text:
\section{Introduction}
\label{sec:intro}
Short-arc orbit determination is a very important step when an
asteroid is first discovered. In these cases the timing is essential,
because we are interested in a rapid follow-up of a possible imminent
impactor, which is an asteroid impacting the Earth shortly after its
discovery, within the same apparition (interval of observability). The
observations are so few that the standard differential
correction procedure \citep{milani:orbdet} to find an orbit by a
least-squares minimization fails, and other methods need to
be used to extract information on the orbit of the object.

Several initial orbit computation methods have been developed in the
last 25 years. For instance, \cite{muinonen93} define a Gaussian
probability density on the orbital elements space using the Bayesian
inversion theory. In particular, they determine asteroid orbital
elements from optical astrometric observations using both a
  priori and a posteriori densities, the latter computed with
a Monte Carlo method.

The few observations in the short arc constrain the position of the
object in the sky, but they leave almost unknown the distance from the
observer (topocentric range) and the radial velocity (topocentric
range-rate). Thus, ranging methods have been developed over
the years to replace or refine the Monte Carlo approach in
the short arc orbit determination. There are two alternative
approaches to the ranging methods: the statistical and the systematic
ones.

The original statistical ranging method (\cite{virtanen2001},
\cite{muinonen2001}) starts from the selection of a pair of
astrometric observations. Then, the the topocentric ranges at
  the epoch of the observations are randomly sampled. Candidate
orbital elements are included in the sample of accepted elements if
the $\chi^2$ value between the observed and computed observations is
within a pre-defined threshold. \cite{oszkiewicz2009} improve the
statistical ranging using the Markov-Chain Monte Carlo (MCMC)
  to sample the phase space. The MCMC orbital ranging method is based
on a bi-variate Gaussian proposal PDF for the topocentric ranges. Then,
\cite{muinonen2016} develop a random-walk ranging method in
which the orbital-element space is uniformly sampled, up to a $\chi^2$
value, with the use of the MCMC method. The weights of each set of
orbital elements are based on a posteriori probability density value
and the MCMC rejection rate. They have developed this method for the
ESA Gaia mission, in the framework of Gaia alerts on potentially new
discovered objects by Gaia (see \cite{tanga2016}).

On the other side, \cite{chesley2005} and \cite{farnocchia2015}
introduce the so-called systematic ranging, which systematically
explores a raster in the topocentric range and range-rate space
$(\rho, \dot{\rho})$. This technique permits to describe the asteroid
orbital elements as a function of range and range-rate. Then, the
systematic ranging also allows one to determine the subset of the
sampling orbits which leads to an impact with the Earth.

In this paper we describe a new approach to the systematic ranging,
based on the knowledge of the Admissible Region (AR)
\citep{milani2004AR}, and a new method to scan the region. The
process has some main advantages if compared to the other methods
described above.

\begin{enumerate}
\item Our grid is more efficient, for two main reasons:
  \begin{itemize}
  \item[-] we discard all the objects that are not in the AR, saving
    CPU time and making the systematic ranging more accurate
      in finding the region in the $(\rho, \dot{\rho})$ space of the
      possible orbital solutions;
  \item[-] we use two different grids depending on the boundary of the
    AR. The first grid is larger and less dense, the second one is
    based on a refinement using the value of the post-fit $\chi^2$ of
    each point in the first grid (see Section~\ref{sec:adm_reg}).
  \end{itemize}
\item The computation of the probability for the potential impactors
  is a rigorous probability propagation from the astrometric error
  model, without any assumption of a priori probability density
  function on the range/range-rate space (see Section~\ref{sec:PDF}).
\end{enumerate}

In Section~\ref{sec:adm_reg} we summarize the features of the AR, and
describe the sampling of the $(\rho, \dot{\rho})$ space needed for the
creation of the Manifold Of Variations (MOV), an extension of the Line
Of Variations (LOV, \cite{milani2005multsol,milani:orbdet}). In
Section~\ref{sec:cobweb} we present the so-called spider-web
\citep{tommei:phd}, to be used when a nominal orbit is
available. Section~\ref{sec:PDF} describes how to compute the impact
probability when an impactor is found. In Section~\ref{sec:results} we
apply our method to the well known cases $2008~TC_3$ and $2014~AA$,
and we show other examples to explore the capabilities of the new
method. Then in Section~\ref{sec:gaia} we test our method on new
objects discovered by Gaia in the framework of the Gaia alerts, and we
show the points of strength of this approach applied to Gaia
observations. Section~\ref{sec:conclusions} contains our conclusions
and comments.

\section{Sampling of the topocentric range and range-rate space}
\label{sec:sampling}

\subsection{Admissible Region and systematic ranging}
\label{sec:adm_reg}
Even if the observations are too scarce, we can anyway
compute the right ascension $\alpha$, the declination $\delta$, and
their time derivatives $\dot{\alpha}$ and $\dot{\delta}$, by fitting
both angular coordinates as a function of time with a polynomial
model. This four quantities can be assembled together to form the
attributable \citep{milani2005}:
\begin{equation}\label{eq:attributable}
  \mathcal{A} = (\alpha, \delta, \dot{\alpha}, \dot{\delta}) \in
  \mathcal{S}^1 \times (-\pi/2, \pi/2) \times \mathbb{R}^2
\end{equation}
at a chosen time $\bar t$, which could be the time of the first
observation or the mean of the observation times. The information
contained in the attributable leaves completely unknown the
topocentric distance $\rho$ and the radial velocity $\dot{\rho}$. We
would have a full description of the topocentric position and velocity
of the asteroid in the attributable elements $(\alpha, \delta,
\dot{\alpha}, \dot{\delta}, \rho, \dot{\rho})$, if $\rho$ and
$\dot{\rho}$ were known. From now on, we will use
$\mathbf{x}=(\mathcal A, \bm{\rho})$, where
$\bm{\rho}=(\rho,\dot\rho)$, to describe the attributable elements.

Given an attributable, we can define the AR as the set of all the
possible couples $(\rho, \dot{\rho})$ satisfying the following
conditions (see \cite{milani2004AR} for more mathematical details):
\begin{enumerate}
  \item\label{cond:solsis} the object belongs to the Solar System, and
    it is not a too long period comet. We consider only the objects
    for which the value of the heliocentric energy is less than
    $-k^2/(2a_{max})$, where $a_{max}=100$~au and $k=0.01720209895$ is
    the Gauss' constant;
  \item\label{cond:earth} the corresponding object is not a satellite
    of the Earth, i.e. the orbit of the object has a non-negative
    geocentric energy when inside the sphere of influence of
      the Earth, whose radius is
\[
    R_{SI} = a_\Earth \sqrt[3]{\frac{\mu_\Earth}{3\mu_\Sun}} \simeq 0.010044\text{
      au}.
\]

