Entry, Descent, Landing and Ascent

Home , Gravity turn, Perturbation (astronomy)

Politecnico di Torino

SEEDS SpacE Exploration and Development Systems

V Edition 2009 - 10

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 (ϕ = 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 (ω = 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 := {O; xP , yP , zP }, centred in the centre of mass of the planet, O, and with a fixed orientation ˆ ˆ ˆ ˆ ˆ iP , jP , and kP with respect to the fixed stars. Unit vectors iP and jP lie on the equatorial ˆ plane, perpendicular to the planet spin axis, kP . The constant angular velocity vector is thus ˆ equal to ω~ = ωkP . 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 ) ; ωP = (0, 0, ω)

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 := {O; xR, yR, zR}. It should be noted that T ωR = (0, 0, ω) has the same components of ωP . Indicating with λ and ϕ 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 = rˆi = (r0 + h)(cos ϕ cos λ, cos ϕ sin λ, sin ϕ)

The unit vector ˆi = ~r/r, together with jˆ, in the Eastward direction, and kˆ, towards North, deﬁne a third frame, the local vertical – local horizontal frame (LVLH) FT := {P; xT , yT , zT }, centred in the centre of mass CM of the vehicle.2 The rotation sequence that takes the rotating ˆ ˆ planetocentric frame FR onto the LVLH axes FT is a rotation λ about kR ≡ kP , followed by a rotation −ϕ about jˆ. As a consequence, the angular rate of FT with respect to FR is ˙ ˆ Ω~ = λkR − ϕ˙jˆ which can be expressed in FT as

˙ ˙ T ΩT = (λ sin ϕ, −ϕ,˙ λ cos ϕ)

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 ∈ 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 ψ 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 ψ cos γ, cos ψ 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, ϕ in the rotating planetocentric frame:

r˙ = h˙ = v sin γ v cos γ sin ψ λ˙ = r cos ϕ v ϕ˙ = cos γ cos ψ r

1.3.2 Dynamic equations The absolute velocity and acceleration in the rotating frame are expressed respectively as

d~r = vR = r˙ R + ωR × rR = wR + ωR × rR dt R d~v = v˙ R + ωR × vR dt R = r¨R + ω˙ R × rR + 2ωR × r˙ R + ωR × (ωR × rR)

= w˙ R + 2ωR × wR + ωR × (ωR × rR) where wR = r˙ R is the relative velocity and the planet rotation rate ω is assumed constant. The last equation can be rewritten in term of LVLH components,

d~v = T TRw˙ R + 2ωT × wT + ωT × (ωT × rT ) dt T where the relative acceleration T TRw˙ R is given by

T TRw˙ R = w˙ T + ΩT × wT

3 21 Recall that, given two frames F1 and F2 in relative motion, such that ω~ = ω~ is the angular velocity of F2 with respect to F1, the time derivative of a vector variable w~ expressed in the two frames are related by the following relation: „ dw~ « = T 21w˙ 1 = w˙ 2 + ω2 × w2 dt 2 where T 21 is the coordinate transformation matrix, such that w2 = T 21w1. 4 G. Avanzini - Entry, descent, landing, and ascent – 1. Equations of motion

T Letting f T = (fV , fE, fN ) be the resultant of the external forces acting on the vehicle, it is f m˙ w˙ + Ω × w + 2ω × w + ω × (ω × r ) = T − (w + ω × r ) T T T T T T T T m m T T T Carrying out the vector operations indicated in the previous equation and explicitng the resulting relations with respect to the time derivative of the relative velocity magnitudev ˙, and the time derivatives of the heading and climb angles, ψ˙ andγ ˙ , the following ordinary diﬀerential equations are obtained:4 f f f v˙ = ω2r cos ϕ(cos ϕ sin γ − cos γ cos ψ sin ϕ) + V sin γ + N cos ψ + E sin ψ cos γ + m m m m˙ − (v + ωr cos γ cos ϕ sin ψ) m v2 vγ˙ = ω2r cos ϕ(cos γ cos ϕ + cos ψ sin γ sin ϕ) + cos γ + 2ωv cos ϕ sin ψ + r f f f m˙ + V cos γ − sin γ N cos ψ + E sin ψ + ωr cos ϕ sin γ sin ψ m m m m

ω2r cos ϕ sin ϕ sin ψ v2 vψ˙ = + 2ω(sin ϕ − cos ϕ cos ψ tan γ)v + cos γ sin ψ tan ϕ + cos γ r f cos ψ − f sin ψ m˙ cos ϕ cos ψ + E N − ωr m cos γ m cos γ

1.3.3 Forces The vehicle can be acted upon by different forces during different phases of the mission. Atmo- sphere may be absent on the celestial body considered (like the Moon) or the vehicle may be so high above the planet surface that atmosphere is rarified and aerodynamic action negligible (like during the de–orbiting phase). In these cases aerodynamic terms in the equations of motion can be set to zero. The vehicle may be propelled by a rocket engine, like during launch and ascent or during the terminal deceleration before touch–down on a planet with no atmosphere. Thrust is used to vary the energy of the vehicle (lift and accelerate or decelerate it). At present all re–entry techniques in presence of atmosphere rely on aerodynamic forces to decelerate the spacecraft, either along a ballistic (high value of descent angle) or gliding trajectory. The only force which is always present is gravity.

Gravity force Gravity force m~g is directed along the local vertical −kˆ = −~r/r. When dealing with launch or entry problem it is useful to refer gravity acceleration g to its value on the surface. An inverse square gravitational ﬁeld will be assumed, with the gravity acceleration

2 ~g = −g0(r0/r) kˆ

5 where g0 is the gravity acceleration on the planet surface of (constant) radius r0.

4The equations that follows can be obtained by directly writing the acceleration in LVLH coordinates, as „ d~v « = r¨T + (ω˙ T + Ω˙ T ) × rT + 2(ωT + ΩT ) × r˙ R + (ωT + ΩT ) × [(ωT + ΩT ) × rT ] dt T If on one side the compact vector form is obtained more easily, the goniometric manipulations necessary to obtain the rate of variation of v, γ and ψ are much more complex. 5This is equivalent to the assumption of a perfectly spherically symmetric mass distribution for the planet. G. Avanzini - Entry, descent, landing, and ascent – 1. Equations of motion 5

Aerodynamic forces

Aerodynamic forces can be decomposed into three components, drag D, lift L, and side force C. This decomposition is referenced to as the representation of the aerodynamic action in terms of wind axis components. The wind axes, FW := {P; xW , yW , zW }, are centred in the centre of mass of the body, and oriented with the xW axis in the direction of the vehicle velocity with respect to the air mass, the zW axis perpendicular to xW , in the symmetry plane of the vehicle, and yW completes a right–handed triad. Drag acts in a direction opposite to the velocity vector. Lift is perpendiculat to ~v, in the symmetry plane of the vehicle. The side–force is related to a misalignment β (the so–called sideslip angle) of the velocity vector with respect to the plane of symmetry. When the vehicle is axi–symmetric, like most rocket launchers or a reentry capsule, every plane that contains the axis of symmetry is a plane of symmetry, so one can assume that we have only two components of the aerodynamic forces, namely D, along the direction of the relative wind, and L, perpendicular to it. For the sake of simplity, it will be assumed that even when the conﬁguration of the vehicle is simply symmetric with respect to the longitudinal plane (like the space shuttle or Ariane 4 and Ariane 5 launchers), the ﬂight condition remains symmetric, that is, both the sideslip angle and the side–force are zero. Aerodynamic forces can be written as

1 1 L = ρ(h)v2SC (α, M); D = ρ(h)v2SC (α, M) 2 L 2 D where CL and CD are the nondimensional lift and drag coefficients, respectively. Assuming symmetric flight (~v lies in a plane of symmetry of the vehicle =⇒ β = 0), they depend upon the angle–of–attack α, that is the direction of the velocity vector with respect to the zero–lift line, and Mach number M = v/a(h), where a is the so–called speed of sound, that is, the speed of pressure perturbation in the atmosphere.6 This is the simplest possible aerodynamic model for an entry problem. As a matter of fact, aerodynamic coefficients can depend also on angular velocity components and deflection of aerodynamic surfaces (spoiler, elevons), in which case the generation of aerodynamic force is accompanied by generation of moments that affect attitude dynamics, as it will be outlined with more details in the last paragraph of the chapter. The lift–to–drag ratio, E = L/D = CL/CD is called efficiency. It is a parameter of paramount importance when studying re–entry vehicles. It is often assumed to be 0 or close to 0 for ballistic entry (capsules), but its value can vary between 0 and 10 for gliding entry (as for the space shuttle), being very low during the initial, high–α, re–entry phase, increasing up to its maximum values during the low angle–of–attack gliding phase and at landing.

Thrust

Thrust is produced by rocket engines, that usually cannot be throttled, that is, the magnitude of the thrust force is constant as they are on–off devices. Sometimes different engines with different thrust values are employed in order to have a control action that suits different manoeuvre tasks. Thrust can be vectorized, when necessary: the exhaust nozzle is mounted on a spherical bearing and cannot be tilted in order to vary the thrust moment arm with respect to the vehicle centre of mass. This is particularly important for attitude control of rocket launchers. √ 6It can be demonstrate that if the perfect gas approximation holds for the atmosphere, it is a = γRT , where γ is the adiabatic index (the ratio of constant pressure to constant volume specific heat, γ = cp/cv), R is the gas constant (287 J kg K−1 for air) and T its absolute temperature in kelvin. 6 G. Avanzini - Entry, descent, landing, and ascent – 1. Equations of motion

Total force acting on the vehicle For trajectory analysis it will be assumed that ε is the thrust angle of attack, that is, the angle between thrust and velocity vectors.7 It will be also assumed that thrust lies always in the symmetry plane of the vehicle, that is, in the same plane with drag and lift aerodynamic force ~ components and the velocity vector. The vector f T can thus be decomposed into a tangential component (along the velocity vector) and one normal to it,

Tt = T cos ε ; Tn = T sin ε ~ where T = ||f T ||. The resulting sum of aerodynamic and propulsive force expressed in terms of wind–axis components is given by ~ ~ T (f A + f T )W = (ft, 0, fn) where ft is the component along the direction ˆt, parallel to ~v and tangent to the trajectory, while fn lies along the direction nˆ, perpendicular to ~v in the symmetry plane of the vehicle:

ft = T cos ε − D ; fn = T sin ε + L

Indicating with σ the bank angle,8 that is, the angle between nˆ and the vertical plane ~ ~ that contains both ~r and ~v, it is possible to express the sum of f A and f T in terms of LVLH components, that is, ~ ~ T (f A + f T )T = (fV , fE, fN ) Letting ~ ~ ˆ f A + f T = ftt + fnnˆ and writing the unit vectors ˆt and nˆ in terms of LVLH components as

T ˆtT = (sin γ, cos γ sin ψ, cosγ cos ψ) T nˆ T = (cos σ cos γ, − sin σ cos ψ − cos σ sin γ sin ψ, sin σ sin ψ − cos σ sin γ cos ψ) one gets

fV = ft sin γ + fn cos σ cos γ

fE = ft cos γ sin ψ − fn(sin σ cos ψ + cos σ sin γ sin ψ)

fN = ft cos γ cos ψ + fn(sin σ sin ψ − cos σ sin γ cos ψ)

