Aero-Thermal Demise of Reentry Debris: a Computational Model

Aero-thermal Demise of Reentry Debris: A Computational Model

by Troy M. Owens

Bachelor of Science In Aerospace Engineering Florida Institute of Technology 2013

A thesis submitted to the College of Mechanical and Aerospace Engineering at Florida Institute of Technology in partial fulfillment of the requirements for the degree of

Master of Science in Aerospace Engineering

Melbourne, Florida August, 2014

No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the author. For information address: T. M. Owens.

PRINTED IN THE UNITED STATES OF AMERICA

______

We the undersigned committee hereby approve the attached thesis

Aero-Thermal Demise of Reentry Debris: A Computational Model by Troy M. Owens

______Dr. Daniel Kirk Professor, Mechanical and Aerospace Engineering Associate Dean for Research Major Advisor

______Dr. David Fleming Professor, Mechanical and Aerospace Engineering Committee Member

______Dr. Ronnal Reichard Professor, Marine and Environmental Systems Director of Laboratories Coordinator; Senior Design Committee Member

______Dr. Hamid Hefazi Professor, Mechanical and Aerospace Engineering Department Head

Abstract

Title: Aero-thermal Demise of Reentry Debris: A Computational Model Author: Troy Owens Major Advisor: Daniel R. Kirk, Ph.D.

The modeling of fragment debris impact is an important part of any space mission. Planned debris or failure at launch and reentry need to be modeled to understand the hazards to property and populations. With more accurate impact predictions, a greater confidence can be used to close areas for protection and generate destruct criteria for space vehicles. One aspect of impact prediction that is especially difficult to simulate in a simple yet accurate way is the aero-thermal demise of reentry debris. This thesis will attempt to address the problem by using a simple set of inputs and combining models for the earth, atmosphere, impact integration and stagnation-point heating.

Current tools for analyzing reentry demise are either too simplistic or too complex for use in range safety analysis. NASA’s Debris Assessment Software 2.0 (DAS 2.0) has simple inputs that a range safety analyst would understand, but only gives the demise altitude as output and no ability to specify breakup conditions. Object Reentry Survival Analysis Tool (ORSAT), the standard for reentry demise analysis, requires inputs that only the vehicle manufacturer knows and a trained operator. The output from ORSAT gives a full range of fragment properties and for numerous breakup conditions. This thesis details a computational model with simple inputs like DAS 2.0, but an output closer to that of ORSAT, that will be useful in many mission risk analysis scenarios.

This is achieved by using 1) WGS 84, a fourth order spherical harmonic model of the earth’s surface and gravity; 2) the 1976 U.S. Standard Atmosphere; 3) an impact integrator for a spherical rotating earth; and 4) a stagnation-point heating correlation based on the Fay-Riddell theory.

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Contents

Abstract ...... iii

Contents ...... v

List of Keywords and Abbreviations ...... ix

List of Exhibits ...... xi

List of Symbols...... xiii

Symbols for Impact Integration ...... xiii Symbols for Fay-Riddell Stagnation Point Heating ...... xv Symbols for Aero-thermal Demise ...... xvii 1 Introduction ...... 1

1.1 DAS 2.0: Debris Assessment Software 2.0 ...... 2 1.2 ORSAT: Orbital Reentry Survival Analysis Tool ...... 4 1.3 Aerospace Survivability Tables ...... 4 1.4 SCARAB: Spacecraft Atmospheric Re-entry and Aerothermal Break-up ....6 1.5 Computation Model ...... 8 1.6 Risk Analysis ...... 9 2 Impact Integration ...... 11

2.1 Equations of Motion ...... 11 2.1.1 Relative Angular Motion ...... 12

2.1.2 Equations for Flight Over a Rotating Spherical Earth ...... 14

3 Stagnation-Point Heating ...... 20

3.1 Fay and Riddell Theory ...... 20 3.1.1 Laminar Boundary-Layer in Dissociated Gas ...... 21

3.1.2 Boundary Layer Ordinary Differential Equations ...... 23

3.1.3 Heat Transfer Rate ...... 24

3.1.4 Equilibrium Boundary Layer ...... 25

3.2 Detra, Kemp and Riddell Correlation ...... 27 3.2.1 Radiation Heat Balance ...... 29

4 Algorithm ...... 31

4.1 Earth Model ...... 31 4.2 Zonal Harmonic Gravity Vector ...... 32 4.3 Atmospheric Model ...... 34 4.3.1 Lower Atmosphere ...... 34

4.3.2 Upper Atmosphere ...... 35

4.4 Impact Integrator ...... 38 4.5 Aero-thermal Demise ...... 43 4.5.1 Fragment Properties ...... 43

4.5.2 Material Properties ...... 43

4.5.3 Shape Assumptions ...... 44

4.5.4 Stagnation Point Heating ...... 45

4.5.5 Liquid Fraction ...... 47

4.5.6 Fragment Tables ...... 47

5 Results ...... 53

5.1 Understanding Aero-heating ...... 53 5.1.1 Reentry Trajectory, Heat Flux, and Bulk Temperature ...... 53

5.1.2 Varying Breakup Altitude ...... 57

5.1.3 Varying Initial Temperature of Debris Fragment ...... 61

5.1.4 Varying Initial Velocity ...... 63

5.1.5 Varying Flight Path Angle ...... 67

5.1.6 Varying Materials of Debris Fragment ...... 70

5.1.7 Varying Mass of Debris Fragment ...... 72

5.2 Model Comparisons ...... 77 5.2.1 DAS 2.0 ...... 77

5.2.2 Aerospace Survivability Tables ...... 78

5.3 Input and Output Debris Fragment Catalog ...... 82 6 Conclusions ...... 86

6.1 Practical Application ...... 86 6.2 Validation...... 87 6.3 Performance ...... 88 6.4 Possible Future Work ...... 89 References ...... 91

Appendix ...... 94

Appendix A: Material Properties ...... 94 Appendix B: Supplemental Algorithms ...... 98 Alternate Correlations ...... 98

Trajectory Site Direction Cosines ...... 100

ECEF Coordinates to XYZ Coordinates ...... 100

ECEF Coordinates to Aeronautical Coordinates ...... 101

Appendix C: MATLAB Code ...... Error! Bookmark not defined. demiseUtility.m ...... Error! Bookmark not defined.

demise.m ...... Error! Bookmark not defined.

glideDerivatives.m ...... Error! Bookmark not defined.

Atmosphere1976.m ...... Error! Bookmark not defined.

createEarth.m ...... Error! Bookmark not defined.

vii getGravity.m ...... Error! Bookmark not defined. fragmentExporter.m ...... Error! Bookmark not defined. fragmentImporter.m ...... Error! Bookmark not defined.

viii

List of Keywords and Abbreviations

Keyword/Abbreviation Definition

AST Office of Commercial Space Transportation

Orbital Test Vehicle. Unmanned spacecraft, launches Boeing X-37 OTV as rocket lands as a space plane

CAIB Columbia Accident Investigation Board

Area where a debris fragment will cause human Casualty Area casualty, same as hazard area

CFD Computational fluid dynamics

CSV Comma-sepperated values, a data file format

DAS 2.0 Debris Assessment Software 2.0

Estimated or expected casualties, usually measured in Ec micro-casualties

ECEF Earth Centered Earth Fixed, same as EFG

EFG Earth Fixed Geocentric, same as ECEF

ESA European Space Agency

FAA Federal Aviation Adminstration

Area where a debris fragment will cause a human Hazard Area casualty, same as casualty area

JARSS MP Joint Advance Range Safety System: Mission Planning

Programing language and integrated development MATLAB environment (IDE) developed by MathWorks

Risk of human casualty due to debris hazard. Equal to Micro-casualties 1*10-6 casualty per event, see Ec

Someone who performs a flight safety analysis for a Mission Analyst mission, see Risk Analyst

Enthalpy-entropy chart, h-s chart. Plots the total heat Mollier diagram against entropy

NASA National Aeronautics and Space Administration

ODE Ordinary differential equation

ORSAT Orbital Reentry Survival Analysis Tool

Pi Probability of Impact

Planned Debris Debris from staging and other planned events

Someone who performs a flight safety analysis for a Risk Analyst mission, see Mission Analyst

Spacecraft Atmospheric Re-entry and Aerothermal SCARAB Break-up

WGS 84 World geodetic Survey of 1984

List of Exhibits

Figure 1: Casualty Risk from Reentry Debris ...... 3 Figure 2: Aluminum 2024-T8xx Tumbling Hollow Sphere Survivability Table (Ref. 25) ...... 6 Figure 3: Coupling and Dependence for Disintegration Prediction (Ref. 12) ...... 7 Figure 4: Thermal Fragmentation and Mechanical Breakup (Ref. 12) ...... 8 Figure 5: Relative Angular Motion (Ref. 26) ...... 13 Figure 6: Kinematics of Rotation (Ref. 26) ...... 13 Figure 7: Coordinate Systems (Ref. 26) ...... 16 Figure 8: Aerodynamic and Propulsive Forces (Ref. 26) ...... 17 Figure 9: Stagnation Point Heat Transfer Rate vs. Flight Velocity (Ref. 5) ...... 29 Figure 10: Corrected (red) and Uncorrected (blue) Cd Impacts ...... 52 Figure 11: Single Fragment, Altitude vs. Time ...... 55 Figure 12: Single Fragment, Altitude vs. Range ...... 55 Figure 13: Single Fragment, Heat Flux vs. Time ...... 56 Figure 14: Single Fragment, Temperature vs. Time ...... 57 Figure 15: Varying Breakup Altitude, Altitude vs. Time ...... 58 Figure 16: Varying Breakup Altitude, Altitude vs. Range ...... 58 Figure 17: Varying Breakup Altitude, Heat Flux vs. Time ...... 59 Figure 18: Varying Breakup Altitude, Temperature vs. Time...... 60 Figure 19: Varying Breakup Altitude, Liquid Fraction vs. Time ...... 61 Figure 20: Varying Temperature, Temperature vs. Time ...... 62 Figure 21: Varying Temperature, Liquid Fraction vs. Time ...... 62 Figure 22: Varying Temperature, Heat Flux vs. Time ...... 63 Figure 23: Varying Velocity, Altitude vs. Time ...... 64 Figure 24: Varying Velocity, Altitude vs. Range ...... 64 Figure 25: Varying Velocity, Heat Flux vs. Time ...... 65

Figure 26: Varying Velocity, Temperature vs. Time ...... 66 Figure 27: Varying Velocity, Liquid Fraction vs. Time ...... 66 Figure 28: Varying Flight Path Angle, Altitude vs. Time ...... 67 Figure 29: Varying Flight Path Angle, Altitude vs. Range ...... 68 Figure 30: Varying Flight Path Angle, Heat Flux vs Time ...... 69 Figure 31: Varying Flight Path Angle, Temperature vs. Time...... 69 Figure 32: Varying Flight Path Angle, Liquid Fraction vs. Time ...... 70 Figure 33: Varying Materials, Heat Flux vs. Time ...... 71 Figure 34: Varying Materials, Temperature vs. Time ...... 71 Figure 35: Varying Materials, Liquid Fraction vs. Time ...... 72 Figure 36: Varying Mass, Altitude vs Time ...... 73 Figure 37: Varying Mass, Altitude vs. Range ...... 74 Figure 38: Varying Mass, Heat Flux vs. Time ...... 75 Figure 39: Varying Mass, Temperature vs. Time ...... 75 Figure 40: Varying Mass, Liquid Fraction vs. Time ...... 76

Table 1: WGS84 Ellipsoid Derived Geometric Constants ...... 31 Table 2: Local Arrays (1976 Std. Atmosphere) ...... 34 Table 3: Atmosphere Fit Parameters (1976 Std. Atmosphere) ...... 36 Table 4: Debris Fragment Shape Assumptions ...... 44 Table 5: Example Fragment ...... 54 Table 6: DAS 2.0 Debris Fragments ...... 77 Table 7: Computational Model Debris Fragments, Compared to DAS 2.0 ...... 78 Table 8: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder ...... 80 Table 9: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder ...... 81 Table 10: Demise Utility Input...... 82 Table 11: Demise Utility Output, Adjusted Fragment Tables ...... 83

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List of Symbols

Symbols for Impact Integration

Variable Value Units Description

m 풂 Local speed of sound s

풂⊕ m Earth’s semi-major axis

1 휷 Atmospheric decay parameter, scale height m

푪푳 Coefficient of lift

∗ 푪푳 Coefficient of lift at the smallest glide angle

푪푫 Coefficient of drag

∗ 푪푫 Coefficient of drag at the smallest glide angle

푪푫ퟎ Zero lift coefficient of drag

푭푵 N Force normal to the flight path

푭푻 N Force tangential to the flight path

m 품 Local gravitational acceleration s2

m 품 Gravitational acceleration at Earth’s surface ퟎ s2

휸 rad Flight path angle, glide angle of vehicle

푲 Induced drag factor, function of Mach

풎 kg Mass of the vehicle

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풏 Wing loading factor

풏풔 Structural loading limit

rad 흎 Angular velocity of Earth s

흋 rad Glide turn roll angle of vehicle

흓 rad Latitude of the vehicle

휽 rad Longitude of vehicle

흍 rad Heading of vehicle

풓 m EFG magnitude radius to vehicle

푹⊕ m Mean radius of Earth

푺 m2 Plane area of the vehicle

m 푽 Vehicle velocity s

풚 Trajectory aerodynamic coordinate state

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Symbols for Fay-Riddell Stagnation Point Heating

Variable Value Units Description

풄풊 Mass fraction of component 푖

J 풄 Specific heat per unit mass at constant pressure 풑 kgK

m2 푫 Molecular diffusion coefficient s

m2 푫푻 Thermal diffusion coefficient sK J 풆 풆 Internal energy per unit mass, of component 푖 풊 kg

J 풉 Enthalpy per unit mass of mixture kg

J 풉 Perfect gas enthalpy per unit mass of component 푖 풊 kg

J 풉 ퟎ Heat of formation of component 푖 at 0 K per unit mass 풊 kg

J 풉 ퟎ Dissociation energy per unit mass of atomic products 푨 kg

W 풌 Thermal conductivity K

풍 m Characteristic length

Lewis Number: 퐷𝑖휌푐푝̅ ⁄푘 ratio of the rate of thermal 푳 푳풊푻 diffusivity to the mass diffusivity, thermal Lewis number

흁 Pa ∙ s Absolute viscosity

Nusselt Number: 푞푥푐̅ ⁄푘 (ℎ − ℎ ) ratio of 푵풖 푝푤 푤 푠 푤 convective to conductive heat transfer

흂 Pa ∙ s Kinematic viscosity

풑 Pa Pressure

Prandtl Number: 푐̅ 휇⁄푘 ratio of momentum diffusivity 푷풓 0.71 푝 to thermal diffusivity, value for air

휱 Dissipation function

W 풒 Heat flux m2 m 풒⃗⃗ 풒⃗⃗ Vector mass velocity, vector diffusion velocity 풊 s

풓 m Cylindrical radius of body

푹풉 m Body nose radius, radius of heating

Reynolds Number: 푢 푥⁄푣 ratio of the inertial to 푹풆 푒 푤 viscous forces

kg 𝝆 Mass density m3

푻 K Absolute temperature m 풖 푥 component of velocity s m 풗 푦 component of velocity s

kg Mass rate of formation of component 푖 per unit 풘 풊 m3s volume and time m 풙 Distance along meridian profile s m 풚 Distance normal to the surface s

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Symbols for Aero-thermal Demise

Variable Value Units Description

m 풂 Local speed of sound s

2 푨풘 m Wetted area

kg 휷 Hypersonic continuum ballistic coefficient 풉풄 m2

푪푫풉풄 Hypersonic continuum coefficient of drag

푪푫휷 Coefficient of drag corrected for ballistic coefficient

푪푫푴풂풄풉 Coefficient of drag from fragment drag tables

J 푪̅ Mean Specific heat capacity of the fragment 풑풃 kgK

J 푪 1.0045 ∙ 103 Specific heat capacity of air 풑∞ kgK

휹 m Recession length, flat plate

휺풃 휺풘 Surface emissivity of the fragment m 품 9.80665 Standard gravitational acceleration ퟎ s2

휸 radians Flight angle of the fragment

풉 풉풊 m Fragment height and interior height, box

J 풉 Heat of fusion 풇 kg

푯풓 푯풓ퟎ m Hazard radius & user defined hazard radius

Area-averaging factor (a less conservative 0.8 for 풌 0.12 ퟐ composites)

Fragment length and interior length, cylinder, flat plate 풍 풍 m 풊 and box

푳푭 Liquid fraction of the fragment

풎풃 kg Thermal mass of the fragment

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풎풊 kg Mass of the interior of the fragment were it solid

풎푳푭 kg Thermal mass adjust by liquid fraction of the fragment

W 𝝈 5.5670373 ∙ 10−8 Stefan-Boltzmann constant 풔풃풄 m2K4 W 풒 Stagnation heat flux 풔 m2 W 풒 Radiation heat flux 풓풂풅 m2 W 푸̇ Net heat flow s

푸ퟎ W Heat of initial temperature

푸풎풆풍풕 W Heat of melting

푸풂 W Heat of ablation

푸 W Heat content of the fragment

Radius and interior radius of fragment, sphere and 풓 풓 m 풊 cylinder

3 푹⊕ 6378.1 ∙ 10 m Mean radius of Earth

푹풉 m Radius of heating

J 푹 287 Gas constant for air ∗ K kg 𝝆 Density of the fragment material 풃 m3 kg 𝝆 Free stream air density ∞ m3

푺 m2 Aerodynamic reference area of the fragment

풕 m Fragment thickness, flat plate

푻ퟎ K Initial body bulk temperature of the fragment

푻풃 K Body bulk temperature of the fragment

푻풎풆풍풕 K Melting temperature of the fragment

푻풓풆풇 300 K Reference temperature

xviii

푻풔 K Adiabatic stagnation temperature

푻풘 K Hot wall temperature of the fragment m 푼 7924.8 Reference velocity, 26000 ft/s 풓풆풇 s m 푼 Free stream velocity ∞ s

풘 풘풊 m Fragment width and interior width, flat plate and box

풛 m Altitude position of the fragment

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T. M. Owens 1

1 Introduction

In the past few years there has been a steady rise in the number of space launches as well as the privatization of America’s launch capabilities. With the increase in space missions, there is an increase in demand for range safety analysis to address possible risk to the population. Tools used to evaluate this risk rely on accurate vehicle debris fragment models to estimate the probability of a human casualty. By developing simple to use and accurate models for the aero-thermal demise of reentry debris, better predictions of the probability of impact (Pi) and estimated casualties (Ec) can be made.

