9.1.1 Objectives

From the beginning of humankind’s use of space, human-made objects have re-entered the Earth’s atmosphere and experienced the severe aerodynamic heating and loads characteristic of high-speed atmospheric re-entry. Some of these re-entries have generated fragments that survived to impact the Earth’s surface and be hazardous to people and damaging to property. This chapter addresses the safety of both broad types of space hardware re-entries: either controlled, so impact is targeted in a specific area, or uncontrolled, where re-entry can occur anywhere within the latitude band defined by the orbital inclination of the re-entering object.

The overall objective of this chapter is to help prepare safety engineers to answer the ultimate questions involved in the design of safety re-entry operations:

Finding valid answers to these questions often requires significant effort. A thorough understanding of the material presented in this chapter will help answer these questions without undue expenditure of time or other resources.

The remainder of this introductory section presents a roadmap that will help the reader navigate through the complexity and diversity of the technical issues involved in re-entry safety design, followed by a brief overview of the phenomena involved in an uncontrolled re-entry breakup and the nature of a re-entry “footprint.” Subchapter 9.2 presents various analytical models of re-entry trajectories for both controlled and uncontrolled flight above and within the sensible atmosphere, which is conventionally delineated by an altitude of 120 km. Subchapter 9.2 will help the reader model re-entry trajectories, and develop an understanding of the physical phenomena and constraints involved with both controlled and uncontrolled re-entries. Subchapter 9.3 focuses on analysis methods that can be used to predict the breakup and demise during uncontrolled or controlled re-entry. The breakup and demise analysis methods described in Subchapter 9.3 will help the reader model the trajectory of vehicle fragments until demise or ground impact. Subchapter 9.4 provides empirical data on debris from controlled and uncontrolled re-entries, including the results of re-entry testing conducted to provide insights on re-entry breakup and an overview of a device designed to provide definitive data by recording, preserving, and transferring data during actual re-entry events. The data on objects known to have survived re-entry can help validate, refine, or refute models for breakup and demise. In addition, careful study of the empirical data in Subchapter 9.4 can provide insight into potential design techniques to enhance demise. Subchapter 9.5 discusses analysis techniques used to predict the hazards and risks from controlled and uncontrolled re-entries with or without breakup. Many elements of the launch debris analyses described in Chapter 5 are also directly applicable to re-entry. Subchapter 9.5 addresses the elements of a re-entry risk analysis that are unique compared to launch, and includes subsections with examples of re-entry risk and hazard analyses for the following types of re-entry missions:

The field of re-entry safety design is in a highly evolutionary state. As more data becomes available, some existing models are validated and more refined models that make fewer assumptions become possible. The approaches presented in this chapter are typical approaches, but not necessarily the only valid approaches.

9.1.2 Roadmap for Re-Entry Operations Safety Analysis

A roadmap is helpful to navigate through the complexity and diversity of the technical issues involved in re-entry safety design. Figure 9.1.1 is a roadmap that qualitatively shows how the elements of a re-entry safety analysis fit together and helps put the sections of this chapter in context.

A re-entry safety analysis typically begins with the identification of potentially hazardous events. Re-entry debris potentially presents the same five hazards that are associated with launching space vehicles:

Subchapter 9.5 discusses typical re-entry events that are potentially hazardous and describes threshold characteristics for inert and explosive debris. The probability of potentially hazardous re-entry events are estimated using the same methods applied to launch operations.

Establishing realistic state vectors (e.g. the position and velocity) where potentially hazardous debris could be released (i.e. foreseeable Break-Up State Vectors, BUSVs) involves two major elements of a re-entry safety analysis that are highly inter-dependent: modeling the vehicle trajectory using Newton’s laws of motion, such as those described in Subchapter 9.2, and analysis methods to predict vehicle breakup, such as those described in Subchapter 9.3. The output of the trajectory and breakup debris analyses includes the fragment characteristics (e.g. mass, size, shape, and material type) and state vectors where fragments begin to follow essentially ballistic trajectories (lift effects on the debris are treated as a source of dispersion in the predicted impact points). Foreseeable BUSVs and fragment characteristics are critical input to various debris survivability analyses, which are described in Subchapter 9.3 as well. Empirical data on objects recovered after re-entry, such as presented in Subchapter 9.4, can help validate, refine, or refute models for breakup and demise.

Subchapter 9.5 discusses analysis techniques used to predict the hazards and risks from controlled and uncontrolled re-entry breakup.

9.1.3 Overview of an Uncontrolled Re-Entry Breakup Footprint

William Ailor

Figure 9.1.2 is a simple illustration of the breakup process for an uncontrolled re-entry. Uncontrolled re-entries occur as the atmosphere slowly drags an orbiting object deeper into the atmosphere. Moving at over 7 km/sec, the object begins to heat as it encounters increasingly significant atmospheric density below the re-entry interface altitude of about 120 km. The heating increases as gravity and drag lower the altitude, eventually low melting point materials reach a condition where they fail. Heating on the primary object and on released fragments continues to increase, and aerodynamic deceleration loads also begin to build. Magnesium and aluminum structures have been observed to fail consistently at approximately 78 km (42 nm) altitude, causing a catastrophic breakup of the object. This major breakup phenomenon near 78 km altitude is remarkably independent of vehicle attitude and rates, diameter, shape, and entry flight path angle in between –0.3° and –1.5°. High heating rates and aerodynamic forces that produce thermal melting, thermal fragmentation, and mechanical fracture during re-entry are the primary causes of the various external destruction events. Destruction may also be introduced by internal components of the spacecraft (SC), like propellant tanks with residual fuel. Pressurization of the tanks by heating may exceed the critical burst level and the release explosive gases may lead to an explosive destruction of the SC.

Items made from materials with relatively low melting points, such as aluminum, typically fail first, releasing fragments that generally decelerate further and follow their own trajectories. Major fragments such as electronics boxes, propellant and pressurization tanks, and other components are released. Deceleration loads build to seven or more times the acceleration of gravity, potentially causing additional failures due to inertial loads. Each object is heated further until its velocity drops to the point where the heating and loads diminish. At this point, the original orbiting object has been broken into a number of smaller fragments, each falling independently. Much of the structure of the original object, typically aluminum, has melted away; objects made of high melting point materials like titanium, glass, and steel often survive to impact. Some objects made of low melting point materials can survive to impact because they were released very early in the trajectory and decelerated quickly or they were shielded from much of the re-entry heating by other objects. There are competing effects that complicate the prediction of whether a given object will survive to impact or demise. However, re-entry heating rates are approximately proportional to the velocity cubed and inversely related to the radius of curvature. Thus, small objects released early often demise, unless they have low enough density to slow down rapidly.

The “footprint” is the area where debris hazards are predicted to occur given a re-entry breakup. A typical footprint for a re-entering spacecraft of 800 kg or more is approximately 2000 km long, contained within 35 km of the original ground track as illustrated in Figure 9.1.2. For a re-entering launch stage, a typical footprint length is 1000 km (540 nmi), with representative surviving debris listed in Table 9.1.1.

The major re-entry breakup process takes place over a ~5 minute period. Objects that survive the re-entry environment continue to decelerate and most will approach a terminal velocity proportional to the square root of their ballistic coefficient at about 18 km (60,000 ft). (Recall that the ballistic coefficient is defined as β = W/(C_DA), where W is the weight of the object, C_D is its drag coefficient, and A its reference area). From this point, the surviving fragments fall nearly vertically, with their trajectory blown by winds and some additional dispersion potentially due to lift. Unless the object has substantial thrust or lift exerted in the cross-range direction, any surviving re-entry debris will fall in the vicinity of the predicted ground track of the object prior to re-entry as shown in Figure 9.1.2.

Spacecraft represent a different category as far as the quantity of surviving debris. Spacecraft are generally complex objects with solar panels, extensive internal equipment, mechanisms, electrical motors, batteries, complex fittings, pressurization bottles, and propellant tanks imbedded within a structure encased by an external skin and thermal blankets. Because some spacecraft utilize titanium bolts, fasteners, and structures, and may have instruments with glass and other high-melting-point materials, fragments of these might survive in hazardous sizes. It is also possible that solar panels and lightweight or flimsy elements that are separated early in the re-entry might survive to impact, as a result of their shape and density. Table 9.1.2 lists the estimated fragment distribution that survives re-entry for a generic spacecraft with a dry mass of 800 kg or more.