1.3.4 Equations of motion for flight over a spherical rotating Earth Upon substitution of the above expressions of total aerodynamic, propulsive and gravity force components into the dynamic equation presented in the previous paragraph, coupled with the navigation equations, one gets a set of 6 first order ordinary differential equations: T cos ε − D v˙ = − g sin γ + ω2r cos ϕ (cosϕ sin γ − cos γ cos ψ sin ϕ) + m m˙ − (v + ωr cos γ cos ϕ sin ψ) m

T sin ε + L v2 vγ˙ = cos σ − g − cos γ + 2ωv cos ϕ sin ψ + m r

7Note that for axi–symmetric body it is ε = α, unless the nozzle can be tilted by an angle δ, in which case it is ε = α + δ. 8This angle is also known as the velocity roll angle. G. Avanzini - Entry, descent, landing, and ascent – 1. Equations of motion 7

m˙ + ω2r cos ϕ (cos γ cos ϕ + cos ψ sin γ sin ϕ) + ωr cos ϕ sin γ sin ψ m

T sin ε + L v2 ω2r vψ˙ = − sin σ + cos γ sin ψ tan ϕ + cos ϕ sin ϕ sin ψ + m cos γ r cos γ m˙ cos ϕ cos ψ + 2ωv (sin ϕ − cos ϕ cos ψ tan γ) − ωr m cos γ

r˙ = h˙ = v sin γ

v cos γ sin ψ λ˙ = r cos ϕ

v ϕ˙ = cos γ cos ψ r 1.4 Particular cases

1.4.1 Free–ﬂight (T = 0, m = const) When no thrust force is present, as in gliding and ballistic entry, the mass is also constant, as no propellant is used during this phase. Minor mass changes can occur when parachutes or pieces of the thermal protection system are released, but this is usually a negligible variation. In this case the equations of motion achieve a slightly simpler expression:

D v˙ = − − g sin γ + ω2r cos ϕ (cosϕ sin γ − cos γ cos ψ sin ϕ) m

L v2 vγ˙ = cos σ − g − cos γ + 2ωv cos ϕ sin ψ + ω2r cos ϕ (cos γ cos ϕ + cos ψ sin γ sin ϕ) m r

L sin σ v2 ω2r vψ˙ = − + cos γ sin ψ tan ϕ + cos ϕ sin ϕ sin ψ + 2ωv (sin ϕ − cos ϕ cos ψ tan γ) m cos γ r cos γ Kinematic equations retain the usual formulation.

1.4.2 Launch and entry in the equatorial plane (ϕ = 0) When launch (or descent on a planet with no atmosphere) is performed in the equatorial plane, ϕ = 0 and ψ = π/2, so that sin ϕ = cos ψ = 0 and cos ϕ = sin ψ = 1. In order to maintain such a condition, it is also necessary that no component of the lift is projected out of the motion plane, so that it is also σ = 0 ⇒ cos σ = 1 and sin σ = 0. In this case the equations of motion are written as T cos ε − D m˙ v˙ = − g sin γ + ω2r sin γ − (v + ωr cos γ) m m

T sin ε + L v2 m˙ vγ˙ = − g − cos γ + 2ωv + ω2r cos γ + ωr sin γ m r m

r˙ = h˙ = v sin γ

v λ˙ = cos γ r 8 G. Avanzini - Entry, descent, landing, and ascent – 1. Equations of motion

100 7 2 2 ! r ! r 2 6 2 80 v /r v /r 2!v 5 2!v g −g g −g 0 0 60 4 a/g [%]

a/g [%] 40 3 2 20 1

0 0 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 v/v v/v orb orb Figure 1.1: Relative size of acceleration terms for suborbital velocities at 20,000 m (% of g).

If entry is performed with T = 0 (and m = const), the equations of motion can be further simpliﬁed: D v˙ = − − g sin γ + ω2r sin γ m

L v2 vγ˙ = − g − cos γ + 2ωv + ω2r cos γ m r

r˙ = h˙ = v sin γ

v λ˙ = cos γ r

1.4.3 Order of magnitude of diﬀerent terms in the E.o.M. −5 −1 For the Earth case (r⊕ = 6, 371 km, ω⊕ = 2π/24 h = 7.292 10 rad s ), the relative size of diﬀerent acceleration terms in the previous equations is analyzed in Fig. 1.1, where the values of acceleration terms induced by Earth motion (namely transport ω2r and Coriolis 2ωv acceleration) and Earth’s curvature (the centrifugal acceleration v2/r) are compared. Note that the variation of gravity acceleration g with respect to its value on Earth’s surface g0 and transport acceleration are below 1% and can be neglected. Coriolis acceleration is sizable, if it achieves its maximum value for motion on the equatorial plane, as ~v and ω~ are perpendicular. But in general its magnitude depends on the orientation of the velocity vector with respect to the planet surface (angles γ and ψ), while the centrifugal term v2/r does not. For high suborbital speed, only the centrifugal term is of the same order of magnitude of g, equating g itself for v = vorb.

1.4.4 Motion over a spherical, non–rotating Earth (ω = 0)

Although for low suborbital speed (v < 0.1vorb, as in supersonic ﬂight at high altitude) Coriolis acceleration is the most important term, it is relatively customary to neglect it, when dealing with preliminary entry or launch calculations. Neglecting both Coriolis and transport acceleration it is like assuming that the planet is non–rotating. This assumption greatly simpliﬁes the equations of motion and allows for an analytical evaluation of entry performance. In this case it is possible to set ω = 0 in every equation, and one obtains the following set of equations of motion: T cos ε − D m˙ v˙ = − g sin γ − v m m G. Avanzini - Entry, descent, landing, and ascent – 1. Equations of motion 9

T sin ε + L v2 vγ˙ = − g − cos γ m r

r˙ = h˙ = v sin γ

v λ˙ = cos γ r It should be noted that, although these equations were derived starting from those written for the equatorial plane, any term depending upon the direction ψ of the velocity vector with respect to the local meridian disappeared, so that the above equations are valid for motion constrained over any plane containing a great circle.

1.4.5 Motion over a ﬂat, non–rotating planet For relatively low speed (low supersonic and subsonic range, M < 3) at low altitude (h < 15, 000 m) all the apparent forces become negligible, that is less than 1% of g. In this case motion can be described as taking place over a ﬂat, non–rotating Earth, where also v2/r → 0:

T cos ε − D v˙ = − g sin γ m

T sin ε + L vγ˙ = cos σ − g cos γ m

T sin ε + L vψ˙ = − sin σ m cos γ

r˙ = h˙ = v sin γ

x˙ = v cos γ sin ψ

y˙ = v cos γ cos ψ

This model can be used for terminal phases of ﬂight, right after launch or in the last phase before touch–down. In this case x and y are cross ranges in the East and North direction, respectively.

1.5 The problem of attitude

When dealing with descent, landing and ascent the trajectory of the centre of mass of the vehicle is strongly coupled with vehicle attitude: direction and size of forces acting on it strongly depend upon the angular displacement from the direction of the velocity vector (angles α and ε). This means that in order to control the trajectory it is necessary to properly aim the sum of aerodynamic force and thrust (or only one of them, if the other one is absent in the particular case) by controlling vehicle attitude. Attitude dynamics can be analyzed starting from ﬁrst principles, namely angular momentum (h~ ) balance. Assuming that the vehicle is a rigid body acted upon by a total torque m~ , it is possible to write the vector equation dh~ = m~ dt 10 G. Avanzini - Entry, descent, landing, and ascent – 1. Equations of motion in a set of body axes, that is a set of axes ﬁxed with respect to the body and centred in its centre of mass: a a a Jω˙ B + ωB × JωB = mB where J is the inertia tensor, and the absolute angular velocity

a r ωB = ωB + ΩB + ωB is obtained as the sum of the angular velocity of the Earth with respect to the inertial frame FI , the angular velocity of the LVLH frame with respect to the Earth and, ﬁnally, the relative angular velocity of the vehicle with respect to LVLH axes. a As a matter of fact, ||ωB|| is usually close to zero during (quasi) steady phases of motion, when the vehicle must maintain a certain attitude, so that it is possible to substitute the dynamic equation with an equilibrium condition mB = 0. At the same time, when attitude maneuver are necessary, the relative angular velocity is several orders of magnitude higher than the other a contributions to ωB, that can thus be neglected. For this reason attitude motion is usually referred directly to the LVLH frame,

r r r Jω˙ B + ωB × JωB = mB where either Euler angles or quaternions are used for describing the attitude. Aerodynamic torques acting on the spacecraft, when present, depend on the relative velocity of the vehicle with respect to the atmosphere, on its angular motion and the displacement of control surfaces, such as spoilers or elevons, the use of the latter ones being limited to the final phase of gliding entry when the lower, more dense layers of the atmosphere are reached. Aerodynamic torque can be created also by mass displacement: a ballast is moved from its nominal position so as to shift the centre of mass of the vehicle and change the moment arm of aerodynamic forces with respect to the CM. In this case it is necessary to develop an ad hoc model for systems with a moving CM.9 The propulsive system can deliver control torques, either tilting the main exhaust nozzle of the vehicle or using a reaction control system (RCS) made with pairs of thrusters that produce a net torque about a given axis. RCS is always used during the first phase of planetary entry, when atmosphere density is very low, and it is practically the only viable option when there is no atmosphere at all. In the first case, attitude control is of paramount importance in order to keep the vehicle in the correct position during entry at hypersonic velocity. The available tolerance is tiny and any error can compromise the efficiency of the thermal protection system. When there is no atmosphere, as landing on the Moon, RCS is used in order to correctly point the thrust vector delivered by the main engine.

In order to avoid an excessive burden in the mathematical description of entry, descent and launch manoeuvres, attitude motion will not be considered in the models used in the sequel, by assuming that a control system succesfully maintain the prescribed attitude during the ﬂight. As a consequence, the angle–of–attack and the bank angle will be considered as control variables for the trajectory problem. This approach separates the time–scales of the evolution of trajectory on one side (slow variables), and attitude on the other one (fast variables).

9This can be done by using the generalized Euler equation. See P.C. Hughes, Spacecraft attitude dynamics, J. Wiley, New York, 1986. Chapter 2

Atmospheric entry

Under quite general assumptions, the equations of motion over a spherical planet can be simplified and some analytical results can be derived for the case of entry in a planet with an atmosphere. In this case drag is used for slowing down the vehicle, thrust is zero and the mass remains constant. Two techniques will be considered, ballistic entry at high flight path angles and gliding entry at small flight path angles. In both cases peak values of the heat flux and total heat absorbed must be kept under control.

