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Satellite Fragmentation Modeling with IMPACT

Marlon E. Sorge1 The Aerospace Corporation, El Segundo, CA, 90245

The software program IMPACT developed by The Aerospace Corporation has been used for more than 20 years to model explosions and hypervelocity collisions at orbital altitudes. IMPACT has been used extensively for intercept test range and orbital safety studies, flight test planning, on-orbit event analyses, debris mitigation and spacecraft design studies, warfare simulations, and space safety and risk analyses. This paper discusses the approach and algorithms used in the development of the IMPACT collision model.

Nomenclature A = scaling coefficient of power law Ab = beta distribution normalization coefficient a,b = beta distribution parameters CN = cumulative number of fragments of a mass M and larger M = mass of fragment M1 = mass of the largest mass bin Mi = mass of the ith mass bin V1 = characteristic velocity of largest mass bin Vi = characteristic velocity of ith mass bin

I. Introduction he software program IMPACT, developed by The Aerospace Corporation, has been used for more than 20 years to model explosions and hypervelocity collisions at orbital altitudes. This paper discusses the approach and

algorithms used in the IMPACT collision model. IMPACT is a semi-empirical model combining both conservation laws and empirically derived relationships

from ground and orbital event data to model a range of hypervelocity collision scenarios. The model is designed to account for some basic characteristics of the colliding objects but does not consider details of their designs. For a number of cases in which the model is used, details of the designs are not available and often the exact engagement conditions are not known.

Collisions and explosion events have been occurring in orbit since the early years of the space program. These events can produce a significant amount of debris including thousands or even tens of thousands of fragments that are capable of ending a satellite’s mission if they were to collide at low Earth orbital velocities. Shortly after a fragmentation event, the debris forms a cloud that encircles the region of the fragmenting object’s orbit. During this early phase, which may last weeks in low Earth orbit, the cloud can present a significant collision hazard to satellites passing through it. As the debris fragments’ orbits are dispersed due to natural orbit perturbations, the debris fragments spread until they are approximately randomly distributed in right ascension of ascending node (RAAN) forming a shell around the earth and becoming part of the background debris environment. They then pose a long-term hazard to satellites. Since most of the explosion or collision debris fragments which can be a hazard to satellites will not be trackable, it is necessary to model the fragmentation events to assess possible risks and provide mitigation options if needed. This requires the use of a fragmentation model. The initial algorithms for IMPACT were developed in 1984 to support safety analyses for the US Air Force P-78 Solwind anti-satellite test. The algorithms were converted into the first version of IMPACT to support safety analyses for the SDIO Delta 180 test in 1986. IMPACT has been used since then to support intercept and flight test planning, on-orbit event analysis and

1 Senior Project Engineer, Technology Development, Test and Demonstration Directorate, 2155 Louisiana Blvd., NE, Suite 5000, Albuquerque, NM, 87110-5425, Senior member AIAA, Member Directed Energy Professional Society.

T

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Copyright © 2008 by The Aerospace Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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risk mitigation, satellite design studies, warfare simulations, space test hazard modeling and debris mitigation and minimization studies.

II. Algorithms IMPACT models both explosions and hypervelocity collisions. This paper will discuss the IMPACT

hypervelocity collision model. IMPACT is a semi-empirical model combining a number of empirical distributions derived from orbital and ground test data with enforcement of mass, momentum and energy conservation. The imposition of conserving mass, momentum, and energy constrains the empirical results to prevent certain physically impossible combinations of outputs and make results internally consistent and constrained by observations. Empirical relationships are used to model the fragment mass distribution (number of fragments of different masses), fragment velocity distributions, and relationships between fragment mass, size, area and material density.

A semi-empirical approach was chosen to allow for the creation of a fast-running model that does not require very detailed inputs. In most of the events studied using IMPACT, detailed information on the target and/or projectile is not available. In some cases, detailed information on the exact kinematics of the collision event is also not available, especially for unexpected events. As a result, it is often necessary to study a number of variations or different scenarios requiring the model to be run many times. A short execution time is needed to meet this requirement.

