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Introduction to Aerospace Engineering Introduction to Aerospace Engineering Lecture slides Challenge the future 1 Introduction to Aerospace Engineering AE1-102 Dept. Space Engineering Astrodynamics & Space Missions (AS) • Prof. ir. B.A.C. Ambrosius • Ir. R. Noomen Delft University of Technology Challenge the future Part of the lecture material for this chapter originates from B.A.C. Ambrosius, R.J. Hamann, R. Scharroo, P.N.A.M. Visser and K.F. Wakker. References to “”Introduction to Flight” by J.D. Anderson will be given in footnotes where relevant. 5 - 6 Orbital mechanics: satellite orbits (1) AE1102 Introduction to Aerospace Engineering 2 | This topic is (to a large extent) covered by Chapter 8 of “Introduction to Flight” by Anderson, although notations (see next sheet) and approach can be quite different. General remarks Two aspects are important to note when working with Anderson’s “Introduction to Flight” and these lecture notes: • The derivations in these sheets are done per unit of mass, whereas in the text book (p. 603 and further) this is not the case. • Some parameter conventions are different (see table below). parameter notation in customary “Introduction to Flight” notation gravitational parameter k2 GM, or µ angular momentum h H AE1102 Introduction to Aerospace Engineering 3 | Overview • Fundamentals (equations of motion) • Elliptical orbit • Circular orbit • Parabolic orbit • Hyperbolic orbit AE1102 Introduction to Aerospace Engineering 4 | The gravity field overlaps with lectures 27 and 28 (”space environment”) of the course ae1-101, but is repeated for the relevant part here since it forms the basis of orbital dynamics. Learning goals The student should be able to: • classify satellite orbits and describe them with Kepler elements • describe and explain the planetary laws of Kepler • describe and explain the laws of Newton • compute relevant parameters (direction, range, velocity, energy, time) for the various types of Kepler orbits Lecture material: • these slides (incl. footnotes) AE1102 Introduction to Aerospace Engineering 5 | Anderson’s “Introduction to Flight” (at least the chapters on orbital mechanics) is NOT part of the material to be studied for the exam; it is “just” reference material, for further reading. Introduction Why orbital mechanics ? • Because the trajectory of a satellite is (primarily) determined by its initial position and velocity after launch • Because it is necessary to know the position (and velocity) of a satellite at any instant of time • Because the satellite orbit and its mission are intimately related AE1102 Introduction to Aerospace Engineering 6 | Satellites can perform remote-sensing (”observing from a distance”), with unparalleled coverage characteristics, and measure specific phenomena “in-situ”. If possible, the measurements have to be benchmarked/calibrated with “ground truth” observations. LEO: typically at an altitude between 200 and 2000 km. GEO: at an altitude of about 36600 km, in equatorial plane. Introduction (cnt’d) Which questions can be addressed through orbital mechanics? • What are the parameters with which one can describe a satellite orbit? • What are typical values for a Low Earth Orbit? • In what sense do they differ from those of an escape orbit? • What are the requirements on a Geostationary Earth Orbit, and what are the consequences for the orbital parameters? • What are the main differences between a LEO and a GEO, both from an orbit point of view and for the instrument? • Where is my satellite at a specific moment in time? • When can I download measurements from my satellite to my ground station? • How much time do I have available for this? • ………. AE1102 Introduction to Aerospace Engineering 7 | Some examples of relevant questions that you should be able to answer after having mastered the topics of these lectures. Fundamentals [Scienceweb, 2009] Kepler’s Laws of Planetary Motion: 1. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. 2. The line joining a planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. 3. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semi-major axes. AE1102 Introduction to Aerospace Engineering 8 | The German Johannes Kepler (1571-1630) derived these empirical relations based on observations done by Tycho Brahe, a Danish astronomer. The mathematical foundation/explanation of these 3 laws were given by Sir Isaac Newton (next sheet). Fundamentals (cnt’d) Newton’s Laws of Motion: 1. In the absence of a force, a body either is at rest or moves in a straight line with constant speed. 