2.1 Assumptions and formulation of the equations of motion

As stated in the previous chapter, it is usually possible to simplify the equations of motion for flight over a spherical, rotating planet neglecting the terms ω2r. For preliminary analysis purposes it is actually possible to neglect Coriolis acceleration as well, so that a simplifying assumption of flight over a non–rotating spherical planet can be introduced. Moreover, for atmospheric entry thrust is not used to control the vehicle trajectory, inasmuch as the velocity is reduced by means of energy dissipation between the highest layers of the atmosphere, where velocity is still in the range of orbital speed, and the lowest layers, where velocity must already be small enough to allow the vehicle structure to stand the dynamic pressure induced by the more dense gas surrounding it. For entry in the Earth atmosphere this means a variation between initial velocities in the order of several km s−1 and terminal speed around 5 m s−1, for reentry capsules, and 100 m s−1, for the Space Shuttle. It should be remembered that, for the capsule case, the last portion of the deceleration is achieved by use of braking systems such as drouges and parachutes. If finally the motion takes place in a plane containing a great circle, the set of nonlinear ordinary differential equations that describe the motion are given by

ρSC v2 v˙ = − D − g sin γ 2m

ρSC v2 v2 vγ˙ = L − g − cos γ 2m r

r˙ = h˙ = v sin γ

v θ˙ = cos γ r with the usual meaning of the symbols, where the generic angular variable θ replaced the longitude λ for the sake of generality.

11 12 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

Table 2.1: Layers of Earth’s atmosphere

h Name Temperature Lapse rate [km] [◦C] [◦C/km] 0 Sea level 15.0 ———– ↓ Troposphere ↓ −6.5 11 Tropopause -56.5 ———– ↓ Stratosphere (I) ↓ 0.0 20 -56.5 ———– ↓ Stratosphere (II) ↓ +1.0 32 -44.5 ———– . Stratosphere (III) ↓ +2.8 47 → 51 Stratopause -2.5 — 0.0 — . Mesosphere (I) ↓ −2.8 71 -58.5 ———– ↓ Mesosphere (II) ↓ −2.0 84.852 Mesopause -86.2 ———– ↓ Thermosphere ↓ > 0

2.2 The atmosphere

The density of the atmosphere varies significantly from the planet surface, where it is maximum, up to the outermost layers, where it is close to 0. The outer edge of the atmosphere cannot be defined as it is a blurred boundary. In what follows, this boundary will be assumed to be the height of the reentry trajectory where deceleration due to drag becomes sizable, a data that for reentry in the Earth atmosphere is approximately 100 km above our planet’s surface. Assuming an atmosphere in hydrostatic equilibrium made by a perfect gas (so that p/ρ = RΘ, where p is static pressure, R is the gas constant, and Θ its absolute temperature), it is dp dp g = −ρg ⇒ = − dr dr p RΘ The latter equation is at the basis of the definition of the atmosphere model. For the Earth case, an International Standard Atmosphere (ISA) model is defined where the temperature profile Θ(h) is assumed to be piecewise linear. The piecewise constant derivative ∂Θ∂h is called the lapse rate (Tab.2.1). 1 By integrating the hydrostatic equation from the reference ground level, r = r0 ⇒ h = 0, up to the assumed atmosphere boundary, the pressure profile p(h) is obtained, and the density profile ρ(h) is then determined from the ideal gas state law.

A simpler expression for the variation of ρ can be obtained assuming a locally exponential law. From the gas state law it is dp dρ dΘ = + p ρ Θ By substitution of the above diﬀerential relation into hydrostatic law one gets dρ dΘ g = − + dr ρ Θ RΘ

1Sea level is assumed, for the Earth case. G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 13 that is dρ dr = − ρ H where 1 dΘ g −1 H = + Θ dr RΘ is the scale height of density variation. In the most general case, H depends on the altitude above the planet surface (that is, equivalently, on the magnitude of the position vector), but it can be considered constant over a suﬃciently small altitude range, so that in this range it is ρ(h) ≈ ρ¯exp(−(h − h¯)/H) where h¯ is a reference altitude andρ ¯ is the exact density at that altitude.

Assuming an exponential density variation over a wide range of altitude (i.e. H constant) results in an oversimpliﬁed model. If, on the converse, an isothermal atmosphere is assumed, the product gH is exactly constant.

2.3 Equation of motion in nondimensional form

By application of the chain rule, it is possible to express the derivatives of v2, γ and r with respect to θ as d(v2) dv dv dt dγ dγ dt dr dr dt = 2v = 2v ; = ; = dθ dθ dt dθ dθ dt dθ dθ dt dθ Upon substitution of the time derivatives of v, γ and r as obtained from the equations of motion in the last expressions, by taking into account the inverse of the fourth one,

dt dθ −1 r = = dθ dt v cos γ one gets

d(v2) ρSC rv2 = − D − 2gr tan γ dθ m cos γ

dγ ρSC r gr = L + 1 − dθ 2m cos γ v2

dr = r tan γ dθ In order to derive some analytical results about the problem of atmospheric entry, it is convenient to reformulate the equations of motion in nondimensional form, by deﬁning two nondimensional variables, namely

ρSC √ v2 Z = D rH ; u = 2m gr

The variation of Z is ruled by the (approximately exponential) variation of ρ, so that it can be considered as the altitude variable.2 The nondimensional speed u is given by the square of the ratio between the current vehicle velocity and orbital speed at the considered altitude.

2Chapman, D.R., An Approximate Analytical Method for Studying Entry into Planetary Atmosphere, NASA TR R-11, 1959. 14 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

The derivative of the nondimensional parameters u and Z with respect to θ can now be 2 determined. Taking into account that gr = g0r0/r, for the velocity variable one gets 2 2 2 2 du 1 d(v ) v dr ρSCDr v v = + 2 = − − 2 tan γ + tan γ dθ gr dθ g0r0 dθ m cos γ gr gr By making use of the deﬁnitions of Z and u, it is du 2Zu r r = − − (2 − u) tan γ dθ cos γ H

Assuming that CD is constant and expressing the variation of ρ according to the differential law dρ/ρ = dr/H, the derivative od Z with respect to θ is given by dZ d ρSC √ dρ dr SC √ ρSC 1 dr dH dr = D rH = D rH + D √ H + r dθ dθ 2m dr dθ 2m 2m 2 rH dθ dr dθ Taking into account the definition of Z, the expression of dr/dθ and that the derivative of ρ with respect to the radius can be expressed in the form dρ/dr = −ρ/H, one gets dZ r H 1 dH = −Z tan γ 1 − − dθ H 2r 2 dr For a strictly exponential atmosphere H is constant and dH/dr = 0, but this is too a rough an approximation. For a (at least locally) isothermal atmosphere it is H = RΘ/g, so that the product gH is constant and dH/dr = 2H/r, so that one can write dZ r 3H = −Z tan γ 1 − dθ H 2r In all the considered cases, the variables H and r always appear together as r/H. This ratio varies as a function of the altitude. For the Earth’s atmosphere its value lies between 750 and 1300, with an average value of 900. Using this average value results in a better approximation, than by assuming an exponential atmosphere. Moreover, r/H is usually a large number, so that its inverse can be neglected with respect to 1 in the expression of dZ/dθ, which thus becomes dZ r = − Z tan γ dθ H Finally, it is easy to write the equation for dγ/dθ in terms of nondimensional variables, as √ dγ ρSC r C rH gr r r C Z 1 L D√ L = + 1 − 2 = + 1 − dθ 2m cos γ CD rH v H CD cos γ u Thus, a simple, nondimensional model is now available for the analysis of reentry trajectories: du 2Zu r r = − − (2 − u) tan γ dθ cos γ H dZ r = − Z tan γ dθ H dγ r r C Z 1 = L + 1 − dθ H CD cos γ u Note that this equations can be applied to any vehicle, regardless of its actual mass and size, provided that an estimate of its aerodynamic efficiency and a reasonable value for the ratio r/H are available. Moreover, some important physical quantities can now be expressed in terms of the nondimensional variables Z and u. G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 15

Deceleration and load factor From the expression of dv/dt, it is possible to derive the value of the tangential acceleration, scaled with respect to gravity acceleration,3

1 dv r r − = Zu + sin γ g dt H p For small ﬂight–path angles, the second term is negligible and (at/g) ≈ Zu r/H. This means that the tangential deceleration, that is the rate od reduction of the velocity, is simply proportional to the product Zu. The load factor acting on the vehicle, that is the ratio between the total aerodynamic force and vehicle weight mg, is given by

s v 2 2 u " 2# a L D u r CL = + = Zut 1 + g mg mg H CD and, for√ small ﬂight–path angles, is thus equal to the tangential acceleration multiplied by a factor 1 + E2.

Dynamic pressure Dynamic pressure, 0.5ρv2, is also a quantity of interest, as loads on the vehicle structure can usually be expressed in terms of dynamic pressure times some form factor. Looking at the equation of dv/dt, it is quite easy to derive the following expression,

1 mg r r ρv2 = Zu 2 SCD H where it is apparent that dynamic pressure is proportional to vehicle deceleration.

Heat flux and heat absorbed Together with mechanical loads acting on the structure, the estimate of thermal loads during reentry is of paramount importance. According to the similarity law, the heat flux q on the body is a fraction of the heat flux at the stagnation point, qs, so that

q = kqs with k < 1 and K ρ n v m qs = √ √ R ρ0 gr where R is the radius of curvature at the stagnation point and the constants K, n, and m depends on the flow characteristics in the boundary layer. For laminar flow it is possible to assume n = 1/2 while a simple power law with m = 3 can be used for the speed variable. By using a strictly exponential approximation for density variation and the definition of Z it is possible to express the density ratio as

ρ 2m Z 2m Z = √ ≈ √ ρ0 SCDρ0 rH SCDρ0 r0H

3That is, expressing it in number of g’s. 16 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry where the last approximation is justified by the fact that over a reentry trajectory the variation of r within the atmosphere is a small fraction of the planet radius r0. It is then possible to express the heat flux as K 2m Z 1/2 q = k √ √ u3/2 R SCDρ0 r0H The last equation can be reorganised in the following form: " # K 2m 1/2 q = k Z1/2u3/2 2 1/4 (ρ0r0H) RSCD where the heat flux is obtained as the product of three terms. The first term represents the effects of the characteristics of the planet’s atmosphere, while the second one depends on physical characteristics of the vehicle (namely mass, shape and dimensions). The third factor represents the effects of the particular trajectory flown, so that, for a given vehicle and a given planet, the only parameter necessary for a qualitative analysis of heat fluxes is √ q¯ = Zu3

If the vehicle operates at radiation equilibrium temperature (that is, the temperature is allowed to increase until the amount of heat radiated becomes equal to that absorbed because of convection with the outer flow), the heat flux equation is sufficient for the analysis. If on the converse the vehicle is allowed to absorb only a finite quantity of heat (heat–sink type), than it is also necessary to evaluate the total heat absorbed, Q. This quantity is given by the double integral Z t Z Z t Z Q = q(t, P )dA dt = kqs(t, P )dA dt tE S tE S where S is the surface exposed to the flow and tE is the atmosphere entry and and t < tTD is any time instant prior to touch–down time. It is possible to define a shape factor c, 1 Z 1 Z q c = kdA = dA A S A S qs where A is the wetted area. For a hemisphere, it is c ≈ 0.5. The heat absorbed can now be expressed in terms of a simpler time integral Z t Q = cA qsdt tE where the previous expression of q can be used for qs = q/k. By making this last substitution one gets " # K 2m 1/2 Z t √ Q = cA Zu3dt 2 1/4 (ρ0r0H) RSCD tE For small flight–path angles the integration can be performed with respect to the speed variable, as it is du 2Zu r r = − dθ cos γ H At the same time, it is dθ v √ rg = cos γ = u cos γ dt r r so that du 2Zu r r √ rg r g = − u cos γ = − 2 Zu3/2 dt cos γ H r H G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 17

It is then possible to operate a change in the integration variable, by substituting the time increment dt with its value expressed in terms of speed variable increment, s 1 H dt = − du 2Zu3/2 g so that the equation for the heat absorbed can be written in the form " #" # K H 1/4 2m 1/2 Z uE du Q = 2 2 cA √ 2 g0ρ0r0 RSCD u Z so that the nondimensional quantity

Z uE du Q¯ = √ u Z is suﬃcient for a parametric study.

