Aero-thermal Demise of Reentry Debris€¦ · The output from ORSAT gives a full range of fragment...

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

All rights reserved.

Copyright © 2014 by T. M. Owens.

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

iii

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.

v

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

vi

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

vii

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.

viii

getGravity.m ........................................................ Error! Bookmark not defined.

fragmentExporter.m ............................................ Error! Bookmark not defined.

fragmentImporter.m ............................................ Error! Bookmark not defined.

ix

List of Keywords and Abbreviations

Keyword/Abbreviation Definition

AST Office of Commercial Space Transportation

Boeing X-37 OTV Orbital Test Vehicle. Unmanned spacecraft, launches

as rocket lands as a space plane

CAIB Columbia Accident Investigation Board

Casualty Area Area where a debris fragment will cause human

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

Ec Estimated or expected casualties, usually measured in

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

Hazard Area Area where a debris fragment will cause a human

casualty, same as casualty area

JARSS MP Joint Advance Range Safety System: Mission Planning

MATLAB Programing language and integrated development

environment (IDE) developed by MathWorks

x

Micro-casualties Risk of human casualty due to debris hazard. Equal to

1*10-6 casualty per event, see Ec

Mission Analyst Someone who performs a flight safety analysis for a

mission, see Risk Analyst

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

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

Risk Analyst Someone who performs a flight safety analysis for a

mission, see Mission Analyst

SCARAB Spacecraft Atmospheric Re-entry and Aerothermal

Break-up

WGS 84 World geodetic Survey of 1984

xi

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

xii

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

xiii

List of Symbols

Symbols for Impact Integration

Variable Value Units Description

𝒂 m

s Local speed of sound

𝒂⊕ m Earth’s semi-major axis

𝜷 1

m Atmospheric decay parameter, scale height

𝑪𝑳 Coefficient of lift

𝑪𝑳∗ Coefficient of lift at the smallest glide angle

𝑪𝑫 Coefficient of drag

𝑪𝑫∗ Coefficient of drag at the smallest glide angle

𝑪𝑫𝟎 Zero lift coefficient of drag

𝑭𝑵 N Force normal to the flight path

𝑭𝑻 N Force tangential to the flight path

𝒈 m

s2 Local gravitational acceleration

𝒈𝟎 m

s2 Gravitational acceleration at Earth’s surface

𝜸 rad Flight path angle, glide angle of vehicle

𝑲 Induced drag factor, function of Mach

𝒎 kg Mass of the vehicle

xiv

𝒏 Wing loading factor

𝒏𝒔 Structural loading limit

𝝎 rad

s Angular velocity of Earth

𝝋 rad Glide turn roll angle of vehicle

𝝓 rad Latitude of the vehicle

𝜽 rad Longitude of vehicle

𝝍 rad Heading of vehicle

𝒓 m EFG magnitude radius to vehicle

𝑹⊕ m Mean radius of Earth

𝑺 m2 Plane area of the vehicle

𝑽 m

s Vehicle velocity

𝒚 Trajectory aerodynamic coordinate state

xv

Symbols for Fay-Riddell Stagnation Point Heating


𝒄𝒊 Mass fraction of component 𝑖

𝒄𝒑 J

kgK Specific heat per unit mass at constant pressure

𝑫 m2

s Molecular diffusion coefficient

𝑫𝑻 m2

sK Thermal diffusion coefficient

𝒆 𝒆𝒊 J

kg Internal energy per unit mass, of component 𝑖

𝒉 J

kg Enthalpy per unit mass of mixture

𝒉𝒊 J

kg Perfect gas enthalpy per unit mass of component 𝑖

𝒉𝒊𝟎 J

kg Heat of formation of component 𝑖 at 0 K per unit mass

𝒉𝑨𝟎 J

kg Dissociation energy per unit mass of atomic products

𝒌 W

K Thermal conductivity

𝒍 m Characteristic length

𝑳 𝑳𝒊𝑻

Lewis Number: 𝐷𝑖𝜌𝑐�̅� 𝑘⁄ ratio of the rate of thermal

diffusivity to the mass diffusivity, thermal Lewis number

𝝁 Pa ∙ s Absolute viscosity

𝑵𝒖 Nusselt Number: 𝑞𝑥𝑐�̅�𝑤 𝑘𝑤(ℎ𝑠 − ℎ𝑤)⁄ ratio of

convective to conductive heat transfer

𝝂 Pa ∙ s Kinematic viscosity

𝒑 Pa Pressure

xvi

𝑷𝒓 0.71 Prandtl Number: 𝑐�̅�𝜇 𝑘⁄ ratio of momentum diffusivity

to thermal diffusivity, value for air

𝜱 Dissipation function

𝒒 W

m2 Heat flux

�⃗⃗� �⃗⃗� 𝒊 m

s Vector mass velocity, vector diffusion velocity

𝒓 m Cylindrical radius of body

𝑹𝒉 m Body nose radius, radius of heating

𝑹𝒆 Reynolds Number: 𝑢𝑒𝑥 𝑣𝑤⁄ ratio of the inertial to viscous forces

𝝆 kg

m3 Mass density

𝑻 K Absolute temperature

𝒖 m

s 𝑥 component of velocity

𝒗 m

s 𝑦 component of velocity

𝒘𝒊 kg

m3s

Mass rate of formation of component 𝑖 per unit volume and time

𝒙 m

s Distance along meridian profile

𝒚 m

s Distance normal to the surface

xvii

Symbols for Aero-thermal Demise


𝒂 m

s Local speed of sound

𝑨𝒘 m2 Wetted area

𝜷𝒉𝒄 kg

m2 Hypersonic continuum ballistic coefficient

𝑪𝑫𝒉𝒄 Hypersonic continuum coefficient of drag

𝑪𝑫𝜷 Coefficient of drag corrected for ballistic coefficient

𝑪𝑫𝑴𝒂𝒄𝒉 Coefficient of drag from fragment drag tables

�̅�𝒑𝒃 J

kgK Mean Specific heat capacity of the fragment

𝑪𝒑∞ 1.0045 ∙ 103 J

kgK Specific heat capacity of air

𝜹 m Recession length, flat plate

𝜺𝒃 𝜺𝒘 Surface emissivity of the fragment

𝒈𝟎 9.80665 m

s2 Standard gravitational acceleration

𝜸 radians Flight angle of the fragment

𝒉 𝒉𝒊 m Fragment height and interior height, box

𝒉𝒇 J

kg Heat of fusion

𝑯𝒓 𝑯𝒓𝟎 m Hazard radius & user defined hazard radius

𝒌𝟐 0.12 Area-averaging factor (a less conservative 0.8 for composites)

𝒍 𝒍𝒊 m Fragment length and interior length, cylinder, flat plate and box

𝑳𝑭 Liquid fraction of the fragment

𝒎𝒃 kg Thermal mass of the fragment

xviii

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

𝒎𝑳𝑭 kg Thermal mass adjust by liquid fraction of the fragment

𝝈𝒔𝒃𝒄 5.5670373 ∙ 10−8 W

m2K4 Stefan-Boltzmann constant

𝒒𝒔 W

m2 Stagnation heat flux

𝒒𝒓𝒂𝒅 W

m2 Radiation heat flux

�̇� W

s Net heat flow

𝑸𝟎 W Heat of initial temperature

𝑸𝒎𝒆𝒍𝒕 W Heat of melting

𝑸𝒂 W Heat of ablation

𝑸 W Heat content of the fragment

𝒓 𝒓𝒊 m Radius and interior radius of fragment, sphere and cylinder

𝑹⊕ 6378.1 ∙ 103 m Mean radius of Earth

𝑹𝒉 m Radius of heating

𝑹∗ 287 J

K Gas constant for air

𝝆𝒃 kg

m3 Density of the fragment material

𝝆∞ kg

m3 Free stream air density

𝑺 m2 Aerodynamic reference area of the fragment

𝒕 m Fragment thickness, flat plate

𝑻𝟎 K Initial body bulk temperature of the fragment

𝑻𝒃 K Body bulk temperature of the fragment

𝑻𝒎𝒆𝒍𝒕 K Melting temperature of the fragment

𝑻𝒓𝒆𝒇 300 K Reference temperature

xix

𝑻𝒔 K Adiabatic stagnation temperature

𝑻𝒘 K Hot wall temperature of the fragment

𝑼𝒓𝒆𝒇 7924.8 m

s Reference velocity, 26000 ft/s

𝑼∞ m

s Free stream velocity

𝒘 𝒘𝒊 m Fragment width and interior width, flat plate and box

𝒛 m Altitude position of the fragment

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


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.