To model a collision, IMPACT requires information on both the target, or more massive of the two colliding objects, and the projectile, or less massive of the two colliding objects. The state vectors (position, velocity, and epoch) for both objects at the time of the collision are needed to define the geometry of the collision and the relative velocity between the target and projectile. Relative velocity is one of the critical parameters defining the energy of the collision. Mass, object type, and material information are also needed for both the target and projectile. Mass is defined as both a dry mass, or amount of mass that can create fragments, and a mass of liquids that represents material that will affect the kinematics of the collision but will not generate fragments. Two basic types of objects are defined in IMPACT: “boosters” or objects that are largely hollow, and “satellites” that are more densely constructed. The fragmentation characteristics of these two types of objects are somewhat different. The major material constituents of each object must also be defined in terms of material density and mass fraction of material to total mass of the object. This information is used to generate the relationships between fragment mass, size and area, as ground test data shows noticeable associations between fragment material density and mass–size relationships. An additional option is to define certain specific pieces of debris that are expected to remain intact during the collision and that will not be included in the standard fragmentation process. Such objects might include satellite solar panels, motor assemblies, or other appendages.

The basic program flow for the model can be seen Figure 1. In the initial step, the kinematic changes in the debris cloud motion from the pre-collision state vectors to the post-collision debris clouds are determined. The kinematic changes are due to energy loss and momentum transfer and are calculated using the target and projectile state vectors and total masses. The amount of mass involved in the standard fragmentation process is identified by removing any intact objects that were defined as well as any large fragments if the object is a “booster”. Using the standard fragmentation mass and accounting for the total mass, a mass distribution is generated based on empirical relationships. This results in a distribution of the number of fragments in a range of mass bins. The relationship between fragment size and mass is determined from empirical relationships and the material density information provided as input. The kinetic energy of the fragments relative to their respective debris cloud centers of mass is distributed between the fragment mass bins. Within each mass bin the distribution of fragment velocities relative to the center-of-mass of their respective debris cloud, or spread velocities, is determined based on the spreading energy allotted to that bin. Finally, individual fragment characteristics are generated based on the distributions previously derived. These individual characteristics include position and velocity vectors, mass, and average cross-sectional area used to propagate the fragments’ orbits or for other post-collision analysis. Details of the steps in this process will be discussed in the following sections.

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Remove IntactObjects

Remove LargeFragments

Determine MassDistribution

Determine SpreadingEnergy DistributionBetween Mass Bins

Determine Post-Collision

States

Calculate IndividualFragment Information

Determine Sizes

Output ResultsDetermine SpreadingVelocity DistributionWithin Mass Bins

Figure 1. IMPACT program flow.

III. Energy Distribution The initial step in the IMPACT collision modeling process is to determine the energy redistribution that occurs

during the collision. The energy used to drive a collision, including fragmenting the objects, spreading the fragments, and generating heat, light, and material phase changes, comes from the kinetic energy of the target and projectile relative to the combined center of mass of the entire target and projectile system. As a result of this energy loss, there are shifts in the velocity vectors of the centers of mass of the two debris clouds relative to the pre-collision velocity vectors of the target and projectile. The methodology of this energy transfer is illustrated in Figure 2.

Vcm (ECI)

Vp (ECI)

Vt (c.m. relative)Vt (ECI)

Vp (c.m. relative)

Vcm – System c.m. velocityVt – Target velocityVp – Projectile velocityVf – Fragment velocity(reference frame of vector)

Vcm (ECI)

Vp (ECI)

Vtcm (system c.m. relative)

Vt (ECI)

Vpcm (system c.m. relative)

Vf (ECI)

Vf (ECI)

Vf (projectile relative)

Vf (target relative)

‘Non-kinetic’ energy:Fragmentation energyHeatLightPhase Changes

Pre-Collision Energy Distribution Post-Collision Energy Distribution

Figure 2. Energy Redistribution during Collision.

