The Shape of the Earth

The Earth is not a perfect sphere

• Equatorial diameter slightly greater than polar diameter

• Earth is an oblate ellipsoid–slightly flattened

• The geoid exaggerates small departures from spherical

The Earth’s Rotation

Earth rotates on its axis:

•Counterclockwise at North Pole

•Left to right (eastward) at Equator

•One rotation is a solar day (24 hours)

Axis: an imaginary straight line through the center of the Earth around which the Earth rotates

Poles: the two points on the Earth’s surface where the axis of rotation emerges

The Earth’s Rotation

Environmental Effects of the Earth’s Rotation:

•Day and night

•Fluctuating air temperature

•Tides

The Geographic Grid

Parallels and MeridiansGeographic grid: network of parallels and meridians used to fix location on the Earth

Parallel: east-west circle on the Earth’s surface, lying on a plane parallel to the equator

Meridian: north-south line on the Earth’s surface, connecting the poles

The Geographic Grid

Parallels and MeridiansEquator: Parallel of latitude lying midway between the Earth’s poles; it is designated latitude 0º• Longest parallel of latitude• Midway between poles• Fundamental reference line for measuring position

Latitude: arc of a meridian between the equator and a given point on the globe

Longitude: arc of a parallel between the prime meridian and a given point on the globe

The Geographic Grid

Latitude and Longitude

Latitude is measured north and south of the equator, up to 90º

Longitude is measured east and west of the Prime Meridian—meridian that passes through Greenwich, England—up to 180º

Visualizing Physical GeographyCopyright © 2008 John Wiley and Sons

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Earth’s Revolution Around the Sun

Tilt of the Earth’s AxisEarth has seasons because of the tilt of the axis.

• Axis aims toward Polaris (North Star)

• Axis tilted at an angle of 23 ½ ° from a right angle to plane of the ecliptic

Plane of the Ecliptic: plane of the Earth’s orbit around the Sun

Earth’s Revolution Around the Sun

The Four SeasonsEarth’s axis tilted toward North Star throughout Earth’s orbit.• December 22: N hemisphere tilted away from the sun at the maximum angle• June 21: N hemisphere tilted toward the sun at the maximum angle

Earth’s Revolution Around the Sun

The Four Seasons

Summer solstice: solstice occurring on June 21 or 22, when the subsolar point is at 23 1/2° N; June Solstice

Winter solstice: solstice occurring on December 21 or 22, when the subsolar point is at 23 1/2° S; December Solstice

Equinox: time when subsolar point falls on equator and circle of illumination passes through both poles

Circle of illumination: separates day hemisphere from night hemisphere

Earth’s Revolution Around the Sun

Equinox Conditions

Subsolar point: point on the Earth’s surface where the sun is directly overhead at noon

• Circle of illumination passes through both poles

• Subsolar point at equator

• Day and night of equal length everywhere on the globe

•Occurs twice per year

•Vernal Equinox: March 21

•Autumnal Equinox: September 23

Earth’s Revolution Around the Sun

Solstice Conditions• Circle of illumination grazes Arctic and Antarctic Circles• June Solstice: north pole has 24 hours of daylight; daylength increases from equator to north

pole• December Solstice: south pole has 24 hours of daylight; daylength increases from equator to

south pole

Earth’s Revolution Around the Sun

Earth revolves around the sun every 365.242 days• Orbit is an ellipse• Leap year corrects for the extra quarter day• Orbit is counterclockwise• Perihelion: point in orbit when Earth is closest to Sun• Aphelion: point in orbit when Earth is farthest from Sun

A great-circle arc, on the sphere, is the analogue of a straight line, on the plane.

Where two such arcs intersect, we can define the spherical angle either as angle between the tangents to the two arcs, at the point of intersection, or as the angle between the planes of the two great circles where they intersect at the centre of the sphere.

(Spherical angle is only defined where arcs of great circles meet.)

A spherical triangle is made up of three arcs of great circles, all less than 180°. The sum of the angles is not fixed, but will always be greater than 180°. If any side of the triangle is exactly 90°, the triangle is called quadrantal.

Set up a system of rectangular axes OXYZ:

O is at the centre of the sphere; OZ passes through A; OX passes through arc AB (or the extension of it); OY is perpendicular to both.

Find the coordinates of C in this system: x = sin(b) cos(A) y = sin(b) sin(A) z = cos(b)

sine rule:   sin(a)/sin(A) = sin(b)/sin(B) = sin(c)/sin(C)

cosine rule:    cos(a) = cos(b) cos(c) + sin(b) sin(c) cos(A)     cos(b) = cos(c) cos(a) + sin(c) sin(a) cos(B)     cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C)

Napier's Rules for a spherical right triangle

1. The sine of an angle is equal to the product of cosines of the opposite two angles. 2. The sine of an angle is equal to the product of tangents of the two adjacent angles.

Nautical Mile

It is the distance measured along the great circle joining the points which subtends one minute of arc at the centre of earth

Exercise:A point A, has longitude 2°W, latitude 50°N. And another place B, has longitude 97°W, latitude 50°N. How far apart are they, in nautical miles, along a great circle arc?

Use the cosine rule:cos AW = cos WP cos AP + sin WP sin AP cos P = cos240° + sin240° cos 95°= 0.5508 So AW = 56.58° = 3395 nautical miles

(This is 7% shorter than the route along a parallel of latitude).If you set off from Alderney on a great-circle route to Winnipeg, in what direction (towards what azimuth) would you head?Use the sine rule:sin A / sin WP = sin P / sin WAso sin x = sin 40° sin 95° / sin 56.58° = 0.77so x = 50.1° or 129.9° .Common sense says 50.1° (or check using cosine rule to get PW).