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.


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 𝑛 possible events is as followsRef. 1,


𝐸𝑐 =∑𝑃𝑖𝐴𝑐𝑖𝐷𝑝𝑖

𝑛

𝑖=1

Eq. 1

Where the 𝑃𝑖 is the impact probability, 𝐴𝑐𝑖 is the casualty or hazard area of the

debris and 𝐷𝑝𝑖 is the population density of the area at risk for the 𝑖𝑡ℎ 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, 𝑟(𝑡), velocity

vector, 𝑉(𝑡), and mass, 𝑚(𝑡). The total force, 𝐹, at each instance is a sum of the

gravitational, 𝑚 ∙ 𝑔, aerodynamic, 𝐴, and propulsion thrust forces, 𝑇.

𝑭 = 𝑻 + 𝑨 +𝑚𝒈 Eq. 2


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,

𝑑𝒓

𝑑𝑡= 𝑽 Eq. 3

Newton’s second law gives the force equation.

𝑚𝑑𝑽

𝑑𝑡= 𝑭 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, 𝑂𝑋1𝑌1𝑍1

and 𝑂𝑥𝑦𝑧 respectively. The system 𝑂𝑥𝑦𝑧 rotates with respect to the fixed system

𝑂𝑋1𝑌1𝑍1with angular velocity 𝜔. Let 𝐴 be an arbitrary vector in the rotating

system as seen in Figure 5,

𝑨 = 𝐴𝑥𝑖̂ + 𝐴𝑦𝑗̂ + 𝐴𝑧�̂� Eq. 5

T. M. Owens 13

Figure 5: Relative Angular Motion (Ref. 25)

The 𝑖̂, 𝑗̂ and �̂� unit vectors in the rotating reference system, 𝑂𝑥𝑦𝑧, are functions of

time. Therefore, the time derivative of 𝐴 with respect to the fixed system 𝑂𝑋1𝑌1𝑍1

is,

𝑑𝑨

𝑑𝑡= (

𝑑𝐴𝑥𝑑𝑡

𝑖̂ +𝑑𝐴𝑦

𝑑𝑡𝑗̂ +

𝑑𝐴𝑧𝑑𝑡

�̂�) + (𝐴𝑥𝑑𝑖̂

𝑑𝑡+ 𝐴𝑦

𝑑𝑗̂

𝑑𝑡+ 𝐴𝑧

𝑑�̂�

𝑑𝑡) Eq. 6

At point 𝑃, the linear velocity in a fixed system rotating with angular velocity 𝜔 at

position vector 𝑟 as seen in Figure 6 is,

𝑽 =𝑑𝒓

𝑑𝑡= 𝝎 × 𝒓 Eq. 7

Figure 6: Kinematics of Rotation (Ref. 25)


Then, according to Poisson’s formulas,

𝑑𝑖̂

𝑑𝑡= 𝝎 × 𝑖̂

𝑑𝑗̂

𝑑𝑡= 𝝎 × 𝑗̂

𝑑�̂�

𝑑𝑡= 𝝎 × �̂�

Eq. 8

Using this with the definition of vector 𝐴 in the equation of the time derivative of

𝐴 the following equation is found,

𝐴𝑥𝑑𝑖̂

𝑑𝑡+ 𝐴𝑦

𝑑𝑗̂

𝑑𝑡+ 𝐴𝑧

𝑑�̂�

𝑑𝑡= 𝝎 × 𝑨 Eq. 9

Then in the rotating system 𝑂𝑥𝑦𝑧 the time derivative of 𝐴 is,

𝛿𝑨

𝛿𝑡=𝑑𝐴𝑥𝑑𝑡

𝑖̂ +𝑑𝐴𝑦

𝑑𝑡𝑗̂ +

𝑑𝐴𝑧𝑑𝑡

�̂� Eq. 10

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

be rewritten as,

𝑑𝑨

𝑑𝑡=𝛿𝑨

𝛿𝑡+ 𝝎 × 𝑨 Eq. 11

2.1.2 Equations for Flight Over a Rotating Spherical Earth

𝑂𝑋1𝑌1𝑍1, is the inertial reference frame, with the origin at the center of a

spherical earth’s gravitation field. The plane 𝑂𝑋1𝑌1 is the equatorial plane. The

reference frame 𝑂𝑋𝑌𝑍 is fixed with respect to earth with 𝑂𝑍 and 𝑂𝑍1 coincident.

The atmosphere is assumed to rotate at the same constant angular acceleration,

𝜔.

T. M. Owens 15

The equation for absolution acceleration is found by setting the position vector

𝐴 = 𝑟 and taking the time derivative of Eq. 11 in the earth fixed frame 𝑂𝑋𝑌𝑍.

𝑑𝑽

𝑑𝑡=𝛿2𝒓

𝛿𝑡2+ 2𝝎 ×

𝛿𝒓

𝛿𝑡+ 𝝎 × (𝝎 × 𝒓) Eq. 12

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

𝑚𝛿2𝒓

𝛿𝑡2= 𝑭 − 2𝑚𝝎×

𝛿𝒓

𝛿𝑡− 𝑚𝝎× (𝝎 × 𝒓) Eq. 13

Or,

𝑚𝑑𝑽

𝑑𝑡= 𝑭 − 2𝑚𝝎 × 𝑽 −𝑚𝝎× (𝝎 × 𝒓) Eq. 14

The velocity, 𝑉, 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,

−2𝝎 × 𝑽, and the transport acceleration, −𝝎× (𝝎 × 𝒓). The Coriolis

acceleration is zero when the flight path angle is parallel to the earth’s pole and

reaches a maximum of 2𝜔𝑉 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, 𝜔2𝑟, when the body is on the equatorial plane.

The fixed coordinate system, 𝑂𝑋𝑌𝑍, and rotating coordinate system, 𝑂𝑥𝑦𝑧, can be

seen in Figure 7. The longitude is angle 𝜃 and latitude is 𝜙.The angle 𝛾 is the flight

path angle and 𝛹, 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 𝑥 axis.


Figure 7: Coordinate Systems (Ref. 25)

Thereby the velocity vector in the rotating system 𝑂𝑥𝑦𝑧 is,

𝑽 = 𝑉 sin 𝛾 𝑖̂ + 𝑉 cos 𝛾 cos𝛹 𝑗̂ + 𝑉 cos 𝛾 sin𝛹 �̂� Eq. 15

The angular velocity in the 𝑂𝑥𝑦 plane is,

𝝎 = 𝜔 sin𝜙 𝑖̂ + 𝜔 cos𝜙 �̂� Eq. 16

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

the unit vectors,

𝝎× 𝑽 = −𝜔𝑉 cos 𝛾 cos𝜙 cos𝛹 𝑖̂ + 𝜔𝑉(sin 𝛾 cos𝜙 − cos 𝛾 sin𝛹)𝑗̂

+ 𝜔𝑉 cos 𝛾 sin 𝜙 cos𝛹 �̂� Eq. 17

𝝎× (𝝎 × 𝒓) = −𝜔2 cos2 𝜙 𝑖̂ + 𝜔2𝑟 sin𝜙 cos𝜙 �̂� Eq. 18

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

T. M. Owens 17

𝑚𝒈 = −𝑚𝑔(𝑟)𝑖 ̂ 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,

𝐹𝑇 = 𝑇 cos 휀 − 𝐷𝐹𝑁 = 𝑇 sin 휀 + 𝐿

Eq. 20

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

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

𝑭𝑇 = 𝐹𝑇 sin 𝛾 𝑖̂ + 𝐹𝑇 cos 𝛾 cos𝛹 𝑗̂ + 𝐹𝑇 cos 𝛾 sin𝛹 �̂� 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 𝜎. This results in the equation,

