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VOLUME 8

Special Report

Optical Signatures Code

Volume 8

OPTISIG Algorithm Notes

June 2016

SPONSORED BY THE MISSILE DEFENSE AGENCY

NOTICE – The views, opinions, and/or findings contained in this document are those of the author(s) and should not be construed as an official Department of Defense position, policy, or decision unless designated by other official documentation.

DISTRIBUTION STATEMENT C – Distribution authorized to U.S. Government Agencies and their contractors; Critical Technology, Export Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224 Martin Road, Redstone Arsenal, AL 35898 or higher DoD authority.

WARNING – This document contains technical data whose export is restricted by the Arms Export Control Act (Title 22, U.S.C. sec 2751 et seq.) or the Export Administration Act of 1979, as amended, Title 50, U.S.C., App. 2401 et seq.

Violations of these export laws are subject to severe criminal penalties.

Disseminate in accordance with the provisions of DoD Directive 5230.25.

DESTRUCTION NOTICE – For classified documents follow the procedures in DoD 5200.22-M, Industrial Security Manual, Section 11-19 ORD or 5200.1-R, Information Security Program Regulation Chapter IX. For unclassified, limited documents destroy by any method that will prevent disclosure of contents or reconstruction of the document.

NOTICE – This document contains information that may be exempt from mandatory disclosure under the Freedom of Information Act.

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Optical Signatures Code Volume 8 – OPTISIG Algorithm Notes OSF Contract No.:

HQ0147-11-D-0015

Task Order 9

CDRL A031

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900RPT2255A

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Redstone Arsenal, Alabama 35898

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13. ABSTRACT (Maximum 200 words)

The OPTISIG Algorithm Notes manual provides background information on the numerical methods used to simulate physical processes in OPTIcal Signature In-line Generator (OPTISIG). OPTISIG simulates target geometry, trajectories, and thermal response to compute optical signatures for sensor pass bands from visible through long wavelength infrared. OPTISIG is part of the Optical Signatures Code (OSC) suite of tools developed by Teledyne Brown Engineering, Inc. for the Missile Defense

Agency, the U.S. Army Space and Missile Defense Command, and the Air Force Research Laboratory in support of ballistic missile defense programs.

14. SUBJECT TERMS

OPTISIG, OSC, optical signatures, infrared, IR, missile defense, algorithm, discrimination

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SPECIAL REPORT

900RPT2255A

Optical Signatures Code

Volume 8

OPTISIG Algorithm Notes

June 2016

NOTICE – The views, opinions, and/or findings contained in this document are those of the author(s) and should not be construed as an official Department of Defense position, policy, or decision unless designated by other official documentation.

DISTRIBUTION STATEMENT C – Distribution authorized to U.S. Government Agencies and their contractors; Critical Technology, Export Controlled; 30 November 2015. Other requests for this document shall be referred to MDA/ESP (Attn: Mark Underwood or active program COTR), 5224 Martin Road, Redstone Arsenal, AL 35898 or higher DoD authority.

WARNING – This document contains technical data whose export is restricted by the Arms Export Control Act (Title 22, U.S.C. sec 2751 et seq.) or the Export Administration Act of 1979, as amended, Title 50, U.S.C., App. 2401 et seq.

Violations of these export laws are subject to severe criminal penalties.

Disseminate in accordance with the provisions of DoD Directive 5230.25.

DESTRUCTION NOTICE – For classified documents follow the procedures in DoD 5200.22-M, Industrial Security Manual, Section 11-19 ORD or 5200.1-R, Information Security Program Regulation Chapter IX. For unclassified, limited documents destroy by any method that will prevent disclosure of contents or reconstruction of the document.

NOTICE – This document contains information that may be exempt from mandatory disclosure under the Freedom of Information Act.

Prepared For

Missile Defense Agency

Redstone Arsenal, Alabama

Under Contract Number

HQ0147-11-D-0015, Task Order 9

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This document was prepared and released in accordance with CDRL A039.

A 27 June 2016 Cover, i-vi, 7-1 Updated distribution markings, task order number, sponsor, and references 6 and 7.

Changed CDRL number from A039 to A031 v

ABSTRACT

OPTISIG is a high-speed missile threat optical signatures code that provides visible through long wavelength infrared (visible - LWIR) optical signatures for missile defense applications.

This manual provides the user with a technical discussion of the algorithms and approximations used in the code.

APPROVED BY:

vi

FOREWORD

The OPTIcal Signature In-line Generator (OPTISIG) is incorporated in the Optical Signatures

Code (OSC) software suite as a stand-alone program and as a subroutine that can be integrated into other simulations. OSC and OPTISIG are prepared as part of the Operation, Maintenance, and Sustainment of the Hardbody Toolset of the MDA Electro-Optical/Infrared

(EO/IR) Toolkit for Contract Number HQ0147-11-D-0015, under the technical direction of Dr.

Mark Underwood (MDA/ESP) for the Missile Defense Agency. Recent previous releases were issued under the technical direction of Mr. Cliff Burson (MDA/ESP) and Ms. Carol Barclay

(USASMDC/ARSTRAT).

