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
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.
iii
iv
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
2
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
Impact Integration ............................................................................................11
2.1
3
Equations of Motion ..................................................................................11
2.1.1
Relative Angular Motion ....................................................................12
2.1.2
Equations for Flight Over a Rotating Spherical Earth ........................14
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
v
3.1.4
3.2
Detra, Kemp and Riddell Correlation ........................................................27
3.2.1
4
5
Equilibrium Boundary Layer ...............................................................25
Radiation Heat Balance ......................................................................29
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
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
vi
5.1.6
Varying Materials of Debris Fragment ...............................................70
5.1.7
Varying Mass of Debris Fragment ......................................................72
5.2
5.2.1
DAS 2.0 ...............................................................................................77
5.2.2
Aerospace Survivability Tables...........................................................78
5.3
6
Model Comparisons ..................................................................................77
Input and Output Debris Fragment Catalog ..............................................82
Conclusions .......................................................................................................86
6.1
Practical Application ..................................................................................86
6.2
Validation...................................................................................................87
6.3
Performance ..............................................................................................88
6.4
Possible Future Work ................................................................................89
References................................................................................................................91
Appendix ..................................................................................................................94
Appendix A: Material Properties ..........................................................................94
Appendix B: Supplemental Algorithms ................................................................98
Alternate Correlations ......................................................................................98
Trajectory Site Direction Cosines ................................................................... 100
ECEF Coordinates to XYZ Coordinates ........................................................... 100
ECEF Coordinates to Aeronautical Coordinates ............................................ 101
Appendix C: MATLAB Code...................................... Error! Bookmark not defined.
demiseUtility.m.................................................... Error! Bookmark not defined.
demise.m ............................................................. Error! Bookmark not defined.
glideDerivatives.m ............................................... Error! Bookmark not defined.
Atmosphere1976.m ............................................. Error! Bookmark not defined.
createEarth.m ...................................................... Error! Bookmark not defined.
vii
getGravity.m ........................................................ Error! Bookmark not defined.
fragmentExporter.m ............................................ Error! Bookmark not defined.
fragmentImporter.m............................................ Error! Bookmark not defined.
viii
List of Keywords and Abbreviations
Keyword/Abbreviation
Definition
AST
Office of Commercial Space Transportation
Boeing X-37 OTV
CAIB
Casualty Area
Orbital Test Vehicle. Unmanned spacecraft, launches
as rocket lands as a space plane
Columbia Accident Investigation Board
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
JARSS MP
MATLAB
Area where a debris fragment will cause a human
casualty, same as casualty area
Joint Advance Range Safety System: Mission Planning
Programing language and integrated development
environment (IDE) developed by MathWorks
ix
Micro-casualties
Mission Analyst
Mollier diagram
Risk of human casualty due to debris hazard. Equal to
1*10-6 casualty per event, see Ec
Someone who performs a flight safety analysis for a
mission, see Risk Analyst
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
SCARAB
WGS 84
Someone who performs a flight safety analysis for a
mission, see Mission Analyst
Spacecraft Atmospheric Re-entry and Aerothermal
Break-up
World geodetic Survey of 1984
x
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
xi
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
xii
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
𝑪∗𝑳
Coefficient of lift at the smallest glide angle
𝑪𝑫
Coefficient of drag
𝑪∗𝑫
Coefficient of drag at the smallest glide angle
𝑪𝑫 𝟎
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
𝒎
Induced drag factor, function of Mach
kg
Mass of the vehicle
xiii
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
xiv
Symbols for Fay-Riddell Stagnation Point Heating
Variable
Value
Units
Description
Mass fraction of component 𝑖
𝒄𝒊
𝒄𝒑
J
kgK
Specific heat per unit mass at constant pressure
𝑫
m2
s
Molecular diffusion coefficient
𝑫𝑻
m2
sK
Thermal diffusion coefficient
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
𝒆
𝑳
𝒆𝒊
Lewis Number: 𝐷𝑖 𝜌𝑐̅𝑝 ⁄𝑘 ratio of the rate of thermal
diffusivity to the mass diffusivity, thermal Lewis
number
𝑳𝒊 𝑻
𝝁
Pa ∙ s
Absolute viscosity
Nusselt Number: 𝑞𝑥𝑐̅𝑝𝑤 ⁄𝑘𝑤 (ℎ𝑠 − ℎ𝑤 ) ratio of
convective to conductive heat transfer
𝑵𝒖
𝝂
Pa ∙ s
𝒑
Pa
Kinematic viscosity
Pressure
xv
𝑷𝒓
Prandtl Number: 𝑐̅𝑝 𝜇 ⁄𝑘 ratio of momentum diffusivity
to thermal diffusivity, value for air
0.71
𝜱
Dissipation function
W
m2
Heat flux
m
s
Vector mass velocity, vector diffusion velocity
𝒓
m
Cylindrical radius of body
𝑹𝒉
m
Body nose radius, radius of heating
𝒒
⃗
𝒒
⃗𝒊
𝒒
Reynolds Number: 𝑢𝑒 𝑥 ⁄𝑣𝑤 ratio of the inertial to
viscous forces
𝑹𝒆
𝝆
kg
m3
𝑻
K
Absolute temperature
𝒖
m
s
𝑥 component of velocity
𝒗
m
s
𝑦 component of velocity
𝒘𝒊
kg
m3 s
𝒙
m
s
Distance along meridian profile
𝒚
m
s
Distance normal to the surface
Mass density
Mass rate of formation of component 𝑖 per unit
volume and time
xvi
Symbols for Aero-thermal Demise
Variable
Value
Units
Description
𝒂
m
s
Local speed of sound
𝑨𝒘
m2
Wetted area
𝜷𝒉𝒄
kg
m2
Hypersonic continuum ballistic coefficient
𝑪𝑫𝒉𝒄
Hypersonic continuum coefficient of drag
𝑪 𝑫𝜷
Coefficient of drag corrected for ballistic coefficient
𝑪𝑫𝑴𝒂𝒄𝒉
Coefficient of drag from fragment drag tables
̅𝒑
𝑪
𝒃
𝑪𝒑∞
1.0045 ∙ 103
𝜹
𝜺𝒃
9.80665
Specific heat capacity of air
m
Recession length, flat plate
m
s2
radians
𝒉𝒊
𝒉𝒇
𝑯𝒓 𝟎
𝒌𝟐
𝒍
J
kgK
Surface emissivity of the fragment
𝜸
𝑯𝒓
Mean Specific heat capacity of the fragment
𝜺𝒘
𝒈𝟎
𝒉
J
kgK
𝒍𝒊
Flight angle of the fragment
m
Fragment height and interior height, box
J
kg
Heat of fusion
m
Hazard radius & user defined hazard radius
Area-averaging factor (a less conservative 0.8 for
composites)
0.12
m
𝑳𝑭
𝒎𝒃
Standard gravitational acceleration
Fragment length and interior length, cylinder, flat plate
and box
Liquid fraction of the fragment
kg
Thermal mass of the fragment
xvii
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
m2 K 4
Stefan-Boltzmann constant
𝒒𝒔
W
m2
Stagnation heat flux
𝒒𝒓𝒂𝒅
W
m2
Radiation heat flux
𝑸̇
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
m
Radius and interior radius of fragment, sphere and
cylinder
m
Mean radius of Earth
m
Radius of heating
J
K
Gas constant for air
𝒓
𝒓𝒊
𝑹⊕
6378.1 ∙ 103
𝑹𝒉
𝑹∗
287
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
K
Reference temperature
𝑻𝒓𝒆𝒇
300
xviii
𝑻𝒔
K
Adiabatic stagnation temperature
𝑻𝒘
K
Hot wall temperature of the fragment
m
s
Reference velocity, 26000 ft/s
m
s
Free stream velocity
m
Fragment width and interior width, flat plate and box
m
Altitude position of the fragment
𝑼𝒓𝒆𝒇
𝑼∞
𝒘
𝒘𝒊
𝒛
7924.8
xix
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 aerothermal demise of reentry debris. With simple inputs the model generates a
debris catalog that can be used by other risk analysis tools. There is also the
possibility that the impact integrator could be used within an existing tool in order
to consider demise of an existing debris catalog.
To understand what the work in this thesis is attempting to accomplish, it helps to
understand how current tools work to estimate reentry debris survivability and
casualty risk. The Debris Assessment Software 2.0 (DAS 2.0) and Orbital Reentry
Survival Analysis Tool (ORSAT) are used by National Aeronautics and Space
Adminstration (NASA) and other American launch providersRef. 17, Ref. 2. Aerospace
Survivability Tables were developed for the Federal Aviation Authority (FAA) and
Office of Commercial Space Transportation (AST)Ref. 24. The Spacecraft
Atmospheric Re-entry and Aerothermal Break-up (SCARAB) tool is used by the
European Space Agency (ESA)Ref. 11. DAS 2.0 is a simplistic first order solution, like
this thesis, whereas ORSAT and SCARAB are pseudo-CFD programs with much
2
Aero-thermal Demise
greater complexity. The computational model is based in part on the algorithm
used to build the survivability table.
1.1
DAS 2.0: Debris Assessment Software 2.0
The Debris Assessment Software is a NASA utility built to perform a variety of
orbital debris assessments according to the NASA Technical Standard 8719.14,
Process for Limiting Orbital DebrisRef. 13. The technical standards are a set of
mission requirements which can govern the acceptance of a mission for launch.
The reentry-survivability model, which checks requirement 4.7-1 from the
technical standard, is the portion relevant to this thesis. The safety guideline is in
the NASA Safety Standard 1740.14, Guidelines and Assessment Procedures for
Limiting Orbital Debris, and it states that "the total debris casualty area for
components and structural fragments will not exceed 8 m2." This equates to 100
micro-casualties per reentry event or 1:10,000Ref. 16.
Figure 1 shows the output from DAS2.0’s assessment of requirement 4.7-1 for the
example missions. The top portion summarizes the inputs. The mission LEO1 has a
number of sub-components or debris fragments with a variety of different
materials and shapes. The output shows that the mission LEO1 is non-compliant.
Several of the debris fragments survive to impact giving a total casualty area of
10.35 m2, just over the limit of 8 m2. The components each have a demise altitude,
casualty area and impact energy. The demise altitude is when the debris fragment
is fully ablated. If the demise altitude is 0, the fragment has survived to impact and
has a casualty area and impact energy.
As the simplest of the models, DAS 2.0 has an advantage in that it does not
require the user to have a detailed knowledge of the spacecraft’s geometry. Just
the overall shape (sphere, flat plate, cylinder or box), rough dimensions and
material for each of the fragment classes are required. However, DAS 2.0 is limited
T. M. Owens
3
in that its output does not allow for the creation of a demise modified debris
catalog because the output only has the impact mass and energy. Also DAS 2.0 is
not particularly useful for missions with planned reentry as it can only have a
single failure event at an altitude of 78 km. Its ease of use serves as a benchmark
for the computational model developed in this thesis.
Figure 1: Casualty Risk from Reentry Debris
4
1.2
Aero-thermal Demise
ORSAT: Orbital Reentry Survival Analysis Tool
The Orbital Reentry Survival Analysis Tool (ORSAT) is a much more complex and
higher fidelity tool for analyzing the thermal breakup of spacecraft during
reentryRef. 2. Like DAS 2.0, it is was developed to meet the requirements of NASA
standards, specifically NASA Technical Standard 8719.14, A Process for Limiting
Orbital Debris. Like DAS 2.0, the casualty risk from all reentry debris should be less
than 1:10,000. ORSAT uses integrated trajectory, atmospheric, aerodynamic,
aerothermodynamic and ablation algorithms to find the impact risk.
It is able to use three different atmospheres, 1976 U.S. standard, MSISe-90 and
GRAM-99 atmosphere. It can model either spinning or tumbling modes for the
fragment debris. The drag coefficients are found from the kinetic energy at
impact. The stagnation point continuum heating rates are found for spheres and
correlated for other geometries and flight regimes. To find the surface
temperature, it is able to use both a lumped mass and 1-D conduction. Demise is
assumed once the net heat absorbed reaches the heat of ablation for the
material.
Unfortunately, ORSAT is only available to the Orbital Debris Program Office at
Johnson Space Center so a true comparison cannot be made in this thesis.
However, there are some capabilities that ORSAT obviously has that this
computational model will not. Probably the most significant is ORSAT’s ability to
define more complex geometries and predict aerodynamic breakup.
1.3
Aerospace Survivability Tables
The Aerospace Survivability Tables is a set of tabular data on the demise of various
fragments that is part of a larger tool to estimate the total casualty expectation
made by The Aerospace CorporationRef. 24. The model has been validated against
T. M. Owens
5
the Columbia Accident Investigation Board (CAIB) report for casualty expectation
and impact probability and The Aerospace Corporation’s higher fidelity model
Atmospheric Heating and Breakup tool (AHaB) for survivability of debris. Their
model does not change the fragment properties as they impact nor does it
account for the wall gradient temperature. It uses the Detra-Kemp-Riddell
stagnation point heating correlation with a radiation heat balance to determine
the amount of ablation for the fragments. The algorithm sits nicely between
simple tools like DAS 2.0 and pseudo CFD tools like ORSAT which is why it was
chosen as a starting point for the computational model outlined in this thesis.
The tables cover three materials, aluminum 2024-T8xx, stainless steel 21-6-9 and
titanium (6 Al-4 V), and three hollow shapes, spheres, cylinders and flat plates.
There is the choice between 1541°R and 540°R as breakup temperature of the
debris. The tables also vary the breakup flight conditions. It covers from 46 to 30
Nmi in altitude with the flight path angles of -0.5, -3.5 and -5.5 degrees. The
velocities at each altitude are based on what is to be expected from a reentry
trajectory. As an example at 42 Nmi there is a choice between 25,000, 23,000 and
21,000 ft/s.
Figure 2 shows an example table from the Aerospace Survivability Tables for an
aluminum sphere. The rows are for radius in feet and the columns for weight in
pounds. The values in the tables are liquid fractions, how much of the mass of the
fragment has ablated, with one being fully demised and zero meaning the debris
fragment has survived intact to impact. The s in the table indicates that the
fragment has skipped off the atmosphere.
6
Aero-thermal Demise
Figure 2: Aluminum 2024-T8xx Tumbling Hollow Sphere Survivability Table (Ref. 24)
The obvious disadvantage to the tables is the limitation of having to choose the
fragment and breakup conditions that best fit for the survivability analysis instead
of computing it. The accuracy only to the first decimal is not a major concern as
the Detra-Kemp-Riddell stagnation point heating correlation is only accurate to
10% of the heating rate, at bestRef. 4.
1.4
SCARAB: Spacecraft Atmospheric Re-entry and
Aerothermal Break-up
The SCARAB tool is very similar to ORSAT. It was developed primarily for use by
the European Space Agency and partnersRef. 11. The program is broken into five
disciplines with different dependencies and couplings, flight dynamics,
T. M. Owens
7
aerodynamics, aerothermodynamics, structural analysis, thermal analysis and
deformation/disintegration/melting as seen in Figure 3. A spacecraft is defined by
geometric modeling, materials and physical modeling. SCARAB can use a variety of
panelized geometric primitives to construct more complex shapes and volume
elements.
