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## Background (fictional)

In the not so far future, space technology has become much more affordable and the number of objects in GEO have increased dramatically. The interest in launching new satellites for businesses has never been higher, but the risk of collision with space debris is also a larger threat. Any production of space debris has large repercussions as future legislation holds the business owners of a satellite responsible and forces them to pay for deorbiting of their debris before any damage to the GEO environment can occur.

So far, this new law was effective to prevent the worst case and encouraged some businesses to start paying for complete clean-ups of their assets. Other businesses however refused to pay the additional cost and left their satellites for long periods more or less uncontrolled in overcrowded graveyard orbits, in the hope that nothing bad would happen.

Because of this carelessness, it took not long for the first "rogue debris" to make their appearance and cause severe damage that could not be accounted to anybody. Whoever was responsible for their generation was either unaware of the fact or simply tried to escape the law. To prevent such incidents in the future, you have been appointed to a special task force of ESA whose mission is to track all the rogue debris and infer who is accountable for their generation.

## Finding the Origin

The data given to you describes the orbital elements of 100 space debris in GEO that have been discovered and tracked in recent years by ground based observation. It is clear that each of those debris must have originated from defunct satellites of which you have 100 trajectories traced over the last 30 years. Within this timespan, each debris must have detached from those satellites (i.e. at some point in the past the debris had the same position and velocity as its originator satellite).

As part of the taskforce, you need to determine two things for each of the debris given to you: What is the id of the defunct satellite and what is the value of the effective area over mass ratio $$C_r(A/m)$$ of the debris? In this expression, $$C_r$$ is the reflectivity coeffcient and $$A/m$$ is the area to mass ratio of the object in question. Knowing the effective area over mass ratio as accurately as possible is crucial: it allows you to send the correct invoice to the owner of the satellite and select an appropriate deorbiting technology to prevent further disaster.

Determing the exact value from ground is difficult, but you know that

$$-0.5 \leq \log_{10}\left[C_r(A/m)\right] \leq 1.8.$$

must hold for each debris, where in the above equation the effective area over mass ratio is in $$m^2/kg$$.

The orbits of the debris are subject to several forces which are outlined in the equations of motion below.

The final score of this challenge based on the fraction of correctly identified defunct satellite origins and the error in your estimation on the effective area over mass ratio. You will find further details on the Scoring and the Data provided in their corresponding sections.

## Getting started

To help you get over the first difficulties of this challenge, you may download and use the Python code which we have written for this challenge using the Heyoka integrator [1]. Our code encapsulates all equations of motion for this particular challenge and thus allows you to reliably propagate the orbits of all objects forward and backward in time. While we sincerely hope that this will be a useful resource to you, we give no warranties and what you do with it is completely up to you. If you prefer to write your code from scratch you can find all necessary equations and constants detailed below.

## References

[1] Biscani, Francesco, and Dario Izzo. "Revisiting high-order Taylor methods for astrodynamics and celestial mechanics." Monthly Notices of the Royal Astronomical Society 504.2 (2021): 2614-2628.

## Equations of motion

Let $$(X,Y,Z)$$ be EME2000 fixed frame Cartesian co-ordinates. The orbit of an object is governed by the following equations of motion:

$$\ddot{X} = f_{Kep,X}(X,Y,Z)+f_{J2,X}(X,Y,Z)+f_{C22,X}(X,Y,Z,t)+f_{S22,X}(X,Y,Z,t)\\ +f_{Moon,X}(X,Y,Z,t)+f_{Sun,X}(X,Y,Z,t)+f_{SRP,X}(X,Y,Z,t) \\ \ddot{Y} = f_{Kep,Y}(X,Y,Z)+f_{J2,Y}(X,Y,Z)+f_{C22,Y}(X,Y,Z,t)+f_{S22,Y}(X,Y,Z,t)\\ +f_{Moon,Y}(X,Y,Z,t)+f_{Sun,Y}(X,Y,Z,t)+f_{SRP,Y}(X,Y,Z,t) \\ \ddot{Z} = f_{Kep,Z}(X,Y,Z)+f_{J2,Z}(X,Y,Z)+f_{C22,Z}(X,Y,Z,t)+f_{S22,Z}(X,Y,Z,t)\\ +f_{Moon,Z}(X,Y,Z,t)+f_{Sun,Z}(X,Y,Z,t)+f_{SRP,Z}(X,Y,Z,t)$$