The following people have contributed, over time, to OPTISIG’s programming and documentation: John W. Robinson, Robert A. Bradford, Andrew Edwards, Jeffery L. King, and

W. Brown Wellman.

vii

Table of Contents

1. Introduction ...................................................................................................................................... 1-1

2. Initialization Quantities .................................................................................................................... 2-1

2.1 Time, Timesteps, and Events .................................................................................................. 2-1

2.2 Coordinate Systems ................................................................................................................. 2-2

2.2.1 Important Triplets of Axes ................................................................................................ 2-4

2.2.2 Kinematic Vectors ............................................................................................................ 2-4

2.2.3 Environmental Vector....................................................................................................... 2-4

2.2.4 Intrabody Vectors ............................................................................................................. 2-4

2.3 Trajectory and Dynamics ......................................................................................................... 2-5

2.3.1 Ballistic Trajectories ......................................................................................................... 2-5

2.3.2 Aircraft/Stationary Trajectories ....................................................................................... 2-5

2.3.3 Exoatmospheric Motion About the Center of Mass ....................................................... 2-5

2.3.4 Endoatmospheric Dynamics ............................................................................................ 2-5

2.3.5 Angle of Attack Convergence for a Precessing RV ......................................................... 2-6

3. Optical Signature Algorithms ........................................................................................................... 3-1

3.1 Signature Formulation Overview ............................................................................................. 3-1

3.2 Data Initialization Considerations ........................................................................................... 3-2

3.2.1 Units .................................................................................................................................. 3-2

3.2.2 Spectral Earthshines ........................................................................................................ 3-2

3.2.3 Planck Radiance Tables .................................................................................................. 3-3

3.2.4 Directional Emissivity Behavior ....................................................................................... 3-3

3.2.5 Air Density/Atmospheric Properties ................................................................................ 3-5

3.2.6 Material Properties .......................................................................................................... 3-5

3.2.7 Material Properties Temperature Dependence ............................................................. 3-5

3.3 Geometrical Approximations ................................................................................................... 3-7

3.3.1 Roll-Averaging Conical Normals ...................................................................................... 3-8

3.3.2 Cone Emissive Area ......................................................................................................... 3-9

3.3.3 Nose Cap Projected Area ............................................................................................... 3-10

3.3.4 Sun, Earth, Albedo Incidences ...................................................................................... 3-11

3.3.5 Obscuration: Intrabody and Penumbra Solar Flux ....................................................... 3-14

3.3.6 Reflections ...................................................................................................................... 3-20

3.3.7 OPTISIG Albedo Specular Reflection Algorithm ............................................................ 3-29

3.3.8 Approximating Solar BRDF Reflection With Specular/Diffuse Weighting ................... 3-32

4. Thermal Response Algorithms ........................................................................................................ 4-1

4.1 Thermal Response ................................................................................................................... 4-1

4.1.1 Single-Layer Heating (Subroutine THRSP) ...................................................................... 4-1 viii

4.1.2 Multilayer Thermal Response (subroutine INDPTH) ...................................................... 4-4

4.1.3 Thermally Insulated Objects ............................................................................................ 4-7

4.1.4 Fragment Thermal Response .......................................................................................... 4-8

4.2 Ascent and Reentry Heating Rates ....................................................................................... 4-10

4.2.1 Atmospheric Properties ................................................................................................. 4-10

4.2.2 Free Molecular Heating ................................................................................................. 4-10

4.2.3 Convective Heating Overview ........................................................................................ 4-11

4.2.4 Nosetips/Spheres .......................................................................................................... 4-16

4.2.5 Conic Elements .............................................................................................................. 4-17

4.2.6 Flat Plate ......................................................................................................................... 4-25

4.3 Newton’s Law Heating ........................................................................................................... 4-29

5. Database-Driven Ascent Heating Algorithms for OPTISIG ............................................................. 5-1

5.1 Description of Approach .......................................................................................................... 5-1

5.2 Verification of Approach .......................................................................................................... 5-3

5.3 Generation of Interpolatable Ascent Databases .................................................................... 5-4

6. Boost Phase Trajectory and Dynamics Modeling ........................................................................... 6-1

7. References ....................................................................................................................................... 7-1 ix

List of Tables

Table 2-1: Temporal Variables (seconds) .................................................................................. 2-1

Table 2-2: Correspondence Table .............................................................................................. 2-6

Table 3-1: Clear and Cloudy Constants, Earthshine Integrals ................................................. 3-2

Table 3-2: One Parameter Thermal Dependence Model Parameter and Comparison .......... 3-7

Table 3-3: Solution Regions of µ1 + µ2 cos ′ = 0 .................................................................... 3-8

Table 3-4: Roll Averaged Quantities .......................................................................................... 3-9

Table 3-5: Body Parameters .................................................................................................... 3-18

Table 3-6: Correspondence Table ........................................................................................... 3-20

Table 3-7: Diffuse Cone Coefficients ...................................................................................... 3-24

Table 3-8: Sphere Evaluation .................................................................................................. 3-25

Table 4-1: Thermal Response Variables ................................................................................... 4-1

Table 4-2: Requirements ......................................................................................................... 4-10

Table 4-3: Notation .................................................................................................................. 4-14

Table 4-4: Cylindrical Form Factors ........................................................................................ 4-23 x

List of Figures

Figure 2-1: Coordinate Transformation to Locate the Body Axes ............................................ 2-3

Figure 2-2: The Allowed θ Region Centered at θ0 and of Extent 2∆θ. ..................................... 2-8

Figure 2-3: The Relationship of ∆ to ∆ ................................................................................... 2-9

Figure 3-1: Clear and Cloudy Spectra ........................................................................................ 3-2

Figure 3-2: Directional Emissivity Profile ................................................................................... 3-3

Figure 3-3: Directional Emissivity (OPTISIG includes FASTSIG Method) ................................. 3-4

Figure 3-4: Specific Heat Temperature Dependencies ............................................................ 3-6

Figure 3-5: Mean Emissivity Versus Aspect for a Cylinder for the Emissivity Profile Adopted . 3-

Figure 3-6: Albedo Model Facet Arrangement ....................................................................... 3-12

Figure 3-7: Parameter Definitions for Flat Plate Angular Dependence ................................ 3-14

