# Tuning of NASA Standard Breakup Model for Fragmentation Events Modelling

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{1}and M

_{2}, and the impact velocity v

_{imp}(i.e., the norm of the relative velocity). In the case of explosions, instead, an important role is played by the scale factor of the SBM, assuming a default value of 1 for upper stages with mass between 600 kg and 1000 kg [12]. Several works [4,25,26,27], however, demonstrated that, giving such inputs, the number and the characteristics of the simulated fragments are different to the ones generated by a real breakup event. In particular, Anz–Meador and Matney [25] and Braun et al. [4] demonstrated that several debris clouds derived from explosions of nominal upper stages are represented by values of the scale factor significantly different from 1. Similarly, Pardini and Anselmo [27] show that out of four on-orbit accidental collisions, all having an energy-to-mass ratio greater than 40 J/g, only one generated a number of fragments similar to the one predicted by the SBM. The reason is that the breakup model is purely energy-based and does not consider the relative attitude between the two colliding objects nor the point of collision. Indeed, the basic implementation of the SBM assumes that collisions will always involve the entire body of each parent. This does not account for spacecraft consisting of multiple connected structures; for instance, the presence of appendages such as gravity gradient booms or deployed solar panels. In this respect, the effect of a collision is clearly different if the projectile hits the main body of a satellite or only a part of it. In the latter case, the fragmented mass of the two objects will be probably lower and the breakup model would overestimate the number of the debris produced. Similarly, for explosions, the uncertainty regards the energy driving the breakup and, hence, the actual exploded mass. Therefore, a good estimation of the fragmented masses of the parents is critical to correctly set the input parameters of the SBM.

_{c}equal to 6 cm.

## 2. Review of the Standard Breakup Model

_{targ}and m

_{proj}are the masses of the target and projectile, respectively, while v

_{imp}is the impact velocity. Specifically, the collision is defined as catastrophic if the value of E

_{p}is greater than 40 J/g. In the case of non-catastrophic collisions, i.e., E

_{p}is less than 40 J/g, the fragmented mass of the parents is computed as follows [12],

_{1}and m

_{2}are the fragmented masses of the target and the projectile, respectively.

_{c}, is estimated using Equation (4) for collisions and Equation (5) for explosions.

_{frag}is the fragmented mass, and s is the scale factor. Then, the distribution of the number of fragments with respect to the A/M, for each characteristic length interval, is computed. For each pair of L

_{c}-A/M intervals, values of characteristic length and area/mass ratio are extracted and then assigned to each fragment. It is assumed that within each interval there is a uniform probability distribution. The value of the area (A) for each fragment is computed as a function of its length L

_{c}, using the following equation, valid both for collisions and explosions.

## 3. Breakup Model Sensitivity Analysis and Tuning Approach

_{imp}), normally computed from the state vectors of the parents, generally affects the fragmented mass during a collision. However, considering that the typical uncertainty in the knowledge of the impact velocity is in the order of m/s [34], this effect is very limited. Moreover, as long as the number of catalogued fragments is known, the impact velocity does not influence the iterative estimation of the fragmented mass but its computational cost. In fact, v

_{imp}is used only for the computation of the specific kinetic energy and, hence, for the definition of the type of collision (i.e., catastrophic or non-catastrophic). Such definition determines the initial values with which the iterative algorithm starts. Therefore, the impact velocity does not affect the final value of the estimated fragmented masses but only the number of iterations needed to reach convergence. In particular, if the collision is catastrophic, the impact velocity has no effect on the number of iterations, since the initial value of the iterative approach would always be equal to the total mass of the parents, regardless of the value of v

_{imp}. In case of a non-catastrophic collision, the impact velocity instead enters directly in Equations (2) and (3) to compute the estimated fragmented mass according to the SBM. This could lead to an increase of number of iterations, e.g., from 2 to 5. Based on these considerations, the masses of the parents are the only parameters to tune the model. The tuning of the masses can be done based on the accurate information provided by the catalogued fragments: namely, their number and orbital parameters. The orbital parameters available for the catalogued fragments can provide useful information on the ΔV distribution. However, such distribution seems to be rather insensitive to the input masses. To support this statement, a sensitivity analysis was carried out. For a fixed value of the projectile mass, set equal to 100 kg, the SBM was run for three different values of the target mass M