The computational model detailed in this thesis combines several well established algorithms for modeling earth geometry, gravitational acceleration, atmospheric properties, impact propagation and aero-thermal demise to model the aero- thermal demise of reentry debris. With simple inputs the model generates a debris catalog that can be used by other risk analysis tools. There is also the possibility that the impact integrator could be used within an existing tool in order to consider demise of an existing debris catalog.

To understand what the work in this thesis is attempting to accomplish, it helps to understand how current tools work to estimate reentry debris survivability and casualty risk. The Debris Assessment Software 2.0 (DAS 2.0) and Orbital Reentry Survival Analysis Tool (ORSAT) are used by National Aeronautics and Space Adminstration (NASA) and other American launch providersRef. 17, Ref. 2. Aerospace Survivability Tables were developed for the Federal Aviation Authority (FAA) and Office of Commercial Space Transportation (AST)Ref. 24. The Spacecraft Atmospheric Re-entry and Aerothermal Break-up (SCARAB) tool is used by the European Space Agency (ESA)Ref. 11. DAS 2.0 is a simplistic first order solution, like this thesis, whereas ORSAT and SCARAB are pseudo-CFD programs with much

2 Aero-thermal Demise

greater complexity. The computational model is based in part on the algorithm used to build the survivability table.

1.1 DAS 2.0: Debris Assessment Software 2.0

The Debris Assessment Software is a NASA utility built to perform a variety of orbital debris assessments according to the NASA Technical Standard 8719.14, Process for Limiting Orbital DebrisRef. 13. The technical standards are a set of mission requirements which can govern the acceptance of a mission for launch. The reentry-survivability model, which checks requirement 4.7-1 from the technical standard, is the portion relevant to this thesis. The safety guideline is in the NASA Safety Standard 1740.14, Guidelines and Assessment Procedures for Limiting Orbital Debris, and it states that "the total debris casualty area for components and structural fragments will not exceed 8 m2." This equates to 100 micro-casualties per reentry event or 1:10,000Ref. 16.

Figure 1 shows the output from DAS2.0’s assessment of requirement 4.7-1 for the example missions. The top portion summarizes the inputs. The mission LEO1 has a number of sub-components or debris fragments with a variety of different materials and shapes. The output shows that the mission LEO1 is non-compliant. Several of the debris fragments survive to impact giving a total casualty area of 10.35 m2, just over the limit of 8 m2. The components each have a demise altitude, casualty area and impact energy. The demise altitude is when the debris fragment is fully ablated. If the demise altitude is 0, the fragment has survived to impact and has a casualty area and impact energy.

As the simplest of the models, DAS 2.0 has an advantage in that it does not require the user to have a detailed knowledge of the spacecraft’s geometry. Just the overall shape (sphere, flat plate, cylinder or box), rough dimensions and material for each of the fragment classes are required. However, DAS 2.0 is limited

T. M. Owens 3

in that its output does not allow for the creation of a demise modified debris catalog because the output only has the impact mass and energy. Also DAS 2.0 is not particularly useful for missions with planned reentry as it can only have a single failure event at an altitude of 78 km. Its ease of use serves as a benchmark for the computational model developed in this thesis.

Figure 1: Casualty Risk from Reentry Debris

4 Aero-thermal Demise

1.2 ORSAT: Orbital Reentry Survival Analysis Tool

The Orbital Reentry Survival Analysis Tool (ORSAT) is a much more complex and higher fidelity tool for analyzing the thermal breakup of spacecraft during reentryRef. 2. Like DAS 2.0, it is was developed to meet the requirements of NASA standards, specifically NASA Technical Standard 8719.14, A Process for Limiting Orbital Debris. Like DAS 2.0, the casualty risk from all reentry debris should be less than 1:10,000. ORSAT uses integrated trajectory, atmospheric, aerodynamic, aerothermodynamic and ablation algorithms to find the impact risk.

It is able to use three different atmospheres, 1976 U.S. standard, MSISe-90 and GRAM-99 atmosphere. It can model either spinning or tumbling modes for the fragment debris. The drag coefficients are found from the kinetic energy at impact. The stagnation point continuum heating rates are found for spheres and correlated for other geometries and flight regimes. To find the surface temperature, it is able to use both a lumped mass and 1-D conduction. Demise is assumed once the net heat absorbed reaches the heat of ablation for the material.

Unfortunately, ORSAT is only available to the Orbital Debris Program Office at Johnson Space Center so a true comparison cannot be made in this thesis. However, there are some capabilities that ORSAT obviously has that this computational model will not. Probably the most significant is ORSAT’s ability to define more complex geometries and predict aerodynamic breakup.

1.3 Aerospace Survivability Tables

The Aerospace Survivability Tables is a set of tabular data on the demise of various fragments that is part of a larger tool to estimate the total casualty expectation made by The Aerospace CorporationRef. 24. The model has been validated against

T. M. Owens 5

the Columbia Accident Investigation Board (CAIB) report for casualty expectation and impact probability and The Aerospace Corporation’s higher fidelity model Atmospheric Heating and Breakup tool (AHaB) for survivability of debris. Their model does not change the fragment properties as they impact nor does it account for the wall gradient temperature. It uses the Detra-Kemp-Riddell stagnation point heating correlation with a radiation heat balance to determine the amount of ablation for the fragments. The algorithm sits nicely between simple tools like DAS 2.0 and pseudo CFD tools like ORSAT which is why it was chosen as a starting point for the computational model outlined in this thesis.

The tables cover three materials, aluminum 2024-T8xx, stainless steel 21-6-9 and titanium (6 Al-4 V), and three hollow shapes, spheres, cylinders and flat plates. There is the choice between 1541°R and 540°R as breakup temperature of the debris. The tables also vary the breakup flight conditions. It covers from 46 to 30 Nmi in altitude with the flight path angles of -0.5, -3.5 and -5.5 degrees. The velocities at each altitude are based on what is to be expected from a reentry trajectory. As an example at 42 Nmi there is a choice between 25,000, 23,000 and 21,000 ft/s.

Figure 2 shows an example table from the Aerospace Survivability Tables for an aluminum sphere. The rows are for radius in feet and the columns for weight in pounds. The values in the tables are liquid fractions, how much of the mass of the fragment has ablated, with one being fully demised and zero meaning the debris fragment has survived intact to impact. The s in the table indicates that the fragment has skipped off the atmosphere.

6 Aero-thermal Demise

Figure 2: Aluminum 2024-T8xx Tumbling Hollow Sphere Survivability Table (Ref. 24)

The obvious disadvantage to the tables is the limitation of having to choose the fragment and breakup conditions that best fit for the survivability analysis instead of computing it. The accuracy only to the first decimal is not a major concern as the Detra-Kemp-Riddell stagnation point heating correlation is only accurate to 10% of the heating rate, at bestRef. 4.

1.4 SCARAB: Spacecraft Atmospheric Re-entry and Aerothermal Break-up

The SCARAB tool is very similar to ORSAT. It was developed primarily for use by the European Space Agency and partnersRef. 11. The program is broken into five disciplines with different dependencies and couplings, flight dynamics,

T. M. Owens 7

aerodynamics, aerothermodynamics, structural analysis, thermal analysis and deformation/disintegration/melting as seen in Figure 3. A spacecraft is defined by geometric modeling, materials and physical modeling. SCARAB can use a variety of panelized geometric primitives to construct more complex shapes and volume elements.

Figure 3: Coupling and Dependence for Disintegration Prediction (Ref. 11)

SCARAB uses a materials database with 20 parameters that can be extended to a three phase model (gas, liquid and solid). The aerodynamic and aero-heating analysis is broken into free molecular, hypersonic continuum and rarefied transitional flow regimes. The thermal analysis uses thin and thick thermal heating which allows layered melting of solids with low heat conductivity. The latest versions of SCARAB can also perform a finite element analysis to find the stress resulting from inertial and aerodynamic forces. An example of the thermal fragmentation and mechanical breakup as computed by SCARAB can be seen in Figure 4.

As with ORSAT the main difference between this tool and the computational model developed for this thesis is the complexity of geometry and the ability to predict structural failure.

8 Aero-thermal Demise

Figure 4: Thermal Fragmentation and Mechanical Breakup (Ref. 11)

1.5 Computation Model

There is an obvious gap in complexity level between the computational model developed in this thesis and those of tools like ORSAT and SCARAB. However most of these deficiencies are not critical to performing a flight risk analysis.

The model in this thesis only takes into consideration the aero-thermal analysis of the hypersonic continuum flow regime. This is adequate for approximating demise as the reentry flight speeds are typically on the order of Mach 10 and maximum aero-heating occurs from 80 to 50 kms of altitude. The ORSAT and SCARAB tools are also designed for impact prediction of spacecraft and not designed for landing reentry, whereas the model in this thesis is mainly concerned with the failure of launch and reentry vehicles. Therefore, the very high altitude flight regimes are of little importance because minimal heat transfer takes place at supersonic and

T. M. Owens 9

subsonic flight speeds; there is no real value added to include them in this analysis.

While the inability to predict a breakup event may seem like a major weakness of the code developed for this thesis, it is of minimal importance at the stage of risk analysis for this computation model and its expected use. Typically a debris catalog will be provided to the mission analysis that may have several failure modes such as an intact crash, partial breakup and full breakup. The probability of each of these outcomes is determined through some other analysis, perhaps by the vehicle manufacture itself. Thus, there is no way a complex geometry could be constructed from the debris catalog provided. The tools that the mission analyst will use to predict risk typically assume failure at every point in the trajectory at the failure rates from the probability of outcomes. Therefore, knowing precisely when a part will fail is not as important as knowing where it will land if it failed at that trajectory time and what sort of casualty risk can be expected.

The advantage this utility will have over some other reentry demise analysis tools is that it will take in a standard debris catalog and return the standard debris catalog with values adjusted for aero-thermal demise. This allows a risk analyst to use the existing workflow and simply run the computational model developed in this thesis before other risk tools to account for the demise casualty reduction.

1.6 Risk Analysis

Understanding the desire for a tool that can predict aero-thermal demise requires some understanding of how a mission risk analysis is performed. The typical main risk criterion is 1:10,000 or 100 micro-casualties. To determine this, the casualty expectation is found by summing the probability of every possible event and the casualty consequences at each mission event. The general form of the casualty expectation equation for 푛 possible events is as followsRef. 1,

10 Aero-thermal Demise

푛

퐸푐 = ∑ 푃𝑖퐴푐𝑖퐷푝𝑖 Eq. 1 𝑖=1

Where the 푃𝑖 is the impact probability, 퐴푐𝑖 is the casualty or hazard area of the 푡ℎ debris and 퐷푝𝑖 is the population density of the area at risk for the 푖 event. The population in an area is partially under the control of the launch provider as they can close portions of the launch area. This, however, is restrictive to other work and may not be possible in public areas. The probability of impact could possibly be changed by increasing reliability of the trajectory; however, this is generally not an option open to the mission analyst. Thus, the only real way to find a reduction in the expected casualties is to change the hazard area of the debris.

One possible method is to introduce the effects of sheltering. Sheltering is the effect that buildings will have on the risk of human casualty. This is dependent on the time of week and day as well as the mass and speed of the impacting debris fragment. It does not always reduce the expected casualties however. Some large debris is considered to cause building collapses which will cause more casualties than if the debris were to impact an open area.

Therefore, the aero-thermal demise of reentry debris should be considered. Partially demising fragments will not greatly reduce the casualty area; however, the mass loss can mean greater benefit from the effects of sheltering. Fully demising debris has no casualty area. As a result, there will be a clear reduction in expected casualties. This and the accurate prediction of where the debris will impact due to the changing ballistic coefficient give a clear benefit to performing demise analysis along with risk analysis.

T. M. Owens 11

2 Impact Integration

The impact integration relies on the derivation of the equations of motion over a rotating spherical earth. Because of the very high initial speeds and altitudes of the debris fragments, many terms that would be otherwise ignored for impact or ballistic calculations must be included.

The probably of impact is not explored. This is a separate and distinct problem that involves creating a bivariate normal distribution of impact probability defined by an impact covariance. The covariance should take into account factors such as explosion velocity, position, velocity, wind and drag uncertainties. The probability distribution can be built through a Monte Carlo set of impact propagations, typically on the order of 10,000 or more. Other approximations of Gaussian distributions such as Jacobian-based techniques or a Julier-Ulhmann method can be used for much faster propagationRef. 12.

2.1 Equations of Motion

The most important component of the impact integration is the derivation of force equations. The following is a derivation of the equations of motion that will result in three force equations for velocity, heading and flight path angle from Ch. 2 of VinhRef. 25. These general equations of flight over a spherical, rotating earth allow for use with high performance reentry vehicles like the Boeing X-37, capsules like the Apollo Command Module or a piece of reentry debris.

Consider a body with a point mass defined by a position vector, 푟(푡), velocity vector, 푉(푡), and mass, 푚(푡). The total force, 퐹, at each instance is a sum of the gravitational, 푚 ∙ 푔, aerodynamic, 퐴, and propulsion thrust forces, 푇.

푭 = 푻 + 푨 + 푚품 Eq. 2

12 Aero-thermal Demise

For reentry debris fragments, the propulsive force is zero, and in a vacuum, the aerodynamic forces are zero.

The kinematic vector equation for the inertial system defined by the position, velocity and mass is,

푑풓 = 푽 푑푡 Eq. 3

Newton’s second law gives the force equation.

푑푽 푚 = 푭 푑푡 Eq. 4

Because Newton’s second law requires a fixed system, and the earth’s system is rotating, a transformation is required.

2.1.1 Relative Angular Motion

Consider a fixed inertial reference frame system and a rotating system, 푂푋1푌1푍1 and 푂푥푦푧 respectively. The system 푂푥푦푧 rotates with respect to the fixed system

푂푋1푌1푍1with angular velocity 휔. Let 퐴 be an arbitrary vector in the rotating system as seen in Figure 5,

̂ 푨 = 퐴푥푖̂ + 퐴푦푗̂ + 퐴푧푘 Eq. 5

T. M. Owens 13

Figure 5: Relative Angular Motion (Ref. 25)

The 푖,̂ 푗 ̂ and 푘̂ unit vectors in the rotating reference system, 푂푥푦푧, are functions of time. Therefore, the time derivative of 퐴 with respect to the fixed system 푂푋1푌1푍1 is,

̂ 푑푨 푑퐴푥 푑퐴푦 푑퐴푧 푑푖̂ 푑푗̂ 푑푘 = ( 푖̂ + 푗̂ + 푘̂) + (퐴 + 퐴 + 퐴 ) 푑푡 푑푡 푑푡 푑푡 푥 푑푡 푦 푑푡 푧 푑푡 Eq. 6

At point 푃, the linear velocity in a fixed system rotating with angular velocity 휔 at position vector 푟 as seen in Figure 6 is,

푑풓 푽 = = 흎 × 풓 푑푡 Eq. 7

Figure 6: Kinematics of Rotation (Ref. 25)

14 Aero-thermal Demise

Then, according to Poisson’s formulas,

푑푖̂ = 흎 × 푖̂ 푑푡

푑푗̂ = 흎 × 푗̂ 푑푡 Eq. 8

푑푘̂ = 흎 × 푘̂ 푑푡

Using this with the definition of vector 퐴 in the equation of the time derivative of 퐴 the following equation is found,

푑푖̂ 푑푗̂ 푑푘̂ 퐴 + 퐴 + 퐴 = 흎 × 푨 Eq. 9 푥 푑푡 푦 푑푡 푧 푑푡

Then in the rotating system 푂푥푦푧 the time derivative of 퐴 is,

훿푨 푑퐴푥 푑퐴푦 푑퐴푧 = 푖̂ + 푗̂ + 푘̂ 훿푡 푑푡 푑푡 푑푡 Eq. 10

The equation for the time derivative with respect to the fixed system, Eq. 6, can be rewritten as,

푑푨 훿푨 = + 흎 × 푨 푑푡 훿푡 Eq. 11

2.1.2 Equations for Flight Over a Rotating Spherical Earth

푂푋1푌1푍1, is the inertial reference frame, with the origin at the center of a spherical earth’s gravitation field. The plane 푂푋1푌1 is the equatorial plane. The reference frame 푂푋푌푍 is fixed with respect to earth with 푂푍 and 푂푍1 coincident. The atmosphere is assumed to rotate at the same constant angular acceleration, 휔.

T. M. Owens 15

The equation for absolution acceleration is found by setting the position vector 퐴 = 푟 and taking the time derivative of Eq. 11 in the earth fixed frame 푂푋푌푍.

푑푽 훿2풓 훿풓 = + 2흎 × + 흎 × (흎 × 풓) Eq. 12 푑푡 훿푡2 훿푡

The equation Eq. 4 can then be put in the earth-fixed system,

훿2풓 훿풓 푚 = 푭 − 2푚흎 × − 푚흎 × (흎 × 풓) Eq. 13 훿푡2 훿푡

Or,

푑푽 푚 = 푭 − 2푚흎 × 푽 − 푚흎 × (흎 × 풓) 푑푡 Eq. 14

The velocity, 푉, is the velocity relative to the earth-fixed system. From this there are two acceleration forces as the earth rotates. They are the Coriolis acceleration, −2흎 × 푽, and the transport acceleration, −흎 × (흎 × 풓). The Coriolis acceleration is zero when the flight path angle is parallel to the earth’s pole and reaches a maximum of 2휔푉 when the flight path angle is perpendicular to the polar axis. The transport acceleration is zero when the body is at the poles and at its maximum, 휔2푟, when the body is on the equatorial plane.

The fixed coordinate system, 푂푋푌푍, and rotating coordinate system, 푂푥푦푧, can be seen in Figure 7. The longitude is angle 휃 and latitude is 휙.The angle 훾 is the flight path angle and 훹, the heading. The flight path angle is positive for a launch trajectory and negative for a reentry trajectory. The heading is the angle between the local parallel of the latitude and the projection of the velocity vector on earth’s surface with right hand positive about the 푥 axis.