Figure 3-8: Comparison of Rectangular Horse and Exact Observation: a) Geometry definition,

b) Comparison .......................................................................................................................... 3-15

Figure 3-9: The Rectangular Horse Approximation ................................................................ 3-17

Figure 3-10: Centroid Location ............................................................................................... 3-17

Figure 3-11: Body Locations ................................................................................................... 3-18

Figure 3-12: Sensor View ........................................................................................................ 3-19

Figure 3-13: OSC BDR Database Peak Width Frequency ..................................................... 3-22

Figure 3-14: Bidirectional Signature Contributions ............................................................... 3-27

Figure 3-15: Specular Reflection - Earth Source ................................................................... 3-29

Figure 3-16: Contributing Source Elements ........................................................................... 3-30

Figure 3-17: (a) Earth Facet Model; (b) Contributing Source Elements ............................... 3-30

Figure 3-18: Reflected Area on Target ................................................................................... 3-31

Figure 3-19: Ray Trace from Sensor to Reflected Area on Target ........................................ 3-31

Figure 3-20: Ray Trace from Reflected Area on Target to Source ........................................ 3-32

Figure 3-21: Total Reflected Intensity for Conic Target ......................................................... 3-33

Figure 4-1: Single-Layer Exact and Linearized Thermal Solutions .......................................... 4-3

Figure 4-2: Rvtemp Calculations of Shock Angle on Streamline to Edge of Boundary Layer for

Midfrustum Stations on a Slender, Triconic RV ..................................................................... 4-12

Figure 4-3: Algorithm Comparison at α = 0 for a Triconic .................................................. 4-13

Figure 4-4: Air Property Relations ........................................................................................... 4-15

Figure 4-5: Cone Algorithm Velocity Dependence Regions ................................................... 4-16

Figure 4-6: Effective Length on a Flare Conic Section ( < prev) .......................................... 4-19

Figure 4-7: Comparison of Algorithms for a Triconic ............................................................. 4-20 xi

Figure 4-8: Cone-Cylinder Correspondence ........................................................................... 4-23

Figure 4-9: Cylinder Algorithm Angle-of-Attack Dependence ................................................ 4-24

Figure 4-10: Cone Algorithms: a) Slender cone, 7° Half Angle; b) 35° Half Angle............... 4-25

Figure 4-11: Elliptical Stagnation Scale ................................................................................. 4-27

Figure 4-12: Plate Model ......................................................................................................... 4-28

Figure 5-1: Interpolation Can Work for both Laminar and Turbulent Flow ............................. 5-4

Figure 5-2: Description Of Database Generation Process ....................................................... 5-5

1-1

1. Introduction

OPTISIG is a high-speed optical signatures code that provide visible through long wavelength infrared (visible - LWIR) optical signatures for BMD applications. They are capable of modeling target vehicle (e.g., RV) and sensor flight trajectories and target thermal response due to environmental fluxes and reentry heating. They use an analytical function solution to calculate signatures (as opposed to a discretized or finite difference approach).

The purpose of this manual is to describe the algorithms and approximations used in the codes and to indicate the region of validity of critical algorithms and their relationship to the more exact, or more general, OSC algorithms. A secondary purpose is to provide the user with insight into the overall modeling philosophy and structure. Where appropriate, efforts have been expended to correlate the programs performing a function (and variables used in the program code) to the algorithm descriptions.

Section 2 describes initialization quantities (time, timesteps, and events), coordinate systems, and trajectory and dynamics. Section 3 describes the optical signature algorithms.

Section 4 describes the exo and endoatmospheric thermal response algorithms. Section 5 describes ascent heating algorithms, and Section 6 contains a brief note on boost phase trajectory and dynamics modeling.

2-1

2. Initialization Quantities

This section describes the times, timesteps, and events used in driving the simulation. The physics applied in modeling the trajectory and dynamics is also discussed.

2.1 Time, Timesteps, and Events

The time origin is assumed to be the time when the rotating Earth axes coincide with the program Earth-centered inertial (ECI) axes. The time variables shown in Table 2-1 are used to drive the codes.

Table 2-1: Temporal Variables (seconds)

Temporal Variables

Variable Initial Value Description

TALO TSHEAT Current simulation time

TSHEAT Start time for heating (= deployment time of Body 1)

EXOMAX 10 Maximum exo time step

STARTT(I) Input First signature time for sensor

JSIG(I) Input Number of signature intervals for sensor

STOPT(I) Input Last signature time for sensor

DTENDO 1 Maximum endo time step

DTSTEP Conditional Time step dependent on

Exo of endo (Body 1)

Precession period (Body 1)

Length to next signature time

TDPLOY(I) Input Deployment time of Body 1

Other timesteps are used to subdivide the DTSTEP time interval in the thermal response and trajectory algorithms as required by particular algorithm convergence requirements. Use of the precession period to initially determine DTSTEP ensures that body 1 thermal flux calculations are a sufficient temporal representation for valid thermal responses.

Closely related to times and temporal steps are discrete events the target undergoes and the transition logic to go from one regime to another. These are summarized as follows:

Exo-endo Transition: Drag deceleration equal to 1% of sea-level gravity. In the program, the dynamic pressure (½ V2) is tested against 0.01 g0, where is the ballistic coefficient and g0 is the gravitational constant at sea level.

Thermal Demise: If permitted by the body status flag (IDEMIS(I)=2), thermal demise occurs when a body station temperature reaches ablation temperature of the material.

Detachment/Separation from Body 1: Occurs at deployment time or at altitude

(appendages).

2-2

Day/Night Illumination: Occurs when the target/Sun axis is tangent to the Earth.

Deployment: Deployment time.

