Politecnico di Torino SEEDS SpacE Exploration and Development Systems V Edition 2009 - 10 Entry, Descent, Landing and Ascent Lecture notes - Ver. 1.0.3 Author: Giulio Avanzini Dipartimento di Ingegneria Aeronautica e Spaziale e-mail: [email protected] Contents 1 Equations of motion for trajectory analysis 1 1.1 Introduction and nomenclature . 1 1.2 Reference Frames . 2 1.3 Equations of motion . 3 1.3.1 Kinematic equations . 3 1.3.2 Dynamic equations . 3 1.3.3 Forces . 4 1.3.4 Equations of motion for flight over a spherical rotating Earth . 6 1.4 Particular cases . 7 1.4.1 Free–flight (T = 0; m = const) . 7 1.4.2 Launch and entry in the equatorial plane (' = 0) . 7 1.4.3 Order of magnitude of different terms in the E.o.M. 8 1.4.4 Motion over a spherical, non{rotating Earth (! = 0) . 8 1.4.5 Motion over a flat, non{rotating planet . 9 1.5 The problem of attitude . 9 2 Atmospheric entry 11 2.1 Assumptions and formulation of the equations of motion . 11 2.2 The atmosphere . 12 2.3 Equation of motion in nondimensional form . 13 2.4 Phases of atmospheric entry . 17 2.5 Gliding Entry . 18 2.6 Ballistic entry . 21 2.7 Thermal control . 23 2.7.1 Gliding entry . 24 2.7.2 Ballistic entry . 25 2.8 Critical analysis of gliding and ballistic entry . 27 2.8.1 Gliding vs ballistic entry: historical facts . 27 2.8.2 Gliding vs ballistic entry: a comparison . 27 2.8.3 Limits of the findings . 28 3 Launch and ascent 31 3.1 Basic facts about rocket launchers . 31 3.2 Motion in free space . 33 3.2.1 Tsiolkovsky Equation . 33 3.2.2 Rocket mass{parameters . 34 3.2.3 Rocket engine performance parameters . 35 3.2.4 Burnout range . 35 3.3 Rocket equations of motion over a spherical non{rotating planet . 36 3.4 Simplified estimate of rocket performance . 37 i ii CONTENTS 3.4.1 Assumptions . 38 3.4.2 Gravity turn . 38 3.4.3 Preliminary estimate of the ∆v budget . 39 3.5 Staging . 40 Chapter 1 Equations of motion for trajectory analysis In this chapter the equations of motion that describe the dynamics of a body flying over a spherical, rotating Earth will be derived. Simplifications as a function of velocity and altitude will be discussed. 1.1 Introduction and nomenclature Both a rocket launcher and a reentry vehicle fly in a flight envelope that ranges from low{speed at low altitude up to suborbital and close-to-orbital velocities at very high altitude. In presence of an atmosphere this may result in subsonic, supersonic, or hypersonic flow regimes, in a rarefied gas at very high altitude or a dense one close to the planet surface. For this reason the equations of motion usually adopted for describing the flight of vehicles in the atmosphere may not be suitable to describe the entire trajectory, as these equations are derived under the simplifying assumptions of flight over a flat, non{rotating Earth. In this paragraph, the equations of motion will be derived in the most general framework of flight over a spherical, rotating planet with an atmosphere. In the particular case of flight (be it descent and landing or launch) in vacuum, the aerodynamic terms in the equations of motion will be dropped. Also, the terminal phases (rocket take–off and approach and landing) can be described dropping all the terms related to the so{called apparent forces and a suitable pseudo{inertial frame fixed with respect to the planet surface can be adopted. When considering a vehicle as a point mass, its motion is simply described by its position and velocity vectors, namely ~r(t) and ~v(t) = d~r(t)=dt. In what follows the position of the vehicle will always be described with respect to the centre of mass of the primary massive body (the planet). In general the mass of the vehicle will not be constant, and it will be indicated as m(t). This is the case of rocket launchers or descent vehicles for approaches to bodies without an atmosphere, where aerodynamic deceleration and gliding force are absent or negligible. Thrust is the primary driving force for accelerating the vehicle to or decelerating it from orbital velocity and this requires a considerable fuel mass rate for generating the required force. Newton's Second Law of dynamics states that the rate of variation of momentum is equal to the applied force. In mathematical terms this is expressed by the fundamental equation d~v dm m + ~v = f~ dt dt where the applied force f~ is given by the sum of three terms, 1 2 G. Avanzini - Entry, descent, landing, and ascent { 1. Equations of motion ~ ~ ~ f = f A + f T + m~g namely the aerodynamic force (subscript A), the thrust force (subscript T ), and the vehicle weight m~g. 1.2 Reference Frames Both position and velocity vectors can be expressed in a set of fixed planetocentric coordinates FP := fO; xP ; yP ; zP g, centred in the centre of mass of the planet, O, and with a fixed orientation ^ ^ ^ ^ ^ iP , jP , and kP with respect to the fixed stars. Unit vectors iP and jP lie on the equatorial ^ plane, perpendicular to the planet spin axis, kP . The constant angular velocity vector is thus ^ equal to !