16 Aero-thermal Demise

Figure 7: Coordinate Systems (Ref. 25)

Thereby the velocity vector in the rotating system 푂푥푦푧 is,

푽 = 푉 sin 훾 푖̂ + 푉 cos 훾 cos 훹 푗̂ + 푉 cos 훾 sin 훹 푘̂ Eq. 15

The angular velocity in the 푂푥푦 plane is,

흎 = 휔 sin 휙 푖̂ + 휔 cos 휙 푘̂ Eq. 16

This can then be used to find the Coriolis and transport accelerations in terms of the unit vectors,

흎 × 푽 = −휔푉 cos 훾 cos 휙 cos 훹 푖̂ + 휔푉(sin 훾 cos 휙 − cos 훾 sin 훹)푗̂ Eq. 17 + 휔푉 cos 훾 sin 휙 cos 훹 푘̂

2 2 2 흎 × (흎 × 풓) = −휔 cos 휙 푖̂ + 휔 푟 sin 휙 cos 휙 푘̂ Eq. 18

The force of gravitational acceleration in the total force 퐹 is,

T. M. Owens 17

푚품 = −푚푔(푟)푖 ̂ Eq. 19

The aerodynamic forces, lift and drag, can be put into terms of tangential and normal forces to the flight plane. The angle between thrust and the velocity vector is angle 휀 as seen in Figure 8. The propulsive and aerodynamic forces can be grouped,

퐹푇 = 푇 cos 휀 − 퐷 Eq. 20 퐹푁 = 푇 sin 휀 + 퐿

Figure 8: Aerodynamic and Propulsive Forces (Ref. 25)

In the unit vector form, the normal force can be defined as,

̂ 푭푇 = 퐹푇 sin 훾 푖̂ + 퐹푇 cos 훾 cos 훹 푗̂ + 퐹푇 cos 훾 sin 훹 푘 Eq. 21

The normal force in vector form requires a coordinate transformation between the rotating reference frame and the flight plane, as well as accounting for the roll angle 휎. This results in the equation,

푭푁 = 퐹푁 cos 휎 cos 훾 푖̂ − 퐹푁(cos 휎 sin 훾 cos 훹 + sin 휎 sin 훹)푗̂

̂ Eq. 22 + 퐹푇(cos 휎 sin 훾 sin 훹 − sin 휎 cos 훹)푘

Then, the equation Eq. 8 can be put in terms of the latitude and longitude.

18 Aero-thermal Demise

푑푖̂ 푑휃 푑휙 = cos 휙 푗̂ + 푘̂ 푑푡 푑푡 푑푡 푑푗̂ 푑휃 푑휃 = − cos 휙 푖̂ + sin 휙 푘̂ 푑푡 푑푡 푑푡 Eq. 23 푑푘̂ 푑휙 푑휃 = − 푖̂ − sin 휙 푗̂ 푑푡 푑푡 푑푡

Then, using the equation Eq. 23 in Eq. 15,

푑푟 푑휃 푑휙 푽 = 푖̂ + 푟 cos 휙 푗̂ + 푟 푘̂ 푑푡 푑푡 푑푡 Eq. 24

The derivative of velocity is,

푑푽 푑푉 푑훾 푉2 = [sin 훾 + 푉 cos 훾 − cos2 훾] 푖̂ 푑푡 푑푡 푑푡 푟 푑푉 푑훾 푑훹 + [cos 훾 cos 훹 − 푉 sin 훾 cos 훹 − 푉 cos 훾 sin 훹 푑푡 푑푡 푑푡 푉2 + cos 훾 cos 훹 (sin 훾 − cos 훾 sin 훹 tan 휙)] 푗̂ 푟 Eq. 25 푑푉 푑훾 푑훹 + [cos 훾 sin 훹 − 푉 sin 훾 sin 훹 + 푉 cos 훾 cos 훹 푑푡 푑푡 푑푡 푉2 + cos 훾 (sin 훾 sin 훹 − cos 훾 cos2 훹 tan 휙)] 푘̂ 푟

Next, by substituting equation Eq. 25 into Eq. 14, the scalar equations of motion are found to be,

푑푉 푑훾 푉2 퐹 퐹 sin 훾 + 푉 cos 훾 − cos2 훾 = 푇 sin 훾 + 푁 cos 휎 cos 훾 푑푡 푑푡 푟 푚 푚 Eq. 26 − 푔 + 2휔푉 cos 훾 cos 훹 cos 휙 + 휔2푟 cos 휙

T. M. Owens 19

푑푉 푑훾 푉2 퐹 퐹 sin 훾 + 푉 cos 훾 − cos2 훾 = 푇 sin 훾 + 푁 cos 휎 cos 훾 푑푡 푑푡 푟 푚 푚 Eq. 27 − 푔 + 2휔푉 cos 훾 cos 훹 cos 휙 + 휔2푟 cos 휙

푑푉 푑훾 푉2 퐹 퐹 sin 훾 + 푉 cos 훾 − cos2 훾 = 푇 sin 훾 + 푁 cos 휎 cos 훾 푑푡 푑푡 푟 푚 푚 Eq. 28 − 푔 + 2휔푉 cos 훾 cos 훹 cos 휙 + 휔2푟 cos 휙

푑푉 푑훾 푑휓 Solving for the derivatives , , and , 푑푡 푑푡 푑푡

푑푉 퐹푇 = − 푔 sin 훾 + 휔2 푟 cos 휙 (sin 훾 cos 휙 − cos 훾 sin 휓 sin 휙) 푑푡 푚 Eq. 29

푑훾 퐹 푉2 푉 = 푁 cos 휑 − 푔 cos 훾 + cos 훾 푑푡 푚 푟 Eq. 30 + 2휔푉 cos 휓 cos 휙 + 휔2 푟 cos 휙 (cos 훾 cos 휙 − sin 훾 sin 휓 sin 휙)

푑휓 퐹 sin 휑 푉2 푉 = 푁 − cos 훾 cos 휓 tan 휙 + 2휔푉(tan 훾 sin 휓 cos 휙 − sin 휙) 푑푡 푚 cos 훾 푟 Eq. 31 휔2푟 − cos 휓 sin 휙 cos 휙 cos 훾

The 휔2푟 term is the transport acceleration and the 2휔푉 term is the Coriolis acceleration. If the speeds are much less than orbital, then the equations could be simplified further to not include the Coriolis and transport accelerations; however, for the purposes of this thesis, they are necessary terms.

20 Aero-thermal Demise

3 Stagnation-Point Heating

Stagnation point heating is the main mode of heat transfer for bodies reentering an atmosphere. Convective heat transfer depends on the properties of the atmosphere, planet and reentering bodies. Radiative heat transfer balances the convective heat transfer in the net heat flux.

3.1 Fay and Riddell Theory

The Theory of Stagnation Point Heat Transfer in Dissociated Air by Fay and RiddellRef. 6 is probably the seminal work on stagnation point heating theory. Many of the correlations for stagnation point heating find their roots in Fay and Riddell’s theory and the work of others at Avco Research Laboratory in the 1950’s. The Fay- Riddell theory reduces a set of general boundary-layer equations for stagnation point heating into nonlinear ordinary differential equations for a broad flight regime.

The derivation starts with the equation for the heat flux in a quiescent dissociated gas where ℎ퐴0 is the dissociation energy per unit mass, 퐷 is the diffusion coefficient and 푐퐴 is the atomic mass fraction.

푞 = 푘∇푇 + ℎ퐴0퐷휌∇푐퐴 Eq. 32

The first term on the right of equation Eq. 32 is the transport of kinetic energy and the second term is the potential recombination energy of the dissociated gas. This can be simplified by neglecting the process of dissociation and recombination as well as substituting for the temperature gradient and assuming a Lewis number of unity.

T. M. Owens 21

푘 푞 = ∇(ℎ + 푐퐴ℎ퐴0) Eq. 33 푐푝

3.1.1 Laminar Boundary-Layer in Dissociated Gas

Stagnation point heat transfer is a combination of thermal and aerodynamic effects. Thereafter, the partial differential equations associated with the boundary layer need to be found. The general continuity equation for the mass rate of formation of the species 푖 per unit volume and time is,

∇ ∙ [휌(푞 + 푞 𝑖)푐𝑖] = 푤𝑖 Eq. 34

The mass average velocity, 푞 𝑖,of the species 푖 can be found by,

퐷𝑖 퐷𝑖푇 푞 𝑖 = ∇푐𝑖 − ∇푇 Eq. 35 푐𝑖 푇

The first term on the right hand side is the concentration diffusion, the second is the thermal diffusion and the pressure diffusion is assumed negligible. The continuity equation is summed for all species so it takes the form,

∇ ∙ (휌푞 ) = 0 Eq. 36

In addition, the energy equation for a fluid element is required,

휌푞 ∆ ∑ 푐𝑖푒𝑖 = ∇ ∙ (푘∆푇) − ∇ ∙ (∑ 휌푞 𝑖 푐𝑖ℎ𝑖) + ∑ 푤𝑖 ℎ𝑖0 + 푝∇ ∙ 푞 + 훷 Eq. 37

With 훷 being the dissipation function, the steady-state energy equation can be rewritten using the idea gas assumption, conservation of mass, continuity equation and relationship of enthalpy to internal energy.

22 Aero-thermal Demise

휌푞 ∇ ∑ 푐𝑖(ℎ𝑖 − ℎ𝑖0 ) = ∇ ∙ [푘∇푇 − ∑ 휌푞 𝑖 푐𝑖(ℎ𝑖 − ℎ𝑖0 )] + 푞 ∇푝 + 훷 Eq. 38

Taking into account the boundary layer assumptions, the equations Eq. 34, Eq. 36, and Eq. 38 can be rewritten as partial differentials. The centrifugal forces are neglected assuming the boundary-layer thickness is much less than the radius of curvature of the body. The 푥 is tangential and 푦 is normal to the surface with 푢 and 푣 being the velocity components respectively.

(휌푟푢)푥 + (휌푟푣)푦 = 0 Eq. 39

푇푦 휌푢푐𝑖푥 + 휌푣푐𝑖푦 = (퐷𝑖휌푐𝑖푦 + 퐷𝑖푇 휌푐𝑖 ) + 푤𝑖 Eq. 40 푇 푦

휌푢ℎ + 휌푣ℎ = (푘푇 ) + 푢푝 + 휇푢 2 푥 푦 푦 푦 푥 푦

푇푦 Eq. 41 + [∑ 퐷𝑖휌(ℎ𝑖 − ℎ𝑖0)푐𝑖푦 + ∑ 퐷𝑖푇휌푐𝑖(ℎ𝑖 − ℎ𝑖0) ] 푇 푦

The equation of motion is,

휌푢푢 + 휌푣푢 = −푝 + (휇푢 ) 푥 푦 푥 푦 푦 Eq. 42

The equation Eq. 41 can be rewritten in terms of temperature instead of enthalpy for simpler use with transport coefficients.

푐̅ (휌푢푇 + 휌푣푇 ) = (푘푇 ) + 푢푝 + 휇푢 2 푝 푥 푦 푦 푦 푥 푦

푇푦 Eq. 43 + ∑ 푤𝑖(ℎ𝑖 − ℎ𝑖0) + [∑ 퐷𝑖휌푐𝑖푦 + ∑ 퐷𝑖푇 휌푐𝑖 ] 푇 푦

Similarly, equation Eq. 41 can be rewritten to simply be in terms of the enthalpy,

T. M. Owens 23

푢2 푢2 푘 푢2 휌푢 (ℎ + ) + 휌푣 (ℎ + ) = [ (ℎ + ) ] + 푢푝푥 2 2 푐푝̅ 2 푥 푦 푦 푦 Eq. 44 푇 2 푘 푦 + 휇푢 + [∑ (퐷 휌 − ) (ℎ − ℎ 0 )푐 + ∑ 퐷 푇 휌푐 (ℎ − ℎ 0) ] 푦 𝑖 푐̅ 𝑖 𝑖 𝑖푦 𝑖 𝑖 𝑖 𝑖 푇 푝 푦

The equations Eq. 39, Eq. 40, Eq. 42 and Eq. 43 or Eq. 61 form a system of partial differential equations that must be solved.

3.1.2 Boundary Layer Ordinary Differential Equations

In order to simplify the solution, the partial differential equations are reduced to ordinary differential equations. An exact solution can only exist when the boundary layer is considered to be frozen or in thermodynamic equilibrium. The first step is to set transformations of the independent variables and dimensionless independent variables.

푦 푟푢∞ 휂 ≡ ( ) ∫ 휌푑푦 Eq. 45 √2휉 0

푥 2 휉 ≡ ∫ 휌푤휇푤푢∞푟 푑푥 Eq. 46 0

휕푓 푢 휂 휕푓 ≡ ; 푓 = ∫ 푑휂 Eq. 47 휕휂 푢∞ 0 휕휂

푢2 (ℎ + ) 2 푔 = Eq. 48 ℎ푠

푇 휃 = Eq. 49 푇∞

24 Aero-thermal Demise

푐𝑖 푠𝑖 = Eq. 50 푐𝑖∞

The subscript ∞ is the free stream condition and 푤 is for the condition at the wall.

At the stagnation point, the equations for 푓, 푔, 휃 and 푠𝑖 are functions of 휂 as 휉 increases. Also, assuming a thermodynamic equilibrium, the equations Eq. 39, Eq. 40, Eq. 42 and Eq. 43 or Eq. 61 can be written as,

푓 휌푣 = − [(√2휉푓휉 + ) 휉푥 + √2휉푓푦휂푥]⁄푟 Eq. 51 √2휉

−1 푙 퐿𝑖푇푠𝑖푇휂 푤𝑖 푑푢∞ [ (퐿𝑖푠𝑖휂 + )] + 푓푠𝑖휂 + [(2 ) ] = 0 Eq. 52 푃푟 푇 휂 휌푐𝑖푠 푑푥 푠

1 휌푠 (푙푓 ) + 푓푓 + ( − 푓 2) = 0 휂휂 휂 휂휂 2 휌 휂 Eq. 53

−1 푐푝̅ 푙 푐푝̅ 푑푢 푤 (ℎ − ℎ 0) ( ) + 푓휃 + [(2 ∞) ] ∑ 𝑖 𝑖 𝑖 푐̅ 푃푟 푐̅ 휂 푑푥 휌 푐̅ 푇 푝푤 휂 푝푤 푠 푝푤 푠 Eq. 54 푐푝𝑖̅ 푐𝑖푠푙 퐿𝑖푇푠𝑖휃휂 + ∑ (퐿𝑖푠𝑖휂 + ) 휃휂 = 0 푐푝푤̅ 푃푟 휃 휂

푙 푙 (ℎ𝑖 − ℎ 0 ) 퐿 푇 푠𝑖휃휂 ( 푔 ) + 푓푔 + { ∑ [푐 𝑖 ] [(퐿 − 1)푠 + 𝑖 ]} = 0 푃푟 휂 휂 푃푟 𝑖푠 ℎ 𝑖 𝑖휂 휃 Eq. 55 휂 푠 휂

3.1.3 Heat Transfer Rate

The local heat transfer rate, which is a sum of the conduction and diffusion transports at the wall, is given by the equation,

푇 휕푇 휕푐𝑖 퐷𝑖 푐𝑖 휕푇 푞 = (푘 ) + [∑ 휌(ℎ𝑖 − ℎ 0) (퐷𝑖 + )] 휕푦 𝑖 휕푦 푇 휕푦 Eq. 56 푦=0 푦=0

T. M. Owens 25

The dimensionless terms from the previous section can be used to get the equation,

푟푘푤휌푤푢∞푇∞ (ℎ𝑖 − ℎ𝑖0) 퐿𝑖푇 푠𝑖휃휂 푞 = ( ) [휃휂 + ∑ 푐𝑖 (퐿𝑖푠𝑖휂 + )] 푐̅ 푇 휃 Eq. 57 √2휉 푝 ∞ 휂=0

At the stagnation point,

푟휌푤푢∞ 2 푑푢∞ = √ ( ) Eq. 58 √2휉 휈푤 푑푥 푠

Thereby allowing the heat transfer equation to be rewritten as,

푁푢 푑푢∞ (ℎ푠 − ℎ푤) 푞 = √휌푤푢푤 ( ) Eq. 59 √푅푒 푑푥 푠 푃푟

3.1.4 Equilibrium Boundary Layer

The equilibrium boundary layer is found through a numerical solution of equations Eq. 52, Eq. 53, Eq. 54 and Eq. 55, as explained in Fay-RiddellRef. 6. Because a catalytic wall is assumed it is not necessary to find the frozen heat transfer rate. For a Lewis number of unity the heat transfer parameter relies only on the variation of 휌휇 across the boundary-layer giving the equation,

0.4 푁푢 휌푠휇푠 = 0.67 ( ) Eq. 60 √푅푒 휌푤휇푤

A further simplification can be made if only a single species ‘air’ is considered with an average heat of formation from atomic oxygen and hydrogen found by,

26 Aero-thermal Demise

ℎ 0 = ∑ 푐 (−ℎ 0 )⁄ ∑ 푐 퐴 𝑖푠 𝑖 𝑖푠 Eq. 61 푎푡표푚푠 푎푡표푚푠

Also, the numerical solution effect of the Lewis number can be taken into account by the equation,

푁푢 푁푢 0.52 ℎ퐷 ( ) = 1 + (퐿 − 1) Eq. 62 √푅푒 √푅푒 퐿=1 ℎ푠

From equations Eq. 60 and Eq. 62, with the Prandtl number set to 0.71, the stagnation point heat transfer rate equation Eq. 59 is found to be,

0.1 0.4 0.52 ℎ퐷 푑푢∞ 푞 = 0.94(휌푠휇푠) (휌푠푙휇푠푙) [1 + (퐿 − 1) ] √( ) Eq. 63 ℎ푠 푑푥 푠

The velocity gradient as defined by a modified Newtonian flow is,

푑푢∞ 1 2(푝푠 − 푝∞) ( ) = √ Eq. 64 푑푥 푠 푅 휌푠

So, with these various correlations, the stagnation point heat transfer rate can be developed.