2.2 Coordinate Systems

With few exceptions, all vectors are in the ECI representation where at time equals zero one has:

(Re,0,0) passing through the Greenwich meridian and the equator.

(0,Re,0) passing through latitude 0° and longitude 90° East.

(0,0,Re) passing through the North Pole.

All calculations are performed in inertial reference systems. Sines and cosines are calculated from projections of a vector onto relevant triplets of orthogonal unit vectors. This ensures covariance of calculations and invariance of results. Figure 2-1 shows the axes relations.

2-3

The input reference axes are constructed from position and velocity according to various options. Z may lie along reentry velocity, range-insensitive axis, or local vertical. α, β, γ, are draws from distributions to locate the space axes: A0, B0, C0. The body axes (A, B, C) rotate about the space axes according to the time evolution of the Euler Angles: , , .

Figure 2-1: Coordinate Transformation to Locate the Body Axes z

AO

BO

CO

y x

a) REFERENCE AXES: x, y, z

SPACE AXES: AO, BO, CO

2-4

2.2.1 Important Triplets of Axes

These are updated at each time point:

Body Axes: A(*,I), B(*,I), C(*,I) for body I. The instantaneous location of axes “frozen” into the body, with A(*,I) being the body longitudinal axis.

Space Axes: A0(*,I), B0(*,I), C0(*,I). For exo targets, these are inertially fixed, and A, B, C execute rotations about these axes. Euler rotations connect A and A0 and so on.

Viewing Axes: X0(*), Y0(*), Z0(*) are defined in subroutine OBSC and used in this context:

X0 points from body 1 to the sensor.

Y0 is the projection of A(*,1) into the plane perpendicular to X0.

Z0 is the cross product of X0 and Y0.

2.2.2 Kinematic Vectors

These are summarized as follows:

XT(*,I): ECI body I position; magnitude XTMAG(I)

VT(*,I): ECI body I velocity; magnitude VTMAG(I)

XS(*,I): ECI sensor I position

V(*,I): ECI sensor I velocity (ballistic), ECR sensor velocity (aircraft/ stationary)

UTGSEN(*,I): unit vector from body 1 to sensor I

2.2.3 Environmental Vector

These are as follows:

UEARTH(*): local down unit vector from body 1

UWIND(*): unit velocity vector for body 1

USUN(*): unit vector to the Sun.

2.2.4 Intrabody Vectors

The intrabody separation vector d(*,I) = XT(*,I) - XT(*,1) has the following representations:

Body 1 Axes: d(*,I) = XLA(I)*A(*,1) + XLB(I)*B(*,1) + XLC(I)*C(*,I) which defines XLA, XLB, XLC as the components of d relative to the body 1 axes.

2-5

Viewing Axes: d(*,I) = XLX(I)*XO(*) + XLY(I)*YO(*) + XLZ(I)*ZO(*) which defines XLX, XLY, and XLZ as the components of d relative to the viewing axes.

2.3 Trajectory and Dynamics

2.3.1 Ballistic Trajectories

Ballistic target and sensor motion is calculated by integrating the equations of motion

(assuming no drag on sensors) with a Fourth Order Runge Kutta algorithm. This same type of algorithm is employed by the OSC BALLIS3 program. Round and oblate earth models are available in OPTISIG.

2.3.2 Aircraft/Stationary Trajectories

These trajectories have motion only relative to a fixed point on Earth and are allowed only for sensors. Constant altitude motion is the only option and, consequently, motion relative to the Earth is accomplished by rotating the ECR initial position through an angle vt/r about an axis (r, x, v)

ECR

and rotating the resultant to the ECI frame. OPTISIG does not propagate sensor trajectories.

2.3.3 Exoatmospheric Motion About the Center of Mass

Euler’s torque-free solution in terms of Euler angles (, , ) and rotation matrices Rij(, , ) is used. The matrices and angles are defined as in standard American mechanics textbooks.

The directions of the space axes correspond to the standard axes as:

and similarly for the body axes:

In the usual active sense, A is the result of applying the Euler rotations to A0, and the same relationship connects B with B0 and C to C0.

2.3.4 Endoatmospheric Dynamics

An algorithm applicable to small precessional exo -motion ),0( exo is used to dampen large angles of attack upon reentry. Basically, the space axes (A0, B0, C0) are rotated so that A0 approaches the velocity vector. The motion about the new space axes is unchanged.

(A0, B0, C0) (Z, X, Y)

(A, B, C) (z, x, y)

2-6

2.3.5 Angle of Attack Convergence for a Precessing RV

2.3.5.1 Relationship to a Nutating Top with One Point Fixed

In the space reference system, the equations of motion are vv ˆˆ2 xA hMd J where the sign assumes that the center of pressure (CP) lies behind the center of mass (CM) and d - moment arm: |CP - CM|

M v2/2drag force

Â, 𝑣 - body axis, velocity unit vector.

This parallels the treatment of a top with one point fixed in a gravity field with correspondence given in Table 2.2.

Table 2-2: Correspondence Table

Correspondence Table

Quantity Reentry Top

Force �̂�𝑀𝜌𝑣2 2𝛽⁄ Mg

Reference point CM Fixed Point

Moment arm -d A 𝓵A

Angular Momentum About CM About Fixed Point

Main space axis: Â0 �̂� −�̂�

Torque |Force||d|Â×Â0 -|M g 𝓵| Â×Â0

Consequently, the reentry problem looks like the top problem with:

Gravity reversed: Positive feedback rather than negative feedback in the torque equation; 𝐽 → |𝐽|�̂�0 rather than a purely periodic motion.

“Gravity” variable with time g → ρv2/2β ~ exp (αt) initially with α ≅ -h/atmospheric scale height.