The terms in the right hand side of the above equation are given by:

$$\begin{array}{ll} f_{Kep,X}(X,Y,Z) &=& -{GM_E X\over (X^2+Y^2+Z^2)^{3/2}}\\ f_{Kep,Y}(X,Y,Z) &=& -{GM_E Y\over (X^2+Y^2+Z^2)^{3/2}}\\ f_{Kep,Z}(X,Y,Z) &=& -{GM_E Z\over (X^2+Y^2+Z^2)^{3/2}} \end{array}$$

### J2-terms

$$\begin{array}{ll} f_{J2,X}(X,Y,Z) &=& {GM_ER_E^2\sqrt{5}C_{20} X\over 2(X^2+Y^2+Z^2)^{1/2}} \left({3\over(X^2+Y^2+Z^2)^2}-{15Z^2\over(X^2+Y^2+Z^2)^3}\right) \\ f_{J2,Y}(X,Y,Z) &=& {GM_ER_E^2\sqrt{5}C_{20} Y\over 2(X^2+Y^2+Z^2)^{1/2}} \left({3\over(X^2+Y^2+Z^2)^2}-{15Z^2\over(X^2+Y^2+Z^2)^3}\right) \\ f_{J2,Z}(X,Y,Z) &=& {GM_ER_E^2\sqrt{5}C_{20} Z\over 2(X^2+Y^2+Z^2)^{1/2}} \left({9\over(X^2+Y^2+Z^2)^2}-{15Z^2\over(X^2+Y^2+Z^2)^3}\right) \end{array}$$

### C22-terms and S22-terms

$$\begin{array}{ll} f_{C22,X}(X,Y,Z,t) &=& f_{C22,x}(x,y,z)\cos(\theta_G+\nu_Et) - f_{C22,y}(x,y,z)\sin(\theta_G+\nu_Et) \\ f_{C22,Y}(X,Y,Z,t) &=& f_{C22,x}(x,y,z)\sin(\theta_G+\nu_E t) + f_{C22,y}(x,y,z)\cos(\theta_G+\nu_E t) \\ f_{C22,Z}(X,Y,Z,t) &=& f_{C22,z}(x,y,z)\\ f_{S22,X}(X,Y,Z,t) &=& f_{S22,x}(x,y,z)\cos(\theta_G+\nu_Et) - f_{S22,y}(x,y,z)\sin(\theta_G+\nu_Et) \\ f_{S22,Y}(X,Y,Z,t) &=& f_{S22,x}(x,y,z)\sin(\theta_G+\nu_E t) + f_{S22,y}(x,y,z)\cos(\theta_G+\nu_E t) \\ f_{S22,Z}(X,Y,Z,t) &=& f_{S22,z}(x,y,z) \end{array}$$

with

$$\begin{array}{ll} x&=&~X\cos(\theta_G+\nu_Et)+Y\sin(\theta_G+\nu_Et) \\ y&=&-X\sin(\theta_G+\nu_Et)+Y\cos(\theta_G+\nu_Et) \\ z&=&~Z \end{array}$$

