9.8 Fragmentation Guidelines Based on Historical Data.
The FAA has conducted an analysis of historical fragment data to provide guidelines for applicants as to typical results. There are two key metrics. The first is the number of fragments greater than 300 grams, a critical input for the assessment of risk to commercial aircraft. The second is the total basic casualty area of inert debris, which is computed as the sum of the basic casualty area, 𝐴𝐴𝐶𝐶 = 𝜋𝜋�𝑒𝑒𝑟𝑟𝑟𝑟𝑚𝑚𝑓𝑓 + 1 ft� 2 , of all fragments with ground impact kinetic energy greater than 11 ft-lbf, assuming terminal velocity at sea level.
9.8.1 Available Data.
There are sufficient historical data to provide guidance for explosions of liquid fueled rockets and for re-entry vehicles. For liquid fueled rockets, aerodynamic breakup and explosive breakup are generally similar, except with different breakup-induced velocities. For re-entry vehicles, there is sufficient historical data to provide guidance on debris from aerothermal breakup. The guidelines are based on data from previous debris analysis results. This the data shows more variation in the number of fragments than the casualty area. The guidelines below are subject to revision with the incorporation of additional data.
9.8.2 Liquid Fueled Rocket Guidelines.
For the explosive breakup of a fully loaded propulsion system (exclusive of other stages, boosters, interstages, payloads, etc.,) with dry weight between 1,000 and 50,000 lbs, the following ranges of the number of fragments and basic casualty area are typical:
2.5𝑊𝑊0.52 < 𝑁𝑁𝑟𝑟𝑟𝑟𝑚𝑚𝑓𝑓 < 7.5𝑊𝑊0.52 48𝑊𝑊0.52 < 𝐴𝐴𝐶𝐶 < 72𝑊𝑊0.52 where weight is the dry weight in pounds, and casualty area in square feet. The degree of fragmentation is often predicted to decrease when there is less propellant remaining.
This may reduce the predicted number of fragments by a factor of three, and the estimated casualty area by one-third.
9.8.3 Re-entry Vehicle Guidelines.
For aerothermal breakup of a vehicle designed to re-enter with weight between 10,000 and 100,000 lbs, the following ranges of the number of fragments and basic casualty area are typical:
6𝑊𝑊2.64 < 109𝑁𝑁𝑟𝑟𝑟𝑟𝑚𝑚𝑓𝑓 < 12𝑊𝑊2.64 80𝑊𝑊2.64 < 109𝐴𝐴𝐶𝐶 < 120𝑊𝑊2.64 where W is the weight in pounds, and casualty area in square feet, and these represent fragments surviving to ground impact.
10 PROPAGATION OF HAZARDOUS DEBRIS TO IMPACT.
A high-fidelity evaluation should compute the impact probability distribution for each class of fragments at the locations of hazarded people in accordance with § 450.135(b), and for specific assets that the operator is requested to evaluate. For this evaluation, the hazardous debris must be propagated following the vehicle termination event, or a jettisoned body from the moment of release, until it is no longer a hazard, as indicated under § 450.121(c). Each hazardous debris propagation under § 450.121(c) creates a trajectory that must account for significant forces acting on bodies in accordance with § 450.121(c)(1), and there should be a sufficient number of trajectories to account for uncertainty in breakup conditions and in external forces that effect the impact probability distribution. The set of fragment trajectories from each failure and planned event state vector may be computed to generate sample impact points that can later be fit with a functional distribution. The succeeding paragraphs discuss the data and parameters that should be accounted for, and general approaches for defining the propagator algorithm.
10.1 Atmospheric Data for Propagation.
The impact probability distribution for each class of fragments must account for the atmospheric conditions, in accordance with § 450.135(b)(4)(ii). The hazardous debris trajectories are influenced by wind and drag, in proportion to air density ρ, near the surface. Wind conditions can carry hazardous debris towards populated areas. The likelihood of high wind magnitudes can lead to high risk to surrounding populations or result in significant constraints on when a launch can occur. The minimum set of parameters that should be accounted for are air density, wind, and Mach number.