Altitude and time–of–flight As a by–products of the previous analysis, it is possible to derive the expressions for altitude variation and time–of–flight between two velocities. Altitude is obtained under the hypothesis of strictly exponential atmosphere, where given the altitude vatiable Z it is ρ h 2m Z Z = exp − ≈ √ = 0 ρ0 H SCDρ0 r0H Z By taking the logarithm of the last expression, one gets Z h = H log 0 Z Time–of–flight can be obtained from the integration of the time increment dt expressed in terms of speed variable increment, that is, s 1 H Z uE du ∆t = 3/2 2 g u Zu

2.4 Phases of atmospheric entry

Strictly speaking, an atmospheric reentry trajectory is the terminal part of a trajectory aimed at slowing down a vehicle from suborbital speed to a velocity compatible with either the use of aerodynamic decelerators (parachutes and drogues) or a gliding phase for a controlled, airplane– like landing. The sequence of events usually starts well outside of the atmosphere with a de–orbiting manoeuvre, that takes the vehicle on a ballistic trajectory with a periapsis smaller than the planet radius. This means that at a certain point the vehicle will enter the planet atmosphere with a certain flight–path angle. The actual atmospheric entry conventionally starts at an altitude where the density of the atmosphere is sufficient to produce a sizable deceleration on the vehicle (for the Earth atmosphere, a height between 100 and 120 km). Two techniques can be used for deceleration: (i) a gliding entry at small flight–path angle, which means that together with the drag, a non negligible lift force is produced by the flow on the vehicle, or (ii) a ballistic entry at large flight–path angles, where the lift force is almost zero. 18 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

Depending on the technique adopted for the reentry, it is possible to perform some further simplifications on the equations of motion, so that first order, analytical solution can be derived, which are useful for preliminary design purposes, with the evaluation of physical quantities such as maximum deceleration and heat fluxes or dynamic pressure peak. The final part of the reentry trajectory depends on the type of trajectory flown before. For lifting bodies, gliding flight is ended by a controlled landing manoeuvre on a runway similar to that of a conventional airplane (although touch–down velocities are usually relatively high). Non–lifting bodies are decelerated by a one or more parachutes down to a velocity of few meters per second (5 to 7 m s−1), and the trajectory ends with a non–controlled hard manding on the ground (Soyuz case) or a splashdown (Apollo missions). The impact velocity is maintained relatively high, in order to avoid an excessive unpredictable effect of wind on the touch–down point.

2.5 Gliding Entry

For the gliding entry case, the small ﬂight–path angle assumption allows to simplify the equations of motion, by writing cos γ ≈ 1 and sin γ ≈ tan γ ≈ γ. Moreover, the term (2 − u) tan γ in the equation of the velocity variable is assumed negligible. The equations of motion are thus written as

du r r = −2Zu − (2 − u)γ dθ H dZ r = − Zγ dθ H dγ r r C 1 = L Z + 1 − dθ H CD u where the aerodynamic eﬃciency E = CL/CD is also assumed constant. Also, the initial value of the altitude variable Z is assumed to be close to 0, as the density is very small. The variation of the ﬂight–path angle is so small and slow that its time derivative is almost zero. This is equivalent to consider the vehicle in quasi–equilibrium along the direction normal to the trajectory. Assuming dγ/dθ ≈ 0 it is possible to solve the third equation for the altitude variable Z, r r r C 1 1 − u H C L Z + 1 − ≈ 0 ⇒ Z¯(u) = D H CD u u r CL The last relation can be written in a form valid for any planet, as

r 2 ¯ r 1 − (v/vc) CD Z = 2 H (v/vc) CL √ √ where vc = gr ≈ g0r0 is the orbital velocity. As the variation of r is relatively small, the orbital velocity at the planet surface can be assumed as the reference velocity. The variation of the altitude variable Zpr/H as a function of velocity is reported in Fig 2.1, where it is clear that speed monotonically decreases with altitude (remembering from the definition of Z that the higher Z, the lower is the altitude). In this respect, for u → 0 the equilibrium glide causes Z to diverge towards infinity for a constant value of the aerodynamic efficiency E. Also, higher E provides a higher gliding altitude, for the same velocity. Finally note that for a lifting vehicle the reference area S is usually the wing planform area. As a consequence, a higher wing loading mg/S will cause a faster altitude loss, since for a given value of Z, a higher ρ is obtained, which means a lower flight altitude. G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 19

30 1/2

Z(r/H) 20 E =1 2

10 3 4 5 0 0 0.2 0.4 0.6 0.8 1 v/v c Figure 2.1: Variation of the altitude variable Z with velocity.

An estimate of the actual variation of altitude above the planet surface can be obtained under the assumption of an exponential atmosphere from the equation h = H log (Z0/Z). At the same time, the (slow) variation of the ﬂight–path angle can be obtained by writing

du dZ¯ −1 C r r = = L u2 dZ du CD H The same derivative can be evaluated from the equations of motion as

du du dθ 2Z¯(u)upr/H − (2 − u)γ = = dZ dθ dZ (r/H)Z¯(u)γ where the gliding equilibrium value of Z¯ = Z¯(u) can be introduced. By equating the two expression, the following equation is obtained r H 2 2 − u C C r r u + L = L u2 r γ 1 − u CD CD H which can be solved with respect to γ, yelding

C 2(1 − u) γ = − D CL (r/H)u(1 − u) + (2 − u) When the velocity is a non negligible fraction of the orbital speed and u = O(1), the fligh–path angle remains very small, because of the large value of the parameter r/H. When flying at low speed close to the planet surface, u becomes very small, u ≈ 0, and the flight–path angle become 1 1 γ ≈ − = CL/DD E the classical results obtained in flight–mechanics textbooks for gliding flight. In such a case, for slow speed and small values of aerodynamic efficiency, the small flight–path angle assumption does not hold any more (Fig. 2.2). Both this latter fact and the infinite growth of Z for u → 0 20 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

5 4 −5 3 2 E = 1 −10 [deg] !

−15

−20 0 0.2 0.4 0.6 0.8 1 v/v c Figure 2.2: Variation of the ﬂight–path angle γ with velocity. indicates that a more accurate model is necessary to describe the terminal phase of the reentry trajectory. The range can be estimated by assuming that the term (2 − u)γ is negligible in the equation of du/dθ, that is du r r ≈ −2Zu dθ H Using the solution Z¯(u) in the previous equation, one gets

dθ 1 C 1 = − L du 2 CD 1 − u which can be easily integrated between the initial entry velocity uE and the current one, u, so that 1 C 1 − u θ(u) = L log 2 CD 1 − uE From this equation it is apparent that the glide range increases with aerodynamic eﬃciency. Deceleration at/g and load factor a/g achieve also simple expression, by use of the equilibrium value of the altitude variable Z¯(u), where

a r r 1 − u 1 − v2/(g r ) t = Z¯(u)u = = 0 0 g H CL/CD CL/CD v u " 2# s 2 s a ¯ u r CL 1 v 1 = Z(u)ut 1 + = (1 − u) 1 + 2 = 1 − 1 + 2 g H CD (CL/CD) g0r0 (CL/CD)

Both deceleration and load factor are minimumum during the entry phase, when 1−u is close to zero (v is still in the high suborbital speed range), being steadily increasing during the reentry, and maximum at the end. The maximum deceleration and load factor are reduced by using maximum lift–to–drag ratio. G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 21

2.6 Ballistic entry

For ballistic entry we assume that the vehicle does not generate a lift force, so that CL = 0, and the drag force is assumed to be signiﬁcantly greater than the gravity force. The pertinent equations of motion thus become

du 2Zu r r 2Zu r r = − − (2 − u) tan γ ≈ − dθ cos γ H cos γ H dZ r = − Z tan γ dθ H dγ u − 1 = dθ u In the sequel the altitude variable Z will be used as the independent variable, while the initial ﬂight–path angle γE will be the free parameter that characterize the subsequent trajectory. From the second and third equation one gets dγ dγ dθ 1 − u H cos γ = = dZ dθ dZ Zu r sin γ This equation can be reorganized as d 1 − u H [log(cos γ)] = dZ Zu r Although initially Z ≈ 0, and the left hand term is consequently large, as soon as Z becomes non–negligible, because of the increase in ρ as the altitude rapidly decrease during the steep entry at high velocity, the large value of the ratio r/H makes the right hand side negligibly small, which means that a ﬁrst order solution for the ballistic entry dictates that

cos γ ≈ cos γE From the physical point of view, the very initial phase of the deceleration is not correctly represented, but after this, the trajectory is essentially an arc of a logarithmic spiral.

As for the velocity variable u, it is possible to define its variation as a function of Z by taking the ratio of the first two equations of motion, r du du dθ 2u H = = dZ dθ dZ sin γ r which assuming a constant flight–path angle sin γ = sin γE can be integrated analytically, yelding r ! r ! u 2Z H v Z H = exp ⇒ = exp uE sin γE r vE sin γE r where ZE ≈ 0 was assumed. This equation completes the first–order approximation of the trajectory for a ballistic reentry. Note that for steeper trajectories, the deceleration is faster, since the (negative) argument√ of the exponential becomes higher in modulus. Finally, from the definition of Z = ρSCD/(2m) rH, it is clear that, for a given value of altitude (that is, ρ) Z is larger for higher drag force and smaller for higher aerodynamic load mg/S, so that speed decreases more rapidly for higher drag coefficients and smaller loads. Figure 2.3 represents grafically the variation of√ speed as a function of the altitude variable. Note that knowing the value of Z0 = ρ0SCD/(2m) r0H (that is, the value of Z on the surface) allows one to determine the velocity in proximity of the planet surface. 22 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

1 10 −30 −40 −60 −89

0 10

1/2 −20 −10 −1 10 ! = −5 deg Z(r/H)

−2 10

−3 10 0 0.2 0.4 0.6 0.8 1 v/v c Figure 2.3: Variation of the altitude variable Z with velocity for ballistic reentry.