and

$$\begin{array}{ll} f_{C22,x}(x,y,z) &=& {5GM_ER_E^2\sqrt{15}C_{22}x(y^2-x^2)\over 2(x^2+y^2+z^2)^{7/2}}+{GM_ER_E^2\sqrt{15}C_{22}x\over(x^2+y^2+z^2)^{5/2}} \\ f_{C22,y}(x,y,z) &=& {5GM_ER_E^2\sqrt{15}C_{22}y(y^2-x^2)\over 2(x^2+y^2+z^2)^{7/2}}-{GM_ER_E^2\sqrt{15}C_{22}y\over(x^2+y^2+z^2)^{5/2}} \\ f_{C22,z}(x,y,z) &=& {5GM_ER_E^2\sqrt{15}C_{22}z(y^2-x^2)\over 2(x^2+y^2+z^2)^{7/2}}\\ f_{S22,x}(x,y,z) &=& -{5GM_ER_E^2\sqrt{15}S_{22}x^2y\over (x^2+y^2+z^2)^{7/2}}+{GM_ER_E^2\sqrt{15}S_{22}y\over(x^2+y^2+z^2)^{5/2}} \\ f_{S22,y}(x,y,z) &=& -{5GM_ER_E^2\sqrt{15}S_{22}xy^2\over (x^2+y^2+z^2)^{7/2}}+{GM_ER_E^2\sqrt{15}S_{22}x\over(x^2+y^2+z^2)^{5/2}} \\ f_{S22,z}(x,y,z) &=& -{5GM_ER_E^2\sqrt{15}S_{22}xyz\over (x^2+y^2+z^2)^{7/2}} \end{array}$$

### Solar tide

$$\begin{array}{ll} f_{Sun,X}(X,Y,Z,t) &=&-G M_\odot\left( {(X-X_\odot)\over [(X-X_\odot)^2+(Y-Y_\odot)^2+(Z-Z_\odot)^2]^{3/2}} +{X_\odot\over(X_\odot^2+Y_\odot^2+Z_\odot^2)^{3/2}} \right)\\ f_{Sun,Y}(X,Y,Z,t) &=&-G M_\odot\left( {(Y-Y_\odot)\over [(X-X_\odot)^2+(Y-Y_\odot)^2+(Z-Z_\odot)^2]^{3/2}} +{Y_\odot\over(X_\odot^2+Y_\odot^2+Z_\odot^2)^{3/2}} \right)\\ f_{Sun,Z}(X,Y,Z,t) &=&-G M_\odot\left( {(Z-Z_\odot)\over [(X-X_\odot)^2+(Y-Y_\odot)^2+(Z-Z_\odot)^2]^{3/2}} +{Z_\odot\over(X_\odot^2+Y_\odot^2+Z_\odot^2)^{3/2}} \right) \end{array}$$

where

$$\begin{pmatrix} X_\odot\\ Y_\odot\\ Z_\odot \end{pmatrix} = \begin{pmatrix} r_\odot \cos \lambda_\odot\\ r_\odot \sin \lambda_\odot \cos \varepsilon\\ r_\odot \sin \lambda_\odot \sin \varepsilon \end{pmatrix}$$

with

$$\begin{array}{ll} \lambda_\odot &=& \Omega_\odot+\omega_\odot+\ell_{\odot}+\frac{\pi}{180}\left( {6892\over3600} \sin \ell_{\odot} + {72\over 3600} \sin 2\ell_{\odot}\right)\\ r_\odot[10^6\mbox{km}] &=&149.619-2.499 \cos \ell_{\odot} - 0.021 \cos 2\ell_{\odot}\\ \ell_{\odot} &=& \varphi_{\odot,0}+\nu_\odot t \end{array}$$

### Lunar tide

$$\begin{array}{ll} f_{Moon,X}(X,Y,Z,t) &=&-G M_{\mathcal M}\left({(X-X_{\mathcal M})\over [(X-X_{\mathcal M})^2+(Y-Y_{\mathcal M})^2+(Z-Z_{\mathcal M})^2]^{3/2}} +{X_{\mathcal M}\over(X_{\mathcal M}^2+Y_{\mathcal M}^2+Z_{\mathcal M}^2)^{3/2}} \right)\\ f_{Moon,Y}(X,Y,Z,t) &=&-G M_{\mathcal M}\left({(Y-Y_{\mathcal M})\over [(X-X_{\mathcal M})^2+(Y-Y_{\mathcal M})^2+(Z-Z_{\mathcal M})^2]^{3/2}} +{Y_{\mathcal M}\over(X_{\mathcal M}^2+Y_{\mathcal M}^2+Z_{\mathcal M}^2)^{3/2}} \right)\\ f_{Moon,Z}(X,Y,Z,t) &=&-G M_{\mathcal M}\left({(Z-Z_{\mathcal M})\over [(X-X_{\mathcal M})^2+(Y-Y_{\mathcal M})^2+(Z-Z_{\mathcal M})^2]^{3/2}} +{Z_{\mathcal M}\over(X_{\mathcal M}^2+Y_{\mathcal M}^2+Z_{\mathcal M}^2)^{3/2}} \right) \end{array}$$