\end{enumerate}
The AR is a compact set, and it can have at most two connected
components, which means that it could be represented as the union of
no more than two disjoint regions in the $(\rho,\dot{\rho})$
space. The AR has usually one component, and the case with two
components indicates the possibility for the object to be distant
(perihelion $q>28$ au). As shown in ~\cite{milani2004AR}, the number
of connected components depends on the number of the roots of a
polynomial resulting from condition number~\ref{cond:solsis}. This
polynomial cannot have more than three distinct real positive roots:
the AR has two connected components if the roots are
three, and it has one component if there is only one root. It is worth
noting that the region defined by condition number~\ref{cond:solsis}
could contain points with arbitrarily small values of $\rho$. The
boundary of the region given by condition~\ref{cond:earth} turns out
to have two different shapes: it can be formed just by the line of
geocentric energy equal to $0$ (if it is entirely contained in the
region $0<\rho<R_{SI}$), or by a segment of the straight vertical line
$\rho=R_{SI}$ and two arcs of the zero-curve of the geocentric energy
(for $0<\rho<R_{SI}$). We also discard the orbits corresponding to
meteors too small to be source of meteorites, using the condition
$H\leq H_{max}$, where $H_{max} = 34.5$ is the \textit{shooting star
  limit}~\citep{milani2004AR}, and $H$ is the absolute
magnitude. Given all these conditions, we can sample the AR with a
finite number of points.

If a nominal solution does not exist, we make use
of the systematic ranging to scan the AR. We sample the
AR in two different ways, depending on the number of connected
components, and the values of the roots ($r_1$, $r_2$ and $r_3$, in
ascending order). Table~\ref{tab:summary_AR} summarizes the conditions
and the grids used in the $(\rho,\dot{\rho})$ space. In particular, we
compute a rectangular grid in the range/range-rate space, with range
as in Table~\ref{tab:summary_AR}, and range-rate controlled by the AR
equations~\citep{milani2004AR}. Nevertheless, since the AR has a shape
dictated by a polynomial equation and it is not a rectangle, we check
the value of the heliocentric energy for each grid point, and we
discard those not satisfying condition~\ref{cond:solsis}. Orbits not
satisfying condition~\ref{cond:earth} are discarded as well, except
when we compute the probability for the asteroid to be a satellite of
the Earth\footnote{The object could be either an artificial satellite
  or an interplanetary orbit in a temporary Earth satellite
  capture~\citep{granvik2012}.}.

\begin{table}[h]
  \begin{center}
    \caption{The table shows different methods we use to sample
      the AR in the $(\rho,\dot{\rho})$ space with
      respect to the values of the roots, and the connected components
      of the AR. The sampling in $\dot\rho$ is always uniform.}
     \resizebox{\columnwidth}{!}{\begin{tabular}{l|c|c|l}
      \hline
      \multirow{2}*{\textbf{Roots}} & \textbf{AR} &  \multirow{2}*{\textbf{Grid}} & \multirow{2}*{\textbf{Sampling in} $\mathbf{\rho}$}\\
      & \textbf{components} &\\
      \hline
      $r_1 < \sqrt{10}$ au & 1 & $50\times 50$ & Unif. in $\log_{10}(\rho)$\\
      $r_1 \ge \sqrt{10}$ au & 1 & $50\times 50$ & Unif. in $\rho$\\
      $r_1 > 0$, $r_2 > 0$, $r_3 > 0$ & 2 & $100\times 100$ & Unif. in $\rho$\\
      \hline
    \end{tabular}}
    \label{tab:summary_AR}
  \end{center}
\end{table}

The target function is defined by
\[
    Q(\mathbf{x}) = \frac{1}{m} \bm{\xi}(\mathbf x)^T W \bm{\xi}(\mathbf x),
\]
where $\mathbf{x}=(\mathcal A,\bm\rho)$ are the fit parameters, $m$ is
the number of observations used in the least squares fit, $\bm{\xi}$
is the vector of the observed-computed debiased astrometric
residuals\footnote{In case there is a bias in the
  observations~\citep{farnocchia:fcct}, the residuals are computed
  following the classical definition of the residuals as
  observed-computed, and also subtracting the biases vector.}, and $W$
is the weight matrix. The choice of the weights for each observatory
is fundamental, and it has to take into account the debiasing of the
star catalog systematic errors, unless the astrometric reduction has
already been performed with an essentially bias-free star catalog,
e.g. the Gaia DR1~\citep{gaia_astrometry_2016}.

Given a subset $K$ of the AR, we define the Manifold Of
Variations $\mathcal M$ as the set of the points
$(\mathcal A^*(\bm\rho_0),\bm\rho_0)$ such that $\bm\rho_0\in K$ and
$\mathcal A^*(\bm\rho_0)$ is the local minimum of the function
$\left.Q\right|_{\bm\rho=\bm\rho_0}$. In addition, the value of the
minimum RMS of the residuals is less than a given threshold
$\Sigma$. In general, the MOV is a 2-dimensional manifold, such that
the differential of the map from the sampling space to $\mathcal M$
has rank $2$.

In the case of the systematic ranging, $K$ is the AR,
scanned with a regular semi-logarithmic or uniform grid. For each
sample point $\bm\rho_0=(\rho_0,\dot\rho_0)$ we fix $\rho=\rho_0$ and
$\dot\rho=\dot\rho_0$ in the target function, and then we search
$\mathcal A^*(\bm\rho_0)$ by means of an iterative procedure, the
doubly constrained differential corrections. The normal equation is
\[
    C_{\mathcal A} \Delta \mathcal A = D_{\mathcal A},
\]
where
\[
    C_{\mathcal A} = B_{\mathcal A}^T W B_{\mathcal A}\,,\quad
    D_{\mathcal A} = -B_{\mathcal A}^T W \bm{\xi}\,, \quad
    B_{\mathcal A} = \frac{\partial{\bm{\xi}}}{\partial{\mathcal A}}.
\]
We indicate as $K'$ the subset of $K$ on which the doubly constrained
differential corrections converges. In this way, the sampling of the
MOV is done over $K'$.

For each point $\mathbf{x}$ on the MOV, we also compute a $\chi$ value
\begin{equation}{\label{chimov}}
    \chi(\mathbf x) = \sqrt{m(Q(\mathbf x)- Q^*)},
\end{equation}
where $Q^*$ is the minimum value of the target function:
$Q(\mathbf{x}^*)$ if a reliable nominal solution exists, or the
minimum value of $Q(\mathbf{x})$ over $K'$ otherwise.