𝑭𝑁 = 𝐹𝑁 cos 𝜎 cos 𝛾 𝑖̂ − 𝐹𝑁(cos 𝜎 sin 𝛾 cos𝛹 + sin 𝜎 sin𝛹)𝑗̂

+ 𝐹𝑇(cos 𝜎 sin 𝛾 sin𝛹 − sin 𝜎 cos𝛹)�̂� Eq. 22

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


𝑑𝑖̂

𝑑𝑡= cos𝜙

𝑑𝜃

𝑑𝑡𝑗̂ +

𝑑𝜙

𝑑𝑡�̂�

𝑑𝑗̂

𝑑𝑡= −cos𝜙

𝑑𝜃

𝑑𝑡𝑖̂ + sin𝜙

𝑑𝜃

𝑑𝑡�̂�

𝑑�̂�

𝑑𝑡= −

𝑑𝜙

𝑑𝑡𝑖̂ − sin𝜙

𝑑𝜃

𝑑𝑡𝑗̂

Eq. 23

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

𝑽 =𝑑𝑟

𝑑𝑡𝑖̂ + 𝑟 cos𝜙

𝑑𝜃

𝑑𝑡𝑗̂ + 𝑟

𝑑𝜙

𝑑𝑡�̂� Eq. 24

The derivative of velocity is,

𝑑𝑽

𝑑𝑡= [sin 𝛾

𝑑𝑉

𝑑𝑡+ 𝑉 cos 𝛾

𝑑𝛾

𝑑𝑡−𝑉2

𝑟cos2 𝛾] 𝑖̂

+ [cos 𝛾 cos𝛹𝑑𝑉

𝑑𝑡− 𝑉 sin 𝛾 cos𝛹

𝑑𝛾

𝑑𝑡− 𝑉 cos 𝛾 sin𝛹

𝑑𝛹

𝑑𝑡

+𝑉2

𝑟cos 𝛾 cos𝛹 (sin 𝛾 − cos 𝛾 sin𝛹 tan𝜙)] 𝑗̂

+ [cos 𝛾 sin𝛹𝑑𝑉

𝑑𝑡− 𝑉 sin 𝛾 sin𝛹

𝑑𝛾

𝑑𝑡+ 𝑉 cos 𝛾 cos𝛹

𝑑𝛹

𝑑𝑡

+𝑉2

𝑟cos 𝛾 (sin 𝛾 sin𝛹 − cos 𝛾 cos2𝛹 tan𝜙)] �̂�

Eq. 25

Next, by substituting equation Eq. 25 into Eq. 14, the scalar equations of motion

are found to be,

sin 𝛾𝑑𝑉


𝑑𝛾

𝑑𝑡−𝑉2

𝑟cos2 𝛾 =

𝐹𝑇𝑚sin 𝛾 +

𝐹𝑁𝑚cos𝜎 cos 𝛾

− 𝑔 + 2𝜔𝑉 cos 𝛾 cos𝛹 cos𝜙 + 𝜔2𝑟 cos𝜙

Eq. 26

T. M. Owens 19

sin 𝛾𝑑𝑉


𝑑𝛾

𝑑𝑡−𝑉2

𝑟cos2 𝛾 =




Eq. 27

sin 𝛾𝑑𝑉


𝑑𝛾

𝑑𝑡−𝑉2

𝑟cos2 𝛾 =




Eq. 28

Solving for the derivatives 𝑑𝑉

𝑑𝑡, 𝑑𝛾

𝑑𝑡, and

𝑑𝜓

𝑑𝑡,

𝑑𝑉

𝑑𝑡=𝐹𝑇𝑚− 𝑔 sin 𝛾 + 𝜔2 𝑟 cos𝜙 (sin 𝛾 cos𝜙 − cos 𝛾 sin𝜓 sin 𝜙) Eq. 29

𝑉𝑑𝛾

𝑑𝑡=𝐹𝑁𝑚cos𝜑 − 𝑔 cos 𝛾 +

𝑉2

𝑟cos 𝛾

+ 2𝜔𝑉 cos𝜓 cos𝜙 + 𝜔2 𝑟 cos𝜙 (cos 𝛾 cos𝜙 − sin 𝛾 sin𝜓 sin𝜙)

Eq. 30

𝑉𝑑𝜓

𝑑𝑡=𝐹𝑁𝑚

sin𝜑

cos 𝛾−𝑉2

𝑟cos 𝛾 cos𝜓 tan𝜙 + 2𝜔𝑉(tan𝛾 sin𝜓 cos𝜙 − sin 𝜙)

−𝜔2𝑟

cos 𝛾cos𝜓 sin 𝜙 cos𝜙

Eq. 31

The 𝜔2𝑟 term is the transport acceleration and the 2𝜔𝑉 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.


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 ℎ𝐴0 is the dissociation energy per unit mass, 𝐷 is the diffusion

coefficient and 𝑐𝐴 is the atomic mass fraction.

𝑞 = 𝑘∇𝑇 + ℎ𝐴0𝐷𝜌∇𝑐𝐴 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

𝑞 =𝑘

𝑐𝑝∇(ℎ + 𝑐𝐴ℎ𝐴0) Eq. 33

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 𝑖 per unit volume and time is,

∇ ∙ [𝜌(𝑞 + 𝑞 𝑖)𝑐𝑖] = 𝑤𝑖 Eq. 34

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

𝑞 𝑖 =𝐷𝑖𝑐𝑖∇𝑐𝑖 −

𝐷𝑖𝑇

𝑇∇𝑇 Eq. 35

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,

∇ ∙ (𝜌𝑞 ) = 0 Eq. 36

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

𝜌𝑞 ∆∑𝑐𝑖𝑒𝑖 = ∇ ∙ (𝑘∆𝑇) − ∇ ∙ (∑𝜌𝑞 𝑖 𝑐𝑖ℎ𝑖) +∑𝑤𝑖 ℎ𝑖0 + 𝑝∇ ∙ 𝑞 + 𝛷 Eq. 37

With 𝛷 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.


𝜌𝑞 ∇∑𝑐𝑖(ℎ𝑖 − ℎ𝑖0) = ∇ ∙ [𝑘∇𝑇 −∑𝜌𝑞 𝑖 𝑐𝑖(ℎ𝑖 − ℎ𝑖0)] + 𝑞 ∇𝑝 + 𝛷 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 𝑥 is tangential and 𝑦 is normal to the surface with 𝑢

and 𝑣 being the velocity components respectively.

(𝜌𝑟𝑢)𝑥 + (𝜌𝑟𝑣)𝑦 = 0 Eq. 39

𝜌𝑢𝑐𝑖𝑥 + 𝜌𝑣𝑐𝑖𝑦 = (𝐷𝑖𝜌𝑐𝑖𝑦 + 𝐷𝑖𝑇𝜌𝑐𝑖𝑇𝑦

𝑇)𝑦+ 𝑤𝑖 Eq. 40

𝜌𝑢ℎ𝑥 + 𝜌𝑣ℎ𝑦 = (𝑘𝑇𝑦)𝑦 + 𝑢𝑝𝑥 + 𝜇𝑢𝑦2

+ [∑𝐷𝑖𝜌(ℎ𝑖 − ℎ𝑖0)𝑐𝑖𝑦 +∑𝐷𝑖𝑇𝜌𝑐𝑖(ℎ𝑖 − ℎ𝑖0)𝑇𝑦

𝑇]𝑦

Eq. 41

The equation of motion is,

𝜌𝑢𝑢𝑥 + 𝜌𝑣𝑢𝑦 = −𝑝𝑥 + (𝜇𝑢𝑦)𝑦 Eq. 42

The equation Eq. 41 can be rewritten in terms of temperature instead of enthalpy

for simpler use with transport coefficients.

