12.215 Modern NavigationToday’s Class Latitude and LongitudeSimple Geocentric Latitude and LongitudeGeocentric quantitiesGeocentric relationship to XYZProblem with GeocentricGeocentric quantitiesEllipsoidal quantitiesGeodetic LatitudeRelationship between fg and XYZInverse relationshipOther itemsAstronomical latitude and longitudeCoordinate axes directionsMotion of rotation axisMotion of rotation axis 1993-2001Rotation rate of the EarthRotation rate of the EarthConventional definitions of coordinate systemsSummary12.215 Modern NavigationThomas Herring09/18/2006 12.215 Modern Naviation L02 2Today’s ClassLatitude and Longitude• Simple spherical definitions• Geodetic definition: For an ellipsoid• Astronomical definition: Based on direction of gravity• Relationships between the types• Coordinate systems to which systems are referred• Temporal variations in systems09/18/2006 12.215 Modern Naviation L02 3Simple Geocentric Latitude and Longitude• The easiest form of latitude and longitude to understand is the spherical system:• Latitude: Angle between the equatorial plane and the point. Symbol φc(in this class)• Latitude is also the angle between the normal to the sphere and the equatorial plane• Related term: co-latitude = 90o-latitude. Symbol θc(in this class). Angle from the Z-axis• Longitude: Angle between the Greenwich meridian and meridian of the location. Symbol λc09/18/2006 12.215 Modern Naviation L02 4Geocentric quantities• Geocentric Latitude and Longitude• Note: Vector to P is also normal to the sphere.ZXYGreenwichMeridianPλφMeridian of point PEquatorHθR09/18/2006 12.215 Modern Naviation L02 5Geocentric relationship to XYZ• One of the advantages of geocentric angles is that the relationship to XYZ is easy. R is taken to be radius of the sphere and H the height above this radiusφc= tan−1(Z / X2+ Y2)λc= tan−1(Y /X)R + Hc= X2+ Y2+ Z2X=(R+Hc)cosφccosλcY = (R + Hc)cosφcsinλcZ = (R + Hc)sinφc09/18/2006 12.215 Modern Naviation L02 6Problem with Geocentric• Geocentric measures are easy to work with but they have several serious problems• The shape of the Earth is close to an bi-axial ellipsoid (i.e., an ellipse rotated around the Z-axis)• The flattening of the ellipsoid is ~1/300 (1/298.257222101 is the defined value for the GPS ellipsoid WGS-84).• Flattening is (a-b)/a where a is the semi-major axis and b is the semi-minor axis. • Since a=6378.137 km (WGS-84), a-b=21.384 km09/18/2006 12.215 Modern Naviation L02 7Geocentric quantities• If the radius of the Earth is taken as b (the smallest radius), then Hcfor a site at sea-level on the equator would be 21km (compare with Mt. Everest 28,000feet~8.5km).• Geocentric quantities are never used in any large scale maps and geocentric heights are never used.• We discuss heights in more in next class and when we do spherical trigonometry we will use geocentric quantities.09/18/2006 12.215 Modern Naviation L02 8Ellipsoidal quantities• The most common latitude type seen is geodetic latitude which is defined as the angle between the normal to the ellipsoid and the equatorial plane. We denote with subscript g.• Because the Earth is very close to a biaxial ellipsoid, geodetic longitude is the same as geocentric longitude (the deviation from circular in the equator is only a few hundred meters: Computed from the gravity field of the Earth).09/18/2006 12.215 Modern Naviation L02 9Geodetic LatitudeNorthEquatorGeoidgravity directionNormal to ellipsoidφgφaLocal equipotenital surfaceEarth's surfacePAstronomical Latitude also shown09/18/2006 12.215 Modern Naviation L02 10Relationship between φgand XYZ• This conversion is more complex than for the spherical case. X= (N+ hg)cos(φg)cos(λg)Y = (N + hg)cos(φg)sin(λg)Z = [(1− e2)N + hg]sin(φg)where e2= 2 f − f2 and N (North -South radius of curvature) isN2= a2/[1− e2sin2(φg)]09/18/2006 12.215 Modern Naviation L02 11Inverse relationship• The inverse relationship between XYZ and geodetic latitude is more complex (mainly because to compute the radius of curvature, you need to know the latitude).• A common scheme is iterative: a N'= a/1− e2sinφ'r'= X2+ Y2[1 − e2N'/(N'+h')]φ'= tan−1(Z /r')h'= X2+ Y2/cosφ'−N' or h'=Z/sinφ'−(1 − e2)N'iterate to a until h' change is small09/18/2006 12.215 Modern Naviation L02 12From http://www.colorado.edu/geography/gcraft/notes/datum/gif/xyzllh.gifClosed form expression for small heights09/18/2006 12.215 Modern Naviation L02 13Other items• A discussion of geodetic datum and coordinate systems can be found at:http://www.colorado.edu/geography/gcraft/notes/datum/datum.html• Geodetic longitude can be computed in that same way as for geocentric longitude• Any book on geodesy will discuss these quantities in more detail (also web searching on geodetic latitude will return many hits).• The difference between astronomical and geodetic latitude and longitude is called “deflection of the vertical”09/18/2006 12.215 Modern Naviation L02 14Astronomical latitude and longitude• These have similar definitions to geodetic latitude and longitude except that the vector used is the direction of gravity and not the normal to the ellipsoid (see earlier figure).• There is not direct relationship between XYZ and astronomical latitude and longitude because of the complex shape of the Earth’s equipotential surface.• In theory, multiple places could have the same astronomical latitude and longitude.• As with the other measures, the values of depend on the directions of the XYZ coordinate axes.09/18/2006 12.215 Modern Naviation L02 15Coordinate axes directions• The origin of the XYZ system these days is near the center of mass of the Earth deduced from the gravity field determined from the orbits of geodetic satellites (especially the LAGEOS I and II satellites).• The direction of Z-axis by convention is near the mean location of the rotation axis between 1900-1905. At the time, it was approximately aligned with the maximum moments of inertia of the Earth. (review:http://dept.physics.upenn.edu/courses/gladney/mathphys/java/sect4/subsubsection4_1_4_2.html09/18/2006 12.215 Modern Naviation L02 16Motion of rotation axis• The rotation axis has moved about 10 m on average since 1900 (thought to be due to post-glacial rebound).• It also moves in circle with a 10 m diameter with two strong periods: Annual due to atmospheric mass movements and 433-days which is a natural