Finally, the deceleration can be analytically determined in the framework of this ﬁrst order approximation as r ! a r r r r 2Z H = Zu = ZuE exp g H H sin γE r From the analysis of the sign of the derivative r !" r # d a r r 2Z H 2Z H = uE exp 1 + dZ g H sin γE r sin γE r it is possible to see that the acceleration steadily increases until it becomes maximum for the critical value of the altitude variable, r H 1 Z∗ = − sin γ r 2 E that corresponds to a maximum for the deceleration,

a u sin γ r r = − E E g max 2e H The velocity at this point is

∗ u 1 ∗ 1 = ⇒ v = √ vE ≈ 0.607vE uE e e This means that the maximumu deceleration is achieved when the vehicle has lost 40% of its initial speed. In order to be feasible, the critical value of the altitude variable must correspond to a positive ∗ value of altitude h, that is ρ(h ) < ρ0, where ρ0 is the value of atmospheric density on the planet surface. Taking into account the deﬁnition of Z this is true only if ρ SC H − sin γ ≤ 0 D E m G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 23

1 10

! = −89 deg 0 10 −40 −60 −20 −30 −10 1/2 −1 −5 10 Z(r/H)

−2 10

−3 10 0 0.05 0.1 0.15 0.2 aH/(u gr) E Figure 2.4: Variation of deceleration parameter with the altitude variable Z for ballistic reentry.

It this inequality is not satisfied, the acceleration grows steadily, and it is maximum right before impact on the planet surface. Figure 2.4 reports the deceleration profile as a function of the altitude variable for various entry flight–path angles. The dotted line indicates the envelope of the peak deceleration values.

2.7 Thermal control

Thermal control represents one of the major concern during a reentry trajectory, especially for manned vehicles. In this latter case, together with the usual problem of vehicle integrity, severe cabin temperature constraints need also to be considered. For this reason, a reentry vehicle is always equipped with a thermal protection system (TPS), which usually represent a significant portion of the total vehicle weight. Sizing of the TPS is of paramount importance, as it is necessary to keep its weight as low as possible, in oder not to penalize the payload weight available on board of the vehicle, while maintaining a sufficient safety margin. The trade–off is usually not trivial, and it is important to evaluate as soon as possible the requirement for the TPS during the mission design phase. Three quantities need to be kept under control:

• local heat ﬂux, q, and in particular the peak value qmax reached on any point of the body during the trajectory;

• average heat ﬂux, qav on the body and its maximum value;

• heat load, Q, that is, the total heat absorbed.

Values for q and Q as a function of the non–dimensional parameters Z and u were derived in Section 2.3, and are recalled here: " # K 2m 1/2 q = k Z1/2u3/2 2 1/4 (ρ0r0H) RSCD 24 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

" #" # K H 1/4 2m 1/2 Z uE du Q = 2 2 cA √ 2 g0ρ0r0 RSCD u Z

Note that both terms are written as the product of three factors, the first depending on the planet physical propertied, the second one on vehicle characteristic, and the third one is a non– dimensional performance index, that can be used in order to perform comparisons. It is then possible to define a non–dimensional heat flux

q¯ = Z1/2u3/2 and a non–dimensional heat absorbed

Z uE du Q¯ = √ u Z

As for the average heat flux, qav, it is possible to write it as 1 q = C ρv3 av 4 F where CF is the equivalent skin friction coefficient. This latter parameter is proportional to the non–dimensional quantity 3/2 q¯av = Zu In what follows, these parameters will be evaluated for both gliding and ballistic entry trajectories, in the framework of the first–order solutions derived in the previous Sections of this chapter. In this way it will be possible to compare the performance in terms of thermal load for each class of vehicle.

2.7.1 Gliding entry By making use of the equilibrium condition for the gliding descent, r 1 − u H C Z¯(u) = D u r CL the nondimensional heat ﬂux is

" r #1/2 1 − u H C u(1 − u)1/2 q¯ = D u3/2 = 1/2 1/4 u r CL (CL/CD) (r/H) It is easy to show thatq ¯ is maximum when

d √ √ u u 1 − u = 1 − u − √ = 0 ⇒ u = u∗ = 2/3 du 2 1 − u

The maximum heat flux experienced during the reentry is achieved when the velocity is p2/3 of the orbital speed (that is, v = 0.816vc), and it is equal to 2 q¯ = √ max 1/2 1/4 3 3(CL/CD) (r/H) The peak value of the heat flux can be reduced by incrementing the aerodynamic efficiency, in order to reduce the deceleration, that is, energy dissipation rate. G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 25

Using the same quasi–steady solution, Z¯(u) in the equation of the average heat ﬂux, one gets r 1 − u H C (1 − u)u1/2 q¯ = D u3/2 = av 1/2 u r CL (CL/CD)(r/H) which is maximumm for d √ √ 1 − u (1 − u) u = − u + √ = 0 ⇒ u = u∗ = 1/3 du 2 u av which means that the peak of average heat absorbption occurs later along the trajectory, when v = 0.577vc, the peak value being proportional to 2 q¯ = √ av,max 1/2 3 3(CL/CD)(r/H)

Finally, the total heat absorbed during the reentry trajectory, under the hypothesis of a heat–sink type reentry vehicle, is proporional to the nondimensional quantity

Z uE du r 1/4 C 1/2 Z uE u1/2du Q¯ = = L p ¯ 1/2 uf Z(u) H CD uf (1 − u)

Assuming that the reentry trajectory begins at a speed close to orbital velocity (uE ≈ 1) and terminates at a velocity close to zero (uf ≈ 0), it is possible to perform the integration explicitly, thus obtaining the expression for the (non–dimensional) total heat absorbed, that is,4

π r 1/4 C 1/2 Q¯ = L 2 H CD Note that an increase in aerodynamic eﬃciency causes a higher heat load, in terms of heat absorbed during the trajectory, as a consequence of the longer gliding path and slower deceleration, that increases the time spent by the vehicle ﬂying at hypervelocity.

2.7.2 Ballistic entry For ballistic reentry trajectories, the first order solution exploit the altitude/density variable Z as the independent variable. Remembering that r ! 2Z H u = uE exp sin γE r non–dimensional heat fluxq ¯ and average heat fluxq ¯av are given, respectively, by r ! 1/2 3/2 3Z H q¯ = Z uE exp sin γE r

r ! 3/2 3Z H q¯av = ZuE exp sin γE r

4The integration of the function f(x) = px/(1 − x) is far from trivial. The indeﬁnite integral of f is given by

Z r x r 1 √ √ √ F (x) = dx = ˆ(x − 1) x + x − 1 sinh−1( x − 1)˜ 1 − x 1 − x which is singular for x = 1, but converges towards 0 for x → 1. 26 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

p Letting κ = (3/ sin γE) H/r, the heat ﬂux is maximum when

d h i exp(κZ) Z1/2 exp(κZ) = 0 ⇒ (1 + 2κZ) = 0 dZ 2Z1/2

This means thatq ¯max is achieved for an altitude

1 sin γ r r Z∗ = − = − E 2κ 6 H and its value is r 1/4 sin γ 1/2 q¯ = u 3/2 − E max E H 6e Similarly, the maximum value of the average heat ﬂux over the vehicle is achieved when

d [Z exp(κZ)] = 0 ⇒ exp(κZ) (1 + κZ) = 0 dZ that is, for 1 sin γ r r Z∗ = − = − E av κ 3 H

The corresponding value ofq ¯av,max is

r 1/2 sin γ q¯ = −u 3/2 E av,max E H 3e

In both case, the local heat ﬂux and its average value increases for higher entry velocity uE and steeper ﬂight–path angles.

Unfortunately, no closed form solution for the total heat absorbed is available for the ballistic entry case, since the non–dimensional value is given by the following integral

Z uE du Q¯ = √ u Z where it is necessary to perform a variable change using the ﬁrst order solutio for u = u(Z) to obtain r r ! 2u H Z 0 2Z H Q¯ = E Z1/2 exp dZ sin γE r ZF sin γE r where it is assumed that Z ≈ 0 upon entering the planet atmosphere. By introducing the auxiliary, strictly positive variable r 2Z H y2 = − sin γE r it is possible to rearrange the expression in the following form: s r Z yf 2π H 2 2 Q¯ = uE √ exp(−η )dη − sin γE r π 0 where the term between parentheses is the error function, that is tabulated or the value of which can be determined by numerical integration techniques. G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 27

2.8 Critical analysis of gliding and ballistic entry

2.8.1 Gliding vs ballistic entry: historical facts Yuri Gagarin completed his single orbit on April 12, 1961 and ejected from the reentry capsule 108 minutes after launch, hard–landing being considered too dangerous. Exactly 20 years after this historical event, the space shuttle Columbia took off from Kennedy space center, on April 12, 1981. Two days and 6 hours later the Columbia was the first space vehicle to performe a gliding entry in the Earth’s atmosphere, landing on the runway of Edwards Air Force Base, in California. For 20 years ballistic trajectories were the only option for reentry capsules. Mercury, Gemini and Apollo missions and still nowadays all Russian manned missions relied upon this technique. Personnell from the International Space Station (ISS) get back to the Earth on russian reentry capsules. Also, all exploration mission aimed at landing a probe on the surface of other planets employed ballistic entry capsules. The STS fleet (Space Transportation System) is still nowadays the only group of vehicles that perform gliding entries, in spite of the fact that the design was frozen in 1973 (that is, from the technological point of view this class of vehicles is more than 30 years old). The European Space Agency (ESA) designed a reentry shuttle for the ISS named Hermes, using the more advanced concept of lifting body, but this program was cancelled as the development of a completely new reusable vehicle was not economically convenient when compared to the use of more conventional, yet highly reliable russian Soyuz capsules. As a major advantage of gliding entry, the range is longer and controllability of the vehicle much better, so that alternate landing sites can be chosen in case of bad weather on the nominal one. Reentry capsules are less flexible, from this point of view and, moreover, there is a certain level of uncertainty on the landing spot, since touch–down can take place on a relatively wide area (the so–called touch–down footprint), dispersion being caused by wind. In order to limit dispersion it is necessary not to allow speed to drop below a given value (several meters per second), which in turn causes the impact to be quite violent for the astronauts inside the capsule. On the converse, complexity of the vehicle makes its reliability still an issue, and the two tragedies of Challenger and Columbia dramatically confirmed the vulnerability of the vehicle.

2.8.2 Gliding vs ballistic entry: a comparison As far as aerodynamic heating is concerned, for both classes of vehicles kinetic energy is dis- sipated during the hypervelocity phase, receiving in exchange a great amount of heat. For a ballistic capsule the heat flux can be reduced by adopting a blunted shape that realize a large pressure drag, with low curvature surfaces. On the converse, winged and lifting bodies have more constraint on the aerodynamic shape of the vehicle, that is usually a relatively slender body. This greatly increases the local heat flux, so that it may be necessary to avoid a potentially dangerous thermal stress on the vehicle structure by designing a TPS based on thermal equilibrium, where the heat aborbed by convection from the atmosphere is radiated back because of the high temperature reached by the thermal shield. In this respect, the longer flight time of a radiating gliding vehicle is a positive feature. The analysis proposed in the previous sections was performed assuming that the vehicle motion is constrained on a great circle plane, a simplification that can be reasonable for a reentry capsule. Gliding lifting bodies, on the converse, perform airplane–like turns, not only for directing the course towards the landing runway but also to increase the course length and dissipate excessive heat, by performing a series of S–turns with bank–to–bank roll manoeuvres. Together with heat loads, also mechanical stress and maximum deceleration are factors that 28 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry

Table 2.2: Maxima for heat and mechanical loads in gliding entry.