_{targ}(i.e., 100 kg, 500 kg and 1000 kg), and three different values of the impact velocity v

_{imp}(i.e., 1 km/s, 5 km/s and 10 km/s). For each couple (M

_{targ}, v

_{imp}), 100 runs were executed. For each run the mean μ and the standard deviation σ of the ΔV distribution were computed for all the fragments generated by the breakup (hereinafter called population) and for the fragments with a characteristic length greater than 10 cm (hereinafter called sample). Therefore, for each couple (M

_{targ}, v

_{imp}), a set of 100 means (μ

_{ΔV}) and standard deviations (σ

_{ΔV}) is obtained. Figure 1, Figure 2 and Figure 3 show the boxplots of μ

_{ΔV}and σ

_{ΔV}in the different cases for the sample fragments. On each box, the central mark (i.e., the red line) indicates the median of the statistics parameter (i.e., μ

_{ΔV}and σ

_{ΔV}), the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. The difference between the 75th and 25th percentiles (i.e., the height of the box) is called interquartile range (IQR). The whiskers extend to a value equal to 1.5*IQR above the 75th percentile and 1.5*IQR below the 25th percentile. Outliers are the values beyond the whiskers and are plotted individually using the ‘+’ symbol. By looking at the mean and standard deviation distributions, for a fixed impact velocity, the ΔV distributions are statistically similar regardless of the target mass. In fact, applying a two-sample t-test to the ΔV distributions obtained with the same impact velocity and different target mass, the null hypothesis (i.e., the sample data are from populations whose expected means are the same) is accepted with 95% of confidence in all the cases. Moreover, the percentage variation of the median of the μ

_{ΔV}and σ

_{ΔV}distributions was computed for target mass equal to 500 kg and 1000 kg with respect to the case in which the target mass is equal to 100 kg. The results of this analysis are reported in Table 1 considering different values of the impact velocity. The percentage variation is less than 5% for the median of μ

_{ΔV}and less than 10% for the median of σ

_{ΔV}.

_{min}, can be detected and tracked (e.g., L

_{min}of order of 10 cm size at LEO and 1 m at GEO). Secondly, after a short time from the fragmentation event (FEV), the cloud of debris is not dispersed enough, and it is hard to correctly catalogue the different fragments of the cloud. Thirdly, especially for high energy collisions or for breakup events at lower orbits, a significant number of fragments decay due to the atmospheric drag. Thus, the epoch at which the fragments TLEs are known is important.

- The type of input file to load (i.e., ephemerides or TLEs). In both cases, the software extracts the state vector of the parent(s),
**x,**either directly, in the case of ephemerides, or converting the orbital parameters to position and velocity, in case of TLEs. Moreover, the impact velocity is computed as the norm of the relative velocity difference between the two colliding objects. - Mass and size of the parents

_{tot}) and the mass of the parent is lower than a tolerance threshold (i.e., 5%). If the extraction procedure leads to a value of M

_{tot}greater than the mass of the parent by more than 5%, the fragments with minimum A/M probability in the current L

_{c}bin are removed. Then the ΔV distribution is estimated. The SBM only provides the distribution of the ΔV module for the fragments generated by the breakup event. To analyze the fragments’ evolution, a direction must be assigned to each ΔV. This task must be done assuring that the conservation of momentum is met. Hence, an icosahedral grid was created whose nodes correspond to directions in a three-dimensional space [26]. It is assumed that each node has the same probability to be associated to each fragment. The ΔV, multiplied by the direction cosines corresponding to each direction, gives the velocity components of each fragment. Each fragment is therefore characterized by length, mass, area, and area/mass ratio, position at the fragmentation event (FEV), coinciding with the position of the parents, speed at FEV, computed as the sum between the speed of the parent and the ΔV. The obtained set of fragments is then filtered by removing objects characterized by:

