Spacecraft orbits and missions

General Astrophysics and Space Research Course 210142, Space Physics Module Spring 2009, Joachim Vogt Spacecraft orbits and missions Topics of this lecture Basics of celestial mechanics Geocentric orbits from LEO to GEO SSO, Lagrange points, gravity assists Appendix Review questions and further reading Additional problems and sample solutions Basics of celestial mechanics Celestial mechanics: motions and gravitational effects of celestial objects (stars, planets, moons,... ). The motion of planets around the Sun is described by Kepler s laws. Orbital mechanics or astrodynamics: motions of rockets, man-made satellites and spacecraft. Special attention is given to orbits around the Earth, but the field also deals with propulsion, transition between orbits, and interplanetary missions. Satellites: objects in orbit around a celestial body (star, planet, moon). Distinguish between natural satellites: planets around the Sun, moons around planets, moonlets around asteroids, and artificial satellites: spacecrafts in orbit around a celestial body. Orbital motion is also called revolution that has to be distinguished from planetary rotation: the Earth revolves around the Sun, and rotates around an axis that intersects the surface at the geographic poles. 1 2 Kepler I : The Law of Orbits Kepler II : The Law of Areas Planetary orbits are elliptical, with the Sun at one focus. An (imaginary) line connecting the planet to the Sun sweeps out equal areas in equal times. [Hyperphysics (1)] Important terms: foci of the ellipse, semi-major axis, semi-minor axis, eccentricity, perihelion, aphelion. The special case of zero eccentricity yields a circular orbit. [Hyperphysics (1)] The second law follows from the conservation of angular momentum L = m r v in central force fields such as the gravitational field. Planetary motion is fastest at the perihelion, and slowest at the aphelion. 3 4

Kepler III : The Law of Periods Exercise: The mass of the Sun [Hyperphysics (1)] For a planet revolving around the Sun, the square of its orbital period T is proportional to the cube of the semi-major axis a of its elliptical orbit: T 2 a 3. More precisely, we write T 2 = 4π2 a 3. GM Here GM is the product of the gravitational constant G and the mass M of the Sun. The planetary parameters T and a allow to deduce the solar mass M. Sample question: Using the Earth s orbital parameters, compute the mass of the Sun. Note that the orbit is almost circular (eccentricity e =0.0167), so you may write a =1AU=1.496 10 8 km. The Earth s orbital period is T/s = 365.25 24 3600 = 3.156 10 7, and the numerical value of the gravitational constant is G =6.673 10 11 m 3 kg 1 s 2. Hence we get for the mass of the Sun M = 4π 2 a 3 /(GT 2 ) in SI units, i.e., M kg = 4 9.87 (1.496 1011 ) 3 6.673 10 11 (3.156 10 7 ) = 1.99 2 1030. Question A: Repeat this exercise with the orbital parameters of other planets to verify the result. 5 6 General two-body problem Kepler s laws apply also to objects that orbit celestial bodies other than the Sun (e.g., satellites around planets) as long as the central body is much more massive than the satellite. General case: two bodies of masses M 1 and M 2 orbit around the common barycenter (center-of-mass), and the distance parameters are measured with respect to it. The general form of the third law is given by Further terminology T 2 = 4π 2 G(M 1 + M 2 ) a3. Other terms for the points of closest approach (farthest excursion) on an elliptical orbit are as follows. General: periaspsis or pericenter (apoapsis or apocenter). Earth: perigee (apogee). Star: periastron (apoastron). Moon: periselene or perilune (aposelene or apolune). Special terms also for other planets, galaxy, black holes,... 7 Exercise: Barycenter of the Earth-Moon system Sample question: Compute the distance D bc of the barycenter of the Earth-Moon system from the Earth s center-of-mass. Note that for two masses M 1 and M 2 located on a line at distances D 1 and D 2, D bc =(M 1 D 1 + M 2 D 2 )/(M 1 + M 2 ). Let the subscripts 1 and 2 belong to Earth and Moon parameters, respectively. If we measure distances from the Earth s center, then D 1 = 0, and the formula can be divided by M 2 to yield D bc = D 2 /(μ + 1) where μ = M 1 /M 2 is the Earth-Moon mass ratio. With μ = 81, and D 2 = 3.84 10 5 km, we obtain D bc = 4680 km. The barycenter of the Earth-Moon system is inside the Earth about 1700 km below the surface. Question B: Repeat this exercise for other planet-moon pairs in the solar system. 8