9.2.1 Low Earth Orbit Evolution

Jean-François Goester

This section presents a simple analytical model for orbit evolution of an uncontrolled spacecraft at altitudes above 120 km. Modern computer equipment can perform numerical simulations using complex models without consuming an inordinate amount of computer time. However, a safety engineer can gain valuable insight into the qualitative aspects of low Earth orbital and atmospheric re-entry trajectory evolution by studying relatively simple analytical relationships, such as those presented in this and the following subchapters.

We begin with an examination of why the orbit of a low altitude object will inevitably decay such that the object eventually falls toward the Earth. From the study of orbital mechanics, we understand that an object’s orbit is determined by a balance of the centrifugal force and gravity, such that if the object slows down the semi-major axis (i.e. the perigee) of the orbit will be reduced.

The terrestrial atmosphere does not suddenly end at a particular altitude, but instead its density can be calculated at more than 1000 km from the surface of the planet. Of course, it is not comparable to that encountered by a conventional aircraft, but considering the tremendous speed of satellites (about 7700 m/s for a circular orbit at an altitude of 350 km) and the length of time that they spend there (several days), the effect of the atmosphere cannot be ignored. Atmospheric friction induces, among others, a force that is directly opposite to the velocity (drag) and thus leads to a reduction of the semi-major axis of the orbit, which sooner or later causes the object to fall to the ground.

Often the object will burn up when entering the dense layers of the atmosphere. However, debris from some satellites and launch vehicle stages can survive to impact. This is what happened, for example, with SKYLAB, COSMOS 1402, COSMOS 1900 and SALYUT 7, and others as described later in this chapter. These events showed how difficult it is to predict the place and time of impact. The best ways to approximate the atmosphere and aerodynamic forces using numerical simulations is discussed below. Deriving analytical models is also interesting, even if they are imperfect, since they help us understand the evolution of the orbit and the atmospheric trajectory.

Now consider a simplified analytical model for orbit evolution that reveals both its interest and limitations compared to a numerical model. In order to simplify the problem, the following hypotheses are used:

Since the perturbing acceleration is non-gravitational, the most suitable equation of motion corresponds to Gauss’s equations. In addition, since this acceleration is collinear with the velocity, the equations are expressed in the orbital reference frame illustrated below:

Expressed in this reference frame, the components of the acceleration are:

(2)

The Gauss equations can be written as:

(3)

(4)

(5)

(6)

(7)

(8)

It can already be seen that the two variables that govern the motion of the orbital plane (i, Ω) remain constant. The drag does not therefore act on the inclination or on the longitude of the ascending node.

The evolution of the shape of the orbit given by equations (3) and (4) will now be examined. Replacing n, V and T by their values as functions of the orbital parameters a, e, E and assuming that the variations in the semi-major axis and the eccentricity are small, one finally obtains:

(9)

(10)

The terms with the index “0” correspond to the initial conditions. Note that e₀ has been set to zero (initial circular orbit).

Over one period (ν ∈ [0, 2π]), considering only the variations in the anomaly, one finds:

(11)

(12)

To obtain the time of impact, simply use equation (9) and by integration:

(13)

Since (a_f – a₀) is more than 100 km, while H₀ is of the order of 10 km, the exponential term can be neglected compared to 1:

(14)

As an example, take a satellite with the following characteristics:

Taking a density value of 6.79E–11 kg/m³ at this altitude and a scale height of 40.24 km (JACCHIA model for an exospheric temperature of 1000 K), one obtains a survival time of about 60 days. Similarly, for an initial altitude of 300 km, the lifetime is about 220 days.

While it is true that this analytical model provides information quickly, it is very inaccurate and incomplete. For example, only quasi-circular orbits can be processed, i.e. those with small eccentricities.

As example, here are the results obtained with a numerical trajectory propagator and with the same hypotheses as previously except, naturally, for the drag coefficient and the modeling of the atmosphere. The atmosphere model used is DTM BARLIER + CIRA88 (with the two models combined) and the drag coefficient corresponds to the “average” curve detailed in Figure 9.2.1. In order to see the influence of atmospheric conditions, different values of the intensity of the solar flux and the geomagnetic index have been set. This leads to differences in the impact date of more than two months!

9.2.2 Ballistic Trajectory Below 120 km Altitude

Georg Koppenwallner

The essential trajectory features are clearly demonstrated if we use the ballistic re-entry of an object with constant ballistic coefficient (B = m/C_DA_ref) as sketched in Figure 9.2.2. With a simple exponential atmosphere model and a locally flat Earth, Allen and Eggers (1957) derived the following analytical expressions for the trajectory data and for flow conditions along the trajectory.

Maximum heating rate is given by:

Allen/Eggers analytic analysis reveals that ballistic re-entry trajectories in terms of V(h)/V_E are in principle similar, with an altitude shift determined by B sin(θ_E).

The main deceleration occurs in a narrow altitude span of Δh ~20 km, which with decreasing B sin(θ_E) is shifted to higher altitudes. The solution also shows that the re-entering object experiences first the high heating phase at V/V_E = e^–1/3 and then a phase of maximum deceleration loads at V/V_E = e^–1/2.

Figure 9.2.3 shows trajectories for B = 0.01, B = 1 and B = 100 kg/m², which cover extremely small debris particles and larger fragments. It is evident in this figure that extremely small particles experience maximum loads at relatively high altitudes compared to larger objects.

The right side of this figure indicates the increase of the molecular mean free λ path with altitude, which is important to discriminate between the different aerodynamic flow regimes as described below.

9.2.3 Aerodynamic Flow Regimes

Georg Koppenwallner

Viscous and rarefied flow regimes determine how aerodynamic forces and aero-heating should be analyzed. The non-dimensional Knudsen number Kn = λ/L provides the best means to distinguish between three main aerodynamic flow regimes relevant to re-entry analyses.

In this case the molecular mean free path λ is much larger than the characteristic body length L. The free stream molecules hit the body without collisions with wall-reflected particles. In this regime, the free molecular theory applies and the aerodynamic coefficients are independent of the Knudsen number. In the free molecular flow regime, the aerodynamic coefficients C_F depend only on shape, attitude, and Mach number.

Mutual collisions between wall reflected and free stream molecules occur; as the Knudsen number approaches the lower limit, molecular collisions lead to formation of shock and boundary layers. The aerodynamic coefficients in this flow regime also depend on the Knudsen number.

This regime is dominated by molecular collisions with wall reflected particles that produce shock waves and a viscous boundary layer along the wall. The relevant non-dimensional parameters are Mach number M and Reynolds number Re = ρVL_ref/μ. Via mean free path and viscosity relationships, Mach and Reynolds number can be related to Kn as shown below:

In principle the following basic relation for aerodynamic force coefficients exists.

In hypersonic continuum flow for blunt bodies the Mach number independence principle is valid and the pressure forces dominate over the frictional forces. With simplified methods like the Newtonian flow model the aerodynamic force coefficients can be determined in an approximate way and the functional dependence can be reduced to:

Aero-heating analysis is the critical part in hypersonic continuum flow. A simple and fast approach is to relate the local heating to the stagnation point heating rate, for which analytical formulations exist, from Lees (1956); Fay and Ridell (1958) and Detra, Kemp and Ridell (1957).

For an aerodynamic cold wall with Tw << T₀ the heat transfer Stanton number is given by relations of the following form.

In this equation R_N is the effective nose radius at the stagnation point and μ(T₀) is the viscosity at stagnation point total temperature. Aero-heating on other parts of the body with smaller flow inclination angles is often related via the local flow inclination to the stagnation point heating rate. For small objects or objects with small ballistic coefficient the free molecular heating rate is used to obtain the limiting values of the stagnation point Stanton number with St_s = 1α_E;

With α_E the energy accommodation coefficient:

Due to the complexity of spacecraft structures, the continuous change of attitude, and the change of shape due to melting, advanced aerodynamic computational fluid dynamic tools like Navier–Stokes Solver or direct simulation Monte Carlo (DSMC) methods are not feasible for aerodynamic analysis. Thus only methods reduced to the essential aerodynamic influence parameters in the different flow regimes can be applied.