10.1.1 Air Density.
Air density is a function of altitude, geographic location, and time. For debris analysis, the spatial variation is important, especially the vertical profile of density. In the lower atmosphere, the temporal variation of density has no significant effect on impact locations, and the geographic variation effect is relatively small. Thus, average density models as a function of geographic location are generally sufficient up to around 300 km altitude. The high-altitude (above 300 km) density model is very important when the vehicle is re-entering at a shallow angle (e.g., less than a few degrees from horizontal), and especially for very small reentry angles. In these cases, a model that extends to one (1) million feet should be used. For steeper angles, the importance of density models above 300 kilofeet depends on the fragment characteristics. The density model at these altitudes also is more important as the fragment ballistic coefficient is smaller. For non-shallow trajectories for fragments relevant to debris risk (above 1 psf), the air density profile should extend from the surface up to about 400 kilofeet. Also, when implementing the air density model in the ballistic trajectory propagator, the potential for erroneous skipping at the “top” of a model should be considered (skipping can physically occur, but can also be a numerical aberration).
10.1.2 Air Density Models.
Since temporal variations in air density have insignificant effect on debris analysis (except for very shallow reentries), standard air density models can be used. For the lower atmosphere, a range reference atmosphere is commonly used in the launch or landing area. Below the exosphere, the Earth Global Reference Atmospheric Model from NASA (see reference [11] of paragraph 3.4 of this AC) should be used, as it provides sufficient data for the atmosphere. This dataset provides mean monthly historical values up to about 400 kilofeet, on grids of latitude and longitude that span the entire Earth. It has the accuracy and resolution adequate for debris analysis, but equivalent alternatives can also be used. For the exosphere, the density can vary significantly as a function of time due to space weather, but the COSPAR International Reference Atmosphere Model should be used as an average when temporal variability is not needed. It also contains references to models that account for temporal variation.
When combining models, the result should be faired over a range of approximately 25 percent in altitude. Where air density is provided in more localized wind data or models (see section 9.1.5) the density from those sources may be used. A range reference atmosphere may also be used in the local area.
10.1.3 Mach number.
Mach number M is defined as 𝑀𝑀 = 𝑣𝑣/𝑐𝑐, where v is the ambient fragment or intact body speed with respect to the wind, and c is the speed of sound. If the speed of sound is not provided, then M can be computed using the air temperature T, e.g. 𝑀𝑀 = 𝑣𝑣 / �𝛾𝛾𝑅𝑅𝑇𝑇, where 𝛾𝛾 is the specific heat ratio, and 𝑅𝑅 is the specific gas constant. For a good approximation, the atmosphere obeys the Ideal Gas Law, which is a direct relation between temperature, density, and pressure, for all altitudes for which drag should be accounted. Temperature data can be obtained from the same sources as air density data.
10.1.4 Wind.
Wind effects are very important for debris analysis, but wind above the jet stream (i.e., above 60,000 feet) usually has little effect. Appropriate application of wind is essential for compliance with § 450.135(a), as it is common for operations to be acceptable in some wind conditions, but not others. Section 450.135(a) provides two options: an analysis in planning that identifies limits on what conditions are acceptable for the operation or an analysis in the countdown that uses the best available data. For the launch or landing area (where the vehicle is traveling slower than two miles per second), local time-appropriate wind data or models should be used. Outside the launch or landing area, a three-dimensional statistical model should be used. Local data should only be used within 100 miles of the location it represents and should be faired with three-dimensional statistical model over approximately 50 miles horizontal distance and approximately 25 percent in altitude.
10.1.5 Local Wind Data for Countdown Analysis.
During the operation countdown, there are two options for wind data for areas where the vehicle is flying below hypersonic speeds.
10.1.5.1 The first option is to use measured wind data, from various measuring systems such as Jimsphere, Rawinsonde and Windsonde soundings, and Doppler Radar Profiler measurements. Data should be obtained within six hours of launch. The uncertainty in the wind forecast should account for the time delay between the forecast and the flight. This data is typically centered about a specific measurement location and thus depends only on altitude. This data should extend to at least above the top of the jet stream.