T. M. Owens 27

3.2 Detra, Kemp and Riddell Correlation

The correlation for stagnation point heating in a continuum flow developed by Detra, Kemp and RiddellRef. 4 is an exact formulation that takes into account the high temperature dissociation phenomena. It starts with the formula from the Fay-Riddell theory equation,

ℎ푠 0.1 0.4 푑푢∞ 푞푠 = 0.94 (1 − ) (휌푠휇푠) (휌푠푙휇푠푙) ℎ푠푙√( ) [1 ℎ푠푙 푑푥 푠 Eq. 65 ℎ퐷 + 0.45(퐿 − 1) ] ℎ푠푙

The viscosity is extrapolated using Sutherland’s law, the Prandtl number is made a constant 0.71, and the Lewis number is also taken as a constant. The assumption is that there is a thermodynamic equilibrium; however, the correlation can be used with a nonequalibrium boundary layer as long as the surface is catalytic.

This formula can be reduced to a function of density and velocity. This is done using a Mollier diagram of the National Bureau of Standards’ dataRef. 10. The solution of the shock wave equations is found through iteration, and the inviscid flow properties are used to find the stagnation point velocity gradient. Assuming a Newtonian pressure distribution gives,

푑푢푒 1 2푝푠푙 ( ) = √ Eq. 66 푑푥 푠 푅ℎ 휌푠푙

The variation of the stagnation point heating should vary with respect to density and velocity by approximately √휌 푢3 (this can be seen in many of the other similarly derived correlations examples in Appendix B: Supplemental Algorithms). By using this velocity distribution and correlating the equation Eq. 65 to

28 Aero-thermal Demise

experimental data from hypersonic shock tubes, the equation for the stagnation point heating flux isRef. 4,

3.15 17600 휌∞ 푈∞ ℎ푠 − ℎ푏 푞푠 = √ ( ) ( ) Eq. 67 √푅ℎ 휌푠푙 푈푟푒푓 ℎ푠푙 − ℎ푟푒푓

This is for units of Btu/ft2-sec which can be converted by multiplying by a factor of 11,364 to W/m2. The reference enthalpy is the enthalpy at 300 K. The equation is accurate ± 10 % over a range of 7,000 to 25,000 fps from sea level to 250,000 ft (2,134 to 7,620 m/s from sea level to 76,200 m).

A plot of experimental data versus the correlation can be seen in Figure 9. Equation Eq. 67 is Eq. (2) in the plot. These are the results from a shock tube experiment using air by Avoco Research Laboratory to measure the stagnation point heat transfer rate. The experiments simulated three flight altitudes of roughly 111,00 to 127,000 ft; 64,000 to 80,000 ft; and 11,000 to 31,000 ft (33,833 to 38,710 m; 19,510 to 24384 m; and 3,353 to 9,449 m)Ref. 4.

T. M. Owens 29

Figure 9: Stagnation Point Heat Transfer Rate vs. Flight Velocity (Ref. 4)

Some similarly derived correlations from the Fay-Riddell theory can be found in Appendix B: Supplemental Algorithms, Alternate Correlations. The reason for choosing this particular correlation over the others is that it is directly relates to the work performed by Fay and Riddell, as well as that used in the Reentry Hazard Analysis HandbookRef. 24, upon which the stagnation point heating algorithm is in part based.

3.2.1 Radiation Heat Balance

The other major source of heating is radiation through emission. The radiative cooling is accounted for by the Stefan-Boltzmann law assuming a lumped-mass

30 Aero-thermal Demise

node. Like the stagnation point heating flux, the radiation energy flux is in units of W/m2.

4 푞푟푎푑 = 휀휎푠푏푐푇 Eq. 68

The cold-wall heat flux is averaged over the surface of the debris fragment by the fraction of instantaneous cold-wall flux at the stagnation point as seen in equation Ref. 8 Eq. 69 . This fraction, the area averaging factor (0 < 푘2 < 1), is assumed to have a value of 0.12 for a reasonable match to past data of tumbling reentry debrisRef. 24. For composites like graphite reinforced epoxy, this value can be set to 0.8 for a more accurate, though less conservative, mass loss rateRef. 8.

푞푟푎푑 = 푘2푞푠 Eq. 69

This can then be put into the heat energy balance or net heat flow equation. The heat input less the heat output is equal to the heat absorbed.

4 푄̇ = (푘2푞푠 − 휀푏휎푠푏푐푇푏 )퐴푤 Eq. 70

The wall temperature, because of the lumped-mass assumption, is the temperature of the body. This is found by using the following,

푄 , 푓표푟 푄 < 푚 퐶̅ 푇 ̅ 푏 푝푏 푚푒푙푡 푚푏퐶푝푏 푇푏 = Eq. 71

푇 , 푓표푟 푄 ≥ 푚 퐶̅ 푇 { 푚푒푙푡 푏 푝푏 푚푒푙푡

T. M. Owens 31

4 Algorithm

The following algorithm has been implemented in MATLAB code. See Error! Reference source not found.. Coordinate transforms and other supplemental algorithms can be found in Appendix B: Supplemental Algorithms. Coordinate transformations are modified from the function libraries outlined in the Joint Advanced Range Safety System Mathematics and Algorithms documentRef. 12 .

4.1 Earth Model

The model of earth employed was developed from the Department of Defense (DoD) World Geodetic System 1984 (WGS84). The WGS84 earth model was created by the National Imagery and Mapping Agency (NIMA) in order to define a common, simple and accessible 3-dimensional coordinate system. The WGS84 also has a method for finding gravity using ellipsoidal zonal harmonicsRef. 14.

Table 1 is a collection of the derived geometric constants. The values were obtained through precise GPS ephemeris estimation process. The method can potentially be used for other planetary bodies if the geometric constants are known. These are set in the createEarth function.

Table 1: WGS84 Ellipsoid Derived Geometric Constants

Variable Value Units Description

풂⊕ 6378137.0 푚 Semi-major axis

풃⊕ 6356752.3142 푚 Semi-minor axis

풇 1/298.257223563 Flattening

−2 풆⊕ 8.1819190842622 ∙ 10 First Eccentricity

ퟐ −3 풆⊕ 6.69437999014 ∙ 10 First Eccentricity Squared

32 Aero-thermal Demise

−2 풆⊕́ 8.2094437949696 ∙ 10 Second Eccentricity

ퟐ −3 풆⊕́ 6.73949674228 ∙ 10 Second Eccentricity Squared

푚2 흁 3.986004418 ∙ 1014 Gravitational Constant 푠2

흎 7292115 ∙ 10−11 Angular Velocity

−3 푱ퟐ 1.082629989 ∙ 10 Second Degree Zonal Harmonic

−6 푱ퟑ −2.53881 ∙ 10 Third Degree Zonal Harmonic

−6 푱ퟒ −1.61 ∙ 10 Fourth Degree Zonal Harmonic

4.2 Zonal Harmonic Gravity Vector

The values from the model are used to find the acceleration due to gravity using the fourth order zonal harmonic (J4) in the getGravity MATLAB function. The zonal harmonic coefficients could extend out hundreds of terms; however, other forces like lift and drag are dominant and only the first few terms are required. The components of gravitational acceleration in ECEF coordinates areRef. 12,

2 2 2 푟 = √푒 + 푓 + 푔 Eq. 72

푒 3푎 2퐽 5푒푔2 푒 5푎 3퐽 7푒푔3 3푒푔 퐺 = −휇 [ − ⊕ 2 ( − ) − ⊕ 3 ( − ) 푒 푟3 2 푟7 푟5 2 푟9 푟7 Eq. 73 15푎 4퐽 21푒푔4 14푒푔2 푒 − ⊕ 4 ( − + )] 8 푟11 푟9 푟7

T. M. Owens 33

푓 3푎 2퐽 5푓푔2 푓 5푎 3퐽 7푓푔3 3푓푔 퐺 = −휇 [ − ⊕ 2 ( − ) − ⊕ 3 ( − ) 푓 푟3 2 푟7 푟5 2 푟9 푟7 Eq. 74 15푎 4퐽 21푓푔4 14푓푔2 푓 − ⊕ 4 ( − + )] 8 푟11 푟9 푟7

푔 3푎 2퐽 5푔3 3푔3 푎 3퐽 35푔4 30푔2 3 퐺 = −휇 [ − ⊕ 2 ( − ) − ⊕ 3 ( − + ) 푔 푟3 2 푟7 푟5 2 푟9 푟7 푟5 Eq. 75 5푎 4퐽 63푔5 70푔3 15푔 − ⊕ 4 ( − + )] 8 푟11 푟9 푟7

Of import, the ECEF components of gravity can be used to find the magnitude of the local gravity,

2 2 2 푔 = √퐺푒 + 퐺푓 + 퐺푔 Eq. 76

34 Aero-thermal Demise

4.3 Atmospheric Model

The atmosphere is modeled using the 1976 U.S. Standard AtmosphereRef. 14. The model is built from experimental rocket data and theory for the mesosphere and lower thermosphere as well as satellite data. The model is fit to the mean of a range of solar activity and weather conditions.

The MATLAB function is based atmo.f90 Fortran code from Public Domain Aeronautical Software (PDAS)Ref. 3. It is broken into two parts, a lower atmosphere algorithm for below 86 km and an upper atmosphere portion that is valid from 86 to 1000 km with constant values for altitudes above 1000 km. The atmospheric model is queried throughout the impact integrator to get the current state.

4.3.1 Lower Atmosphere

The first part of the atmospheric model is a table of properties at various altitude bands.

Table 2: Local Arrays (1976 Std. Atmosphere)

Altitude [km] Temperature [K] Pressure [ATM] Gradient

0.000 288.150 1.0 -6.5

11.000 216.650 0.2233611 0.0

20.000 216.650 0.05403295 1.0

32.000 228.650 8.5666784 ∙ 10−3 2.8

47.000 270.650 1.0945601 ∙ 10−3 0.0

51.000 270.650 6.6063531 ∙ 10−4 -2.8

71.000 214.650 3.9046834 ∙ 10−5 -2.0

84.852 186.946 3.68501 ∙ 10−6 0.0

T. M. Owens 35

Next, the geometric altitude is converted to geopotential altitude,

푧푅⊕ 푧푔 = Eq. 77 푧 + 푅⊕

The values from the row of Table 2 matching the altitude less than or equal to the geodetic altitude will be used; indicated by subscript푡. First, the local temperature is found by the following with values from the table having subscript t,

푇∞ = 푇푡 + 퐺푡(푧푔 − 푧푡) Eq. 78

Next, the local pressure,

−퐺푀∙(푧푔−푧푡) 푇 푃푡푒푥푝 0 , 푓표푟 퐺푡 = 0 훿 = Eq. 79 퐺푀 푇 퐺푡 푃 ( 0) , 푓표푟 퐺 ≠ 0 { 푡 푇 푡

푃0 푃 = ∞ 훿 Eq. 80

Then, the local density,

훿 휌∞ = 휌0 Eq. 81 푇∞⁄푇0

Finally, the local speed of sound,

푎 = √훾푅∗푇∞ Eq. 82

4.3.2 Upper Atmosphere

The upper atmosphere table is parameters of a polynomial rather than a table to interpolate values. The temperature is the kinetic temperature.

36 Aero-thermal Demise

Table 3: Atmosphere Fit Parameters (1976 Std. Atmosphere)

Altitude 풅 풅 퐥퐨퐠 풑 퐥퐨퐠 𝝆 [km] 퐥퐨퐠 풑 풅풛 퐥퐨퐠 𝝆 풅풛 86 -0.985159 -11.875633 -0.177700 -0.177900 93 -2.225531 -13.122514 -0.176950 -0.180782 100 -3.441676 -14.394597 -0.167294 -0.178528 107 -4.532756 -15.621816 -0.142686 -0.176236 114 -5.415458 -16.816216 -0.107868 -0.154366 121 -6.057519 -17.739201 -0.079313 -0.113750 128 -6.558296 -18.449358 -0.064668 -0.090551 135 -6.974194 -19.024864 -0.054876 -0.075044 142 -7.333980 -19.511921 -0.048264 -0.064657 150 -7.696929 -19.992968 -0.042767 -0.056087 160 -8.098581 -20.513653 -0.037847 -0.048485 170 -8.458359 -20.969742 -0.034273 -0.043005 180 -8.786839 -21.378269 -0.031539 -0.038879 190 -9.091047 -21.750265 -0.029378 -0.035637 200 -9.375888 -22.093332 -0.027663 -0.033094 250 -10.605998 -23.524549 -0.022218 -0.025162 300 -11.644128 -24.678196 -0.019561 -0.021349 400 -13.442706 -26.600296 -0.016734 -0.017682 500 -15.011647 -28.281895 -0.014530 -0.016035 600 -16.314962 -29.805302 -0.011315 -0.014330 700 -17.260408 -31.114578 -0.007673 -0.011626 800 -17.887938 -32.108589 -0.005181 -0.008265 1000 -18.706524 -33.268623 -0.003500 -0.004200

T. M. Owens 37

The pressure and density are found by evaluating the cubic polynomial for the band the geopotential altitude from which equation Eq. 77 lies. Using pressure as an example,

log 푝 − log 푝 푖 = 푡+1 푡 푧푡+1 − 푧푡

푧푔 − 푧푡 푗 = 푧푡+1 − 푧푡

푘 = 1 − 푗 Eq. 83

푝 = 푘 log 푝푡 + 푗 log 푝푡+1 푑 푑 − 푘푗(푧 − 푧 ) [푘 (푖 − log 푝 ) − 푗 (푖 − log 푝 )] 푡+1 푡 푑푧 푡 푑푧 푡+1

The kinetic temperature can be found by,

186.8673, 푓표푟 86 < 푧 ≤ 86 km 푔

2 푧푔 − 91 263.1905 + 12√1 − ( ) , 푓표푟 91 < 푧 < 110 km 19.9429 푔 푇∞ = Eq. 84

240 + 12(푧 − 110), 푓표푟 110 ≤ 푧 ≤ 120 km 푔 푔

푅⊕ + 120 1000 − (1000 − 120) exp [−0.01875(푧푔 − 120) ] , 푓표푟 110 < 푧푔 ≤ 1000 km { 푅⊕ + 푧푔

However, if the altitude is over 1000 km, the constant values are used,

푇∞ = 1000 −20 푝 = 1 ∙ 10 푝0 Eq. 85 −21 휌 = 1 ∙ 10 휌0

38 Aero-thermal Demise

4.4 Impact Integrator

In order to find the rate of change of the debris fragment or vehicle’s state in the impact trajectory, the effects of the vehicle’s aerodynamics, gravity and earth’s rotation must be taken into consideration. This is done in the body reference frame. The aerodynamic glide derivatives need to be found in order to solve the ODE and find the glide trajectory of a vehicle or fragments. The following is a summary of a general algorithm used to find those derivatives. Simplifications can be made for non-lifting bodies.

The local density and speed of sound are obtained from the atmospheric data. The local acceleration due to gravity is computed using the J4 gravity model with the getGravity function. The semi major axis, 푎⊕, is from the earth model created using the function createEarth. The induced drag factor, K, is found by evaluating the polynomial at the flight Mach. The coefficients are known from the induced drag turn model. It follows, the coefficient of lift and the coefficient of drag at the maximum lift-to-drag and smallest glide angle are found by,

퐶퐷0 퐶∗ = √ Eq. 86 퐿 퐾

∗ 퐶퐷 = 2퐶퐷0 Eq. 87

The glide coefficient of lift is found through a series of computations. The first being to find ratio of the square of the flight speed to the square of the circular orbital speed,

푉2 푉2 2 = Eq. 88 푉푐 푔0푎⊕

T. M. Owens 39

Next, the final glide flight path angle needs to be calculated. The atmospheric decay parameter or inverse of the scale height for earth, 훽, times the mean radius of earth, 푅⊕, is assumed to be 900. This is a mean value for altitudes under 120 kilometers (note, a scale height is the distance a value decreases by a factor of 푒; in the case of earth, it is roughly 8.5 km for the isothermal pressure gradient).

푉2 2 (1 − 2) 푉푐 sin 훾 = ∗ 2 2 Eq. 89 퐶퐿 푉 푉 ∗ [훽푅⊕ (1 − 2) + (2 − 2)] 퐶퐷 푉푐 푉푐

−1 ≤ sin 훾 ≤ 1 Eq. 90

The glide coefficient of lift is then computed. The radius, 푟, in this case is the magnitude of the EFG position coordinates for the vehicle.

푉2 2푚 퐶 = (푔 − ) 퐿 푟 휌푉2 cos 휑 푆 Eq. 91

The coefficient of lift is scaled by the difference in the final flight path angle and the initial flight path angle, 푑훾.

[ | |] 푠푐푎푙푒 = min 1, 푑훾 Eq. 92

∗ ∗ −퐶퐿 < 퐶퐿(1 + 푠푖푔푛(푑훾) ∙ 0.1 ∙ 푠푐푎푙푒) < 퐶퐿 Eq. 93

Next, the coefficient of lift has to be checked to see if it exceeds the structural load limit of the vehicle. If it does, the alternate formulation for the coefficient of lift is used. This is only valid for vehicles, typically the loading limit of a debris fragment will not be known so this step can be skipped.

40 Aero-thermal Demise

2 휌|퐶퐿|푉 푆 푛 = 2푔푚 Eq. 94

2푔푚 퐶 = 푠푖푔푛(퐶 )푛 퐿 퐿 푠 휌푉2 Eq. 95

The parabolic drag polar or coefficient of drag is found by,

2 퐶퐷 = 퐶퐷0 + 퐾퐶퐿 Eq. 96

During glide, there are no thrust forces so the combined aerodynamic and propulsive forces tangential and normal to the velocity vector can be found from the coefficients of lift and drag, the dynamic pressure and the mass-area.

2 퐹푇 푉 휌푆 = −퐶 Eq. 97 푚 푑 2푚

2 퐹푁 푉 휌푆 = 퐶 Eq. 98 푚 푙 2푚

The force equations are used to find the changes in velocity, flight path angle and heading.