9.2.4 Modeling Trajectories with Lift Below 120 km: The Atmospheric Arc

Jean-François Goester

This section examines the atmospheric portion of the trajectory (i.e. for altitudes less than about 120 km) for a re-entry vehicle where both lift and drag forces can be significant. Below 120 km altitude the drag will become as large as or even larger than the force due to gravity. The ratio of friction to gravity forces can be of the order of 10 (i.e. a drag deceleration that is ten times larger than gravity). So an orbital type model is no longer appropriate.

In order to work with variables that are more representative of the motion, rather than using the six “position/velocity” parameters or the orbital parameters, the parameters (r, L, l, V, γ, χ), which are defined in the following diagram, will be used:

represents a unit vector along the local vertical and the local horizontal plane.

In the non-inertial frame , the fundamental equation of dynamics is written as follows:

(15)

where:

Acceleration Expressions

The relative acceleration is given by

By expressing the derivative of the vector in the frame , we obtain:

(16)

Then, centrifugal acceleration can be expressed as: where represent the rotation vector of the Earth.

By expressing the equations again in the frame :

(17)

At last, CORIOLIS acceleration equals to

(18)

Force Expressions

Only aerodynamic (lift and drag) and gravitational forces are discussed here. It is clear that the other terms, such as solar radiation pressure or lunisolar potential, are negligible. In addition, in order to simplify the equations in this book, only the central term of the terrestrial gravity potential is taken into account. Adding the term in J₂ would, for instance, modify the weight equation by adding, in particular, another term along the axis. For controlled re-entries, one should also consider propulsive forces. Here the three remaining forces can be described as follows.

Assuming that the atmosphere moves at the same speed as the Earth, so V_a = V and defined as follows:

where μ, the bank angle, i.e. defining the rotation around the velocity axis (not to be confused with roll, which is rotation around the longitudinal axis of the vehicle).

Thus, the lift component may be expressed as:

(19)

The aerodynamic coefficients are customarily associated with a constant reference surface, but depend on the Mach number as well as the angle of attack (AOA or α), which corresponds to the angle formed by the direction of the vehicle (centerline) and the direction of the velocity relative to the air flow.

Equations of Motion

The two vector equations and lead to the six following scalar ones:

(20)

(21)

(22)

(23)

(24)

(25)

Equilibrium Hypothesis

In order to obtain analytical expressions for the various characteristic parameters of the trajectory, the equation of motion should be simplified.

First, the terms in Ω_T and especially Ω_T² due to the rotation of the Earth can be neglected. Indeed, the velocity of a body re-entering the dense layers of the atmosphere is much larger than that of the Earth (about 500 m/s at the Equator).

The second approximation concerns the flight path angle (FPA). It will be considered to be very small, and its derivative assumed negligible compared to the other terms in equation (7). These assumptions are fairly realistic both for natural re-entries (from a low orbit, the FPA is actually very small because most of the time, the eccentricity of the orbit before re-entering is nearly zero) and for controlled or manned flights (even for lunar APOLLO missions where the entry FPA was ~ –6 degrees). Thus, the equation (7) can be written as follows.

(26)

In this equation, the name of “equilibrium” hypothesis is evident because the lift balances the weight (mg) and the centrifugal force (due to the angular velocity). This relationship is quite useful for understanding the influence of each term for the case of a “rebound” in the atmosphere. Indeed, it could be due to the lift (when |μ| is lower than 90 degrees) or conversely, to a “perigee effect”: a perigee high enough so that there is no capture by the atmosphere.

Assuming that r and g are constants, since |r - r_eq| << r_eq (equatorial radius):

(27)

Next, the atmospheric density is modeled by an exponential function:

(28)

After some calculations and other approximations, which are too long to explain here, equations (7) and (28) can be put in the form:

(29)

(30)

By eliminating ρ, an approximate expression for γ can be obtained:

(31)

Equations (44) and (10), which describe the evolution of the deceleration and the FPA, can be used to deduce the analytical expressions for the time to impact (t), the lateral (ś) and longitudinal (s) ranges, and , the azimuth of the velocity vector with respect to the East, i.e. with respect to the Equator:

(32)

(33)

(34)

(35)

V_e, t_e, ψ_e, , s_e are the initial conditions at the entry of the vehicle.

These relationships are very interesting because even though they are not as accurate as the more complete numerical models that use atmospheric models or aerodynamic coefficients, the influence of the various parameters can immediately be examined. Specifically, it can therefore be seen that:

First Rebound Hypothesis

The following analytical solution of equations (40) to (8) includes the same approximations above (fixed Earth, small FPA), but also considers that the predominant force acting on the vehicle is the aerodynamic lift. Because of this, a reduced FPA (in absolute value) is observed, and then a rebound on the atmosphere occurs.

Equations (40), (63) and (7) thus give these following three equations, respectively:

(36)

(37)

(38)

The other hypotheses/approximations used in this analytical model are:

The two following equations are then obtained:

(39)

(40)

Comparison of Numerical and Analytical Models for Re-entry Trajectories

The following curve shows the trajectories obtained with the equilibrium and first rebound hypotheses as well as a numeric simulation by integrating the equations of motion. The aerodynamic coefficients are taken as constants and the atmospheric model is exponential with a height scale (H = 1/β) of 7000 m, which is characteristic of these low altitudes. The following case is considered:

Figure 9.2.4 clearly shows that the first rebound hypothesis is only valid at the beginning of the trajectory, whereas the “equilibrium” hypothesis applies better afterwards. Note however, that these trajectories correspond to a particular re-entry case, particularly with respect to the L/D ratio and the entry flight path angle. Indeed, for a trajectory with a smaller entry FPA and especially a larger L/D ratio, the equilibrium hypothesis takes over earlier and better, as can be seen in Figure 9.2.5, once the various rebounds on the atmosphere have been eliminated.¹

However, a noticeable difference in the initial (altitude, velocity) couple can be noted. Indeed, by fixing the initial altitude (120 km), the initial velocity is also fixed, which does not correspond directly to the initial velocity on the two other curves. This difference already exists in Figure 9.2.4, but is less visible.

The Effect of the Ballistic Coefficient

Examination of the equations of motion shows that the term S/m is always in front of the aerodynamic coefficients. This is why the term SC_D/m – called the ballistic coefficient – is a key vehicle parameter affecting the trajectory. The lift coefficient C_L is accounted by the L/D ratio (C_L/C_D) as shown in the next section.

Figure 9.2.6 shows the effect of the ballistic coefficient for a particular case. Instead of an “altitude–velocity” diagram, a more geometrical representation with the longitudinal range on the abscissa axis is used to reveal the large differences in range associated with differences in the ballistic coefficient. A diamond marks the place where the maximum load factor is attained, as well as its value, which increases with the ballistic coefficient, but only slightly (5% for a 100-fold increase in SC_D/m).

The Effect of the L/D Ratio

Returning to the case in the previous section (SC_D/m = 0.002 m²/kg), this time the L/D ratio is varied. Figure 9.2.7 shows the strong sensitivity of the longitudinal range to the L/D ratio: range increases by a factor of two for L/D = 0.3. But more important – especially for manned flight – is the large drop in the maximum load factor encountered during the descent. For the same L/D ratio value of 0.3, the load factor goes down to 2.24 g.

There is also a plot of the trajectory for an L/D ratio of −0.3, which does not represent a negative C_L, but rather a bank angle of 180 degrees (flying upside down so that the lift force is in the direction of gravity). The difference in longitudinal range with respect to a ballistic re-entry (L/D = 0) is less pronounced (i.e. it is more difficult to bend the trajectory down than to make it “float” more), but with nearly same the maximum load factor as for the case where the lift vector remains pointed up.