Figure 3: Coupling and Dependence for Disintegration Prediction (Ref. 11)
SCARAB uses a materials database with 20 parameters that can be extended to a
three phase model (gas, liquid and solid). The aerodynamic and aero-heating
analysis is broken into free molecular, hypersonic continuum and rarefied
transitional flow regimes. The thermal analysis uses thin and thick thermal heating
which allows layered melting of solids with low heat conductivity. The latest
versions of SCARAB can also perform a finite element analysis to find the stress
resulting from inertial and aerodynamic forces. An example of the thermal
fragmentation and mechanical breakup as computed by SCARAB can be seen in
Figure 4.
As with ORSAT the main difference between this tool and the computational
model developed for this thesis is the complexity of geometry and the ability to
predict structural failure.
8
Aero-thermal Demise
Figure 4: Thermal Fragmentation and Mechanical Breakup (Ref. 11)
1.5
Computation Model
There is an obvious gap in complexity level between the computational model
developed in this thesis and those of tools like ORSAT and SCARAB. However most
of these deficiencies are not critical to performing a flight risk analysis.
The model in this thesis only takes into consideration the aero-thermal analysis of
the hypersonic continuum flow regime. This is adequate for approximating demise
as the reentry flight speeds are typically on the order of Mach 10 and maximum
aero-heating occurs from 80 to 50 kms of altitude. The ORSAT and SCARAB tools
are also designed for impact prediction of spacecraft and not designed for landing
reentry, whereas the model in this thesis is mainly concerned with the failure of
launch and reentry vehicles. Therefore, the very high altitude flight regimes are of
little importance because minimal heat transfer takes place at supersonic and
T. M. Owens
9
subsonic flight speeds; there is no real value added to include them in this
analysis.
While the inability to predict a breakup event may seem like a major weakness of
the code developed for this thesis, it is of minimal importance at the stage of risk
analysis for this computation model and its expected use. Typically a debris
catalog will be provided to the mission analysis that may have several failure
modes such as an intact crash, partial breakup and full breakup. The probability of
each of these outcomes is determined through some other analysis, perhaps by
the vehicle manufacture itself. Thus, there is no way a complex geometry could be
constructed from the debris catalog provided. The tools that the mission analyst
will use to predict risk typically assume failure at every point in the trajectory at
the failure rates from the probability of outcomes. Therefore, knowing precisely
when a part will fail is not as important as knowing where it will land if it failed at
that trajectory time and what sort of casualty risk can be expected.
The advantage this utility will have over some other reentry demise analysis tools
is that it will take in a standard debris catalog and return the standard debris
catalog with values adjusted for aero-thermal demise. This allows a risk analyst to
use the existing workflow and simply run the computational model developed in
this thesis before other risk tools to account for the demise casualty reduction.
1.6
Risk Analysis
Understanding the desire for a tool that can predict aero-thermal demise requires
some understanding of how a mission risk analysis is performed. The typical main
risk criterion is 1:10,000 or 100 micro-casualties. To determine this, the casualty
expectation is found by summing the probability of every possible event and the
casualty consequences at each mission event. The general form of the casualty
expectation equation for 𝑛 possible events is as followsRef. 1,
10
Aero-thermal Demise
𝑛
𝐸𝑐 = ∑ 𝑃𝑖 𝐴𝑐𝑖 𝐷𝑝𝑖
Eq. 1
𝑖=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 𝑖 𝑡ℎ 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
2
11
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
12
Aero-thermal Demise
For reentry debris fragments, the propulsive force is zero, and in a vacuum, the
aerodynamic forces are zero.
The kinematic vector equation for the inertial system defined by the position,
velocity and mass is,
𝑑𝒓
=𝑽
𝑑𝑡
Eq. 3
Newton’s second law gives the force equation.
𝑚
𝑑𝑽
=𝑭
𝑑𝑡
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, 𝑂𝑋1 𝑌1 𝑍1
and 𝑂𝑥𝑦𝑧 respectively. The system 𝑂𝑥𝑦𝑧 rotates with respect to the fixed system
𝑂𝑋1 𝑌1 𝑍1with angular velocity 𝜔. Let 𝐴 be an arbitrary vector in the rotating
system as seen in Figure 5,
𝑨 = 𝐴𝑥 𝑖̂ + 𝐴𝑦 𝑗̂ + 𝐴𝑧 𝑘̂
Eq. 5
T. M. Owens
13
Figure 5: Relative Angular Motion (Ref. 25)
The 𝑖̂, 𝑗̂ and 𝑘̂ unit vectors in the rotating reference system, 𝑂𝑥𝑦𝑧, are functions of
time. Therefore, the time derivative of 𝐴 with respect to the fixed system 𝑂𝑋1 𝑌1 𝑍1
is,
𝑑𝐴𝑦
𝑑𝑨
𝑑𝐴𝑥
𝑑𝐴𝑧
𝑑𝑖̂
𝑑𝑗̂
𝑑𝑘̂
=(
𝑖̂ +
𝑗̂ +
𝑘̂) + (𝐴𝑥 + 𝐴𝑦 + 𝐴𝑧 )
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
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
=𝝎×𝒓
𝑑𝑡
Figure 6: Kinematics of Rotation (Ref. 25)
Eq. 7
14
Aero-thermal Demise
Then, according to Poisson’s formulas,
𝑑𝑖̂
= 𝝎 × 𝑖̂
𝑑𝑡
𝑑𝑗̂
= 𝝎 × 𝑗̂
𝑑𝑡
Eq. 8
𝑑𝑘̂
= 𝝎 × 𝑘̂
𝑑𝑡
Using this with the definition of vector 𝐴 in the equation of the time derivative of
𝐴 the following equation is found,
𝐴𝑥
𝑑𝑖̂
𝑑𝑗̂
𝑑𝑘̂
+ 𝐴𝑦 + 𝐴𝑧
=𝝎×𝑨
𝑑𝑡
𝑑𝑡
𝑑𝑡
Eq. 9
Then in the rotating system 𝑂𝑥𝑦𝑧 the time derivative of 𝐴 is,
𝑑𝐴𝑦
𝛿𝑨 𝑑𝐴𝑥
𝑑𝐴𝑧
=
𝑖̂ +
𝑗̂ +
𝑘̂
𝛿𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
Eq. 10
The equation for the time derivative with respect to the fixed system, Eq. 6, can
be rewritten as,
𝑑𝑨 𝛿𝑨
=
+𝝎×𝑨
𝑑𝑡
𝛿𝑡
Eq. 11
2.1.2 Equations for Flight Over a Rotating Spherical Earth
𝑂𝑋1 𝑌1 𝑍1, is the inertial reference frame, with the origin at the center of a
spherical earth’s gravitation field. The plane 𝑂𝑋1 𝑌1 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 𝛿 2 𝒓
𝛿𝒓
= 2 + 2𝝎 ×
+ 𝝎 × (𝝎 × 𝒓)
𝑑𝑡 𝛿𝑡
𝛿𝑡
Eq. 12
The equation Eq. 4 can then be put in the earth-fixed system,
𝑚
𝛿 2𝒓
𝛿𝒓
= 𝑭 − 2𝑚𝝎 ×
− 𝑚𝝎 × (𝝎 × 𝒓)
2
𝛿𝑡
𝛿𝑡
Eq. 13
𝑑𝑽
= 𝑭 − 2𝑚𝝎 × 𝑽 − 𝑚𝝎 × (𝝎 × 𝒓)
𝑑𝑡
Eq. 14
Or,
𝑚
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, 𝜔2 𝑟, 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.
16
Aero-thermal Demise
Figure 7: Coordinate Systems (Ref. 25)
Thereby the velocity vector in the rotating system 𝑂𝑥𝑦𝑧 is,
𝑽 = 𝑉 sin 𝛾 𝑖̂ + 𝑉 cos 𝛾 cos 𝛹 𝑗̂ + 𝑉 cos 𝛾 sin 𝛹 𝑘̂
Eq. 15
The angular velocity in the 𝑂𝑥𝑦 plane is,
𝝎 = 𝜔 sin 𝜙 𝑖̂ + 𝜔 cos 𝜙 𝑘̂
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 cos 𝜙 cos 𝛹 𝑖̂ + 𝜔𝑉(sin 𝛾 cos 𝜙 − cos 𝛾 sin 𝛹)𝑗̂
+ 𝜔𝑉 cos 𝛾 sin 𝜙 cos 𝛹 𝑘̂
𝝎 × (𝝎 × 𝒓) = −𝜔2 cos 2 𝜙 𝑖̂ + 𝜔2 𝑟 sin 𝜙 cos 𝜙 𝑘̂
The force of gravitational acceleration in the total force 𝐹 is,
Eq. 17
Eq. 18
T. M. Owens
17
𝑚𝒈 = −𝑚𝑔(𝑟)𝑖̂
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 𝜀 as seen in Figure 8. The propulsive and aerodynamic forces can be
grouped,
𝐹𝑇 = 𝑇 cos 𝜀 − 𝐷
𝐹𝑁 = 𝑇 sin 𝜀 + 𝐿
Eq. 20
Figure 8: Aerodynamic and Propulsive Forces (Ref. 25)
In the unit vector form, the normal force can be defined as,
𝑭 𝑇 = 𝐹𝑇 sin 𝛾 𝑖̂ + 𝐹𝑇 cos 𝛾 cos 𝛹 𝑗̂ + 𝐹𝑇 cos 𝛾 sin 𝛹 𝑘̂
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 cos 𝛹 + sin 𝜎 sin 𝛹)𝑗̂
+ 𝐹𝑇 (cos 𝜎 sin 𝛾 sin 𝛹 − sin 𝜎 cos 𝛹)𝑘̂
Then, the equation Eq. 8 can be put in terms of the latitude and longitude.
Eq. 22
18
Aero-thermal Demise
𝑑𝑖̂
𝑑𝜃
𝑑𝜙
= cos 𝜙
𝑗̂ +
𝑘̂
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑗̂
𝑑𝜃
𝑑𝜃
= − cos 𝜙
𝑖̂ + sin 𝜙
𝑘̂
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑘̂
𝑑𝜙
𝑑𝜃
=−
𝑖̂ − sin 𝜙
𝑗̂
𝑑𝑡
𝑑𝑡
𝑑𝑡
Eq. 23
Then, using the equation Eq. 23 in Eq. 15,
𝑽=
𝑑𝑟
𝑑𝜃
𝑑𝜙
𝑖̂ + 𝑟 cos 𝜙
𝑗̂ + 𝑟
𝑘̂
𝑑𝑡
𝑑𝑡
𝑑𝑡
Eq. 24
The derivative of velocity is,
𝑑𝑽
𝑑𝑉
𝑑𝛾 𝑉 2
= [sin 𝛾
+ 𝑉 cos 𝛾
−
cos2 𝛾] 𝑖̂
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑟
+ [cos 𝛾 cos 𝛹
𝑑𝑉
𝑑𝛾
𝑑𝛹
− 𝑉 sin 𝛾 cos 𝛹
− 𝑉 cos 𝛾 sin 𝛹
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑉2
+
cos 𝛾 cos 𝛹 (sin 𝛾 − cos 𝛾 sin 𝛹 tan 𝜙)] 𝑗̂
𝑟
+ [cos 𝛾 sin 𝛹
+
Eq. 25
𝑑𝑉
𝑑𝛾
𝑑𝛹
− 𝑉 sin 𝛾 sin 𝛹
+ 𝑉 cos 𝛾 cos 𝛹
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑉2
cos 𝛾 (sin 𝛾 sin 𝛹 − cos 𝛾 cos 2 𝛹 tan 𝜙)] 𝑘̂
𝑟
Next, by substituting equation Eq. 25 into Eq. 14, the scalar equations of motion
are found to be,
sin 𝛾
𝑑𝑉
𝑑𝛾 𝑉 2
𝐹𝑇
𝐹𝑁
+ 𝑉 cos 𝛾
−
cos 2 𝛾 = sin 𝛾 + cos 𝜎 cos 𝛾
𝑑𝑡
𝑑𝑡
𝑟
𝑚
𝑚
2
− 𝑔 + 2𝜔𝑉 cos 𝛾 cos 𝛹 cos 𝜙 + 𝜔 𝑟 cos 𝜙
Eq. 26
T. M. Owens
sin 𝛾
19
𝑑𝑉
𝑑𝛾 𝑉 2
𝐹𝑇
𝐹𝑁
+ 𝑉 cos 𝛾
−
cos 2 𝛾 = sin 𝛾 + cos 𝜎 cos 𝛾
𝑑𝑡
𝑑𝑡
𝑟
𝑚
𝑚
Eq. 27
2
− 𝑔 + 2𝜔𝑉 cos 𝛾 cos 𝛹 cos 𝜙 + 𝜔 𝑟 cos 𝜙
sin 𝛾
𝑑𝑉
𝑑𝛾 𝑉 2
𝐹𝑇
𝐹𝑁
+ 𝑉 cos 𝛾
−
cos 2 𝛾 = sin 𝛾 + cos 𝜎 cos 𝛾
𝑑𝑡
𝑑𝑡
𝑟
𝑚
𝑚
Eq. 28
2
− 𝑔 + 2𝜔𝑉 cos 𝛾 cos 𝛹 cos 𝜙 + 𝜔 𝑟 cos 𝜙
Solving for the derivatives
𝑑𝑉 𝑑𝛾
,
𝑑𝑡 𝑑𝑡
, and
𝑑𝜓
𝑑𝑡
,
𝑑𝑉 𝐹𝑇
=
− 𝑔 sin 𝛾 + 𝜔2 𝑟 cos 𝜙 (sin 𝛾 cos 𝜙 − cos 𝛾 sin 𝜓 sin 𝜙)
𝑑𝑡
𝑚
𝑑𝛾 𝐹𝑁
𝑉2
𝑉
=
cos 𝜑 − 𝑔 cos 𝛾 + cos 𝛾
𝑑𝑡
𝑚
𝑟
Eq. 29
Eq. 30
2
+ 2𝜔𝑉 cos 𝜓 cos 𝜙 + 𝜔 𝑟 cos 𝜙 (cos 𝛾 cos 𝜙 − sin 𝛾 sin 𝜓 sin 𝜙)
𝑉
𝑑𝜓 𝐹𝑁 sin 𝜑 𝑉 2
=
−
cos 𝛾 cos 𝜓 tan 𝜙 + 2𝜔𝑉(tan 𝛾 sin 𝜓 cos 𝜙 − sin 𝜙)
𝑑𝑡
𝑚 cos 𝛾
𝑟
𝜔2 𝑟
−
cos 𝜓 sin 𝜙 cos 𝜙
cos 𝛾
Eq. 31
The 𝜔2 𝑟 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.