푑푉 퐹푇 = − 푔 sin 훾 + 휔2 푟 cos 휙 (sin 훾 cos 휙 − cos 훾 sin 휓 sin 휙) 푑푡 푚 Eq. 99

푑훾 퐹 푉2 푉 = 푁 cos 휑 − 푔 cos 훾 + cos 훾 푑푡 푚 푟 Eq. 100 + 2휔푉 cos 휓 cos 휙 + 휔2 푟 cos 휙 (cos 훾 cos 휙 − sin 훾 sin 휓 sin 휙)

푑휓 퐹 sin 휑 푉2 푉 = 푁 − cos 훾 cos 휓 tan 휙 + 2휔푉(tan 훾 sin 휓 cos 휙 − sin 휙) 푑푡 푚 cos 훾 푟 Eq. 101 휔2푟 − cos 휓 sin 휙 cos 휙 cos 훾

T. M. Owens 41

The rate of change of position along the east and north axis can be found by,

푑푋 = 푉 cos 훾 cos 휓 푑푡 Eq. 102

푑푌 = 푉 cos 훾 sin 휓 푑푡 Eq. 103

The change in the state vector can then be expressed by the following matrix. The terms are the geodetic latitude, longitude, height above the ellipsoid, vehicle heading (clockwise from North), ground speed (in the X-Y plane), and vertical speed, respectively. The rate of change of the state vector must be real.

푑푌 ⁄푎⊕ 푑푡

푑푋 ⁄(푎 cos 휙) 푑푡 ⊕

푉 sin 훾

푦̇ = 푑휓 Eq. 104

푑푡

푑푉 푑훾 cos 훾 − 푉 sin 훾 푑푡 푑푡

푑푉 푑훾 sin 훾 + 푉 ∙ cos 훾 [ 푑푡 푑푡]

The differential equations expressed in 푦̇ can be solved using a differential equation solver such as MATLAB’s Runge-Kuta methods ode23 and ode45. The time span is taken from the integration limit and the initial state is the trajectory state at the malfunction time.

42 Aero-thermal Demise

The fragment debris studied in this thesis are tumbling with no defined head or tail, so there is no lift. The equations can be simplified from there more general form, however, the approach is the same.

T. M. Owens 43

4.5 Aero-thermal Demise

The aero-thermal demise algorithm follows roughly that of the algorithm outlined in the Reentry Hazard Analysis HandbookRef. 24.

4.5.1 Fragment Properties

A fragment’s mass, shape, material, and dimensions are the required properties in order to perform a demise analysis. A fragment may also have a parent fragment and will not start ablating until that parent has demised (this case was not included in the utility). An initial temperature of the fragment can also be set; for most analysis though, the reference temperature of 300 K is appropriate unless the risk analyst has knowledge of fragment preheating.

4.5.2 Material Properties

The material properties used in the demise utility are from the Debris Assessment Software 2.0’s material database. The material database can be found in Appendix A: Material Properties. The material properties that are required to perform a demise analysis are density, specific heat, heat of fusion and the melting temperature. The specific heat capacities of the fragments with a range of specific heats are taken as the mean of the range. All material properties are assumed to be constant. Each fragment is assigned a material and uses properties from the material database. It is possible to have user defined materials, but the material database should be adequate for most analyses.

44 Aero-thermal Demise

4.5.3 Shape Assumptions

The geometry of the demising fragments is simplified into one of four shapes: spheres, cylinders, flat plates and boxes. Flat plates should have a thickness less than one twentieth of the width. The box shape is approximated as an equivalent cylinder. Each of the shapes is considered to be hollow and tumbling with uniform ablation over the surface.

Table 4: Debris Fragment Shape Assumptions

Variable Sphere Cylinder Flat Plate Box

푙 > 푤 > ℎ 푙 > 푤 ≫ 푡 Misc… 푚푏 푤ℎ 푡 = 푟 = √ 휌푏푙푤 휋

2 Wetted area, Aw 4휋푟 2휋(푟 + 푙) 2푙푤 + 2푙푡 + 2푤푡 2휋푟(푟 + 푙)

Hypersonic continuum 2푟 2푟 0.92 0.720 + 0.326 ( ) 1.84 0.720 + 0.326 ( ) drag coefficient, CDhc 푙 푙

Aerodynamic 휋푟2 2푟푙 푙푤 2푟푙 reference area, S

푤 Heating radius, Rh 푟 푟 푟 2

For all cases, the hypersonic continuum ballistic coefficient is,

푚푏 훽ℎ푐 = Eq. 105 푆퐶퐷ℎ푐

T. M. Owens 45

4.5.4 Stagnation Point Heating

The formula for the specific heat capacity of airRef. 9,

1373, 푓표푟 푇 ≥ 2000 K 푏

−5 2 퐶푝 = 959.9 + 0.15377푇푏 + 2.636 ∙ 10 푇푏 , 푓표푟 300 < 푇푏 < 2000 K Eq. 106 ∞

{ 1004.7, 푓표푟 푇푏 ≤ 300 K

Adiabatic stagnation temperature,

2 푈∞ 푇푠 = 푇∞ + Eq. 107 2퐶푝∞

Heat of initial temperature is the heat required to raise the body bulk temperature from absolute zero to initial temperature. This is the heat energy present in the fragment at breakupRef. 24.

̅ 푄0 = 푚푏퐶푝푏푇0 Eq. 108

Heat of melting is the heat required to raise the body bulk temperature from the initial temperature to the melt temperature.

̅ ( ) 푄푚푒푙푡 = 푚푏퐶푝푏 푇푚푒푙푡 − 푇0 Eq. 109

Heat of ablation is the heat required to melt the entire body.

̅ ( ) 푄푎 = 푚푏퐶푝푏 푇푚푒푙푡 − 푇0 + 푚푏ℎ푓 Eq. 110

Equation Eq. 111 is the Detra-Kemp-Riddell stagnation point heating correlation whose derivation is explained in section 3.2 Detra, Kemp and Riddell Correlation. Here the reference temperature is 300 K and the reference velocity is 7924.8 m/s (26000 ft/s). Of note, 0.3048 is the reference radius of 1 foot converted to meters.

46 Aero-thermal Demise

Note that the ratio of enthalpies has been converted to a ratio of temperatures to simplify the equation as all enthalpies are the enthalpy of air.

3.15 0.3048휌∞ 푇푠 − 푇푏 푈∞ 푞푠 = 200006400√ ( ) ( ) Eq. 111 푅ℎ휌푠푙 푇푠 − 푇푟푒푓 푈푟푒푓

For the net het heat flow equation, the surface emissivity is taken to be one because the emissivity approaches unity after a quick char build-up. The variable k2, the area averaging factor, is taken to be 0.12. For composites, a value of 0.8 can be used to account for a greater mass loss rate, but for the purposes of simplification and conservancy, the value of 0.12 is used for all materialsRef. 8.

4 푄̇ = (푘2푞푠 − 휀푏휎푠푏푐푇푏 )퐴푤 Eq. 112

Heat content of the fragment body at time t,

푡 푄(푡) = 푄0 + ∫ 푄̇ 푑푡 Eq. 113 0

Body bulk temperature, the right hand side of the inequality is the heat of melting from the initial temperature of absolute zero.

푄 , 푓표푟 푄 < 푚 퐶̅ 푇 ̅ 푏 푝푏 푚푒푙푡 푚푏퐶푝푏 푇푏 = Eq. 114

푇 , 푓표푟 푄 ≥ 푚 퐶̅ 푇 { 푚푒푙푡 푏 푝푏 푚푒푙푡

The body bulk temperature is put back into the equation for stagnation point heating and then iterated until the error in the net heat flux reaches an acceptable level. There is a discontinuity in the solution as the stagnation temperature approaches the reference temperature; this does not affect the result of the analysis as it occurs in the region of aero-cooling.

T. M. Owens 47

4.5.5 Liquid Fraction

Liquid fraction is a measure of the fraction of debris that has melted. If the maximum amount of heating is less than the heat of melting, none of the fragment can be liquid. If the maximum amount of heating is greater than the heat of ablation, the entire fragment is considered liquid and fully demisedRef. 24.

0, 푓표푟 푄 ≤ 푄 푚푎푥 푚푒푙푡

푄푚푎푥 − 푄푚푒푙푡 퐿퐹 = , 푓표푟 푄푚푒푙푡 < 푄푚푎푥 < 푄푎 Eq. 115 푄푎 − 푄푚푒푙푡

{ 1, 푓표푟 푄푚푎푥 ≥ 푄푎

4.5.6 Fragment Tables

4.5.6.1 Mass Table

The mass of the demised fragment at a given time can be found simply by,

푚퐿퐹 = 푚푏(1 − 퐿퐹) Eq. 116

Any liquid part of debris fragment is considered to be blown away and not counted in the mass of the fragment. This is typical of other demise analysis tools such as SCARABRef. 11.

4.5.6.2 Area Table

To recalculate the aerodynamic reference area table, the utility uses the MATLAB fzero function. The function is a combination of bisection, secant and inverse quadratic interpolation methods. Alternatively, a Newtonian search method could be employed. The functions find the reduction in thickness of hollow bodies or the recession length of flat plates.

48 Aero-thermal Demise

For the sphere, just what would be ‘interior’ mass needs to be calculated before using the time varying liquid fraction mass to get the time varying areas.

4휋푟3 푚 = 휌 − 푚 Eq. 117 𝑖 3

2 ⁄3 3(푚퐿퐹 + 푚𝑖) 푆 = 휋 ( ) Eq. 118 4휋휌

For the cylinder case, the interior mass is found. It is then used to obtain the initial thickness by solving the next equation for 푡. Then, the internal dimensions are used in the next equation solved for 푡 with the time varying demised mass. The thicknesses are then used in the time varying area calculation.

2 푚𝑖 = 휋푟 푙휌 − 푚 Eq. 119

푚𝑖 0 = −푡3(2푟푙)푡2 − (푟2 + 2푟푙)푡 + 푟2푙 − 휋휌 Eq. 120

푟𝑖 = 푟 − 푡 Eq. 121

푙𝑖 = 푙 − 푡 Eq. 122

푚퐿퐹 0 = 푡3(2푟 푙 )푡2 + (푟 2 + 2푟 푙 )푡 + 푟 2푙 − 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 휋휌 Eq. 123

푆 = 2(푟𝑖 + 푡)(푙𝑖 + 푡) Eq. 124

In the flat plate case, the recession length, 훿, is solved by finding the zeroes of the function with the time varying mass. It is then used in the area calculation.

푚퐿퐹 0 = 훿2 − 훿(푙 + 푤) + 푙푤 − 푡휌 Eq. 125

T. M. Owens 49

푆 = (푙 − 훿)(푤 − 훿) Eq. 126

Solving for the box area starts with finding the interior mass and using that in the next equation being solved for 푡. This is used to find the interior dimensions. Then solving for 푡 with the liquid fraction masses the time varying thickness can be used to find the area.

푚𝑖 = 푙푤ℎ휌 − 푚 Eq. 127

푚𝑖 0 = −푡3(푙 + 푤 + ℎ)푡2 − (푙ℎ + 푤ℎ + 푙푤)푡 + 푙푤ℎ − 휌 Eq. 128

푙𝑖 = 푙 − 푡 Eq. 129

푤𝑖 = 푤 − 푡 Eq. 130

ℎ𝑖 = ℎ − 푡 Eq. 131

푚퐿퐹 0 = 푡3(푙 + 푤 + ℎ )푡2 + (푙 ℎ + 푤 ℎ + 푙 푤 )푡 + 푙 푤 ℎ − 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 휌 Eq. 132

(푤𝑖 + 푡)(ℎ𝑖 + 푡) 푆 = (푙 + 푡)√ Eq. 133 𝑖 휋

4.5.6.3 Hazard Area Table

From the new aerodynamic reference areas the hazard radius is calculated by the equation,

푆 퐻 = √ Eq. 134 푟 휋

50 Aero-thermal Demise

If the user supplied a hazard radius in the fragment creator, then the hazard radius will be calculated using the maximum from that input and adjusting it proportionally to the demise aerodynamic reference area. This allows the user to force a smaller or larger hazard area using the following equation,

푆 퐻 = 푚푎푥{퐻 }√ 푟 푟0 푚푎푥{푆} Eq. 135

4.5.6.4 Count Table

The count table is only adjusted if the fragment fully demises, in which case the time when the liquid fraction equals one the count is set to zero.

4.5.6.5 Drag Table

The drag table for demised is created to simulate the ballistic coefficient of a fragment with time varying mass and area. Because the fragment tables are built with only the impact mass and area, the uncorrected ballistic coefficient can lead to error in impact prediction. This is not a significant issue in reentry trajectories where the majority of fragments of interest are breaking up at lower altitudes with lower velocities (under Mach 10 the ballistic coefficients are roughly constant); however, with a launch trajectory with overflight risk, the ballistic coefficient corrections become necessary.

The corrected 퐶퐷 table is only calculated for the fragment breakup at the highest altitude. This approximation correlates nicely across all breakup times as the flight Mach of the fragment is roughly proportional to altitude.

The first step in the calculation is to use the existing drag table for the fragment and find the coefficient of drag vs. Mach for the demised fragment. If the Mach is higher than the highest in the drag table, the 퐶퐷 associated with the highest Mach

T. M. Owens 51

in the table is used, and vice versa for Mach lower than those in the table. The equation below is used to find the ballistic coefficient as a function of Mach with time varying mass and area.

푚푏 β푀푎푐ℎ = Eq. 136 푆퐶퐷푀푎푐ℎ

This is then used with the impact mass and area of the fragment to generate the corrected Mach dependent coefficient of drag values.

푚𝑖푚푝푎푐푡 퐶퐷훽 = Eq. 137 푆𝑖푚푝푎푐푡β푀푎푐ℎ

The drag table is trimmed so that the repeated fragment descent Mach values at high altitudes are excluded.

The reasoning for the drag table correction is probably best illustrated in Figure 10 generated using the Single Debris Field tool in JARSS MP and plotted in FSACAD (Flight Safety Analyst CAD). The green line is the vacuum impact trace from a rocket launch with European overflight risk. The blue boxes represent the impacts without the drag coefficient corrected and the red boxes are with the corrected drag tables. As the trajectory reaches near orbital speeds, it is obvious there is a significant difference in impact prediction. Because of the mass loss in the demising debris, the ballistic coefficient is adjusted lower and the impact point is not as far down range as the uncorrected case.

52 Aero-thermal Demise

Figure 10: Corrected (red) and Uncorrected (blue) Cd Impacts

T. M. Owens 53

5 Results

The following are the results from the computational model outlined in this thesis. The first section will explain how changing breakup conditions and fragment properties can affect the aero-thermal demise. The second is a comparison against the established reentry debris analysis tools DAS 2.0 and the Aerospace Survivability Tables.

5.1 Understanding Aero-heating

The plots generated for this section are included in an effort to help with the understanding of the mechanism of aero-heating and how the computational model simulates them. A reentering body begins to interface with the atmosphere at about 122 km with aero-heating starting at 80 km and aero-cooling after about 50 km. A fragment’s liquid fraction or demise is affected by many different parameters such as initial temperature, breakup altitude or time, material properties, geometry, aerodynamic stability, etc. A few of these will be examined in this section. Some simplifications have to be made for analysis, but the results of that analysis should be consistent with empirical data and other validated methods.

5.1.1 Reentry Trajectory, Heat Flux, and Bulk Temperature

The initial conditions for the following plots are a break up altitude of 120 km over the intersection of the prime meridian and equator with a flight path angle of -0.5 degrees, a 0 degree heading, a velocity of 7600 m/s and a debris starting temperature of 300 K.

For these test cases, an example debris fragment that is known to experience partial demise at a variety of initial breakup conditions was used.

54 Aero-thermal Demise

Table 5: Example Fragment

Name Shape Radius [m] Length [m] Material Aluminum 2.6 Cylinder 0.85344 0.85344 (generic)

Mach Cd Time [s] Mass [kg] Area [m2] Beta [kg/m^2] Hazard Radius [m] 1 0.81 0 50 1.45672 42.37493 0.1215 0.9999 0.45

Figure 11 shows the altitude versus time and Figure 12 shows the altitude versus range as it is calculated by the impact integrator. From the figures, it is evident that below about the 80 km mark, where aero-heating is highest, the most significant drag is experienced. The fragment reaches a range of about 3100 km before falling on a nearly vertical trajectory. Ground impact is at about 1200 seconds, just off the end of the plot.

T. M. Owens 55

Altitude [km] vs. Time [s] 120

100

60 Altitude [km] 40

0 0 200 400 600 800 1000 Time [s]

Figure 11: Single Fragment, Altitude vs. Time

Altitude [km] vs. Range [km] 120

100

60 Altitude [km] 40

0 0 500 1000 1500 2000 2500 3000 3500 4000 Range [km]

Figure 12: Single Fragment, Altitude vs. Range

56 Aero-thermal Demise

Figure 13 shows the heat flux versus time. The figure captures both the aero- heating and aero-cooling regimes. The slight bump in the aero-heating regime is from the switch between upper and lower altitude models at 86 km.

The calculation is stopped when the velocity is less than 2,134 m/s. This keeps the heat flux calculation within the valid range of the stagnation point heating correlation from section 3.2 Detra, Kemp and Riddell Correlation. In this case the computation is stopped at the altitude of about 53 km.

5 x 10 Heat Flux [W/m2-s] vs. Time [s] 3

2.5

1.5

0.5

0 Heat Flux [W/m2-s] -0.5

-1

-1.5

-2 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 13: Single Fragment, Heat Flux vs. Time

Figure 14 shows the bulk temperature of the fragment through time. In this case, it is below the melting point at all times. As the fragment descends further cooling than what is shown in the plot would take place.

T. M. Owens 57

Fragment Bulk Temperature [K] vs. Time [s] 900

800

700

600

500 Fragment Bulk Temperature [K] 400

300 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 14: Single Fragment, Temperature vs. Time

5.1.2 Varying Breakup Altitude

Now, to see the effects varying the breakup altitude, the same debris fragment was given an initial breakup 120 and 50 km altitude in 10 km steps. Figure 15 and Figure 16 do not reveal unexpected results. The debris fragments with initial states at lower altitudes impact and decelerate more quickly in the denser atmosphere. This particular fragment, because of its relatively low overall density as a hollow cylinder, has a roughly vertical trajectory after 40 km of altitude.