9.2.5 Control of Re-Entry Trajectories

Jean-François Goester

After having treated the de-orbiting maneuver(s), the atmospheric phase, which by convention begins below 120 km altitude (about 40,000 ft) is now discussed. In contrast to the preceding phase, the two principal forces to be taken into account will be the attraction of the Earth and the atmospheric friction (lift + drag). The atmospheric effect can even be larger in magnitude than gravity. Propulsive atmospheric trajectories will not be considered here. The motion of the vehicle will only be affected by modulating the aerodynamic forces (as for a glider).

First, the domain in which the spacecraft can evolve is defined, given its kinematic, structural and thermal constraints. Then, some general points about optimizing this type of trajectory will be discussed, with a way to derive suboptimal reference trajectories, which will be easier to study. Finally some examples of guidance strategies that have been used will be presented.

The equations of motion that were defined in section 9.2.4 will be used in what follows.

Flight Envelope

During its re-entry into the atmosphere, the space vehicle must evolve in a domain called the “flight envelope”. This sets the range of altitudes between which it can move at each moment.² Since the parameter “time” is not very significant from the mechanical or energy point of view, it is preferable to reason in terms of velocity.

Thus, the flight envelope will be defined in general in an altitude/velocity diagram that takes into account the kinetic (velocity) and potential (altitude) energy.

The limits of the domain are usually defined within the section that follows.

Equilibrium Trajectory

This is the lift ceiling, that is, the maximum altitude that the vehicle can have for a given velocity taking into account its aerodynamic behavior (lift in particular) and gravitational attraction. It corresponds to:

(41)

which is like saying that the lift balances the weight.³

The equations giving the equilibrium trajectory comes from the equations of motion (7):

(42)

If the rotation of the Earth is neglected and after multiplying by m.V, this becomes:

(43)

By using the two following simplifying hypothesis in addition:

The following equation for the altitude of the lift ceiling is obtained:

(45)

It can be seen that the lift ceiling depends on the control parameters of the vehicle: the aerodynamic bank angle (μ) and the angle of attack (α) via the lift coefficient C_L.

The maximal ceiling is, of course, obtained for zero bank angle.

In what follows, it will be seen that in general, the angle of incidence follows a more-or-less well defined profile, which can then be taken into account when calculating this limit (as for the following limits).

Finally, to allow some margin, a non-zero bank angle can be considered (for example of the order of 30 degrees).

Dynamic Pressure Limit

This is a structural limit on the re-entering vehicle especially when it is equipped with aerodynamic flaps. In this case, the hinge torque can be expressed in the form:

(46)

Thus maximum torque in the hinges corresponds to maximum dynamic pressure.

The expression of the dynamic pressure limit (max) is:

(47)

Now, this expression depends only on the atmosphere (β) and on the velocity, but not on the control parameters of the vehicle (α and μ).

Thermal Limit (In Flux)

To find the bounds of the flight envelop from the thermal point of view, a maximal value of the reference flux is usually used, i.e. the flux, which a sphere whose radius is equal to the radius of curvature that the “nose” of the vehicle (R_n) would experience. The equation for the flux is:

(48)

The temperature (at least on the “nose”) can be obtained using Stefan’s Law (T is expressed in degrees Kelvin):

(49)

The altitude corresponding to this thermal limit is then:

(50)

Load Factor Limit

The re-entry vehicle must not exceed a given maximal load factor, which depends on its size and the possible presence of crew. Most of the time, this value is expressed either in vehicle axes, especially with respect to the “z” axis (extrados/intrados), or as a factor of the total load. The expression for these load factors is:

(51)

By making the same approximations as for the lift ceiling, the total load factor limit is:

(52)

Flight Envelope in Altitude/Velocity

In order to numerically illustrate all these limits, the two following cases (Figure 9.2.8) were studied with a total load factor of 5 g and a maximum flux of 600 kW/m² (the dynamic pressure is not active):

On the right side of each plot in Figure 9.2.8, it can be seen that the area between the thermal limit and the equilibrium limit, which is called the “re-entry corridor” is rather small.

Flight Envelope in Deceleration/Velocity

To work in altitude/velocity diagrams raises the problem of the in-flight determination of the altitude. Indeed, it is rather difficult to determine exactly this parameter because to use external means of measurement with very high speed is impossible because of the temperatures that are met. Even the global positioning system (GPS) can be unusable during certain periods, called “blackout”, because of an ionization of the plasma which forms around the vehicle.

On the other hand, the drag deceleration is something directly measurable with accelerometers or inertial measurement units. Furthermore, to convert a altitude/velocity diagram in a deceleration/velocity one is relatively easy by considering some simplifying hypotheses:

9.2.6 Principles of Guidance

Jean-François Goester

It is not the purpose of this section to discuss the navigation/guidance/control chain in general. However, it is useful to consider the guidance function in the context of the navigation/guidance/control chain. As a summary:

It can be seen in Figure 9.2.10 that these functions are closely related. Navigation must give information to the guidance/control that is stable enough for the control system to react properly. In the same way, the guidance must send realistic nominal values.

The following sections review some current or former (especially for American vehicles) re-entry guidance algorithms. It will be seen that there are four guidance principles: the first two were used on GEMINI, the third one was used on SOYUZ, while the last one corresponds to APOLLO and later on the US SPACE SHUTTLE or Atmospheric Re-entry Demonstrator (ARD).

Roll Guidance

This technique was used on the GEMINI III, IV, VIII, IX-A, X, XI and XII flights. It is based on a reference trajectory with zero L/D ratio. It is mainly applied to re-entries which do not require a lateral displacement.

Figure 9.2.11 illustrates the logic of guidance: by working in the direction of the lift, one tries to follow a lift profile until the intersection with a ballistic trajectory that intersects the target. After this, a constant roll speed is set to cancel out the natural lift of the vehicle.

This speed can be cancelled from time to time in order to use the lift again if the vehicle gets too far from the reference ballistic trajectory, which is moreover slightly biased for an impact point in front of the target. This is done so that afterwards, only positive lift will be used, i.e. upward (bank less than 90 degrees in absolute value). As for lateral guidance, this is done by intermittent bank commands (for example of 90 degrees).

Constant Bank Angle Guidance

This technique was also used on GEMINI, specifically on flights V, VI-A and VII. Once more, the distance to the target with respect to a reference distance is involved, but this time it is based on half-lift trajectory.

If these two ranges are equal, a bank angle of 60 degrees is commanded. On the other hand, if the target is too far, a bank angle lower than 60 degrees is requested and conversely, if the target is too close, a bank angle greater than 60 degrees is requested in order to force the trajectory to go lower. The lateral guidance is obtained by changing the sign of the bank.

SOYUZ-TM Guidance

The kind of guidance used on the vehicles of the type SOYUZ-TM is rather “rudimentary” by comparison with the lift to drag guidance (see below). Nevertheless, it does not apply to the same type of vehicle (the lift to drag ratio of the SOYUZ in hypersonic phase does not exceed 0.25) and does not pursue a high landing accuracy thanks to available zones on Kazakhstan.

The guided atmospheric phase begins towards an altitude of 80 km. In fact the criterion for starting up of the guidance is mainly based on the variable V_S called “apparent velocity”, which is defined as follows:

(53)

with: tsep = date of the separation from orbital and service modules (the separation occurring towards 140 km in altitude).

The guidance acts then only on the bank angle (named γ in Russian equations!) of a 60 degree average value modulated according to the atmospheric situation:

γ_com = (1- 2S)∗(60^o+δγ_cor), S = 0 or 1 according to the initial sign that we want to give to the bank angle.

The value of γ_com is updated every 201.8 m/s in V_S by comparison to a profile of reference time according to V_S.

When V_S = 4300.8 m/s a roll-reversal is commanded, that is a change of sign of the bank angle. This event is fixed, which implies the absence of lateral guidance, strictly speaking.

The guidance ends towards 21 km in altitude (V_S = 7372.78 m/s) and finally, the parachute phase begins towards 10.7 km in altitude for a duration of about 15 minutes.