Geocentric orbits from LEO to GEO Orbital elements To completely characterize a Keplerian (unperturbed) orbit, six parameters must be specified. Keplerian elements: semi-major axis a, eccentricity e, inclination i, argument of the periapsis ω, longitude of ascending node Ω, mean anomaly M 0 (at epoch). Important categories of satellite orbits around Earth LEO Low Earth Orbits MEO Medium Earth Orbits GSO/GEO Geosynchronous/Geostationary Earth Orbits Distinguish also between polar orbits and equatorial orbits, circular orbits (small eccentricity) and highly elliptical orbits (large eccentricity). [Wikipedia (2)] Instead of M 0, other parameters are in use also. 9 10 Low Earth Orbit (LEO) Exercise: Orbital periods of LEO satellites [NASA (3)] Altitude range: altitudes below about 2000 km, practical upper limit of about 1000 km due to the increased radiation exposure (Van Allen belts), practical lower limit of about 160 km due to atmospheric drag (significant at altitudes below 500 km). LEO satellites: space stations (ISS), astronomy (Hubble ST), weather monitoring, communication (Iridium), reconnaissance (spy) missions (Keyhole, SAR-Lupe). 11 Sample question: How large is the orbital period T of an Earth satellite on a circular orbit at an altitude of h = 500 km? We apply Kepler s third law T 2 = 4π2 GM a3 (SI units) with central body mass M = M E =5.98 10 24 kg and semi-major axis a = R E + h =6.87 10 6 mto get T s = 2π 6.673 10 11 5.98 10 (6.87 24 106 ) 3/2 = 5664. The orbital period of the satellite is about 94 minutes. Question C: Repeat this exercise for an Earth satellite on a circular orbit at altitude h = 800 km. 12

Equatorial and polar LEOs Equatorial LEO: low inclination, least energy requirements. Polar LEO: high inclination, used for Earth monitoring and surveillance. Geosynchronous Orbit and Geostationary Orbit Geosynchronous Orbit (GSO) Orbital periods are exactly one sidereal Earth day: T =23 h 56 m. Ground paths are repeated once per day. At the farthest point of a GSO, the distance from the Earth s center (semi-major axis) is a =6.6 R E. GSO satellites are often used for telecommunication purposes. Geostationary Orbit (GEO) or Clarke Orbit: = circular equatorial GSO, i.e., a GSO with zero eccentricity and zero inclination. From the ground, a satellite at a GEO always occupies the same point in the sky. [GFZ/CHAMP (4)] Supersynchronous orbits: above GSO/GEO, westward drift, are used for disposal (graveyard orbits) or storage of satellites. Subsynchronous orbits: below GSO/GEO, eastward drift, can be used for station changes in an eastern direction. 13 14 Special GSOs: Tundra orbits Tundra orbits are examples of GSOs with high values of inclination and eccentricity. Geostationary Transfer Orbits (GTO) Direct insertion into GEO only by Heavy Lift Launch Vehicles (e.g., Delta IV, Space Shuttle, Proton, Ariane 5). [Chris Peat (5)] [Wikipedia (6)] Launch vehicles of smaller capacity use a geostationary transfer orbit (GTO): launch into a (circular) LEO, upper stage of the launch vehicle fires a rocket (tangent to the orbit), increase of velocity (delta-v) lifts the apogee, once in GTO, the satellite itself fires at the agogee (apogee motor) to lift the perigee. Such a procedure is also called a Hohmann transfer. 15 16

Medium Earth Orbit (MEO) The term Medium Earth Orbit (MEO) refers to the region in space between LEO and GEO. Special MEOs: Molniya Orbits A Semi-Synchronous Orbit has an orbital period of half a sidereal day (i.e., T = 11 h 58 m ). Examples are the GPS constellation, and also the Russian Molniya satellites. [Wikipedia (6)] The most common use for satellites in MEO is navigation (e.g., GPS, Glonass, Galileo). Typical example: GPS circular orbits, orbital radius 26 600 km (altitude 20 200 km), each satellite completes two orbits each day. [Wikipedia (6)] 17 18 SSO, Lagrange points, gravity assist Orbits and missions: SSO, Lagrange points, gravity assist Kepler s laws describe the motion of an object in the gravitational force field of a single idealized (spherically symmetric) celestial body. Not considered in the case of geocentric satellites are, e.g., (a) atmospheric drag, (b) non-spherically symmetric components of the Earth s gravitational field, (c) gravitational attraction of other celestial bodies (Sun, Moon,... ). Such perturbations give rise to variations of the orbital elements which are monitored and corrected in the process of station-keeping. A Sun-Synchronous Orbit (SSO) is a special polar LEO that takes advantage of an orbit perturbation imposed by (b). Lagrange points are special positions in a configuration of two bodies in circular revolution around each other, e.g., the Earth-Moon system or the Sun-Earth system. Gravity assists at the Moon or other planets are important to gain sufficient energy for interplanetary missions. Sun-Synchronous Orbits Earth s deviation from spherical symmetry is mainly due to its equatorial bulge caused by centrifugal action (which makes the planet an oblate ellipsoid rather than a perfect spheroid). For inclined (and not exactly polar) satellite orbits, this leads to a precession (i.e., a slow rotation) of the orbital plane. The precession rate matches the effect of Earth s revolution around the Sun for special choices of the altitude: typically 600 800 km, and the inclination: about 98 (i.e., an almost polar orbit but slightly retrograde revolution with respect to the sense of the Earth s rotation). Then the orbital plane is fixed with respect to the Sun which explains the term Sun-Synchronous Orbit. Caution: Do not confuse with the term Heliosynchronous Orbit : that refers to a orbital motion around the Sun (!) with a period that matches the solar rotation period. 19 20