When a nominal solution does not exist, the systematic ranging is
performed by a two-step procedure. The first iteration is to compute a
grid following the conditions in Table~\ref{tab:summary_AR}. Once we
obtain a first preliminary grid, we densify for a higher
resolution. We select the minimum and the maximum value of $\rho$ and
$\dot{\rho}$ among all the values of the points for which we have a
convergence of the $4$-dimension differential correction, and the
value of $\chi$ is less than $5$. We also compute the score with
respect to the first grid, and we use the value to select the grid for
the second step. The score gives us a first insight into the
  nature of the object, even when the asteroid is not a potential
  impactor. We define the score as a probability of the object to
  belong to different classes (NEO, MBO, DO, and SO), where each class
  is defined by the following conditions:

\begin{itemize}
\item \textbf{NEO}: Near Earth Object, an object with $q < 1.3$ au,
  where $q$ is the perihelion distance (in au).
\item \textbf{MBO}: Main Belt Object, belonging either to the Main
  Belt or to the Jupiter Trojans. In particular we choose the
  conditions
  \[
    \left\{\begin{array}{ll} 1.7\text{ au} < a < 4.5\text{ au}\\ e<0.4\end{array}\right.
    \quad\text{or}\quad
    \left\{\begin{array}{ll} 4.5\text{ au} < a < 5.5\text{ au}\\ e<0.3\end{array}\right.
  \]
  where $a$ is the semi-major axis (in au), and $e$ is the
  eccentricity.
\item \textbf{DO}: Distant Object, characterized by $q > 28$ au
  (e.g. a Kuiper Belt Object (KBO)).
\item \textbf{SO}: Scattered Object, not belonging to any of the
  previous classes.
\end{itemize}

If the object is close to be a NEO (the NEO-score is more than
$50\%$), we use a uniform grid in $\log_{10}(\rho)$, otherwise we use
a uniform grid in $\rho$. Then we compute again the Manifold Of
Variations on the new and denser $100 \times 100$ grid.

\subsection{Spider web}
\label{sec:cobweb}
Let us suppose that a nominal orbit has been obtained by unconstrained
differential corrections, starting from a preliminary orbit as first
guess (for instance using the Gauss' method,
~\citep{milani:orbdet}). Then, we can use the nominal solution as the
center of the subset of the MOV we are interested in, and we can adopt
a different procedure to scan the AR. If a nominal orbit exists, and
the value of the geodesic curvature signal-to-noise ratio
(SNR)~\citep{milani2008} is greater than $3$, instead of using a grid
(Section~\ref{sec:adm_reg}), we compute a spider web sampling in a
neighborhood of the nominal solution \citep{tommei:phd}. This is
obtained by following the level curves of the quadratic approximation
of the target function used to minimize the RMS of the observational
residuals. The advantage of the use of the cobweb is that, firstly, it
is faster than the systematic ranging, and secondly it is more
accurate in the cases for which we have already a reliable nominal
solution.

Let $\mathbf{x}^*$ be the nominal solution with its uncertainty,
represented by the $6\times 6$ covariance matrix $\Gamma$. In a
neighborhood of $\mathbf{x}^*$, the target function can be well
approximated by means of the quadratic form defined by the normal
matrix $C=\Gamma^{-1}$. The matrix $C$ is positive definite, hence the
level curves of the target function are concentric 5-dimensional
ellipsoids in the 6-dimensional orbital elements space. The level
curves on the $(\rho,\dot\rho)$ space are represented by the marginal
ellipsoids, defined by the normal matrix
\[
C^{\bm\rho\bm\rho}=\Gamma_{\bm\rho\bm\rho}^{-1},
\]
where $\Gamma_{\bm\rho\bm\rho}$ is the restriction of $\Gamma$ to the
$(\rho,\dot\rho)$ space. To sample these curves we choose the maximum
value $\sigma_{max} = 5$ for the confidence parameter. Then, for each
level curve within the confidence level $\sigma_{max}$, we select the
points corresponding to some fixed directions. We initially create a
regular grid of points in the space of polar elliptic coordinates
$(R,\theta)$, where $0\leq \theta < 2\pi$ and $0\leq R \leq
\sigma_{max}$\footnote{We assume that the nominal solution
  corresponds to the point $(0,0)$.}. Then we apply to each point of
the grid the following transformation, depending on the covariance
matrix of the nominal orbit and on the orbit itself:
\begin{equation}\label{eq:cobweb}
  \begin{pmatrix}
    \rho \\
    \dot{\rho}
  \end{pmatrix}
  = R
  \begin{pmatrix}
    \sqrt{\lambda_1} \cos{\theta} & -\sqrt{\lambda_2} \sin{\theta}\\
    \sqrt{\lambda_2} \sin{\theta} & \phantom{-}\sqrt{\lambda_1} \cos{\theta}
  \end{pmatrix}
  \mathbf{v_1} +
  \begin{pmatrix}
    \rho^* \\
    \dot{\rho}^*
  \end{pmatrix},
\end{equation}
where $\lambda_1>\lambda_2$ are the eigenvalues of the $2 \times 2$
matrix $\Gamma_{\bm\rho\bm\rho}$, $\mathbf{v_1}$ is the eigenvector
corresponding to the greatest eigenvalue $\lambda_1$, and $\rho^*$ and
$\dot{\rho}^*$ are the range and range-rate values of the nominal
solution. Figure \ref{fig:cobweb} shows an example of spider web
sampling on the $(\rho,\dot\rho)$ plane.

\begin{figure}[h]
  \centering
  \includegraphics[width=\columnwidth]{cobweb.pdf}
  \caption{The figure shows an example of spider web around the
    nominal solution $\bm \rho^*=(\rho^*,\dot\rho^*)$. The points follow
    concentric ellipses, corresponding to different values of the
    parameter $R$. For each fixed direction (identified by $\theta$),
    there is one point on each level curve.}
  \label{fig:cobweb}
\end{figure}

\section{Probability density computation}
\label{sec:PDF}
We obtain a probability distribution on the sampling space to be used
for several applications, such as the computation of the impact
probability or the score. We begin assuming that the residuals are a
Gaussian random variable $\bm\Xi$, with zero mean and covariance
$\Gamma_{\bm\xi}=W^{-1}$. Hence the probability density function on
the residuals space is
\begin{align}\label{eq:res_gauss}
    p_{\bm\Xi}(\bm \xi) &= N(\bm 0,\Gamma_{\bm \xi})(\bm \xi)=
    \frac{\sqrt{\det W}}{(2\pi)^{m/2}}
    \exp\left(-\frac{mQ(\bm\xi)}{2}\right)=\nonumber\\ &=\frac{\sqrt{\det
        W}}{(2\pi)^{m/2}} \exp\left(-\frac 12 \bm\xi^TW\bm\xi\right)
\end{align}

A possible approach to propagate the density \eqref{eq:res_gauss} to
the sampling space uses the Bayesian theory to combine the density
coming from the residuals with a prior distribution. The {\em a
  posteriori} probability density function for $(\rho, \dot{\rho})$ is
given in \cite{muinonen93} as
\begin{displaymath}
  p_{post}(\rho, \dot{\rho}) \propto p(\bm{\xi}(\rho, \dot{\rho}))
  \cdot p_{prior}(\rho, \dot{\rho})
\end{displaymath}
where $p_{prior}$ is a prior distribution on the sampled
space. Hereinafter we report some possible choices for the prior
probability.

\begin{itemize}
\item \emph{Jeffreys' prior}. It has been used for the first time in
  \cite{muinonen2001}. It takes into account the partial derivatives
  of the vector of the residuals with respect to the coordinates
  $(\rho, \dot{\rho})$. Jeffreys' prior tends to favor orbits where
  the object is close to the observer, because of the sensitivity of
  the residuals for small topocentric distances. \cite{granvik2009}
  makes versions of the code (which utilizes Jeffreys' prior) publicly
  available for the first time.
\item \emph{Prior based on a population model}. This approach requires
  the computation of the prior distribution as posterior of another
  prior, which is selected as proportional to $\rho^2$ by geometric
  considerations on the Cartesian space of position and velocity.
\item \emph{Uniform distribution}. Uniform distribution in the
  $(\rho,\dot\rho)$ space.
\end{itemize}
\cite{farnocchia2015} gives a detailed description of all these
different possible choices, and they also analyze how the impact
probabilities change according to different prior distributions. They
conclude that the uniform distribution is a good choice for an a
  priori probability density function, because it represents a good
compromise between a simple approach and the identification of
potential impactors.

