Figure 3.9: Controlled re-entry orbital parameters.
The previous parameters define the characteristics of the re-entry orbit, shown in Figure 3.9: it is an ellipse, with the perigee on the Earth (h = 0 km) and an apogee altitude at 250 km, leading to a flight path angle of −1.1◦. The ∆V budget for the re-entry manoeuvre is evaluated at the apogee of the re-entry orbit, according to the typical equation of astrodynamics, and it results about 75 m/s:
Vc= r µ RA = 7.755km/s (3.54) Vt= ht RA = 7.680km/s (3.55) ∆V = |Vc− Vt| = 75m/s (3.56)
The hybrid unit for the controlled re-entry will be determined through the graphs reported in section 2.3.4, which give the total mass of the propulsive unit, sized to provide the required ∆V , in function of the total mass of the de-orbited system at the end of the de-orbiting phase (so only inert mass is considered, while the propellant mass for the de-orbiting is excluded).
3.4
Model for multiple transfers optimization
This section is focused on the procedure implemented to determine the preliminary size of the propul- sive system needed for the space tug to accomplish multiple transfers in a single mission. Some basic assumptions were made. A removal trend of at least five objects per year was considered, being the trend suggested by previous researches [11] as the minimum number to make ADR effective. Since it is well known that orbital changes in inclination require high ∆V budgets, with consequent high propellant mass for the propulsion system, the manoeuvres were supposed to take place between objects orbiting in the same orbital band as regards the inclination; in the specific case, the procedure was implemented for debris in the SSO, where the inclination range is 99◦± 1◦. A chemical propulsion technology, based on liquid bipropellants N2O4/MMH, was considered for the space tug. This system was selected since it
Chapter 3
3.4. Model for multiple transfers optimization
appeared to be one of the better solutions, among those available in chemical propulsion, for the specific impulse and the inert lass fraction. Electric propulsion could be an alternative solution, advantageous for the propellant savings that implies thanks to its high specific impulses; however longer transfer times should be expected, increasing the mission duration and decreasing the annual removal rate. In our case, according to [57], the system was characterised by a specific impulse Isp of 300 s and an inert mass
fraction finert of 0.17. The total mass budget was estimated through a routine implemented in Matab.
The equations used were derived by the basic equations of chemical propulsion:
M R = eIspg0∆v (3.57) mprop mpay =(M R − 1)(1 − finert) (1 − finertM R) (3.58) finert= minert minert+ mprop (3.59)
where ∆V is the velocity required for the orbital transfer, mprop is the propellant mass, mpay is
the mass of the entire system of de-orbiting kits trasnfered to each debris. A safety margin of 10% was considered in the total mass budget. The procedure implemented was articulated in the following steps: once proper de-orbiting kits are determined for each target debris and the removal sequence is defined, and so all the ∆V required in each transfer, the set of equations is solved considering as payload mass the mass of the de-orbiting kits that are effectively transfered in each manoeuvre. Starting from the last trasnfer, mpay is the mass of the last de-orbiting kit that is released; known the ∆V the required
propellant mass for the manoeuvre is determined through equation 3.58 and then the inert mass through equation 3.59. The total mass is the sum of these two contributions. The procedure is then repeated backward, considering in each step the mass of the payload as:
(mpay)i= 1
X
i=4
[(mDK)5+ (mDK)i+ (mprop)i+1] (3.60)
The final mass of the propulsive system is the sum of the inert mass and the propellant mass obtained in each step.
In order to minimize the total mass odf the space tug, the sequqnce of orbital transfers should be optimized, which is translated in the minimization of the ∆V budget necessary to accomplish the entire mission. Since all the targets lied in the same SSO region, the altitude and inclination variations did not affect significantly the total ∆V ; more critical were the changes in terms of RAAN. Several sets of five objects were considered and a Matlab code was developed to determine the best sequence of multiple orbital transfers. This part of the activity was developed in collaboration with Prof. Lorenzini.
The structure of the procedure is schematically presented in Figure 3.10. The sequence should be com- pleted within one year. It was set, for each target, a time interval for each rendezvous and capture manoeuvre of about 15 days. The remaining time for the orbital transfers was then 290 days. In order to consider the advantage derived by the natural alignment of the RAAN, that could allowed significant propellant savings in the final mass budget, it was fixed a maximum time limit within the alignment should take place. If the time of the natural alignment is exceeded, an impulsive manoeuvre is foreseen. Considering that an impulsive manouvre is usually fast, and in the best case scenario where natural alignment is possible for all the debris, the maximum time allowed for this operation, for each debris, is 72 days. The total ∆V budget for each orbital sequence is the sum of three main components: 1) the ∆V for the RAAN alignment, when required; 2) the ∆V for the orbital transfer to approach the debris orbit; 3) the ∆V to rendezvous with the target debris. The total number of possible combination of five objects, and then the number of possible sequences, is 5!. The output of the code is the minimum ∆V budget, among all the possible combination evaluated. This value, and the single ∆V evaluated in the correspondent sequence of orbital transfers, are then used to estimate the mass budget for the space tug.
Chapter 3
3.4. Model for multiple transfers optimization
Figure 3.10: Scheme of the procedure implemented to determine the optimum sequence of orbital transfers that minimize the total ∆V budget, and so, the propellant mass required to de-orbit a set of five selected debris within 1 year. For each set of 5 debris, the total number of possible combination is 5!. For each sequence i (√seq.i in the figure) four orbital transfers are performed. For each orbital transfer j (√j in the figure) it is determined the time required for the RAAN alignment and the consequent ∆V , that is the sum of the ∆V for the RAAN alignment, if required, for the orbital transfer and the rendezvous maneouvre. The total ∆V budget for the i-th sequence is the sum of the ∆V determined for each transfer j, as well as the total maneouvre time is the sum of the time for the RAAN alignment, the otbital transfer and the rendezvous manoeuvre of each transfer j.