Let us note that multiple different back-up modes are planned on board, in particular re-entries of ballistic type (by a spin mode of 13 deg/s which nullifies the lift capacity) in degraded cases (errors on de-orbit impulse, wrong behavior of the nominal guidance …).

Lift to Drag Ratio Guidance

This principle was first used on APOLLO, and then improved for use on the US SPACE SHUTTLE. It was also selected for the ARD.

First of all, it is necessary to define a reference trajectory (Figure 9.2.13) on which we shall try to be enslaved. This trajectory will have to be inside a flight envelope that we shall establish in a diagram deceleration/velocity. This trajectory will generally have the following characteristics:

The most direct and efficient way of being locked on this deceleration profile (shown in Figure 2.1.15) would be by an angle of attack command. However, on space vehicles, it is rather difficult to work with this angle, which is very sensitive to any instability and is directly tied to possible thermal constraints. Remember that the flight envelopes covered here are cast compared to a “classic” aircraft, and that the atmosphere is not really the preferred domain of a space vehicle.

Since it is preferable not to work with the side-slip angle, the bank angle is the only command parameter left. Going back to the equation of dynamics (7), it can be seen that the derivative of the FPA (γ) is a function of the cosine of the bank angle.

Changing the value of this cosine changes the derivative of the FPA, and thus, at the following step, the FPA, and later the altitude, and as a consequence the density. With this approach, it is possible to act on the drag.

The bank angle can be calculated with the following equation:

(54)

with: L being the lift, D being the drag, L_v being the vertical lift, μ being the bank angle.

The control is achieved with three corrective terms, proportional, integral and derivative (PID) in the form:

(55)

The commanded bank is deduced from:

(56)

Note that the derivative of D can be replaced by the vertical velocity V_z, which is a more “physical” term.

In order to better control the accuracy in longitudinal range, the reference deceleration (D_ref) can be replaced by a commanded deceleration (D_com) taking into account the distance of the point of impact (dimp):

(57)

Moreover, in order to improve the performance of this type of guidance, a slight modulation of the angle of attack can be introduced:

(58)

and:

(59)

Sometimes, it is also preferred to use the “total velocity” defined as below rather than the relative velocity in order to take into account the gravitational effects in a better way.

(60)

Finally, the lateral guidance is obtained by the choice of the sign of the bank angle (since it follows the arccosine, we have the choice of the sign) via the determination of the road angle (angle between the speed of the vehicle and the direction of the target point) with respect to a road corridor. An example of this type of lateral guidance is given Figure 9.2.14.

This guidance logic is more sophisticated that the previous ones. Nevertheless, its level of complexity can easily be approached. In addition, it seems to be the best suited for lifting bodies.

References

1. Allen HJ, Eggers Jr AJ. A study of the motion and aerodynamic Heating of Missiles entering the Earth’s Atmosphere at High Supersonic Speeds NACA TN-4047. Washington USA, 1957 1957.

2. Detra RW, Kemp NH, Ridell FR. Addendum to Heat Transfer to Satellite Vehicles Re-entering the Atmosphere. Jet Propulsion. 1957;Vol. 27(no. 12):1256–1257 Dec. 1957.

3. Fay JA, Ridell FR. Theory of stagnation point heat transfer in Dissociated Air Journal of Aeronautitical Sciences Vol 24 pp 351-356, 1957. Reprint in AIAA Journal. 1958;Vol. 41(No. 7a):373–386 2003.

4. Lees L. Laminar Heat Transfer over Blunt-Nosed Bodies at Hypersonic Speeds. Jet Propulsion. 1956;Vol. 26(no.4):259–269 1956.

9.3.1 Introduction to Destructive Re-Entry Phenomena and Modeling

Destructive re-entry analyses are generally for SC designed to perform an orbital mission, and not to survive re-entry like a ballistic or lifting re-entry vehicle. The similarities and differences between the two cases are considered in the development process of the analysis methods. Table 9.3.1 summarizes the main differences between re-entry vehicles designed to survive and the destructive re-entry of an orbital SC.

The unstable motion associated with destructive re-entry requires that the analysis account for tumbling SC without an a priori fixed head and tail. A destructive re-entry analysis should also account for the change of shape and mass properties due to melting and fragmentation, as well as the increased number of objects present after each fragmentation. Figure 9.3.1 shows the main events during a destructive re-entry.

In summary the main events during a destructive re-entry are:

In addition the following additional destruction phenomena need to be considered:

Analysis of the destruction of a SC along its trajectory involves the following elements:

The main challenges in treating these different subjects are:

Solutions of this complex, multidisciplinary process can be achieved by the following approaches:

Material Properties Effect on Destructive Re-Entry

Complete demise of an object requires enough energy to heat it up to the melting temperature T_m plus the heat of fusion (also referred to as specific heat of melting, h_m) to complete the melting process. One can then assume that the melted liquid layer is continuously blown off from the object.

Thus, the specific heat capacity , the melting temperature T_m and the specific heat of melting h_m are important material properties that influence thermal destruction. With a mean value of between 293 K and the melting temperature one obtains:

High melting temperatures T_M allow high surface temperatures and therefore also high radiant heat losses given by: .

Figure 9.3.2 shows melting temperatures of typical materials with tungsten, molybdenum and titanium having the highest melting temperatures. Figure 9.3.3 shows the heat storage capacities necessary to reach the melting state (a prevalent failure condition) and the complete melting (demise condition). Titanium shows the highest resistance against thermal destruction. The heat of fusion for aluminum is equivalent to Inconel and stainless steel, however due to its low melting temperature it has significantly less capability for re-radiation of the aerodynamic heat input.

Introduction to Object Oriented Analysis Methods

Object oriented numerical models are typical examples of simplified numerical analysis approaches. Since satellites during a tumbling re-entry generally do not produce significant average lift forces acting against gravity, a simplified ballistic re-entry approach can be used to model the trajectory and flight dynamics. Thus, only a 3-degree of freedom (DOF) flight dynamics analysis can be applied, which reduces the aerodynamics into a simple drag analysis. In addition the SC can be modeled as simple geometric objects with specified mass and material. The object analysis method can, in principle, also be applied to complete satellites. The SC is then represented by a container, which contains an assembly of geometric objects representing the main components of the original SC as shown in Figure 9.3.4. The idealized container size with object assembly must reproduce the overall mass and ballistic coefficient of the SC at the beginning of re-entry. The container SC starts the re-entry at the entry interface with prescribed initial conditions. At a prescribed breakup altitude the container releases its content of objects, which are then individually traced until ground impact or demise. The breakup altitude is often fixed between 75 and 85 km.

In this type of model, the heating of the objects generally starts after release at the breakup altitude, and the resulting foot print of the ground impact will not show any lateral dispersion.

Advantages of this method are:

Disadvantages are:

Introduction to Spacecraft Oriented Analysis Methods

These numerical models use a more detailed physics-based model of the SC during its re-entry, often including an appropriate computer-aided design (CAD) type model of the SC itself. Figure 9.3.5 explains this approach with a panelized 3D model of the SC.

More detailed analysis methods are necessary to treat trajectory and 6DOF flight dynamics. The 6DOF flight dynamics requires an aerodynamic analysis for forces and moments related to the actual state in flight, including the orientation. Aero-heating is analyzed over the vehicle surface and input to the thermal analysis tool. Thus, the whole vehicle is generally modeled and analyzed with a special type finite element model that is compatible with the analysis methods.

Advantages of the method are:

Main disadvantages:

9.3.2 Object Analysis Methods for Destructive Re-Entry

Analysis methods for simple objects provide a general insight into the dependence of re-entry demise on object material, object size and ballistic coefficient. Object analysis methods can be used for the following purposes:

Trajectory Analysis, Earth Gravity and Atmospheric Density

Trajectory analysis is based on a 3-degree of freedom trajectory of a point mass. Thus only aerodynamic drag forces and gravity forces are acting on the body. If impact points in an Earth-fixed coordinate system are required one may assume a spherical rotating Earth and four equations need to be numerically integrated as function of time:

Acting forces are the aerodynamic drag force and the Earth gravity force.