58 Aero-thermal Demise

Altitude [km] vs. Time [s] 120

100

60 Altitude [km] 40

0 0 200 400 600 800 1000 Time [s]

Figure 15: Varying Breakup Altitude, Altitude vs. Time

Altitude [km] vs. Range [km] 120

100

60 Altitude [km] 40

0 0 500 1000 1500 2000 2500 3000 3500 4000 Range [km]

Figure 16: Varying Breakup Altitude, Altitude vs. Range

T. M. Owens 59

The heat flux for varying altitudes is shown in Figure 17. These results are slightly counterintuitive for an actual reentry. At the lower altitudes, the flight speed would not be expected to be near orbital. As such, the low altitude breakup conditions have an immediate peak heat flux and then cool rapidly in the denser atmosphere.

5 x 10 Heat Flux [W/m2-s] vs. Time [s] 5 120 km 110 km 4 100 km 90 km 80 km 3 70 km 60 km 50 km 2

1 Heat Flux [W/m2-s] 0

-1

-2 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 17: Varying Breakup Altitude, Heat Flux vs. Time

The bulk temperature of the fragment versus time in Figure 18 shows how the first four breakup altitudes, 120, 110, 100 and 90 km, all have a similar peak temperature. They are all below the melting temperature of 850 K, so no fragments are fully demised. The temperature plot, if below the melting point, does not tell us much about the demise of fragments. For that, the liquid fraction is the best indicator.

60 Aero-thermal Demise

Fragment Bulk Temperature [K] vs. Time [s] 900

800

700

600

120 km 500 110 km 100 km

Fragment Bulk Temperature [K] 90 km 80 km 400 70 km 60 km 50 km 300 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 18: Varying Breakup Altitude, Temperature vs. Time

The liquid fraction is the single quantity of the aero-thermal demise calculations that is used to adjust the mass and dimensions of the fragment as it goes through the impact integration. Figure 19 shows the liquid fraction over time for each of the breakup altitudes. The high breakup altitude cases experience about 85% mass loss or ablation, and the two low altitudes, 50 and 60 km, have no ablation.

T. M. Owens 61

Liquid Fraction vs. Time [s] 1

0.9

0.8

0.7

0.6

0.5

0.4 Liquid Fraction 120 km 110 km 0.3 100 km 90 km 0.2 80 km 70 km 0.1 60 km 50 km 0 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 19: Varying Breakup Altitude, Liquid Fraction vs. Time

5.1.3 Varying Initial Temperature of Debris Fragment

Typically, the initial temperature of a debris fragment in aero-thermal demise is set to 300 K unless preheating is known. Preheating could be from an external component of a vehicle that has broken up or from a child of a parent fragment that has broken into pieces. A temperature range from 300 to 600 K with 100 K steps for the debris fragment is used. The breakup altitude is set at 100 km. The altitude and range plots for each case are nearly overlapping, so there is no need to compare them for the case of varying initial debris temperature.

Looking at the temperature in Figure 20, it is clear that several of the fragments with a higher initial temperature reach the melting temperature of 850 K for aluminum. In Figure 21, the three higher temperature cases, 400, 500 and 600 K, all reach the liquid fraction of 1 and are fully demised. The 600 K case demises almost immediately; the curve is only just visible at the upper left of the figure.

62 Aero-thermal Demise

Fragment Bulk Temperature [K] vs. Time [s] 900

800

700

600

500 Fragment Bulk Temperature [K] 400

300 0 50 100 150 200 250 300 Time [s]

Figure 20: Varying Temperature, Temperature vs. Time

Liquid Fraction vs. Time [s] 1

0.9

0.8

0.7

0.6

0.5

0.4 Liquid Fraction

0.3

0.2 300 K 400 K 0.1 500 K 600 K 0 0 50 100 150 200 250 300 Time [s]

Figure 21: Varying Temperature, Liquid Fraction vs. Time

T. M. Owens 63

Figure 22 shows how the initial temperature affects the heat flux. Counterintuitively, the lower initial temperature debris fragment has the highest curve and peak. This is due to the radiative cooling effect in the upper atmosphere before significant aero-heating is encountered. The heat flux plot would suggest that the fragments with lower initial temperature. However, the heat energy in the body of the high initial temperature fragments lowers the required heat energy to ablate the fragment.

5 x 10 Heat Flux [W/m2-s] vs. Time [s] 3

2.5

1.5

0.5

0 Heat Flux [W/m2-s] -0.5

-1 300 K 400 K -1.5 500 K 600 K -2 0 50 100 150 200 250 300 Time [s]

Figure 22: Varying Temperature, Heat Flux vs. Time

5.1.4 Varying Initial Velocity

Another important factor to the demise of debris fragments is the flight speed. The speed is varied from 7500 to 5500 m/s in 500 m/s steps at a breakup altitude of 100 km. The variation in flight speed has an obvious effect on the rate of altitude and cross range as seen in Figure 23 and Figure 24.

64 Aero-thermal Demise

Altitude [km] vs. Time [s] 120 7,500 km/s 7,000 km/s 6,500 km/s 100 6,000 km/s 5,500 km/s

60 Altitude [km] 40

0 0 200 400 600 800 1000 Time [s]

Figure 23: Varying Velocity, Altitude vs. Time

Altitude [km] vs. Range [km] 120 7,500 km/s 7,000 km/s 6,500 km/s 100 6,000 km/s 5,500 km/s

60 Altitude [km] 40

0 0 500 1000 1500 2000 Range [km]

Figure 24: Varying Velocity, Altitude vs. Range

T. M. Owens 65

There is nothing unexpected in the heating of the fragments shown in Figure 25, Figure 26 and Figure 27. The lower heating and liquid fraction of the slower cases is as expected. The less kinetic energy there is to dissipate as heat energy, the less ablation the debris fragment will experience. The 7000 and 6500 m/s cases have a higher and earlier peak than the 7500 m/s case due to reaching the more dense atmosphere more quickly in the impact trajectory.

5 x 10 Heat Flux [W/m2-s] vs. Time [s] 3

2.5

1.5

0.5

0 Heat Flux [W/m2-s] -0.5 7,500 km/s -1 7,000 km/s 6,500 km/s -1.5 6,000 km/s 5,500 km/s -2 0 50 100 150 200 250 300 Time [s]

Figure 25: Varying Velocity, Heat Flux vs. Time

66 Aero-thermal Demise

Fragment Bulk Temperature [K] vs. Time [s] 900

800

700

600

500

Fragment Bulk Temperature [K] 7,500 km/s 7,000 km/s 400 6,500 km/s 6,000 km/s 5,500 km/s 300 0 50 100 150 200 250 300 Time [s]

Figure 26: Varying Velocity, Temperature vs. Time

Liquid Fraction vs. Time [s] 1

0.9

0.8

0.7

0.6

0.5

0.4 Liquid Fraction

0.3 7,500 km/s 0.2 7,000 km/s 6,500 km/s 0.1 6,000 km/s 5,500 km/s 0 0 50 100 150 200 250 300 Time [s]

Figure 27: Varying Velocity, Liquid Fraction vs. Time

T. M. Owens 67

5.1.5 Varying Flight Path Angle

The other important part of the initial state is the flight path angle. This does not have as large an effect on the heating of the fragment as changing the altitude or the velocity as the net energy in the trajectories is roughly similar. It does, however, have a significant effect on the altitude versus time and cross range as seen in Figure 28 and Figure 29. The flight path angles are -0.5, -3.0 and -5.5 degrees. The shallower angle would be typical of a gliding reentry, and the steeper angle would be for a capsule return from deep space or the moon.

Altitude [km] vs. Time [s] 120 -0.5 deg -3.0 deg 100 -5.5 deg

60 Altitude [km] 40

0 0 200 400 600 800 1000 Time [s]

Figure 28: Varying Flight Path Angle, Altitude vs. Time

68 Aero-thermal Demise

Altitude [km] vs. Range [km] 120 -0.5 deg -3.0 deg 100 -5.5 deg

60 Altitude [km] 40

0 0 500 1000 1500 2000 2500 Range [km]

Figure 29: Varying Flight Path Angle, Altitude vs. Range

In Figure 30, the steeper flight path angle trajectory gives a higher and earlier peak to the heat flux. In Figure 31 and Figure 32 it can be seen that the only significant effect of the change in flight path angle is the earlier ablation for the steeper flight angles. The liquid fraction is within 10% for the three cases. The steeper flight path angle experiences less mass loss because it does not spend as much time in the flight regime where aero-heating is dominant.

T. M. Owens 69

5 x 10 Heat Flux [W/m2-s] vs. Time [s] 5

1 Heat Flux [W/m2-s] 0

-1 -0.5 deg -3.0 deg -5.5 deg -2 0 50 100 150 200 250 300 Time [s]

Figure 30: Varying Flight Path Angle, Heat Flux vs Time

Fragment Bulk Temperature [K] vs. Time [s] 900

800

700

600

500 Fragment Bulk Temperature [K] 400 -0.5 deg -3.0 deg -5.5 deg 300 0 50 100 150 200 250 300 Time [s]

Figure 31: Varying Flight Path Angle, Temperature vs. Time

70 Aero-thermal Demise

Liquid Fraction vs. Time [s] 1

0.9

0.8

0.7

0.6

0.5

0.4 Liquid Fraction

0.3

0.2 -0.5 deg 0.1 -3.0 deg -5.5 deg 0 0 50 100 150 200 250 300 Time [s]

Figure 32: Varying Flight Path Angle, Liquid Fraction vs. Time

5.1.6 Varying Materials of Debris Fragment

For the previous examples aluminum was used as it is very common in spacecraft construction that will often experience partial demise. Other materials such as titanium almost never demise, whereas composites will demise very rapidly. To show this, the example fragment is set with the same properties as in 5.1.1 for aluminum, titanium and graphite reinforced epoxy materials. From the heat flux in Figure 33, it can be seen that material properties have a significant effect on the stagnation point heating.

T. M. Owens 71

5 x 10 Heat Flux [W/m2-s] vs. Time [s] 3 Aluminum 2.5 Titanium Graphite-Epoxy 2

1.5

0.5

0 Heat Flux [W/m2-s] -0.5

-1

-1.5

-2 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 33: Varying Materials, Heat Flux vs. Time

Fragment Bulk Temperature [K] vs. Time [s] 1000 Aluminum Titanium 900 Graphite-Epoxy

800

700

600

500 Fragment Bulk Temperature [K]

400

300 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 34: Varying Materials, Temperature vs. Time

72 Aero-thermal Demise

The liquid fraction in Figure 35 is as expected. The aluminum debris fragment partially demises as in earlier examples. The titanium fragment experiences no significant ablation and the composite fragment demises fully. The graphite reinforced epoxy reaches its melting point so it will have fully demised. The composite fragment also starts demising earlier in the impact trajectory as less heat energy is required to ablate it when compared to aluminum.

Liquid Fraction vs. Time [s] 1 Aluminum 0.9 Titanium Graphite-Epoxy 0.8

0.7

0.6

0.5

0.4 Liquid Fraction

0.3

0.2

0.1

0 0 50 100 150 200 250 300 350 400 450 500 Time [s]

Figure 35: Varying Materials, Liquid Fraction vs. Time

5.1.7 Varying Mass of Debris Fragment

The example fragment being used throughout this section is a thin walled cylinder, so by increasing the mass of the fragment, several different effects on the impact trajectory and demise can be seen. The cases examined are 1, 2, 3, 5 and 8 times the mass of the original fragment. In Figure 36 and Figure 37, the effects of the increase in the ballistic coefficient of the higher mass fragments is that they have a

T. M. Owens 73

larger range to impact and impact earlier as they carry more speed being less effected by aerodynamic forces.

Altitude [km] vs. Time [s] 120 50 kg 100 kg 150 kg 100 250 kg 400 kg

60 Altitude [km] 40

0 0 200 400 600 800 1000 Time [s]

Figure 36: Varying Mass, Altitude vs Time

74 Aero-thermal Demise

Altitude [km] vs. Range [km] 120 50 kg 100 kg 150 kg 100 250 kg 400 kg

60 Altitude [km] 40

0 0 500 1000 1500 2000 2500 3000 Range [km]

Figure 37: Varying Mass, Altitude vs. Range

The peak heat flux as shown in Figure 38 for the larger fragments is also much higher than that of the lighter fragments. As can be seen in Figure 39 and Figure 40, this does not cause the fragment to have a higher bulk temperature or more mass loss. The mass of the fragment requires more energy to heat up than the increase in heat flux accounts for with these fragment properties. Another aspect of the higher ballistic coefficient fragments is they experience peak aero-thermal heating later in the impact trajectory. The lowest mass fragment has peak heating at about 80 km, whereas the heaviest at 70 km. This is due to their ability to carry speed into the denser atmosphere.

T. M. Owens 75

5 x 10 Heat Flux [W/m2-s] vs. Time [s] 10 50 kg 100 kg 150 kg 8 250 kg 400 kg

Heat Flux [W/m2-s] 2

-2 0 50 100 150 200 250 300 350 400 Time [s]

Figure 38: Varying Mass, Heat Flux vs. Time

Fragment Bulk Temperature [K] vs. Time [s] 1000 50 kg 100 kg 900 150 kg 250 kg 400 kg 800

700

600

500 Fragment Bulk Temperature [K]

400

300 0 50 100 150 200 250 300 350 400 Time [s]

Figure 39: Varying Mass, Temperature vs. Time

76 Aero-thermal Demise

Liquid Fraction vs. Time [s] 1 50 kg 0.9 100 kg 150 kg 0.8 250 kg 400 kg 0.7

0.6

0.5

0.4 Liquid Fraction

0.3

0.2

0.1

0 0 50 100 150 200 250 300 350 400 Time [s]

Figure 40: Varying Mass, Liquid Fraction vs. Time

T. M. Owens 77

5.2 Model Comparisons

Only DAS 2.0 and the Aerospace Survivability Tables were chosen for a comparison to the demise model developed in this thesis. Both are freely available and simple to use. Other more detailed comparisons to tools like ORSAT would have to be done before being considered for operational acceptance.

5.2.1 DAS 2.0

DAS 2.0 sets the initial breakup at 78km for all of the first order subcomponents. This is considered the most likely altitude for aero-breakup that will give a conservative estimate to reentry debris survivabilityRef. 16. If a demising component has its own subcomponents, those begin demising once the parent fragment is fully demised. To compare the demise between the computational model and DAS 2.0, four fragments were chosen from the example mission, the properties of which are shown in Table 6.

Table 6: DAS 2.0 Debris Fragments

Name Material Body Type Mass [kg] Diameter Length [m] Height [m] Demise Alt /Width [m] [km] Bottom Panel Graphite Epoxy 1 Flat Plate 12.1 1.66 1.66 76.6

Antenna Aluminum 7075-T6 Flat Plate 6 0.2 1.1 0

Antenna Aluminum 7075-T6 Cylinder 3 0.1 4 76.1 Attachment Transponder Aluminum 7075-T6 Box 2 0.33 0.35 0.1 70.4

The results from the computational model do not exactly match the demise altitudes of DAS, but they are similar as shown in Table 7. The graphite epoxy fragment, bottom panel, survives with a liquid fraction of 0.26. The antenna ends with a 0.93 liquid fraction and the transponder with a 0.85 liquid fraction. The antenna attachment fully demises almost immediately at an altitude of 78 km.

78 Aero-thermal Demise

The difference in the flat plates between the computational model and DAS 2.0 is interesting. The low density graphite epoxy fragment does not experience much heating, however the more compact and dense aluminum antenna does and nearly demises. Testing the suggestion to change the area averaging factor to 0.8 for composites makes the graphite epoxy fragment fully demise at 78 km.

Table 7: Computational Model Debris Fragments, Compared to DAS 2.0

Name Material Area Liquid Computational DAS 2.0 Averaging Fraction Model Demise Alt Demise Alt [km] Factor 퐤ퟐ [km] Bottom Panel Graphite Epoxy 1 0.12 0.26 0 76.6

Bottom Panel Graphite Epoxy 1 0.8 1.0 78.0 76.6

Antenna Aluminum 7075-T6 0.12 0.93 0 0

Antenna Attachment Aluminum 7075-T6 0.12 1.0 78.8 76.1

Transponder Aluminum 7075-T6 0.12 0.85 0 70.4

Overall the results from the computational model are more conservative than those from DAS 2.0. A flight safety analyst may make the judgment that the two fragments that have less than 15% of their original mass may be considered to fully demise because of loss of structural integrity, bringing the results more in line with DAS 2.0 predicting all fragments demise before impact.

5.2.2 Aerospace Survivability Tables

The computational model outlined in this thesis is in part based on the algorithm that builds the Aerospace Survivability TablesRef. 24. The atmosphere, earth model and impact integrator are all different. The computational model also using time varying properties for the fragments as it integrates to impact, whereas the survivability tables use static properties from the initial breakup. Also, being less conservative means a greater potential for reduction in casualty estimation.

T. M. Owens 79

Table 8 and Table 9 make a comparison between the computational model and the survivability tables across a broad range of fragment properties. The fragment is an aluminum cylinder weighing from 5 to 5000 lbs with a length between 1 to 30 ft and radius from 0.1 to 5 ft. All impacts were computed for 42 Nmi of altitude, -0.5 degrees flight path angle and a velocity of 25,000 ft/s. This is roughly equivalent to the 78km breakup altitude case used by DAS 2.0. The tables are such that only certain combinations of initial state and fragment properties are available.

The upper set of tables with the green-red gradient has the liquid fraction as computed in the model and the liquid fraction from the survivability tables with red indicating debris survival to impact and green demise. The blank spots in the table are for fragments that the survivability tables consider physically infeasible. The lower red-blue gradient plots are the difference between the two data sets. There is a significant difference between the two sets of liquid fractions. This is not particularly troubling as the survivability tables are a conservative estimate and use a very different impact integration method. The trend between the two sets of liquid fractions as seen in the upper plot matches rather well however.

A previous simpler impact integrator used with the computational model that did not take into account the mass varying properties of the debris fragment had a better match to the survivability tables which also use a static fragment. It was, however, very poor at predicting the actual point of impact so the new method is preferable even with the poor match to the survivability tables. Also being less conservative means that there is a greater potential for casualty reduction in the risk analysis.