10.1.5.2 The second option is to use a wind forecast authorized by either the Department of Defense or NOAA (such as the North American Mesoscale (NAM) model or Global Forecast System (GFS)). Forecasts are usually sufficiently accurate for debris analysis for up to 72 hours, but should be verified by comparing newer forecasts to the forecast used in the countdown analysis. It is critical that the forecast computed for the planned operation time should be used. If localized weather events (including weather fronts or storms) are anticipated, the applicant must ensure the operational flight conditions are consistent with the forecast.
The uncertainty in the wind forecast should account for the time delay between the forecast and the flight.
10.1.6 Local Wind Data for Availability Study.
If an availability study is performed to satisfy § 450.135(a)(1), then appropriate wind data should be obtained. The data set should include at least 100 wind profiles appropriate to the location of the operation within the same season of the operation
(e.g., within 30 days of the launch date). Either data from the operation location (e.g., range) or from the NOAA Radiosonde database (https://ruc.noaa.gov/raobs/) should be used. An operator should either analyze every profile or a representative sample that includes the worst foreseeable conditions under which an operation might be attempted, as § 450.165(a) requires that the actual operational conditions be within the range of what has been determined to be acceptable. In accordance with § 450.135(c)(1), the applicant must submit a description of the methods used to demonstrate compliance with the safety criteria in § 450.101. In accordance with § 450.115(c), the applicant must include a description of how the operator will account for the conditions immediately prior to enabling the flight of a launch vehicle or the reentry of a reentry vehicle, such as the final trajectory, atmospheric conditions, and the exposure of people.
10.1.7 Non-local Wind Data.
A three-dimensional atmospheric model should be used for areas outside the local area.
The NASA EARTH-GRAM model (discussed above) or its equivalent should be used, and should account for variability within the month of the operation. If doing a countdown analysis, a three-dimensional forecast model can be used.
10.2 Failure and Planned Event State Vectors.
All state vectors corresponding to failure and planned events required by § 450.117(d)(4) and § 450.119(c)(4) must be identified to conduct a hazardous debris risk analysis under § 450.135. Any time and spatial uncertainty in these state vectors must be also specified if it exists in accordance with § 450.117(d)(4)(iii). This information defines the starting points of the hazardous debris trajectories. These state vectors may be associated with intact vehicles, planned deployed objects, or vehicle breakup fragments. State vectors from FSS destruct action of the vehicle are discussed in AC 450.117-1, Normal and Malfunction Trajectory Analysis. The other most common sources are discussed in the subparagraphs below.
10.2.1 Self-Breakup State Vectors.
Vehicles on malfunction trajectories may experience sufficient external aerodynamic loads or internal inertia forces to cause the vehicle to come apart. The same is true for tumbling vehicles on ballistic trajectories when they survive until the end of powered flight since they could immediately start tumbling rather than remaining stable. Launch operators should determine vehicle breakup criteria limit ranges for each phase of flight, which should account for uncertainty in the thresholds. These ranges are often assumed to be represented by Gaussian distributions by default, although better informed understandings of the load limits may allow for other distributions to be used.
10.2.1.1 In general, a breakup state time is the moment when the stress related vehicle structural load criterion limit is exceeded. These breakup state vectors should be computed by interpolation between the time bounding malfunction trajectory initiation points.
10.2.1.2 The uncertainty in the breakup state vectors must be accounted for in the statistical trajectory set (paragraph 10.3 of this AC) in accordance with § 450.117(d)(4)(iii). The uncertainty in the criteria limit will result in a range of breakup state times and their associated breakup state vectors.
Each breakup state vector can be assigned a probability based on the associated structural limit distribution probability value. However, the distribution of the breakup state times may be very non-Gaussian even when the criteria limit distribution is Gaussian. The uncertainty in the breakup state vectors must additionally account for failure state vector uncertainty if the vehicle or malfunction trajectory does not already do so.
10.2.1.3 This may be done by using the mean breakup state vector as the basis for the statistical trajectory. The mean state vector is where the 50 percent threshold is exceeded. The uncertainty in the state vector is then represented by one or more Gaussian covariance matrices that will be sampled during the setup of the based statistical trajectory set.