Orbits and missions: SSO, Lagrange points, gravity assist Orbits and missions: SSO, Lagrange points, gravity assist Sun-Synchronous Orbit (cont d) Lagrange points Advantages: near-constant solar illumination conditions of the satellite s surface footprint, satellite orientation with respect to the Sun can be fixed (e.g., for a constant energy supply by solar panels, solar and astronomical observations), If two celestial bodies are in circular orbits around each other, there are five positions where another (much less massive) object can co-rotate with the configuration. [NASA/Landsat (7)] SSOs are typically used for remote sensing of the Earth surface (Landsat, ERS), meteorological satellites, spy satellites, observation of the Sun (TRACE, Yohkoh, ACRIMSat, Hinode), astronomical telescopes (IAS). [NASA/WMAP (8)] Spacecraft missions at Lagrange points L1: SOHO, ACE. L2: WMAP, and future space observatories like Herschel, Planck, and Gaia. L4 and L5: STEREO. 21 22 Orbits and missions: SSO, Lagrange points, gravity assist Orbits and missions: SSO, Lagrange points, gravity assist Gravity assists or Swing-bys: = close flybys at a planet to alter the velocity of a spacecraft. Gravity assists are also used to slow down space vehicles. Useful for missions to the inner planets, e.g., MESSENGER to Mercury. Missions to the outer planets, e.g., Cassini to Saturn, make use of gravity assists to gain speed. [NASA (9)] [JHU/APL (10)] 23 24

Orbits and missions: SSO, Lagrange points, gravity assist Figures and references Gravity assist analogy: elastic collision Light objects can change their speed substantially when they bounce off a heavy object in motion. [JPL/NASA (11)] In head-on elastic collision between a light object (mass m and velocity v) and a heavy object (mass M and velocity V ), the post-collision velocity of the light object is given by v (m M)v +2MV = m + M which simplifies to v = v +2V in the limit m/m 0. 25 (1) The illustrations of Kepler s laws are taken from the HyperPhysics web page hosted by the Department of Physics and Astronomy at Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html (URL checked on 22 February 2008). (2) Image credit: Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/image:orbit1.svg, URL checked on 22 February 2008). (3) Image credit: NASA (http://spaceflight.nasa.gov/gallery/, URL checked on 22 February 2008). (4) Image credit: Geoforschungszentrum (GFZ) Potsdam, CHAMP mission web pages (http://www.gfz-potsdam.de/champ/systems/index SYSTEMS.html, URL checked on 22 February 2008). (5) Image credit: Heavens Above web pages are developed and maintained by Chris Peat at http://www.heavens-above.com/ (URL checked on 23 February 2008). The diagram shows the orbit of Radionet-3 operated by Sirius Satellite Radio. 26 Orbits and missions: Figures and references Figures and references (cont d) (6) Image credit: Wikipedia Commons, http://commons.wikimedia.org/wiki/. File names of images: Orbits around earth scale diagram.svg, Molniya.jpg, Hohmann transfer orbit.svg (URLs checked on 23 February 2008). (7) Image credit: NASA Landsat Handbook web pages (http://landsathandbook.gsfc.nasa.gov/handbook/handbook toc.html, URL checked on 23 February 2008). (8) Image credit: NASA, public outreach pages of the WMAP mission (http://map.gsfc.nasa.gov/m mm/ob techorbit1.html, URL checked on 23 February 2008). (9) Image credit: JPL/NASA, public outreach web page of the Cassini-Huygens mission to Saturn and Titan (http://cassini-huygens.jpl.nasa.gov/mission/gravity-assists.cfm, URL checked on 24 February 2008). (10) Image taken from the MESSENGER web site at JHU/APL (http://messenger.jhuapl.edu/the mission/trajectory.html, URL checked on 24 February 2008). (11) Cartoon taken from the the JPL web page A Gravity Assist Primer at http://www2.jpl.nasa.gov/basics/grav/primer.html (URL checked on 24 February 2008). 27 Review questions and further reading Review questions Explain the following key terms of celestial and orbital mechanics: astrodynamics, natural satellites, artificial satellites, revolution, planetary rotation. Explain and discuss Kepler s three law of planetary motion. Draw an ellipse and explain the following terms: foci, semi-major and semi-minor axis, eccentricity. Which law determines the velocity along the orbit? How is the orbital period related with the semi-major axis? Which law allows you to work out the mass of the central body, and how? What is the barycenter of a two-body system? Which form does Kepler s third law assume if the two bodies are equal in mass? How is the pericenter (apocenter) of a Kepler orbit called if the central body is (a) the Sun, (b) the Earth, and (c) the Moon? Name and explain three of the six Keplerian orbital elements. What is the altitude range of Low Earth Orbits (LEOs), and which kind of satellites populate LEOs? What are the advantages and disadvantages of equatorial and polar LEOs? 28