20
3
Aero-thermal Demise
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 FayRiddell 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 + ℎ𝐴0 𝐷𝜌∇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
𝑘
∇(ℎ + 𝑐𝐴 ℎ𝐴0 )
𝑐𝑝
𝑞=
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 + 𝑞𝑖 )𝑐𝑖 ] = 𝑤𝑖
Eq. 34
The mass average velocity, 𝑞𝑖 ,of the species 𝑖 can be found by,
𝑞𝑖 =
𝐷𝑖
𝐷𝑇
∇𝑐𝑖 − 𝑖 ∇𝑇
𝑐𝑖
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 ∆ ∑ 𝑐𝑖 𝑒𝑖 = ∇ ∙ (𝑘∆𝑇) − ∇ ∙ (∑ 𝜌𝑞𝑖 𝑐𝑖 ℎ𝑖 ) + ∑ 𝑤𝑖 ℎ𝑖 0 + 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.
22
Aero-thermal Demise
𝜌𝑞 ∇ ∑ 𝑐𝑖 (ℎ𝑖 − ℎ𝑖 0 ) = ∇ ∙ [𝑘∇𝑇 − ∑ 𝜌𝑞𝑖 𝑐𝑖 (ℎ𝑖 − ℎ𝑖 0 )] + 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
𝜌𝑢𝑐𝑖𝑥 + 𝜌𝑣𝑐𝑖𝑦 = (𝐷𝑖 𝜌𝑐𝑖𝑦 + 𝐷𝑖 𝑇 𝜌𝑐𝑖
Eq. 39
𝑇𝑦
) + 𝑤𝑖
𝑇 𝑦
Eq. 40
𝜌𝑢ℎ𝑥 + 𝜌𝑣ℎ𝑦 = (𝑘𝑇𝑦 )𝑦 + 𝑢𝑝𝑥 + 𝜇𝑢𝑦 2
+ [∑ 𝐷𝑖 𝜌(ℎ𝑖 − ℎ𝑖 0 )𝑐𝑖𝑦 + ∑ 𝐷𝑖 𝑇 𝜌𝑐𝑖 (ℎ𝑖 − ℎ𝑖 0 )
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.
𝑐̅𝑝 (𝜌𝑢𝑇𝑥 + 𝜌𝑣𝑇𝑦 ) = (𝑘𝑇𝑦 )𝑦 + 𝑢𝑝𝑥 + 𝜇𝑢𝑦 2
+ ∑ 𝑤𝑖 (ℎ𝑖 − ℎ𝑖 0 ) + [∑ 𝐷𝑖 𝜌𝑐𝑖𝑦 + ∑ 𝐷𝑖 𝑇 𝜌𝑐𝑖
𝑇𝑦
]
𝑇 𝑦
Eq. 43
Similarly, equation Eq. 41 can be rewritten to simply be in terms of the enthalpy,
T. M. Owens
23
𝑢2
𝑢2
𝑘
𝑢2
𝜌𝑢 (ℎ + ) + 𝜌𝑣 (ℎ + ) = [ (ℎ + ) ] + 𝑢𝑝𝑥
2 𝑥
2 𝑦
𝑐̅𝑝
2 𝑦
𝑦
𝑇𝑦
𝑘
+ 𝜇𝑢𝑦 2 + [∑ (𝐷𝑖 𝜌 − ) (ℎ𝑖 − ℎ𝑖 0 )𝑐𝑖𝑦 + ∑ 𝐷𝑖 𝑇 𝜌𝑐𝑖 (ℎ𝑖 − ℎ𝑖 0 ) ]
𝑐̅𝑝
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.
𝑦
𝑟𝑢∞
𝜂≡(
) ∫ 𝜌𝑑𝑦
√2𝜉 0
Eq. 45
𝑥
𝜉 ≡ ∫ 𝜌𝑤 𝜇𝑤 𝑢∞ 𝑟 2 𝑑𝑥
0
𝜂
𝜕𝑓
𝑢
𝜕𝑓
≡
; 𝑓=∫
𝑑𝜂
𝜕𝜂 𝑢∞
0 𝜕𝜂
𝑢2
(ℎ + 2 )
𝑔=
ℎ𝑠
𝜃=
𝑇
𝑇∞
Eq. 46
Eq. 47
Eq. 48
Eq. 49
24
Aero-thermal Demise
𝑠𝑖 =
𝑐𝑖
𝑐𝑖∞
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 + √2𝜉𝑓𝑦 𝜂𝑥 ]⁄𝑟
𝐿𝑖 𝑇 𝑠𝑖 𝑇𝜂
𝑙
𝑤𝑖
𝑑𝑢∞ −1
[ (𝐿𝑖 𝑠𝑖𝜂 +
)] + 𝑓𝑠𝑖𝜂 +
[(2
) ] =0
𝑃𝑟
𝑇
𝜌𝑐𝑖𝑠
𝑑𝑥 𝑠
𝜂
1 𝜌𝑠
(𝑙𝑓𝜂𝜂 ) + 𝑓𝑓𝜂𝜂 + ( − 𝑓𝜂 2 ) = 0
𝜂
2 𝜌
Eq. 51
Eq. 52
Eq. 53
𝑐̅𝑝 𝑙
𝑐̅𝑝
𝑑𝑢∞ −1
𝑤𝑖 (ℎ𝑖 − ℎ𝑖 0 )
(
) +
𝑓𝜃𝜂 + [(2
) ] ∑
𝑐̅𝑝𝑤 𝑃𝑟 𝜂 𝑐̅𝑝𝑤
𝑑𝑥 𝑠
𝜌 𝑐̅𝑝𝑤 𝑇𝑠
𝐿𝑖 𝑇 𝑠𝑖 𝜃𝜂
𝑐̅𝑝𝑖 𝑐𝑖𝑠 𝑙
+∑
(𝐿𝑖 𝑠𝑖𝜂 +
) 𝜃𝜂 = 0
𝑐̅𝑝𝑤 𝑃𝑟
𝜃
𝜂
(ℎ𝑖 − ℎ𝑖 0 )
𝐿𝑖 𝑇 𝑠𝑖 𝜃𝜂
𝑙
𝑙
( 𝑔𝜂 ) + 𝑓𝑔𝜂 + { ∑ [𝑐𝑖𝑠
] [(𝐿𝑖 − 1)𝑠𝑖𝜂 +
]} = 0
𝑃𝑟
𝑃𝑟
ℎ𝑠
𝜃
𝜂
𝜂
Eq. 54
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,
𝜕𝑇
𝜕𝑐𝑖 𝐷𝑖𝑇 𝑐𝑖 𝜕𝑇
𝑞 = (𝑘 )
+ [∑ 𝜌(ℎ𝑖 − ℎ𝑖 0 ) (𝐷𝑖
+
)]
𝜕𝑦 𝑦=0
𝜕𝑦
𝑇 𝜕𝑦 𝑦=0
Eq. 56
T. M. Owens
25
The dimensionless terms from the previous section can be used to get the
equation,
(ℎ𝑖 − ℎ𝑖 0 )
𝐿𝑖 𝑇 𝑠𝑖 𝜃𝜂
𝑟𝑘𝑤 𝜌𝑤 𝑢∞ 𝑇∞
𝑞=(
) [𝜃𝜂 + ∑ 𝑐𝑖
(𝐿𝑖 𝑠𝑖𝜂 +
)]
𝑐̅𝑝 𝑇∞
𝜃
√2𝜉
𝜂=0
Eq. 57
At the stagnation point,
𝑟𝜌𝑤 𝑢∞
√2𝜉
=√
2 𝑑𝑢∞
(
)
𝜈𝑤 𝑑𝑥 𝑠
Eq. 58
Thereby allowing the heat transfer equation to be rewritten as,
𝑞=
𝑑𝑢∞ (ℎ𝑠 − ℎ𝑤 )
√𝜌𝑤 𝑢𝑤 (
)
𝑑𝑥 𝑠
𝑃𝑟
√𝑅𝑒
𝑁𝑢
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,
𝜌𝑠 𝜇𝑠 0.4
= 0.67 (
)
𝜌𝑤 𝜇𝑤
√𝑅𝑒
𝑁𝑢
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,
26
Aero-thermal Demise
ℎ𝐴0 = ∑ 𝑐𝑖𝑠 (−ℎ𝑖 0 )⁄ ∑ 𝑐𝑖𝑠
𝑎𝑡𝑜𝑚𝑠
𝑎𝑡𝑜𝑚𝑠
Eq. 61
Also, the numerical solution effect of the Lewis number can be taken into account
by the equation,
𝑁𝑢
(
𝑁𝑢
√𝑅𝑒 √𝑅𝑒
)
= 1 + (𝐿0.52 − 1)
𝐿=1
ℎ𝐷
ℎ𝑠
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(𝜌𝑠 𝜇𝑠 )0.1 (𝜌𝑠𝑙 𝜇𝑠𝑙 )0.4 [1 + (𝐿0.52 − 1)
ℎ𝐷
𝑑𝑢∞
] √(
)
ℎ𝑠
𝑑𝑥 𝑠
Eq. 63
The velocity gradient as defined by a modified Newtonian flow is,
(
𝑑𝑢∞
1 2(𝑝𝑠 − 𝑝∞ )
) = √
𝑑𝑥 𝑠 𝑅
𝜌𝑠
Eq. 64
So, with these various correlations, the stagnation point heat transfer rate can be
developed.
T. M. Owens
3.2
27
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 −
ℎ𝑠
𝑑𝑢∞
) (𝜌𝑠 𝜇𝑠 )0.1 (𝜌𝑠𝑙 𝜇𝑠𝑙 )0.4 ℎ𝑠𝑙 √(
) [1
ℎ𝑠𝑙
𝑑𝑥 𝑠
ℎ𝐷
+ 0.45(𝐿 − 1) ]
ℎ𝑠𝑙
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,
(
𝑑𝑢𝑒
1 2𝑝𝑠𝑙
√
) =
𝑑𝑥 𝑠 𝑅ℎ 𝜌𝑠𝑙
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
28
Aero-thermal Demise
experimental data from hypersonic shock tubes, the equation for the stagnation
point heating flux isRef. 4,
3.15
17600 𝜌∞ 𝑈∞
𝑞𝑠 =
)
√ (
√𝑅ℎ 𝜌𝑠𝑙 𝑈𝑟𝑒𝑓
ℎ𝑠 − ℎ𝑏
(
)
ℎ𝑠𝑙 − 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
30
Aero-thermal Demise
node. Like the stagnation point heating flux, the radiation energy flux is in units of
W/m2.
𝑞𝑟𝑎𝑑 = 𝜀𝜎𝑠𝑏𝑐 𝑇 4
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 = 𝑘2 𝑞𝑠
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.
𝑄̇ = (𝑘2 𝑞𝑠 − 𝜀𝑏 𝜎𝑠𝑏𝑐 𝑇𝑏 4 )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
,
𝑚𝑏 𝐶𝑝̅ 𝑏
{ 𝑇𝑚𝑒𝑙𝑡 ,
𝑓𝑜𝑟 𝑄 < 𝑚𝑏 𝐶𝑝̅ 𝑏 𝑇𝑚𝑒𝑙𝑡
Eq. 71
𝑓𝑜𝑟 𝑄 ≥ 𝑚𝑏 𝐶𝑝̅ 𝑏 𝑇𝑚𝑒𝑙𝑡
T. M. Owens
4
31
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 document Ref. 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 harmonics Ref. 14.
Table 1 is a collection of the derived geometric constants. The values were
obtained through precise GPS ephemeris estimation process. The method can
potentially be used for other planetary bodies if the geometric constants are
known. These are set in the createEarth function.
Table 1: WGS84 Ellipsoid Derived Geometric Constants
Variable
Value
Units
Description
𝒂⊕
6378137.0
𝑚
Semi-major axis
𝒃⊕
6356752.3142
𝑚
Semi-minor axis
1/298.257223563
Flattening
𝒆⊕
8.1819190842622 ∙ 10−2
First Eccentricity
𝒆⊕ 𝟐
6.69437999014 ∙ 10−3
First Eccentricity Squared
𝒇
32
Aero-thermal Demise
𝒆⊕́
8.2094437949696 ∙ 10−2
Second Eccentricity
𝒆⊕́ 𝟐
6.73949674228 ∙ 10−3
Second Eccentricity Squared
𝑚2
𝑠2
𝝁
3.986004418 ∙ 1014
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
Gravitational Constant
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 are Ref. 12,
𝑟 = √𝑒 2 + 𝑓 2 + 𝑔2
Eq. 72
5𝑎⊕ 3 𝐽3 7𝑒𝑔3 3𝑒𝑔
𝑒 3𝑎⊕ 2 𝐽2 5𝑒𝑔2 𝑒
𝐺𝑒 = −𝜇 [ 3 −
( 7 − 5) −
( 9 − 7)
𝑟
2
𝑟
𝑟
2
𝑟
𝑟
15𝑎⊕ 4 𝐽4 21𝑒𝑔4 14𝑒𝑔2 𝑒
−
( 11 −
+ 7 )]
8
𝑟
𝑟9
𝑟
Eq. 73
T. M. Owens
33
5𝑎⊕ 3 𝐽3 7𝑓𝑔3 3𝑓𝑔
𝑓 3𝑎⊕ 2 𝐽2 5𝑓𝑔2 𝑓
𝐺𝑓 = −𝜇 [ 3 −
( 7 − 5) −
( 9 − 7 )
𝑟
2
𝑟
𝑟
2
𝑟
𝑟
15𝑎⊕ 4 𝐽4 21𝑓𝑔4 14𝑓𝑔2 𝑓
−
( 11 −
+ 7 )]
8
𝑟
𝑟9
𝑟
Eq. 74
𝑎⊕ 3 𝐽3 35𝑔4 30𝑔2
𝑔 3𝑎⊕ 2 𝐽2 5𝑔3 3𝑔3
3
𝐺𝑔 = −𝜇 [ 3 −
( 7 − 5 )−
( 9 − 7 + 5)
𝑟
2
𝑟
𝑟
2
𝑟
𝑟
𝑟
5𝑎⊕ 4 𝐽4 63𝑔5 70𝑔3 15𝑔
−
( 11 − 9 + 7 )]
8
𝑟
𝑟
𝑟
Eq. 75
Of import, the ECEF components of gravity can be used to find the magnitude of
the local gravity,
𝑔 = √𝐺𝑒 2 + 𝐺𝑓 2 + 𝐺𝑔 2
Eq. 76
34
4.3
Aero-thermal Demise
Atmospheric Model
The atmosphere is modeled using the 1976 U.S. Standard Atmosphere Ref. 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,
𝑃𝑡
−𝐺𝑀∙(𝑧𝑔 −𝑧𝑡 )
𝑇0
𝑒𝑥𝑝
,
𝛿=
𝑓𝑜𝑟 𝐺𝑡 = 0
Eq. 79
𝐺𝑀
{
𝑇0 𝐺𝑡
𝑃𝑡 ( ) ,
𝑇
𝑃∞ =
𝑓𝑜𝑟 𝐺𝑡 ≠ 0
𝑃0
𝛿
Eq. 80
𝛿
𝑇∞ ⁄𝑇0
Eq. 81
Then, the local density,
𝜌∞ = 𝜌0
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.