The kinetic energy losses and the decrease in relative kinetic energy cause a shift such that the velocity vectors

of the debris cloud centers of mass tend to be closer to the velocity vector of the system center of mass than were the pre-collision velocity vectors. In actual collisions this shift has generally been a small fraction of the total relative kinetic energy, meaning that the two objects appear to pass through each other and continue on trajectories fairly similar to the pre-collision velocities. This is conceptually similar to an approach in Reference 1. In IMPACT the standard energy loss is 15% of the relative kinetic energy, although this may be varied for non-standard collisions. As a result of the non-linearity of the relationships between energy and velocity, energy losses much larger than 15% will have an increasingly noticeable effect on the center-of-mass velocities of the debris clouds, resulting in unrealistic changes in the post-collision debris cloud trajectories. This kinematic effect limits the amount of energy

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that can go into driving the collision process and still result in post-collision debris clouds consistent with observations. Of the 15% of the relative kinetic energy transferred, one-third of it is distributed to the spreading of the fragments relative to their debris clouds’ centers of mass, and the other two-thirds are assigned to other dissipative forms such as heat, light, phase changes, and the energy required for fragmenting the objects.

IV. Mass Distribution Mass distributions in IMPACT are represented by the numbers of fragments in a pre-defined set of mass bins.

IMPACT considers fragment masses down to a minimum mass bin of 10-6 kg, which corresponds to fragments around 0.5 to 1 mm in diameter for typical material densities.

Before a mass distribution is generated, an initial check must be made to determine whether the collision is energetic enough to produce a “complete” fragmentation, meaning one that will involve enough of the target that the empirical relationships can be generically applied. The standard measure of a “complete” fragmentation is that the kinetic energy of the projectile relative to the target divided by the target mass is greater then 40,000 m2/s2, although this transition point is not distinct in the ground test data. Rather than use this as an absolute cut-off and in order to provide a smooth transition from complete fragmentation to highly incomplete fragmentation, IMPACT stops analysis at a relative kinetic energy to target mass ratio of 10,000 m2/s2. The mass distributions of collisions from slightly above the 40,000 m2/s2 transition to 10,000 m2/s2 are characterized by a sharp decrease in the number of smaller fragments as an increasing amount of the mass is concentrated in larger fragments. This provides a smooth transition from a typical hypervelocity collision with large numbers of small fragments to lower energy collisions where most of the target object is broken into a few larger fragments.

Before the empirical mass distribution relationships are applied, the amount of mass to be fragmented using these relationships must be determined. Although the liquid mass is used as part of the kinematic calculations discussed previously, it is not included in the mass used to generate fragments. Additionally the mass designated for fragments that are to remain intact and the special large fragments from booster-type object distributions are removed from the dry mass leaving the remainder of the mass for fragmentation.

The fragmenting mass distribution is modeled using a power law of the form of equation 1 where CN is the cumulative number of fragments of a given mass, M and larger, and A and B are constants for a given event. The power law form for the cumulative mass distribution matches the data reasonably well over a wide range of collision parameters; this has been consistently demonstrated through debris analysis dating back to the 1970s2. An example of a cumulative mass distribution in this form is shown in Figure 3. It should be noted that a breakup model lower size limit is applied to the distribution to prevent the exponential increase in the number of fragments at very small sizes. At some mass below the lowest value modeled by IMPACT, the production of smaller fragments would be expected to decrease3.

CN=A (M)-B (1)

1

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1000

10000

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1000000

10000000

1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02

Fragment Mass, kg

Cum

ulat

ive

Num

ber o

f Fra

gmen

ts

Figure 3. Example Power Law Mass Distribution.

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The two parameters in the mass distribution equation are determined by the specific characteristics of a collision.

Parameter A determines the mass distribution curve’s position along the y-axis direction, and parameter B determines the slope of the curve on a log-log plot. Parameter B is determined by equation 2 where Em is the relative kinetic energy of the colliding object divided by the mass of the object being fragmented, and C and D are empirically derived constants. The form of the expression for B can be seen in Figure 4. As discussed previously, the sharp drop in the curve (i.e., rapid increase in slope) at the lower end of the relative energy per mass can be seen. Lower slopes in the distribution curve result in a higher proportion of the mass being placed into larger fragments and thus the generation of significantly fewer small fragments. Parameter A is used to adjust the total mass in the distribution to conserve the dry mass of the fragmenting object by scaling the curve along the y-axis.

B = C Log (Em) + D (2)

0.65

0.66

0.67

0.68

0.69

0.7

0.71

0.72

0.73

0.74

0 500000 1000000 1500000 2000000

Relative Energy per Mass, J/kg

Expo

nent

Mag

nitu

de

Figure 4. Mass Distribution Power Law Exponent.

