Satellite geodesy ge-2112 Orbital Mechanics E Schrama

Satellite geodesy (ge-2112) Orbital Mechanics E. Schrama

Orbital Mechanics • • Astronomy Orbital mechanics Astronomic observations Earth rotation and Time systems Kepler orbital elements Groundtracks of satellites Visibility of satellites One step further than Keplerian mechanics

Astronomy • • Structure inner solar system Structure outer solar system Four big giants Difference between stars, planets, comets and galaxies • Some interesting web sites

Inner Solar System • • • Sun Mercury Venus Earth + 1 Moon Mars + 2 Moons (Source: www. fourmilab. ch/cgi-bin/uncgi/Solar)

Full Solar System • • Jupiter + 16 moons +. . . Saturn + 18 moons +. . . Uranus + 21 Moons +. . . Neptune + 8 Moons +. . . • • Pluto + 1 moon Comets Small bodies Astroid belt http: //seds. lpl. arizona. edu/nineplanets/nineplanets. html

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

Difference planet and star Sun Earth Star Planet

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

Comets Kohoutek (1974)

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.

Our galaxy 250 billion stars 100 000 ly diameter center 30000 ly thickness 700 ly

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.

Orbital Mechanics • Different views on the solar system by Nicolaus Copernicus, Tycho Brahe and Johannes Kepler • Keplers laws on orbit motions • Lab Tests on Keplers laws

Copernicus, Brahe and Kepler • In the 16 th 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. Kepler described planetary motion according to three laws.

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: http: //observe. ivv. nasa. gov/nasa/education/reference/orbits/orbit_sim. html

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

In-plane Kepler parameters Apohelium Perihelium

Kepler’s second law A B D O C ABO CDO

Kepler’s Third law The variable n represents the mean motion in rad/s, 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

Lab test • Situation: The Earth has a semi major axis at 1 AU and an excentricity e=0. 01 • Question 1: what are the values of r in the perihelium and apohelium • Question 2: what is the orbital period for a planet at a=1. 5 AU and e=0. 02 • Question 3: plot for the Earth a graph of r as a function of the true anomaly

Astronomic Observations • • • Make a photo of an object against a starry background What type of camera and film do you need? How do you design a tri-pod for the camera? What’s on the photo when an object passes the field of view? Valid objects are: star, planet, comet, galaxy, our moon, other moons, satellite, astroid, aircraft, sun • What are the consequences of precession, nutation and polar motion? • Could you synchronize your clock in this way? • Literature: Seeber p 142 -144, or surf on the web

Telescopes and tripods Declination Right ascension (time controlled)

Examples of “polar mounts”

The sky at night Pole star Horizon Seeber, pg 143 W local meridian N E

Right Ascension and declination (1) Local Meridian H Pole star Star Reference direction Situation for an observer in the field

Right ascension and declination (2) Spheric distance P to S is 90 -declination angle = Spherical distance H to S is right ascension = Pole star Right ascension: Star Declination: Vernal equinox: Aries

Celestial sphere • The position of any star is indicated by en • Right ascension is a rotation in the equatorial plane of the Celestial sphere • Declination is the rotation perpendicular to the local meridian in the celestial sphere • Pole star is on the z-axis of the celestial sphere • The “vernal equinox” defines the x-axis • The observer is in the center of the c. s.

Spherical trigonometry Z P: Pole star Z: Zenith P : Declination u: Hour angle (related to ) : geographic latitude observer S: Star S

Some remarks • The height of the pole star above the horizon is equal to the latitude of the observer • Precession and nutation affect the focal point of circles along which stars move (variations that last 26000 year or 18. 6 year respectively) • At the moment our pole star is close to the focal point, in the past it was another star • However, Polar motion changes the position of the pole star above the local horizon

Lab test • How much time does it take for a star to pass again through the local meridian (we call this a siderial day). • A star passes at noon the local meridian in Amsterdam, when will it pass the local meridian in Rome Italy? • How do you measure the astronomical azimuth of the New Church in Delft from the roof of the geodesy building? • Why do you need an accurate clock for astronomical navigation by sextant? • What is the accuracy of your clock to navigate within a 10 km error margin?

Quiz This is a picture taken with a long exposure time: 1) how long was the exposure time? 2) what is the latitude of the observer? 3) what does the picture look like 4 h later? 4) ditto 30 degrees eastward? 5) ditto 8 months later? 6) ditto 10 degrees north? 7) ditto 24 hours later? 8) The faint stripe is caused by a satellite passing through the field of view: How do you calculate the right ascension of the orbital plane?

