A common way to evaluate the latent long-term environmental impact of an orbiting object, refraining from running thousands of complex simulations based on quite uncertain scenario assumptions, is to devise an indexing scheme grounded on reasonable premises. The main advantage of these heuristic approaches is the adoption of simple to understand and simple to apply rules, while the main drawback is the lack of a validated cause-effect foundation, unavoidable if the uncertain, stochastic and possibly chaotic nature of the long-term debris evolution is taken into account. Notwithstanding this limitation, in order to evaluate the long-term detrimental effects of the objects abandoned in low Earth orbit (LEO), a new indexing scheme was devised.

Regarding its potential long-term adverse effects on the debris environment, the criticality index R of an object in LEO, where a higher index is associated with a higher potential threat, can be obtained as the product of two functions:

R = f ⋅ g (1)

where f depends on the probability of catastrophic breakup Pc due to orbital debris collision and on the number of new "projectiles" Np resulting from the breakup, while g characterizes the long-term impact on the environment as a function of the fragments cloud lifetime, volume of space involved and interaction with the pre-existing debris distribution. After a few simplifying assumptions applicable to the specific problem, the probability of catastrophic breakup can be expressed as

Pc ∼ F(h,i,M) · A · LT (2)

where A is the average collisional cross-section of the target object, F = F(h,i,M) the flux of debris able to catastrophically breakup the target object as a function of its mass M, altitude h and inclination i, and LT is the object residual lifetime, which can be expressed, in terms of the body mass-to-area ratio M/A and "normalized" lifetime function l(h), as:

LT ≅ l(h) · M/A (3)

Eq. (2) can then be written in the following way:

Pc ∼ F(h,i,M) · l(h) · M (4)

According to the NASA standard breakup model, the cumulative number of fragments Np generated in a catastrophic collision and larger than a given characteristic size is proportional to the cumulative mass of the target object and impacting debris, raised to the 0.75th power. However, the cumulative mass is in practice very close to the target mass, being the latter typically much larger (by 3 orders of magnitude in LEO) than the impactor's one. As a result, Np is proportional to M0.75, leading to the expression:

f ≡ F(h,i,M) · l(h) · M1.75 ∼ Pc · M0.75 (5)

In order to characterize the long-term impact on the environment of the resulting debris cloud, we introduced the concept of collisional debris cloud decay of 50% of the catalogable fragments (CDCD50), with typical size d ≥ 10 cm in LEO. Regarding the volume of space affected by the potential breakup and the interaction of the resulting cloud of fragments with the pre-existing debris distribution, we opted for the choice of a function z, defined in the following way:

z(h,i,d≥10cm) ≅ F(h,i,d≥10cm)/F(h,i=0°,d≥10cm) (6)

In LEO, it has a strong dependence on the orbit inclination i, but varies relatively slightly with the height h. The function g in Eq. (1) was therefore defined as follows:

g ≡ CDCD50(h,d≥10cm) · z(h,i,d≥10cm) (7)

From a practical point of view, it was highly desirable deriving from R a criticality index RN both normalized and dimensionless. To do so, we reverted to the average intact object in LEO in mid-2013, defining a reference object with

the following characteristics: M0 = 934 kg, h0 = 800 km, i0 = 98.5°. The normalized and dimensionless criticality index RN then became:

RN ≅ (F/F0) · l(h)/l(ho) · (M/M0)1.75 · CDCD50(h)/CDCD50(h0) · z(h,i)/z(h0,i0) (8)

Where F0 = F(h0,i0,M0) and l(h)/l(h0) ≡ 1 when h > h0. The latter cut off, set at a lifetime around 200 years, was introduced to avoid giving too much relative weight to objects with very long lifetimes, much longer than any reasonable temporal horizon for the current modeling and technology projections.

The criticality index meaning is easy to understand, because RN is referred to an average intact object in LEO placed in the most popular orbital regime, the sun-synchronous one. The number obtained for a specific object "proportionally" compares its latent detrimental effects on the long-term debris environment with those of the reference body. In other words, as an example, RN = 3 would imply, according to the indexing scheme devised, that the object under evaluation would be equivalent, concerning its long-term detrimental effects on the debris environment, to 3 average objects in sun-synchronous orbit.

Notwithstanding the straightforward meaning of RN, its values may span a range of many orders of magnitude, so a logarithmic index RNL might be more functional in certain cases. It was defined in the following way:

RNL ≡ log10(RN) + 1 (9)

in order to obtain RNL = RN = 1 for the reference body, and RNL ≥ 0 when RN ≥ 0.1 (i.e. 1/10 of the criticality index for the reference body).

Reference: Anselmo, L. and Pardini, C., Compliance of the Italian satellites in low Earth orbit with the end-of-life disposal guidelines for space debris mitigation and ranking of their long-term criticality for the environment, Acta Astronautica, 114 (2015) 93−100, doi: 10.1016/j.actaastro.2015.04.024

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