Author(s): Jerôme Berthier, Jean-Marc Petit

Solar System objects exhibit apparent motions that are generally much faster than those of typical stars. As a consequence, SSOs could not be included in the Initial Gaia Source List (see Section 2.2.3), nor identified by the global (IDT) or final (IDU) cross-match processing (see Section 2.4.9). Known and unknown SSOs are thus found among unmatched Gaia detections.

It must be noted that, as time passes, the inventory of known SSOs is growing. Since the time of launch of Gaia, more than one hundred thousand new asteroids have been discovered, bringing the total number of known SSOs to more than 734 000 in mid 2017. In order to link daily SSO detections by Gaia (called transits) to an SSO population in constant evolution, a database of pre-computed ephemerides of all known SSOs is regularly built by CU4/SSO (see Section 1.2.2). Time-dependent Gaia-centric positions of SSOs are arranged by using a HEALPix spatial index (Górski et al. 2005), a grid resolution ${N}_{side}={2}^{10}$, and a time resolution adapted to each object. They are stored into an Apache Cassandra database (https://cassandra.apache.org/). During each processing cycle (described in this Section 4.4), packets of transits have been cross-matched with known SSOs extracted from the database. For this purpose, the pairs {HEALPix, Epoch} of each transit have been used to extract a sample of zero to few tens of SSO candidates. Their Gaia-centric accurate positions at the transit epoch have then been re-computed by means of $n$-body numerical integrations, providing equatorial coordinates which can be directly compared to measured transit coordinates.

The first criterion of candidate selection relies on the accuracy of SSO orbits. For each target, the ephemeris uncertainty at the epoch of each transit is computed based on the 1-$\sigma $ RMS (${\sigma}_{o}$) of its orbit. A candidate is retained if its current ephemeris uncertainty (CEU) satisfies the condition:

$$ | (4.1) |

where $t$ is the observation epoch of transit, ${t}_{0}$ is the reference epoch of the orbital elements, $\dot{{\sigma}_{o}}$ is the rate of change of ${\sigma}_{o}$, both provided by the Asteroid Orbital Elements Database (ASTORB) maintained and made publicly available at the Lowell Observatory web site, and $\u03f5$ is a given threshold. The adopted value $\u03f5=10$ ${}^{\mathrm{\prime \prime}}$ leads to discarding all SSOs having uncertain orbits, which could lead to spurious identifications.

The second criterion takes into account the relative positions of SSO candidates compared to the recorded transit position, as described by Pineau et al. (2011) in their cross-correlation algorithm. The SSO position is projected into a 2D plane centred on the transit position, so that the relative coordinates of the SSO are $x=d,y=0$ where $d$ is the angular distance between the two sources calculated by the Haversine function:

$$d=2\mathrm{arcsin}\sqrt{{\mathrm{sin}}^{2}\frac{{\delta}_{s}-{\delta}_{t}}{2}+{\mathrm{sin}}^{2}\frac{{\alpha}_{s}-{\alpha}_{t}}{2}\mathrm{cos}{\delta}_{t}\mathrm{cos}{\delta}_{s}}$$ | (4.2) |

where ${\alpha}_{t},{\delta}_{t}$ and ${\alpha}_{s},{\delta}_{s}$ are the equatorial coordinates of the transit and the SSO candidate respectively. In this plane, a SSO candidate is retained if its coordinates satisfy the condition:

$$\frac{d}{{\sigma}_{{x}_{c}}\sqrt{1-{({\rho}_{c}{\sigma}_{{x}_{c}}{\sigma}_{{y}_{c}})}^{2}}}\le k$$ | (4.3) |

where $k=3.43935$ is the 2D completeness value for a 3-$\sigma $ criterion (e.g. 99.7%), and where ${\sigma}_{{x}_{c}}=\sqrt{{\sigma}_{{x}_{t}}^{2}+{\sigma}_{{x}_{s}}^{2}}$ and ${\rho}_{c}{\sigma}_{{x}_{c}}{\sigma}_{{y}_{c}}={\rho}_{t}{\sigma}_{{x}_{t}}{\sigma}_{{y}_{t}}+{\rho}_{s}{\sigma}_{{x}_{s}}{\sigma}_{{y}_{s}}$ represent the errors on the positions of the transit ($t$) and the SSO candidate ($s$) expressed by their variance-covariance matrix, assuming Gaussian errors. For SSOs, the positional error is taken as the current ephemeris uncertainty, e.g., ${\sigma}_{{\alpha}_{s}}={\sigma}_{{\delta}_{s}}=$CEU. For transits, the positional error is fixed to $0.5$ ${}^{\mathrm{\prime \prime}}$, as no formal error is known at this step of the processing.

A third criterion based on the difference of magnitudes between the observed transit and SSO candidates might be used in principle to discriminate between different candidates. Nevertheless, the uncertainty on the predicted apparent magnitudes of many SSOs, which can reach in some cases values of the order of 1 magnitude, or even more, limits the scope of this criterion. The reason of this large uncertainty is due to the fact that the magnitude of an SSO at any given epoch depends on the phase angle, namely the angle between the directions to the Sun and to the observer as seen from the object, and on the absolute magnitude of the SSO, namely the magnitude corresponding to zero phase angle (ideal opposition) and unit distance from the Sun and the observer. Unfortunately, SSO absolute magnitudes listed in catalogues are often affected by very large errors, and also the variation of magnitude as a function of phase angle is rarely known with good accuracy. In most cases, finally, also the shape of the object and its rotation period are not known, producing a further magnitude uncertainty due to the periodic modulation of the visible cross section of the object produced by its rotation. Moreover, a magnitude-based criterion could also possibly lead to failed identifications of satellites of some SSOs, for which the difference in magnitude can reach several magnitudes. Therefore, no magnitude-based criterion was used in the Gaia DR2.

In the case that more than one SSO candidate satisfies the identification criteria, there is no obvious method to identify the correct object at this step of the processing. In Gaia DR2 the best candidate has been selected by calculating the quadratic distances between each observed transit and the corresponding SSO candidates, and in the case of more than one possible choice, the object minimizing the above distance was selected. Fortunately, with a grid resolution of $3.44$ ${}^{\prime}$(${N}_{side}={2}^{10}$), this scenario rarely occurs because the likelihood that two SSOs share the same pixel at the same time is close to zero.

The validation of the identification process has shown that the rate of correct identifications is very close to $100\%$, with an uncertainty of less than $1\%$. This mainly comes from uncertainties on the positions of some SSO candidates, and from the presence of some unfiltered contaminants. This will progressively disappear in the forthcoming Gaia data releases. It is therefore expected that the performances of the CU4/SSO processing chain will tend to improve with time.