10.2.1.4 There are three common structural limit cases that should be evaluated.
The first case is from aerodynamic forces. The aerodynamic breakup criteria should account for angle-of-attack α, i.e., the angle between the vehicle’s roll axis and the velocity vector, and the external dynamic 47
pressure q acting on the vehicle. The criteria breakup distribution is based on the dynamic pressure multiplied by total angle of attack (q-α) limits of breakup, or on a q*sin(α) limit. Both of these limits are simplifications; if it is known from the design that more sophisticated quantification of structural limits is appropriate and have a significant effect on the breakup time, then those limits should be used. Structural limits should represent the upper bound of what the vehicle could survive. This type of breakup should be evaluated for vehicle malfunction trajectories, and for vehicle ballistic trajectories.
10.2.1.5 The second case is from inertial forces. One breakup criterion is when a vehicle has reached sufficient rotational speed in a vacuum or low-density atmosphere. The criteria breakup distribution is based on the rotation rate limits of breakup. The rotation rate accounts for yaw and pitch rates. This type of breakup should be evaluated for a vehicle malfunction trajectory.
Another inertial force can be a buoyant force as a result of g-loads from the vehicle. This would present a force on any structures submerged in a fluid and may exceed attachment hardware limits. A third type of inertial force breakup is when g-loads exceed structural capability.
10.2.1.6 The third case is from aerothermal effects. An aerothermal induced breakup occurs when friction buildup leads to melting of portions of the vehicle. Sufficient time for melting can lead to aerothermal breakup during vehicle reentry from orbit, or if vehicle thrust termination occurs at sufficiently high altitudes and velocity. Since melting typically starts at altitudes centered about 75,000 feet when speeds reach about Mach 15, this case should be handled for all events that occur above 100,000 feet at speeds as low as Mach 10. The criteria breakup distribution is based on the altitude limits provided that a vehicle has exceeded a speed threshold.
Although the altitude limits should include uncertainty, the speed threshold may be a single value. This type of breakup should be evaluated for a vehicle ballistic trajectory.
10.2.2 Thrust Termination Failure State Vectors.
The trajectory points of a randomly selected trajectory from the set of normal trajectories, or the nominal trajectory, are used to create a set of failure state vectors corresponding to thrust termination for an intact vehicle. The vehicle follows a ballistic trajectory after failure and self-breakup should be considered prior to surface impact for a vehicle that is tumbling. If self-breakup does occur, then the self-breakup state vector would replace the intact vehicle failure state vector for use in the risk analysis. The state vector uncertainty is added for the nominal trajectory case at thrust termination. The full set of trajectory points, or a subset of sufficient resolution, may be used. Since thrust termination failures can lead to potentially high consequence intact impacts, the analysis should identify the impact conditions in terms of speed and location. Thus, the sample rate of the failure trajectory should be at least four times higher than the tumble rate and high enough that velocity can be interpolated accurately to within 10 percent. Smaller limits down to 10 Hertz (Hz) may be necessary to meet § 450.119(b) for high speed 48
vehicles, such as suborbital vehicles, or during abrupt maneuvers. State times that occur more often do not yield much benefit and so intermediate trajectory points can be filtered out.
10.2.3 Jettisoned Body State Vectors.
Planned deployment events jettison one or more bodies. These may be stages or other attached items, such as small doors or fins, the function of which has ceased. The risk from these bodies should be included in the debris risk analysis, in accordance with § 450.135(b)(3). Each body originates from a planned event state vector that is determined during mission planning.
10.2.3.1 A body may be jettisoned from a trajectory randomly selected out of a set of normal trajectories, to account for state vector uncertainty.
Alternatively, a single jettison state vector may be specified from the nominal trajectory, and its uncertainty accounted for using covariance data derived from the ensemble of normal dispersed trajectories, see paragraph 10.3 of this AC. The planned event state vector is computed through linear interpolation of time between bounding time trajectory points.
10.2.3.2 It is necessary to consider self-breakup of a jettisoned body during its ballistic fall. These bodies may be smaller than the main vehicle, but they can be large main stages that can still have points of weakness.
Self-breakup of unpressurized sections may not be necessary to consider if they are more compact and have much higher structural breakup limits.
Also, since these bodies are not powered in free-fall, they will not reach the speeds needed for inertial forces to be significant.