Once the appropriate power law distribution has been determined, it is used to calculate a discrete number of

fragments for each of the mass bins using the characteristic mass, which is the representative mass, for each bin. Any intact or large booster fragments are reintroduced into the mass distribution. The total mass in the discrete version of the distribution is determined and compared to the total available mass. Minor adjustments are made to the discrete version of the distribution to ensure conservation of the fragmenting mass. Conversion from a curve to a discrete representation of the mass distribution for the large fragments can cause noticeable departures from the original distribution, but this is consistent with test data.

V. Velocity Distribution Spread velocities in IMPACT are defined as the incremental velocities of fragments relative to the post-collision

center of mass velocity of the debris cloud. As discussed in the section on energy distributions, the post-collision velocity of the debris cloud will not be the same as the pre-collision velocity of the object from which the debris cloud was created. This velocity difference arises from the energy lost from the motion of the objects to various sinks involved in the fragmentation process, including heat, light, and the spreading of the fragments.

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In a standard collision modeled with IMPACT, five percent of the kinetic energy of each object relative to the mutual center of mass of the system (target and projectile) is transferred to spreading the fragments. The kinetic energy lost by one object is transferred to the spreading of the other object. By forcing the relationship between the debris cloud center-of-mass shifts and the other forms to which energy is transferred, bounds can be placed on unobservable quantities such as the spreading of very small fragments by their interrelationships with observable parameters such as large fragment spreading and overall energy transfer.

Once the overall spreading energy has been determined by the kinematics of the collision, the energy is partitioned between the fragment mass bins. This is done through the definition of a characteristic velocity for each mass bin. The characteristic velocity is the spread velocity each fragment would have if all of the spreading energy in the mass bin were evenly divided between the fragments in that bin. The characteristic velocities are generated using equation 3.

Vi = (M1/Mi)1/4V1 (3)

M1 and V1 are the characteristic mass and velocity of the largest mass bin and Mi and Vi are the characteristic

mass and velocity of the ith mass bin. V1 is scaled so that the total spreading energy is conserved. A sample characteristic velocity plot is shown in Figure 5.

1

10

100

1000

10000

1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04

Fragment Mass, kg

Char

acte

rist

ic V

eloc

ity, m

/s

Figure 5. Sample Characteristic Velocity Distribution.

Within each mass bin the spreading energy represented by the characteristic velocity is distributed between the

fragments, giving them a range of spreading velocities. Spreading velocities within a bin are distributed using beta distributions. A sample beta distribution is shown in Figure 6. The beta distribution both displays the tail observed in the velocity distributions of on-orbit fragmentation debris such as seen in Ref. 4 and has a defined maximum and minimum. Beta distributions are of the form in equation 4.

F(v) = Abv(a-1)(1-v)(b-1) (4)

The parameter Ab normalizes the distribution. Parameters a and b are determined by the most probable velocity

and energy conservation. The most probable velocity is a function of the relative velocity, energy of the collision and size of the fragment.

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For each mass bin a separate beta distribution is determined, which conserves the spreading energy allocated through the characteristic velocity and the mass and total number of fragments in the bin. Once the distribution is determined and the fragments in the bin allocated to specific spread velocities, the total spreading energy in the bin is calculated. Minor adjustments are made to correct any difference between the allotted energy and the actual kinetic energy in the distribution over discrete fragments. The overall spread velocity distribution is the aggregation of all of the individual mass bin spread velocity distributions.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8 1

v

F(v)

Figure 6. Sample Beta Distributions.

As a result of the distribution of spreading energy to the mass bins and the conservation of that energy, the

magnitudes of the spread velocities in a bin is a function of the number of fragments in that bin as well as the energy. This produces an interaction between the mass distribution and the spread velocity distribution. As collisions become more energetic, there is an increase in the amount of spreading energy available. There is also an increase in the number of fragments, particularly in the smaller mass bins, to which this energy can be distributed; this limits the increase in the average spread velocity magnitude, particularly for the smaller fragments. The resulting total spread velocity magnitude distribution has the general shape of the curve in Figure 6. Most of the fragments are in the lower half of the spread velocity range with significantly fewer fragments in the higher end of the range.