80 Aero-thermal Demise

Table 8: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder

W of 5 lb W of 10 lb W of 25 lb r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30

0.1 1.0 1.0 1.0 1.0 1.0 1.0 0.0 1.0 1.0 1.0 1.0 1.0 0.0 0.0 1.0 1.0 1.0 1.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 1.0 0.9 0.8 1.0 0.5 0.5 1.0 1.0 0.9 0.8 1.0 0.6 1.0 1.0 1.0 1.0 0.9 0.8 1.5 0.7 0.7 0.6 0.5 0.3 0.0 0.7 0.7 0.7 0.6 0.5 0.5 0.8 0.8 0.7 0.7 0.6 1.0 2 0.6 0.5 0.5 0.3 0.1 0.0 0.6 0.6 0.5 0.5 0.3 0.2 0.7 0.6 0.6 0.6 1.0 0.4 3 0.4 0.3 0.2 0.1 0.0 0.0 0.4 0.4 0.3 0.3 0.1 0.0 0.5 0.5 0.4 0.4 0.3 0.2 4 0.2 0.1 0.0 0.0 0.0 0.0 0.3 0.3 0.2 0.1 0.0 0.0 0.4 0.3 0.3 0.3 0.2 0.1

Model 25000 ft/s 25000 Model 5 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.1 0.1 0.0 0.0 0.0 0.3 0.2 0.2 0.2 0.1 0.0

0.1 1.0 1.0 1.0 1.0 1.0 1.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 0.3 0.3 0.2 0.4 0.3 0.3 0.3 0.2 0.4 0.4 0.3 0.3 0.3 0.2 1.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 0.0 0.0 0.0 3 4

Table 25000 ft/s 25000 Table 5 W of 250 lb W of 500 lb W of 750 lb

r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30

0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.3 0.7 1.0 1.0 1.0 1.0 0.0 0.0 0.6 1.0 1.0 1.0 0.0 0.0 0.0 0.7 1.0 1.0 1 0.4 0.6 0.8 1.0 1.0 1.0 0.1 0.3 0.6 0.8 1.0 1.0 0.0 0.2 0.4 0.6 0.9 1.0 1.5 0.5 0.6 0.7 0.8 0.8 0.8 0.2 0.4 0.5 0.7 0.8 0.8 0.1 0.2 0.4 0.6 0.7 0.8 2 0.5 0.6 0.6 0.7 0.7 0.6 0.3 0.4 0.5 0.6 0.7 0.7 0.1 0.2 0.4 0.5 0.6 0.7 3 0.5 0.5 0.5 0.5 0.5 0.5 0.3 0.4 0.4 0.5 0.5 0.5 0.2 0.3 0.3 0.4 0.5 0.5 4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.2 0.3 0.3 0.4 0.4 0.4

Model 25000 ft/s 25000 Model 5 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.3 0.3 0.3 0.3

0.5 deg 0.5 0.1 - 0.5 0.2 0.5 0.8 1.0 0.2 0.5 0.8 0.3 0.6 1 0.1 0.2 0.3 0.4 0.4 0.0 0.0 0.2 0.3 0.4 0.0 0.0 0.1 0.2 0.3 1.5 0.0 0.1 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.1 2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 42 nmi, nmi, 42 3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Table 25000 ft/s 25000 Table 5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

W of 5 lb W of 10 lb W of 25 lb

r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30 0.1 0.0 0.0 0.0 0.0 0.0 0.0

0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1 0.6 0.5 0.8 0.1 0.7 0.6 0.5 0.8 0.2 0.6 0.7 0.7 0.6 0.6 1.5 0.6 0.7 0.6 0.7 0.7 0.6 1.0 2 0.5 0.6 0.6 1.0 3 25000 ft/s 25000 4 5 W of 250 lb W of 500 lb W of 750 lb

r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30 0.1

0.5 0.5 0.5 0.2 0.0 0.4 0.5 0.2 0.4 0.4 1 0.5 0.6 0.7 0.6 0.6 0.3 0.6 0.6 0.7 0.6 0.2 0.4 0.5 0.7 0.7 1.5 0.7 0.7 0.8 0.8 0.5 0.7 0.7 0.7 0.4 0.6 0.7 0.7 2 0.6 0.7 0.7 0.6 0.5 0.6 0.7 0.7 0.4 0.5 0.6 0.7 3 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.5 0.5 25000 ft/s 25000 4 0.4 0.4 0.3 0.4 0.4 0.4 0.4 0.4 0.4 5 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

T. M. Owens 81

Table 9: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder

W of 50 lb W of 75 lb W of 100 lb r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30

0.1 0.0 0.0 0.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 1.0 1.0 0.5 1.0 1.0 1.0 1.0 1.0 1.0 0.9 1.0 1.0 1.0 1.0 1.0 0.8 1.0 1.0 1.0 1.0 1.0 1 1.0 1.0 1.0 1.0 1.0 0.9 0.9 1.0 1.0 1.0 1.0 1.0 0.8 1.0 1.0 1.0 1.0 1.0 1.5 0.9 0.8 0.8 0.7 0.7 0.7 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.8 0.8 0.8 0.7 0.7 2 0.7 0.7 0.6 0.6 0.6 0.5 0.7 0.7 0.7 0.6 0.6 0.6 0.7 0.7 0.7 0.6 0.6 0.6 3 0.5 0.5 0.5 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.4 0.4 0.5 0.5 0.5 0.5 0.4 0.4 4 0.4 0.4 0.3 0.3 0.3 0.2 0.4 0.4 0.4 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.3 0.3

Model 25000 ft/s 25000 Model 5 0.3 0.3 0.3 0.2 0.2 0.1 0.3 0.3 0.3 0.3 0.2 0.2 0.3 0.3 0.3 0.3 0.2 0.2

0.1 0.5 0.7 1.0 1.0 1.0 1.0 0.5 0.8 1.0 1.0 1.0 0.3 0.7 1.0 1.0 1.0 1 0.4 0.4 0.3 0.3 0.3 0.4 0.4 0.4 0.3 0.3 0.3 0.4 0.4 0.3 0.3 1.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4 0.0 0.0 0.0 0.0

Table 25000 ft/s 25000 Table 5 0.0 W of 1000 lb W of 2500 lb W of 5000 lb

r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30

0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.6 0.9 1.0 0.0 0.0 0.0 0.0 0.5 0.7 0.0 0.0 0.0 0.0 0.0 0.0 1 0.0 0.1 0.3 0.5 0.8 0.9 0.0 0.0 0.0 0.2 0.4 0.6 0.0 0.0 0.0 0.0 0.1 0.3 1.5 0.0 0.1 0.3 0.5 0.7 0.7 0.0 0.0 0.0 0.2 0.4 0.5 0.0 0.0 0.0 0.0 0.1 0.3 2 0.0 0.1 0.3 0.4 0.6 0.6 0.0 0.0 0.0 0.2 0.3 0.5 0.0 0.0 0.0 0.0 0.1 0.2 3 0.1 0.2 0.3 0.4 0.5 0.5 0.0 0.0 0.0 0.1 0.3 0.4 0.0 0.0 0.0 0.0 0.1 0.2 4 0.1 0.2 0.3 0.3 0.4 0.4 0.0 0.0 0.0 0.1 0.3 0.3 0.0 0.0 0.0 0.0 0.1 0.2

Model 25000 ft/s 25000 Model 5 0.1 0.2 0.2 0.3 0.3 0.3 0.0 0.0 0.0 0.1 0.2 0.3 0.0 0.0 0.0 0.0 0.1 0.1

0.5 deg 0.5 0.1 - 0.5 0.2 0.5 0.1 1 0.0 0.0 0.0 0.2 0.3 0.0 0.0 0.0 0.0 0.0 1.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 42 nmi, nmi, 42 3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Table 25000 ft/s 25000 Table 5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

W of 50 lb W of 75 lb W of 100 lb

r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30 0.1

0.5 0.3 0.0 0.0 0.0 0.0 0.4 0.2 0.0 0.0 0.0 0.5 0.3 0.0 0.0 0.0 1 0.6 0.6 0.7 0.7 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.6 0.6 0.7 0.7 1.5 0.8 0.7 0.7 0.7 0.8 0.8 0.7 0.7 0.7 0.8 0.7 0.7 2 0.6 0.6 0.6 0.5 0.7 0.6 0.6 0.6 0.7 0.6 0.6 0.6 3 0.4 0.4 0.5 0.4 0.4 0.5 0.4 0.4 25000 ft/s 25000 4 0.3 0.3 0.4 0.3 5 0.3 W of 1000 lb W of 2500 lb W of 5000 lb

r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30 0.1

0.5 0.4 0.4 0.4 1 0.1 0.3 0.5 0.6 0.6 0.2 0.4 0.6 0.1 0.3 1.5 0.3 0.5 0.7 0.7 0.0 0.2 0.4 0.5 0.0 0.1 0.3 2 0.3 0.4 0.6 0.6 0.0 0.2 0.3 0.5 0.0 0.0 0.1 0.2 3 0.4 0.5 0.5 0.1 0.3 0.4 0.0 0.1 0.2 25000 ft/s 25000 4 0.3 0.4 0.4 0.1 0.3 0.3 0.0 0.1 0.2 5 0.3 0.3 0.3 0.1 0.2 0.3 0.0 0.1 0.1

82 Aero-thermal Demise

5.3 Input and Output Debris Fragment Catalog

The demise model is able to ingest a catalog of debris fragments and perform a survivability analysis for an entire trajectory at any malfunction times desired. The trajectory can be for launch or reentry vehicle. This builds a full set of demise adjust fragment tables across the trajectory, allowing other tools to make a complete casualty risk estimation for the mission instead of single failure events.

The MATLAB functions importFragment and exportFragment work to import from a formatted CSV to a MATLAB structure defining the debris fragment and then export the demised fragment set to a CSV file of the same format. This allows for import of the fragment data across a wide range of risk analysis tools.

For a test case, the debris fragment defined in Table 10 from a SpaceX Falcon 9 rocket launch was used as an example. The coefficient of drag tables vary by flight Mach and all others vary by the trajectory time. Only the properties used in the demise analysis are included in the tables. Other properties not included, such as explosion velocity and yield factors, are fragment properties that other risk analysis tools might use.

Table 10: Demise Utility Input

Name Shape Radius [m] Length [m] Material 2nd stage main Aluminum Cylinder 0.064008 0.36576 LOX line (generic)

Mach Cd Time [s] Mass [kg] Area [m2] Beta [kg/m^2] Hazard Radius [m] 1 0.81 0 9.2049 0.046823 242.7021 0.1215 0.9999 0.45

A demise analysis was performed on the fragment between 512 and 550 seconds of the launch trajectory. This details the over-flight of Europe as seen in Figure 10.

T. M. Owens 83

The output generated with the fragment exporter can be seen in Table 11. The first two columns are the Mach versus coefficient of drag table. The last five columns are all time based tables. Each time is a breakup event where the trajectory state is used as the initial state of the demising fragment. The properties in the table are the impact states of the fragment which can be used in risk analysis. Because this is a launch trajectory the later times, which are higher speed and altitude, see the fragment experiencing greater mass loss. One thing to note is the hazard radius, which can be used to compute the casualty or hazard area. As the fragment is mostly hollow there is not a significant reduction in overall area as it ablates. This is the case with many debris fragments, so to get an appreciable reduction in casualty area a fragment must fully demise. The mass does reduce to a less than half of the initial mass after 548 seconds which can give a casualty reduction when considering sheltering effects or risk to aircraft. The reduction in mass means less damage to structures, leading to fewer casualties.

Table 11: Demise Utility Output, Adjusted Fragment Tables

Mach Cd Time [s] Mass [kg] Area [m2] Beta [kg/m^2] Hazard Radius [m] 21.48 0.44 512 9.205 0.046823 447.9701 0.1215 21.21 0.45 513 9.205 0.046823 447.9701 0.1215 20.93 0.45 514 9.205 0.046823 447.9701 0.1215 20.64 0.46 515 9.205 0.046823 447.9701 0.1215 20.34 0.47 516 9.205 0.046823 447.9701 0.1215 20.03 0.48 517 9.205 0.046823 447.9701 0.1215 19.72 0.49 518 9.205 0.046823 447.9701 0.1215 19.39 0.50 519 9.205 0.046823 447.9701 0.1215 19.06 0.50 520 9.1798 0.046773 447.2244 0.12144 18.72 0.51 521 9.0828 0.046579 444.3356 0.12118 18.37 0.52 522 8.9823 0.046378 441.3262 0.12092 18.01 0.53 523 8.8793 0.046171 438.2193 0.12065

84 Aero-thermal Demise

Mach Cd Time [s] Mass [kg] Area [m2] Beta [kg/m^2] Hazard Radius [m] 17.65 0.54 524 8.7731 0.045957 434.9959 0.12037 17.28 0.55 525 8.6632 0.045735 431.6374 0.12008 16.91 0.57 526 8.5507 0.045506 428.1696 0.11978 16.54 0.58 527 8.4338 0.045268 424.543 0.11946 16.18 0.59 528 8.3131 0.04502 420.7661 0.11914 15.82 0.60 529 8.1886 0.044764 416.8372 0.1188 15.47 0.61 530 8.0596 0.044497 412.7308 0.11844 15.10 0.62 531 7.9259 0.044219 408.4359 0.11807 14.74 0.64 532 7.7876 0.04393 403.9496 0.11769 14.36 0.65 533 7.6438 0.043628 399.2335 0.11728 13.99 0.66 534 7.4941 0.043312 394.2739 0.11686 13.62 0.67 535 7.3389 0.042982 389.0744 0.11641 13.26 0.68 536 7.1762 0.042634 383.5531 0.11594 12.90 0.69 537 7.0068 0.042269 377.7339 0.11544 12.54 0.70 538 6.8301 0.041886 371.5778 0.11492 12.18 0.71 539 6.6442 0.041479 365.0059 0.11436 11.82 0.72 540 6.4496 0.04105 358.0165 0.11376 11.46 0.73 541 6.2447 0.040594 350.5357 0.11313 11.10 0.74 542 6.0281 0.040108 342.4799 0.11245 10.74 0.75 543 5.7993 0.039589 333.7988 0.11172 10.39 0.76 544 5.5566 0.039032 324.3892 0.11093 10.03 0.77 545 5.2984 0.038433 314.142 0.11008 9.68 0.77 546 5.0213 0.037781 302.8519 0.10914 9.34 0.78 547 4.724 0.037071 290.3758 0.10811 9.00 0.78 548 4.4031 0.036292 276.4624 0.10697 8.67 0.79 549 4.0536 0.035427 260.73 0.10568 8.35 0.79 550 3.67 0.034457 242.7048 0.10423 8.04 0.80 7.73 0.80

T. M. Owens 85

Mach Cd Time [s] Mass [kg] Area [m2] Beta [kg/m^2] Hazard Radius [m] 7.43 0.80 7.14 0.80 6.86 0.81 6.58 0.81 6.32 0.81 6.06 0.81 5.82 0.81 5.58 0.81 5.36 0.81 5.14 0.81 4.93 0.81 4.73 0.81 4.53 0.81 4.35 0.81 4.17 0.81 4.00 0.81 3.84 0.81 3.69 0.81 3.54 0.81 3.40 0.81 3.26 0.81 3.13 0.81 3.01 0.81 2.89 0.81 2.78 0.81 2.67 0.81 2.57 0.81

86 Aero-thermal Demise

6 Conclusions

For this thesis, a computational model was developed using an earth model defined by WGS 84 with a fourth order harmonic model of gravity, the 1976 U.S. Standard Atmosphere, a general impact integrator for a rotating earth and a stagnation point heating model based on Fay-Riddell theory. The model can be used with a wide range of demise fragments differing in shape, material and breakup state. It is also able to generate a more usable output than that of DAS 2.0 with inputs of similar complexity. DAS 2.0 is only able to consider a single breakup condition from an uncontrolled orbital reentry. The computational model can use any breakup state defined by a reentry or launch trajectory for uncontrolled or controlled flight, making it much more flexible in application. There is still a significant gap in complexity and capability between the model developed in this thesis and tools like ORSAT, however, it should be able to reduce the need for these tools by giving an adequate estimation of aero-thermal demise for many different kinds of mission risk analysis.

6.1 Practical Application

There are numerous applications for a fragment set with demise adjusted properties, and many risk analysis tools will derive benefits from its application. For example a probability of impact tool would benefit from the corrected ballistic coefficient to make more accurate predictions of the impact point. Expected casualty estimation would likewise benefit from the reduction in casualty area in partially and fully demised debris to reduce the overall expected casualties. More specialized tools like ship and aircraft hit predictors, real-time systems and destruct line tools will also be able to use the demise adjusted fragment catalog.

T. M. Owens 87

Of significance, a similar method to the algorithm outlined in this thesis developed by the author for Millennium Engineering and Integration’s Joint Advanced Range Safety System Mission Planning (JARSS MP) has already been used to perform fragment demise analysis on missions for the SpaceX Falcon 9, Boeing X-37 and other vehicles. The work for the SpaceX Falcon9 debris catalog was done as part of a task for the FAA to assess the overflight risk of the Falcon 9-0003 mission. The Boeing X-37 debris catalog was developed as part of the OTV Feasibility Analysis for possible landing at the Cape Canaveral Air Force Station under the 30th and 45th Space Wings. The algorithm in this thesis has several advantages in that it has a more sophisticated impact integrator and takes into account the time varying properties of the debris fragments.

In all cases, the more accurate prediction of impacts and expected casualty risk will give the mission analyst more confidence in how to manage the risk of the mission. A greater confidence in risk analysis would result in the ability to close airspace for less time, allow more ships downrange, not close down facilities around the launch site or even be the difference between acceptable and unacceptable overall mission risk. Notably, the above can reduce the cost and increase the number of launch opportunities.

6.2 Validation

The computational model outlined in this thesis finds itself between DAS 2.0 and the Aerospace Survivability Tables in terms of the conservatism of demise prediction. DAS 2.0 is considered the conservative answer that then triggers a higher fidelity analysis from tools like ORSAT, so being less conservative than survivability tables is not a significant concern. See 5.2 Model Comparisons for a detailed comparison between the computational model and existing tools.