36
Aero-thermal Demise
Table 3: Atmosphere Fit Parameters (1976 Std. Atmosphere)
Altitude
𝐥𝐨𝐠 𝒑
𝒅
𝐥𝐨𝐠 𝒑
𝒅𝒛
𝐥𝐨𝐠 𝝆
𝒅
𝐥𝐨𝐠 𝝆
𝒅𝒛
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
[km]
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
Eq. 83
𝑝 = 𝑘 log 𝑝𝑡 + 𝑗 log 𝑝𝑡+1
− 𝑘𝑗(𝑧𝑡+1 − 𝑧𝑡 ) [𝑘 (𝑖 −
𝑑
𝑑
log 𝑝𝑡 ) − 𝑗 (𝑖 − log 𝑝𝑡+1 )]
𝑑𝑧
𝑑𝑧
The kinetic temperature can be found by,
186.8673,
𝑇∞ =
𝑓𝑜𝑟 86 < 𝑧𝑔 ≤ 86 km
𝑧𝑔 − 91 2
263.1905 + 12√1 − (
) ,
19.9429
𝑓𝑜𝑟 91 < 𝑧𝑔 < 110 km
240 + 12(𝑧𝑔 − 110),
𝑓𝑜𝑟 110 ≤ 𝑧𝑔 ≤ 120 km
Eq. 84
𝑅⊕ + 120
1000 − (1000 − 120) exp [−0.01875(𝑧𝑔 − 120)
] , 𝑓𝑜𝑟 110 < 𝑧𝑔 ≤ 1000 km
𝑅⊕ + 𝑧𝑔
{
However, if the altitude is over 1000 km, the constant values are used,
𝑇∞ = 1000
𝑝 = 1 ∙ 10−20 𝑝0
𝜌 =1∙
10−21 𝜌0
Eq. 85
38
4.4
Aero-thermal Demise
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,
𝐶𝐷 0
𝐾
Eq. 86
𝐶𝐷∗ = 2𝐶𝐷 0
Eq. 87
𝐶𝐿∗ = √
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
𝑉2
=
𝑉𝑐2 𝑔0 𝑎⊕
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).
2 (1 −
sin 𝛾 =
𝑉2
)
𝑉𝑐2
𝐶𝐿∗
𝑉2
𝑉2
[𝛽𝑅
(1
−
)
+
(2
−
)]
⊕
𝐶𝐷∗
𝑉𝑐2
𝑉𝑐2
−1 ≤ sin 𝛾 ≤ 1
Eq. 89
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
2𝑚
) 2
𝑟 𝜌𝑉 cos 𝜑 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.
40
Aero-thermal Demise
𝑛=
𝜌|𝐶𝐿 |𝑉 2 𝑆
2𝑔𝑚
𝐶𝐿 = 𝑠𝑖𝑔𝑛(𝐶𝐿 )𝑛𝑠
2𝑔𝑚
𝜌𝑉 2
Eq. 94
Eq. 95
The parabolic drag polar or coefficient of drag is found by,
𝐶𝐷 = 𝐶𝐷 0 + 𝐾𝐶𝐿 2
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.
𝐹𝑇
𝑉 2 𝜌𝑆
= −𝐶𝑑
𝑚
2𝑚
Eq. 97
𝐹𝑁
𝑉 2 𝜌𝑆
= 𝐶𝑙
𝑚
2𝑚
Eq. 98
The force equations are used to find the changes in velocity, flight path angle and
heading.
𝑑𝑉 𝐹𝑇
=
− 𝑔 sin 𝛾 + 𝜔2 𝑟 cos 𝜙 (sin 𝛾 cos 𝜙 − cos 𝛾 sin 𝜓 sin 𝜙)
𝑑𝑡
𝑚
𝑉
𝑑𝛾 𝐹𝑁
𝑉2
=
cos 𝜑 − 𝑔 cos 𝛾 + cos 𝛾
𝑑𝑡
𝑚
𝑟
Eq. 99
Eq. 100
2
+ 2𝜔𝑉 cos 𝜓 cos 𝜙 + 𝜔 𝑟 cos 𝜙 (cos 𝛾 cos 𝜙 − sin 𝛾 sin 𝜓 sin 𝜙)
𝑉
𝑑𝜓 𝐹𝑁 sin 𝜑 𝑉 2
=
−
cos 𝛾 cos 𝜓 tan 𝜙 + 2𝜔𝑉(tan 𝛾 sin 𝜓 cos 𝜙 − sin 𝜙)
𝑑𝑡
𝑚 cos 𝛾
𝑟
𝜔2 𝑟
−
cos 𝜓 sin 𝜙 cos 𝜙
cos 𝛾
Eq. 101
T. M. Owens
41
The rate of change of position along the east and north axis can be found by,
𝑑𝑋
= 𝑉 cos 𝛾 cos 𝜓
𝑑𝑡
Eq. 102
𝑑𝑌
= 𝑉 cos 𝛾 sin 𝜓
𝑑𝑡
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
⁄𝑎
𝑑𝑡 ⊕
𝑑𝑋
⁄(𝑎⊕ cos 𝜙)
𝑑𝑡
𝑉 sin 𝛾
𝑦̇ =
𝑑𝜓
𝑑𝑡
cos 𝛾
[sin 𝛾
Eq. 104
𝑑𝑉
𝑑𝛾
− 𝑉 sin 𝛾
𝑑𝑡
𝑑𝑡
𝑑𝑉
𝑑𝛾
+ 𝑉 ∙ cos 𝛾 ]
𝑑𝑡
𝑑𝑡
The differential equations expressed in 𝑦̇ can be solved using a differential
equation solver such as MATLAB’s Runge-Kuta methods ode23 and ode45. The
time span is taken from the integration limit and the initial state is the trajectory
state at the malfunction time.
42
Aero-thermal Demise
The fragment debris studied in this thesis are tumbling with no defined head or
tail, so there is no lift. The equations can be simplified from there more general
form, however, the approach is the same.
T. M. Owens
4.5
43
Aero-thermal Demise
The aero-thermal demise algorithm follows roughly that of the algorithm outlined
in the Reentry Hazard Analysis HandbookRef. 24.
4.5.1 Fragment Properties
A fragment’s mass, shape, material, and dimensions are the required properties in
order to perform a demise analysis. A fragment may also have a parent fragment
and will not start ablating until that parent has demised (this case was not
included in the utility). An initial temperature of the fragment can also be set; for
most analysis though, the reference temperature of 300 K is appropriate unless
the risk analyst has knowledge of fragment preheating.
4.5.2 Material Properties
The material properties used in the demise utility are from the Debris Assessment
Software 2.0’s material database. The material database can be found in Appendix
A: Material Properties. The material properties that are required to perform a
demise analysis are density, specific heat, heat of fusion and the melting
temperature. The specific heat capacities of the fragments with a range of specific
heats are taken as the mean of the range. All material properties are assumed to
be constant. Each fragment is assigned a material and uses properties from the
material database. It is possible to have user defined materials, but the material
database should be adequate for most analyses.
44
Aero-thermal Demise
4.5.3 Shape Assumptions
The geometry of the demising fragments is simplified into one of four shapes:
spheres, cylinders, flat plates and boxes. Flat plates should have a thickness less
than one twentieth of the width. The box shape is approximated as an equivalent
cylinder. Each of the shapes is considered to be hollow and tumbling with uniform
ablation over the surface.
Table 4: Debris Fragment Shape Assumptions
Variable
Sphere
Cylinder
Flat Plate
𝑙>𝑤≫𝑡
𝑚𝑏
𝑡=
𝜌𝑏 𝑙𝑤
Misc…
Wetted area, Aw
Hypersonic continuum
drag coefficient, CDhc
Aerodynamic
reference area, S
Heating radius, Rh
Box
𝑙>𝑤>ℎ
𝑟=√
𝑤ℎ
𝜋
4𝜋𝑟 2
2𝜋(𝑟 + 𝑙)
2𝑙𝑤 + 2𝑙𝑡 + 2𝑤𝑡
2𝜋𝑟(𝑟 + 𝑙)
0.92
2𝑟
0.720 + 0.326 ( )
𝑙
1.84
2𝑟
0.720 + 0.326 ( )
𝑙
𝜋𝑟 2
2𝑟𝑙
𝑙𝑤
2𝑟𝑙
𝑟
𝑟
𝑤
2
𝑟
For all cases, the hypersonic continuum ballistic coefficient is,
𝛽ℎ𝑐 =
𝑚𝑏
𝑆𝐶𝐷 ℎ𝑐
Eq. 105
T. M. Owens
45
4.5.4 Stagnation Point Heating
The formula for the specific heat capacity of airRef. 9,
1373,
𝑓𝑜𝑟 𝑇𝑏 ≥ 2000 K
𝐶𝑝∞ = 959.9 + 0.15377𝑇𝑏 + 2.636 ∙ 10−5 𝑇𝑏 2 , 𝑓𝑜𝑟 300 < 𝑇𝑏 < 2000 K Eq. 106
{
1004.7,
𝑓𝑜𝑟 𝑇𝑏 ≤ 300 K
Adiabatic stagnation temperature,
𝑇𝑠 = 𝑇∞ +
2
𝑈∞
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 = 𝑚𝑏 𝐶𝑝̅ 𝑏 𝑇0
Eq. 108
Heat of melting is the heat required to raise the body bulk temperature from the
initial temperature to the melt temperature.
𝑄𝑚𝑒𝑙𝑡 = 𝑚𝑏 𝐶𝑝̅ 𝑏 (𝑇𝑚𝑒𝑙𝑡 − 𝑇0 )
Eq. 109
Heat of ablation is the heat required to melt the entire body.
𝑄𝑎 = 𝑚𝑏 𝐶𝑝̅ 𝑏 (𝑇𝑚𝑒𝑙𝑡 − 𝑇0 ) + 𝑚𝑏 ℎ𝑓
Eq. 110
Equation Eq. 111 is the Detra-Kemp-Riddell stagnation point heating correlation
whose derivation is explained in section 3.2 Detra, Kemp and Riddell Correlation.
Here the reference temperature is 300 K and the reference velocity is 7924.8 m/s
(26000 ft/s). Of note, 0.3048 is the reference radius of 1 foot converted to meters.
46
Aero-thermal Demise
Note that the ratio of enthalpies has been converted to a ratio of temperatures to
simplify the equation as all enthalpies are the enthalpy of air.
3.15
0.3048𝜌∞ 𝑇𝑠 − 𝑇𝑏
𝑈∞
𝑞𝑠 = 200006400√
(
)(
)
𝑅ℎ 𝜌𝑠𝑙
𝑇𝑠 − 𝑇𝑟𝑒𝑓 𝑈𝑟𝑒𝑓
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.
𝑄̇ = (𝑘2 𝑞𝑠 − 𝜀𝑏 𝜎𝑠𝑏𝑐 𝑇𝑏 4 )𝐴𝑤
Eq. 112
Heat content of the fragment body at time t,
𝑡
𝑄(𝑡) = 𝑄0 + ∫ 𝑄̇ 𝑑𝑡
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
,
𝑚𝑏 𝐶𝑝̅ 𝑏
{ 𝑇𝑚𝑒𝑙𝑡 ,
𝑓𝑜𝑟 𝑄 < 𝑚𝑏 𝐶𝑝̅ 𝑏 𝑇𝑚𝑒𝑙𝑡
Eq. 114
𝑓𝑜𝑟 𝑄 ≥ 𝑚𝑏 𝐶𝑝̅ 𝑏 𝑇𝑚𝑒𝑙𝑡
The body bulk temperature is put back into the equation for stagnation point
heating and then iterated until the error in the net heat flux reaches an acceptable
level. There is a discontinuity in the solution as the stagnation temperature
approaches the reference temperature; this does not affect the result of the
analysis as it occurs in the region of aero-cooling.
T. M. Owens
47
4.5.5 Liquid Fraction
Liquid fraction is a measure of the fraction of debris that has melted. If the
maximum amount of heating is less than the heat of melting, none of the
fragment can be liquid. If the maximum amount of heating is greater than the
heat of ablation, the entire fragment is considered liquid and fully demisedRef. 24.
0,
𝑄𝑚𝑎𝑥 − 𝑄𝑚𝑒𝑙𝑡
,
𝑄𝑎 − 𝑄𝑚𝑒𝑙𝑡
𝐿𝐹 =
{
1,
𝑓𝑜𝑟 𝑄𝑚𝑎𝑥 ≤ 𝑄𝑚𝑒𝑙𝑡
𝑓𝑜𝑟 𝑄𝑚𝑒𝑙𝑡 < 𝑄𝑚𝑎𝑥 < 𝑄𝑎
Eq. 115
𝑓𝑜𝑟 𝑄𝑚𝑎𝑥 ≥ 𝑄𝑎
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.
48
Aero-thermal Demise
For the sphere, just what would be ‘interior’ mass needs to be calculated before
using the time varying liquid fraction mass to get the time varying areas.
4𝜋𝑟 3
𝑚𝑖 = 𝜌
−𝑚
3
Eq. 117
2⁄
3
3(𝑚𝐿𝐹 + 𝑚𝑖 )
𝑆 = 𝜋(
)
4𝜋𝜌
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 = 𝜋𝑟 2 𝑙𝜌 − 𝑚
Eq. 119
0 = −𝑡 3 (2𝑟𝑙)𝑡 2 − (𝑟 2 + 2𝑟𝑙)𝑡 + 𝑟 2 𝑙 −
𝑚𝑖
𝜋𝜌
Eq. 120
𝑟𝑖 = 𝑟 − 𝑡
Eq. 121
𝑙𝑖 = 𝑙 − 𝑡
Eq. 122
0 = 𝑡 3 (2𝑟𝑖 𝑙𝑖 )𝑡 2 + (𝑟𝑖 2 + 2𝑟𝑖 𝑙𝑖 )𝑡 + 𝑟𝑖 2 𝑙𝑖 −
𝑚𝐿𝐹
𝜋𝜌
𝑆 = 2(𝑟𝑖 + 𝑡)(𝑙𝑖 + 𝑡)
Eq. 123
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 = 𝛿 2 − 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
0 = −𝑡 3 (𝑙 + 𝑤 + ℎ)𝑡 2 − (𝑙ℎ + 𝑤ℎ + 𝑙𝑤)𝑡 + 𝑙𝑤ℎ −
Eq. 127
𝑚𝑖
𝜌
Eq. 128
𝑙𝑖 = 𝑙 − 𝑡
Eq. 129
𝑤𝑖 = 𝑤 − 𝑡
Eq. 130
ℎ𝑖 = ℎ − 𝑡
Eq. 131
0 = 𝑡 3 (𝑙𝑖 + 𝑤𝑖 + ℎ𝑖 )𝑡 2 + (𝑙𝑖 ℎ𝑖 + 𝑤𝑖 ℎ𝑖 + 𝑙𝑖 𝑤𝑖 )𝑡 + 𝑙𝑖 𝑤𝑖 ℎ𝑖 −
𝑆 = (𝑙𝑖 + 𝑡)√
(𝑤𝑖 + 𝑡)(ℎ𝑖 + 𝑡)
𝜋
𝑚𝐿𝐹
𝜌
Eq. 132
Eq. 133
4.5.6.3 Hazard Area Table
From the new aerodynamic reference areas the hazard radius is calculated by the
equation,
𝑆
𝐻𝑟 = √
𝜋
Eq. 134
50
Aero-thermal Demise
If the user supplied a hazard radius in the fragment creator, then the hazard radius
will be calculated using the maximum from that input and adjusting it
proportionally to the demise aerodynamic reference area. This allows the user to
force a smaller or larger hazard area using the following equation,
𝑆
𝐻𝑟 = 𝑚𝑎𝑥{𝐻𝑟 0 }√
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
𝑆𝐶𝐷 𝑀𝑎𝑐ℎ
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.