Azimuth is measured clockwise from north, so azimuth is 360° - 50.1° = 309.9° (Note that this is 40° north of the “obvious” due-west course.)Back to "Spherical trigonometry".

(Figure 4-2). In the continental United States, longitude is commonly reported as a westlongitude. To convert easterly to westerly referenced longitudes, the easterly longitude must besubtracted from 360 deg.

I. Latitude and Longitude on Spherical EarthLatitude and longitude are the grid lines you see on globes. For a spherical earththese are angles seen from the center of the earth. The angle up from the equator islatitude. In the southern hemisphere is it negative in the convention used in geodesy. Ithas a range of –90 degrees to 90 degrees. The reference for latitude is set by the equator -effectively set by the spin axis of the earth.The angle in the equatorial plane is the longitude. There is no natural reference forlongitude. The zero line, called the prime meridian, is taken, by convention, as the linethrough Greenwich England. (This was set by treaty in 1878. Before that each majornation had its own zero of longitude.)

Because the Earth is not a true sphere, the situation is more complicated than the simple oneoutlined above, though the latter is accurate enough for most purposes.When a plumb-line is suspended by an observer at a point on the Earth’s surface, its directionmakes an angle with the plane of the Earth’s equator. This angle is called the astronomical latitude,φ. The point where the plumb-line’s direction meets the equatorial plane is not, in general, the centreof the Earth. The angle between the line joining the observer to the Earth’s centre and the equatorialplane is the geocentric latitude, φ (see figure 7.3).There is yet a third definition of latitude. Geodetic measurements on the Earth’s surface showlocal irregularities in the direction of gravity due to variations in the density and shape of the Earth’scrust. The direction in which a plumb-line hangs is affected by such anomalies and these are referred toas station error. The geodetic or geographic latitude, φ, of the observer is the astronomical latitudecorrected for station error.The geodetic latitude is, therefore, related to a reference spheroid whose surface is defined by themean ocean level of the Earth. If a and b are the semi-major and semi-minor axes of the ellipse ofrevolution forming the ‘geoid’, the flattening or ellipticity, , is given by

The longitude used in geodesy is positive going east from the prime meridian. Thevalues go from 0 to 360 degrees. A value in the middle United States is therefore about260 degrees east longitude. This is the same as -100 degrees east.In order to make longitudes more convenient, often values in the westernhemisphere are quoted in terms of angles west from the prime meridian. Thus the2longitude of -100 E (E for East) is also 100 W (W for West). Similarly latitudes south ofthe equator are often given as "S" (for south) values to avoid negative numbers.

Latitude and Longitude on Ellipsoidal EarthThe earth is flattened by rotational effects. The cross-section of a meridian is noloner a circle, but an ellipse. The ellipse that best fits the earth is only slightly differentfrom a circle. The flattening, defined in the figure below, is about 1/298.25 for the earth.Latitude and longitude are defined to be "intuitively the same as for a sphericalearth". This loose definition has been made precise in geodesy. The longitude is theexactly the same as for a spherical earth. The way latitude is handled was defined by theFrench in the 17th century after Newton deduced that the world had an elliptical crosssection.

Before satellites latitude was measured by observing the stars. In particularobserving the angle between the horizon and stars. The horizon was taken to beperpendicular to the vertical measured by a plumb bob or spirit level. The "vertical line"of the plumb bob was thought to be perpendicular to the sphere that formed the earth.The extension to an ellipsoidal earth is to use the line perpendicular to the ellipsoid todefine the vertical. This is essentially the same as the plumb bob.1

The figures below show the key effects of rotation on the earth and coordinates.The latitude is defined in both the spherical and ellipsoidal cases from the lineperpendicular to the world model. In the case of the spherical earth, this line hits theorigin of the sphere - the center of the earth. For the ellipsoidal model the up-down linedoes not hit the center of the earth. It does hit the polar axis though

The length of the line to the center of the earth for a spherical model is the radiusof the sphere. For the ellipsoidal model the length from the surface to the polar axis isone of three radii needed to work with angles and distance on the earth. (It is called theradius of curvature in the prime vertical, and denoted RN here. See the note on radii of theearth for details.)There are not two types of latitude that can easily be defined. The angle that theline makes from the center of the earth is called the geocentric latitude. Geocentriclatitude is usually denoted as f¢, or fc . It is commonly used in satellite work. It does notstrike the surface of the ellipsoid at a right angle. The line perpendicular to the ellipsoidmakes an angle with the equatorial plane that is called the geodetic latitude. (“Geodetic"in geodesy usually implies something taken with respect to the ellipsoid.) The latitude onmaps is geodetic latitude. It is usually denoted as g

Geodetic Coordinates. Geodetic coordinate components consist of:· latitude (f),· longitude (l),· ellipsoid height (h).

Geodetic latitude, longitude, and ellipsoid height define the position of a point on the surface of the Earth with respect to the reference ellipsoid.

1) Geodetic latitude (f).

The geodetic latitude of a point is the acute angular distance

between the equatorial plane and the normal through the point on the ellipsoid measured in the meridian plane

Geodetic latitude is positive north of the equator and negative south of the equator.

(2) Geodetic longitude (l).

The geodetic longitude is the angle measured counter-clockwise (east), in the equatorial plane, starting from the prime meridian (Greenwich meridian), to the meridian of the defined point

(3) Ellipsoid Height (h).

The ellipsoid height is the linear distance above the reference ellipsoid measured along the ellipsoidal normal to the point in question. The ellipsoid height is positive if the reference ellipsoid is below the topographic surface and negative if the ellipsoid is above the topographic surface.