88 Aero-thermal Demise

Many parts of the algorithm were chosen based on their previous acceptance in the risk assessment industry. The WGS 84 and 1976 Standard Atmosphere are both commonly used in many risk analysis tools. The impact integration algorithm has been used as part of a glide turn integrator in JARSS MP by the 45th space wing in operations. The aero-thermal demise portion of the algorithm in this thesis is an implementation of the method used to generate the Aerospace Survivability Tables. This method was developed for evaluating risk of FAA-licensed operations and used in the CAIB. In order to fully validate the model, a comparison would need be made against the CFD and pseudo-CFD tools.

6.3 Performance

The performance of a tool implemented using this computational model is also important. Faster running utilities allow for the mission analyst to work more swiftly and through more possible scenarios. Performance is also very important for possible rapid response missions where all of the risk analysis may have to take place in under 24 or 48 hours.

In order to generate the data for Table 8 Survivability Table Liquid Fraction Comparison, Aluminum Cylinder, 576 demise impacts were computed in 180 seconds. The test machine used has an Intel i7-3770K at 3.50 GHz, 32 GB of memory and a 120 GB Intel 520 Series SSD (a typical Intel Xenon equipped workstation should have similar performance abilities). The 500+ impacts would be typical of a 1 Hz launch trajectory data set used by a mission analyst. A landing trajectory would typically have closer to 2000 seconds of trajectory states; therefore, the expected tool run time would be about 720 seconds.

A large performance increase could be made using an ODE solver if the debris fragment's impact state is the only one of interest; this was not used in this implementation as the descent history was required for generating figures.

T. M. Owens 89

Further performance benefits could be made by implementing the code in C++ or other compiled language. Impact integration could also be implemented in multiple threads, distributed computing or GPU computing for real-time systems.

6.4 Possible Future Work

There are many small improvements that could be made to this computational model: a better atmosphere model, temperature dependent materials properties and better predictions of the fragment's coefficient of drag across all flight regimes. There are several more extensive enhancements of this work that could also be of value.

One of the obvious improvements that could be made is the addition of an aero- breakup or thermal fragmentation model. Probably the simplest possible method is to define a loading factor, like wing loading, and assume vehicle breakup when this value is exceeded in malfunction turns. More complex methods involve estimating the strength of a parent structure and calculating the resulting child fragments upon breakup.

While the thrust of this thesis was to estimate the aero-thermal demise of reentry debris, the algorithm could be applied to vehicle trajectory design. The impact integrator is appropriate for either ballistic or gliding reentry, and the stagnation point heating method can be applied to an intact vehicle. The algorithm could be modified to find the heating of the spacecraft and also used to determine if a trajectory has too much heat loading. This would aid in the down selection from several possible reentry scenarios.

Another possibility is to expand to a full mission risk analysis tool. This would require extensive work to develop algorithms for the probability of impact, risk to population, sheltering and et cetera. There are many tools that already

90 Aero-thermal Demise

accomplish this, like JARSS MP, so this computational model is probably more beneficial to fragment demise analysis.

T. M. Owens 91

References

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Range Flight Safety Analysis, NASA & DOD, July 2005. (PPT of slides).

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Program Office, URL: http://orbitaldebris.jsc.nasa.gov/reentry/orsat.html, 08/24/2009.

Ref. 3 Carmichael, R., "Properties of the U.S. Standard Atmosphere 1976," Public

Domain Aeronautical Software, URL: http://www.pdas.com/atmos.html, 13 February 2014.

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Satellite Vehicle Re-entering the Atmosphere," Journal of Jet Propulsion, Vol. 27, No. 12, 1957, 1256-1257.

Ref. 5 Fay, J. A., Kemp, N. H., "Theory of Stagnation-Point Heat Transfer in a

Partially Ionized Diatomic Gas," AIAA Journal, Vol. 1, No. 12, 1963, pp. 2741-2751.

Ref. 6 Fay, J. A., Riddell, F. R., "Theory of Stagnation Point Heat Transfer in

Dissociated Air," Journal of the Aeronautical Sciences, Vol. 25, No. 2, 1958, pp. 73-85, 121.

Ref. 7 Gollan, R. J., Jacobs, P. A., Karl, S., Smith, S. C., "Numerical Modelling of

Radiating Superorbital Flows," ANZIAM Journal, vol. 45, No. E, pp. 248-268.

Ref. 8 Hallman, W. P., Moody, D. M., "Trajectory Reconstruction and Heating

Analysis of Columbia Composite Debris Pieces," Aerospace Report No. ATR- 2005(5138)-1, April 15 2005.

Ref. 9 Kelly, R. L., Rochelle, W. C., "Atmospheric Reentry of a Hydrazine Tank,"

ESCG Jacobs Technology, Huston, TX, 2008.

92 Aero-thermal Demise

Ref. 10 Kemp, N. H., Riddell, F. R. "Heat Transfer to Satellite Vehicles Re-entering

the Atmosphere," Journal of Jet Propulsion, Vol. 27, No. 2, 1957, pp. 132- 137.

Ref. 11 Koppenwallner, G., Fritshce, B., Lips, T., Klinkrad, H., "SCARAB – A Multi-

Disciplinary Code for Destruction Analysis of Space-Craft During Re-Entry," Fifth European Symposium on Aerothermodynamics for Space Vehicles, ESA SP-563, Cologne, Germany, February 2005.

Ref. 12 Millennium Engineering and Integration Company: Range Systems, "Joint

Advanced Range Safety System Mathematics and Algorithms," Satellite Beach, Florida, May 5 2014.

Ref. 13 National Aeronautics and Space Administration, "Process for Limiting

Orbital Debris," National Aeronautics and Space Administration, NASA-STD 8719.14 (Rev. A with Change 1), May 2012.

Ref. 14 National Imagery and Mapping Agency, "Department of Defense World

Geodetic System 1984: Its Definition and Relationships with Local Geodetic Systems," National Imagery and Mapping Agency (NIMA), 3rd Ed., Amendment 2, 23 June 2004.

Ref. 15 National Oceanic and Atmospheric Administration, National Aeronautics

and Space Administration, United States Air Force," U.S. Standard Atmosphere, 1976," NASA-TM-X-74335.

Ref. 16 Office of Safety and Mission Assurance, Johnson space Center Space

Physics Branch, "NASA Safety Standard: Guidelines and Assessment Procedures for Limiting Orbital Debris," National Aeronautics and Space Administration, NSS 1740.14, August 1995.

T. M. Owens 93

Ref. 17 Opiela, J. N., Hillary, E., Whitlock, D. O., Hennigan, M., ESCG, "Debris

Assessment Software Version 2.0 User’s Guide," JSC 64047, NASA Lyndon B. Johnson Space Center, Huston, TX, Jan. 2012.

Ref. 18 Samareh, J. A., "A Multidisciplinary Tool for Systems Analysis of Planetary

Entry, Descent, and Landing (SAPE)," NASA/TM-2009-215950, Langley research Center, Hampton, VA, Nov. 2009.

Ref. 19 Schneider, S. P., Gustafson, W., "Methods for Analysis of Preliminary

Spacecraft Designs," Purdue University, Sept. 19, 2005, pp. 27.

Ref. 20 Scott, C. D., et. al., "Design Study of an Integrated Aerobreaking Orbital

Transfer Vehicle," National Aeronautics and Space Administration, NASA- TM- 58264, Huston, TX, March 1985.

Ref. 21 Sutton, K., Graves, R. A. Jr., "A General Stagnation-Point Convective-

Heating Equation For Arbitrary Gas Mixtures," NASA TR R-376, Langley Research Center, Hampton, VA, Nov. 1971.

Ref. 22 Tauber, M. E., "A Review of High-Speed Convective, Heat-Transfer

Computation Methods," NASA TP-2914, Jul. 1989.

Ref. 23 Tauber, M. E., Menees, G. P., Adelman, H. G., "Aerothermodynamics of

Transatmospheric Vehicles," AIAA Paper 86-1257, Jun. 1, 1986.

Ref. 24 Tooley, J., Habiger, T. M., Bohman, K. R., "Reentry Hazard Analysis

Handbook," Aerospace Report No. ATR-2005(5138)-2, Jan. 28 2005.

Ref. 25 Vinh, N. X., "Flight Mechanics of High-Performance Aircraft," Cambridge

Aerospace Series 4, New York, NY, 1999.

Ref. 26 Weaver, M. A., Baker, R. L., Frank, M. V., "Probalistic Estimation of Reentry

Debris Area," Third European Conference on Space Debris, ESA SP-473, Vol. 2, Darmstadt, Germany, March 2001, pp. 515-520.

94 Aero-thermal Demise

Appendix

Appendix A: Material Properties Material properties from Debris Assessment Software Version 2.0 User’s GuideRef. 17, the DAS 2.0 built-in materials. Specific heats used are the mean specific heat between the reference and melting temperature.

Specific Heat of Melt Density Heat Fusion Temperature Material (kg/m3) (J/kg-K) (J/kg) (K)

Acrylic 1170 1465 0 505

Alumina 3990 1011 106757 2305.4

Aluminum (generic) 2700 1100 390000 850

Aluminum 1145-H19 2697 904 386116 919

Aluminum 2024-T3 2803.2 972.7 386116 856

Aluminum 2024-T8xx 2803.2 972.7 386116 856

Aluminum 2219-T8xx 2812.8 1006.5 386116 867

Aluminum 5052 2684.9 900.2 386116 880

Aluminum 6061-T6 2707 896 386116 867

Aluminum 7075-T6 2787 1012.4 376788 830

Barium Element 3492 285 55824 983

Beryllium Element 1842 2635.1 1093220 1557

Beta Cloth 1581 837.5 232.6 650

Brass- Cartridge 8521.8 406.1 179091 1208

Brass- Muntz 8393.67 412.35 167461 1174

T. M. Owens 95

Specific Heat of Melt Density Heat Fusion Temperature Material (kg/m3) (J/kg-K) (J/kg) (K)

Brass- Red 8746 404 195372 1280

Cobalt 8862 658.45 259600 1768

Copper Alloy 8938 430.6 204921 1356

Cork 261.294 1629.2 2860980 922

Cu/Be (0.5% Beryllium) 8800 397 204921 1320

Cu/Be (1.9% Beryllium) 8248.6 452.5 204921 1199

Fiberfrax 96.1 1130.5 0 2089

Fiberglass 1840.35 1046.8 232.6 1200

FRCI-12 (shuttle tile) 192.22 1978.9 0 1922

Gallium Arsenide (GaAs) 5316 325 0 1510

Germanium 5320 363.7 430282.6 1210.7

Gold Element 19300 139.85 64895 1336

Graphite Epoxy 1 1550.5 879.3 23 700

Graphite Epoxy 2 1550.5 879.3 23 700

Hastelloy 188 8980 498.1 309803 1635

Hastelloy 25 9130 498.1 309803 1643

Hastelloy c 8920.67 596.5 309803 1620

Hastelloy n 8576.4 501.7 309803 1623

Inconel 600 8415 538.45 297206 1683.9

Inconel 601 8057.29 632.9 311664 1659

Inconel 625 8440 410 311664 1593

Inconel 718 8190 435 311664 1571

96 Aero-thermal Demise

Specific Heat of Melt Density Heat Fusion Temperature Material (kg/m3) (J/kg-K) (J/kg) (K)

Inconel X 8297.5 484.05 311664 1683.2

Invar 8050 566.55 2740000 1700

Iron 7865 572.6 272125 1812

Lead Element 11677 134.65 23958 600

Macor Ceramic 2520 790 236850 1300

Magnesium AZ31 1682 1212.8 339574 868

Magnesium HK31A 1794 1184.75 325619 877

MLI 772.48 1046.6 232.6 617

Molybdenum 10219 321.85 293057 2899

MP35N 8430 583 309803 1650

Nickel 8906.26 583.35 309803 1728.2

Niobium (Columbium) 8570 307.65 290000 2741

NOMEX 1380 1256 232.6 572

Platinum 21448.7 138.45 113967 2046.4

Polyamide 1420 1130 232.6 723

Polycarbonate (aka Lexan) 1250 1260 0 573

RCG Coating 1665.91 1224.2 0 1922

Reinforced Carbon-Carbon 1688.47 1257.55 37650 2144

Rene 41 8249 630.9 311664 1728

Silver Element 10492 233.15 105833 1234

Sodium-Iodide 3470 84 290759 924

Stainless Steel (generic) 7800 600 270000 1700

T. M. Owens 97

Specific Heat of Melt Density Heat Fusion Temperature Material (kg/m3) (J/kg-K) (J/kg) (K)

Stainless Steel 17-4 ph 7833.03 666.8 286098 1728

Stainless Steel 21-6-9 7832.8 439.2 286098 1728

Steel A-286 7944.9 460.6 286098 1644

Steel AISI 304 7900 545.1 286098 1700

Steel AISI 316 8026.85 460.6 286098 1644

Steel AISI 321 8026.6 608.2 286098 1672

Steel AISI 347 7960 554.95 286098 1686

Steel AISI 410 7749.5 485.7 286098 1756

Strontium Element 2595 737 95599 1043

Teflon 2162.5 1674 0 533

Titanium (6 Al-4 V) 4437 805.2 393559 1943

Titanium (generic) 4400 600 470000 1950

Tungsten 19300 157.55 220040 3650

Uranium 19099 158.95 52523 1405

Uranium Zirconium 6086.8 418.7 131419 6086.8 Hydride

Water 999 5490.55 0.1 273

Zerodur 2530 2487.1 250000 1424

Zinc 7144.2 405.3 100942 692.6

98 Aero-thermal Demise

Appendix B: Supplemental Algorithms

Alternate Correlations

There are many possible correlations that can be made from the Fay-Riddell theory. They are all quite similar in derivation to the Detra-Kemp-Riddell implemented in this thesis.

Tauber-Menees-Adelman stagnation point heating correlationRef. 23,

1 − (퐶̅ − 푇 ) −4 휌∞ 푝푏 푤 3 푞푠푡푎푔 = 1.83 ∙ 10 √ ( ) 푈∞ Eq. 138 푅ℎ 1 2 2 푈∞

Sutton-Graves stagnation point heating correlationRef. 21, Ref. 18,

−4 휌∞ 3 푞푠푡푎푔 = 1.7623 ∙ 10 √ 푈∞ Eq. 139 푅ℎ

Scott stagnation point heating correlationRef. 20,

3.05 −4 휌∞ 푈∞ 푞푠푡푎푔 = 1.83 ∙ 10 √ ( 4) Eq. 140 푅ℎ 10

Tauber-Bowles-Yang stagnation point heating correlationRef. 19,

1 − (퐶̅ − 푇 ) −8 휌∞ 푝푏 푤 3 푞푠푡푎푔 = 1.83 ∙ 10 √ ( ) 푈∞ Eq. 141 푅ℎ 1 2 2 푈∞

Tauber-Bowles-Yang stagnation point heating correlation for Mars and VenusRef. 19,

T. M. Owens 99

1 − (퐶̅ − 푇 ) −8 휌∞ 푝푏 푤 3.04 푞푠푡푎푔 = 1.35 ∙ 10 √ ( ) 푈∞ Eq. 142 푅ℎ 1 2 2 푈∞

Tauber-Sutton formula for radiative heatingRef. 7,

푎 푏 푞푟푎푑 = 푅ℎ ∙ 휌∞ ∙ 푓(푈∞) Eq. 143

0 ≤ 푎 ≤ 1, 푓표푟 퐸푎푟푡ℎ 푟푒푒푛푡푟푦 { 푎 = 0.526, 푓표푟 푀푎푟푠 푟푒푒푛푡푟푦

푏 = 1.22, 푓표푟 퐸푎푟푡ℎ 푟푒푒푛푡푟푦 { 푏 = 1.19, 푓표푟 푀푎푟푠 푟푒푒푛푡푟푦

7 푓(푈∞) ≅ 푈∞ Eq. 144

Rate of mass loss to ablation,

푄̇ 푚̇푏 = − Eq. 145 ℎ푓

100 Aero-thermal Demise

Trajectory Site Direction Cosines 휋 푥 = − 훷 2 1 0 0 푀푥 = [0 cos 푥 sin 푥] 0 − sin 푥 cos 푥

2휋 Eq. 146 푧 = − 휃 3 cos 푧 − sin 푧 0 푀푧 = [sin 푧 cos 푧 0] 0 0 1

퐷퐶 = 푀푥푀푧

Rotate using the azimuth,

cos 훹 − sin 훹 0 퐷퐶 = 퐷퐶 [sin 훹 cos 훹 0] Eq. 147 0 0 1

ECEF Coordinates to XYZ Coordinates

Translate origin to site location

푒 = 푒 + 푒푠𝑖푡푒

푓 = 푓 + 푓푠𝑖푡푒 Eq. 148

푔 = 푔 + 푔푠𝑖푡푒

Rotate position and velocity,

[푥 푦 푧] = 퐷퐶′[푒 푓 푔] Eq. 149 [푥̇ 푦̇ 푧̇] = 퐷퐶′[푒̇ 푓̇ 푔̇]

T. M. Owens 101

ECEF Coordinates to Aeronautical Coordinates

2 2 푊2 = 푒 + 푓 2 2 푍2 = 푒 + 푓

푅12 = 푊2 + 푍2

푍2 푅22 = 푊2 + ( ) 1 − 푒⊕

√푅12 푅12 푆2 = ( − 1) 푎⊕ 푅22 Eq. 150 푒⊕́ 푠푒2 = 푅12 2 푆 = 푆2(1 + 1.5푆2푊2푍2푠푒2 )

푣1 = 1 + 푆 푔푣1 푔푠 = 1 − 푒⊕́ + 푆

Latitude, 푔푠 휙 = tan−1 Eq. 151 √푊2

Longitude, 푓 휃 = tan−1 푒 Eq. 152

Altitude, √푊 + 푔 2 2 2 푠 푧 = √푊2 + 푔푠 − Eq. 153 푣1

Heading, −1 푥푖̇ 휓𝑖 = tan Eq. 154 푦푖̇

Ground Speed,

102 Aero-thermal Demise

2 2 푣𝑖 = √푥푖̇ + 푦푖̇ Eq. 155

Vertical Speed, 푧푖̇ = 푧푖̇ Eq. 156