�̂� → 𝑣(𝑡), which deviates from a constant direction for all but vertical reentries.

This is a relatively minor change for high-β reentry.

Because of the basic correspondence — torque direction — the solution can be thought of as a modest modification of the fixed-point top treatments given in advanced mechanics books

(References 1 and 2).

2-7

2.3.5.2 Basic Physics Overview

Because the torque has direction x 0, there are two constants of motion:

2). (ReferenceGoldstein respect towith edinterchang are b and a - N.B.

/ˆ

/ˆ

B

Bo

IAJb

IAJa

Incorporation of these constants into the angular momentum results in the following form:

sin

ˆˆ ˆ sin

)cos(ˆ sin

)cos(

AxA A ab o A ba o

I from which and can be expressed as functions of :

2sin

)cos( ba cos)( bI

B which generates a pseudo-potential for :

sin

)cos( ba

II

EnergyKinetic BB

The true potential that generates the torque is given by

V (, t) IBC (t)cos where c t MAIR v2d

2IB

The region of the variation in is given by

0),()(02 tVtE TOT and shown in Figure 2-2.

A A

2-8

If c2(t) is increasing, then θ0 and ∆θ0 both tend to zero.

Figure 2-2: The Allowed θ Region Centered at θ0 and of Extent 2∆θ.

At a given time, (depending on initial conditions) can be anywhere within the line segment given by E - VTOT > 0. The approximate center of this region is given by

VTOT

0 .

As time passes, and ∆ change due to the time dependence of c2(t). If c2(t) increases with t, then and ∆ will decrease as shown. This drift toward 0 is because cos = ( ) the convergence of the angle of attack, , to zero.

Obviously the solution to the problem hinges on solving VTOT / 0 and perturbatively solving for the local motion about . To the order of ∆2, the solution for is given by

(t ) o(t) (t) with

(t) o (t) cos (t) and the basic motion is the variable amplitude and frequency precession of the main body axis, , about a vector with angles , in the space system where

)()( oo dtt

The perturbation ∆ can be obtained geometrically from considering the precessional motion shown in Figure 2-3.

A v

2-9

Figure 2-3: The Relationship of ∆ to ∆

Once and are determined, is determined from the constant of motion b by:

.cos B

A bI

In particular to first order in ∆:

dt ooo t

B

A o

I btI cos)( coso .

The Solution for 0 and ∆

One can obtain the second-order differential equation either from

IB

= c2(t) (Ao x A)

2-10 or from Hamilton’s equations:

TOT

VE

IB

(a b cos) b a cos c o is readily obtained from the solution of

(a bq) (b aq) c

1 q where q = cos o.

Newton’s method is applicable with the process stopping when

|qn 1qn | (1 n 2q ) with small.

To complete the specification at t = h, the rate of change of (t) must be known. Consider d dt a bq b aq c

1 q which yields

22 11 qcqc q

2 sin o provided one models c2(t) as c2(0) et with = ln [c2(h)/ c2(0)]/h.

The initial conditions on ∆ and ∆ at t = 0 are:

)0()0( 01 qn + 1 -qn = (a - bqn) (b - aqn) + c2(1-qn

2) a2 + b2 -2abqn + 4c2qn(1-qn 2)

2-11

By changing variable = + ∆ and expanding the torque, one obtains:

221/2124222

Oqqqcabqba which is of the form (to first order)

0)(2 t which has the approximate solution

A cos dt B sin dt t t which is exact for the two cases

= constant

= constant / (1 – βt)2

Because the frequency, , is increasing, the latter form will be used for local estimates.

In practice, to obtain the solution at t = h given the initial conditions at t = 0, solve for

1 (0), q1 cos (0(0))

2 (h), q2 cos (o(h)) given

Next, and are obtained as follows:

h

2ln c1 2 = c2(0) and c2

2 = c2(h) = e+ahc1 2.

(1 - 1/2) h

.)0()0( 01

2-12

Using the local approximations, solve for:

h dt h 1 2

The constants A′, B′ in the ∆ solution

A cos + Bsin are found from the following straightforward application of the initial values ∆1 = ∆(0)

A 1 1

The solution at t = h is given by

(h) 2 A cos B sin

). cossin (22122 /)( BAh

The coning half angle of the motion, ∆, is given by o(t) A B

2 / for 0 t h which is approximately linear in t and decreasing oo

1 - t = /1.

2-13

2.3.5.4 The Solution for and

is broken into two parts:

The oscillatory part

(t ) o(t )sin( ) sino where

(t) o(t)cos ) is obtained by considering circular precession and straight-forward geometrical arguments.

The final form for ∆ is given by

2)(

The drift term o o dt)( .

Using the expression for q , one can transform (o):

)1( sin cos

2 aqb q aqb qc o ba

Taking the mean value of as as suggested by dt 12o h an approximation to is obtained:

a d(b aq)

(b aq) ln b aq2 b aq1

2-14

The final solution for 22is given by:

2 1 2 1 cos ba is also broken into two parts:

Consideration of the ξ dependence of ∆ and the b constant of motion leads to the oscillatory part solution coso

The drift term

.dtq o h o B

A hbI

An approximate evaluation of the integral is as follows:

dtq aqb baqb aaqb dtqq dto h o h o h q b qq a q dto h o 12

Consequently hbIB

IA

q2 q1 b

2 1 1 2 .

3-1

3. Optical Signature Algorithms

This section describes the optical signature algorithms and approximations used and indicates the region of validity of critical algorithms. The algorithms are also compared with the more exact OSC algorithms.