𝑐�̅�(𝜌𝑢𝑇𝑥 + 𝜌𝑣𝑇𝑦) = (𝑘𝑇𝑦)𝑦 + 𝑢𝑝𝑥 + 𝜇𝑢𝑦2

+∑𝑤𝑖(ℎ𝑖 − ℎ𝑖0) + [∑𝐷𝑖𝜌𝑐𝑖𝑦 +∑𝐷𝑖𝑇𝜌𝑐𝑖𝑇𝑦

𝑇]𝑦

Eq. 43

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

T. M. Owens 23

𝜌𝑢 (ℎ +𝑢2

2)𝑥

+ 𝜌𝑣 (ℎ +𝑢2

2)𝑦

= [𝑘

𝑐�̅�(ℎ +

𝑢2

2)𝑦

]

𝑦

+ 𝑢𝑝𝑥

+ 𝜇𝑢𝑦2 + [∑(𝐷𝑖𝜌 −

𝑘

𝑐�̅�) (ℎ𝑖 − ℎ𝑖0)𝑐𝑖𝑦 +∑𝐷𝑖𝑇𝜌𝑐𝑖(ℎ𝑖 − ℎ𝑖0)

𝑇𝑦

𝑇]𝑦

Eq. 44

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.

𝜂 ≡ (𝑟𝑢∞

√2𝜉)∫ 𝜌𝑑𝑦

𝑦

0

Eq. 45

𝜉 ≡ ∫ 𝜌𝑤𝜇𝑤𝑢∞𝑟2𝑑𝑥

𝑥

0

Eq. 46

𝜕𝑓

𝜕𝜂≡𝑢

𝑢∞; 𝑓 = ∫

𝜕𝑓

𝜕𝜂

𝜂

0

𝑑𝜂 Eq. 47

𝑔 =

(ℎ +𝑢2

2 )

ℎ𝑠 Eq. 48

𝜃 =𝑇

𝑇∞ Eq. 49


𝑠𝑖 =𝑐𝑖𝑐𝑖∞

Eq. 50

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

At the stagnation point, the equations for 𝑓, 𝑔, 𝜃 and 𝑠𝑖 are functions of 𝜂 as 𝜉

increases. Also, assuming a thermodynamic equilibrium, the equations Eq. 39, Eq.

40, Eq. 42 and Eq. 43 or Eq. 61 can be written as,

𝜌𝑣 = − [(√2𝜉𝑓𝜉 +𝑓

√2𝜉) 𝜉𝑥 +√2𝜉𝑓𝑦𝜂𝑥] 𝑟⁄ Eq. 51

[𝑙

𝑃𝑟(𝐿𝑖𝑠𝑖𝜂 +

𝐿𝑖𝑇𝑠𝑖𝑇𝜂

𝑇)]𝜂+ 𝑓𝑠𝑖𝜂 +

𝑤𝑖𝜌𝑐𝑖𝑠

[(2𝑑𝑢∞𝑑𝑥

)𝑠]−1

= 0 Eq. 52

(𝑙𝑓𝜂𝜂)𝜂+ 𝑓𝑓𝜂𝜂 +

1

2(𝜌𝑠𝜌− 𝑓𝜂

2) = 0 Eq. 53

(𝑐�̅�

𝑐�̅�𝑤

𝑙

𝑃𝑟)𝜂

+𝑐�̅�

𝑐�̅�𝑤𝑓𝜃𝜂 + [(2

𝑑𝑢∞𝑑𝑥

)𝑠]−1

∑𝑤𝑖𝜌

(ℎ𝑖 − ℎ𝑖0)

𝑐�̅�𝑤𝑇𝑠

+∑𝑐�̅�𝑖

𝑐�̅�𝑤

𝑐𝑖𝑠𝑙

𝑃𝑟(𝐿𝑖𝑠𝑖𝜂 +

𝐿𝑖𝑇𝑠𝑖𝜃𝜂

𝜃)𝜂𝜃𝜂 = 0

Eq. 54

(𝑙

𝑃𝑟𝑔𝜂)

𝜂+ 𝑓𝑔𝜂 + {

𝑙

𝑃𝑟∑[𝑐𝑖𝑠


ℎ𝑠] [(𝐿𝑖 − 1)𝑠𝑖𝜂 +


𝜃]}𝜂

= 0 Eq. 55

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,

𝑞 = (𝑘𝜕𝑇

𝜕𝑦)𝑦=0

+ [∑𝜌(ℎ𝑖 − ℎ𝑖0) (𝐷𝑖𝜕𝑐𝑖𝜕𝑦

+𝐷𝑖𝑇𝑐𝑖𝑇

𝜕𝑇

𝜕𝑦)]𝑦=0

Eq. 56

T. M. Owens 25

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

equation,

𝑞 = (𝑟𝑘𝑤𝜌𝑤𝑢∞𝑇∞

√2𝜉) [𝜃𝜂 +∑𝑐𝑖


𝑐�̅�𝑇∞(𝐿𝑖𝑠𝑖𝜂 +


𝜃)]𝜂=0

Eq. 57

At the stagnation point,

𝑟𝜌𝑤𝑢∞

√2𝜉= √

2

𝜈𝑤(𝑑𝑢∞𝑑𝑥

)𝑠 Eq. 58

Thereby allowing the heat transfer equation to be rewritten as,

𝑞 =𝑁𝑢

√𝑅𝑒√𝜌𝑤𝑢𝑤 (

𝑑𝑢∞𝑑𝑥

)𝑠

(ℎ𝑠 − ℎ𝑤)

𝑃𝑟 Eq. 59

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 𝜌𝜇 across the boundary-layer giving the equation,

𝑁𝑢

√𝑅𝑒= 0.67 (

𝜌𝑠𝜇𝑠𝜌𝑤𝜇𝑤

)0.4

Eq. 60

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,


ℎ𝐴0 = ∑ 𝑐𝑖𝑠(−ℎ𝑖0)

𝑎𝑡𝑜𝑚𝑠

∑ 𝑐𝑖𝑠𝑎𝑡𝑜𝑚𝑠

⁄ Eq. 61

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

by the equation,

𝑁𝑢

√𝑅𝑒(𝑁𝑢

√𝑅𝑒)𝐿=1

= 1 + (𝐿0.52 − 1)ℎ𝐷ℎ𝑠

Eq. 62

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,

𝑞 = 0.94(𝜌𝑠𝜇𝑠)0.1(𝜌𝑠𝑙𝜇𝑠𝑙)

0.4 [1 + (𝐿0.52 − 1)ℎ𝐷ℎ𝑠]√(

𝑑𝑢∞𝑑𝑥

)𝑠 Eq. 63

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

(𝑑𝑢∞𝑑𝑥

)𝑠=1

𝑅√2(𝑝𝑠 − 𝑝∞)

𝜌𝑠 Eq. 64

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,

𝑞𝑠 = 0.94 (1 −ℎ𝑠ℎ𝑠𝑙) (𝜌𝑠𝜇𝑠)

0.1(𝜌𝑠𝑙𝜇𝑠𝑙)0.4ℎ𝑠𝑙√(

𝑑𝑢∞𝑑𝑥

)𝑠[1

+ 0.45(𝐿 − 1)ℎ𝐷ℎ𝑠𝑙]

Eq. 65

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,

(𝑑𝑢𝑒𝑑𝑥

)𝑠=1

𝑅ℎ√2𝑝𝑠𝑙𝜌𝑠𝑙

Eq. 66

The variation of the stagnation point heating should vary with respect to density

and velocity by approximately √𝜌 𝑢3 (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


experimental data from hypersonic shock tubes, the equation for the stagnation

point heating flux isRef. 4,

𝑞𝑠 =17600

√𝑅ℎ√𝜌∞𝜌𝑠𝑙

(𝑈∞𝑈𝑟𝑒𝑓

)

3.15

(ℎ𝑠 − ℎ𝑏ℎ𝑠𝑙 − ℎ𝑟𝑒𝑓

) Eq. 67

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


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

W/m2.

𝑞𝑟𝑎𝑑 = 휀𝜎𝑠𝑏𝑐𝑇4 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

Eq. 69Ref. 8. This fraction, the area averaging factor (0 < 𝑘2 < 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.

𝑞𝑟𝑎𝑑 = 𝑘2𝑞𝑠 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.