Hereinafter we propose a new method to propagate the probability
density function $p_{\bm\Xi}(\bm\xi)$ back to the sampling space. This
method is a rigorous propagation of the density function according to
the probability theory, and it does not use any a priori
assumption. The main idea is that we propagate the probability density
function from the residual space to the orbital element space (more
precisely, on the Manifold of Variations), and then to the sampling
space, according to the Gaussian random variable transformation
law. The main difference in using this approach instead of the uniform
distribution, is that we compute the Jacobian determinant of the
transformation and it is not equal to 1, which is the value chosen by
\cite{farnocchia2015}.

We define the following spaces:
\begin{itemize}
\item $S$ is the space of the sampling variables. It changes depending
  on the case we are considering: $S=\mathbb{R}^+ \times \mathbb{R}$ if the sampling
  is uniform in $\rho$, $S=\mathbb{R}^2$ if the sampling is uniform in
  $\log_{10}\rho$, and $S=\mathbb{R}^+ \times \mathcal S^1$ in the cobweb
  case.
\item $K'$ is the subset of the points of the AR such
  that the doubly constrained differential corrections give a point on
  the MOV;
\item $\mathcal M$ is the MOV, a 2-dimensional manifold in the
  six-dimensional orbital elements space $\mathcal{X}$;
\item $\mathbb{R}^m$ is the residuals space. The residuals are a function of
  the fit parameters: $\bm \xi=F(\bold x)$, with
  $F:\mathcal{X}\rightarrow \mathbb{R}^m$ differentiable, and we define the
  manifold of possible residuals as $V=F(\mathcal{X})$ \citep[Section
  5.7]{milani:orbdet}.
\end{itemize}
Without loss of generality, we can assume that
\begin{equation}\label{eq:res_norm_gauss}
    p_{\bm\Xi}(\bm \xi) = N(\bm 0,I_m)(\bm \xi)= \frac
    1{(2\pi)^{m/2}}\exp\left(-\frac 12 \bm\xi^T\bm\xi\right),
\end{equation}
where $I_m$ is the $m\times m$ identity matrix. As explained in
\citet[Section 5.7]{milani:orbdet}, this is obtained by using the
normalized residuals in place of the true residuals; biases due to
star catalog can also be removed while forming the normalized
residuals. With this technique, the probability density function
becomes normalized to a standard normal distribution. Thus from now
on, we will use $\bm\xi$ to indicate the normalized residuals, and the
function $F$ maps the orbital elements space to the normalized
residuals space.

Then we consider the following chain of maps (defined in
\ref{app:IPcomp})
\[
S \overset{f_\sigma}{\longrightarrow} \mathbb{R}^+\times \mathbb{R} \supseteq K'
\overset{f_\mu}{\longrightarrow} \mathcal{X} \supseteq \mathcal M
\xrightarrow{\left.F\right|_{\mathcal M}}V
\]
and we use their Jacobian matrices to compute the probability density
function on $S$. Let $\mathbf{s}$ be the variable of the sampling
space $S$, and let $\mathbf{S}$ be the corresponding random
variable. We use the compact notation $\chi^2(\mathbf{s})$ to indicate
$\chi^2(\mathbf{x}(\bm\rho(\mathbf{s})))$. The probability density
function of $\mathbf{S}$ is
\tiny
\begin{equation}\label{eq:p_S}
p_{\mathbf{S}}(\mathbf{s}) = \frac{\displaystyle
  \exp\left(-\frac{\chi^2(\mathbf{s})}{2}\right) \det
  M_\mu(\bm\rho(\mathbf{s}))\det M_\sigma(\bm\rho(\mathbf{s}))}
{\displaystyle\int_{f_{\sigma}^{-1}(K')}
  \exp\left(-\frac{\chi^2(\mathbf{s})}{2}\right)\det
  M_\mu(\bm\rho(\mathbf{s}))\det
  M_\sigma(\bm\rho(\mathbf{s}))\,d\mathbf{s}}
\end{equation}
\normalsize
where $M_\mu$ is the $2\times 2$ matrix associated to $f_\mu$
(considered as tangent map to the surface $\mathcal M$), and
$M_\sigma$ is the $2\times 2$ Jacobian matrix of $f_\sigma$. The
derivation of \eqref{eq:p_S} is given in \ref{app:IPcomp} (explicit
expressions for the Jacobian determinants are provided in
\ref{subsec:MOV-AR} and \ref{subsec:AR-S}, respectively).

It is worth noting that we limit our analysis to Solar System orbits
(condition number~\ref{cond:solsis} of Section~\ref{sec:adm_reg}),
because interstellar objects are very rare. As a consequence, we use a
Bayesian theory with a population limited to the Solar System, and all
the probability computations we describe are actually conditional
probabilities to the AR, which is even a subset of the Solar System.

\subsection{Impact probability computation}
Each point on the MOV can be thought as an orbit compatible with the
observations, and we call it Virtual Asteroid (VA). We propagate the
VAs into the future, currently for 30 days from the date of the
observations, and we search for Virtual Impactors (VIs), which are
connected sets of initial conditions leading to an impact
\citep{milani:clomon2}. If a VI has been found on the Modified Target
Plane (MTP, \cite{milani99}), it is associated to a subset $\mathcal
V\subseteq S$ of the sampling space, and hence its probability is
given by
\tiny
\begin{equation*}
\mathbb P(\mathcal V) = \int_{\mathcal V}
p_{\mathbf{S}}(\mathbf{s})\,d\mathbf{s}=
\end{equation*}
\begin{equation}\label{eq:IPcomp}
  =\frac{\displaystyle
    \int_{\mathcal V}\exp\left(-\frac{\chi^2(\mathbf{s})}{2}\right) \det
    M_\mu(\bm\rho(\mathbf{s}))\det
    M_\sigma(\bm\rho(\mathbf{s}))\,d\mathbf{s}}
  {\displaystyle\int_{f_{\sigma}^{-1}(K')}
    \exp\left(-\frac{\chi^2(\mathbf{s})}{2}\right)\det
    M_\mu(\bm\rho(\mathbf{s}))\det
    M_\sigma(\bm\rho(\mathbf{s}))\,d\mathbf{s}}.
\end{equation}
\normalsize
If for a given object we find impacting solutions, we assign to the
object an impact flag, which is an integer number related to the
computation of the impact probability. It depends on the impact
probability and on the arc curvature, as shown in
Table~\ref{tab:impact_flag}. An arc has significant curvature if
$\chi^2>10$, where $\chi$ is the chi-value of the geodesic curvature
and the acceleration (as defined in \cite{milani2007}). The impact
flag can take the integer values from $0$ to $4$: $0$ indicates a
negligible chance of collision with the Earth, whereas the maximum
value $4$ expresses an elevated impact risk ($\ge 1 \%$). It is
conceived as a simple and direct communication tool to assess the
importance of collision predictions, and to give the priority for the
follow-up activities.