- length smaller than a threshold value (L
_{thr}) defined by the user, - negative semimajor axis (SMA
_{thr}), corresponding to a hyperbolic orbit, not of interest for the analyses of this paper. In fact, the possibility to track these fragments vanishes in time and their number is usually lower than 2% of the fragments over the size threshold. - altitude at perigee lower than 120 km (h
_{thr}), corresponding to a fragment that will re-enter before being catalogued.

#### The Iterative Approach

- The TLEs of the fragments are loaded and the orbital parameters are extracted. Let N
_{cat}be the number of catalogued fragments. - A set of simulated fragments is generated from the breakup model using as default values the mass of the parents and their relative velocity. Let N
_{SBM,Lc}be the number of simulated fragments with a length greater than the predefined size threshold. - The catalogued fragments are backward propagated until the FEV. This helps to distinguish the two clusters of fragments belonging respectively to the target and the projectile out of a TLE file containing mixed fragments generated by the parents. The general principle is to compute the norm of the relative velocity between fragments and parents at FEV, and to assign each fragment to the parent corresponding to the minimum norm. In the case of orbital planes rotated by at least a few degrees with respect to each other, which will typically characterize collision geometries for very low eccentricity orbits, the concept has a direct analytical implementation based on inclinations and right ascension of the ascending node. In fact, it is well-known that a plane change requires a large value of the out-of-plane component of ΔV. This is confirmed also from historic breakups, registering a maximum plane change of 3 degrees for the fragments with respect to their parent’s original orbit [22]. Therefore, it can be assumed that, even with high energy collisions, the breakup fragments will only be dispersed a small amount from the parents’ orbital planes. Using the law of cosines for spherical triangles, the rotation of each fragment’s orbital plane with respect to target and projectile ones is computed:

_{targ}and θ

_{proj}are the angles between the fragment orbit plane and the orbit plane of target and projectile, respectively, i

_{frag}, i

_{targ}, and i

_{proj}are the inclination of the fragment, target and projectile orbit planes, Ω

_{frag}, Ω

_{targ}and Ω

_{proj}are the right ascension of the ascending node of the fragment, target and projectile orbit planes. If θ

_{targ}is smaller (larger) than θ

_{proj}, the fragment is assigned to the target (projectile).

- 4.
- Estimation of decayed fragments: due to a negative ΔV or to the atmospheric drag, some fragments may decay in the time frame, Δt
_{FEV,TLE}, between the FEV epoch and the last available TLE epoch. The number of decayed fragments, N_{dec}, can be negligible in the iterative process when many catalogued fragments, N_{cat}, are available. However, when N_{cat}is small, the number of decayed objects might represent a significant percentage, thus affecting the estimation of the fragmented mass. To consider this aspect in the iterative process, N_{dec}must be estimated. This is done by counting all the N_{SBM,Lc}fragments having an altitude at perigee below 150 km. It is indeed reasonable to consider that the fragments with this characteristic decayed in the time frame Δ_{FEV,TLE}. - 5.
- Iterative algorithm: the iterative algorithm operates until the following relationship is satisfied.

_{Lc,nodec}is the number of not decayed fragments larger than the threshold (i.e., N

_{SBM,Lc}− N

_{dec}). The values for the tolerances depend on the statistical variability of EVOLVE over different runs, which increases as the fragmented mass (and consequently, the number of catalogued fragments) reduces. So, for N

_{cat}less than 50, a minimum tolerance of 0.3 N

_{cat}is set. For N

_{cat}in the range 50–100 and N

_{cat}greater than or equal to 100, the tolerance values were scaled down (i.e., 0.2 N

_{cat}and 0.1 N

_{cat}, respectively). Figure 6 shows a scheme of the iterative logic implemented.

