MOVEMENTS OF THE EARTH (Apparent movement of the sun, time determination and geographic coordinates)

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Environmental. Physics. LESSON 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 :

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Environmental

Physics

LESSON 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)

Environmental

Maximum circle:

It is a circle defined by the intersection of the sphere with a plane dividing it into two equal parts.

Physics

2

1

Real positions

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.

Minor circle:

It is a circle defined by the intersection of the sphere with a plane dividing it into two non-equal parts.

Environmental

N

Spin of the celestial sphere

Spin of the Earth

Physics

S

REFERENCES IN THE CELESTIAL SPHERE

Pole Line (axis of the world)

Right hand rule

Meridians

Maximum circles

perpendicular

to equator

Celestial Equator

Maximum circle perpendicular to the axis of the world

Environmental

 is the latitude of this point

N

Physics

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.

Meridian of reference

L is the longitude of all those points

Environmental

N

L

Physics

S

GEOGRAPHIC COORDINATES: LONGITUDE

Equatorial diameter: 12,756 Km.

Equatorial length: 40,075 Km.

MERIDIAN: Any maximum circle passing throught the poles.

Meridian lenght: 40,008 Km.

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.

... really L is the longitude of any point lying on that meridian!

Environmental

Physics

23º 27’

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).

June 21/22

Summer solstice

Ecliptic plane

March 20/21

Spring equinox

Environmental

 = 23º 27’

23º 27’

 = 0

April 4

23º 27’

January 3

PERIHELION

1 U.A.

1.017 U.A.

0.983 U.A.

July 4

APHELION

1 U.A.

23º 27’

October 5

September 21/22

Fall equinox

Physics

December 21/22

Winter solstice

23º 27’

 = 0

 = -23º 27’

THE ORBIT OF THE EARTH: THE SEASONS

1 A.U.. = (149597890±500) km  1.496108 km

Environmental

Inverse relative distance

J = day of the year

Eccentricity factor

(J = 1 .. 365)

Physics

EARTH-SUN DISTANCE

Spencer Formula

r0 = 1 U.A.

Duffie y Beckman Formula

Environmental

Physics

Earth-Sun distance

Environmental

Physics

EARTH-SUN DISTANCE: XY representation

Calculus from Duffie-Beckman formula

Celestial north pole

Fall equinox

Apparent path of the Sun on the ecliptic plane

Environmental

Declination angle 

Summer solstice

23º27’

23º27’

Winter solstice

Physics

Spring equinox

Celestial equator plane

Celestial south pole

YEARLY APPARENT MOVEMENT OF THE SUN

Declination angle

23º 27’

Environmental

23º 27’

23º 27’

23º 27’

Physics

SOLSTICES

SUMMER

WINTER

Spencer formula for declination

Environmental

 (degrees)

Physics

On equinoxes  = 0

On the summer solstice  = +23º27’

On the winter solstice  = -23º27’

Declination formula (Crop Evapotranspiration/FAO)

 (degrees)

Environmental

Crop Evapotranspiration

Spencer

Day of the year

Physics

http://www.fao.org/docrep/X0490E/x0490e00.htm

Summer solstice

Environmental

Spencer

Crop Evap.

Winter solstice

Spring equinox

Fall equinox

Physics

COMPARING RESULTS FOR DECLINATION FROM SPENCER FORMULA AND CROP FORMULA

 (degrees)

Number of day of the year

Celestial equator

Environmental

Celestial north pole

90-

horizon

Physics

CELESTIAL EQUATOR AND CELESTIAL NORTH POLE

 latitude

Observer in

Northern hemisphere

Zenit

Celestial north pole

Environmental

90-

W

N

S

Physics

E

CELESTIAL EQUATOR AND CELESTIAL NORTH POLE (II)

Observer in northern hemisphere

Polar

Environmental

Celestial NP

Physics

CIRCUMPOLAR STARS

Zenit

Celestial south pole

Environmental

W

90-

Physics

S

N

E

CELESTIAL EQUATOR AND CELESTIAL SOUTH POLE

Observer in southern hemisphere

Celestial equator

Zenit

Celestial north pole

Tropic of Capricorn

Tropic of Cancer

Environmental

23º 27’

W

-23º 27’

