2.2 Atmospheric Models
A huge number of different atmospheric models is nowadays available for the characterization of the atmospheric density variation with orbital altitude. The development of these methods mainly relies on two approaches: the coupling into a single relation of conservation laws and atmospheric components models, and the exploitation of in-situ measurements and satellites observational data [17]. Figure 8 shows a schematic survey of some of the models developed over the years together with their source and development basis.


Figure 8: Atmospheric models evolution over the years with relative origin and derivation [23].
Nowadays, the most used models are: Standard Atmosphere (USSA76), Jacchia-Roberts with its various versions (J71, J77, GRAM-99), COSPAR International Reference Atmosphere (CIRA90) and Mass Spectrometer Incoherent Scatter (NRLMSIS-90) [23].
Some of these models present a static description of the atmosphere, with the density obtained as a function of the only orbital altitude, while many others, like the Jacchia-Roberts model, are time varying models with much higher computational requirements. These demanding calculations are due to the uncertainties related to the forecasted solar activity and the relatively high effect of this on the atmospheric density at different altitudes.
Since our preliminary analysis does not assume a specific mission scenario, neither in terms of beginning of the mission nor in terms of mission duration, it is more reasonable to assume a simple static model for the atmospheric density. One of the simplest static models is the Harris-Priester model [24, 17], which relies on a number of tables listing reference density values obtained from observational data within a complete solar cycle. Table 2 shows the values of the atmospheric density, in the minimum and maximum density case, used in our analyses for the implementation of the Harris-Priester model.



Table 2: Data used for the implementation of the Harris-Priester atmospheric model [24].
According to [23], any atmospheric model has 10-15% of inherent accuracy, and a medium density model (as the CIRA-86 at mean atmospheric conditions, very close to the Harris-Priester model used in this study) has a root mean square accuracy of the order of ±10% [25]. This small lack in accuracy is well compensated by its high computational speed and, despite its limitations with respect to short and long period variations of density, this model has been selected to describe three different phases of the solar cycle, resulting in Minimum Density, Medium Density and Maximum Density scenarios. The medium one has been obtained as an average between the minimum and the maximum densities of Table 2.
In the following, these three different density scenarios have been used to assess the optimal foam ball radius (see Sec. 4) for the DISCOS list but, under the hypothesis of periodic and regular cyclical change of the solar flux [17], the best option for the starting of an active debris removal mission is a high solar flux period. Starting the removal mission during this period, indeed, the deorbiting phase of the foamed debris immediately starts at the highest possible rate. Then, over the years, the change in the solar flux should decrease this deorbiting speed but the time spent in orbit during maximum density and minimum density would always at least be equal. Actually, if the deorbiting time is not an integer multiple of the solar flux cycle, maximum density periods would outnumber minimum density ones. For this reason, the Harris-Priester medium density model, as the one used in [26] for first order estimations, should be considered as a more realistic, and possibly conservative, case.
As an example, the atmospheric density values predicted by this model have been also compared with a different static model, the Standard Atmosphere USSA76 [27], as shown in Fig. 9. This latter model always gives smaller atmospheric density values for orbital altitude above 370 km. Below this threshold the atmospheric density is rather high and the small time period spent by deorbiting objects below this altitude is a very small fraction of the whole deorbiting time. Thus, the resulting final orbital lifetimes should actually be less conservative than the ones provided by our model.


Figure 9: Comparison between the Harris-Priester model and the Standard Atmosphere USSA76 model.

Below: a close up of the upper plot between 500 and 1000 km where the Harris-Priester
model is more conservative.
For the sake of completeness, the lifetime values provided by the method described in Sec. 2.1 have been also compared with the plots, generated with the SatLife program, provided by [39]. These plots show two curves for a set of three ballistic coefficients. One of the curves, representing a deorbiting operation started during a solar minimum “when there will be a low level of decay” always predict, for altitudes below 700 km, orbital lifetimes longer than the others. Indeed, the curve corresponding to deorbitation begun at the start of a solar maximum “when the satellite will decay most rapidly for several years” provides shorter orbital lifetimes. The lifetime values predicted by our approach always lie between the two curves, thus it is reasonable to assume that the computed lifetimes represent reasonable assessments of the actual debris deorbiting time. By way of example, considering an object with a ballistic coefficient (m A-1 Cd-1) of 200 starting from
500 km of altitude, our method predicts an orbital lifetime of 6.8 years, while the plots provide ~3 years (Orbit starting at solar maximum) and 7 years (Orbit starting at solar minimum) of lifetime. In this case, actually, we are closer to the most conservative case. For an object with a ballistic coefficient of 20 starting from 600 km of altitude, our code provides 2.65 years of lifetime as compared with ~1.5 to ~5 years given by the reference plot.