𝐶𝐷 𝛽 =
𝑚𝑖𝑚𝑝𝑎𝑐𝑡
𝑆𝑖𝑚𝑝𝑎𝑐𝑡 β𝑀𝑎𝑐ℎ
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.
52
Aero-thermal Demise
Figure 10: Corrected (red) and Uncorrected (blue) Cd Impacts
T. M. Owens
5
53
Results
The following are the results from the computational model outlined in this thesis.
The first section will explain how changing breakup conditions and fragment
properties can affect the aero-thermal demise. The second is a comparison against
the established reentry debris analysis tools DAS 2.0 and the Aerospace
Survivability Tables.
5.1
Understanding Aero-heating
The plots generated for this section are included in an effort to help with the
understanding of the mechanism of aero-heating and how the computational
model simulates them. A reentering body begins to interface with the atmosphere
at about 122 km with aero-heating starting at 80 km and aero-cooling after about
50 km. A fragment’s liquid fraction or demise is affected by many different
parameters such as initial temperature, breakup altitude or time, material
properties, geometry, aerodynamic stability, etc. A few of these will be examined
in this section. Some simplifications have to be made for analysis, but the results
of that analysis should be consistent with empirical data and other validated
methods.
5.1.1 Reentry Trajectory, Heat Flux, and Bulk Temperature
The initial conditions for the following plots are a break up altitude of 120 km over
the intersection of the prime meridian and equator with a flight path angle of -0.5
degrees, a 0 degree heading, a velocity of 7600 m/s and a debris starting
temperature of 300 K.
For these test cases, an example debris fragment that is known to experience
partial demise at a variety of initial breakup conditions was used.
54
Aero-thermal Demise
Table 5: Example Fragment
Name
Shape
Radius [m]
Length [m]
2.6
Cylinder
0.85344
0.85344
Mach
Cd
1
0.81
0.9999
0.45
Material
Aluminum
(generic)
Time [s]
Mass [kg]
Area [m2]
Beta [kg/m^2]
Hazard Radius [m]
0
50
1.45672
42.37493
0.1215
Figure 11 shows the altitude versus time and Figure 12 shows the altitude versus
range as it is calculated by the impact integrator. From the figures, it is evident
that below about the 80 km mark, where aero-heating is highest, the most
significant drag is experienced. The fragment reaches a range of about 3100 km
before falling on a nearly vertical trajectory. Ground impact is at about 1200
seconds, just off the end of the plot.
T. M. Owens
55
Altitude [km] vs. Time [s]
120
100
Altitude [km]
80
60
40
20
0
0
200
400
600
800
1000
Time [s]
Figure 11: Single Fragment, Altitude vs. Time
Altitude [km] vs. Range [km]
120
100
Altitude [km]
80
60
40
20
0
0
500
1000
1500
2000
2500
Range [km]
3000
Figure 12: Single Fragment, Altitude vs. Range
3500
4000
56
Aero-thermal Demise
Figure 13 shows the heat flux versus time. The figure captures both the aeroheating 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.
3
x 10
5
Heat Flux [W/m2-s] vs. Time [s]
2.5
Heat Flux [W/m2-s]
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
50
100
150
200
250
300
Time [s]
350
400
450
500
Figure 13: Single Fragment, Heat Flux vs. Time
Figure 14 shows the bulk temperature of the fragment through time. In this case,
it is below the melting point at all times. As the fragment descends further cooling
than what is shown in the plot would take place.
T. M. Owens
57
Fragment Bulk Temperature [K] vs. Time [s]
900
Fragment Bulk Temperature [K]
800
700
600
500
400
300
0
50
100
150
200
250
300
Time [s]
350
400
450
500
Figure 14: Single Fragment, Temperature vs. Time
5.1.2 Varying Breakup Altitude
Now, to see the effects varying the breakup altitude, the same debris fragment
was given an initial breakup 120 and 50 km altitude in 10 km steps. Figure 15 and
Figure 16 do not reveal unexpected results. The debris fragments with initial
states at lower altitudes impact and decelerate more quickly in the denser
atmosphere. This particular fragment, because of its relatively low overall density
as a hollow cylinder, has a roughly vertical trajectory after 40 km of altitude.
58
Aero-thermal Demise
Altitude [km] vs. Time [s]
120
100
Altitude [km]
80
60
40
20
0
0
200
400
600
800
1000
Time [s]
Figure 15: Varying Breakup Altitude, Altitude vs. Time
Altitude [km] vs. Range [km]
120
100
Altitude [km]
80
60
40
20
0
0
500
1000
1500
2000
2500
Range [km]
3000
3500
Figure 16: Varying Breakup Altitude, Altitude vs. Range
4000
T. M. Owens
59
The heat flux for varying altitudes is shown in Figure 17. These results are slightly
counterintuitive for an actual reentry. At the lower altitudes, the flight speed
would not be expected to be near orbital. As such, the low altitude breakup
conditions have an immediate peak heat flux and then cool rapidly in the denser
atmosphere.
5
x 10
5
Heat Flux [W/m2-s] vs. Time [s]
120 km
110 km
100 km
90 km
80 km
70 km
60 km
50 km
4
Heat Flux [W/m2-s]
3
2
1
0
-1
-2
0
50
100
150
200
250
300
Time [s]
350
400
450
500
Figure 17: Varying Breakup Altitude, Heat Flux vs. Time
The bulk temperature of the fragment versus time in Figure 18 shows how the
first four breakup altitudes, 120, 110, 100 and 90 km, all have a similar peak
temperature. They are all below the melting temperature of 850 K, so no
fragments are fully demised. The temperature plot, if below the melting point,
does not tell us much about the demise of fragments. For that, the liquid fraction
is the best indicator.
60
Aero-thermal Demise
Fragment Bulk Temperature [K] vs. Time [s]
900
Fragment Bulk Temperature [K]
800
700
600
120 km
110 km
100 km
90 km
80 km
70 km
60 km
50 km
500
400
300
0
50
100
150
200
250
300
Time [s]
350
400
450
500
Figure 18: Varying Breakup Altitude, Temperature vs. Time
The liquid fraction is the single quantity of the aero-thermal demise calculations
that is used to adjust the mass and dimensions of the fragment as it goes through
the impact integration. Figure 19 shows the liquid fraction over time for each of
the breakup altitudes. The high breakup altitude cases experience about 85%
mass loss or ablation, and the two low altitudes, 50 and 60 km, have no ablation.
T. M. Owens
61
Liquid Fraction vs. Time [s]
1
0.9
0.8
Liquid Fraction
0.7
0.6
0.5
0.4
120 km
110 km
100 km
90 km
80 km
70 km
60 km
50 km
0.3
0.2
0.1
0
0
50
100
150
200
250
300
Time [s]
350
400
450
500
Figure 19: Varying Breakup Altitude, Liquid Fraction vs. Time
5.1.3 Varying Initial Temperature of Debris Fragment
Typically, the initial temperature of a debris fragment in aero-thermal demise is
set to 300 K unless preheating is known. Preheating could be from an external
component of a vehicle that has broken up or from a child of a parent fragment
that has broken into pieces. A temperature range from 300 to 600 K with 100 K
steps for the debris fragment is used. The breakup altitude is set at 100 km. The
altitude and range plots for each case are nearly overlapping, so there is no need
to compare them for the case of varying initial debris temperature.
Looking at the temperature in Figure 20, it is clear that several of the fragments
with a higher initial temperature reach the melting temperature of 850 K for
aluminum. In Figure 21, the three higher temperature cases, 400, 500 and 600 K,
all reach the liquid fraction of 1 and are fully demised. The 600 K case demises
almost immediately; the curve is only just visible at the upper left of the figure.
62
Aero-thermal Demise
Fragment Bulk Temperature [K] vs. Time [s]
900
Fragment Bulk Temperature [K]
800
700
600
500
400
300
0
50
100
150
Time [s]
200
250
300
Figure 20: Varying Temperature, Temperature vs. Time
Liquid Fraction vs. Time [s]
1
0.9
0.8
Liquid Fraction
0.7
0.6
0.5
0.4
0.3
0.2
300 K
400 K
500 K
600 K
0.1
0
0
50
100
150
Time [s]
200
250
Figure 21: Varying Temperature, Liquid Fraction vs. Time
300
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.
3
x 10
5
Heat Flux [W/m2-s] vs. Time [s]
2.5
Heat Flux [W/m2-s]
2
1.5
1
0.5
0
-0.5
-1
300 K
400 K
500 K
600 K
-1.5
-2
0
50
100
150
Time [s]
200
250
300
Figure 22: Varying Temperature, Heat Flux vs. Time
5.1.4 Varying Initial Velocity
Another important factor to the demise of debris fragments is the flight speed.
The speed is varied from 7500 to 5500 m/s in 500 m/s steps at a breakup altitude
of 100 km. The variation in flight speed has an obvious effect on the rate of
altitude and cross range as seen in Figure 23 and Figure 24.
64
Aero-thermal Demise
Altitude [km] vs. Time [s]
120
7,500 km/s
7,000 km/s
6,500 km/s
6,000 km/s
5,500 km/s
100
Altitude [km]
80
60
40
20
0
0
200
400
600
800
1000
Time [s]
Figure 23: Varying Velocity, Altitude vs. Time
Altitude [km] vs. Range [km]
120
7,500 km/s
7,000 km/s
6,500 km/s
6,000 km/s
5,500 km/s
100
Altitude [km]
80
60
40
20
0
0
500
1000
Range [km]
1500
Figure 24: Varying Velocity, Altitude vs. Range
2000
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.
3
x 10
5
Heat Flux [W/m2-s] vs. Time [s]
2.5
Heat Flux [W/m2-s]
2
1.5
1
0.5
0
-0.5
7,500 km/s
7,000 km/s
6,500 km/s
6,000 km/s
5,500 km/s
-1
-1.5
-2
0
50
100
150
Time [s]
200
Figure 25: Varying Velocity, Heat Flux vs. Time
250
300
66
Aero-thermal Demise
Fragment Bulk Temperature [K] vs. Time [s]
900
Fragment Bulk Temperature [K]
800
700
600
500
7,500 km/s
7,000 km/s
6,500 km/s
6,000 km/s
5,500 km/s
400
300
0
50
100
150
Time [s]
200
250
300
Figure 26: Varying Velocity, Temperature vs. Time
Liquid Fraction vs. Time [s]
1
0.9
0.8
Liquid Fraction
0.7
0.6
0.5
0.4
0.3
7,500 km/s
7,000 km/s
6,500 km/s
6,000 km/s
5,500 km/s
0.2
0.1
0
0
50
100
150
Time [s]
200
250
Figure 27: Varying Velocity, Liquid Fraction vs. Time
300
T. M. Owens
67
5.1.5 Varying Flight Path Angle
The other important part of the initial state is the flight path angle. This does not
have as large an effect on the heating of the fragment as changing the altitude or
the velocity as the net energy in the trajectories is roughly similar. It does,
however, have a significant effect on the altitude versus time and cross range as
seen in Figure 28 and Figure 29. The flight path angles are -0.5, -3.0 and -5.5
degrees. The shallower angle would be typical of a gliding reentry, and the steeper
angle would be for a capsule return from deep space or the moon.
Altitude [km] vs. Time [s]
120
-0.5 deg
-3.0 deg
-5.5 deg
100
Altitude [km]
80
60
40
20
0
0
200
400
600
800
Time [s]
Figure 28: Varying Flight Path Angle, Altitude vs. Time
1000
68
Aero-thermal Demise
Altitude [km] vs. Range [km]
120
-0.5 deg
-3.0 deg
-5.5 deg
100
Altitude [km]
80
60
40
20
0
0
500
1000
1500
Range [km]
2000
2500
Figure 29: Varying Flight Path Angle, Altitude vs. Range
In Figure 30, the steeper flight path angle trajectory gives a higher and earlier peak
to the heat flux. In Figure 31 and Figure 32 it can be seen that the only significant
effect of the change in flight path angle is the earlier ablation for the steeper flight
angles. The liquid fraction is within 10% for the three cases. The steeper flight
path angle experiences less mass loss because it does not spend as much time in
the flight regime where aero-heating is dominant.
T. M. Owens
69
5
x 10
5
Heat Flux [W/m2-s] vs. Time [s]
4
Heat Flux [W/m2-s]
3
2
1
0
-0.5 deg
-3.0 deg
-5.5 deg
-1
-2
0
50
100
150
Time [s]
200
250
300
Figure 30: Varying Flight Path Angle, Heat Flux vs Time
Fragment Bulk Temperature [K] vs. Time [s]
900
Fragment Bulk Temperature [K]
800
700
600
500
400
300
-0.5 deg
-3.0 deg
-5.5 deg
0
50
100
150
Time [s]
200
250
Figure 31: Varying Flight Path Angle, Temperature vs. Time
300
70
Aero-thermal Demise
Liquid Fraction vs. Time [s]
1
0.9
0.8
Liquid Fraction
0.7
0.6
0.5
0.4
0.3
0.2
-0.5 deg
-3.0 deg
-5.5 deg
0.1
0
0
50
100
150
Time [s]
200
250
300
Figure 32: Varying Flight Path Angle, Liquid Fraction vs. Time
5.1.6 Varying Materials of Debris Fragment
For the previous examples aluminum was used as it is very common in spacecraft
construction that will often experience partial demise. Other materials such as
titanium almost never demise, whereas composites will demise very rapidly. To
show this, the example fragment is set with the same properties as in 5.1.1 for
aluminum, titanium and graphite reinforced epoxy materials. From the heat flux in
Figure 33, it can be seen that material properties have a significant effect on the
stagnation point heating.