2. A body experiencing a force F experiences an acceleration a related to F by F = m×a, where m is the mass of the body. Alternatively, the force is proportional to the time derivative of momentum. 3. Whenever a first body exerts a force F on a second body, the second body exerts a force −F on the first body. F and −F are equal in magnitude and opposite in direction. AE1102 Introduction to Aerospace Engineering 9 | Sir Isaac Newton (1643-1727), England. Note: force F and acceleration a are written in bold, i.e. they are vectors (magnitude + direction). Fundamentals (cnt’d) Newton’s Law of Universal Gravitation: Every point mass attracts every single other point mass by a force pointing along the line connecting both points. The force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses: m m F= G 1 2 r2 AE1102 Introduction to Aerospace Engineering 10 | Note 1: so, F has a magnitude and a direction it should be written, treated as a vector. Note 2: parameter “G” represents the universal gravitational constant; G = 6.6732 × 10 -20 km 3/kg/s 2. Fundamentals (cnt’d) 3D coordinate system: • Partial overlap with “space environment” (check those sheets for conversions) • Coordinates: systems and parameters -90° ≤ δ ≤ +90° 0° ≤ λ ≤ 360° AE1102 Introduction to Aerospace Engineering 11 | Selecting a proper reference system and a set of parameters that describe a position in 3 dimensions is crucial to quantify most of the phenomena treated in this chapter, and to determine what a satellite mission will experience. Option 1: cartesian coordinates, with components x, y and z. Option 2: polar coordinates, with components r (radius, measured w.r.t. the center-of-mass of the central object; not to be confused with the altitude over its surface), δ (latitude) and λ (longitude). Fundamentals (cnt’d) Gravitational attraction: Gmρ dv Elementary force: dF = sat i r2 Total acceleration GM due to symmetrical ɺɺr = − earth Earth: r2 AE1102 Introduction to Aerospace Engineering 12 | Parameter “G” is the universal gravitational constant (6.67259×10 -11 m3/kg/s 2), “msat ” represents the mass of the satellite, “r” is the distance between the satellite and a mass element of the Earth (1 st equation) or between the satellite and the center-of-mass of the Earth (2 nd equation), “ρ” is the mass density of an element 3 24 “dv” of the Earth [kg/m ], “Mearth ” is the total mass of the Earth (5.9737×10 kg). The product of G and Mearth is commonly denoted as “µ” , which is called the 9 3 2 gravitational parameter of the Earth (=G×Mearth =398600.44×10 m /s ). Gravitational acceleration for different “planets” “planet” mass radius radial acceleration [kg] [km] [m/s 2] at surface at h=1000 km Sun 1.99 × 10 30 695990 274.15 273.36 Mercury 3.33 × 10 23 2432 3.76 3.47 Venus 4.87 × 10 24 6052 8.87 6.53 Earth 5.98 × 10 24 6378 9.80 7.33 Moon 7.35 × 10 22 1738 1.62 0.65 Mars 6.42 × 10 23 3402 3.70 2.21 Jupiter 1.90 × 10 27 70850 25.26 24.56 Saturn 5.69 × 10 26 60000 10.54 10.20 Uranus 8.74 × 10 25 25400 9.04 8.37 Neptune 1.03 × 10 26 25100AE1102 Introduction10.91 to Aerospace Engineering10.09 13 | G = 6.6732 × 10 -20 km 3/kg/s 2. Accelerations listed here are due to the central ( i.e. main) term of the gravity field only. Numerical example acceleration Question: Consider the Earth. What is the radial acceleration? 1. at sea surface 2. for an earth-observation satellite at 800 km altitude 3. for a GPS satellite at 20200 km altitude 4. for a geostationary satellite at 35800 km altitude Answers: see footnotes below (BUT TRY YOURSELF FIRST!) AE1102 Introduction to Aerospace Engineering 14 | Answers (DID YOU TRY?): 1. 9.80 m/s 2 2. 7.74 m/s 2 3. 0.56 m/s 2 4. 0.22 m/s 2 Numerical example acceleration Question: 1. Consider the situation of the Earth, the Sun and a satellite somewhere on the line connecting the two main bodies. Where is the point where the attracting forces of the Earth and the Sun, acting on the satellite, are in equilibrium? 3 2 11 Data: µEarth = 398600.4415 km /s , µSun = 1.327178×10 km 3/s 2, 1 AU (average distance Earth-Sun) = 149.6×10 6 km. Hint: trial-and-error. 2. The Moon orbits the Earth at a distance w.r.t. the center-of- mass of the Earth of about 384000 km. Still, the Sun does not pull it away from the Earth. Why? Answers: see footnotes below (BUT TRY YOURSELF FIRST!) AE1102 Introduction to Aerospace Engineering 15 | Answers (DID YOU TRY?): 1. at a distance of 258811 km form the center of the Earth 2. two reasons: the Sun not only attracts the satellite in between, but also the Earth itself, so one needs to take the difference between the two; also, the centrifugal acceleration needs to be taken into account.
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