�̇� = (𝑘2𝑞𝑠 − 휀𝑏𝜎𝑠𝑏𝑐𝑇𝑏4)𝐴𝑤 Eq. 70

The wall temperature, because of the lumped-mass assumption, is the

temperature of the body. This is found by using the following,

𝑇𝑏 =

{

𝑄

𝑚𝑏𝐶�̅�𝑏, 𝑓𝑜𝑟 𝑄 < 𝑚𝑏𝐶�̅�𝑏𝑇𝑚𝑒𝑙𝑡

𝑇𝑚𝑒𝑙𝑡, 𝑓𝑜𝑟 𝑄 ≥ 𝑚𝑏𝐶�̅�𝑏𝑇𝑚𝑒𝑙𝑡

Eq. 71

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


𝒂⊕ 6378137.0 𝑚 Semi-major axis

𝒃⊕ 6356752.3142 𝑚 Semi-minor axis

𝒇 1/298.257223563 Flattening

𝒆⊕ 8.1819190842622 ∙ 10−2 First Eccentricity

𝒆⊕𝟐 6.69437999014 ∙ 10−3 First Eccentricity Squared


𝒆⊕́ 8.2094437949696 ∙ 10−2 Second Eccentricity

𝒆⊕́𝟐 6.73949674228 ∙ 10−3 Second Eccentricity Squared

𝝁 3.986004418 ∙ 1014 𝑚2

𝑠2 Gravitational Constant

𝝎 7292115 ∙ 10−11 Angular Velocity

𝑱𝟐 1.082629989 ∙ 10−3 Second Degree Zonal Harmonic

𝑱𝟑 −2.53881 ∙ 10−6 Third Degree Zonal Harmonic

𝑱𝟒 −1.61 ∙ 10−6 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,

𝑟 = √𝑒2 + 𝑓2 + 𝑔2 Eq. 72

𝐺𝑒 = −𝜇 [𝑒

𝑟3−3𝑎⊕

2𝐽2

2(5𝑒𝑔2

𝑟7−𝑒

𝑟5) −

5𝑎⊕3𝐽3

2(7𝑒𝑔3

𝑟9−3𝑒𝑔

𝑟7)

−15𝑎⊕

4𝐽4

8(21𝑒𝑔4

𝑟11−14𝑒𝑔2

𝑟9+𝑒

𝑟7)]

Eq. 73

T. M. Owens 33

𝐺𝑓 = −𝜇 [𝑓

𝑟3−3𝑎⊕

2𝐽2

2(5𝑓𝑔2

𝑟7−𝑓

𝑟5) −

5𝑎⊕3𝐽3

2(7𝑓𝑔3

𝑟9−3𝑓𝑔

𝑟7)

−15𝑎⊕

4𝐽4

8(21𝑓𝑔4

𝑟11−14𝑓𝑔2

𝑟9+𝑓

𝑟7)]

Eq. 74

𝐺𝑔 = −𝜇 [𝑔

𝑟3−3𝑎⊕

2𝐽2

2(5𝑔3

𝑟7−3𝑔3

𝑟5) −

𝑎⊕3𝐽3

2(35𝑔4

𝑟9−30𝑔2

𝑟7+3

𝑟5)

−5𝑎⊕

4𝐽4

8(63𝑔5

𝑟11−70𝑔3

𝑟9+15𝑔

𝑟7)]

Eq. 75

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

the local gravity,

𝑔 = √𝐺𝑒2 + 𝐺𝑓

2 + 𝐺𝑔2 Eq. 76


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,

𝑧𝑔 =𝑧𝑅⊕𝑧 + 𝑅⊕

Eq. 77

The values from the row of Table 2 matching the altitude less than or equal to the

geodetic altitude will be used; indicated by subscript𝑡. First, the local temperature

is found by the following with values from the table having subscript t,

𝑇∞ = 𝑇𝑡 + 𝐺𝑡(𝑧𝑔 − 𝑧𝑡) Eq. 78

Next, the local pressure,

𝛿 =

{

𝑃𝑡𝑒𝑥𝑝

−𝐺𝑀∙(𝑧𝑔−𝑧𝑡)

𝑇0 , 𝑓𝑜𝑟 𝐺𝑡 = 0

𝑃𝑡 (𝑇0𝑇)

𝐺𝑀𝐺𝑡, 𝑓𝑜𝑟 𝐺𝑡 ≠ 0

Eq. 79

𝑃∞ =𝑃0𝛿

Eq. 80

Then, the local density,

𝜌∞ = 𝜌0𝛿

𝑇∞ 𝑇0⁄ Eq. 81

Finally, the local speed of sound,

𝑎 = √𝛾𝑅∗𝑇∞ 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.


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

Altitude

[km] 𝐥𝐨𝐠 𝒑 𝒅

𝒅𝒛𝐥𝐨𝐠𝒑 𝐥𝐨𝐠 𝝆

𝒅

𝒅𝒛𝐥𝐨𝐠 𝝆

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,

𝑖 =log 𝑝𝑡+1 − log 𝑝𝑡

𝑧𝑡+1 − 𝑧𝑡

𝑗 =𝑧𝑔 − 𝑧𝑡

𝑧𝑡+1 − 𝑧𝑡

𝑘 = 1 − 𝑗

𝑝 = 𝑘 log 𝑝𝑡 + 𝑗 log 𝑝𝑡+1

− 𝑘𝑗(𝑧𝑡+1 − 𝑧𝑡) [𝑘 (𝑖 −𝑑

𝑑𝑧log 𝑝𝑡) − 𝑗 (𝑖 −

𝑑

𝑑𝑧log 𝑝𝑡+1)]

Eq. 83

The kinetic temperature can be found by,

𝑇∞ =

{

186.8673, 𝑓𝑜𝑟 86 < 𝑧𝑔 ≤ 86 km

263.1905 + 12√1 − (𝑧𝑔 − 91

19.9429)

2

, 𝑓𝑜𝑟 91 < 𝑧𝑔 < 110 km

240 + 12(𝑧𝑔 − 110), 𝑓𝑜𝑟 110 ≤ 𝑧𝑔 ≤ 120 km

1000 − (1000 − 120) exp [−0.01875(𝑧𝑔 − 120)𝑅⊕ + 120

𝑅⊕ + 𝑧𝑔] , 𝑓𝑜𝑟 110 < 𝑧𝑔 ≤ 1000 km

Eq. 84

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

𝑇∞ = 1000

𝑝 = 1 ∙ 10−20𝑝0

𝜌 = 1 ∙ 10−21𝜌0

Eq. 85


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, 𝑎⊕, 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,

𝐶𝐿∗ = √

𝐶𝐷0𝐾

Eq. 86

𝐶𝐷∗ = 2𝐶𝐷0 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,

𝑉2

𝑉𝑐2=

𝑉2

𝑔0𝑎⊕ Eq. 88

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, 𝛽, times the mean radius

of earth, 𝑅⊕, 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 𝑒;

in the case of earth, it is roughly 8.5 km for the isothermal pressure gradient).

sin 𝛾 =2 (1 −

𝑉2

𝑉𝑐2)

𝐶𝐿∗

𝐶𝐷∗ [𝛽𝑅⊕ (1 −

𝑉2

𝑉𝑐2) + (2 −

𝑉2

𝑉𝑐2)]

Eq. 89

−1 ≤ sin 𝛾 ≤ 1 Eq. 90

The glide coefficient of lift is then computed. The radius, 𝑟, in this case is the

magnitude of the EFG position coordinates for the vehicle.

𝐶𝐿 = (𝑔 −𝑉2

𝑟)

2𝑚

𝜌𝑉2 cos𝜑 𝑆 Eq. 91

The coefficient of lift is scaled by the difference in the final flight path angle and

the initial flight path angle, 𝑑𝛾.

𝑠𝑐𝑎𝑙𝑒 = min[1, |𝑑𝛾|] Eq. 92

−𝐶𝐿∗ < 𝐶𝐿(1 + 𝑠𝑖𝑔𝑛(𝑑𝛾) ∙ 0.1 ∙ 𝑠𝑐𝑎𝑙𝑒) < 𝐶𝐿

∗ 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.


𝑛 = 𝜌|𝐶𝐿|𝑉

2𝑆

2𝑔𝑚 Eq. 94

𝐶𝐿 = 𝑠𝑖𝑔𝑛(𝐶𝐿)𝑛𝑠2𝑔𝑚

𝜌𝑉2 Eq. 95

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

𝐶𝐷 = 𝐶𝐷0 + 𝐾𝐶𝐿2 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.

𝐹𝑇𝑚= −𝐶𝑑

𝑉2𝜌𝑆

2𝑚 Eq. 97

𝐹𝑁𝑚= 𝐶𝑙

𝑉2𝜌𝑆

2𝑚 Eq. 98

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

heading.