10.2.4 Intact Impact State Vectors.
A vehicle will impact the surface during powered flight if it survives a malfunction flight and does not achieve an orbital condition. Intact impacts only occur for a limited set of Monte Carlo simulations and may result in a sparse set of impacts in the tails of bivariate distributions. Uncertainty about the impact points differs for each impact point and should be evaluated. This should be computed in a way that ensures sufficient resolution to produce smooth and continuous individual risk contours, but should not be artificially smoothed. Intact impacts can be a significant contributor to risk results and should be accounted for using a statistically significant sample size (e.g., at least 30).
10.2.4.1 Kernel Density Estimation Procedure of Impact Location Uncertainty.
A Kernel Density Estimation (KDE) procedure is an example of an acceptable approach, which considers the distances between the neighboring impact points to determine the degree of smoothing needed.
KDE works better with more impact points, but can be applied with as few as thirty. The points that are used in the KDE should come from the same type of events (i.e., the same failure mode and breakup type) and from 49
failure trajectories within the same set. A KDE creates a smooth distribution from a collection of samples by applying a distribution about each sample (a kernel). The total distribution is then the sum of the individual distributions. Typically, a Gaussian distribution is used for each kernel.
10.2.4.2 Impact Vector Based Uncertainty.
An alternative approach is to compute uncertainty based on the velocity vector at impact. The magnitude of the horizontal component of the velocity vector can be scaled by a reference time (e.g., one to two seconds) to estimate an uncertainty in the impact location.
10.3 Statistical Trajectory Set.
Tumbling fragments, planned jettisoned hazardous debris, or an intact non-powered vehicle follow a ballistic trajectory. A ballistic trajectory’s initial point is one of the failure or planned event state vectors discussed in paragraph 10.2 of this AC. A statistical risk analysis should employ a set of ballistic trajectories for each fragment class, which accounts for a set of fragments with similar characteristics, or a single body. Each trajectory in a set should account for all relevant sources of uncertainty of the event’s state vector, and additional sources that affect the free-fall motion. At given times or locations, a trajectory set is described by a probabilistic distribution for use in a statistical risk analysis, or a hazard bound for a containment analysis. The trajectory set should account for uncertainty in ballistic coefficient β (i.e., drag), breakup-induced velocity (delta-v), lift, and wind.
10.3.1 Sampling Approach.
A trajectory set associated with an event state vector can be obtained from explicit rules of sampling. An example is the covariance sampling presented in reference [13] of paragraph 3.4 of this AC. This sampling method is not random and follows specific rules and is appropriate for creating trajectories if the sources of uncertainty and subsequent trajectories nearly obey Gaussian statistics, (i.e., the skewness of the sampled points does not exceed roughly 0.15 throughout ballistic fall.) There are also other types of correlated random samples without the assumption of Gaussian uncertainty distributions. For more general statistics, the trajectory set should be created by performing a random Monte Carlo sampling of the state vector’s sources of uncertainty. A typical number of Monte Carlo for ballistic trajectories for a given class of fragments is usually about 300-500 to create impact dispersions.
10.3.2 Uncertainty in Initial State Vectors and Ballistic Coefficient.
Event state vectors account for state vector uncertainty from the guidance and performance and any uncertainty involving its selection, when the event state vectors are set up (see paragraph 10.2 of this AC). The uncertainty in β typically can be characterized using a log-normal distribution. This sampling is done at the start of the ballistic trajectory. Guidelines for assigning beta are discussed in chapter 10 of this AC for usage when precise values are not known for a given fragment.
10.3.3 Uncertainty in Breakup-induced Velocity.
The most common modeling of breakup-induced velocity assumes that there is no preferred direction. The uncertainty is then Gaussian in each direction, leading to a Maxwellian distribution in 3-dimensions. This approach is usually valid for propellant tank explosions. There are situations where the breakup-induced velocity has a preferred directionality. At a given time, sampled fragments may form, for instance, a forward cone, hollow sphere, or a lateral torus or ring. An indication of what type of shape may form could be surmised by examining a plot of initial fragment speed vs directed angle off the vehicle’s x-axis, or a computer-generated animation of the breakup. To define the sampling from a directed velocity distribution, in a precise manner, may require a fair amount of effort when there is no simple hazardous debris cloud shape, such as setting up a detailed finite element analysis program. This case may require describing the set of sampled fragments by a sum of many simple distributions. This sampling is done at the start of the ballistic trajectory. However, the uncertainty in vehicle orientation during that directional-hazardous debris event can also cause hazardous debris to occur in any direction, and so this uncertainty of final hazardous debris dispersion needs to be considered in accordance with § 450.121(c)(2)(iv).