The four trajectory equations are numerically integrated using a Runge–Kutta method with prescribed time step Δt.

Simple Object Modeling Approaches and Material Properties

In general these methods use simple object shapes with prescribed flow orientation or with prescribed rotational motions. In order to fully describe the model geometry data, motion type specifications, mass and some material data are needed. Table 9.3.2 shows the relevant data describing the objects.

The geometric parameters do not contain the wall thickness of the object. Wall thickness, s, is in principle determined by mass and material density ρ and object surface area, with s = m/(ρA_S).

The destruction analysis is greatly simplified by the assumption of infinite heat conduction of the material. This assumption and the prescribed motions facilitate an approximation of the aerodynamic heat transfer distribution over the surface by a mean aerodynamic heating.

Simple Object Aerodynamics

The bodies to be treated are in principle blunt and the destructive re-entry occurs at hypersonic Mach values. Therefore the Mach number independence principle applies for the drag coefficient. The drag coefficient, C_D, only depends on the geometric parameters in continuum and free molecular flow, but in the rarefied transitional flow a bridging method is used.

Table 9.3.3 summarizes for a sphere and for cylinders the drag coefficient dependence on geometry and attitude motion.

Rarefied transitional flow regime is usually covered by a bridging relation, which gives for Kn<<1 the continuum drag and for Kn >>1 the free molecular drag coefficient. If one uses the normalized drag coefficient definition as shown in the following equation the bridging relation has the following limiting values: continuum flow F_bridge = 0, and free molecular flow F_bridge = 1.

Simple Object Aero-Heating Rate

The principal approach is similar to drag calculation, namely to determine for typical body shapes and tumbling motions the heat transfer rates for continuum flow and for free molecular flow, using a bridging method in the transitional flow regime.

Thermal Response Analysis

Net heating rate for a tumbling object with infinite heat conductivity is given by the following energy balance

with Q_rerad the energy lost by re-radiation_,

The resulting thermal response of the object is given by:

This equation is valid for T < T_melt. For T = T_melt the failure case is reached with mass loss by melting.

For T = T_melt isothermal melting with mass loss of the liquid layer will occur

Demise of the object occurs if all mass is molten.

The necessary time integrated heat input must reach the following limit:

Numerical Solution of the Re-Entry Analysis

Initial conditions for integration are usually fixed for the atmospheric re-entry phase, which usually is defined to begin at 120 km altitude. Numerical solution requires solving first the four trajectory equations by a Runge Kutta method with a prescribed time step Δt. The gravity field model and atmospheric models provide the necessary inputs as function of altitude and time. For each time step of solution the net heating rate Q′_net is determined and the thermal response is numerically integrated. The numerical thermal response analysis has to distinguish between the heating and melting which are separated by the condition T_object = T_melt.

9.3.3 Analytical Re-Entry Destruction Analysis

Approaches for analytical re-entry and destruction analysis have been reported by Rachel et al. (1999); Baker et al. (1999); Koppenwallner, Fritsche & Lips (2001) and Koppenwallner & Lips (2008). The analytical approximate solutions have the advantage that they show directly the parameters which influence the destruction or survival process of an object.

The thermal destruction process of an object will depend on the heat input profile during re-entry and on the thermal response of the object. The thermal response of the object combines heat storage and heat loss due to radiation from the surface. If infinite heat conduction within the body is assumed, then the following energy balance equation applies for a local wall element:

The first term denotes the energy stored in the object and the second term the energy lost by radiation. In principle we can distinguish the following limiting cases leading to the re-entry survival of an object.

Radiative Energy Balance and Radiative Survival

This is applicable for thin-walled bodies with higher melting temperatures. The object has the capability to re-radiate the aero-thermal heating rate:

(61)

(62)

Calorimetric Energy Balance and Calorimetric Survival

This is applicable to thick walled or solid bodies. The object has such a high heat storage capability that during re-entry the melting temperature will not be reached. For a thermally thick body the heat storage capacity is dominant, radiative heat losses are neglected. Failure will occur in case the integral heat transferred to the body exceeds the critical heat storage capacity. We then obtain:

(63)

(64)

(65)

To show the usefulness of the criteria we use as example a spherical shape.

The resulting parameter for the re-entry trajectory is the ballistic coefficient B which is shown in the following table for a solid spherical body and a spherical shell:

For a solid sphere the ballistic coefficient is only dependent on the radius whereas for a spherical shell only the wall thickness s is important. The same rule applies for all solid and hollow shells of identical external shape. Thus a large spherical thin-walled shell can have identical ballistic coefficient and identical trajectory as a small spherical solid object.

Demise and Survival of Simple Solid Bodies

This subject has been studied by Koppenwallner, Fritsche & Lips (2001; 2008) in several publications. The following general results can be given.

Figure 9.3.6 shows for solid spherical objects the span of radii for demise (R_min < R < R_max) and the survival conditions as function of the material for typical metals.

Body shapes like solid discs and cylinders with diameter to length ratios of 0.2 have been studied by Lips and show a similar behavior.

Demise and Survival of Simple Hollow Shells

For simple hollow shells a similar analysis can be conducted as for solid bodies. The important point is that the ballistic coefficient of hollow thin-walled bodies is independent of body size and only dependent on wall thickness. The criteria for radiative survival of hollow spherical shells is given by the following relations:

The criteria for calorimetric survival or demise are:

We have on the left side of these equations geometric parameters like in terms of wall thickness s and shell radius R namely either s/R or s∗R and on the right side material data of the object and entry conditions. If we plot these relations for selected materials in a chart with wall thickness as ordinate and shell diameter as abscissa one obtains the specific demise and survival region of specific objects.

Figure 9.3.7 shows the demise and survival regions for titanium and aluminum spheres, a function diameter (d) and wall thickness (s); the solid line dark line represents the massive limit with s = d/2. For titanium the demise region is small whereas for aluminum the demise region is quite large.

The straight lines represent the analytical borders, with definition of why survival or demise occurs.

The curved boundary envelopes were obtained by numerical 2D re-entry integration of a large variety of selected objects with different wall thickness and diameter. The combination of analytic and numerical results complements each other because numeric results are more precise, but the analytic results provide insight into the main reason for demise and survival.

Similar results can be derived for other geometric shapes.

9.3.4 SC-Oriented Codes

These numerical models try to reproduce the SC with its main physical quantities by a panellized model similar to a finite element model. The systems have to combine the SC modelling with all relevant analysis modules. A classic SC-oriented code is the ESA SCARAB software (Space-Craft Atmospheric Re-entry and Aerothermal Breakup) developed by HTG, see e.g. Koppenwallner, G., Fritsche B., Lips T., Klinkrad H. (2004) and Lips, T., Fritsche, B., (2004).

Flight Dynamic Analysis

The 6DOF trajectory analysis requires aerodynamic force and moment vectors calculated for each state vector during flight and the SC geometry, which may change during the destruction process.

Flight dynamics requires in addition the actual SC-mass, the actual center of mass (CoM) and the mass moment of inertia (MoI) data. During the destruction process these data will change and the code has therefore to update these data continuously. The governing equations for the flight dynamics are based on Newton’s second law as follows.

Aerodynamics and Aero-Heating

Aerodynamics and aero-heating are calculated on the local panel level for the actual geometric vehicle state and for the actual free stream and vehicle orientation. This means that for each elementary surface panel pressure, shear stress and heat transfer rates are calculated. The integration over all surface elements gives the integral aerodynamic forces and moments.

The panel aerodynamics is applied in the hypersonic regime (Ma > 5) and it distinguishes between free molecular flow, rarefied transitional flow and continuum flow. In continuum flow the local pressure coefficient is calculated with a modified Newtonian method considering also the mutual shadowing of surface panels. In free molecular flow the exact free molecular formula for convex surface elements are used for pressure and skin friction calculation. In the rarefied transitional flow a special local bridging method is used which distinguish between pressure and shear stress.