Critical Non–dimensional Glide case condition parameter Crit. point Max value

√ √ ∗ 1/2 1/2 3/2 v 3 6 CD H 1/4 qmax Z u √ = g0ro 6 9 CL r √ √ ∗ 3/2 v 3 6 CD H 1/2 qav,max Zu √ = g0ro 3 9 CL r ∗ (a /g) Zu √v = 0 CD t max g0r0 CL

need careful consideration. Load factor is of paramount importance for manned mission, since crew members can tolerate a high levels of g’s for a limited time. Mechanical load on the structure is roughly proportional to dynamic pressure which, in turn, is approximately proportional to the non–dimensional factor Zu, just like deceleration. As a consequence, Zu = max represents a condition of maximum stress for both structure and crew. Tables 2.2 and 2.3 reports the critical conditions for gliding and ballistic reentry cases, respectively. Each critical condition can be determined by maximising a suitable non–dimensional factor (second column). Note that the independent variable used for gliding reentry is velocity, while nondimensional altitude/density parameter is useed for the ballistic case.

Table 2.3: Maxima for heat and mechanical loads in ballistic entry.

Critical Non–dimensional Ballistic case condition parameter Crit. point Max value

1/4 1/2 q Z1/2u3/2 ∗ sin γE p r 3/2 r sin γE max Z = − 6 H uE H − 6e 1/2 q Zu3/2 ∗ sin γE p r 3/2 r sin γE av,max Z = − 3 H −uE H 3e 1/2 (a /g) Zu ∗ sin γE p r r sin γE t max Z = − 2 H −uE H 2e

In both cases the maximum local heat ﬂux is the ﬁrst critical condtion encountered, followed by the maximum average heating rate. The peak deceleration occurs at a lower altitude (that is, higher Z) for ballistic entry, while the deceleration is steadily increasing for gliding trajectories, so that the peak deceleration occurs at the terminal phase of the reentry.

2.8.3 Limits of the findings In this chapter performance in ballistic and gliding flight were determined on the basis of a first–order solution of a simplified model. Together with possible deviations from flight–path on a great–circle plane, a further important hypothesis was the assumption of constant drag coefficient for ballistic flight and aerodynamic efficiency for gliding entry. As a matter of fact drag coefficient and efficiency change significantly with Mach number and angle–of–attack α. As recalled previously, vehicle velocity will vary in a very wide range, and also the angle–of–attack is unlike to remain constant throughout the trajectory. As an example, G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry 29 in order to limit the final deceleration peak in gliding flight, the first portion of the reentry trajectory of lifting bodies is actually covered in ballistic mode at high–α to maximize drag when density is still very small. Only after the velocity has been sizably reduced, the vehicle is pitched down to a low–α, high–efficiency attitude. A more complete picture of reentry trajectories must rely on more complete models, where also heat exchange is considered in its dynamical aspects. This means that a more realistic trajectory design must employ optimisation techniques where both lift and bank modulation are considered as control variables for managing both trajectory and heat fluxes. Nonetheless the previous analysis offers a good starting point for the understanding of the phenomena related to atmospheric entry in ballistic and gliding modes. 30 G. Avanzini - Entry, descent, landing, and ascent – 2. Atmospheric entry Chapter 3

Launch and ascent

3.1 Basic facts about rocket launchers

Injection into orbit starting from a launch site on the planet surface require a great increment of energy, inasmuch as it is necessary to increase both spacecraft potential energy (in order to increment its altitude and push the vehicle out of the planet atmosphere) and its kinetic energy (in order to reach orbit velocity). Rocket launchers are at present the only option to launch a spacecraft into orbit. They consist of a rocket engine and tanks for fuel and oxidizer. Fuel is burned in a combustion chamber and accelerated through a nozzle, thus providing a net thrust on the vehicle because of conservation of total linear momentum. As fuel gets burned the rocket mass gets smaller and provided that the force delivered by the jet flow remains approximately constant, the resulting acceleration becomes higher. The very early phase of the take–off is nearly vertical, but quite soon the trajectory is bended by gravity (the so–called gravity turn). Angle–of–attack must be kept as small as possible, in order to limit aerodynamic loads on the structure. Passive stabilisation by aerodynamic appendages (stabilising fins) is nowadays replaced by attitude active stabilisation through thrust vector control (TVC), that is, tilting the thrust vector out of the axis of symmetry of the rocket, thus creating a net torque because of a non–zero moment arm of the thrust force with respect to the rocket centre of mass. TVC can be achieved by three systems:

• ﬂow deﬂection by spoilers or jet vanes;

• ﬂuid injection inside the nozzle, close to small ﬁns placed inside the nozzle;

• nozzle deﬂection, that is rotating the nozzle in the desired direction.

All the system causes a certain loss of thrust. Higher losses occurs with jet deflection. At the same time, only the first two techniques allow for the generation of roll control moments. When nozzle deflection is used, torques about the roll axis are generated by use of thrusters. Thrust magnitude control (TMC) is also possible, but it is seldom implemented as it adds considerable complexity to the system, thus reducing its reliability. As an example, an extend- able exit cone was designed for the space shuttle main engine, but was never installed. In order to optimise the exploitation of energy produced by the combustion, it is possible to divide the rocket in stages, that is units made of fuel tank and rocket engine, that are detached from the rest of the rocket as soon as they run out of fuel. This technique has the advantage of releasing during the launch and acceleration manoeuvre the unnecessary mass of empty tanks, thus improving the exploitation of energy in the subsequent phase of the trajectory. Moreover, the rocket engine and nozzle(s) of the following stage can be optimised for the mass and expected

31 32 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent

Figure 3.1: Saturn V and Ariane 5 rocket launchers. environmental condition at the altitude achieved. At the same time a penalty is paid because it is necessary to have a rocket engine for each stage and hardware for detachment of the empty stages, which constitutes a weight penalty for the launcher during the initial phases of the acceleration. For this reason staging is convenient only for large and very large rocket launchers.

Several studies were performed in the last 20 years of the last century for sizing and designing SSTO launchers (Single–Stage–To–Orbit), and their more complex variant, the Two–Stage–To– Orbit launch vehicle, where an air–breathing propulsion system is used to accelerate a smaller rocket up to a certain altitude in the upper stratosphere. The possibility of exploiting air– breathing propulsion is appealing, as it would save the weight of oxidizer and tank to store it, yet too many technological breakthroughs are still necessary nowadays, before a new, revolutionary generation of launchers becomes available. In spite of this great amount of studies, staged rockets are stil the most reliable and affordable solution. The major variation in rocket launchers configuration took place in the 80’s, where the vertical stack of stages was replaced by a combined use of boosters with a two–stage rocket. The boosters provide the initial thrust force for lifting and accelerating the rocket during the first minute or so. Then they detach from the main body and are usually recovered. The last stage of the rocket takes the spacecraft up to orbit altitude and provides the terminal ∆v for orbit injection at perigee. The orbit is then circularized with the kick–off engine attached to the spacecraft itself. For small spacecraft, circularization may be achieved by the orbit manoeuvre engine.

In this chapter the equations of motion for determining the launch trajectory from a spherical non–rotating planet will be considered for the sake of simplicity. A planar motion will be assumed with the thrust force always parallel to the velocity vector. Finally the problem of staging will be considered, in order to provide guidelines for choosing the number and the size of launcher stages. But first of all a quick overview to the easiest model, motion in free space, will allow for the definition of some figures of merit. G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent 33

3.2 Motion in free space

If no force is acting on the rocket moving in vacuum (environmental pressure pa = 0), conservation of linear momentum dictates that

(m + ∆mf )v = m(v + ∆v) − ∆mf (ve − v) ⇒ m∆v = ∆mf ve where ∆mf is the mass of fuel and oxidizer burnt and ejected at velocity (relative to the vehicle) ve through the nozzle. By replacing finite increments with differential terms it is dv m =m ˙ v + p A dt f e e e where the time–derivative of vehicle mass is the opposite of the fuel–mass flow rate,m ˙ = −m˙ f < 0, while the additiona term peAe is due to the difference between pressure pe at the nozzle exit 1 section of area Ae and environmental pressure, pa = 0 in vacuum. The total thrust delivered by the rocket engine can thus be written as

T = peAe +m ˙ f ve =m ˙ f c =m ˙ f g0Isp where c is the effective exhaust velocity, g0 is the nominal gravity acceleration on the planet surface and Isp is the specific impulse of the rocket engine, which is measured in seconds. The specific impulse is one of the most important performance parameters of a rocket engine, and it can be considered roughly constant during engine operations.

3.2.1 Tsiolkovsky Equation From the previous equation it is possible to write the acceleration as

dv c c dm = −m˙ = − dt m m dt which can be integrated by variable separation from the initial condition v = v0 and m = m0, yielding the Tsiolkovsky Equation:

m m ∆v = c log 0 = c log 0 m m0 − mf where mf is the mass of fuel burned and ejected through the nozzle. The equation can be expressed in terms of speciﬁc impulse: m0 ∆v ∆v = g0Isp log ⇒ mf = m0 1 − exp − m0 − mf g0Isp

The total useful propellant mass is the mass of fuel that can be burned in the engine,

mf = m0 − me where me is the empty weight. The burn time, tb, is thus given by the integral equation

Z tb mf = m˙ f (t)dt 0

1A good introduction to rocket propulsion is given in the book written by J.W. Cornelliss, H.F.R. Sch¨oyer and K.F. Wakker, Rocket propulsion and Spacecraft Dynamics, Pitman, London, 1979. 34 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent

which, for a constant ﬂow rate becomes simply tb = mf /m˙ f . The mass ratio of a rocket is given by m Λ = 0 > 1 me As a consequence, the velocity increment at burnout will be

∆vid = g0Isp log Λ

This is the ideal velocity increment as it depends only on the characteristics of the engine (its speciﬁc impulse) and on the mass ratio, without being inﬂuenced by the actual thrust time– hystory and by the presence of other forces (gravity or aerodynamic drag).

3.2.2 Rocket mass–parameters The total initial mass of the rocket

m0 = mf + ms + mpl can be divided into three major terms:

• mf , the useful propellant mass (which is diﬀerent from the total propellant mass, which includes also the residual propellant left at engine shut–oﬀ);

• ms, the structural mass (including not only the rocket structure, but also all the on–board systems, piping, engine, mounting, insulaion, etc.);

• mpl, the payload mass (which can be a satellite, but also a complete rocket, in which case a staged rocket is being dealt with).