10.3.4 Wind Uncertainty.
Wind uncertainty should be incorporated when using three-dimensional models of historical data and when using measured data in the launch countdown. For historical data, the uncertainty should represent the wind variability that corresponds to monthly statistics. For measurement data, the uncertainty should statistically represent the potential change in wind conditions between the measurement and the time of the flight operation. Wind uncertainty should not be applied when using historical data samples in availability studies. When used, uncertainty data should be specified in altitude bands, with uncertainty given as a two-dimensional covariance. If correlation of uncertainty between altitudes is available, this data should be used. The time that the sampled fragment or intact body exists in the altitude band is used to convert wind uncertainty to the net position uncertainty for each band. The total uncertainty at a given time is the sum over all bands through which the sampled fragment or body passes.
10.3.5 Lift Uncertainty.
Uncertainty due to lift may be computed by a circular Gaussian distribution. The 1-σ radius is the difference in locations with and without lift force. Additional details are provided in section 8.4.
10.4 Ballistic Trajectory Generation.
Each ballistic trajectory should be created using a physics-based model that then utilizes the equations of motion accounting for the applied forces. Separate propagation models can be used for when fragments are in a vacuum, (i.e., above the given air density profile data), and when they are in the atmosphere. For motion completely in the vacuum, a fast method is to use Kepler’s solution for a spherical Earth. The solution will propagate the fragment or intact body from its initial location to a desired lower 51
altitude in one step. Additional iterations may be performed to achieve increased accuracy. The final location should be corrected for Earth oblations.
10.4.1 Otherwise, propagator algorithms are designed to propagate fragments in a series of small time steps. Position and velocity components are computed using acceleration and some of its higher derivatives. The size of the steps involves a tradeoff between runtime and accuracy. The time steps should be no larger than one second and should be adjustable to smaller values to account for rapid changes in direction and speed. There are two types of propagators: predictor, and predictor-corrector. A predictor will only move forward in time, without any knowledge of how much error is being introduced.
A predictor-corrector will compute the error buildup between steps, and if too large will reduce the time step and restart the step. For ballistic trajectory generation, time steps of less than a tenth of a second lead to about the same results and runtimes for both types.
Exceptions are reentry cases where trajectories span large distances within the atmosphere before impact. For these, the error adjustment capability of the predictorcorrector usually shows a clear advantage of accuracy. For time steps closer to one second, predictors will run faster at a cost of accuracy compared to predictor-correctors, but are found to be sufficient for launch-to-orbit and non-orbital missions. These statements assume that the codes in question are robust and have been thoroughly tested to yield desired results.
10.4.2 Most propagators employ a version of a Taylor series expansion or Runge-Kutta algorithm. There are many versions of the Runge-Kutta algorithm as depicted in reference [17] of paragraph 3.4 of this AC. The various predictor-correctors have different starting mechanisms, step sizing logic, polynomial order, and error testing logic and thresholds. Both Taylor and Runge-Kutta series have an unbounded number of terms. For ballistic trajectories, the propagator should include at least up to fourth order terms.
10.4.3 All ballistic propagators should account for gravity, which in turn needs an Earth model to define the gravitational constant and Earth’s shape. The WGS84 model should be used for all Earth constants. That model is described in greater detail in the reference [8] of paragraph 3.4 of this AC. For short range trajectories near the launch point, out to about 200 miles, the Earth can be treated as a sphere. Otherwise, the oblateness of the Earth should be accounted for, which is specified by the J2 Earth moments. Neglecting the J2 term can lead to an error of several miles over tens of degrees of span. The next higher even moment, J4, tends to have a non-negligible contribution only for a highly elliptical ballistic trajectory that makes one or more passes around the Earth. The radius of the Earth at a local surface point should be computed using the radius of the Earth at the equator, at the poles, and applying the Earth model eccentricity correction for the local latitude.