Orbits and missions: Review questions and further reading Review questions (cont d) Define and discuss: Geosynchronous Orbit (GSO), Geostationary Orbit (GEO), Geostationary Transfer Orbit (GTO). What is the altitude range of Medium Earth Orbits (MEOs), and which kind of satellites populate MEOs? What is a Semi-Synchronous Orbit? What causes the precession of the orbital plane of satellites at Sun- Synchronous Orbits (SSOs)? Which possibilities do SSOs offer, and what kind of satellites are placed at SSOs? What are Lagrange points, and which spacecraft missions take advantage of them? What is meant by a gravity assist? Which spacecraft missions make use of this maneuvers and how? Textbooks Celestial mechanics is covered in textbooks on classical mechanics. Orbital mechanics is also addressed in printed and electronic material on spaceflight, rockets, and propulsion systems. 29 Orbits and missions: Review questions and further reading Web resources The Jet Propulsion Laboratary maintains a suite of web pages that provides a comprehensive introduction to the subjects discussed in this lecture. Visit Basics of Space Flight at http://www2.jpl.nasa.gov/basics/ David Stern s educational web pages at http://www.phy6.org/ provide information on a variety of space-related topics. In particular, have a look at the suite of pages entitled From Stargazers to Starships at http://www.phy6.org/stargaze/sintro.htm. HyperPhysics web page hosted by the Department of Physics and Astronomy at Georgia State University: http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html. A public outreach programme on space physics is coordinated at the Rice University (Project Manager: Patricia Reiff), see http://earth.rice.edu/connected/spaceupdate.html. A very comprehensive list of links is http://space.rice.edu/istp/. A nice glossary can be found on the web site of the IMAGE spacecraft, see http://pluto.space.swri.edu/image/glossary/glossary intro.html. The Open House web site of the Space Physics & Aeronomy section of the American Geophysical Union also provides useful information, see http://www.windows.ucar.edu/openhouse/open house.html. To reach the Oulu Space Physics Textbook, see http://www.oulu.fi/~spaceweb/textbook/. 30 Additional questions and problems Sample solutions of the problems Problem 1 Compute the first cosmic velocity. It is defined as the (hypothetical) orbital velocity of an Earth satellite on a circular orbit at zero altitude, i.e., with semi-major axis a =1R E (one Earth radius). Problem 2 How does the first cosmic velocity relate to the escape velocity? The latter is also called the second cosmic velocity. Problem 3 Verify that the semi-major axis of a geosynchronous orbit (GSO) is a =6.6 R E. Problem 4 How long does it take to go from Earth to Mars by means of a Hohmann tranfer orbit? Where should Mars be located when the maneuver starts? (This is a somewhat longer exercise.) Sample solution of problem 1 On a circular orbit, the velocity V 1 is given by the ratio of the circumference of the circle and the orbital period T, i.e., V 1 =2πa/T, and therefore, V1 2 = 4π2 a 2. T 2 Here a denotes the radius of the circle. We insert Kepler s third law T 2 = 4π 2 a 3 /(GM) to obtain V1 2 = 4π2 a 2 4π 2 a GM = GM 3 a and, finally, V 1 = GM/a. Inserting the values M = M E = 5.98 10 24 kg, G =6.673 10 11 m 3 kg 1 s 2, and a = R E =6.371 10 6 m yields V 1 = 7.9km/s for the first cosmic velocity. 31 32

Orbits and missions: Sample solutions Sample solution of problem 2 The escape velocity is given by V 2 = 2GM a = 2 V 1 and thus differs from the first cosmic velocity by a factor of 2=1.414. Sample solution of problem 3 We apply Kepler s third law T 2 =4π 2 a 3 /(GM) with central body mass M = M E =5.98 10 24 kg. The orbital period is given (T =23 h 56 m = 86 160 s), and we solve for the semi-major axis a and to yield a m = and a =6.6 R E. ( ) 6.673 10 11 5.98 10 24 (86160) 2 1/3 = 4.22 10 7 Sample solution of problem 4 4π 2 The solution of the problem can be found on the web page From Stargazers to Starships #21b Flight to Mars: How Long? Along what Path? by David Stern: http://www-spof.gsfc.nasa.gov/stargaze/smars1.htm. 33