VI. Size/Area/Mass Distributions Fragment size and area information are secondary characteristics in the IMPACT model and are derived from

fragment mass and material properties inputs. In IMPACT the target and projectile may be made out of single or multiple materials. For each material its total mass is defined as a fraction of the total dry mass. Each material also has a density that is used to relate the mass of a fragment made of this material to its size and area. When fragments from ground tests were measured, it was found that their dimensions were noticeably affected by the density of the material from which they were made5. By including this information in the fragmentation model, effects of material density makeup could be represented in the resulting fragment distributions.

Initially the dry mass of the object is distributed between the materials of which it is made. This involves determining the largest mass bin for which there is a sufficient amount of mass of the material to make two fragments. The materials are then distributed into the progressively smaller bins based on the amount of available mass of the material and the proportion of that mass to the available mass of the other materials. At the smallest mass bins, the materials are generally distributed in fractions similar to their originally input mass fractions. In reality, material distributions may be somewhat more disproportionate based on how brittle the materials are, but IMPACT does not currently take such effects into account.

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The characteristic fragment size for each mass bin is calculated as the average apparent size of the most probable fragment as defined by the average of the two largest perpendicular dimensions and based on the average material density of the mass in the bin. Thus the characteristic or “typical” size of a fragment in a particular mass bin is not only a function of the mass of the bin but also the materials with which the fragments in the bin are made. An example of this relationship can be seen in Figure 7.

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02

Fragment Mass, kg

Frag

men

t Cha

ract

eris

tic S

ize,

m

Figure 7. Characteristic Mass vs Size Relationship.

Average cross-sectional area is an important parameter when propagating fragments at altitudes where

atmospheric drag is significant. Long-term propagations are often done to determine the orbital lifetimes of collision fragments and thus the time that they will be a collision hazard to other objects in orbit. IMPACT calculates average cross-sectional areas for individually generated fragments. These are fragments for which state vectors and other characteristics are statistically generated based on the derived mass, material, and velocity distributions.

The process for generating an average cross-sectional area for an individual fragment starts with statistically choosing a primary material. The material choice is based on the distribution of material mass in the fragment’s mass bin. The results will be that the distributions of materials for individually generated fragments in a given mass bin will be the same as that determined by the original material distribution for the bin. From the material density and the fragment mass, a volume can be derived. A length-to-width ratio is statistically chosen from an empirically derived Gaussian distribution. A fragment thickness is assigned that represents a fundamental size of one of the dimensions of the source object (target or projectile) for the fragment. The thickness is a function of the fragment’s mass bin. A length and width are calculated based on the statistically determined length-to-width ratio, fragment volume and thickness. The length represents the longest dimension of the fragment and the width the longest dimension perpendicular to the length. A thickness-to-apparent-thickness ratio is pulled from an empirical distribution. The apparent thickness represents how large the fragment’s third dimension appears due to “crumpling” of the fragment and other effects. The apparent thickness is larger than the actual thickness that would be calculated only from material density. The apparent thickness is included to make the modeled fragments consistent with the observation that fragments, particularly larger ones, usually appear much less dense than would be expected from assuming that they were solid. The average cross-sectional area is then calculated from the length, width and apparent thickness. The practical consequence of this technique is that the fragments’ average cross-sectional areas vary significantly based on the materials from which they are made (Figure 8); this helps account for the large variations in area-to-mass ratios observed with debris from on-orbit breakup events6. Although “dimensions” for the individual fragments are generated in the process of producing an average cross-sectional area, the intent of the calculation is to produce representative areas for the fragments. The dimensional information will not necessarily correspond directly to actual fragment dimensions.

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1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02

Fragments Mass Bin, kg

Aver

age

Cros

s-Se

ctio

nal A

rea,

m2

AluminumSteel

Figure 8. Sample Average Cross-Sectional Area Distribution

VII. Object Types IMPACT represents the target and projectile in two basic classes, boosters and satellites. These categories are

distinguished by their basic structure. Booster objects such as a launch vehicle upper stage are considered largely hollow, whereas satellites are relatively solid with structure and components throughout their volume. Satellite objects are broken up using the standard fragmentation procedures, which may include the definition of intact objects, but boosters have a specific addition to their mass distribution: the “large fragments” that have been mentioned previously in this paper. It has been observed in tests7 that hollow booster-like objects tend to produce several disproportionately large fragments compared to what would be expected from a standard power law distribution.