_{cat,targ}and N

_{cat,proj}), the number of simulated fragments previously generated, with altitude of perigee greater than 150 km at FEV (N

_{Lc,nodec}) and the mass M

_{SBM,i}

_{,}with i = 1,2 (1 for the target and 2 for the projectile). M

_{SBM,i}

_{,}is equal to the total parent mass (for catastrophic event) or the estimated fragmented mass by the SBM, according to Equations (2) and (3). If the number of simulated fragments is equal (within a certain percentage) to the number of real fragments, the algorithm is not executed, and the number of simulated fragments is the one generated by the SBM. Otherwise, the algorithm is executed. Let M

_{i}be the initial mass of the i-th parent (i.e., i = 1 for the target and i = 2 for the projectile); m

_{i,j}be the estimated fragmented mass for the i-th parent at the j-th iteration; m

_{i,0}= M

_{SBM,i}, be the initial mass the bisection method starts with; [a

_{j}b

_{j}] be the interval in which the bisection method is applied. For j = 1:

_{Lc,nodec}> N

_{cat}+ tol

_{:}

_{Lc,nodec}< N

_{cat}− tol

_{:}

_{Lc,nodec}> N

_{cat}+ tol:

_{Lc,nodec}< N

_{cat}− tol

_{:}

## 4. Results and Discussions

#### 4.1. Simulated Collision Events

_{1}and M

_{2}, the impact velocity and the corresponding fragmented masses (i.e., m

_{1}and m

_{2}). By running the SBM using m

_{1}and m

_{2}as input masses, the reference values for the number of generated fragments, i.e., N

_{1}and N

_{2}, can be obtained.

_{1}and m

_{2}assuming that all the observable fragments, i.e., those with characteristic length larger than 10 cm, are catalogued. To this aim, the iterative approach is applied using the total masses of the parents, i.e., M

_{1}and M

_{2}, as initial guess. The SBM is then iterated until the convergence condition given by Equation (10) is met. At convergence, the number of generated fragments is indicated by N

_{1,est}and N

_{2,est}, respectively, while the corresponding fragmented masses is indicated by m

_{1,est}and m

_{2,est}, respectively.

_{1}= 1000 kg, M

_{2}= 800 kg and the impact velocity is equal to 14 km/s. It is assumed that only the following masses are involved in the fragment generation: m

_{1}= 850 kg, m

_{2}= 490 kg. With these values of masses, the SBM generates a set of N

_{1}= 693 fragments from the target and N

_{2}= 453 from the projectile.

_{1}= 1000 kg, M

_{2}= 800 kg, while v

_{imp}is set to 14 km/s. The application of the standard SBM would produce a number of fragments, N

_{SBM}, about 14% and 58% more than N

_{1}and N

_{2}, respectively. However, after 3 iterations, the proposed approach reduces this overestimate to 0.9% and 2.2%, respectively, while allowing the estimation of the fragmented masses with an error with respect to the assumed values of about 3% and 2%, respectively. Simulation results are summarized in Table 3.

_{nf}and the standard deviation σ

_{nf}of the number of fragments greater than 10 cm, as well as the mean μ

_{mf}and the standard deviation σ

_{mf}of the estimated fragmented mass, were computed. Figure 12a shows the percentage error associated to μ

_{nf}(solid line), μ

_{nf}− σ

_{nf}and μ

_{nf}+ σ

_{nf}(error bars) for the target (in black) and the projectile (in red). Similarly, Figure 12b shows the percentage error associated to μ

_{mf}(solid line), μ

_{mf}− σ

_{mf}and μ

_{mf}+ σ

_{mf}(error bars) for the target (in black) and the projectile (in red). It must be pointed out that, for assignment errors lower than 10%, the output error is not necessarily due to the erroneous number of catalogued fragments, but also depends on the statistical variability of EVOLVE.