where

$$\begin{pmatrix} X_{\mathcal M}\\ Y_{\mathcal M}\\ Z_{\mathcal M} \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\varepsilon & -\sin\varepsilon\\ 0 & \sin\varepsilon & \cos\varepsilon \end{pmatrix} \cdot \begin{pmatrix} r_{{\mathcal M}} \cos \lambda_{{\mathcal M}} \cos \beta_{{\mathcal M}}\\ r_{{\mathcal M}} \sin \lambda_{{\mathcal M}} \cos \beta_{{\mathcal M}}\\ r_{{\mathcal M}} \sin \beta_{{\mathcal M}} \end{pmatrix}$$

with

$$\begin{array}{ll} r_{{\mathcal M}}[\mbox{km}] &=& 385000-20905 \cos(l_{{{\mathcal M}}})-3699 \cos(2 D_{{{\mathcal M}}}-l_{{{\mathcal M}}})\nonumber\\ &-&2956 \cos(2 D_{{{\mathcal M}}})-570 \cos(2 l_{{{\mathcal M}}})\\ &+&246\cos(2 l_{{{\mathcal M}}}-2 D_{{{\mathcal M}}})-205 \cos(l'_{{{\mathcal M}}}-2 D_{{{\mathcal M}}})\nonumber\\ &-&171 \cos(l_{{{\mathcal M}}}+2 D_{{{\mathcal M}}})\nonumber\\ &-&152 \cos(l_{{{\mathcal M}}}+l'_{{{\mathcal M}}}-2 D_{{{\mathcal M}}}) \end{array}$$

$$\begin{array}{ll} \lambda_{\mathcal M} &= L_0+ \frac{\pi}{180} ( {22640\over 3600} \sin(l_{\mathcal M}) + {769\over 3600} \sin(2 l_{\mathcal M})\\ &- {4856\over 3600} \sin(l_{\mathcal M} - 2 D_{\mathcal M}) + {2370\over 3600} \sin(2 D_{\mathcal M})\\ &- {668\over 3600} \sin(l'_{\mathcal M}) - {412\over 3600} \sin(2 F_{\mathcal M})\\ &- {212\over 3600} \sin(2 l_{\mathcal M} - 2 D_{\mathcal M}) - {206\over 3600} \sin(l_{\mathcal M} + l'_{\mathcal M} - 2 D_{\mathcal M})\\ &+ {192\over 3600} \sin(l_{\mathcal M} + 2 D_{\mathcal M}) - {165\over 3600} \sin(l'_{\mathcal M} - 2 D_{\mathcal M})\\ &+ {148\over 3600} \sin(l_{\mathcal M} - l'_{\mathcal M}) - {125\over 3600} \sin(D_{\mathcal M})\\ &- {110\over 3600} \sin(l_{\mathcal M} + l'_{\mathcal M}) - {55\over 3600} \sin(2 F_{\mathcal M} - 2 D_{\mathcal M}) ) \end{array}$$

$$\begin{array}{ll} \beta_{{\mathcal M}} &=& \frac{\pi}{180} ( {18520\over 3600} \sin\left(F_{{\mathcal M}} + \lambda_{{\mathcal M}}-L_0 + \frac{\pi}{180} ( {412\over 3600} \sin(2F_{{\mathcal M}}) + {541\over 3600} \sin(l'_{{\mathcal M}})) \right)\\ &-& {526\over 3600} \sin(F_{{\mathcal M}} - 2D_{{\mathcal M}})\\ &+& {44\over 3600} \sin(l_{{\mathcal M}} + F_{{\mathcal M}} - 2D_{{\mathcal M}}) - {31\over 3600} \sin(-l_{{\mathcal M}} + F_{{\mathcal M}} - 2D_{{\mathcal M}})\\ &-& {25\over 3600} \sin(-2l_{{\mathcal M}} + F_{\mathcal M}) - {23\over 3600} \sin(l'_{\mathcal M} + F_{\mathcal M} - 2D_{\mathcal M})\\ &+& {21\over 3600} \sin(-l_{\mathcal M} + F_{\mathcal M}) + {11\over 3600} \sin(-l'_{\mathcal M} + F_{\mathcal M} - 2D_{\mathcal M}) ) \end{array}$$