T. M. Owens
71
3
x 10
5
Heat Flux [W/m2-s] vs. Time [s]
Aluminum
Titanium
Graphite-Epoxy
2.5
Heat Flux [W/m2-s]
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
50
100
150
200
250
300
Time [s]
350
400
450
500
450
500
Figure 33: Varying Materials, Heat Flux vs. Time
Fragment Bulk Temperature [K] vs. Time [s]
1000
Aluminum
Titanium
Graphite-Epoxy
Fragment Bulk Temperature [K]
900
800
700
600
500
400
300
0
50
100
150
200
250
300
Time [s]
350
400
Figure 34: Varying Materials, Temperature vs. Time
72
Aero-thermal Demise
The liquid fraction in Figure 35 is as expected. The aluminum debris fragment
partially demises as in earlier examples. The titanium fragment experiences no
significant ablation and the composite fragment demises fully. The graphite
reinforced epoxy reaches its melting point so it will have fully demised. The
composite fragment also starts demising earlier in the impact trajectory as less
heat energy is required to ablate it when compared to aluminum.
Liquid Fraction vs. Time [s]
1
Aluminum
Titanium
Graphite-Epoxy
0.9
0.8
Liquid Fraction
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
250
300
Time [s]
350
400
450
500
Figure 35: Varying Materials, Liquid Fraction vs. Time
5.1.7 Varying Mass of Debris Fragment
The example fragment being used throughout this section is a thin walled cylinder,
so by increasing the mass of the fragment, several different effects on the impact
trajectory and demise can be seen. The cases examined are 1, 2, 3, 5 and 8 times
the mass of the original fragment. In Figure 36 and Figure 37, the effects of the
increase in the ballistic coefficient of the higher mass fragments is that they have a
T. M. Owens
73
larger range to impact and impact earlier as they carry more speed being less
effected by aerodynamic forces.
Altitude [km] vs. Time [s]
120
50 kg
100 kg
150 kg
250 kg
400 kg
100
Altitude [km]
80
60
40
20
0
0
200
400
600
Time [s]
Figure 36: Varying Mass, Altitude vs Time
800
1000
74
Aero-thermal Demise
Altitude [km] vs. Range [km]
120
50 kg
100 kg
150 kg
250 kg
400 kg
100
Altitude [km]
80
60
40
20
0
0
500
1000
1500
Range [km]
2000
2500
3000
Figure 37: Varying Mass, Altitude vs. Range
The peak heat flux as shown in Figure 38 for the larger fragments is also much
higher than that of the lighter fragments. As can be seen in Figure 39 and Figure
40, this does not cause the fragment to have a higher bulk temperature or more
mass loss. The mass of the fragment requires more energy to heat up than the
increase in heat flux accounts for with these fragment properties. Another aspect
of the higher ballistic coefficient fragments is they experience peak aero-thermal
heating later in the impact trajectory. The lowest mass fragment has peak heating
at about 80 km, whereas the heaviest at 70 km. This is due to their ability to carry
speed into the denser atmosphere.
T. M. Owens
75
10
x 10
Heat Flux [W/m2-s] vs. Time [s]
50 kg
100 kg
150 kg
250 kg
400 kg
8
Heat Flux [W/m2-s]
5
6
4
2
0
-2
0
50
100
150
200
Time [s]
250
300
350
400
350
400
Figure 38: Varying Mass, Heat Flux vs. Time
Fragment Bulk Temperature [K] vs. Time [s]
1000
50 kg
100 kg
150 kg
250 kg
400 kg
Fragment Bulk Temperature [K]
900
800
700
600
500
400
300
0
50
100
150
200
Time [s]
250
300
Figure 39: Varying Mass, Temperature vs. Time
76
Aero-thermal Demise
Liquid Fraction vs. Time [s]
1
50 kg
100 kg
150 kg
250 kg
400 kg
0.9
0.8
Liquid Fraction
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
200
Time [s]
250
300
Figure 40: Varying Mass, Liquid Fraction vs. Time
350
400
T. M. Owens
5.2
77
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.
78
Aero-thermal Demise
The difference in the flat plates between the computational model and DAS 2.0 is
interesting. The low density graphite epoxy fragment does not experience much
heating, however the more compact and dense aluminum antenna does and
nearly demises. Testing the suggestion to change the area averaging factor to 0.8
for composites makes the graphite epoxy fragment fully demise at 78 km.
Table 7: Computational Model Debris Fragments, Compared to DAS 2.0
Name
Material
Area
Averaging
Factor 𝐤 𝟐
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
0.12
0.85
0
70.4
Aluminum 7075-T6
Overall the results from the computational model are more conservative than
those from DAS 2.0. A flight safety analyst may make the judgment that the two
fragments that have less than 15% of their original mass may be considered to
fully demise because of loss of structural integrity, bringing the results more in line
with DAS 2.0 predicting all fragments demise before impact.
5.2.2 Aerospace Survivability Tables
The computational model outlined in this thesis is in part based on the algorithm
that builds the Aerospace Survivability TablesRef. 24. The atmosphere, earth model
and impact integrator are all different. The computational model also using time
varying properties for the fragments as it integrates to impact, whereas the
survivability tables use static properties from the initial breakup. Also, being less
conservative means a greater potential for reduction in casualty estimation.
T. M. Owens
79
Table 8 and Table 9 make a comparison between the computational model and
the survivability tables across a broad range of fragment properties. The fragment
is an aluminum cylinder weighing from 5 to 5000 lbs with a length between 1 to
30 ft and radius from 0.1 to 5 ft. All impacts were computed for 42 Nmi of altitude,
-0.5 degrees flight path angle and a velocity of 25,000 ft/s. This is roughly
equivalent to the 78km breakup altitude case used by DAS 2.0. The tables are such
that only certain combinations of initial state and fragment properties are
available.
The upper set of tables with the green-red gradient has the liquid fraction as
computed in the model and the liquid fraction from the survivability tables with
red indicating debris survival to impact and green demise. The blank spots in the
table are for fragments that the survivability tables consider physically infeasible.
The lower red-blue gradient plots are the difference between the two data sets.
There is a significant difference between the two sets of liquid fractions. This is
not particularly troubling as the survivability tables are a conservative estimate
and use a very different impact integration method. The trend between the two
sets of liquid fractions as seen in the upper plot matches rather well however.
A previous simpler impact integrator used with the computational model that did
not take into account the mass varying properties of the debris fragment had a
better match to the survivability tables which also use a static fragment. It was,
however, very poor at predicting the actual point of impact so the new method is
preferable even with the poor match to the survivability tables. Also being less
conservative means that there is a greater potential for casualty reduction in the
risk analysis.
80
Aero-thermal Demise
Model 25000 ft/s
Table 25000 ft/s
25000 ft/s
25000 ft/s
42 nmi, -0.5 deg
Table 25000 ft/s
Model 25000 ft/s
Table 8: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder
r\l
0.1
0.5
1
1.5
2
3
4
5
0.1
0.5
1
1.5
2
3
4
5
r\l
0.1
0.5
1
1.5
2
3
4
5
0.1
0.5
1
1.5
2
3
4
5
W of 5 lb
1
2.5
1.0 1.0
1.0 1.0
1.0 0.9
0.7 0.7
0.6 0.5
0.4 0.3
0.2 0.1
0.0 0.0
1.0 1.0
1.0 1.0
0.3
5
1.0
1.0
0.8
0.6
0.5
0.2
0.0
0.0
1.0
1.0
0.3
0.0
W of 250 lb
1
2.5 5
0.0 0.0 0.0
0.3 0.7 1.0
0.4 0.6 0.8
0.5 0.6 0.7
0.5 0.6 0.6
0.5 0.5 0.5
0.4 0.4 0.4
0.3 0.3 0.3
10
1.0
1.0
1.0
0.5
0.3
0.1
0.0
0.0
20
1.0
1.0
0.5
0.3
0.1
0.0
0.0
0.0
30
1.0
1.0
0.5
0.0
0.0
0.0
0.0
0.0
1.0 1.0
0.2 0.4
10
0.0
1.0
1.0
0.8
0.7
0.5
0.4
0.3
20
0.0
1.0
1.0
0.8
0.7
0.5
0.4
0.3
30
0.0
1.0
1.0
0.8
0.6
0.5
0.3
0.3
0.2 0.5 0.8
0.1 0.2 0.3
0.0 0.1
0.0 0.0
0.0
0.0
0.0
1.0
0.4
0.0
0.0
0.0
0.0
0.0
0.4
0.0
0.0
0.0
0.0
0.0
20
30
W of 5 lb
r\l 1
2.5 5
0.1 0.0 0.0 0.0
0.5 0.0 0.0 0.0
1
0.6 0.5
1.5
0.6
2
3
4
5
W of 250 lb
r\l 1
2.5 5
0.1
0.5
0.5 0.5
1
0.5 0.6
1.5
0.7
2
0.6
3
4
5
10
0.0 0.0
0.8 0.1
10
20
30
0.2
0.7
0.7
0.7
0.5
0.4
0.3
0.0
0.6
0.8
0.7
0.5
0.4
0.3
0.6
0.8
0.6
0.5
0.3
0.3
W of 10 lb
1
2.5
0.0
1.0
1.0
1.0
1.0
1.0
0.7
0.7
0.6
0.6
0.4
0.4
0.3
0.3
0.2
0.1
1.0
1.0
1.0
0.3
W of 500 lb
1
2.5
0.0
0.0
0.0
0.0
0.1
0.3
0.2
0.4
0.3
0.4
0.3
0.4
0.3
0.3
0.3
0.3
0.0
W of 25 lb
1 2.5 5
0.0 0.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0
0.8 0.8 0.7
0.7 0.6 0.6
0.5 0.5 0.4
0.4 0.3 0.3
0.3 0.2 0.2
1.0
1.0 1.0
1.0 1.0 1.0
0.3 0.2 0.4
0.4 0.3
0.0
0.0
0.0
5
1.0
1.0
0.9
0.7
0.5
0.3
0.2
0.1
1.0
1.0
0.3
0.0
0.0
10
1.0
1.0
0.8
0.6
0.5
0.3
0.1
0.0
20
1.0
1.0
1.0
0.5
0.3
0.1
0.0
0.0
30
1.0
1.0
0.6
0.5
0.2
0.0
0.0
0.0
5
0.0
0.6
0.6
0.5
0.5
0.4
0.4
0.3
10
0.0
1.0
0.8
0.7
0.6
0.5
0.4
0.3
20
0.0
1.0
1.0
0.8
0.7
0.5
0.4
0.3
30
0.0
1.0
1.0
0.8
0.7
0.5
0.4
0.3
0.2
0.0
0.0
0.0
0.5
0.2
0.0
0.0
0.0
0.0
0.0
0.8
0.3
0.1
0.0
0.0
0.0
0.0
0.4
0.1
0.0
0.0
0.0
0.0
W of 750
1 2.5
0.0 0.0
0.0 0.0
0.0 0.2
0.1 0.2
0.1 0.2
0.2 0.3
0.2 0.3
0.2 0.2
lb
5
0.0
0.0
0.4
0.4
0.4
0.3
0.3
0.3
10
1.0
1.0
1.0
0.7
0.6
0.4
0.3
0.2
20
1.0
1.0
0.9
0.6
1.0
0.3
0.2
0.1
30
1.0
1.0
0.8
1.0
0.4
0.2
0.1
0.0
1.0
0.3
0.0
0.0
1.0
0.3 0.2
0.0 0.0
0.0
10
0.0
0.7
0.6
0.6
0.5
0.4
0.4
0.3
20
0.0
1.0
0.9
0.7
0.6
0.5
0.4
0.3
30
0.0
1.0
1.0
0.8
0.7
0.5
0.4
0.3
0.3 0.6
0.0 0.0 0.1 0.2 0.3
0.0 0.0 0.0 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
W of 10 lb
W of 25 lb
1
2.5
5 10 20 30 1 2.5 5 10
0.0 0.0
0.0
0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.7 0.6 0.5 0.8 0.2
0.6 0.7 0.7
0.7 0.6
0.7 0.7
0.5
0.6 0.6
0.0
0.6 0.6
0.6 1.0
1.0
W of 500 lb
1
2.5
5
20
0.3
0.4
0.6
0.5
0.5
10
20
30
0.5
0.6
0.7
0.6
0.5
0.4
0.3
0.2
0.7
0.7
0.7
0.5
0.4
0.3
0.6
0.7
0.7
0.5
0.4
0.3
W of 750 lb
1 2.5 5
10
20
30
30
0.4 0.4
0.2 0.4 0.5 0.7 0.7
0.4 0.6 0.7 0.7
0.4 0.5 0.6 0.7
0.4 0.5 0.5
0.4 0.4 0.4
0.3 0.3 0.3
T. M. Owens
81
Model 25000 ft/s
Table 25000 ft/s
25000 ft/s
25000 ft/s
42 nmi, -0.5 deg
Table 25000 ft/s
Model 25000 ft/s
Table 9: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder
r\l
0.1
0.5
1
1.5
2
3
4
5
0.1
0.5
1
1.5
2
3
4
5
r\l
0.1
0.5
1
1.5
2
3
4
5
0.1
0.5
1
1.5
2
3
4
5
W of
1
0.0
1.0
1.0
0.9
0.7
0.5
0.4
0.3
50 lb
2.5
5
0.0 0.0
1.0 1.0
1.0 1.0
0.8 0.8
0.7 0.6
0.5 0.5
0.4 0.3
0.3 0.3
10
1.0
1.0
1.0
0.7
0.6
0.4
0.3
0.2
0.7 1.0 1.0 1.0
0.4 0.4 0.3
0.0 0.0
0.0 0.0
0.0
0.0
W of
1
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.1
1000
2.5
0.0
0.0
0.1
0.1
0.1
0.2
0.2
0.2
lb
5
0.0
0.0
0.3
0.3
0.3
0.3
0.3
0.2
10
0.0
0.6
0.5
0.5
0.4
0.4
0.3
0.3
20
1.0
1.0
1.0
0.7
0.6
0.4
0.3
0.2
W of
1
0.0
0.9
0.9
0.8
0.7
0.5
0.4
0.3
75 lb
2.5
5
0.0 0.0
1.0 1.0
1.0 1.0
0.8 0.8
0.7 0.7
0.5 0.5
0.4 0.4
0.3 0.3
20
1.0
1.0
1.0
0.7
0.6
0.4
0.3
0.2
30
1.0
1.0
1.0
0.7
0.6
0.4
0.3
0.2
1.0
0.5 0.8 1.0 1.0 1.0
0.3 0.3
0.4 0.4 0.4 0.3
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
0.3
0.0
0.0
0.0
20
0.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
30
1.0
1.0
0.9
0.7
0.5
0.4
0.2
0.1
30
0.0
1.0
0.9
0.7
0.6
0.5
0.4
0.3
0.2 0.5
0.0 0.0 0.0 0.2 0.3
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
W of 50 lb
r\l 1
2.5
5
10 20
0.1
0.5 0.3 0.0 0.0 0.0 0.0
1
0.6 0.6 0.7 0.7
1.5
0.8 0.7 0.7
2
0.6 0.6 0.6
3
0.4 0.4
4
0.3
5
W of 1000 lb
r\l 1
2.5
5
10 20
0.1
0.5
0.4 0.4
1
0.1 0.3 0.5 0.6
1.5
0.3 0.5 0.7
2
0.3 0.4 0.6
3
0.4 0.5
4
0.3 0.4
5
0.3 0.3
30
0.6
0.7
0.5
30
0.6
0.7
0.6
0.5
0.4
0.3
10
0.0
1.0
1.0
0.8
0.6
0.5
0.3
0.3
W of
1
0.0
0.8
0.8
0.7
0.7
0.5
0.4
0.3
100
2.5
0.0
1.0
1.0
0.8
0.7
0.5
0.4
0.3
lb
5
0.0
1.0
1.0
0.8
0.7
0.5
0.4
0.3
10
0.0
1.0
1.0
0.8
0.6
0.5
0.4
0.3
20
1.0
1.0
1.0
0.7
0.6
0.4
0.3
0.2
30
1.0
1.0
1.0
0.7
0.6
0.4
0.3
0.2
1.0
0.3
0.0
0.0
0.0
0.0
0.3
0.0
0.0
0.0
20
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.1
30
0.0
0.0
0.3
0.3
0.2
0.2
0.2
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
10
20
30
0.0
0.7
0.7
0.6
0.4
0.3
0.7
0.7
0.6
0.4
20
30
0.1
0.1
0.1
0.1
0.1
0.1
0.3
0.3
0.2
0.2
0.2
0.1
10
0.0
0.0
0.2
0.2
0.2
0.1
0.1
0.1
20
0.0
0.5
0.4
0.4
0.3
0.3
0.3
0.2
30
0.0
0.7
0.6
0.5
0.5
0.4
0.3
0.3
0.3 0.7 1.0 1.0
0.3 0.4 0.4
0.1 0.0
0.0 0.0
0.0
0.0
0.0
W of 5000 lb
1
2.5
5
10
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 0.0 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
0.0
0.0
0.0
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
10
20
30
0.4 0.2 0.0 0.0
0.6 0.6 0.6
0.8 0.8
0.7 0.6
0.5
0.3
0.0
0.7
0.7
0.6
0.4
0.7
0.7
0.6
0.4
W of
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
2500
2.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
lb
5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
W of 75 lb
1
2.5
5
W of 100 lb
1
2.5
5
10
20
0.5 0.3 0.0 0.0
0.7 0.6 0.6
0.7 0.8
0.7 0.6
0.5
0.4
0.3
W of 5000 lb
30
1
2.5
5
10
0.2
0.0 0.2
0.0 0.2
0.1
0.1
0.1
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.6
0.5
0.5
0.4
0.3
0.3
W of 2500 lb
1
2.5
5
0.0
0.0 0.0
0.0
0.0
0.0
82
Aero-thermal Demise
5.3
Input and Output Debris Fragment Catalog
The demise model is able to ingest a catalog of debris fragments and perform a
survivability analysis for an entire trajectory at any malfunction times desired. The
trajectory can be for launch or reentry vehicle. This builds a full set of demise
adjust fragment tables across the trajectory, allowing other tools to make a
complete casualty risk estimation for the mission instead of single failure events.
