1 Satellite observation systems and reference systems (ae4-e01) Orbit Mechanics 1 E. Schrama

2 Orbital Mechanics Reference material is Seeber, chapter 3, study only the material covered in this lecture. Basic Astronomy (this is where orbital mechanics originates from, so it is a good starting point, it is also directly related to reference systems). Orbital mechanics and Kepler orbital elements: briefly summarizes the main results of the BSc aerospace space lectures. Ground tracks of satellites and Visibility of satellites: this is what you must understand for satellite observation from Earth.

3 Basic Astronomy Planetary sciences –Structure inner solar system –Structure outer solar system –Four big giants Terminology: –Difference between stars, planets, comets and galaxies Relation to reference systems Relation to other lectures: –Physics of the Earth (ae4-876): –Planetary sciences (ae4-890)

4 Inner Solar System Sun Mercury Venus Earth + 1 Moon Mars + 2 Moons (Source:

5 Full Solar System Jupiter + 16 moons +... Saturn + 18 moons +... Uranus + 21 Moons +... Neptune + 8 Moons +... Pluto + 1 moon (dwarf planet) Comets Meteorites Asteroid belt Kuiper Belt Oort Cloud Interplanetary medium

6 Four big giants: Neptune, Uranus, Saturn, Jupiter Courtesy: JPL

7 Difference planet and star Earth Planet Star Definition parsec

8 Difference planet and star For an Earth bound observer stars follow daily circular paths relative to the pole star Planets are wandering between stars Comets are faint spots wandering between stars. Fixed faint spots are galaxies, there is a Messier catalog

9 Path of Mars in Opposition

10 Comets Kohoutek (1974)

11 Nearest galaxy: Andromeda Spiral The largest galaxy in our group is called the Andromeda Spiral. A large spiral similar to the Milky Way. It is about 2.3 million light years from Earth and contains about 400 billion stars.

12 Our galaxy 250 billion stars ly diameter center ly thickness 700 ly

13 Web links Click on any of the following links: Powers of ten University of Arizona Lunar and Planetary Laboratory Fourmilab Switzerland JPL web site ESA web site CNES web site The above links are just examples, there are many more astronomy web sites.

14 Relation to reference systems There is a direct relation to motions of the stars and reference systems With the knowledge acquired in previous lectures you should be able to draw paths of stars in the sky at night And you should be able to explain the consequences of precession, nutation, polar motion, right ascension and declination. In addition you should be able to draw the local meridian, the zero meridian, the zenith, your local latitude and longitude and the horizon.

15 Orbital Mechanics Different views on the solar system by: –Nicolaus Copernicus, –Tycho Brahe –Johannes Kepler Final conclusion was: –orbits of planets are just like any other object accelerating in a gravitational field (gun bullet physics) Kepler’s laws on orbit motions summarize the findings for small particles in a central force field It is a good starting point in discussions of satellite observation systems In the second lecture will become more complicated

16 Copernicus, Brahe and Kepler In the 16th century, the Polish astronomer Nicolaus Copernicus replaced the traditional Earth-centered view of planetary motion with one in which the Sun is at the center and the planets move around it in circles. Although the Copernican model came quite close to correctly predicting planetary motion, discrepancies existed. This became particularly evident in the case of the planet Mars, whose orbit was very accurately measured by the Danish astronomer Tycho Brahe The problem was solved by the German mathematician Johannes Kepler, who found that planetary orbits are not circles, but ellipses.

17 Physics of bullets It doesn’t seem to look like Kepler’s solution

18 Physics of satellite orbits g v

19 Kepler’s Laws Law I: Each planet revolves around the Sun in an elliptical path, with the Sun occupying one of the foci of the ellipse. Law II: The straight line joining the Sun and a planet sweeps out equal areas in equal intervals of time. Law III: The squares of the planets' orbital periods are proportional to the cubes of the semimajor axes of their orbits. Reference:

20 Kepler’s first law There are 4 cases: e=0, circle 0<e<1, ellipse e=1, parabola e>1, hyperbola Focal point ellipse

21 In-plane Kepler parameters Perihelium Apohelium

22 Kepler’s second law D B O C A ABO  CDO

23 Kepler’s Third law The variable n represents the mean motion in radians per time unit, a is the semi-major axis, G is the gravitational constant, M is the mass of the Sun, T is the orbital period of the satellite You should be able to scale this equation in different length and mass units.

24 Keplerian orbit elements Equations of motions for the Kepler problem Why is there an orbital plane? Position and velocity in the orbital plane Orientation of orbital plane in 3D inertial space Kepler equation Kepler elements, computational scheme Once again: reference systems.

25 Mechanics of Kepler orbits (1) Note: x,y and z are inertial coordinates

26 Mechanics and Kepler orbits (2)

27 Mechanics and Kepler orbits (3) A particle moves in a central force field The motion takes place within an orbital plane The solution of the equation of motion is represented in the orbital plane Substitution 1: polar coordinates in the orbital plane Substitution 2: replace r by 1/u Characteristic solution of a mathematical pendulum See also Seeber page 54 t/m 66 Eventually: transformation orbital plane to 3D

28 XiXi YiYi ZiZi Perigee Right ascension Nodal line I See Seeber p 69 Orientation ellipse in inertial coordinate system Satellite XYZ: inertial cs  : right ascension  : argument van perigee  : true anomaly I: Inclination orbit plane H: angular momentum vector r: position vector satellite v: velocity satellite

29 Velocity and Position radius r velocity v Note: in this case only , or E or M depend on time.

30 Kepler’s equation There is a difference between the definition of the true anomaly , the eccentric anomaly E and the mean anomaly M Note: do not confuse E and the eccentricity parameter e  E See also Seeber pg 62 ev: M = E - e sin(E) M = n (t - t 0 ) This is Kepler’s equation Perihelium Second relation: Virtual circle ellipse Focus Center

31 Keplerian elements Position and velocity are fully described by: –The semi major axis a –The eccentricity e –The inclination of the orbital plane I –The right ascension of the ascending node  –The argument van perigee  –An anomalistic angle in the orbit plane (mean anomaly M, Eccentric anomaly E or true anomaly  ) (Memorize a drawing)

32 Plate 3.10 from Seeber  Note: v in Seeber is  here

33 Computational scheme 1.It is almost ALWAYS possible to solve this problem 2.The method is explained in Seeber p Make use of the in-plane solution Use rotation matrices to orient orbital plane Solve Kepler’s equation to relate time to theta 3.There may be singularities in this problem and you should be smart enough to find them yourself 4.There are so-called non-singular solutions to this problem

34 XeXe YeYe ZeZe Perigee Right ascension Nodal line I Zie Seeber p 69 Orientation in an Earth fixed coordinate system Satellite (XYZ) e : Earth fixed cs  : right ascension  : G.A.S.T.  : argument of perigee  : true anomaly I: Inclination H: angular moment vector r: position vector v: velocity satellite

35 Groundtracks A ground track is a projection of a satellite on the Earth’s surface, usually we get to see sinus like patterns that slowly propagate to the West because of Earth rotation.

36 Visibility In general: a satellite is visible when it is above the local horizon N E S W N E Z TopocentricGeografic