AT737 Satellite Orbits and Navigation 1

AT737 Satellite Orbits and Navigation2 Newton’s Laws 1.Every body will continue in its state of rest or of uniform motion in a straight line except insofar as it is compelled to change that state by an impressed force. 2.The rate of change of momentum is proportional to the impressed force and takes place in the line in which the force acts. 3.Action and reaction are equal and opposite.

AT737 Satellite Orbits and Navigation3 Newton's Second Law is the familiar where F is force, m is mass, a is acceleration, v is velocity, and t is time. Newton’s Laws (continued)

AT737 Satellite Orbits and Navigation4 Newton’s Law of Universal Gravitation The force of attraction between two point masses m 1 and m 2 separated by a distance r is where G is the Newtonian (or universal) gravitation constant ( x N m 2 kg -2 ).

AT737 Satellite Orbits and Navigation5 Circular Orbit Example Centripetal Force Gravitational Force Period The NOAA satellites orbit at about 850 km above the surface (r = 7228 km) and therefore have a period of about 102 minutes.

AT737 Satellite Orbits and Navigation6 Kepler’s Laws 1.All planets travel in elliptical paths with the sun at one focus. 2.The radius vector from the sun to a planet sweeps out equal areas in equal times. 3.The ratio of the square of the period of revolution of a planet to the cube of its semimajor axis is the same for all planets revolving around the sun. The same laws apply if we substitute satellite for planet and earth for sun, but the proportionality constant is different.

AT737 Satellite Orbits and Navigation7 Ellipse Geometry a = semimajor axis  = eccentricity (0-1)  = true anomaly r = radius Equation of an Ellipse

AT737 Satellite Orbits and Navigation8 Kepler’s Equation M = Mean anomaly n = mean motion constant t p = time of perigeal passage e = eccentric anomaly  = eccentricity Angles M, e, and θ are zero at perigee. NOTE: All angles in radians.

AT737 Satellite Orbits and Navigation9 Right Ascension & Declination  = declination  = right ascension Need a coordinate system to orient orbital plane in space

AT737 Satellite Orbits and Navigation10 Orientation Angles i = inclination angle  = argument of perigee  = right ascension of ascending node i < 90°  prograde i > 90°  retrograde

AT737 Satellite Orbits and Navigation11 Classical Orbital Elements ElementSymbol Semimajor axis*a Eccentricity  Inclinationi Argument of perigee oo Right ascension of ascending node oo Mean anomaly**MoMo Epoch timetoto *Two-line elements give orbits per day instead of a **ESA uses true anomaly instead of mean anomaly

AT737 Satellite Orbits and Navigation12 Sources of Orbital Elements NOAA Satellite Information System (current TBUS and TLEs for GOES and NOAA satellites) T.S. Kelso’s CelesTrak site (TLEs for a lot of satellites—still in business in spite of Space Track) New Government Site Established by Public Law , Section Public Law , Section 913

AT737 Satellite Orbits and Navigation13 Keplerian Orbits Viewed from space, Keplerian orbits are constant and simple. Viewed from a point rotating with the earth, Keplerian orbits are complex.

AT737 Satellite Orbits and Navigation14 Orbit Perturbing Forces ForceSource Nonspherical gravitational field Nonspherical, nonhomogeneous Earth Gravitational attraction of other bodies Sun, moon, planets Radiation pressureSolar radiation Particle fluxSolar wind Lift and dragResidual atmosphere Electromagnetic forces Interaction of electrical currents in the satellite with Earth’s magnetic field

AT737 Satellite Orbits and Navigation15 Perturbation Equations U is the gravitational potential energy (g = -  U) r ee = equatorial radius of Earth = 6,378,137 m J 2 = x a, , and i are unperturbed Anomalistic mean motion constant

AT737 Satellite Orbits and Navigation16 …and more equations Anomalistic period—the time from perigee to moving perigee The reciprocal of this is what you get in two-line elements in place of the semimajor axis Synodic or nodal period—the time from ascending node to ascending node

AT737 Satellite Orbits and Navigation17 Where is that satellite? A step-by-step calculation guide 1.Find the orbital elements of the satellite you are interested in. 2.Update the variable elements (M, , &  ) to the time (t ) that you are interested in: M = M o + (t – t o )(dM/dt), etc. 3.Use Kepler’s equation to calculate the true anomaly (  ). 4.Use ellipse equation to calculate r, the distance of the satellite from the center of the earth.

AT737 Satellite Orbits and Navigation18 Calculations continued… 5.Calculate the argument of latitude:    +  (measures angular distance from equator). 6.Calculate latitude:  = sin -1 (sin  sin i ) 7.Calculate the right ascension of the satellite at time t :  = right ascension of ascending node as calculated in step 2

AT737 Satellite Orbits and Navigation19 Calculations completed 8.Calculate the right ascension of Greenwich (the prime meridian) at time t :  Greenwich = °  t where  t is the time in days (and fraction) between time t and 0000 UTC 1 January Calculate the longitude of the satellite: =  sat -  Greenwich Homework