2.3 Space Debris Models
The assessment of impact probability for orbiting objects due to the proposed method and for the foam balls themselves is a binding task for the estimation both of hazards and risks related to the method (see Sec. 11) and of a suitable foam ball size (see Sec. 4). We only focus in the following on possible space debris impacts, neglecting those due to micrometeoroids. This is a rather reasonable assumption, since below 2000 km the orbital debris environment represents the major threat for space flight if compared with the meteoroid environment [28].

In order to obtain a representative estimation of potential impacts, over the years many space debris flux models have been implemented and refined, using data obtained from the post-flight analysis of spacecrafts as well as from observational data [29]. Some space debris environment models,together with their main characteristics, are listed below:
• NASA90 model (see Sec. 2.4) provides a simple and very fast debris flux calculation for
orbital altitudes below 1000 km, but it does not take into account the existence of a large
number of particles on eccentric orbits. Since this model has been the first more or less
detailed description of the debris environment, it can not be really considered up to date
[28]. In spite that, it remains one of the most valid options for preliminary analyses and
impact probability estimations.
• ORDEM96 model, also known as NASA96, model is the successor of the NASA90 model.
This model, unlike NASA90 model, basically identifies six different inclination domains
and for each orbit performs a numerical collision analysis obtaining the spatial debris
density around the target. Then the sum of the various contributions needs to be numerically converted to obtain the fluxes on the specific target orbit [30].
• ORDEM 2000 model is an evolution of the ORDEM96 model and it is suited for orbit
regions between 200 and 2000 km of altitude. It relies on a completely different approach
compared to the NASA90 and ORDEM96 (NASA96) models. It is based, indeed, on
observational data and analytical techniques to obtain the debris population probability
distribution functions. These functions then, from the debris environment, provide the
presumed space debris flux [12].
• The MASTER 2001 model provides the debris population distribution, both for the past and the future, starting from the numerical modelling of all known fragmentation events as well as the generation of debris particles. The propagation of the particle orbits allows the flux calculation also considering the asymmetry induced by the particle orbits argument of
perigee [31].
• The MASTER 2005 model is the successor of MASTER 2001 with some more refined and more updated features. Both the breakup and fragmentation models have been improved together with the update of some reference data as the reference population [31].
As already pointed out, since our analysis requires a fast and preliminary assessment of the space debris flux and the consequent impact probability, we identified the NASA90 model as the best one for our purposes. It is briefly described in the next section.

2.4 The NASA90 Model
The NASA90 model computes the debris flux F versus the impactor diameter d by means of an analytical formulation. The flux F is defined as the cumulative number of impacts on a spacecraft in circular orbit per square meter and year on the surface of an object randomly rotating around its centre of mass [32]. This value is obtained as function of the minimum impactor diameter d, the considered orbit altitude and inclination (h, i), the mission epoch and the solar radio flux S. It is obtained by means of [32]:


In this equation, the first two terms may be obtained by means of:


with d measured in cm, h in km and S in 104 Jy.
The terms in the square brackets also depend on the mission date, t, expressed in years, on the expected annual growth rate of mass in orbit, p, assumed by default equal to 0.05 [32] and on the growth rate of fragments, q, conservatively assumed 0.04 [32] for t>2011. These terms can be obtained by means of:



The remaining term Y is the discrete inclination dependent function tabulated in Table 3 [32].


The exact value of the Y term in Eq.(8) is obtained by the linear interpolation with respect to the orbital inclination.
The NASA90 model is exploited in Sec. 4 to obtain the foam ball impact probability as function of the foam ball size. The impact probability, of course, increases with the ball radius, thus increasing the risk of random collisions and the cascade effect. On the other hand, decreasing the ball radius,the deorbiting lifetime increases thus increasing the permanence in orbit and, accordingly, also the risk of additional collisions. The ball radius, indeed, is estimated considering the minimum of the curve given by the sum of these two contributions (see Sec. 4).


Authors: M. Andrenucci, P. Pergola, A. Ruggiero /2011
Affiliation: University of Pisa – Aerospace Engineering Department – Italy
ACT researcher(s): J. Olympio, L. Summerer

University of Pisa
Tel: +39-050-967211
Fax: +39-050-974094
Advanced Concepts Team

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