\begin{table}[h]
  \begin{center}
    \caption{The table shows the conditions on the Impact Probability
      (IP) and on the arc quality to assign the impact flag to an
      object.} \small{
    \bigskip
    \begin{tabular}{c|l} \hline
           {\textbf{Impact flag}} & \textbf{Condition} \\
           \hline
           0 & $IP\le 10^{-6}$\\
           1 & $10^{-6} < IP \le 10^{-3}$\\
           2 & $10^{-3} < IP \le 10^{-2}$\\
           3 & $IP > 10^{-2}$ and no significant curvature\\
           4 & $IP > 10^{-2}$ and significant curvature\\
           \hline
    \end{tabular}
  }
    \label{tab:impact_flag}
  \end{center}
\end{table}

The goal for a system dedicated to imminent impactors is to detect all
the possible VIs down to a probability level of about $10^{-3}$,
called completeness level~\citep{delvigna2018}.
To reach the completeness level of $10^{-3}$ for our system, we can
neglect the terms which are of smaller order of magnitude in
equation~\eqref{eq:IPcomp}. This allows us to consider the VAs with a
$\chi$-value less than 5. If $\chi=5$ then $\exp(-\chi^2/2)\simeq
10^{-5.4}$, and the corresponding term is negligible. Moreover, the
choice $\chi<5$ is valid for the score computation, because we are
interested in a score accuracy of about $10^{-2}$.

\section{Results}
\label{sec:results}

We are setting up a service dedicated to the scan of the Minor Planet
Center NEO Confirmation
Page\footnote{\url{http://www.minorplanetcenter.net/iau/NEO/toconfirm_tabular.html}}
(NEOCP). The goal is to identify asteroids as NEOs, MBOs or distant
objects to be confirmed or removed from the NEOCP, and to give early
warning of imminent impactors, to trigger immediately follow-up
observations. The software used to produce these results is a new
version of the OrbFit Software version
5.0\footnote{\url{http://adams.dm.unipi.it/orbfit/}}.

The service involves the following steps, based on the algorithm
presented in Sections~\ref{sec:cobweb} and \ref{sec:PDF}.
\begin{itemize}
\item Scanning of the NEOCP every $2$ minutes. New cases or old cases
  just updated are immediately run.
\item Computation and sampling of the AR using a
  $2$-dimensional representation in the $(\rho, \dot{\rho})$ plane
  with a either grid or a spider web.
\item Computation of the MOV, obtaining a set of VAs.
\item Propagation of the VAs in the future (currently for 30 days).
\item Projection on the Modified Target Plane, searching for VIs.
\item If VIs exist, computation of the Impact Probability.
\item Computation of the score.
\end{itemize}

The time required to run one target strictly depend on the
characteristics of the object, but usually it is between $15$ and $20$
minutes. When predicting possible imminent impacts, one of the most
important requirements to fulfil is to minimize the number of
unjustified alarms. We mark as non-significant cases the objects for
which there are less than 3 observations or the arc length is
less than 30 minutes (see also Sec.~\ref{sec:gaia}), unless there
exists a nominal solution with a geodesic curvature SNR greater than
1. The classification of a case as non-significant does not
mean we skip the computation: we anyway perform all the steps of the
algorithm, and assign the score and the impact flag. Nevertheless,
being non-significant automatically decreases the priority of the
object in case of an alarm.

Unfortunately, these techniques are not enough to remove all the
spurious cases. They usually occur when the astrometry is either known
to be erroneous or noisy, or anyway not reliable. We cannot solve this
problem, and we acknowledge that the astrometric error models based on
large number statistic are not enough to distinguish erroneous and
accurate astrometry in a small sample (see comments in
Section~\ref{sec:conclusions}).

We test our algorithm on the two well known cases of NEAs that have
impacted the Earth few hours after the discovery, namely $2008~TC_{3}$
and $2014~AA$. We have already pointed out that the choice of the
weights is very important in these cases: to be able to compare the
results with \cite{farnocchia2015}, we choose the same
weights. Furthermore, we also select some cases among the objects that
won't impact to also show the importance of the score computation.

\subsection{Graphical representation of the results}
\label{sec:results_graphical}
We use plots showing the AR and its sampling to present
our results. Hereinafter we describe the color code present in our
figures. Concerning the AR, we make use of the following lines.
\begin{itemize}
  \item The red solid line represents the level curve of the
    heliocentric energy equal to $-k^2/(2a_{max})$. Namely, it is the
    outer boundary of the AR, corresponding to the boundary of the region
    defined by condition~\ref{cond:solsis} in Section~\ref{sec:adm_reg}.
  \item The green dashed line shows where the geocentric energy is
    equal to $0$, also taking into account the condition about the
    radius of the Earth sphere of influence, as discussed in
    Section~\ref{sec:adm_reg}
  \item The magenta dashed line (which is parallel to the range-rate
    axis) represents the shooting star limit condition.
  \item The magenta solid lines (which are parallel to the range-rate
    axis) represent different values of the absolute magnitude.
\end{itemize}
We now provide a description of the colors used for the sampling
points. No point is marked if the $4$-dimensional differential
corrections does not converge, because the point does not belong to
the MOV.
\begin{itemize}
  \item The dots are blue if $\chi \le 2$, and green if $2<\chi\le5$.
  \item The dots are black if $\chi > 5$.
  \item In case a VI has been found, we mark with red circles the
    points representing possible impacting orbits.
  \item The orange star represents the point with the minimum $\chi^2$
    value.
\end{itemize}

\subsection{Asteroid $2008~TC_{3}$}
$2008~TC_{3}$ has been discovered by Richard A. Kowalski at the
Catalina Sky Survey on October 7, 2008. The object was spotted 19
hours before the impact, and it is the first body to be observed and
tracked prior to falling on the Earth. After the discovery, hundreds
of astrometric observations were submitted to the Minor Planet
Center (MPC) and these observations allowed the computation of the
orbit and the prediction of the impact. We use the first tracklet
composed by $4$ observations, and then the first two tracklets ($7$
observations) to ascertain whether we could predict the impact.