10.4.4 In the atmosphere, propagators should account for drag force. For a tumbling body, drag is directed opposite to the direction of motion. If the position components are evaluated in the Earth’s rotating frame, then the propagator should also account for the Coriolis and centrifugal pseudo-forces. To evaluate these forces, the Earth model rotation rate is needed.
State vectors consisting of position and velocity can be represented in many different coordinate systems. Propagator codes are generally in an Earth-centered system, which may or may not be rotating with the Earth. At a given state time tj, a propagator will start with position �𝐸𝐸�𝑆𝑆𝑗𝑗�, 𝐹𝐹�𝑆𝑆𝑗𝑗�, 𝐺𝐺�𝑆𝑆𝑗𝑗�� and velocity �𝐸𝐸̇ �𝑆𝑆𝑗𝑗�, 𝐹𝐹̇ �𝑆𝑆𝑗𝑗�, 𝐺𝐺̇ �𝑆𝑆𝑗𝑗�� and project them forward to time tj+1.
10.4.5 The drag acceleration is given by 1𝑆𝑆 ⃗𝑑𝑑𝑟𝑟𝑚𝑚𝑓𝑓�𝑆𝑆𝑗𝑗� = − 𝐶𝐶𝐷𝐷𝜌𝜌𝑣𝑣𝑟𝑟𝑓𝑓𝑠𝑠𝑚𝑚𝑑𝑑𝑣𝑣 ⃗𝑟𝑟𝑓𝑓𝑠𝑠𝑚𝑚𝑑𝑑𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟 2𝑚𝑚�𝑆𝑆𝑗𝑗� where the fluid speed 𝑣𝑣𝑟𝑟𝑓𝑓𝑠𝑠𝑚𝑚𝑑𝑑 is the fragment’s speed with respect to the local winds, 𝜌𝜌 is the local air density, and 𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟 is the aerodynamic reference area. The gravity acceleration, with the J2 correction, is given by 3 𝐴𝐴2 𝐺𝐺2�𝑆𝑆𝑗𝑗� 𝑒𝑒 𝑆𝑆𝑓𝑓𝑟𝑟𝑚𝑚𝑔𝑔𝑚𝑚𝑔𝑔𝑔𝑔�𝑆𝑆𝑗𝑗� = −𝑔𝑔 �1 − 𝐽𝐽2 �5 − 1�� 𝑒𝑒 = 𝐸𝐸�𝑆𝑆𝑗𝑗� 𝑜𝑜𝑒𝑒 𝐹𝐹�𝑆𝑆𝑗𝑗�2 𝑒𝑒2�𝑆𝑆𝑗𝑗� 𝑒𝑒2�𝑆𝑆𝑗𝑗� 𝑒𝑒(𝑆𝑆) 3 𝐴𝐴2 𝐺𝐺2�𝑆𝑆𝑗𝑗� 𝑒𝑒 𝑆𝑆𝑓𝑓𝑟𝑟𝑚𝑚𝑔𝑔𝑚𝑚𝑔𝑔𝑔𝑔�𝑆𝑆𝑗𝑗� = −𝑔𝑔 �1 − 𝐽𝐽2 �5 − 3�� 𝑒𝑒 = 𝐺𝐺�𝑆𝑆𝑗𝑗�2 𝑒𝑒2�𝑆𝑆𝑗𝑗� 𝑒𝑒2�𝑆𝑆𝑗𝑗� 𝑒𝑒(𝑆𝑆) The factor A is the Earth’s equatorial radius, r, which is the distance from the gravitational center of the Earth to the center of gravity of the fragment. For a rotating frame, centrifugal acceleration is given by 𝑆𝑆 ⃗𝑐𝑐𝑟𝑟𝑚𝑚𝑔𝑔𝑟𝑟𝑚𝑚𝑟𝑟𝑠𝑠𝑓𝑓𝑚𝑚𝑓𝑓�𝑆𝑆𝑗𝑗� = −𝜔𝜔��⃗ × �𝜔𝜔��⃗ × 𝑒𝑒 ⃗(𝑆𝑆)� where 𝜔𝜔��⃗ is the Earth's rotation rate about the polar axis, and the Coriolis acceleration is given by 𝑆𝑆 ⃗𝐶𝐶𝐶𝐶𝑟𝑟𝑚𝑚𝐶𝐶𝑓𝑓𝑚𝑚𝑠𝑠�𝑆𝑆𝑗𝑗� = −2𝜔𝜔��⃗ × 𝑣𝑣 ⃗(𝑆𝑆)