~ = !kP . The components of the position, velocity and angular velocity vectors in FP are indiceted by the symbols1 T T T rP = (xP ; yP ; zP ) ; vP = (vxP ; vyP ; vzP ) ; !P = (0; 0;!) The planetocentric frame represents a reasonable approximation of an inertial frame, and in this frame Newton's law achieves a compact form. At the same time, launch and reentry trajectories are more conveniently described in terms of variables that represent the vehicle motion with respect to the (rotating) surface of the planet, such as relative velocity, climb angle and heading, especially if an atmosphere is present, which for the sake of simplicity is assumed to rotate with the planet at the same angular speed. A rotating planetocentric frame can be assumed as frame of reference for the motion, FR := fO; xR; yR; zRg. It should be noted that T !R = (0; 0;!) has the same components of !P . Indicating with λ and ' the longitude and latitude of the vehicle, respectively, and with h its altitude above the surface, the position vector is expressed in FR as T rR = r^i = (r0 + h)(cos ' cos λ, cos ' sin λ, sin ') The unit vector ^i = ~r=r, together with j^, in the Eastward direction, and k^, towards North, define a third frame, the local vertical { local horizontal frame (LVLH) FT := fP; xT ; yT ; zT g, centred in the centre of mass CM of the vehicle.2 The rotation sequence that takes the rotating ^ ^ planetocentric frame FR onto the LVLH axes FT is a rotation λ about kR ≡ kP , followed by a rotation −' about j^. As a consequence, the angular rate of FT with respect to FR is _ ^ Ω~ = λkR − '_j^ which can be expressed in FT as _ _ T ΩT = (λ sin '; −';_ λ cos ') T The position vector in the LVLH frame is simply given by rT = (r; 0; 0) . 1In what follows the subscript to a boldface character indicates the reference frame in which the components 3 of the corresponding vector variable are considered. E.g., vP 2 R is the column vector made up with the components of ~v in FP . 2Most of the classical textbooks on flight dynamics choose as the LVLH frame the so called NED frame, where the x{axis points North, the y axis is directed Eastwards and the z axis lies along the direction of the local gravity acceleration. In the present case a different definition is preferred, especially because the vertical coordinate is positive upwards, a more intuitive feature when dealing with a reentry or launch problem, where altitude varies rapidly and plays a role of paramount importance in the definition of the environment surrounding the vehicle. G. Avanzini - Entry, descent, landing, and ascent { 1. Equations of motion 3 1.3 Equations of motion 1.3.1 Kinematic equations The heading angle is defined as the angle between the local meridian and the projection of the velocity vector on the local horizontal plane (0 towards North, positive towards East), while the climb angle γ represents the inclination of the velocity vector with respect to the horizontal plane (positive up). The velocity vector relative to the planet surface can thus be expressed in LVLH coordinates as T wT = v(sin γ; sin cos γ; cos cos γ) T 3 But in the LVLH frame it is also rT = (r; 0; 0) and wT = r_ T + ΩT × rT Equating the two expressions of wT , it is possible to derive the kinematic equations that provide the evolution of the position of the vehicle in terms of spherical coordinates r; λ, ' in the rotating planetocentric frame: r_ = h_ = v sin γ v cos γ sin λ_ = r cos ' v '_ = cos γ cos r 1.3.2 Dynamic equations The absolute velocity and acceleration in the rotating frame are expressed respectively as d~r = vR = r_ R + !R × rR = wR + !R × rR dt R d~v = v_ R + !R × vR dt R = r¨R + !_ R × rR + 2!R × r_ R + !R × (!R × rR) = w_ R + 2!R × wR + !R × (!R × rR) where wR = r_ R is the relative velocity and the planet rotation rate ! is assumed constant.