and

$$\begin{array}{ll} \varphi_{M}&=&\nu_\odot t\\ \varphi_{M_a}&=&\nu_{M_a}t\\ \varphi_{M_p}&=&\nu_{M_p}t\\ \varphi_{M_S}&=&\nu_{M_s}t\\ L_0&=&\varphi_{M_p}+\varphi_{M_a}+ \frac{\pi}{180}(218.31617)\nonumber\\ l_{\mathcal M}&=&\varphi_{M_a}+ \frac{\pi}{180}(134.96292)\nonumber\\ l'_{\mathcal M}&=&\ell_{\odot}=\varphi_M+ \frac{\pi}{180} (357.5256)\\ F_{\mathcal M}&=&\varphi_{M_p}+\varphi_{M_a}+\varphi_{M_S}+ \frac{\pi}{180} (93.27283)\nonumber\\ D_{\mathcal M}&=&\varphi_{M_p}+\varphi_{M_a}-\varphi_{M}+ \frac{\pi}{180} (297.85027)\nonumber \end{array}$$

$$\begin{array}{ll} f_{SRP,X}(X,Y,Z,t) &=& C_r(A/m) {P_{SRP}a_\odot^2(X-X_\odot)\over [(X-X_\odot)^2+(Y-Y_\odot)^2+(Z-Z_\odot)^2]^{3/2}}\\ f_{SRP,Y}(X,Y,Z,t) &=& C_r(A/m) {P_{SRP}a_\odot^2(Y-Y_\odot)\over [(X-X_\odot)^2+(Y-Y_\odot)^2+(Z-Z_\odot)^2]^{3/2}}\\ f_{SRP,Z}(X,Y,Z,t) &=& C_r(A/m) {P_{SRP}a_\odot^2(Z-Z_\odot)\over [(X-X_\odot)^2+(Y-Y_\odot)^2+(Z-Z_\odot)^2]^{3/2}} \end{array}$$
where $$AOM$$ denotes the effective area over mass ratio.
$$\begin{array}{ll} GM_E &=&3.986004407799724\times 10^5 km^3 sec^{-2} \\ GM_\odot &=& 1.32712440018\times 10^{11}km^3 sec^{-2} \\ GM_{\mathcal M} &=&4.9028\times 10^{3}km^3 sec^{-2} \\ R_E &=&6378.1363 km\\ C_{20} &=& -4.84165371736\times 10^{-4}\\ C_{22} &=& 2.43914352398\times 10^{-6}\\ S_{22} &=&-1.40016683654\times 10^{-6}\\ \theta_G &=& \frac{\pi}{180} (280.4606)\\ \nu_E &=& \frac{\pi}{180} (4.178074622024230\times 10^{-3})\\ \nu_\odot &=& \frac{\pi}{180} (1.1407410259335311\times 10^{-5})\\ \nu_{M_a} &=& \frac{\pi}{180} (1.512151961904581\times 10^{-4})\\ \nu_{M_p} &=& \frac{\pi}{180} (1.2893925235125941\times 10^{-6})\\ \nu_{M_s} &=& \frac{\pi}{180} (6.128913003523574\times 10^{-7})\\ a_\odot &=& 1.49619 \times 10^8 km\\ \varepsilon&=& \frac{\pi}{180}(23.4392911)\\ \varphi_{\odot,0} &=& \frac{\pi}{180} (357.5256)\\ \Omega_\odot+\omega_\odot &=& \frac{\pi}{180} (282.94)\\ P_{SRP} &=&4.56\times 10^{-6} \end{array}$$