The local free molecular Stanton number StFM is calculated with the standard formulation equivalent to pressure and shear stress. Continuum heating is related to the stagnation point Reynolds number and to the local inclination θ of the surface element.

with r_N the effective nose radius of the object and μ(T₀) the stagnation point viscosity. SC oriented codes have used a power law viscosity dependence on temperature T with

For the stagnation point the above formula reduces then to an equivalent expression of Detra, Kemp & Ridell.

The local transitional heating is calculated with the following bridging relation.

The panel wise response analysis is one of the important features necessary for a realistic aerodynamic, model, but also for heating, the thermal response and the destruction process. This requires the use of true volume panels. A neighbour-checking algorithm is necessary to perform a heat conduction analysis. There are adjacent panels within one primitive and touching panels between different elements. A discrimination between inside surfaces not exposed to the flow and external aerodynamic active surfaces is required. Due to the local thermal response molten external volume is removed and internal surfaces start to become active for aerodynamics and external heating. Also mass properties need to be updated after molten panels are lost.

Fragmentation Mechanism and Fragmentation Analysis

Fragmentation means, in principle, that from one object two or more new objects are generated. In principle, two potential modes of fragmentation need to be considered: fragmentation due thermal loads that produce melting, and fragmentation due mechanical loads evaluated at prescribed locations.

A fragment may be generated by melting of a closed chain of surface panels. This means that a closed set of panels loses the connectivity to the main object by melting of panels which originally established the connectivity. Figure 9.3.8 shows an SC with lost panels just before fragmentation due to melting.

Figure 9.3.9 shows the loss of a solar array by mechanical fracture at the support arm acting as joint.

Fragmentation by mechanical loads can be computed by an analysis of structural and dynamic loads on prescribed positions or load bearing elements of the SC, which establish connection between SC components. A typical example is when a solar array breaks off at the array support arm. The SC model usually uses a cut plane passing through a load bearing element. The cut plane separates distinct components that form the rest of the SC. The code has first to analyse the inertia and aerodynamic loads acting on the cut plane through the load bearing joint. The strength of the joint can either be prescribed or calculated by its resistance moment. When the actual load exceeds the joint strength, the SC will be fragmented in two parts at the prescribed position. As alternative to fracture by loads, a fracture at a critical temperature limit may be introduced. This situation can occur, for example, with glued connections that typically loose strength at critical temperature levels.

9.3.5 Introduction to Tank Burst Analysis

This section provides an introduction to the analysis of various rupture scenarios for tanks exposed to re-entry heating. Separate subsections below provide examples of detailed analysis methods applied to tanks containing liquid or frozen propellants.

Tank bursting may occur due to heating of tank content and a follow on pressurization. This may occur for high pressure gas tanks and for tanks with liquid fuel and oxidizer content. A tank is expected to burst if the tank pressure P_T exceeds the burst pressure of tank P_burst (T), which usually depends also on temperature of the tank walls.

For high pressure gas tank the analysis is relatively simple because only the pressurisation due to heating of the gaseous content needs to be accounted for.

where dQ/dt is the heat input into the tank content.

The analysis is more complicated for tanks containing liquid and gaseous content, which is typical for pressurized propellant tanks. Gaseous helium is commonly used for pressurization. The analysis of tank pressurization by external heating requires accounting for the potential vaporization of the liquid content. Thus thermo-physical data of the propellant in the liquid and gaseous state and the phase transition properties are required.

Figure 9.3.10 shows the principle phenomena involved in the pressurization process for a tank containing liquid and gaseous content. The vapor pressure relationship to the heat of vaporization is given in a classical simplification.

Starting from the initial condition, denoted by a 0 in Figure 9.3.10, one can distinguish the following pressurization phases:

Bursting produces local cracks through which the tank contents are released. Locations of the openings and directions of discharge are a priori unknown. Thus only the loss of the tank content can be predicted using with any confidence using these analyses models.

Tank Content Discharge after Bursting

For the release process it is important if bursting occurs with the contents in gaseous or liquid state.

Gaseous discharge will occur as transient sonic free jet, expanding into the low pressure ambient environment. Figure 9.3.11 sketches the resulting plume-type discharge of gaseous contents.

After tank bursting any liquid contents are suddenly exposed to an environment of extreme low ambient pressure in the gaseous phase state. Thus, virtually instantaneous flush evaporation may occur in the tank and in the discharged liquid. This is indicated as isothermal pressure reduction 3a–4 in the phase diagram of Figure 9.3.12. Due to the fast evaporation process, the heat of evaporation will primarily be taken from the liquid which then will cool down, and may lead to freezing.

On the other side of Figure 9.3.12, fast evaporation with numerous bubbles generated in the liquid may result in a large volume increase, which also could result in a vapor bubble explosion. No satisfactory solution has been found until now for the flush evaporation process of liquid gas content after tank bursting.

9.3.6 Explosive Destruction during Re-Entry

Explosive destructions were definitely observed during the ATV – Jules Verne re-entry Hatton J.P., Jenniskens P. (2009). The latest SCARAB developments, ESA’s SC-oriented code, include the prediction of explosive events and the explosive destruction. There are two main questions:

Generation of Explosion Environment and Explosion Likelihood

During re-entry special events may contribute to generation of an explosive environment. These events are for example:

To model these events it is necessary to define the conditions for their occurrence. For example, fuel gas release may be initiated by leaks of valves and fittings if they have reached a critical temperature. The following potential explosion initiating events are typically considered:

For each of the above effects a contribution to an explosion likelihood L can be defined. In case the integrated likelihood L exceeds a critical value an explosion could be initiated.

As not all critical events can be conceived a priori it may in some cases be adequate to conduct first a normal destructive re-entry analysis and review a posteriori the destruction process that may contribute to the explosion likelihood. Figure 9.3.13 illustrates the development of the explosion likelihood along a trajectory.

Re-Entry Explosion Phenomenology and Destruction Process

Figure 9.3.14 sketches the transient phenomena of an explosion. From a gas dynamics perspective, an explosion is essentially an instantaneous release of energy and mass that produces a spherical shock front (initially at least in an ideal case) that propagates with supersonic speed (in a resting environment). In order to generate this transient shock front a critical amount of energy or mass needs to be released. During re-entry, the energy will generally be provided by an instantaneous “explosive combustion” of the released fuel. The explosive combustion is a self-accelerating process due to reaction rate increase with temperature. It is the kinetic energy and the sudden pressure jump that cause the destructive nature of an explosion. If this explosion occurs during re-entry the explosion blast wave will change the complete flow field and the explosive gas release will be blown into a strongly increased wake. The time for explosive energy release is usually assumed to be infinitely short; however the distribution of this energy by the blast wave has a definite time scale that is given primarily by the amount of energy released. The flow field exchange generated by an explosion induced during re-entry is sketched in Figure 9.3.15.

A classical paper of the explosive detonation goes back to Taylor (1950). Additional information can also be found in Oppenheim A.K. (1972). In case of an explosive detonation of fuel, the heat addition is the dominant process, which was addressed by Taylor’s analysis. G.I. Taylor used similarity assumptions for the flow field and derived basic relations for the radial propagation of a spherical shock wave and the transient flow field behind this shock.

Taylor gives the following relation for the shock speed and for shock position for energy release in air:

With B having a numerical value of B = 5.36 for a thermal explosion in air.

Observation of the shock speed or the shock position as function of time allows the explosive energy E to be determined with the above relations. Taylor also showed that a distinct fraction E_therm of the explosive energy E is used to thermally heat the ambient gas and the other fraction E_kin is used to accelerate the ambient gas, with fixed proportions given by the following relation.

For the destructive actions by forces and for acceleration of generated fragments only the kinetic energy fraction E_kin = 0.22 E of the explosion energy is available. Destructive action occurs by the shock pressure passing over an object, which is directly related to the explosive energy and shock position r:

The Probabilistic Explosive Destruction Process

The explosive destruction is usually modeled in a probabilistic way with the constraints set by some conservation laws. One dominant conservation law concerns conservation of the original SC mass in the fragment cloud. The NASA explosion model, developed for the debris environment prediction tool EVOLVE, has been used for this application. The NASA explosion model is based on observed fragment characteristics of various upper-stages in orbit explosions. The size of one fragment in this model is defined by its characteristic length L_C and its area-to-mass ratio A/M (Johnson et al., 2001). Figure 9.3.16 shows the correlation found between A/M and L_C for 1780 observed fragments.