N

S

Physics

E

APPARENT PATH OF THE SUN IN THE NORTHERN HEMISPHERE SKY

Equinoxes

Summer solstice

Winter solstice

Zenit

Celestial north pole

Environmental

W

N

S

Physics

E

APPARENT PATH OF THE SUN

Any day

Observer in northern hemisphere

 declination

 latitude

Season: spring/summer

POSITION OF THE SUN RELATED TO A HORIZONTAL SURFACE

Zenit

Celestial north pole

Environmental

z

W

N

S

15º/hour

Physics

E

 latitude

 declination

Season: spring/summer

Observer in northern hemisphere

 solar altitude

z zenith angle

 solar azimut

 Hour angle

COORDINATES measured

from the center of sun disc

Zenit

Celestial north pole

Environmental

 = 0

W

max

N

S

Physics

E

MAXIMUM SOLAR ALTITUDE

 latitude

 declination

Observer in northern hemisphere

Season Spring / Summer

Zenit

Celestial north pole

Environmental

W

sHour angle

at the sunrise

z = 90º

N

S

s

Physics

E

HOUR ANGLE AT THE SUNRISE

 latitude

 declination

 solar altitude

z zenith angle

 solar azimut

Season Spring / Summer

Observer in northern hemisphere

 = 0

Environmental

shour angle

at the sunrise

Physics

SIGN CRITERION

It varies from 0º (horizon) to 90º (zenit)

 solar altitude

It varies from 0º (cénit) to 90º (horizonte)

z zenith angle

It varies from 0º (south) to 180º (north).

Sign: + towards E, - towards W

 solar azimut

 hour angle

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.

Environmental

Physics

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?

Environmental

Physics

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!).

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!

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 duration of the solar day varies throughout the year for the two main following reasons:

The axis of the Earth is tilted a constant angle onto the ecliptic plane.

Environmental

Cenit

Celestial equator

W

N

S

Physics

E

MEAN SOLAR DAY

Mean Solar Dayis 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

Environmental

Physics

We can obtain data for each day of the year from this one or some similar formula

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.

TIME EQUATION = SOLAR MEAN TIME - SOLAR APPARENT TIME

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

J number of the day of the year

Daily angle

0 to 365, or 0 to 366 for leap year

Environmental

Physics

TIME EQUATION: GRAPHICS

TIME EQUATION = SOLAR MEAN TIME - SOLAR APPARENT TIME

http://averroes.cec.junta-andalucia.es/ies_gaviota/ fisiqui/relojsol/horas.htm

Environmental

www.greenwichmeantime.com

Physics

DETERMINATION OF TIME: GMT

GMT = Greenwich Mean Time

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

Environmental

Physics

http://www.hyperdictionary.com/search.aspx?Dict=&define=UTC&search.x=32&search.y=10

Definition UTC

http://www.its.bldrdoc.gov/fs-1037/dir-009/_1277.htm

DETERMINATION OF TIME: UNIVERSAL TIME

UTC = Universal Time Coordinated

UT = Universal Time

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.

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.

Environmental

Physics

DETERMINACIÓN OF TIME: GREENWICH OBSERVATORY

Zenit

UTC = 12:00:00

GMT = 12:00:00

Environmental

W

N

S

Physics

E

UTC = 00:00:00

GMT = 00:00:00

DETERMINATION OF TIME: GMT y UTC

Cenit

Environmental

Sun apparent movement:

W

N

S

Physics

HSL = 10:00:00

E

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.

HSL = 12:00:00

 = 0º

 = 30º

Example:

Environmental

Physics

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

Environmental

Time equation

Longitude correction

Ls Standard meridian longitude

Le Local meridian longitude

4 min / degree

LAT = LST + 4·(Ls-Le) + Et

Physics

>0 towards W

Degrees

Longitude correction in minutes

Ls, Le

<0 towards E

DETERMINATION OF TIME: LOCAL STANDARD TIME (LST) (II)

Relationship between local apparent time (LAT, HSL) and local standard time (LST, HSE)

LST = LAT

- 4·(Ls-Le)

- Et

(minutes)

Sun apparent movement

15 degrees / hour

Environmental

Physics

>0 towards W

Ls, Le

<0 towards E

www.greenwichmeantime.com

1º52’

4·(-1.867) =- 7.47 min = -7 min 28 s

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.