_{1}= 1000 kg, M

_{2}= 50 kg, and the impact velocity is equal to 1 km/s. It is assumed that only the following masses are involved in the collision: m

_{1}= 30 kg, m

_{2}= 30 kg. With these values of masses, the SBM generates a set of N

_{1}= 37 fragments from the target and N

_{2}= 38 from the projectile (as illustrated in Table 6), considered as reference values. The input data are then set equal to the actual parent masses, i.e., M

_{1}= 1000 kg, M

_{2}= 50 kg while v

_{imp}is still equal to 1 km/s. Without the iterative approach, the SBM estimates fragmented masses, M

_{SBM}, of 50 kg for both the target and the projectile (about 67% more than the assumed involved mass), while the number of generated fragments is equal to 68 and 74. Instead, the iterative approach allows reducing the fragmented mass estimation error down to about 17% of the assumed values. With regards to the number of fragments the SBM commits an error of 84% for the target and 94% for the projectile, while these values are reduced to about 8% and 13% by applying the iterative approach. These results are in accordance with Pardini and Anselmo [27] that highlight how the outputs of the non-catastrophic collisions are worse than catastrophic collisions in the estimation of the number of fragments. The output of the second test case is summarized in Table 6.

#### 4.2. Real Collision Event

_{1,frag}= 6 and N

_{2,frag}= 0 in accordance with the catalogue. Table 7 summarizes all the data.

#### 4.3. Computational Cost Analysis

_{cat}greater than 100 (i.e., 500), between 50 and 100 (i.e., 60), and below 50 (i.e., 20) for both the parents. Figure 18 shows the number of iterations required, for the target (solid line) and the projectile (dashed line), to reach the convergence versus a tolerance value on N

_{cat}equal to 5%, 10%, 20% and 30%. In all cases, a relatively small number of iterations is required to reach convergence.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Mean (

**a**) and standard deviation (

**b**) of velocity distribution of sample fragments for different values of target mass for an impact velocity of 1 km/s. Mass of projectile is assumed to be 100 kg.

**Figure 2.**Mean (

**a**) and standard deviation (

**b**) of velocity distribution of sample fragments for different values of target mass for an impact velocity of 5 km/s. Mass of projectile is assumed to be 100 kg.

**Figure 3.**Mean (

**a**) and standard deviation (

**b**) of velocity distribution of sample fragments for different values of target mass for an impact velocity of 10 km/s. Mass of projectile is assumed to be 100 kg.

**Figure 4.**Median of target (

**a**) and projectile (

**b**) fragments sample velocities for different values of impact velocity and target mass. Mass of projectile is assumed to be 100 kg.

**Figure 5.**Scheme of SBM implementation. Each fragment is characterised by physical characteristics (i.e., size, mass, A/M) and a state vector (i.e., position and velocity).

**Figure 8.**Histograms of reference (

**a**) and estimated (

**b**) fragments in range [75°, 105°] of inclination for first test case.

**Figure 9.**Histograms of reference (

**a**) and estimated (

**b**) fragments in range [6900, 12,000] km of semimajor axis for first test case.

**Figure 10.**Gabbard diagrams for the target (

**a**) and the projectile (

**b**), obtained with input masses equal to m

_{1}= 850 kg and m

_{2}= 490 kg.

**Figure 11.**Gabbard diagrams for the target (

**a**) and the projectile (

**b**), obtained with fragmented masses estimated by iterative approach for first test case.

**Figure 12.**(

**a**) Mean and standard deviation errors for number of fragments greater than 10 cm, both for target and projectile. (

**b**) Mean and standard deviation errors for estimated fragmented mass, both for target and projectile.

**Figure 14.**Histograms of reference (

**a**) and estimated (

**b**) fragments in range [80°, 100°] of inclination for second test case.

**Figure 15.**Histograms of reference (

**a**) and estimated (

**b**) fragments in range [6900, 12,000] km of semimajor axis for second test case.

**Figure 16.**Gabbard diagrams for the target (

**a**) and the projectile (

**b**), obtained with input masses equal to m

_{1}= 30 kg and m

_{2}= 30 kg.

