#### Timeline

The competition is over.

Design $$n$$ missions able to cumulatively remove all the 123 orbiting debris defined in the file debris.csv. One mission is a multiple-rendezvous spacecraft trajectory where a subset of size $$N$$ of the 123 orbiting debris is removed by the delivery and activation of $$N$$ de-orbit packages. The following cost function has to be minimized: $$J = \sum_{i=1}^n C_i = \sum_{i=1}^n \left[c_i + \alpha \left(m_{0_i} - m_{dry}\right)^2\right]$$ where $$C_i$$ is the cost charged by the contracted launcher supplier for the $$i$$-th mission and it is composed of a base cost $$c_i$$ (increasing linearly during the competition time frame) and a term $$\alpha \left(m_{0_i} - m_{dry}\right)^2$$ favouring a lighter spacecraft. At the beginning of the $$i$$-th mission $$m_{0_i}$$ is the spacecraft mass and $$m_{dry}$$ its dry mass. The $$\alpha$$ parameter is set to be $$\alpha = 2.0 \cdot 10^{-6}$$ [MEUR / Kg$$^2$$]: each Kg of launch mass saved results in a discount over the mission cost (but also in less $$\Delta V$$ capability).

The $$i$$-th mission starting epoch is denoted with $$t_{i}^s$$ and its end epoch with $$t_i^f$$. A mission starts with a launch delivering, at $$t_i^s$$, one spacecraft at a chosen debris and ends when all the $$N$$ de-orbit packages on-board have been delivered and activated. An orbiting debris is considered as removed if: a) its position and velocity at some epoch $$t$$ coincides with the spacecraft position and velocity vector and b) for the following $$t_w \ge 5$$ days the spacecraft stays in proximity of the debris while delivering and activating a de-orbit package of mass $$m_{de} = 30$$ [kg].

After, the spacecraft is free to ignite its propulsion system again and leave towards the next debris (note that only in-between debris transfers the spacecraft is subject to the full $$J_2$$ perturbation and its dynamics is described by the equations of motion. During the removal operations (i.e. for a time $$t_w$$) the position and velocity of the spacecraft will, instead, be considered to be those of the debris as computed by the ephemerides)

The basic cost $$c_i$$ of each mission (i.e. not including the $$\alpha$$ term), increases linearly during the competition month and is computed as follows: $$c_i = c_m + \frac{t_{submission} - t_{start}}{t_{end} - t_{start}} (c_M - c_m)$$ where $$t_{submission}$$ is the epoch at which the $$i$$-th mission is validated (via the Kelvins web-site) and $$t_{end}$$ and $$t_{start}$$ are the end and the beginning epochs of the GTOC9 competition. The exact value of $$c_i$$ will be visible at all time on the website. The minimal basic cost $$c_m$$ is $$45$$ MEUR, while the maximum cost $$c_M$$ is $$55$$ MEUR.

Each orbiting debris that is not removed by any of the missions submitted by one team will be considered, at the end of the competition, removed by a dedicated launch costing 55.0018 MEUR


## Orbital Manoeuvres

The only manoeuvres allowed to control the spacecraft trajectory are instantaneous changes of the spacecraft velocity (its magnitude being denoted by $$\Delta V$$. After each such manoeuvre, the spacecraft mass is to be updated using Tsiolkovsky equation: $$m_f = m_i \exp \left(-\frac{\Delta V}{v_e}\right),$$ where $$v_e = I_{sp} g_0$$. A maximum of 5 impulsive velocity changes is allowed during each transfer between two successive debris. These do not include the departure and arrival impulse.

## The Spacecraft

Each spacecraft initial mass $$m_0$$ is the sum of its dry mass, the weights of the $$N \ge 1$$ de-orbit packages to be used and the propellant mass: $$m_0 = m_{dry} + Nm_{de} + m_p$$. All spacecraft have a dry mass of $$m_{dry} = 2000$$ [kg] and a maximum initial propellant mass of $$m_p = 5000$$ [kg] (less propellant may be used, in which case the launch costs will decrease). Each de-orbit package has a weight of $$m_{de} = 30$$ [kg]

## Operational Constraints

The debris removal operations during each of the multiple-rendezvous trajectories are complex and demand some control over the schedule of the debris visits:

• The overall time between two successive debris rendezvous, within the same mission, must not exceed 30 days. So that if the arrival epoch to the debris $$\mathcal D_a$$is $$t_{\mathcal D_a}$$ and the arrival to the next debris $$\mathcal D_b$$ is $$t_{\mathcal D_b}$$, then $$t_{\mathcal D_b}-t_{\mathcal D_a} \le 30$$ [days].

• In order to avoid operating different missions in parallel, a time of at least 30 days must be accounted for between any two missions so that if $$t_i^s > t_j^s$$ then $$t_j^f + 30 \le t_i^s$$ [days], $$\forall i \neq j$$.

• All mission events (arrivals, departures, etc..) must take place in an allowed window, so that indicating with $$t_{event}$$ the epoch of any event, then $$23467 \le t_{event} \le 26419$$ [MJD2000].

• The osculating orbital pericenter $$r_p$$ cannot be smaller than $$r_{m} = 6600000$$ [m] (this is only checked at each event, not in-between)