3.1 Signature Formulation Overview

Each body, i, is assumed to be composed of isothermal elements, j, the signature of which is given by ifJ projA N T l1

D l s l A 1 l A lE

T element temperature l inband optical properties projA projected area s lA , D lA specular and diffuse reflected areas

N , E

Planck and reflective source radiances inband specular weight factor

The point source body signature is given by ij iji JJ

Obscuration is treated by visibility factors accounting for sensor resolution and intrabody invisibility from the sensor position:

.v i ij ii JJ

3-2

3.2 Data Initialization Considerations

3.2.1 Units

All calculations are performed in the meter-kilogram-second (mks) system.

3.2.2 Spectral Earthshines

Clear and cloudy spectra are tabulated from LOWTRAN calculations and are shown in Figure

3-1. For wavelengths beyond 24 µm, λ-4 extrapolations are used. The integrals of these spectra appear in Table 3-1.

Figure 3-1: Clear and Cloudy Spectra

Table 3-1: Clear and Cloudy Constants, Earthshine Integrals

Earthshine Model

Clear (W/m2)

⅓ Tropopause Clouds (W/m2)

⅔ Tropopause Clouds (W/m2)

1 Tropopause Clouds (W/m2)

Tropical 90.25 71.74 42.25 23.78

Mid-Latitude Summer 88.20 78.31 56.03 35.52

Mid-Latitude Winter 70.22 63.09 49.92 36.58

Arctic Summer 81.58 72.26 59.69 41.87

Arctic Winter 59.61 55.61 45.31 35.12

U.S. 62 Standard 80.94 67.01 47.94 35.50

OPTISIG computes the in-band radiance integrals of these spectra and builds the Planck in-band tables. The in-band integrals are called:

EH(I) in-band earthshine radiances (W/m2 sr), with

I band index ≤ 4 (set by parameter NBNMX)

CLOUD ALTITUDE

1/3 TROPAUSE

0 5 10 15 20 25

WAVELENGTH (µm)

RADIANCE, E (W/m 2 sr)

CO 2

H 2 O

O 3

CO 2

H 2 O

3-3

3.2.3 Planck Radiance Tables

In-band radiance tables are generated upon initialization of the code. Integrations are performed using a 12-point Laguerre integration scheme, and results are tabulated in log (T) and log (radiance). OPTISIG interpolates these tables in the logarithm of the required temperature. The table variables are:

EM(100,*) log (in-band radiance)

RADJE(*) in-band radiance (real, W/m2 sr)

3.2.4 Directional Emissivity Behavior

To treat directional emissivity signature effects, OPTISIG, as a default, has a built-in directional emissivity normalized profile based on unpolarized Fresnel reflectance with the index of refraction real and equal to 1.1. Although this is used operationally as a function of the cosine of the emission angle, Figure 3-2 shows the angular dependence. The hemispherical average of this profile is 0.982. This profile is used for the calculation of the physical emissive area for cones, plates, and cylinders when no other profile has been specified.

Figure 3-2: Directional Emissivity Profile

Figure 3-3 compares the OPTISIG default directional emissivity, ε() = εH (directional profile), with that of a Fresnel dielectric with the same hemispherical emissivity, εH.

DIRECTIONAL

EMISSIVITY

CHARACTERISTIC

0 30 60 90

(deg)

(0

3-4

Figure 3-3: Directional Emissivity (OPTISIG includes FASTSIG Method)

The OPTISIG user may elect to use directional emissivity data for each target material, provided directional emissivity data are available in the OSC directional reflectance data base, OPTPROP. The method for generating in-band directional profiles ϵ∆λ(cos) from this data is detailed below.

Directional emissivity (in-band):

𝜖Δ𝜆(cos 𝜃) = ∫ 𝑃(𝜆, 𝑇)𝜖(𝜆, cos 𝜃)𝑑𝜆

Δ𝜆

∫ 𝑃(𝜆, 𝑇)𝑑𝜆

Δ𝜆

P(λ,T) the Planck black body radiation function evaluated at a typical target temperature

∆λ the waveband of interest ϵ(λ, cos) the directional emissivity data

Directional profile (E∆λ(cos)):

𝐸Δ𝜆(cos 𝜃) = 𝜖Δ𝜆(cos 𝜃) 𝜖Δ𝜆 where ϵ∆λ is the hemispherical average of ϵ∆λ (cos) defined as follows:

𝜖Δ𝜆 =

∫ 𝜖Δ𝜆(cos 𝜃) sin 𝜃 cos 𝜃 𝑑𝜃 𝜋

FASTSIG

FRESNEL DIELECTRIC

(deg)

300 60 90

H = 0.895

3-5

OPTISIG generates the directional profile at 11 equally spaced values of cos . These are cos = 0, 0.1, 0.2, ..... 1.0.

3.2.5 Air Density/Atmospheric Properties

OPTISIG uses a set of 33 atmospheric models common to OSC and OPTISIG. These are listed in the separate OPTISIG Users’ Manual and Interface Control Document (Ref. 7).

OPTISIG uses the air density, pressure, and molecular temperature. Sea level density is

1.225 kg/m3. OPTISIG also provides support for a “user-supplied” model atmosphere in addition to the basic 33 standard models. Refer to the Users’ Manual and Interface Control

Document for details of the atmospheric modeling options.

3.2.6 Material Properties

OPTISIG does not use material properties per se, but it uses combinations of material properties (e.g., thermal mass = ρCp ∆x) that are written to a target model file.

3.2.7 Material Properties Temperature Dependence

Empirically, it was found that a wide variety of materials have specific heats that obey the form

This is the generic shape shown in Figure 3-4.

At T = T0, one obtains the relationship between a0 and b0 in its most compact form:

a0 = 1/T0 + b0.