To derive an expression for the number of fragments as function of L_C and A/M, the L_C–A/M space was discretized, the number of fragments per bin was counted, and the resulting histograms were fitted by bimodal normal distributions. Figure 9.3.17 illustrates the discretization process used.

Fragment Length Number of Fragments within Length Element

The EVOLVE model starts with cumulative number of objects with N(L_C) with size greater as L_C. This number N(L_C) is described by a power function of L_C with S a scale factor and a₁ and a₂ fixed numerical values.

The distribution density of fragments within length interval dN/dL is obtained by the length derivative of N(L_C)

The cumulative Number N(L >L_C) and the distribution density dN/dL are shown in Figure 9.3.18 as function of L_C. The figure reveals the problem of the statistical models, namely N(L > L_C) has the value 1 at L_C = 3 and the distribution density drops to values below n_c = 1. This means that the upper length limit of a fragment remains an undetermined length if L_C > 3 m. For actual applications only integer numbers of fragments are possible. As the fragment cloud will have to reproduce the mass of the original object and larger fragments will have higher mass the upper length limit will introduce a statistical scatter in the number of generated cloud objects.

The A/M distribution density of fragments is described by a bimodal normal distribution.

D_A/M(L_C,A/M) is given by a bimodal Gaussian distribution, with distinction between explosion of a SC and of a rocket body.

EVOLVE assumes that the cross-sectional area A is only a function of L_C with the following dependence.

All fragments with size Lc will thus have the same cross-sectional area independent of the A/M ratio. The mass M of a fragment is then given by:

The ejection velocity ΔV is given by a standard distribution with the A/M value as independent parameter. Thus fragments with identical A/M value will obtain the same ΔV. The fragment cloud has to reproduce the mass of the original object. The conservation of mass constraint for the fragment cloud is usually achieved by proper length restriction of the larger objects.

Destructive Re-Entry Analysis of Fragments Generated by the Explosion

Because fragments generated by an explosion may also demise or survive, the re-entry analysis imposes additional demands on the explosion model as follows:

If (a) and (b) are achieved and all generated fragments are further specified, the individual cloud generated cloud fragments need to be analyzed by a fast object oriented code.

Thus, the treatment of an explosion and the follow on re-entry analysis of the fragments needs a set of specific analysis tools to provide the relevant input and output data for a sequential analysis approach. Figure 9.3.19 shows how this was for example solved in the SCARAB system.

Table 9.3.4 shows that a large amount of specific data must be prepared and provided by the individual subsystem to the follow on system in an integrated analysis with explosion event prediction, explosion analysis and explosion fragment analysis.

References

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2. Baker RL, et al. Orbital Spacecraft Re-entry Breakup. IAA-99-IAA 6.7.04, 50th Int Astronautical Congress, 1999 Amsterdam 1999.

3. Johnson NL, Krisko PH, Liou J-C, Anz-Meador PD. NASA’S New Breakup Model of EVOLVE 4.0. Adv Space Res. 2001;Vol. 23(No. 9):1377–1384.

4. Koppenwallner G, Fritsche B, Lips T. Survivability and Ground Risk Potential of Screws and Bolts of Disintegrating Space-craft during Re-entry Proc. 3rd European Space Debris Conference ESA SP-473. 2001;Vol. 2:533–539.

5. Koppenwallner G, Fritsche B, Lips T, Klinkrad H. SCARAB a Multi-Disciplinary Code for Destruction Analysis of Space-Craft during Re-entry, 5th European Symposium on Aerothermodynamics for Space Vehicles. Cologne, Nov 2004; ESA SP-563 2004;281–288.

6. Koppenwallner G, Lips T. Influence of Rarefied Gas Flow on Re-entry Survivability of Space Debris Rarefied Gas Dynamics Proc 26th Int Symposium, July 2008, Kyoto, Japan. In: Abe T, ed. AIP Conference Proceedings, Vol 1084, 209. 2008;740–747.

7. Lees L. Laminar Heat Transfer over Blunt-Nosed Bodies at Hypersonic Speeds. Jet Propulsion. 1956;Vol. 26(no.4):259–269 1956.

8. Matting FW. Approximate Bridging Relations in the Transitional Regime between Free Molecular and Continuum Flow, AIAA. Journal of Spacecraft. 1971;Vol. 8(No.1):35–40.

9. Oppenheim AK. Introduction to Gasdynamics of Explosions. New York-Wien: International Centre for Mechanical Sciences, Course and Lectures N.48, CISM Undine. Springer Verlag; 1972; 1972, ISBN 0-387-81083-8.

10. Rachel W, et al. Modelling of Space Debris Re-entry Survivability and Comparison of Analytical Methods. Amsterdam: IAA-99-IAA. 6.7.03. 50th Int. Astronautical Congress; 1999.

11. Taylor GI. The Formation of Blast Wave by very Intensive Explosion. Proc Roy Soc London, Series A. 1950;Vol. 201(No. 1065):159–174 and Part II, pp. 175–186, March 1950.

12. Schaaf SA, Chambre PL. Flow of Rarefied Gases. Princeton Aeronautical Paperbacks, Princeton University Press 1961.

Further Reading

1. Chapman DR. An approximate analytical method for studying entry into planetary atmospheres. Washington 1959: NACA TN 4276 Washington 1958; NASA TR-11; 1959.

2. Fritsche B, Lips T, Koppenwallner G. Analytical and numerical Re-Entry Analysis of simple objects. Darmstadt, Germany: Proceedings of the Fourth European Conference on Space Debris; 2005; April 2005, ESA SP-587, August 2005.

3. Fritsche B, Koppenwallner G, Lips T. Modeling of Spacecraft Explosions during Re-entry. IAC-04-IAA.5 12.2.08 2008.

4. Koppenwallner G, Fritsche B, Lips T, Martin T, Francillout L, De Pasquale E. Analysis of ATV Destructive Re-entry including Explosion Events. Darmstadt: Proc. 4th European Conference on Space Debris; 2005; April 18-20, 2005, ESA SP-587, pp. 545–550.

5. Reynolds RC, Sato A. DAS, Debris Assessment Software Operations Manual – Version 1. JSC-28437 August 1998 1998.

6. Rochelle WC, Kirk BS, Ting BC. Users Guide for Object Re-entry Survival Analysis Tool (ORSAT), Version 5, SC-28742, 1999. NASA, Lyndon B. Johnson Space Center, 1999 1999.

9.4.1 Case 1: Delta II Stage 2 – Texas

The second stage of a Delta II launch vehicle, used to place a Missile Defence Agency satellite into orbit on 24 April 1996, re-entered over Canada and the United States on 22 January 1997. Figure 9.4.1 is representative of the configuration of the stage prior to re-entry. The propellant tank has a cylindrical sidewall with hemispherical caps attached to each end. An interior hemisphere (not visible) joins with the aft end-cap to form a spherical tank for nitrogen tetroxide oxidizer. The remainder of the internal volume above the oxidizer tank holds Aerozine-50 fuel. The entire propellant tank assembly was constructed of AISI 410 stainless steel. A fuel depletion burn was performed following spacecraft separation, so the tank was empty at re-entry. Structural hardware was aluminum. Total dry weight of the stage was 920 kg. Figure 9.4.2 gives approximate dimensions of the stage prior to re-entry.

The propellant system was pressurized with gaseous helium and nitrogen, contained in four spherical pressure vessels. There were two large and two small pressure spheres, all made from Ti-6Al-4V titanium alloy. One small sphere contained nitrogen, with the remaining three containing helium. A single large sphere was recovered after re-entry. Tables 9.4.3 and 9.4.4 give materials and dimensions of major components.

Recovered Debris

Figure 9.4.2 shows the four debris items that were recovered after the re-entry. Impact locations of each item are given in Table 9.4.5. Table 9.4.6 lists the wind data for this re-entry.

Results of Laboratory Analysis