10.5 Residual Thrust.
A non-ballistic trajectory should be used for motors or engines with residual thrust.
Although the requirements do not explicitly mention residual thrust, it must be dealt with to meet requirement § 450.121(c)(1). There are several types of situations where a thrusting motor or engine can survive vehicle breakup or termination. When a motor or engine has non-negligible residual thrust, then a 5-DOF simulation should be used for essentially axisymmetric vehicles and a 6-DOF simulation for non-axisymmetric vehicles.
10.5.1 Free Fliers.
The first case is when an intact propulsion system flies separately from the main body.
This can occur when a propulsion system detaches from its core vehicle and continues under full power until self-breakup or intact impact of the propulsion system. A similar situation occur when an upper stage breaks up without destroying a lower stage. These should use the same type of analysis as with the powered core vehicle. This case is included in AC 450.119-1, High-Fidelity Malfunction Trajectory Analysis. In some 53
cases, residual thrust may present a negligible risk for casualty expectation, due to conditional probabilities which are discussed in AC 450.131-1, Probability of Failure.
However, it should always be accounted for when performing assessments for flight abort, in accordance with § 450.108 and AC 450.108-1, Flight Abort Rule Development.
10.5.2 Partially Intact SRM.
A partially intact SRM can also have residual thrust. This can occur when vehicle destruct system does not result in complete breakup of an attached SRM. The destruct system is usually designed to break off the rear nozzle and front end of an SRM, such that both ends have thrust but in opposite directions with a small net force from the remaining burning propellant. A partial destruct could yield a larger imbalance in thrust resulting in a net residual thrust 𝑇𝑇. A partially intact SRM can result when the nozzle throat fails and is ejected in the exhaust stream. The wider orifice diminishes the exhaust velocity and thus reduces the thrust. The SRM propagates in a forward direction if the residual thrust acts along the central axis towards the nose-end.
10.5.3 Stability of Unguided Body.
For a thrusting body to continue in stable flight and not tumble might require sufficient thrust force and stabilizing fins, or a thrust offset that produces a moment matched by aerodynamic moments – this is often achievable even for bodies otherwise considered unstable at shallow thrust offset angles. A residual thrust that acts significantly off-axis usually leads to a spiraling motion of the body. This is not as likely to be stable and may rapidly reach a tumbling state, especially if aerodynamic loads are negligible.
10.6 Directed Lift.
Some breakup scenarios can lead to release of objects with significant lift where a stable orientation can be maintained or reached during ballistic fall. An applicant should consider the directed lift of a hazardous debris body in a high-fidelity analysis when computing flight hazard areas in accordance with chapter 12 of this AC. Although the requirements do not otherwise explicitly mention lift, it should be dealt with to meet requirement § 450.121(c)(1). A body may also have oscillatory motion that shifts between a stable lift vector and a condition of instability. Neglecting the stable lift regimes can cause the analysis to estimate mean impact point significantly shifted from the true mean (if the body remains on a known heading), or significantly increase the dispersion around the mean impact point (if the heading of the body is uncertain). This effect is usually only relevant for large components designed for aerodynamic stability.
Vehicle breakup could lead to SRMs or liquid fueled rockets that contain opposing pairs of wings or fins and experience a stable lift force during free-fall. The stability is likely to exist only for sufficiently high speeds, after which a transition to tumbling motion occurs. The lift vector need not be upwards, but could also be orientated laterally for a body not traveling horizontally (perpendicular to drag). A 6-DOF simulation should be used for a hazardous debris body that is stable and has non-negligible lift.