The parameter b0 is determined from the specific heat ratio, = Cp(T1) /Cp(T0), as follows:

CP(T)/CP(To) aoT/(1 + boT) .

b o = 1 T1

(T1/To-)

(-1)

3-6

Figure 3-4: Specific Heat Temperature Dependencies

As an example, for carbon phenolic with T0 = 500 R and T1 = 3,000 R, one obtains b0 -1 =

871 R, and the comparison of the modeled Cp ratio at T = 900 R is as follows:

ρmodel = 1.393 ρdatabase = 1.386.

To change reference temperatures, T0 T0′ b0′ = b0 a0′ = 1/T0′ + b0′.

For many materials of interest, the thermal conductivity tends to track the temperature dependence of the specific heat (exceptions are graphite and carbon cloth). To reduce

3-7 computational time, the temperature dependence of the thermal conductivity is taken as that of the specific heat:

Implementation:

Reference temperature: 290 K b0 -1 is on the model input file and on the materials database

If b0 -1 ≤ 0, Cp(T)/Cp(T0) = 1

Otherwise, Cp(T)/Cp(T0) = a0 T/(1 + b0T)

If Cp and k have the same temperature dependence, then the heat equation is solved by normalization of the surface terms Q - ϵσTs for the thermal response subroutine.

The results are shown for representative materials in Table 3-2. Also shown are worst-case comparisons. Typically, b0 -1 was obtained with T0 ~ 500 R, T1 ~ 2,000 R, and worst case near the midpoint of the T0 – T1 range.

Table 3-2: One Parameter Thermal Dependence Model Parameter and Comparison

Material b-1

(K)

Worst Case

Cp(T)/Cp(T0) Model Cp(T)/Cp(T0) Data k(T)/k(T0) Data

CP 484 1.393 1.396 1.380

SP 332 1.32 1.4 1.21

TFE 271 1.16 1.26 1.12

FS 293 1.29 1.41 1.05

AL 171 1.19 1.0+ 1.39

GR 866 1.51 1.87 0.67

LDMρ 9,333 1.32 1.35 1.12

3.3 Geometrical Approximations

To obtain faster program execution, simple closed-form approximate expressions rather than exact expressions are employed. Errors induced by this method tend to have accuracies (in a root-mean-sense over all possible target orientations) of a few percent.

Because internal temporal sampling is geared to precession periods (∆t ≤ Tprec/8), “roll” or azimuthal averages of geometric quantities are used extensively in the overall algorithm strategy. An example of this is the following estimate (discussed in more detail in Subsection

3.3.2) of the emissive area of a cone:

k(T)/k(To) = CP(T)/CP(To) .

Q - T s

CP(To)

CP(T1)

(Q - T s

3-8 coscoscoscos 0cos dAdAA

AVE

The key to this and other related applications of the mean value theorem is the calculation of the average value of the cosine of the exitance angle for cones and cylinders.

3.3.1 Roll-Averaging Conical Normals

This is performed in subroutine RAVCOS. Consider surface normals distributed about the body axes ˆ A , ˆ B , ˆ C as follows:

with fixed and varying between - and Quantities of interest are related to the roots of the expression cos ˆ n ˆ s 0 where ŝ is an arbitrary unit vector as follows:

cos cos 0 d cos projectedarea Atotal cos where Atotal is the total surface area of the frustum.

Expansion of the dot product n̂() ∙ ŝ leads to the expression cos = 1 + 2cos ( -s) with

1 cos ˆ A ˆ s and2 sin 1 ˆ A ˆ s

As shown in Table 3-3, there are three solution regions for the roots of cos = 0.

Table 3-3: Solution Regions of µ1 + µ2 cos ′ = 0

Region Root: |′| Comments µ1 - µ2 > 0 s∙A > cos δ. Small aspects µ1 - µ2 < 0 0; no solution s∙A < cos δ. Large aspects

Otherwise cos -1

(µ1µ2) -cos δ < s∙A < cos δ. Moderate aspects

Quantities calculated in the subroutine RAVCOS are shown in Table 3-4.

n() = cosA + sin (B cos + C sin )

3-9

Table 3-4: Roll Averaged Quantities

Symbol Variable Value Comment

|′| U5 See

Table 3-3

Root of µ1 + µ2 cos ′ = 0

(cos′)AVE U3 ∫ cos 𝜃 𝑑𝜓 𝜋⁄ 𝜓′

APROJ = ATOT(cos′)AVE

(cos′)AVE U4 U3/U5 Argument of ϵ[(cos′)AVE]

3.3.2 Cone Emissive Area

The emissive area of a cone of total surface area, A

TOT

, in the notation of the previous section where s is the unit vector in the direction of the sensor, is given by:

A ATOT

cos cosd

The evaluation of A is given by

A ATOT

i 1

N cos AVEi cos

AVE

i where the sum is over the N visible azimuthal optical subdivisions. Our interest is in the accuracy of this expression for two extreme relevant limits:

Large cone angles (~ ) for flat plates

Small cone angles (~ 0) for RVs and tanks.

3.3.2.1 Flat Plates (=

his case is of academic interest, but is included to show the global validity of the approximations. For this case, n̂(ψ) = Â independent of ψ. Therefore, the flat plate results are given by:

A ATOT ˆ n ˆ s ˆ n ˆ s /2 ˆ n ˆ s .

3.3.2.2 Small Cone Angles (≈/2)

This is the relevant case for RVs and tanks. An indication of the validity of the approximation used in OPTISIG (and its predecessor FASTSIG) is shown in Figure 3-5, where for a cylinder

(/2)and the emissivity directional dependence of Section 3.2.4, the following quantities are compared cos AVE

3-10

A exact/Aproj as functions of aspect angle of the cylinder axis to the viewer.