We compute a uniform densified grid in $\log_{10}(\rho)$
(Fig.~\ref{fig:2008TC3_att}, top panel) where we consider only the
first $4$ observations, while we are able to compute a reliable
nominal orbit, and the consequent spider web using $7$ observations
(Fig.~\ref{fig:2008TC3_att}, down panel). Table~\ref{tab:ip} shows
that with $4$ observations and using the grid we are able to predict a
possible impact of the object with the Earth with an impact
probability of $\simeq 3.6\%$, and the score of the object to be
classified as a NEA is $100\%$. This would have produced an alert for
the observers that could have immediately followed-up the object. With
$7$ observations we can confirm the certainty that the asteroid is a
NEA (score $=100\%$), and the impact probability grows to $99.7\%$.

\begin{figure}[t!]
    \hspace{-0.3cm}
    \subfloat{%
      \includegraphics[width=0.47\textwidth]{2008TC3_att_4obs.pdf}%
    }%
    \hfill%
    \subfloat{%
      \includegraphics[width=0.48\textwidth]{2008TC3_att_zoom.pdf}%
    }%
    \caption{Grid sample of the $(\rho,\dot{\rho})$ space for the first
    $4$ observations (top panel) and spider web for the first $7$
    observations (down panel) of $2008~TC_3$.}
    \label{fig:2008TC3_att}
\end{figure}

\subsection{Asteroid $2014~AA$}
$2014~AA$ has been discovered by Richard A. Kowalski at the Catalina
Sky Survey on the new year's eve of $2014$. The object was discovered
21 hours before the impact, but it has not been followed-up as
$2008~TC_3$ because of the exceptional night in which it has been
spotted. We use first the first tracklet composed by $3$ observations,
and then the whole set of $7$ observations to test whether we could
have predicted the impact with our method.

\begin{figure}[h!]
    \hspace{-0.3cm}
    \subfloat{%
      \includegraphics[width=0.47\textwidth]{2014AA_att_3obs.pdf}%
    }%
    \hfill%
    \subfloat{%
      \includegraphics[width=0.48\textwidth]{2014AA_att_zoom.pdf}%
    }%
    \caption{Grid sample of the $(\rho,\dot{\rho})$ space for the first
      $3$ observations (top panel) and spider web for the whole set of
      $7$ observations (down panel) of $2014~AA$.}
    \label{fig:2014AA_att}
\end{figure}

\begin{table}[h!]
  \begin{center}
    \caption{The table shows the results of the systematic ranging
      applied to $2008~TC_3$ and $2014~AA$. The columns contain the
      name of the object, the number of observations used, the time
      span covered by the observations, the characteristic of the
      sampling used to compute the MOV (grid or spider web), the score
      of the object (NEA, MBA or Distant), and the impact probability
      (IP).}
    \resizebox{\columnwidth}{!}{\begin{tabular}{l|c|c|c|rrr|r}
        \hline
        \multirow{2}*{\textbf{Name}} & \multirow{2}*{\textbf{$\#$ Obs.}} & \textbf{Time span}& \textbf{Sampling}      &               &\textbf{Score}    &                  & \multirow{2}*{\textbf{IP}}     \\
        &                                                                      & \textbf{(min)}   & \textbf{Grid/Spider}   & \textbf{NEO}    &\textbf{MBO}  & \textbf{DO} &                         \\
        \hline
        $2008~TC_3$   &  4              &    43             &  $\log_{10}(\rho)$ -  grid &  $100\%$        &  0               &      0           &  $3.6\%$\\
        $2008~TC_3$   &  7              &    99             &  Spider                   &  $100\%$        &  0               &      0           &  $99.7\%$\\
        $2014~AA$     &  3              &    28             &  $\log_{10}(\rho)$ -  grid &  $100\%$        &  0               &      0           &  $3.0\%$\\
        $2014~AA$     &  7              &    28             &  Spider                   &  $100\%$        &  0               &      0           &  $100.0\%$\\
        \hline
    \end{tabular}}
    \label{tab:ip}
  \end{center}
\end{table}

These two examples have several analogies: we compute a uniform
densified grid in $\log_{10}(\rho)$ with the first tracklet, that is
only $3$ observations (Fig.~\ref{fig:2014AA_att}, top panel), and we
are able to compute a reliable orbit and the consequent spider web
only with $7$ observations (Fig.~\ref{fig:2014AA_att}, down panel).
Table~\ref{tab:ip} shows that using the first tracklet only, we are
able to predict a possible impact with the Earth with an impact
probability of $\simeq 3.0\%$, and the NEA score of the object is
$100\%$. Again, this would have produced an alert for the observers
that could have immediately followed-up the object. As in the previous
case, with the second tracklet we confirm both that the asteroid is a
NEA (score $=100\%$) and the collision, since the impact probability
grows up reaching the value of $100\%$.

\subsection{Asteroid $2014~QF_{433}$}
The previous examples show how the systematic ranging is capable of
identifying imminent impactors. Although this is one of the most
important applications of this technique, the systematic ranging is
also essential in the first short arc orbit determination
process.

Asteroid $2014~QF_{433}$ has been discovered by F51 - Pan-STARRS 1,
Haleakala on 2014 August 26. The first four observations have been
posted on the NEO Confirmation Page, with the temporary designation
TVPS7NV. It has been on the NEOCP until 2014 September 5. On that day
(with 18 observations) it has been confirmed to be a distant object by the
Minor Planet Center.

\begin{figure}[t!]
    \hspace{-0.3cm}
    \subfloat{%
      \includegraphics[width=0.47\textwidth]{TVPS7NV_att.pdf}%
    }%
    \hfill%
    \subfloat{%
      \includegraphics[width=0.48\textwidth]{TVPS7NV_AR_zoom.pdf}%
    }%
    \caption{Grid sample of the $(\rho,\dot\rho)$ space for the first 4
      observations of $2014~QF_{433}$ (top panel) and an enlargement of
      the second component (down panel). The black star in the
      second component represents the orbit of the object computed with
      all the available observations.}
    \label{fig:TVPS7NV}
\end{figure}

Figure~\ref{fig:TVPS7NV} shows the results of the systematic ranging
on this asteroid, with the four discovery observations only, and 51
minutes of arc length. In this case the AR has two connected
components, indicating the possibility for the object to be distant. The
values of the three positive roots of equation of degree $6$ are
$r_1=1.103$ au, $r_2=40.072$ au, and $r_3=59.786$ au. The
attributable is
\tiny
\[
\mathcal A = (\alpha,\delta,\dot\alpha,\dot\delta)=
(5.7358902, -0.3008327, -3.35275\cdot 10^{-4}, -9.94065\cdot 10^{-5}),
\]
\normalsize
with $\alpha$ and $\delta$ in radians and $\dot\alpha$ and
$\dot\delta$ in radians per day.  The two plots in
Figure~\ref{fig:TVPS7NV} clearly show that the object is distant,
since almost all the grid points corresponding to the MOV lie in the
second connected component. As a consequence, the cumulative score for
the Distant and Scattered classes is $99\%$.

