(BEING CONTINUED FROM 13/05/16)
1.3 Mission Scenario
The above described method can be thought as composed of different phases. Actually these are the complete mission scenario phases:
a) Launch: The platform in charge of targeting and deorbiting the debris has to be launched into an initial orbit. The most suitable choice for the launch orbit of the spacecraft is the one of a specific debris, see Sec. 10. The platform has just to perform the final approach manoeuvre to reach the first target debris.
b) Target Debris Interception: This phase consists in a set of orbital manoeuvres aimed at the acquisition of the same orbital elements of the target debris, i.e. debris rendezvous.
c) Foaming process: In this phase the target debris has been reached and the actual foaming process takes place. During this stage the foam has to:
1. be ejected from the platform and reach the target debris (see Sec. 9)
2. stick to the debris surface
3. grow in volume (see Sec. 6)
4. cover the targeted debris
d) Debris Deorbiting: The debris is now contained within the foam and the natural deorbiting of the system begins.
e) Targeting of next debris: The platform can now target another debris operating its (electric) thrusters to reach a new interception orbit.
f) Platform self-disposal: Once the platform has completed its mission, the thrusters can be
finally used to lower the orbit perigee to deorbit the spacecraft within the 25 years limit, as
stated by the IADC guidelines.
Figure 7 shows the most important phases of the mission: target debris interception, the foaming process and debris deorbiting.
2 SPACE ENVIRONMENT MODELS
Before starting the analysis of the proposed foam-based method, it is mandatory to provide methodologies to compute the behaviour of the deorbiting object. In particular, in this chapter some models, required to assess the deorbiting time and the impact probability, are provided. Since the specific atmospheric model is of fundamental importance to estimate the deorbiting time, the specific one here considered has been chosen as it offers the chance to model different density regimes, from low to high. Moreover, considering only the atmospheric drag, it is clear that the larger the foamed debris, the better this scenario behaves. This does not hold anymore if also the
impact probability is considered. The NASA90 impact probability model is here chosen, among several options (see Sec. 2.3), as it can be easily implemented without relying on specific libraries.
Furthermore, also the numerical methodology by which the deorbiting time is computed through the work is given.
2.1 Deorbiting Time Estimation
The estimation of the decay time for a generic orbiting body is a challenging problem due to the huge number of unknown quantities related to its orbit, shape and the actual atmospheric density. In order to obtain a fast and realistic assessment of the deorbiting times, some perturbation effects acting on the body may be neglected while the most important ones can be averaged over one or more orbital revolutions .
More in detail, let us consider the instantaneous acceleration vector of a generic body on an
elliptical orbit around the Earth as the sum of the acceleration given by Newton gravity law  plus a perturbation component. This latter part changes along the orbit due to the different contributions of the various terms composing this acceleration as Sun and Moon third body acceleration, atmospheric drag, Earth non spherical gravitational field, solar radiation pressure and thruster acceleration . As the typical orbital altitude of the objects we are targeting is between 500 km and 1000 km, it is possible to assume that the perturbation mainly affecting the body is the atmospheric drag, which can be expressed as :
where ρ is the atmospheric density value, Cd a dimensionless number reflecting the object
configuration sensitivity to drag force, A and m respectively the object cross-sectional area and mass and V its relative velocity vector with respect to Earth atmosphere.
Considering a direct method, as the one described by P.H. Cowell in the early 20th century , the integration of this acceleration value for a very long time, as the one needed by an object to reach high density atmospheric layer burning out, could be very expensive in terms of computation time.
An alternative for the solution of this problem is the implementation of the Encke’s method  based on the assumption that the integrated variables are small and so it is expected to be the integration error. This method needs however a very careful rectification to avoid numerical instabilities or large loss of precision , thus nowadays it tends to be avoided. An additional valid alternative to model the perturbation effects is represented by the Gauss form of the Lagrange Planetary Equations . These equations model the time evolution of the classical orbital parameters (semi-major axis a, eccentricity e, inclination i, right ascension of the ascending node,argument of pericenter w and M0 [M0 =M-n(t-t0)]) under the influence of a non-conservative perturbation. These read:
where n is the orbital mean motion, p the semilatus rectum, and h the orbital angular momentum while the three different accelerations ar, aq and ah are respectively the perturbation acceleration along the radial direction, along the normal to the radius vector in the direction of motion (orthoradial) and along the angular momentum vector direction . The three directions here described identify the radial, transverse and normal (RSW) reference frame in which Eqs.(3) are expressed .
Once again, the accurate numerical integration of these equations entails the same issues of the Cowell method but, averaging the resulting time derivative of the orbital elements over one or more orbital revolutions, it is possible to implement a fast and reliable technique for the estimation of deorbiting time.
In order to combine the first of Eqs.(3) with the atmospheric drag expression of Eq.(2), a
transformation from the RSW reference frame to the tangential, normal, omega orbital frame (NTW) is now necessary. This system is identified by the T axis, tangential to the orbit and aligned with the velocity vector, the N axis, normal to the velocity vector in the orbital plane, and the W axis, normal to the orbital plane.
After this reference frame change, the atmospheric drag, acting along the tangential direction, is considered the only non-zero acceleration component. Simplifying Eqs.(3), it is possible to find that inclination and right ascension of the ascending node (RAAN) do not change in time. Semi-major axis and eccentricity are, instead, the only two elements experiencing its secular influence (the argument of pericenter relation is periodic). Focusing on the semi-major axis change, the first of Eqs.(3), under these assumptions, reads :
where Cd, ρ, A and m are the same symbols introduced in Eq.(2).
By means of some simple substitutions for the relative velocity and the true anomaly rate of change,it is now possible to obtain an expression for the instantaneous semi-major axis change due to the atmospheric drag with respect to true anomaly:
The simultaneous integration of Eqs. (6) and (7) has to be carried out numerically. This has been done through the adaptive Lobatto quadrature of the atmospheric force over the eccentric anomaly . This quadrature is based on a four-points Gauss-Lobatto formula, i.e. a quadrature rule approximating a definite integral, by means of a weighted sum of function values at specified points within the domain of integration. Accordingly, a specific model for the atmospheric density variation with the orbital altitude, as described in the following section, is required for these integrations.
(TO BE CONTINUED)
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
Advanced Concepts Team