AT 737 Satellite Orbits and Navigation 1 AT

AT 737 Satellite Orbits and Navigation 1 AT 737 Satellite Orbits and Navigation

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. AT 737 Satellite Orbits and Navigation 2

Newton’s Laws (continued) Newton's Second Law is the familiar where F is force, m is mass, a is acceleration, v is velocity, and t is time. AT 737 Satellite Orbits and Navigation 3

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 (6. 67259 x 10 -11 N m 2 kg-2). AT 737 Satellite Orbits and Navigation 4

Circular Orbit Example Centripetal Force Period AT 737 Satellite Orbits and Navigation Gravitational Force The NOAA satellites orbit at about 850 km above the surface (r = 7228 km) and therefore have a period of about 102 minutes. 5

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. AT 737 Satellite Orbits and Navigation 6

Ellipse Geometry a = semimajor axis = eccentricity (0 -1) = true anomaly r = radius AT 737 Satellite Orbits and Navigation Equation of an Ellipse 7

Kepler’s Equation NOTE: All angles in radians. M = Mean anomaly n = mean motion constant tp = time of perigeal passage e = eccentric anomaly = eccentricity Angles M, e, and θ are zero at perigee. AT 737 Satellite Orbits and Navigation 8

Right Ascension & Declination Need a coordinate system to orient orbital plane in space d = declination = right ascension AT 737 Satellite Orbits and Navigation 9

Orientation Angles i = inclination angle i < 90° prograde i > 90° retrograde = argument of perigee = right ascension of ascending node AT 737 Satellite Orbits and Navigation 10

Classical Orbital Elements Element Semimajor axis* Symbol a Eccentricity Inclination i Argument of perigee o Right ascension of ascending node o Mean anomaly** Mo Epoch time to *Two-line elements give orbits per day instead of a **ESA uses true anomaly instead of mean anomaly AT 737 Satellite Orbits and Navigation 11

Sources of Orbital Elements NOAA Satellite Information System http: //noaasis. noaa. gov/NOAASIS/ml/quicklook. html (current TBUS and TLEs for GOES and NOAA satellites) T. S. Kelso’s Celes. Trak site http: //celestrak. com (TLEs for a lot of satellites—still in business in spite of Space Track) New Government Site http: //www. space-track. org Established by Public Law 108 -136, Section 913 AT 737 Satellite Orbits and Navigation 12

Keplerian Orbits Viewed from space, Keplerian orbits are constant and simple. AT 737 Satellite Orbits and Navigation Viewed from a point rotating with the earth, Keplerian orbits are complex. 13

Orbit Perturbing Forces Force Source Nonspherical gravitational field Nonspherical, nonhomogeneous Earth Gravitational attraction of other bodies Sun, moon, planets Radiation pressure Solar radiation Particle flux Solar wind Lift and drag Residual atmosphere Electromagnetic forces Interaction of electrical currents in the satellite with Earth’s magnetic field AT 737 Satellite Orbits and Navigation 14

Perturbation Equations U is the gravitational potential energy (g = - U) Anomalistic mean motion constant ree = equatorial radius of Earth = 6, 378, 137 m J 2 = 1. 08263 x 10 -3 a, e, and i are unperturbed AT 737 Satellite Orbits and Navigation 15

…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 AT 737 Satellite Orbits and Navigation 16

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 = Mo + (t – to)(d. M/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. AT 737 Satellite Orbits and Navigation 17

Calculations continued… 5. Calculate the argument of latitude: + (measures angular distance from equator). 6. Calculate latitude: = sin-1(sin i) 7. Calculate the right ascension of the satellite at time t: = right ascension of ascending node as calculated in step 2 AT 737 Satellite Orbits and Navigation 18

Calculations completed 8. Calculate the right ascension of Greenwich (the prime meridian) at time t: Greenwich = 99. 965° + 360. 985645 Dt where Dt is the time in days (and fraction) between time t and 0000 UTC 1 January 2000. 9. Calculate the longitude of the satellite: l = sat - Greenwich Homework AT 737 Satellite Orbits and Navigation 19