As a further validation, we take the orbital elements of this asteroid
from the AstDyS database\footnote{Asteroid Dynamics Site, available at
  \url{http://hamilton.dm.unipi.it/astdys/}}, and we compute the range
and the range-rate at the epoch of the attributable. The result is
shown in the down panel of Figure~\ref{fig:TVPS7NV}: the black star
represents the orbit of $2014~QF_{433}$ computed with all the
available observations, and it is in perfect agreement with the
systematic ranging sampling.

\subsection{Asteroid $2017~AE_{21}$}
The case of $2017~AE_{21}$ shows the importance of the score
computation. An object is worthy of attention and has to be
followed-up even though it is not an impactor: for instance, it could
be a potential NEA.

Asteroid $2017~AE_{21}$ has been discovered by F51 - Pan-STARRS 1,
Haleakala on 2017, January 3. It appeared on the NEOCP as a tracklet
of 3 observations spanning 30 minutes, with the temporary designation
P10yBuc. It has been confirmed to be a NEA on 2017 January 24, when it
had 5 observations. With the first tracklet, our system produces an
impact flag of 2, indicating a modest impact risk, with an impact
probability $IP = 2\cdot 10^{-3}$. Moreover, the NEO score is $92\%$,
encouraging some follow-up to confirm. The top panel of
Figure~\ref{fig:P10yBuc} shows the result when using the first
tracklet only. We do not have any reliable nominal orbit to use, and
as a consequence we adopt the grid sampling. The portion of the grid
corresponding to low $\chi$ values (blue points) is quite wide,
indicating a great uncertainty in the orbit determination, and the
uncertainty region contains also impacting solutions.

With just two additional observations, the differential corrections
still fail in computing a reliable nominal orbit, but now the good
portion of the grid is located in a small subregion of the AR (see
Figure~\ref{fig:P10yBuc}, down panel). In this case the uncertainty
region does not contain impacting orbits, thus we get an impact flag
of 0, with $IP = 0$, whereas the NEO score grows to $100\%$. As a
consequence, the low probability VI has been contradicted by the new
observation, but the follow-up suggestion coming from the high $98\%$
NEO score of the first run was reliable.

\begin{figure}[t!]
    \hspace{-0.3cm}
    \subfloat{%
      \includegraphics[width=0.47\textwidth]{P10yBuc_3_att.pdf}%
    }%
    \hfill%
    \subfloat{%
      \includegraphics[width=0.48\textwidth]{P10yBuc_AR_zoom_5obs.pdf}%
    }%
    \caption{Grid sample of the $(\rho,\dot{\rho})$ space for the first
      $3$ observations (top panel) and for the whole set of $5$
      observations (down panel) of $2017~AE_{21}$. In both cases we do
      not have any reliable nominal orbit to use, and as a consequence
      we adopt the grid sampling.}
    \label{fig:P10yBuc}
\end{figure}

\subsection{NEOCP object $P10vxCt$}
As we stated in the introduction of this section, noisy astrometry can
be the cause for unjustified alarms. In fact, if an object has a
single tracklet of few observations and one of them is erroneous, the
arc usually shows a significant curvature, implying that the object
seems very close and fast moving. Most likely, it could be classified
as an immediate impactor with very high impact probability.

Object $P10vxCt$ has been spotted by F51 - Pan-STARRS, Haleakala on
2016 June 8. The first time it appeared on the NEOCP it had a tracklet
with 3 observations spanning about 44 minutes
(Table~\ref{tab:P10vxCt}, above). It has never been confirmed, but it
is anyway an important example to show the risk posed by noisy
astrometric data. With the first 3 observations, our system computes a
nominal solution compatible with a very close orbit, resulting in a
spider web sampling over a small subset of the AR (see
Fig.~\ref{fig:P10vxCt}, top panel). A very large fraction of the MOV
orbits are impacting solutions, and it results in an impact
probability of 99.2\% and impact flag 4, considering the significance
of the curvature. The second batch of observations consists of 4
positions: 3 of them are a remeasurement of the first tracklet
obtained from the discovery images of the object, plus an additional
observation. With this new astrometry, the impact has been ruled out
and the object has been removed from the NEOCP.

\begin{table}[t!]
  \begin{center}
    \caption{Astrometric data for NEOCP object $P10vxCt$.  First
      tracklet with 3 observations (above), and remeasurement of the
      first tracklet from the discovery images (below).}
    \vspace{0.5cm}
     \resizebox{\columnwidth}{!}{\begin{tabular}{l c c c l}
      \hline
      \textbf{Date (UTC)} & $\alpha$ & $\delta$ & $R$ & \textbf{Code} \\
      \hline
      2016-06-08.29327 & 13 13 16.962 & $-20$ 25 56.90 & 21.0 & F51\\
      2016-06-08.30357 & 13 13 12.688 & $-20$ 28 31.36 & 20.9 & F51\\
      2016-06-08.32416 & 13 13 04.699 & $-20$ 33 46.35 & 21.0 & F51\\
      \hline
    \end{tabular}
    \label{tab:P10vxCt}
     }\\
    \vspace{0.5cm}
    \resizebox{\columnwidth}{!}{\begin{tabular}{l c c c l}
      \hline
      \textbf{Date (UTC)} & $\alpha$ & $\delta$ & $R$ & \textbf{Code} \\
      \hline
      2016-06-08.293273 & 13 13 16.963 & $-20$ 25 56.53 & 20.3 & F51\\
      2016-06-08.303571 & 13 13 12.856 & $-20$ 28 33.24 & 20.5 & F51\\
      2016-06-08.324159 & 13 13 04.683 & $-20$ 33 46.33 & 20.4 & F51\\
      \hline
    \end{tabular}}
  \end{center}
\end{table}

\begin{figure}[t!]

    \subfloat{%
      \includegraphics[width=0.45\textwidth]{P10vxCt_3_zoom.pdf}%
    }%
    \hfill%
    \subfloat{%
      \includegraphics[width=0.44\textwidth]{P10vxCt_R_att.pdf}%
    }%
    \caption{Grid sample of the $(\rho,\dot{\rho})$ space for the first
      $3$ observations (top panel) and for their remeasurement (down
      panel) of $P10vxCt$.}
    \label{fig:P10vxCt}
\end{figure}

\begin{figure}[h]
  \centering
  \includegraphics[width=0.9\columnwidth]{P10vxCt_ra_dec.pdf}
  \caption{Plot of the two batches of observations reported in
    Table~\ref{tab:P10vxCt}, for NEOCP object $P10vxCt$. The red line
    represents the originally submitted tracklet, whereas the blue one
    the remeasured tracklet. The higher curvature of the first arc
    with respect to the second is clear.}
  \label{fig:curv_P10vxCt}
\end{figure}