As a consequence the empty weight is

me = ms + mpl

Three non–dimensional parameters can be defined in terms of ratii of masses: m payload ratio λ = pl m0 m structural efficiency ε = s ms + mf m propellant ratio ϕ = f m0 It should be noted that the mass ratio Λ is always greated than one, while λ, ε and ϕ are always less than 1. They are not independent and by applying their definition, it is easy to show that

ϕ = (1 − ε)(1 − λ) 1 1 Λ = = 1 − ϕ ε(1 − λ) + λ) In general the range of these parameters for single–stage rockets lies in the following intervals:

2 < Λ < 10 0.08 < ε < 0.5 0.5 < ϕ < 0.9 0.01 < λ < 0.2 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent 35

) 2.5 sp I 0 /(g id

v 2 # = 0.1 "

1.5 0.2

0.3 1 0.4

0.5 Ideal velocity increment 0.5 0 0.05 0.1 0.15 0.2 Payload ratio ! Figure 3.2: Nondimensional ideal velocity increment as a function of payload ratio and structural eﬃciency.

Note that the minimum values of structural eﬃciency can be obtained only for very large rockets, where the weight of the engine remains a small fraction of the total mass. By making use of the above deﬁnition, the ideal velocity increment can be written as

∆vid = −g0Isp log [ε(1 − λ) + λ]

The behaviour of the nondimensional ideal velocity increment ∆vid/(g0Isp) is reported in Fig. 3.2 as a function of payload ratio and structural eﬃciency. The maximum attainable velocity increment is for zero payload (although the usefulness of a zero payload rocket is questionable). On the converse, it is not possible to reduce the structural eﬃciency down to zero.

3.2.3 Rocket engine performance parameters

Togethet with the specific impulse, Isp, it is possible to define performance parameters for the rocket engine. The first two have very similar mathematical expressions, namely

F speciﬁc thrust β = g0m F thrust–to–weight ratio (T/W )0 = g0m0 where β is the thrust produced per unit weight at a given time, while ((T/W )0) is the same parameter referred to the initial total weight of the rocket. The inertial acceleration is given by (T/m), that is, it is the thrust delivered per unit mass, and it is equivalent to the nominal acceleration in vacuum.

3.2.4 Burnout range

The burnout range in vecuum ∆sb is deﬁned as the distance covered between ignition and engine cut–oﬀ, that is Z tb ∆sb = v(t)dt 0 36 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent where the instantaneous velocity can be evaluated according to Tsiolkovsky equation as

m v(t) = g I log 0 0 sp m(t)

Assuming constant thrust, the mass of the rocket decreases linearly with time,

m = m0 − m˙ f t so that m0 v(t) = g0Isp log m0 − m˙ f t and the burnout time is given by m0 − me Isp 1 tb = = 1 − m˙ f (T/W )0 Λ

Carring out the integration between 0 and tb one gets

2 g0Isp 1 ∆sb = 1 − (log Λ + 1) (T/W )0 Λ

After burnout the velocity in vaccum remains constant and the space increases linearly with time.

3.3 Rocket equations of motion over a spherical non–rotating planet

As stated in Section 1.4.4 most of the apparent forces are negligible with respect to the eﬀect of gravity and centrigugal accelerations over a large speed and altitude range, so that motion can be described as taking place over a spherical, non–rotating planet. Assuming also that motion takes place on a plane, the dynamic equation can be written in the form:

T cos ε − D v˙ = − g sin γ m

T sin ε + L v2 vγ˙ = − g − cos γ m r

r˙ = h˙ = v sin γ

v θ˙ = cos γ r

T m˙ = −m˙ f = − (3.1) g0Isp where the thrust is given by T =m ˙ f ve + (pe − pa)Ae = g0Ispm˙ f , and for a variable mass system the mass of the vehicle itself becomes a state variable (see the last equation). Note that attitude dynamics was neglected, and it is assumed that the prescribed angle–of–attack is tracked by a suitable automatic control system. It should be noted that usually α must be kept close to zero so that L = 0. G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent 37

In compact terms, the system dynamic is described by a system of 5 first order differential equations, x˙ = f(x, u) where the state vector is x = (v, γ, h, θ, m)T while, in the most general case, the control vector is given by u = (T, α, δ)T , where δ is the thrust deflection angle with respect to the vehicle symmetry axis and it is ε = α + δ. This set of ordinary differential equations can be integratet numerically, so that the time– history of the state variable is obtained as a function of the values of thrust magnitude, angle of attack and thrust deflection angle. Usually it is possible to consider T ≈ const, α ≈ δ ≈ 0 and the trajectory design is performed as a function of initial conditions only, that is the values of the state variables at the initial time. This initial time is usually chosen some seconds after take off, when the velocity is still very small but not zero. Zero initial velocity woul make thegamma ˙ equation singular. Note that in order to perform a gravity turn2 it is necessary to assume a small angular displacement from the perfectly vertical take–off trajectory, i.e. the initial climb angle must be slightly less than 90 deg. Since launch to low–Earth–orbits usually requires two or three stages, it is necessary to account for discontinuities in the values of m, T , and D during the integration. As an example, at burnout of the i–th stage T drops to 0 and a ballistic segment is started. When the stage is detached, m is suddenly reduced (but without a “reaction effect”, drag will be slightly reduced, and when the following stage rocket engine is started, T will achieve a new value, as the following engine is sized for a lighter vehicle flying at higher altitude.

3.4 Simpliﬁed estimate of rocket performance

Although throttling is possible, it is seldom used, as it is simpler to conveniently size each stage for the altitude and velocity range of interest. Thrust and speciﬁc impulse will gradually increas with altitude, but considering an average value usually allows for a reasonable estimate of trajectory and booster performance. If T and Isp are constant, also the mass–ﬂow rate is constant and the mass is linearly varying with time, so that it can be removed from the state vector (m(t) = m0 − m˙ f t. In this case the inertial acceleration is given by

T T T/m g (T/W ) = = 0 = 0 0 m m0 [1 − (m ˙ f /m)t] 1 − [T/(mg0Isp)]t 1 − [(T/W )0/Isp]t

Inertial acceleration grows hyperbolically, becoming infinite at time tω = Isp/(T/W )0, when an ideal rocket made entirely by propellant burns the last mass element. As a metter of fact, together with the unfeasibility of such a vehicle, a practical constraints is the maximum acceleration that the structure (and the human crew, for manned mission) can tolerate. From this point of view the maximum load factor for the Apollo missions was 5, while it is 4 for the space–shuttle and Ariane 5, this latter data being a consequence of the expected use of Ariane 5 launchers for injecting the Eropean shuttle Hermes. On the Saturn V excessive acceleration was avoided by switching off the central engine of the first stage while the other four kept on running for several seconds. This technique provided a means for a discontniuous thrust modulation.

2That is, gravity bends the trajectory in the desired direction, in order to achieve as early as possible a tangential thrusting at small climb angle, to limit altitude losses. The gravity turn will be discussed in more details in the next Section. 38 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent

3.4.1 Assumptions As stated before, angle–of–attack and angle–of–sideslip are usually kept as small as possible, in order to limit lateral airloads on the rocket structure. Moreover the atmosphere surrounding the rocket is assumed at rest, so that the velocity of the vehicle with respect to the surfac is equal to the velocity with respect to the air mass. As a consequence, side force and lift (that is, aerodynamic force component in the direction normal to the trajectory) are negligible3 and it is possible to set L ≈ 0 in the equations of motion. Perturbations of aerodynamic angles are compensated for by the attitude control system, that usually employ TVC. As the thrust force delivered is very high, small deflection angles δ are sufficient to provide the required control moments. This means that we can consider also δ ≈ 0 and T has only the tangential component. As a major approximation, also drag D is considered “small” with respect to the other forces and is dropped. This last assumption has more serious consequences on the evaluation of booster peroformance, but a correction will be introduced later. Finally, as far as the angular position variable is decoupled from the others, it is possible to drop the fourth equation. The order of the system is then reduced to three, where the known variation with time of vehicle mass is introduced in order to drop the fifth equation: T v˙ = − g sin γ m(t)

v2 vγ˙ = − g − cos γ r

r˙ = h˙ = v sin γ Upon substitution of the expression deﬁned previously for the T/m(t) inertial acceleration 2 2 and remembering that the gravity acceleration g can be written as g(r) = g0(r0/r) = k/r , the equations of motion achieve the form

g0(T/W )0 k sin γ v˙ = − 2 1 − [(T/W )0/Isp]t r

v k γ˙ = − cos γ r vr2

r˙ = h˙ = v sin γ It should be noted that, for ascent from a celestial body without an atmosphere (like the Moon), aerodynamic forces are zero and the above equations of motion represent a relatively accurate model for describing a planar launch trajectory. Note also that the same set of ordinary differential equation can be used for describing a powered landing on the surface of a planet without atmosphere. In this case the landed mass and touch–down point are the known final conditions. It is then necesasry to integrate the equations of motion backwards in time up to the parking orbit, in order to find the required initial condition for the descent.

3.4.2 Gravity turn The angular velocity of the vehicle in the absolute frame is given byγ ˙ − θ˙. Remembering the expression of θ˙ = (v/r) cos γ, the normal component of the acceleration is an = v(γ ˙ − θ˙) = 3The presence of side forces would also harm the planar motion assumption under which the equations of motion were derived. G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent 39

−g cos γ, that is, it is provided by gravity force as soon as the ﬂight path leaves the vertical above the launch pad. During this gravity turn the booster falls towards the horizontal and the net force experienced by crew and/or payload is purely tangential, parallel to the longitudinal axis of the vehicle.

3.4.3 Preliminary estimate of the ∆v budget From Tsiolkovsky equation it is possible to evaluate the effect of forces in terms of ∆v, which in turn is related to the amount of fuel necessary. For this reason, the preliminary estimate of the amount of fuel necessary for achieving a certain final condition is usually performed in terms of ∆v budget, that is, the contribution of each force acting on the vehicle is evaluated in terms of resulting ∆v, which turns out to be an estimate of a portion of the total fuel necessary. In order to obtain a more realistic estimate of the ∆v budget, it is necessary to include among the forces the aerodynamic drag, if an atmosphere surrounds the planet, as the overall effect on the total ∆v is not negligible, in spite of the fact that the instanteous value of D/m is actually small, compared to other acceleration terms in the equations of motion. Letting 2 D = 0.5ρ(h)SCD(M)v , the acceleration term is given by D ρ(h)SC (M)v2 − = − D m 2m0[1 − (T/W )0/Isp t]

An exponential model of the atmosphere ρ(h) = ρ0 exp(−h/H) is usually reasonable for preliminary calculation purposes, and the variation of drag coeﬃcient CD with drag number is assumed known.

Rearranging the first equation of motion, it is T k D =v ˙ + sin γ + m r2 m Integration with respect to time of all this acceleration terms provides the desired ∆v budget. For a single–stage–to–orbit vehicle, integration is carried out between the initial condition of rest on the launch pad (t = 0 and v = 0) and the final orbit injection, at t = torb with √ √ p √ vorb = gr = g0r0 r0/r. Remembering the definition of the first cosmic velocity vI = g0r0 p p p and taking into account that r0/r = r0/(r0 + h = 1/(1 + h/r0 ≈ 1 − h/(2r0), the final velocity must be vorb = vI [1 − h/(2r0)]. For a multi–stage rocket, the initial condition of the i–th stage is the final condition for the (i − 1)–th stage, which means that integration must be divided into several intervals, one for each stage of the rocket.