**Figure 17.**Gabbard diagrams for the target (

**a**) and the projectile (

**b**), obtained with fragmented masses estimated by iterative approach for second test case.

**Figure 18.**Number of iterations against tolerance on number of catalogued fragments for different values of N

_{cat}, both for target (solid line) and projectile (dashed line).

**Table 1.**Absolute percentage variation of μ

_{ΔV}and σ

_{ΔV}distributions for target mass equal to 500 kg and 1000 kg with respect to target mass equal to 100 kg.

Absolute Percentage Variation of μ_{ΔV} (%) | Absolute Percentage Variation of σ_{ΔV} (%) | ||||
---|---|---|---|---|---|

M_{p} | 500 kg | 1000 kg | 500 kg | 1000 kg | |

v_{imp} | |||||

1 km/s | 0.3 × 10^{−2} | 2.8 | 0.6 | 0.7 | |

5 km/s | 1.4 | 0.9 | 6.6 | 9.2 | |

10 km/s | 2.1 | 4.9 | 2.5 | 2.0 |

Target | |||||

a = 7359 km | e = 0.00348 | i = 83° | Ω = 8.63° | ω = 237° | ν = 100° |

Projectile | |||||

a = 7461 km | e = 0.01459 | i = 100° | Ω = 199.8° | ω = 240° | ν = 25° |

Parents Mass | Assumed Fragmented Mass | N_{cat} | m_{est} | N_{est} | N_{SBM} | |
---|---|---|---|---|---|---|

Target | 1000 kg | 850 kg | 693 | 875 kg | 699 | 793 |

Projectile | 800 kg | 490 kg | 453 | 500 kg | 463 | 716 |

Percentage Assignment Error | 0% | 2% | 4% | 10% | 15% | 20% |

Target | 693 | 703 | 712 | 739 | 761 | 784 |

Projectile | 453 | 443 | 434 | 407 | 385 | 362 |

Target | |||||

a = 7361 km | e = 8.9 × 10^{−5} | i = 90° | Ω = 45° | ω = 90° | ν = 0° |

Projectile | |||||

a = 7361 km | e = 8.9 × 10^{−5} | i = 90° | Ω = 53° | ω = 90° | ν = 0° |

Parents Mass | Assumed Fragmented Mass | N_{cat} | M_{SBM} | m_{est} | N_{SBM} | N_{est} | |
---|---|---|---|---|---|---|---|

Target | 1000 kg | 30 kg | 37 | 50 kg | 25 kg | 68 | 34 |

Projectile | 50 kg | 30 kg | 38 | 50 kg | 25 kg | 74 | 33 |

Parents Mass | M_{SBM} | m_{est} | N_{SBM} | N_{est} | |
---|---|---|---|---|---|

Target | 50 kg | 50 kg | 18.75 kg | 64 | 6 |

Projectile | 2.1 kg | 2.1 kg | 2.1 kg | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Cimmino, N.; Isoletta, G.; Opromolla, R.; Fasano, G.; Basile, A.; Romano, A.; Peroni, M.; Panico, A.; Cecchini, A.
Tuning of NASA Standard Breakup Model for Fragmentation Events Modelling. *Aerospace* **2021**, *8*, 185.
https://doi.org/10.3390/aerospace8070185

**AMA Style**

Cimmino N, Isoletta G, Opromolla R, Fasano G, Basile A, Romano A, Peroni M, Panico A, Cecchini A.
Tuning of NASA Standard Breakup Model for Fragmentation Events Modelling. *Aerospace*. 2021; 8(7):185.
https://doi.org/10.3390/aerospace8070185

**Chicago/Turabian Style**

Cimmino, Nicola, Giorgio Isoletta, Roberto Opromolla, Giancarmine Fasano, Aniello Basile, Antonio Romano, Moreno Peroni, Alessandro Panico, and Andrea Cecchini.
2021. "Tuning of NASA Standard Breakup Model for Fragmentation Events Modelling" *Aerospace* 8, no. 7: 185.
https://doi.org/10.3390/aerospace8070185