Earth rotation and Time systems • Earth rotation dynamics is fully described by a system of ordinary differential equations (ODE). • Newton/Euler: Assumes the Earth to be a gyroscope • Liouville: takes into account flexible behavior of the matter that makes up the gyroscope • Rotation about 1 axis is far larger than for other axes • Solution of the ODE in this case is a precession of the rotation axis along a cone • Precession and nutation are caused by differential gravity effects of Moon and Sun on the Earth. • Some timesystems are linked to earth rotation

Precession and nutation Precession cone Ecliptic Equator Rotational axis : Nutation

Polar motion and LOD variations The direction of this axis is fixed in inertial space Polar Motion: The Earth moves relative to the rotation axis LOD: variations in rotation speed

Time systems • Definition of seasons and calendars • What type of clocks do we have: – Star passes through the local meridian • Earth rotation is a natural clock • Siderial day length from meridian transits • Civil 24 hour clock: 365/366 * siderial clock (explain why) – Sun passing through the local meridian • Local meridian or another reference meridian • Local time and mean time • Equation of time, and sun dials – Man made clocks: • Physics of a man made clock (oscillator and counter) • Definition of UT 1, UT 1 R, TAI, UTC, GPS

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 3 D inertial space Kepler equation Kepler elements, computational scheme Once again: reference systems.

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

Mechanics and Kepler orbits (2)

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, or “Ruimtegeodesie 2” lecture notes • Eventually: transformation orbital plane to 3 D

Orientation ellipse in inertial coordinate system XYZ: inertial cs Satellite Zi : right ascension Perigee : argument van perigee : true anomaly I: Inclination orbit plane H: angular momentum vector I Xi Right ascension Yi r: position vector satellite v: velocity satellite Nodal line Seeber p 69

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

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 See also Seeber pg 62 ev: Virtual circle M = E - e sin(E) M = n (t - t 0) This is Kepler’s equation ellipse Center E Second relation: Focus Perihelium

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)

Plate 3. 10 from Seeber Note: v in Seeber is here

Computational scheme Lab test 1: • Design an algorithm to implement this scheme (Seeber p 96 -101) • Do first all calculations in the orbital plane • Make use of the definition of the angular momentum vector • Use rotation matrices • Release the algorithm on a non trivial example Lab test 2: • Solve Kepler’s equation by successive iterations

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

Earth fixed (CTS) vs. Inertial (CIS) See also Seeber pg: 10 -17

Other coordinate systems (Seeber p 17 -25) • Topocentric: Azimuth and Zenith of a remote object in a local system (East-North-Up oriented) • Global ellipsoidal coord. systems (eg WGS-84) • Local ellipsoidal coord. systems (eg Bessel) • Heliocentric pseudo inertial systems (in case you work in the solar system) • Vertical reference systems (Geoid, Ellipsoid, etc)

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.

Visibility In general: a satellite is visible when it is above the local horizon N W E N Z E S Topocentric Geografic Lab test: compute azimuth and elevation for a station/satellite combination

One step further than Kepler • Please remember that the Kepler problem assumes a central force field with U=GM/r • In reality the gravity potential U is more difficult than that and spherical harmonics are involved. • Moreover there are other conservative and non-conservative forces that determine the motion of a spacecraft

Orbit determination Keplers theory happens to be a very good approximation to describe the motion of small particles in a gravity field as a result of the presence of a large body like the Earth or the Sun. In reality there are higher order multipoles in the gravity field and other accelerations play a role. The more complete equations of motion are therefore: Questions: - what coord system do we use? - explain all symbols - what is the range of n, m, a - calculate the gradient of V

What other accelerations? • Tidal forces cause by Sun and Moon • Gravity effect of air, water in motion etc • Radiation pressure as a result of sun light and light reflected from Earth (Albedo) • Heat radiating away from the spacecraft • Atmospheric drag • Relativistic mechanics

Solution equations of motion • Analytic – – Lagrange planetary equations Gravity Potential in Kepler elements Isolate first order solution Approximate higher order perturbations • Numeric – Conversion to system of first order ODE – Integration of system of equations

Perturbed Kepler orbits The table below lists various acceleration terms that act on the orbit of a GPS satellite, gravitational flattening is by far the largest contributer. Ref: Seeber table 3. 4

Linear perturbations by C 20 not normalised, n: mean motion Ref: Seeber p 84

Remaining perturbations • The remaining perturbations always result in oscillating functions. There are cos/sin series from which the amplitudes and phases are defined • Numerical integration is the way to go, all orbit determination s/w uses this method. • Required is an initial state vector and an acceleration model for the satellite. • To classify satellite orbits a first-order analytical solution can be used.

Classification of orbits • Sun synchronous orbits: runs as fast as the Earth’s rotation around the Sun. This is possible by tuning the a, e and I. • Golden inclination: Perigee is frozen in time • Repeating: Ground tracks reoccupy the same geographic locations after a certain time (a cycle) • Polar orbits: the orbit plane is fixed in inertial space despite the presence of gravitational flattening.