The MATLAB functions importFragment and exportFragment work to import from
a formatted CSV to a MATLAB structure defining the debris fragment and then
export the demised fragment set to a CSV file of the same format. This allows for
import of the fragment data across a wide range of risk analysis tools.
For a test case, the debris fragment defined in Table 10 from a SpaceX Falcon 9
rocket launch was used as an example. The coefficient of drag tables vary by flight
Mach and all others vary by the trajectory time. Only the properties used in the
demise analysis are included in the tables. Other properties not included, such as
explosion velocity and yield factors, are fragment properties that other risk
analysis tools might use.
Table 10: Demise Utility Input
Name
2nd stage main
LOX line
Mach
Cd
1
0.81
0.9999
0.45
Shape
Radius [m]
Length [m]
Cylinder
0.064008
0.36576
Material
Aluminum
(generic)
Time [s]
Mass [kg]
Area [m2]
Beta [kg/m^2]
Hazard Radius [m]
0
9.2049
0.046823
242.7021
0.1215
A demise analysis was performed on the fragment between 512 and 550 seconds
of the launch trajectory. This details the over-flight of Europe as seen in Figure 10.
T. M. Owens
83
The output generated with the fragment exporter can be seen in Table 11. The
first two columns are the Mach versus coefficient of drag table. The last five
columns are all time based tables. Each time is a breakup event where the
trajectory state is used as the initial state of the demising fragment. The
properties in the table are the impact states of the fragment which can be used in
risk analysis. Because this is a launch trajectory the later times, which are higher
speed and altitude, see the fragment experiencing greater mass loss. One thing to
note is the hazard radius, which can be used to compute the casualty or hazard
area. As the fragment is mostly hollow there is not a significant reduction in
overall area as it ablates. This is the case with many debris fragments, so to get an
appreciable reduction in casualty area a fragment must fully demise. The mass
does reduce to a less than half of the initial mass after 548 seconds which can give
a casualty reduction when considering sheltering effects or risk to aircraft. The
reduction in mass means less damage to structures, leading to fewer casualties.
Table 11: Demise Utility Output, Adjusted Fragment Tables
Mach
Cd
Time [s]
Mass [kg]
Area [m2]
Beta [kg/m^2]
Hazard Radius [m]
21.48
0.44
512
9.205
0.046823
447.9701
0.1215
21.21
0.45
513
9.205
0.046823
447.9701
0.1215
20.93
0.45
514
9.205
0.046823
447.9701
0.1215
20.64
0.46
515
9.205
0.046823
447.9701
0.1215
20.34
0.47
516
9.205
0.046823
447.9701
0.1215
20.03
0.48
517
9.205
0.046823
447.9701
0.1215
19.72
0.49
518
9.205
0.046823
447.9701
0.1215
19.39
0.50
519
9.205
0.046823
447.9701
0.1215
19.06
0.50
520
9.1798
0.046773
447.2244
0.12144
18.72
0.51
521
9.0828
0.046579
444.3356
0.12118
18.37
0.52
522
8.9823
0.046378
441.3262
0.12092
18.01
0.53
523
8.8793
0.046171
438.2193
0.12065
84
Aero-thermal Demise
Mach
Cd
Time [s]
Mass [kg]
Area [m2]
Beta [kg/m^2]
Hazard Radius [m]
17.65
0.54
524
8.7731
0.045957
434.9959
0.12037
17.28
0.55
525
8.6632
0.045735
431.6374
0.12008
16.91
0.57
526
8.5507
0.045506
428.1696
0.11978
16.54
0.58
527
8.4338
0.045268
424.543
0.11946
16.18
0.59
528
8.3131
0.04502
420.7661
0.11914
15.82
0.60
529
8.1886
0.044764
416.8372
0.1188
15.47
0.61
530
8.0596
0.044497
412.7308
0.11844
15.10
0.62
531
7.9259
0.044219
408.4359
0.11807
14.74
0.64
532
7.7876
0.04393
403.9496
0.11769
14.36
0.65
533
7.6438
0.043628
399.2335
0.11728
13.99
0.66
534
7.4941
0.043312
394.2739
0.11686
13.62
0.67
535
7.3389
0.042982
389.0744
0.11641
13.26
0.68
536
7.1762
0.042634
383.5531
0.11594
12.90
0.69
537
7.0068
0.042269
377.7339
0.11544
12.54
0.70
538
6.8301
0.041886
371.5778
0.11492
12.18
0.71
539
6.6442
0.041479
365.0059
0.11436
11.82
0.72
540
6.4496
0.04105
358.0165
0.11376
11.46
0.73
541
6.2447
0.040594
350.5357
0.11313
11.10
0.74
542
6.0281
0.040108
342.4799
0.11245
10.74
0.75
543
5.7993
0.039589
333.7988
0.11172
10.39
0.76
544
5.5566
0.039032
324.3892
0.11093
10.03
0.77
545
5.2984
0.038433
314.142
0.11008
9.68
0.77
546
5.0213
0.037781
302.8519
0.10914
9.34
0.78
547
4.724
0.037071
290.3758
0.10811
9.00
0.78
548
4.4031
0.036292
276.4624
0.10697
8.67
0.79
549
4.0536
0.035427
260.73
0.10568
8.35
0.79
550
3.67
0.034457
242.7048
0.10423
8.04
0.80
7.73
0.80
T. M. Owens
Mach
Cd
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
85
Time [s]
Mass [kg]
Area [m2]
Beta [kg/m^2]
Hazard Radius [m]
86
6
Aero-thermal Demise
Conclusions
For this thesis, a computational model was developed using an earth model
defined by WGS 84 with a fourth order harmonic model of gravity, the 1976 U.S.
Standard Atmosphere, a general impact integrator for a rotating earth and a
stagnation point heating model based on Fay-Riddell theory. The model can be
used with a wide range of demise fragments differing in shape, material and
breakup state. It is also able to generate a more usable output than that of DAS
2.0 with inputs of similar complexity. DAS 2.0 is only able to consider a single
breakup condition from an uncontrolled orbital reentry. The computational model
can use any breakup state defined by a reentry or launch trajectory for
uncontrolled or controlled flight, making it much more flexible in application.
There is still a significant gap in complexity and capability between the model
developed in this thesis and tools like ORSAT, however, it should be able to reduce
the need for these tools by giving an adequate estimation of aero-thermal demise
for many different kinds of mission risk analysis.
6.1
Practical Application
There are numerous applications for a fragment set with demise adjusted
properties, and many risk analysis tools will derive benefits from its application.
For example a probability of impact tool would benefit from the corrected ballistic
coefficient to make more accurate predictions of the impact point. Expected
casualty estimation would likewise benefit from the reduction in casualty area in
partially and fully demised debris to reduce the overall expected casualties. More
specialized tools like ship and aircraft hit predictors, real-time systems and
destruct line tools will also be able to use the demise adjusted fragment catalog.
T. M. Owens
87
Of significance, a similar method to the algorithm outlined in this thesis developed
by the author for Millennium Engineering and Integration’s Joint Advanced Range
Safety System Mission Planning (JARSS MP) has already been used to perform
fragment demise analysis on missions for the SpaceX Falcon 9, Boeing X-37 and
other vehicles. The work for the SpaceX Falcon9 debris catalog was done as part of
a task for the FAA to assess the overflight risk of the Falcon 9-0003 mission. The
Boeing X-37 debris catalog was developed as part of the OTV Feasibility Analysis
for possible landing at the Cape Canaveral Air Force Station under the 30th and
45th Space Wings. The algorithm in this thesis has several advantages in that it has
a more sophisticated impact integrator and takes into account the time varying
properties of the debris fragments.
In all cases, the more accurate prediction of impacts and expected casualty risk
will give the mission analyst more confidence in how to manage the risk of the
mission. A greater confidence in risk analysis would result in the ability to close
airspace for less time, allow more ships downrange, not close down facilities
around the launch site or even be the difference between acceptable and
unacceptable overall mission risk. Notably, the above can reduce the cost and
increase the number of launch opportunities.
6.2
Validation
The computational model outlined in this thesis finds itself between DAS 2.0 and
the Aerospace Survivability Tables in terms of the conservatism of demise
prediction. DAS 2.0 is considered the conservative answer that then triggers a
higher fidelity analysis from tools like ORSAT, so being less conservative than
survivability tables is not a significant concern. See 5.2 Model Comparisons for a
detailed comparison between the computational model and existing tools.
88
Aero-thermal Demise
Many parts of the algorithm were chosen based on their previous acceptance in
the risk assessment industry. The WGS 84 and 1976 Standard Atmosphere are
both commonly used in many risk analysis tools. The impact integration algorithm
has been used as part of a glide turn integrator in JARSS MP by the 45 th 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 aerobreakup or thermal fragmentation model. Probably the simplest possible method
is to define a loading factor, like wing loading, and assume vehicle breakup when
this value is exceeded in malfunction turns. More complex methods involve
estimating the strength of a parent structure and calculating the resulting child
fragments upon breakup.
While the thrust of this thesis was to estimate the aero-thermal demise of reentry
debris, the algorithm could be applied to vehicle trajectory design. The impact
integrator is appropriate for either ballistic or gliding reentry, and the stagnation
point heating method can be applied to an intact vehicle. The algorithm could be
modified to find the heating of the spacecraft and also used to determine if a
trajectory has too much heat loading. This would aid in the down selection from
several possible reentry scenarios.
Another possibility is to expand to a full mission risk analysis tool. This would
require extensive work to develop algorithms for the probability of impact, risk to
population, sheltering and et cetera. There are many tools that already
90
Aero-thermal Demise
accomplish this, like JARSS MP, so this computational model is probably more
beneficial to fragment demise analysis.
T. M. Owens
91
References
Ref. 1
"Final Flight Plan Approval (Module 4.4)," Range Safety Training Program:
Range Flight Safety Analysis, NASA & DOD, July 2005. (PPT of slides).
Ref. 2
Brown, M., Stansbery, E., "Orbital Debris: ORSAT," NASA Orbital Debris
Program Office, URL: http://orbitaldebris.jsc.nasa.gov/reentry/orsat.html,
08/24/2009.
Ref. 3
Carmichael, R., "Properties of the U.S. Standard Atmosphere 1976," Public
Domain Aeronautical Software, URL: http://www.pdas.com/atmos.html,
13 February 2014.
Ref. 4
Detra, R. W., Kemp, N.H., Riddell, F. R., "Addendum to Heat Transfer to
Satellite Vehicle Re-entering the Atmosphere," Journal of Jet Propulsion,
Vol. 27, No. 12, 1957, 1256-1257.
Ref. 5
Fay, J. A., Kemp, N. H., "Theory of Stagnation-Point Heat Transfer in a
Partially Ionized Diatomic Gas," AIAA Journal, Vol. 1, No. 12, 1963, pp.
2741-2751.
Ref. 6
Fay, J. A., Riddell, F. R., "Theory of Stagnation Point Heat Transfer in
Dissociated Air," Journal of the Aeronautical Sciences, Vol. 25, No. 2, 1958,
pp. 73-85, 121.
Ref. 7
Gollan, R. J., Jacobs, P. A., Karl, S., Smith, S. C., "Numerical Modelling of
Radiating Superorbital Flows," ANZIAM Journal, vol. 45, No. E, pp. 248-268.
Ref. 8
Hallman, W. P., Moody, D. M., "Trajectory Reconstruction and Heating
Analysis of Columbia Composite Debris Pieces," Aerospace Report No. ATR2005(5138)-1, April 15 2005.
Ref. 9
Kelly, R. L., Rochelle, W. C., "Atmospheric Reentry of a Hydrazine Tank,"
ESCG Jacobs Technology, Huston, TX, 2008.
92
Ref. 10
Aero-thermal Demise
Kemp, N. H., Riddell, F. R. "Heat Transfer to Satellite Vehicles Re-entering
the Atmosphere," Journal of Jet Propulsion, Vol. 27, No. 2, 1957, pp. 132137.
