1. EE 495 Modern Navigation Systems Navigation Mathematics Earth Surface and Gravity Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Slide 1 of 14

2. Navigation Mathematics : Earth surface and Gravity - Earth Modeling Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems The Earth can be modeled as an oblate spheroid  A circular cross section when view from the polar axis (top view)  An elliptical cross-section when viewed perpendicular to the polar axis (side view) This ellipsoid (i.e., oblate spheroid) is an approximation to the “geoid” The geoid is a gravitational equipotential surface which “best” fits (in a least square sense) the mean sea level Ratio exaggerated www.nrcan.gc.ca Max variation betw. ellipsoid and geoid is +3 to -51 meters. Slide 2 of 14

3. Navigation Mathematics : Earth surface and Gravity - Earth Modeling Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Equator Slide 3 of 14

4. Navigation Mathematics : Earth surface and Gravity - Earth Modeling Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems We can define a position “near” the Earth’s surface in terms of latitude, longitude, and height  Geocentric latitude intersects the center of mass of the Earth  Geodetic latitude (L) is the angle between the normal to the ellipsoid and the equatorial plane Geodetic Latitude Surface Normal Equatorial Plane Geocentric Latitude Reference Ellipsoid Slide 4 of 14

5. Navigation Mathematics : Earth surface and Gravity - Earth Modeling Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Equatorial Plane Geodetic Latitude Reference Ellipsoid Geodetic height Slide 5 of 14 Body

6. Navigation Mathematics : Earth surface and Gravity - Earth Modeling EE 495 Modern Navigation Systems R E = Transverse radius of curvature LbLb hbhb RERE zeze X e /Y e plane xexe yeye Disk of constant Latitude (L b ) Wednesday, Feb 4 2014 R N = Meridian radius of curvature Slide 6 of 14

7. Navigation Mathematics : Earth surface and Gravity - Earth Modeling R E = Transverse radius of curvature R E (1-e 2 ) LbLb hbhb RERE zeze (R E +h b )Cos(L b ) (R E (1-e 2 )+h b )Sin(L b ) xexe yeye Disk of constant Latitude (L b ) (R E +h b )Cos(L b ) (R E +h b )Cos(L b )Sin( b ) (R E +h b )Cos(L b )Cos( b ) X e /Y e plane EE 495 Modern Navigation Systems Wednesday, Feb 4 2014Slide 7 of 14

8. Navigation Mathematics : Earth surface and Gravity - Gravity Models Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Slide 8 of 14

9. Navigation Mathematics : Earth surface and Gravity - Gravity Models Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Relationship between specific force, inertial acceleration, and gravitational attraction When stationary on the surface of the Earth  Recall case 1: A fixed point in a rotating frame o Considering frame {0} to be the {i} frame, {1} = {e}, and {2} ={b} gives o Coordinatizing in the e-frame gives Slide 9 of 14

10. Reaction to specific force Navigation Mathematics: Earth surface and Gravity - Gravity Models Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Thus, when stationary on the surface of the Earth the acceleration is due to Centrifugal force Therefore, the acceleration due to gravity is Slide 10 of 14

11. Navigation Mathematics: Earth surface and Gravity - Gravity Models Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Slide 11 of 14 Vector diagram ~3mg~1g1g

12. Navigation Mathematics : Earth surface and Gravity - Gravity Models Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems Slide 12 of 14

13. Navigation Mathematics : Earth surface and Gravity - Gravity Models Wednesday, Feb 4 2014 EE 495 Modern Navigation Systems On March 17, 2002 NASA launched the Gravity Recovery and Climate Experiment (GRACE) which led to the development of some of the most precise Earth gravity models NASA's Grace Gravity Model Slide 13 of 14