To model a booster-type object’s tendency to generate large fragments, IMPACT removes a percentage of the structural mass from the standard fragmentation process and assigns it to several large fragments. The structural mass is defined as the total mass of all of the materials making up the object that are primarily structural versus being components such as electronics. The fraction of the structural mass assigned to large fragments is determined by the ratio of the target mass to the projectile mass. For projectile-to-target mass ratios less than 0.01, nine-tenths of the target structural mass is assigned to large fragments. For projectile-to-target mass ratios greater than 0.01, the fraction of the structural mass assigned to large fragments is determined by the curve in Figure 9. As the projectile mass approaches that of the target, the fraction of mass going to large fragments decreases due to the greater fraction of the target that is directly interacting with the projectile during the collision. The mass assigned to “large fragments” is distributed to several large fragments, the mass of which are determined by pairing fragments in the largest mass bins for which there is sufficiently large fragment mass. The remaining mass of the object is fragmented based on the standard fragmentation procedure. All of the fragments are combined to perform a final assessment and correction to balance the overall mass.

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0

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1

0 0.2 0.4 0.6 0.8 1

Projectile Mass / Target Mass

Larg

e Fr

agm

ent S

truct

ural

Mas

s Fr

actio

n

Figure 9. Booster-Type Object Large Fragment Mass Fraction.

The effect that the booster procedure is imitating is seen most easily in a collision where a proportionally small

projectile impacts a much larger booster-type target. Because of the disparity in sizes, the projectile interacts directly with only a small portion of the target. Due to the distributed nature of the mass of the target, the effects of the impact do not propagate as efficiently as in a more compact target, leaving pieces of the target more distant from the impact point largely intact or only slightly fragmented. The result is a proportionally greater amount of mass at the larger mass end of the distribution. In the case of a projectile, or the smaller object, being a booster, the effect is not noticeable since the larger size of the target causes it to interact with the entire projectile.

In addition to the standard kinetic energy used in the IMPACT collision model, it is also possible to add other sources of energy into the fragmentation process. These energy sources represent any other significant source of energy that would be expected to be released during the collision process; typical examples include propellant or pressurized gasses in tanks. The additional energy can then contribute to the resulting mass and velocity distributions from the breakup event. In real on-orbit collision events, it is frequently the case that the additional energy sources are small compared to the kinetic energy of the collision due to the high relative velocities between orbiting objects

VIII. Conclusion The IMPACT fragmentation model has been used to model orbital altitude collisions for more than 20 years. Its

combination of empirical relationships for mass, spread velocity, and size and area are combined with and constrained by imposition of mass, momentum, and energy conservation. The compromise of including input details such as state vectors and mass and material composition of the target and projectile while not specifically handling finer details such as impact point and specific target and projectile designs has resulted in a model that can represent particular events over a range of conditions while maintaining low computational requirements and inputs at a level of detail that is practical for real-world applications. Due to the semi-empirical structure of the model, as much data as possible should be included to corroborate or correct the empirical relationships so that improvements to the model can continue as new data become available

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Acknowledgments The author wishes to thank several individuals at The Aerospace Corporation for their support of this work and

assistance in preparing this paper. Donald Lewis provided programmatic review and support of the paper. Spencer Campbell and Alan Jenkin provided thorough technical review of the manuscript. The Aerospace Office of Technical Relations provided publication clearance review.

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TR-93-7025, July 1993. 4Badhwar, G. D., Tan, A., “Velocity Perturbation Distributions in the Breakup of Artificial Satellites,” Journal of Spacecraft

and Rockets, Vol. 27, No. 3, May-June 1990. 5Culp, R. D. (ed.), Analysis of Hypervelocity Impact on Satellites, USAF FO 8635-89-K-0227, 1990. 6Anz-Meador, P. D., Potter, A. E., “Density and Mass Distributions of Orbital Debris,” IAA-94-IAA.6.4.689, 45th Congress

of the International Astronautical Federation, Jerusalem, Israel, 9-14 Oct 1994. 7Loftus, J. P. (ed.), Orbital Debris from Upper-Stage Breakup, Progress in Astronautics and Aeronautics, AIAA, New York,

1989, pp. 25-37.