Ref. 11
Koppenwallner, G., Fritshce, B., Lips, T., Klinkrad, H., "SCARAB – A MultiDisciplinary Code for Destruction Analysis of Space-Craft During Re-Entry,"
Fifth European Symposium on Aerothermodynamics for Space Vehicles,
ESA SP-563, Cologne, Germany, February 2005.
Ref. 12
Millennium Engineering and Integration Company: Range Systems, "Joint
Advanced Range Safety System Mathematics and Algorithms," Satellite
Beach, Florida, May 5 2014.
Ref. 13
National Aeronautics and Space Administration, "Process for Limiting
Orbital Debris," National Aeronautics and Space Administration, NASA-STD
8719.14 (Rev. A with Change 1), May 2012.
Ref. 14
National Imagery and Mapping Agency, "Department of Defense World
Geodetic System 1984: Its Definition and Relationships with Local Geodetic
Systems," National Imagery and Mapping Agency (NIMA), 3rd Ed.,
Amendment 2, 23 June 2004.
Ref. 15
National Oceanic and Atmospheric Administration, National Aeronautics
and Space Administration, United States Air Force," U.S. Standard
Atmosphere, 1976," NASA-TM-X-74335.
Ref. 16
Office of Safety and Mission Assurance, Johnson space Center Space
Physics Branch, "NASA Safety Standard: Guidelines and Assessment
Procedures for Limiting Orbital Debris," National Aeronautics and Space
Administration, NSS 1740.14, August 1995.
T. M. Owens
Ref. 17
93
Opiela, J. N., Hillary, E., Whitlock, D. O., Hennigan, M., ESCG, "Debris
Assessment Software Version 2.0 User’s Guide," JSC 64047, NASA Lyndon
B. Johnson Space Center, Huston, TX, Jan. 2012.
Ref. 18
Samareh, J. A., "A Multidisciplinary Tool for Systems Analysis of Planetary
Entry, Descent, and Landing (SAPE)," NASA/TM-2009-215950, Langley
research Center, Hampton, VA, Nov. 2009.
Ref. 19
Schneider, S. P., Gustafson, W., "Methods for Analysis of Preliminary
Spacecraft Designs," Purdue University, Sept. 19, 2005, pp. 27.
Ref. 20
Scott, C. D., et. al., "Design Study of an Integrated Aerobreaking Orbital
Transfer Vehicle," National Aeronautics and Space Administration, NASATM- 58264, Huston, TX, March 1985.
Ref. 21
Sutton, K., Graves, R. A. Jr., "A General Stagnation-Point ConvectiveHeating Equation For Arbitrary Gas Mixtures," NASA TR R-376, Langley
Research Center, Hampton, VA, Nov. 1971.
Ref. 22
Tauber, M. E., "A Review of High-Speed Convective, Heat-Transfer
Computation Methods," NASA TP-2914, Jul. 1989.
Ref. 23
Tauber, M. E., Menees, G. P., Adelman, H. G., "Aerothermodynamics of
Transatmospheric Vehicles," AIAA Paper 86-1257, Jun. 1, 1986.
Ref. 24
Tooley, J., Habiger, T. M., Bohman, K. R., "Reentry Hazard Analysis
Handbook," Aerospace Report No. ATR-2005(5138)-2, Jan. 28 2005.
Ref. 25
Vinh, N. X., "Flight Mechanics of High-Performance Aircraft," Cambridge
Aerospace Series 4, New York, NY, 1999.
Ref. 26
Weaver, M. A., Baker, R. L., Frank, M. V., "Probalistic Estimation of Reentry
Debris Area," Third European Conference on Space Debris, ESA SP-473,
Vol. 2, Darmstadt, Germany, March 2001, pp. 515-520.
94
Aero-thermal Demise
Appendix
Appendix A: Material Properties
Material properties from Debris Assessment Software Version 2.0 User’s GuideRef.
17
, the DAS 2.0 built-in materials. Specific heats used are the mean specific heat
between the reference and melting temperature.
Specific
Heat of
Melt
Density
Heat
Fusion
Temperature
Material
(kg/m3)
(J/kg-K)
(J/kg)
(K)
Acrylic
1170
1465
0
505
Alumina
3990
1011
106757
2305.4
Aluminum (generic)
2700
1100
390000
850
Aluminum 1145-H19
2697
904
386116
919
Aluminum 2024-T3
2803.2
972.7
386116
856
Aluminum 2024-T8xx
2803.2
972.7
386116
856
Aluminum 2219-T8xx
2812.8
1006.5
386116
867
Aluminum 5052
2684.9
900.2
386116
880
Aluminum 6061-T6
2707
896
386116
867
Aluminum 7075-T6
2787
1012.4
376788
830
Barium Element
3492
285
55824
983
Beryllium Element
1842
2635.1
1093220
1557
Beta Cloth
1581
837.5
232.6
650
Brass- Cartridge
8521.8
406.1
179091
1208
Brass- Muntz
8393.67
412.35
167461
1174
T. M. Owens
95
Specific
Heat of
Melt
Density
Heat
Fusion
Temperature
Material
(kg/m3)
(J/kg-K)
(J/kg)
(K)
Brass- Red
8746
404
195372
1280
Cobalt
8862
658.45
259600
1768
Copper Alloy
8938
430.6
204921
1356
Cork
261.294
1629.2
2860980
922
Cu/Be (0.5% Beryllium)
8800
397
204921
1320
Cu/Be (1.9% Beryllium)
8248.6
452.5
204921
1199
Fiberfrax
96.1
1130.5
0
2089
Fiberglass
1840.35
1046.8
232.6
1200
FRCI-12 (shuttle tile)
192.22
1978.9
0
1922
Gallium Arsenide (GaAs)
5316
325
0
1510
Germanium
5320
363.7
430282.6
1210.7
Gold Element
19300
139.85
64895
1336
Graphite Epoxy 1
1550.5
879.3
23
700
Graphite Epoxy 2
1550.5
879.3
23
700
Hastelloy 188
8980
498.1
309803
1635
Hastelloy 25
9130
498.1
309803
1643
Hastelloy c
8920.67
596.5
309803
1620
Hastelloy n
8576.4
501.7
309803
1623
Inconel 600
8415
538.45
297206
1683.9
Inconel 601
8057.29
632.9
311664
1659
Inconel 625
8440
410
311664
1593
Inconel 718
8190
435
311664
1571
96
Aero-thermal Demise
Specific
Heat of
Melt
Density
Heat
Fusion
Temperature
Material
(kg/m3)
(J/kg-K)
(J/kg)
(K)
Inconel X
8297.5
484.05
311664
1683.2
Invar
8050
566.55
2740000
1700
Iron
7865
572.6
272125
1812
Lead Element
11677
134.65
23958
600
Macor Ceramic
2520
790
236850
1300
Magnesium AZ31
1682
1212.8
339574
868
Magnesium HK31A
1794
1184.75
325619
877
MLI
772.48
1046.6
232.6
617
Molybdenum
10219
321.85
293057
2899
MP35N
8430
583
309803
1650
Nickel
8906.26
583.35
309803
1728.2
Niobium (Columbium)
8570
307.65
290000
2741
NOMEX
1380
1256
232.6
572
Platinum
21448.7
138.45
113967
2046.4
Polyamide
1420
1130
232.6
723
Polycarbonate (aka Lexan)
1250
1260
0
573
RCG Coating
1665.91
1224.2
0
1922
Reinforced Carbon-Carbon
1688.47
1257.55
37650
2144
Rene 41
8249
630.9
311664
1728
Silver Element
10492
233.15
105833
1234
Sodium-Iodide
3470
84
290759
924
Stainless Steel (generic)
7800
600
270000
1700
T. M. Owens
97
Specific
Heat of
Melt
Density
Heat
Fusion
Temperature
Material
(kg/m3)
(J/kg-K)
(J/kg)
(K)
Stainless Steel 17-4 ph
7833.03
666.8
286098
1728
Stainless Steel 21-6-9
7832.8
439.2
286098
1728
Steel A-286
7944.9
460.6
286098
1644
Steel AISI 304
7900
545.1
286098
1700
Steel AISI 316
8026.85
460.6
286098
1644
Steel AISI 321
8026.6
608.2
286098
1672
Steel AISI 347
7960
554.95
286098
1686
Steel AISI 410
7749.5
485.7
286098
1756
Strontium Element
2595
737
95599
1043
Teflon
2162.5
1674
0
533
Titanium (6 Al-4 V)
4437
805.2
393559
1943
Titanium (generic)
4400
600
470000
1950
Tungsten
19300
157.55
220040
3650
Uranium
19099
158.95
52523
1405
Uranium Zirconium
6086.8
418.7
131419
6086.8
Water
999
5490.55
0.1
273
Zerodur
2530
2487.1
250000
1424
Zinc
7144.2
405.3
100942
692.6
Hydride
98
Aero-thermal Demise
Appendix B: Supplemental Algorithms
Alternate Correlations
There are many possible correlations that can be made from the Fay-Riddell
theory. They are all quite similar in derivation to the Detra-Kemp-Riddell
implemented in this thesis.
Tauber-Menees-Adelman stagnation point heating correlationRef. 23,
𝑞𝑠𝑡𝑎𝑔 = 1.83 ∙ 10−4 √
𝜌∞ 1 − (𝐶𝑝̅ 𝑏 − 𝑇𝑤 )
3
(
) 𝑈∞
1 2
𝑅ℎ
2 𝑈∞
Eq. 138
Sutton-Graves stagnation point heating correlationRef. 21, Ref. 18,
𝑞𝑠𝑡𝑎𝑔 = 1.7623 ∙ 10−4 √
𝜌∞ 3
𝑈
𝑅ℎ ∞
Eq. 139
Scott stagnation point heating correlationRef. 20,
𝑞𝑠𝑡𝑎𝑔
𝜌∞ 𝑈∞ 3.05
= 1.83 ∙ 10 √ ( 4 )
𝑅ℎ 10
−4
Eq. 140
Tauber-Bowles-Yang stagnation point heating correlationRef. 19,
𝑞𝑠𝑡𝑎𝑔 = 1.83 ∙ 10−8 √
𝜌∞ 1 − (𝐶𝑝̅ 𝑏 − 𝑇𝑤 )
3
(
) 𝑈∞
1 2
𝑅ℎ
2 𝑈∞
Eq. 141
Tauber-Bowles-Yang stagnation point heating correlation for Mars and VenusRef. 19,
T. M. Owens
99
𝜌∞ 1 − (𝐶𝑝̅ 𝑏 − 𝑇𝑤 )
3.04
𝑞𝑠𝑡𝑎𝑔 = 1.35 ∙ 10−8 √ (
) 𝑈∞
1 2
𝑅ℎ
2 𝑈∞
Eq. 142
Tauber-Sutton formula for radiative heatingRef. 7,
𝑏
𝑞𝑟𝑎𝑑 = 𝑅ℎ 𝑎 ∙ 𝜌∞
∙ 𝑓(𝑈∞ )
{
0 ≤ 𝑎 ≤ 1,
𝑎 = 0.526,
𝑓𝑜𝑟 𝐸𝑎𝑟𝑡ℎ 𝑟𝑒𝑒𝑛𝑡𝑟𝑦
𝑓𝑜𝑟 𝑀𝑎𝑟𝑠 𝑟𝑒𝑒𝑛𝑡𝑟𝑦
{
𝑏 = 1.22,
𝑏 = 1.19,
𝑓𝑜𝑟 𝐸𝑎𝑟𝑡ℎ 𝑟𝑒𝑒𝑛𝑡𝑟𝑦
𝑓𝑜𝑟 𝑀𝑎𝑟𝑠 𝑟𝑒𝑒𝑛𝑡𝑟𝑦
7
𝑓(𝑈∞ ) ≅ 𝑈∞
Eq. 143
Eq. 144
Rate of mass loss to ablation,
𝑚̇ 𝑏 = −
𝑄̇
ℎ𝑓
Eq. 145
100
Aero-thermal Demise
Trajectory Site Direction Cosines
𝜋
−𝛷
2
1
0
0
𝑀𝑥 = [0 cos 𝑥 sin 𝑥 ]
0 − sin 𝑥 cos 𝑥
𝑥=
2𝜋
𝑧=
−𝜃
3
cos 𝑧 − sin 𝑧
𝑀𝑧 = [ sin 𝑧 cos 𝑧
0
0
Eq. 146
0
0]
1
𝐷𝐶 = 𝑀𝑥 𝑀𝑧
Rotate using the azimuth,
cos 𝛹
𝐷𝐶 = 𝐷𝐶 [ sin 𝛹
0
− sin 𝛹
cos 𝛹
0
0
0]
1
Eq. 147
ECEF Coordinates to XYZ Coordinates
Translate origin to site location
𝑒 = 𝑒 + 𝑒𝑠𝑖𝑡𝑒
𝑓 = 𝑓 + 𝑓𝑠𝑖𝑡𝑒
Eq. 148
𝑔 = 𝑔 + 𝑔𝑠𝑖𝑡𝑒
Rotate position and velocity,
[𝑥 𝑦 𝑧] = 𝐷𝐶 ′ [𝑒 𝑓 𝑔]
[𝑥̇ 𝑦̇ 𝑧̇ ] = 𝐷𝐶 ′ [𝑒̇ 𝑓̇ 𝑔̇ ]
Eq. 149
T. M. Owens
101
ECEF Coordinates to Aeronautical Coordinates
𝑊2 = 𝑒 2 + 𝑓 2
𝑍2 = 𝑒 2 + 𝑓 2
𝑅12 = 𝑊2 + 𝑍2
𝑍2
𝑅22 = 𝑊2 + (
)
1 − 𝑒⊕
𝑅12
√𝑅12
𝑆2 = (
− 1)
𝑎⊕
𝑅22
𝑠𝑒2 =
𝑒⊕́
Eq. 150
𝑅12
𝑆 = 𝑆2 (1 + 1.5𝑆2 𝑊2 𝑍2 𝑠𝑒2 2 )
𝑣1 = 1 + 𝑆
𝑔𝑣1
𝑔𝑠 =
1 − 𝑒⊕́ + 𝑆
Latitude,
𝜙 = tan−1
𝑔𝑠
√𝑊2
Eq. 151
𝑓
𝑒
Eq. 152
Longitude,
𝜃 = tan−1
Altitude,
𝑧 = √𝑊2 + 𝑔𝑠 2 −
√𝑊2 + 𝑔𝑠 2
𝑣1
Heading,
𝜓𝑖 = tan−1
Ground Speed,
𝑥𝑖̇
𝑦𝑖̇
Eq. 153
Eq. 154
102
Aero-thermal Demise
𝑣𝑖 = √𝑥𝑖̇ 2 + 𝑦𝑖̇ 2
Eq. 155
𝑧𝑖̇ = 𝑧𝑖̇
Eq. 156
Vertical Speed,