𝑑𝑉

𝑑𝑡=𝐹𝑇𝑚− 𝑔 sin 𝛾 + 𝜔2 𝑟 cos𝜙 (sin 𝛾 cos𝜙 − cos 𝛾 sin𝜓 sin 𝜙) Eq. 99

𝑉𝑑𝛾

𝑑𝑡=𝐹𝑁𝑚cos𝜑 − 𝑔 cos 𝛾 +

𝑉2

𝑟cos 𝛾

+ 2𝜔𝑉 cos𝜓 cos𝜙 + 𝜔2 𝑟 cos𝜙 (cos 𝛾 cos𝜙 − sin 𝛾 sin𝜓 sin𝜙)

Eq. 100

𝑉𝑑𝜓

𝑑𝑡=𝐹𝑁𝑚

sin𝜑

cos 𝛾−𝑉2

𝑟cos 𝛾 cos𝜓 tan𝜙 + 2𝜔𝑉(tan𝛾 sin𝜓 cos𝜙 − sin𝜙)

−𝜔2𝑟

cos 𝛾cos𝜓 sin 𝜙 cos𝜙

Eq. 101

T. M. Owens 41

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

𝑑𝑋

𝑑𝑡= 𝑉 cos 𝛾 cos𝜓 Eq. 102

𝑑𝑌

𝑑𝑡= 𝑉 cos 𝛾 sin𝜓 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.

�̇� =

[

𝑑𝑌

𝑑𝑡𝑎⊕⁄

𝑑𝑋

𝑑𝑡(𝑎⊕ cos𝜙)⁄

𝑉 sin 𝛾

𝑑𝜓

𝑑𝑡

cos 𝛾𝑑𝑉

𝑑𝑡− 𝑉 sin 𝛾

𝑑𝛾

𝑑𝑡

sin 𝛾𝑑𝑉

𝑑𝑡+ 𝑉 ∙ cos 𝛾

𝑑𝛾

𝑑𝑡]

Eq. 104

The differential equations expressed in �̇� 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.


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.


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

Misc…

𝑙 > 𝑤 ≫ 𝑡

𝑡 =𝑚𝑏

𝜌𝑏𝑙𝑤

𝑙 > 𝑤 > ℎ

𝑟 = √𝑤ℎ

𝜋

Wetted area, Aw 4𝜋𝑟2 2𝜋(𝑟 + 𝑙) 2𝑙𝑤 + 2𝑙𝑡 + 2𝑤𝑡 2𝜋𝑟(𝑟 + 𝑙)

Hypersonic continuum

drag coefficient, CDhc 0.92 0.720 + 0.326 (

2𝑟

𝑙) 1.84 0.720 + 0.326 (

2𝑟

𝑙)

Aerodynamic

reference area, S

𝜋𝑟2 2𝑟𝑙 𝑙𝑤 2𝑟𝑙

Heating radius, Rh 𝑟 𝑟 𝑤

2 𝑟

For all cases, the hypersonic continuum ballistic coefficient is,

𝛽ℎ𝑐 =𝑚𝑏

𝑆𝐶𝐷ℎ𝑐

Eq. 105

T. M. Owens 45

4.5.4 Stagnation Point Heating

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

𝐶𝑝∞ =

{

1373, 𝑓𝑜𝑟 𝑇𝑏 ≥ 2000 K

959.9 + 0.15377𝑇𝑏 + 2.636 ∙ 10−5𝑇𝑏

2, 𝑓𝑜𝑟 300 < 𝑇𝑏 < 2000 K

1004.7, 𝑓𝑜𝑟 𝑇𝑏 ≤ 300 K

Eq. 106

Adiabatic stagnation temperature,

𝑇𝑠 = 𝑇∞ +𝑈∞2

2𝐶𝑝∞ Eq. 107

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.

𝑄0 = 𝑚𝑏𝐶�̅�𝑏𝑇0 Eq. 108

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

initial temperature to the melt temperature.

𝑄𝑚𝑒𝑙𝑡 = 𝑚𝑏𝐶�̅�𝑏(𝑇𝑚𝑒𝑙𝑡 − 𝑇0) Eq. 109

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

𝑄𝑎 = 𝑚𝑏𝐶�̅�𝑏(𝑇𝑚𝑒𝑙𝑡 − 𝑇0) + 𝑚𝑏ℎ𝑓 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.


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.

𝑞𝑠 = 200006400√0.3048𝜌∞𝑅ℎ𝜌𝑠𝑙

(𝑇𝑠 − 𝑇𝑏𝑇𝑠 − 𝑇𝑟𝑒𝑓

)(𝑈∞𝑈𝑟𝑒𝑓

)

3.15

Eq. 111

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.

�̇� = (𝑘2𝑞𝑠 − 휀𝑏𝜎𝑠𝑏𝑐𝑇𝑏4)𝐴𝑤 Eq. 112

Heat content of the fragment body at time t,

𝑄(𝑡) = 𝑄0 +∫ �̇�𝑑𝑡𝑡

0

Eq. 113

Body bulk temperature, the right hand side of the inequality is the heat of melting

from the initial temperature of absolute zero.

𝑇𝑏 =

{

𝑄

𝑚𝑏𝐶�̅�𝑏, 𝑓𝑜𝑟 𝑄 < 𝑚𝑏𝐶�̅�𝑏𝑇𝑚𝑒𝑙𝑡

𝑇𝑚𝑒𝑙𝑡, 𝑓𝑜𝑟 𝑄 ≥ 𝑚𝑏𝐶�̅�𝑏𝑇𝑚𝑒𝑙𝑡

Eq. 114

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.

𝐿𝐹 =

{

0, 𝑓𝑜𝑟 𝑄𝑚𝑎𝑥 ≤ 𝑄𝑚𝑒𝑙𝑡

𝑄𝑚𝑎𝑥 − 𝑄𝑚𝑒𝑙𝑡𝑄𝑎 − 𝑄𝑚𝑒𝑙𝑡

, 𝑓𝑜𝑟 𝑄𝑚𝑒𝑙𝑡 < 𝑄𝑚𝑎𝑥 < 𝑄𝑎

1, 𝑓𝑜𝑟 𝑄𝑚𝑎𝑥 ≥ 𝑄𝑎

Eq. 115

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,

𝑚𝐿𝐹 = 𝑚𝑏(1 − 𝐿𝐹) 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.


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.

𝑚𝑖 = 𝜌4𝜋𝑟3

3−𝑚 Eq. 117

𝑆 = 𝜋 (3(𝑚𝐿𝐹 +𝑚𝑖)

4𝜋𝜌)

23⁄

Eq. 118

For the cylinder case, the interior mass is found. It is then used to obtain the initial

thickness by solving the next equation for 𝑡. Then, the internal dimensions are

used in the next equation solved for 𝑡 with the time varying demised mass. The

thicknesses are then used in the time varying area calculation.

𝑚𝑖 = 𝜋𝑟2𝑙𝜌 − 𝑚 Eq. 119

0 = −𝑡3(2𝑟𝑙)𝑡2 − (𝑟2 + 2𝑟𝑙)𝑡 + 𝑟2𝑙 −𝑚𝑖

𝜋𝜌 Eq. 120

𝑟𝑖 = 𝑟 − 𝑡 Eq. 121

𝑙𝑖 = 𝑙 − 𝑡 Eq. 122

0 = 𝑡3(2𝑟𝑖𝑙𝑖)𝑡2 + (𝑟𝑖

2 + 2𝑟𝑖𝑙𝑖)𝑡 + 𝑟𝑖2𝑙𝑖 −

𝑚𝐿𝐹

𝜋𝜌 Eq. 123

𝑆 = 2(𝑟𝑖 + 𝑡)(𝑙𝑖 + 𝑡) Eq. 124

In the flat plate case, the recession length, 𝛿, is solved by finding the zeroes of the

function with the time varying mass. It is then used in the area calculation.

