36’’)) = 90º-48º = 42º N

= 42º N

23

Calculation of longitude

LAT = GMT + 4(Ls-Le) + Et

LAT = 12:00 h (noon)

GMT = 14:00 h

Et = -13 min

4(Ls-Le) = 12:00 – 14:00 – (-0:13)

Ls= standard meridian longitude

Le= local meridian longitud

Ls= 0º (Greenwich)

4(Ls-Le) = -120 min – (-13 min) = -107 min

-Le = -107 min/4 (min/grado) = -26.75º

Le = +26.75º = 26º45’ W

= 42º N

Le = 26º45’ W

Le = 26º45’ W

Ls, Le

>0 towards W

<0 towards E


This is LST in Geenwich

24

ΦtantanΦcoscosΦsinsin

cos

s

Sunrise time on 31st January at 42º N

28580.024tan)61.17tan(cos s

º39.73)28580.0(cos 1 s

min 53 hours 4hours 89.4hour/º15

º39.73

Sunrise hour angle:

At sunset the hour angle is the same than it was at sunrise, but towards the west.The number of hours passed from noon until sunset is:

Sunset time (LAT): 12:00 + 4:53 = 16:53 hours

Sunset time (GMT)

LAT = GMT + 4(Ls-Le) + Et GMT = LAT - 4(Ls-Le) - Et

GMT = 16h 53 min - 4(0-26.75) – (-13) min = 16 h 53 min + 107 min + 13 min = 18 h 53 min


25

Determine the local apparent time on each of the following cities for each of the below listed days when it is 12:00:00 UTC. Use the Spencer formula for the time equation. It is recomended to use an Excel sheet to perform all calculations.

PROB. 0105 / LOCAL APPARENT TIME

September 21stOctober 22ndNovember 23rdDecember 24thJanuary 25thFebruary 26thMarch 27thApril 28thMay 29thJune 30th

City Latitude Longitude Day

26

LAT = LST + 4 (Ls-Le) + Et

So Ls = 0º0’0’’ and LST = UTC = 12:00:00. We have to calculate a the time equation for each day from Spencer formula.

)18.229)(2sen04089.02cos014615.0

sen032077.0cos001868.0000075.0(

tE

beeing the daily angle 365

12

J (J is the number of the day of the year)

Ciudad Latitud Longitud Día J (rad) Et (min) LAT (HSL)Atlanta 33º45’ N 84º23’ W 21-sep 264 4.5273 6.90 06:29:22Damasco 33º31’ N 36º18’ E 22-oct 295 5.0610 15.66 14:40:51Katmandú 27º49’ N 85º21’ E 23-nov 327 5.6118 13.30 17:54:42Lisboa 38º40’ N 09º10’ W 24-dic 358 6.1455 0.78 11:24:07Madrid 40º24’ N 03º41’ W 25-ene 25 0.4131 -11.70 11:33:18Montevideo 34º50’ S 56º10’ W 26-feb 57 0.9640 -13.39 08:01:57Moscú 55º45’ N 37º37’ E 27-mar 86 1.4632 -5.97 14:24:30Nairobi 01º18’ S 36º47’ E 28-abr 118 2.0141 2.57 14:29:42Pekín 39º55’ N 116º23’ E 29-may 149 2.5477 2.99 19:48:31Tokio 35º41’ N 139º44’ E 30-jun 181 3.0986 -3.26 21:15:41

Since we have to calculate LAT at 12:00:00 UTC, our reference meridian will be that of Greenwich.

PROB. 0105 / LOCAL APPARENT TIME (CONTINUED)

27

An archeological team aboard an scientific ship looking for sunken vessels of historical interest finds on February 13th a wreckage in the Mediterranean sea, near of the Spanish coast. As the GPS aboard is out of order, the captain takes manually some measures to record the position of the shipwreck:

1º) At the sunrise time, the solar azimut is 72.71º.2º) When the sun is crossing the local meridian, the

spanish official time, indicated by the clock of the ship, is 13:17:23.

A) Determine the geographical position of the wreckage. Point out on the map these coordinates.

B) What is the official time for the sunrise this day on that point? What is the solar elevation upon the horizon at 12:00:00 LAT?

36º

38º

40º

0º2º

PROB. 0106 / LOOKING FOR A WRECKAGE

28

Ecuador celeste

S

E

W

N

=72.71º(Sunrise)

13:17:23Official time when the sun crosses the local meridian:

February 13th data from tables:

= -13.63º; Et = -14.26 min

Relationship between azimut, declination, latitude and solar elevation:

ΦcoscossinΦsinsin

Ψcos

PROB. 0106 / LOOKING FOR A WRECKAGE (CONTINUED)

29

At the sunrise time the solar elevation is = 0 79288.0)º71.72cos(

)º63.13sin(Ψcos

sinΦcos

Latitude = cos-1(0.79288) = 37.54º = 37º 32’ 40’’

From this latitude we can obtain inmediately the solar altitude at noon (12:00:00 LAT):

= 90º - + = 90º - 37.54 + (-13.63) = 38.83º = 38º 49’ 48’’

Longitude determination: LST = Official time –1 = 12:17:23 (winter schedule)

LAT = LST + 4 (Ls-Le) + Et 4 (Ls-Le) = LAT – LST – Et

4 (Ls-Le) = 12:00:00 – 12:17:23 – (-00:14:16) = -00:17:23 + 00:14:16 = -00:03:07

4 (Ls-Le) = -3.117 min Ls-Le = -0.779º

Ls = 0º (Greenwich) Le = +0.779º = 0º 46’ 45’’ W

Longitude of the shipwreck


3036º

38º

40º

0º2º

0º 46’ 45’’ W

37º 32’ 40’’ N

Coordinates of the wreckage:

37º 32’ 40’’ N, 0º 46’ 45’’ W

Sunrise hour angle (February 13th):

ΦΦcoscosΦ

cos tantansinsin

s

s = 79.26º = 79.26/15 = 5.284 hours

Sunrise LAT (February 13th):

12-5.284 = 06:42:58

Sunrise LST (February 13th):

LST = LAT - 4 (Ls-Le) - Et = 07:00:21

Sunrise official time at the wreckage position 08:00:21


1 movements of the earth (apparent movement of the sun, time determination and geographic...

Documents

time official

eratostenes time

official summer time

time determination

official winter time

noon official time

sunrise time lst lst

minutes time equation