Figure 3-5: Mean Emissivity Versus Aspect for a Cylinder for the Emissivity Profile Adopted

3.3.3 Nose Cap Projected Area

For a spherical nose cap (0 ≤ ≤ ) the analytic expression for the projected area requires a large number of operations in three distinct geometrical regions:

I. All parts visible:

II. Some self-obscuration:

III. No part visible:

The OPTISIG/FASTSIG solution is as follows:

I. The exact answer

II. Use of an interpolation between I and III, which matches at ˆ A ˆ s 0, sin .

III. The exact answer: 0.

The region of validity is

If Ao and A90 are the projected areas for nose-on and broadside viewing and p is the interpolation parameter, projected area is approximated by:

FASTSIG APPROXIMATION

ACTUAL

ASPECT ANGLE (deg)

(A

S P

E C

T A

N G

L E

30 60 90

A s > sin

-sin < A s < sin

A s < - sin

< /2 .

p = A s/sin , 3-11

I.

II.

III. 0.

3.3.4 Sun, Earth, Albedo Incidences

The treatment of heating fluxes uses approximate expressions that on the average are of small or no bias. For a flat plate with surface normal unit vector, ˆ n , it is assumed that the form of the incident energy is given by where W(n ̂,…) is constrained by the normalization condition:

For zero bias on the average, it is required that q ̅ be calculated exactly. It can be shown that for sources of energy that are incident from direction i, subtense d (i), and “radiance” N(i), and, consequently, a relatively straightforward, simple evaluation of q ̅ can be obtained.

3.3.4.1 Evaluation of Average Fluxes

For point sources (Sun and free molecular heating), q ̅ is simply one-fourth the maximum incidence:

Sun: q ̅s = 0.25 x solar constant

Free molecular: q ̅w = ρatm v3/8

For earthshine at altitudes such that the half subtense of the visible Earth is angle , where is the average radiance of the visible Earth.

For Albedo, the visible earth disk is broken, by default, into 400 equal solid angle regions.

Figure 3-6 shows how these regions are distributed on the earth disk. At altitudes such that the half subtense of the visible Earth is angle .

Ao (A s)

Ao (A s) (1 + p)/2 + A90 (1 - p2) q(n,....) = q W(n,...)

d(n)W(n,....) = 4 .

dq = 0.25 N(i)d(i) qE = N2(1 - cos )

N

3-12

Aq sq E

1 cos facN ˆ s ˆ n i ˆ s ˆ n i i 1 facN where Nfac is the number of Albedo facets, n ̂i is the outward normal for facet i, s ̂ is the unit vector to the Sun, and E is the Earth Albedo.

Figure 3-6: Albedo Model Facet Arrangement

3.3.4.2 Flux Directional Dependence

The expressions for the normalized angular expression W(n ̂,…) will be developed. From these flat plate normal expressions, appropriate expressions are developed for averages over conical surfaces. By definition, a spherical average of W(n̂,…) is unity.

Notation is as follows:

s ̂ - a unit vector pointing to the center of the source

3-13 n ̂ - unit plate normal

- Body axes

, - polar angles of n in the body system.

Point Sources/Small Sources are as follows:

W(n̂, ŝ) = 2(n ̂ ∙ ŝ + |n ̂ ∙ ŝ |) for flat plates

W(n̂, ŝ) = 4(cos θ)AVE for conical frusta, where (cos θ)AVE is given by the roll average algorithm of Section 3.3.1.

Large Sources: Earth and Albedo are taken as follows:

For flat plates:

1for 1

11sin

11cos2 cos cos cos11

1for 0

ˆ,ˆ for snwhereW

= Z/RE

= X/A

See Figure 3-7 for parameter definitions.

The large-source expressions are approximate. The cone expressions are obtained by where W plate has been previously given for both point sources and large sources, so that accuracy considerations are intrinsic to the flat plate expression.

A,B,C

Wcone(n,s) = 1 d

Wplate n(,),s

3-14

Figure 3-7: Parameter Definitions for Flat Plate Angular Dependence

3.3.5 Obscuration: Intrabody and Penumbra Solar Flux

The objective is to treat obscuration in a fast, efficient way with moderate accuracy for all viewing conditions. The “rectangular horse” algorithm in which all bodies are treated as equivalent rectangles, satisfies these criteria. That the method can be quite accurate is illustrated in Figure 3-8 for the projected areas of overlapping circles. The quality of agreement, ~2%, remains even if the direction of approach of the centers of the squares lies along diagonals of the squares.

ˆ n

X

RE

A

Z

3-15

Figure 3-8: Comparison of Rectangular Horse and Exact Observation: a) Geometry definition,

b) Comparison

3.3.5.1 Body Visibilities

There may be up to three bodies that may or may not have overlapping areas. The body visibility calculations are performed in subroutine OBSC in the sensor viewing coordinates.

The radiant intensities for each body include sensor angular resolution considerations.

Extended sources are decoupled from the calculation of the visibilities and, if their intensities are nonzero, have a visibility of 1. Body 1 (RVs) are treated in a special fashion.

EXACT FASTSIG

r d a) r

EXACT

FASTSIG

2 310 d/r

A r2 b)

3-16

The algorithm calculates the visibilities, Vi, of each body given.

Limited body shape description

ˆ s - unit vector pointing to the sensor

Xci - locations of each body

Api - projected areas.

The intended use is that if Ji represents the single body radiant intensities, then the total intensity would be given by

J = V1 J1 + V2 J2 + V3 J3.

The V’s are constrained to always be between 0 and 1.

The basic algorithm is based on the “rectangular horse” approximation, which is not too bad if care…