0 = 𝛿2 − 𝛿(𝑙 + 𝑤) + 𝑙𝑤 −𝑚𝐿𝐹

𝑡𝜌 Eq. 125

T. M. Owens 49

𝑆 = (𝑙 − 𝛿)(𝑤 − 𝛿) Eq. 126

Solving for the box area starts with finding the interior mass and using that in the

next equation being solved for 𝑡. This is used to find the interior dimensions. Then

solving for 𝑡 with the liquid fraction masses the time varying thickness can be used

to find the area.

𝑚𝑖 = 𝑙𝑤ℎ𝜌 −𝑚 Eq. 127

0 = −𝑡3(𝑙 + 𝑤 + ℎ)𝑡2 − (𝑙ℎ + 𝑤ℎ + 𝑙𝑤)𝑡 + 𝑙𝑤ℎ −𝑚𝑖

𝜌 Eq. 128

𝑙𝑖 = 𝑙 − 𝑡 Eq. 129

𝑤𝑖 = 𝑤 − 𝑡 Eq. 130

ℎ𝑖 = ℎ − 𝑡 Eq. 131

0 = 𝑡3(𝑙𝑖 + 𝑤𝑖 + ℎ𝑖)𝑡2 + (𝑙𝑖ℎ𝑖 +𝑤𝑖ℎ𝑖 + 𝑙𝑖𝑤𝑖)𝑡 + 𝑙𝑖𝑤𝑖ℎ𝑖 −

𝑚𝐿𝐹

𝜌 Eq. 132

𝑆 = (𝑙𝑖 + 𝑡)√(𝑤𝑖 + 𝑡)(ℎ𝑖 + 𝑡)

𝜋 Eq. 133

4.5.6.3 Hazard Area Table

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

equation,

𝐻𝑟 = √𝑆

𝜋 Eq. 134


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,

𝐻𝑟 = 𝑚𝑎𝑥{𝐻𝑟0}√𝑆

𝑚𝑎𝑥{𝑆} 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 𝐶𝐷 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 𝐶𝐷 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.

β𝑀𝑎𝑐ℎ =𝑚𝑏

𝑆𝐶𝐷𝑀𝑎𝑐ℎ

Eq. 136

This is then used with the impact mass and area of the fragment to generate the

corrected Mach dependent coefficient of drag values.

𝐶𝐷𝛽 =𝑚𝑖𝑚𝑝𝑎𝑐𝑡

𝑆𝑖𝑚𝑝𝑎𝑐𝑡β𝑀𝑎𝑐ℎ Eq. 137

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.


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.


Table 5: Example Fragment

Name Shape Radius [m] Length [m] Material

2.6 Cylinder 0.85344 0.85344 Aluminum (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

Figure 11: Single Fragment, Altitude vs. Time

Figure 12: Single Fragment, Altitude vs. Range

0 200 400 600 800 10000

20

40

60

80

100

120Altitude [km] vs. Time [s]

Altitu

de

[km

]

Time [s]

0 500 1000 1500 2000 2500 3000 3500 40000

20

40

60

80

100

120Altitude [km] vs. Range [km]

Altitu

de

[km

]

Range [km]


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.

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.

0 50 100 150 200 250 300 350 400 450 500-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

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

He

at

Flu

x [

W/m

2-s

]

Time [s]

T. M. Owens 57

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.

0 50 100 150 200 250 300 350 400 450 500300

400

500

600

700

800

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

Fra

gm

en

t B

ulk

Te

mp

era

ture

[K

]

Time [s]


Figure 15: Varying Breakup Altitude, Altitude vs. Time

Figure 16: Varying Breakup Altitude, Altitude vs. Range

0 200 400 600 800 10000

20

40

60

80

100


Altitu

de

[km

]

Time [s]

0 500 1000 1500 2000 2500 3000 3500 40000

20

40

60

80

100


Altitu

de

[km

]

Range [km]

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.

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.

0 50 100 150 200 250 300 350 400 450 500-2

-1

0

1

2

3

4

5x 10


He

at

Flu

x [

W/m

2-s

]

Time [s]

120 km

110 km

100 km

90 km

80 km

70 km

60 km

50 km


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.

0 50 100 150 200 250 300 350 400 450 500300

400

500

600

700

800


Fra

gm

en

t B

ulk

Te

mp

era

ture

[K

]

Time [s]

120 km

110 km

100 km

90 km

80 km

70 km

60 km

50 km

T. M. Owens 61

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.

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Liquid Fraction vs. Time [s]

Liq

uid

Fra

ctio

n

Time [s]

120 km

110 km

100 km

90 km

80 km

70 km

60 km

50 km


Figure 20: Varying Temperature, Temperature vs. Time

Figure 21: Varying Temperature, Liquid Fraction vs. Time

0 50 100 150 200 250 300300

400

500

600

700

800


Fra

gm

en

t B

ulk

Te

mp

era

ture

[K

]

Time [s]

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Liq

uid

Fra

ctio

n

Time [s]

300 K

400 K

500 K

600 K

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.

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.

0 50 100 150 200 250 300-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10


He

at

Flu

x [

W/m

2-s

]

Time [s]

300 K

400 K

500 K

600 K


Figure 23: Varying Velocity, Altitude vs. Time

Figure 24: Varying Velocity, Altitude vs. Range

0 200 400 600 800 10000

20

40

60

80

100


Altitu

de

[km

]

Time [s]

7,500 km/s

7,000 km/s

6,500 km/s

6,000 km/s

5,500 km/s

0 500 1000 1500 20000

20

40

60

80

100


Altitu

de

[km

]

Range [km]

7,500 km/s

7,000 km/s

6,500 km/s

6,000 km/s

5,500 km/s

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.

Figure 25: Varying Velocity, Heat Flux vs. Time

0 50 100 150 200 250 300-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10


He

at

Flu

x [

W/m

2-s

]

Time [s]

7,500 km/s

7,000 km/s

6,500 km/s

6,000 km/s

5,500 km/s


Figure 26: Varying Velocity, Temperature vs. Time

Figure 27: Varying Velocity, Liquid Fraction vs. Time

0 50 100 150 200 250 300300

400

500

600

700

800


Fra

gm

en

t B

ulk

Te

mp

era

ture

[K

]

Time [s]

7,500 km/s

7,000 km/s

6,500 km/s

6,000 km/s

5,500 km/s

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Liq

uid

Fra

ctio

n

Time [s]

7,500 km/s

7,000 km/s

6,500 km/s

6,000 km/s

5,500 km/s

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.

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

0 200 400 600 800 10000

20

40

60

80

100


Altitu

de

[km

]

Time [s]

-0.5 deg

-3.0 deg

-5.5 deg


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.

0 500 1000 1500 2000 25000

20

40

60

80

100


Altitu

de

[km

]

Range [km]

-0.5 deg

-3.0 deg

-5.5 deg

T. M. Owens 69

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

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

0 50 100 150 200 250 300-2

-1

0

1

2

3

4

5x 10


He

at

Flu

x [

W/m

2-s

]

Time [s]

-0.5 deg

-3.0 deg

-5.5 deg

0 50 100 150 200 250 300300

400

500

600

700

800


Fra

gm

en

t B

ulk

Te

mp

era

ture

[K

]

Time [s]

-0.5 deg

-3.0 deg

-5.5 deg


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.

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Liq

uid

Fra

ctio

n

Time [s]

-0.5 deg

-3.0 deg

-5.5 deg

T. M. Owens 71

Figure 33: Varying Materials, Heat Flux vs. Time

Figure 34: Varying Materials, Temperature vs. Time

0 50 100 150 200 250 300 350 400 450 500-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10


He

at

Flu

x [

W/m

2-s

]

Time [s]

Aluminum

Titanium

Graphite-Epoxy

0 50 100 150 200 250 300 350 400 450 500300

400

500

600

700

800

900


Fra

gm

en

t B

ulk

Te

mp

era

ture

[K

]

Time [s]

Aluminum

Titanium

Graphite-Epoxy


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.

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

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Liq

uid

Fra

ctio

n

Time [s]

Aluminum

Titanium

Graphite-Epoxy

T. M. Owens 73

larger range to impact and impact earlier as they carry more speed being less

effected by aerodynamic forces.

Figure 36: Varying Mass, Altitude vs Time

0 200 400 600 800 10000

20

40

60

80

100


Altitu

de

[km

]

Time [s]

50 kg

100 kg

150 kg

250 kg

400 kg


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.