Thrust and characteristic velocity Under the assumption of constant speciﬁc impulse it is

Z torb Z torb Z torb T m˙ dm m0 dt = −g0Isp dt = −g0Isp = g0Isp log = ∆vchar 0 m 0 m 0 m mF where ∆vchar is the characteristic velocity, that is, the velocity that would be achieved by the rocket moving in a force–free vacuum, a useful ﬁgure of merit for the potential capabilities of the booster.

Actual velocity increment Integration of the tangential acceleration provides the value of the actual velocity increment. For a single stage rocket it is Z torb v˙dt = vorb 0 40 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent

Eﬀect of gravity

The evaluation of the ∆vgrav related to gravity eﬀects

Z torb k ∆vgrav = 2 sin γdt 0 r requires the knowledge of details on the ﬂight path. An upper boundary for its value can be determined by assuming a vertical ascent with constant gravity acceleration, that is

Z torb ∆vgrav < g0dt = g0torb 0

Drag losses The ∆v lost bacause of dissipation is given by

Z torb Z torb 2 D ρ(h)SCD(M)v ∆vdrag = dt = dt 0 m 0 2m0[1 − (T/W )0/Isp t]

∆vdrag can be accurately estimated only if a complete information on vehicle aerodynamic characteristics and its trajectory are available, together with a reliable atmosphere model. In general its value is not negligible, but much smaller than the other ones that come into play within the budget.

Budget The total ∆v budget is made up by adding the last three terms, relative to the ∆v’s necessary to accelerate the vehicle (increase of kinetic energy), while increasing its altitude (that is, its potential energy) and compensating for dissipation due to air resistance. The total thus obtained,

∆vchar, allows for an estimate of the total amount of propellant necessary to reach the prescribed altitude and velocity. For conventional rocket the budget is as follows:

∆vchar = vorb + ∆vgrav + ∆vdrag 9 → 10.5 103 m s−1 7.6 103 m s−1 1.8 → 2.4 103 m s−1 60 m s−1 72 → 84 % 17 → 26 % 0.5 → 0.6 %

It should be noted that the energy spent for sending into orbit a relatively tiny body is huge. In this respect the overall efficiency of launch, defined as the final energy of the spacecraft divided by the total energy of the fuel burned, turns out to be a very small number. One of the major issues that makes direct boosting into space still the only viable solution for launches lies in the above budget, where the drag penalty to be paid is small. Moreover, the absence of aerodynamic surfaces makes drag even smaller but, more important, allows for more structurally efficient launchers, as there is no requirements for structures that bear aerodynamic loads in airborne flight. On the converse a major penalty is the requirement for oxidizer tanks (often accompanied by cooling system and insulation) necessary for rocket engines, which would be substituted by an air intake for the portion of flight that takes place within the atmosphere.

3.5 Staging

Tsiolkovski equation shows how there is an exponential relation between the final weight of the vehicle after all the fuel has been burnt and its take–off weight. This means that it is extremely G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent 41 important to get rid of the unnecessary mass as soon as possible during a launch trajectory, in order not to accelerate pieces of structure or equipment that are no longer useful. This basic requirement led to the concept of staging. A staged launcher is made by a stack of rockets, where payload of the i–th stage is the overall weight of the following stages. It should be noted that, if on one side staging allows for reducing the mass of the vehicle when one stage is jettisoned from the stack, on the other side it increases significantly the initial launch weight, as each stage has its own rocket motor, pumps, and engine control systems. At the same time it is possible to optimise each rocket for the expected environmental conditions (pressure and temperature), while determining its thrust as a function of vehicle mass, so as not to exceed acceleration constraints. It is relatively easy to take into account staging in the simulation of launch trajectories, by introducing discontinuities in vehicle mass and drag parameters, but it is also quite clear from the previous picture that the overall optimisation of stages is far from trivial. In what follows an elementary approcah for sizing the stages will be discussed.

Let as consider a booster made up with n stages, each one weighting Wi, that must launch a payload weighting Wpl. Using Wpl as the reference weight, the non–dimensional weight (and mass) of each stage is deﬁned as wi = Wi/Wpl The total weight at launch is given by n X Wtot1 = Wpl + Wj j=1 that, in non–dimensional terms, becomes n X wtot1 = 1 + wj j=1 The weight of payload plus the last i stages (that is, the total weight at ignition of the i–th stage is given by n X Wtoti = Wpl + Wj j=i and the relative non–dimensional expression becomes n X wtoti = 1 + wj j=i It is possible to measure the eﬃciency of the overall design of the isolated i–th stage as the ratio

Gross weight − Weight of usable propellant Wi − Wf,i βi = = = 1 − ϕi Gross weight Wi The mass ratio between ignition and burnout of the i–th stage is given by Total stack weight before i–th burn µ = i Total stack weight after i–th burn before stage separation n X Wpl + Wi + Wj

j=i+1 wi + wtoti+1 = n = X βwi + wtoti+1 Wpl + βiWi + Wj j=i+1 42 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent

Table 3.1: Combinations of propellants and oxidizer with attainable Isp

Propellant Oxidizer Reaction Isp [s] Hydrogen Oxygen H2 + O2 454 Hydrogen Fluorine H2 + F2 475 Hydrazine Hydrogen peroxide H2 + H2O2 338 RP-1∗ Oxygen 351

∗ Rocket Propellant Type-1 consists mainly of Kerosene (approximately C10H18).

From Tsiolkovski equation, the ideal i–th velocity increment is given by

∆vi = g0Ispi log(µi) where Ispi is the average specific impulse of the i–th stage (data on available specific impulse as a function of propellant type are reported in Tab. 3.1). As a consequence, the total ideal velocity increment for the payload will be n n X X ∆vtot = ∆vi = g0 [Ispi log(µi)] i=1 i=1 For a given class of rocket launchers, the costs increases fairly linearly with its overall weight. At the same time the value of the βi is stongly dependent upon available technology and at present, for sufficiently large rocket launchers, falls in the neighbourhood of 0.1 (this means that 90% of the total stage weight is made up with propellant and oxidizer!). This means that, in the framework of a preliminary design, the problem of stage sizing may be defined as the determination of minimum overall launch weight Wtot1 for a prescribed value of the total velocity increment ∆vtot necessary to reach the desired orbit, type of fuel that is going to be used and expected structural efficiency of each stage. In nondimensional terms, it is necessary to minimise wtot1 for given values of ∆vtot, βi and Ispi . In this respect it is convenient to express wtot1 as n wtot wtot wtotn−1 Y wtot w = 1 · 2 · ... · · w = i tot1 w w w totn w tot2 tot3 totn i=1 toti+1 where Π is the product sign and sn+1 = wpl = 1 is not reported among the factors.

In order to compute the i–th factor of the product, it is possible to solve for wtoti+1 the deﬁnition of µi, that is wi(1 − βiµi) wtoti+1 = µi − 1 Noting that wiµi(1 − βiµi) wtoti = wtoti+1 + wi = µi − 1 the i–th factor of the product that provides wtot1 can be written as w µ (1 − β ) toti = i i wtoti+1 1 − βiµi Upon substitution of this expression in the product one gets n Y µi(1 − βi) w = tot1 1 − β µ i=1 i i G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent 43

In order to solve the optimisation problem,4 it is more convenient to express the performace index in the form n X J(µ) = log(wtot1 ) = log(µi(1 − βi)) − log(1 − βiµi) i=1

T where µ = (µ1, µ2, . . . , mun) are the optimisation variables and J must be minimised while satisﬁng the constraint n X ϕ(µ) = g0 Ispi log(µi) − ∆vtot = 0 i=1 Upon deﬁnition of the augmented performance index

J ∗ = J(µ) + λϕ(µ) where λ is a Lagrange multiplier,5 it is possible to demonstrate, that a necessary condition for µopt to be the optimal set of parameters is that all the partial derivatives of the augmented performance index J ∗ vanishes, that is ∂J ∗ ∂J ∗ = 0 , i = 1, 2, . . . , n ; = 0 ∂µi ∂λ In the present case it is ∂J ∗ 1 + λg I β = 0 spi + i = 0 , i = 1, 2, . . . , n ∂µi µi 1 − µiβi ∗ n ! ∂J X ∆vtot = I log(µ ) − = 0 ∂λ spi i g i=1 0 The ﬁrst n equations can be solved explicitly,

1 + λg0Ispi µi = λβig0Ispi and these solutions can be cast into the (n + 1)–th equation that must be solved to determine the Lagrange multiplier, n X 1 + λg0Isp ∆vtot I log i = spi λβ g I g i=1 i 0 spi 0 In the most general case this last equation must be solved numerically, but under some further simpliﬁng assumptions it is possible to obtain an analytic solution of the problem.

Equal speciﬁc impulse and weight fraction for all stages If each stage has rocket engines with the same speciﬁc impulse and the structural weight is the ¯ ¯ same fraction of the overall weight, it is Ispi = Isp and βi = β, so that the mass ratio between ignition and burnout is also the same for all the stages:

1 + λg0I¯sp µi =µ ¯ = λβg¯ 0I¯sp

4There are several excellent textbooks on optimisation and optimal control problem. In spite of its age, one of the best remains the book by A.E. Bryson and Y.C. Ho, Applied Optimal Control: optimization, estimation and control, Hemisphere, Washington 1975. 5In constrained problems, it is possible to adjoin the performance index with the constraint functions multiplied by a parameter, the Lagrange multipliers. The number of additional terms (and thus of λ’s) is equal to the number of constraints. When a constraint is satisﬁed, ϕ = 0 and the performance index is unaﬀected. 44 G. Avanzini - Entry, descent, landing, and ascent – 3. Launch and ascent

This in turn means that also the weight ratio of two successive stages is the same:

w µ(1 − β) toti = wtoti+1 1 − µβ so that the overall weight of the stages goes in geometric series. As for the value of the mass ratio, the λ equation reduces to the following

∆vtot nI¯sp logµ ¯ = g0 from which it is possible to determine explicitly

∆v µ¯ = exp tot ng0I¯sp

Approximately it is ∆vtot ≈ 3g0I¯sp, so that for a three stage booster it isµ ¯ ≈ e = 2.718. ¯ Assuming a 10% structure weight fraction β, it is wtoti /wtoti+1 ≈ 3.36. The Saturn V rocket launcher used for the Apollo missions was a three stage booster with s1/s2 = 4.1 and s2/s3 = 3.2. Discrepancies may appear significant, but it must be taken into account that the first stage had a significantly lower specific impulse, while the thirs stage was not exhausted at orbit injection, since it carried also the fuel for injection over the Moon transfer orbit.

Optimisation of a two–stage booster If only two stages are used for orbit injection, it is possible to solve for (λ)−1 the two equations

∗ ∂J 1 + g0λIspi βi −1 = + = 0 , i = 1, 2 ⇒ λ = g0Ispi (1 − µiβi) ∂µi µi 1 − µiβi

−1 By equating the two expressions for (λ) , a linear relation between µ1 and µ2 is obtained

g0Isp1 (1 − µ1β1) = g0Isp2 (1 − µ2β2) that must be solved simultaneously with the equation

∆vtot Isp1 log µ1 + Isp2 log µ2 = g0