Le

Ls

Albacete

Find LST in Albacete when it is 12:00:00 LAT on January 1st. (Longitude of Albacete 1º52’ W)

Greenwich

LST = LAT

- Et

- 4·(Ls-Le)

Longitud correction -7.47 min

Jan 1st Et = -2.90 min

LST = 12:00:00 -(-7.47) -(-2.90)

LST = 12:00:00 +10.37 min =

1º52’ = 1.87º

= 12:10:23

Environmental

Reference meridian

4·(60.00-58.48) = 6.08 min

Et = +14.62 min

16th Oct

Physics

10 h + 20.70 min = 10:20:42

DETERMINATION OF TIME: LOCAL APPARENT TIME (EXEMPLE 2)

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.

+ 4·(Ls-Le)

= 10:00:00

+ 6.08

+ Et

LAT = LST

+ 14.62

= 10 h + 20.70 min

Environmental

Physics

DETERMINATION OF TIME: LEGAL TIME

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).

http://nist.time.gov/

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

Environmental

W

Zenit

S

N

E

Physics

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

Environmental

W

Celestial North Pole

Zenit

S

N

E

Physics

GEOGRAPHIC POSITION: LATITUDE DETERMINATION (II)

2º) POLAR STAR ALTITUDE: THIS IS A DIRECT MEASUREMENT OF LATITUDE

Application: at night, only on the northern hemisphere

Environmental

Having in mind the

daily correction

Physics

LAT measurement

on the point

LST on Ls meridian

GEOGRAPHIC POSITION: LONGITUDE DETERMINATION

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 Le from the equation:

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

Environmental

degrees

minutes

minutes

minutes/degree

HSL – HSE – Et

(Ls-Le)

= L

=

4

Le

Le

Ls

Ls

Physics

E

E

W

W

GEOGRAPHIC POSITION: LONGITUDE DETERMINATION. SIGNS

The sun spends 4 minutes to go over a degree

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

4 (Ls-Le) = LAT – LST – Et

L < 0

L > 0

Environmental

LAT – LST – Et

(Ls-Le)

= L

=

4

Le

Ls

(-6.4)

Physics

1

1

E

W

=

=

4

4

GEOGRAPHIC POSITION: LONGITUDE DETERMINATION. EXEMPLE

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: Et = –6.60 m (-6 m 36 s)

= -13 m

Difference LAT-LST

(Ls-Le) = L

(-13 – (-6.60))

L < 0

(Ls-Le) = L = -1.6º = -1º36’

1º 36’

Le = Ls + 1º36’

Lack of some

corrections

Environmental

Cenit

W

The maximum of hours of sun for a day is twice

Lenght of

the day

N

S

Physics

s

E

THE LENGTH OF THE DAY (SUNRISE AND SUNSET TIME)

The time (in hours) the sun takes to reach the local meridian is

(because the sun goes over 15º/hour in its path on the sky)

LAT = 12:00:00

 = 0º

SUNSET

SUNRISE

Environmental

Solar altitude

(solar disc center)

-34’

 = 0

-16’

-50’

Physics

Sunset behind time

-16’

SOLAR ALTITUDE ANGLE CORRECTION ON SUNRISE AND SUNSET

I. CORRECTION BY ATMOSPHERIC REFRACTION

Correction  3-5 minutos

Environmental

Physics

SOLAR ALTITUDE ANGLE CORRECTION ON SUNRISE AND SUNSET (II)

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

Royal Greenwich bservatory

http://greenwichmeantime.com/

Environmental

Physics

BIBLIOGRAFÍA y DOCUMENTACIÓN

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

Environmental

Spain hour zones

Sunrise and sunset corrections

http://www.rediris.es/red/zona_horaria.es.html

http://www.astrored.org/usuarios/xgarciaf/orto1.htm

Glossary and definitions (English)

http://www.sundialsoc.org.uk/glossary/frameset.htm

The problem of the longitude

http://www.sunlitdesign.com/infosearch/hourangle.htm?indexref=3

http://rubens.anu.edu.au/student.projects97/naval/home.htm (no longer available)

Physics

W J H Andrewes, “Crónica de la medición del tiempo”, Investigación y Ciencia, nov 2002

BIBLIOGRAFÍA y DOCUMENTACIÓN (II)