0 500 1000 1500 2000 2500 30000

20

40

60

80

100


Altitu

de

[km

]

Range [km]

50 kg

100 kg

150 kg

250 kg

400 kg

T. M. Owens 75

Figure 38: Varying Mass, Heat Flux vs. Time

Figure 39: Varying Mass, Temperature vs. Time

0 50 100 150 200 250 300 350 400-2

0

2

4

6

8

10x 10


He

at

Flu

x [

W/m

2-s

]

Time [s]

50 kg

100 kg

150 kg

250 kg

400 kg

0 50 100 150 200 250 300 350 400300

400

500

600

700

800

900


Fra

gm

en

t B

ulk

Te

mp

era

ture

[K

]

Time [s]

50 kg

100 kg

150 kg

250 kg

400 kg


Figure 40: Varying Mass, Liquid Fraction vs. Time

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


Liq

uid

Fra

ctio

n

Time [s]

50 kg

100 kg

150 kg

250 kg

400 kg

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 /Width [m]

Length [m] Height [m] Demise Alt [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 Attachment

Aluminum 7075-T6 Cylinder 3 0.1 4 76.1

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.


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 Averaging Factor 𝐤𝟐

Liquid Fraction

Computational Model Demise Alt

[km]

DAS 2.0 Demise Alt [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.


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

42 nmi, -0.5 deg

Model 25000 ft/s 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

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

Table 25000 ft/s 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

5


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

Model 25000 ft/s 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

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

Table 25000 ft/s 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

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

5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0


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

25000 ft/s

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

4

5


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

25000 ft/s

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

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


r\l 1 2.5 5 10 20 30 1 2.5 5 10 20 30 1 2.5 5 10 20 30 42 nmi, -0.5 deg

Model 25000 ft/s 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

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


0.5 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

5 0.0


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

Model 25000 ft/s 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

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 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

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

5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0


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

25000 ft/s

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

4 0.3 0.3 0.4 0.3

5 0.3


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

25000 ft/s

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

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


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 LOX line

Cylinder 0.064008 0.36576 Aluminum (generic)


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


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



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


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


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.


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


accomplish this, like JARSS MP, so this computational model is probably more

beneficial to fragment demise analysis.

T. M. Owens 91

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Ref. 6

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T. M. Owens 93

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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.

Material

Density

(kg/m3)

Specific

Heat

(J/kg-K)

Heat of

Fusion

(J/kg)

Melt

Temperature

(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

Material

Density

(kg/m3)

Specific

Heat

(J/kg-K)

Heat of

Fusion

(J/kg)

Melt

Temperature

(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


Material

Density

(kg/m3)

Specific

Heat

(J/kg-K)

Heat of

Fusion

(J/kg)

Melt

Temperature

(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

Material

Density

(kg/m3)

Specific

Heat

(J/kg-K)

Heat of

Fusion

(J/kg)

Melt

Temperature

(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

Hydride

6086.8 418.7 131419 6086.8

Water 999 5490.55 0.1 273

Zerodur 2530 2487.1 250000 1424

Zinc 7144.2 405.3 100942 692.6


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,

𝑞𝑠𝑡𝑎𝑔 = 1.83 ∙ 10−4√𝜌∞𝑅ℎ(1 − (𝐶�̅�𝑏 − 𝑇𝑤)

12𝑈∞

2)𝑈∞

3 Eq. 138

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

𝑞𝑠𝑡𝑎𝑔 = 1.7623 ∙ 10−4√

𝜌∞𝑅ℎ𝑈∞3 Eq. 139

Scott stagnation point heating correlationRef. 20,

𝑞𝑠𝑡𝑎𝑔 = 1.83 ∙ 10−4√𝜌∞𝑅ℎ(𝑈∞104

)3.05

Eq. 140

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

𝑞𝑠𝑡𝑎𝑔 = 1.83 ∙ 10−8√𝜌∞𝑅ℎ(1 − (𝐶�̅�𝑏 − 𝑇𝑤)

12𝑈∞

2)𝑈∞

3 Eq. 141

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

T. M. Owens 99

𝑞𝑠𝑡𝑎𝑔 = 1.35 ∙ 10−8√

𝜌∞𝑅ℎ(1 − (𝐶�̅�𝑏 − 𝑇𝑤)

12𝑈∞

2)𝑈∞

3.04 Eq. 142

Tauber-Sutton formula for radiative heatingRef. 7,

𝑞𝑟𝑎𝑑 = 𝑅ℎ𝑎 ∙ 𝜌∞

𝑏 ∙ 𝑓(𝑈∞) Eq. 143

{ 0 ≤ 𝑎 ≤ 1, 𝑓𝑜𝑟 𝐸𝑎𝑟𝑡ℎ 𝑟𝑒𝑒𝑛𝑡𝑟𝑦𝑎 = 0.526, 𝑓𝑜𝑟 𝑀𝑎𝑟𝑠 𝑟𝑒𝑒𝑛𝑡𝑟𝑦

{ 𝑏 = 1.22, 𝑓𝑜𝑟 𝐸𝑎𝑟𝑡ℎ 𝑟𝑒𝑒𝑛𝑡𝑟𝑦𝑏 = 1.19, 𝑓𝑜𝑟 𝑀𝑎𝑟𝑠 𝑟𝑒𝑒𝑛𝑡𝑟𝑦

𝑓(𝑈∞) ≅ 𝑈∞7 Eq. 144

Rate of mass loss to ablation,

𝑚𝑏̇ = −�̇�

ℎ𝑓 Eq. 145


Trajectory Site Direction Cosines

𝑥 =𝜋

2− 𝛷

𝑀𝑥 = [1 0 00 cos 𝑥 sin 𝑥0 − sin 𝑥 cos 𝑥

]

𝑧 =2𝜋

3− 𝜃

𝑀𝑧 = [cos 𝑧 − sin 𝑧 0sin 𝑧 cos 𝑧 00 0 1

]

𝐷𝐶 = 𝑀𝑥𝑀𝑧

Eq. 146

Rotate using the azimuth,

𝐷𝐶 = 𝐷𝐶 [cos𝛹 − sin𝛹 0sin𝛹 cos𝛹 00 0 1

] Eq. 147

ECEF Coordinates to XYZ Coordinates

Translate origin to site location

𝑒 = 𝑒 + 𝑒𝑠𝑖𝑡𝑒

𝑓 = 𝑓 + 𝑓𝑠𝑖𝑡𝑒

𝑔 = 𝑔 + 𝑔𝑠𝑖𝑡𝑒

Eq. 148

Rotate position and velocity,

[𝑥 𝑦 𝑧] = 𝐷𝐶′[𝑒 𝑓 𝑔]

[�̇� �̇� �̇�] = 𝐷𝐶′[�̇� 𝑓̇ �̇�] Eq. 149

T. M. Owens 101

ECEF Coordinates to Aeronautical Coordinates

𝑊2 = 𝑒2 + 𝑓2

𝑍2 = 𝑒2 + 𝑓2

𝑅12 = 𝑊2 + 𝑍2

𝑅22 = 𝑊2 + (𝑍2

1 − 𝑒⊕)

𝑆2 = (√𝑅12𝑎⊕

− 1)𝑅12𝑅22

𝑠𝑒2 =𝑒⊕́

𝑅12

𝑆 = 𝑆2(1 + 1.5𝑆2𝑊2𝑍2𝑠𝑒22)

𝑣1 = 1 + 𝑆

𝑔𝑠 =𝑔𝑣1

1 − 𝑒⊕́ + 𝑆

Eq. 150

Latitude,

𝜙 = tan−1𝑔𝑠

√𝑊2

Eq. 151

Longitude,

𝜃 = tan−1𝑓

𝑒 Eq. 152

Altitude,

𝑧 = √𝑊2 + 𝑔𝑠2 −√𝑊2 + 𝑔𝑠2

𝑣1 Eq. 153

Heading,

𝜓𝑖 = tan−1𝑥�̇�𝑦�̇�

Eq. 154

Ground Speed,


𝑣𝑖 = √𝑥�̇�2 + 𝑦�̇�

2 Eq. 155

Vertical Speed, 𝑧�̇� = 𝑧�̇� Eq. 156

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