## 2 Fragmentation Events
### Fragmentation Characteristics of Individual Events  
To explain the properties and model approaches of individual events, the fragmentation characteristics of four exemplary events will be mentioned in the following. These are the fragmentations of Fengyun-1C (2007), Cosmos-2251 (2009), DSMP-F13 (2015), and NOAA-16 (2015). A more detailed representative analysis will be shown for the Fengyun-1C event. The TLE (Two-Line Elements) catalog is a free source for the amount of cataloged debris each related to the parent object [4]. However, only objects enter the catalog that could be tracked, correlated and re-tracked. Objects that were tracked and correlated but never re-tracked do not enter the catalog. When modeling \begin{table}
\begin{tabular}{c c c c c} \hline
**COSPAR ID** & **NORAD ID** & **SAT NAME** & **EVENT EPOCH** & **DEBRIS (2018)** \\ \hline
1999-025A & 25730 & Fengyun-1C & 11 January 2007 & 3438 \\
2006-057A & 29651 & USA-193 & 21 February 2008 & 2191 \({}^{*}\) \\
1993-036A & 22675 & Cosmos-2251 & 10 February 2009 & 1667 \\
2006-006B & 28944 & Breeze-M R/B & 19 February 2007 & 1460 \\ \hline \end{tabular}
\end{table} 
Table 1: Fragmentation events with the highest number of initially generated trackable fragments (as of 2018) [4,7] (\({}^{*}\) assessed number). Figure 3: Age of object on breakup epoch. individual fragmentation events, it is important to consider the total number of detected objects related to the same event. If not, modeling these events could lead to a high underestimation in terms of object numbers. The initial procedure for obtaining the total number of generated fragments are initial publications by the SSN as well as more refined data from the Orbital Debris Quarterly News (orbitaldebris.jsc.nasa.gov). One specific case of missing objects in the catalog is the explosion of the Breeze-M (2012-044C) in 2012. Although there were about 700 reported detections on this event [9], only 112 objects entered the catalog to this day [4]. During the MASTER population generation, detailed research and data acquisition are performed to include the correct number of detected debris originating from each event. One of the most important models is the standard NASA breakup model [10], which is used in a modified version also in MASTER-2009 [2]. Furthermore, the recommendations concerning the implementation given by [PERSON] [11] are considered. Therein, a power law describes the number-size-distribution of a fragmentation cloud right after the event. The NASA breakup model provides two power laws for the cumulated number of fragments due to explosions and collisions:  
\[N(d>L_{c})= \begin{cases}sf\cdot 6\cdot L_{c}^{-1.6},\text{ for explosions},\\ 0.1\cdot M_{tot}^{0.75}\cdot L_{c}^{-1.71},\text{ for collisions},\end{cases}  \tag{1}\]  
with \(N(d>L_{c})\) representing the total number of fragments with a diameter \(d\) larger then a characteristic length \(L_{c}\) (usually the mean diameter of the fragment). Additionally, \(sf\) is an object type depending scaling factor between zero and one and \(M_{tot}\) the mass involved in the collision. However, this approach must not be applied ignoring the number of detected objects because these are shaping the power law. To illustrate this, the number-size distribution of the NASA breakup model for the Fengyun-1C fragmentation is shown in Figure 4a. To match the number of detected objects with the number-size-distribution, the shape has to be moved to intercept the number of detected objects at the sensor minimum trackable diameter. This calibration is performed for each event in the fragmentation database to ensure a consistent modeling approach based on the underlying detection data. However, the number of cataloged TLE fragments is not static but is increasing over the years, which forces reevaluation of clouds with new information on the debris number. The time history on number of TLE fragments for four different events is shown in Figure 5.  
Figure 4: Number-size distributions for the Fengyun-1C cloud.