To show the role of the remeasurements in the impact removal, we
consider the 3 remeasured observations only (see
Table~\ref{tab:P10vxCt}, below). The second observation in the first
tracklet was badly determined, being off by about $3$ arcsec from the
corresponding one in the second batch. The effect of this shift can be
seen in the curvature parameters $\kappa$ (geodesic curvature) and
$\dot\eta$ (acceleration). For the first tracklet we have
\tiny
\[
\kappa_1 = (0.0010073\pm 0.0001015)\quad\text{and}\quad \dot\eta_1
= (0.0003218\pm 0.0001013),
\]
\normalsize
whereas for the remeasured tracklet
\tiny
\[
\kappa_2 = (0.0000649\pm 0.0006749)\quad\text{and}\quad \dot\eta_2
= (-0.0000430\pm 0.0006750),
\]
\normalsize
both significantly lower than the first ones (see
Fig.~\ref{fig:curv_P10vxCt} for a graphic representation of the two
arcs). Moreover, as we can see from the curvature uncertainties, both
curvature components are not significantly different from $0$. As a
consequence, with the remeasured observations only, the impact
solution is sharply downgraded: a nominal solution cannot be computed
anymore, resulting in a grid sampling of the AR, and the impact orbits
are a very small fraction of the MOV orbits (see
Fig.~\ref{fig:P10vxCt}, right panel). Thus the impact probability
lowers to about $IP = 7.5\cdot 10^{-5}$, with an impact flag of 1.

Providing remeasured observations is not the only way to solve the
problem caused by bad astrometry. The second observation is not as
good as the other two, and let us suppose this information were
provided along with the observation itself. In this case, we could
have properly down-weighted the second observation to take into
account the additional information, and the case would have been
solved. To prove this claim, we assign a formal uncertainty of $3$
arcsec to both the right ascension and the declination of the second
observation. With this choice, the impact solution still remain, but
with an impact probability $IP=4.4\cdot 10^{-4}$. Until this
additional metadata will be provided together with the observations,
cases like the one presented here can be solved only by a manual
intervention after all the computations (remeasurement) or by a fast
follow-up (see Section~\ref{sec:conclusions} for general comments on
this issue).

\section{The ESA Gaia mission and short arc orbit determination}
\label{sec:gaia}
The ESA Gaia mission, currently surveying the sky from the Sun-Earth
$L_2$ Lagrangian Point, is providing astrometry of stars and
asteroids, at the sub-milliarcsec accuracy~\citep{prusti2012} down to
magnitude $V=20.7$. The spin of the satellite is $6$ h, and it
operates in a continuous scanning mode.  It has two lines of sight,
separated by an angle of $106.5^{\circ}$ in its scanning
direction. The continuous mode of observation implies targets are not
pointed at, but are rather passing in front of Gaia fields of
view. Such crossings are called transits. Over $5$ years of nominal
mission duration, the objects observed by Gaia will have a coverage of
$80-100$ observations for an average direction ($60-70$ for the
ecliptic~\citep{tanga2016})

Gaia focal plane is a large Giga-pixel array of 106 CCDs. The CCDs are
organized as follows.
\begin{itemize}
\item The first two CCD strips are devoted to the source
  detection. This is the instrument called Sky Mapper (SM).
\item The following 9 strips are astrometric CCDs (Astrometric Field,
  AF).
\item Other CCD strips are devoted to low resolution
  spectro-photometry (red and blue photometry, RP/BP) and high
  resolution spectroscopy (RVS), which is not considered for asteroid
  studies.
\end{itemize}

Each Solar System Object (SSO) transit is composed, at most, by 10
astrometric observations (AF and SM instruments), distributed
over 50 seconds.  The Data Processing and Analysis Consortium (DPAC)
has, as part of its activities, the source identification and further
processing done on the ground. In this context, the Coordination Unit
4 (CU4) performs the analysis of objects deserving a specific
treatment, as Solar System object (PI: P. Tanga). The DPAC CU4 has
implemented two pipelines for Solar System processing
~\citep{tanga2007, mignard2007}:

\begin{itemize}
\item SSO-ST is the Solar System short-term processing, devoted to
  provide a first, approximate orbit for the recovery of
  objects potentially discovered by Gaia. A ground-based
  follow-up network (Gaia-FUN-SSO,~\cite{thuillot2016}) is currently
  operating, realizing follow-up observations of Gaia
  potential discoveries from the ground;
\item SSO-LT is the Solar System long-term processing, which runs
  for the data releases, performing a more sophisticated data
  reduction with the best possible instrument calibration and
  astrometric solution.
\end{itemize}

\subsection{The short-term processing}
\label{sec:Gaia_ST}
The Solar System short-term processing is based on few transits of the
object (at least $3$), covering a time span of few hours (at
  least six). At now, the astrometry for the alerts is based on a
preliminary calibration, which is not the same used for the long-term
processing. As a result, the error model required by these
observations is not different from the one already used for the
ground-based cases with the best astrometry. We uniformly weight
($0.1$ arcsec) both right ascension and declination.

If the detected source is not successfully linked with a known solar
system object, then it is potentially a discovery. It is thus crucial
to predict a possible sample of orbits for the ground-based follow-up
network to certify the discovery. In the framework of the CU4 data
treatment, this is done by random-walk statistical ranging, which has
been developed by \cite{muinonen2016}. This is the Java code which is
currently producing the orbital data for Gaia alerts in the SSO-ST
pipeline.

We take the opportunity of the availability of the method presented in
Sec.~\ref{sec:sampling} to validate independently the results of the
SSO-ST pipeline.

The impact probability computation, which is a key feature when we
look for possible impactors, is not so essential when we have to deal
with Gaia short-term observations. We expect indeed that among the
objects that Gaia will discover, there will be a large fraction of
Main Belt Asteroids, few Near Earth Asteroids, and it would be very
surprising if it could discover even an imminent
impactors~\citep{carry2014_gfs}. On the other side, the use of the
double grid or of the cobweb is essential in this case.

\subsection{Results}
The same service presented in Section~\ref{sec:results} to scan the
NEO Confirmation Page, has been ran on possible Gaia
discoveries. The graphical representation of the results is identical
to the one given in Section~\ref{sec:results_graphical}. We also
use the same version of the OrbFit software cited in
Sec.~\ref{sec:results}, adapted to deal with a different input given
by Gaia observations.

All the alerts are available at \url{https://gaiafunsso.imcce.fr/},
and hereinafter we present two particular objects, that at the time
seemed to be possible Gaia discoveries, but then their observations
have been linked to ground-based observations submitted earlier in
time.

The first object that could have been discovered by Gaia on 2016,
December 29, is a Main Belt Asteroid. It has been identified as
g0T015. It has four Gaia transits which cover a time span of
$\simeq 16$ hours.

We apply the systematic ranging (using the two grids) on the
observations, and the results are summarized in
Fig.~\ref{fig:g0T015_AR}. The object has been then followed-up by the
Observatoire de Haute-Provence (OHP) for two consecutive nights (2017,
January 3-4), and the observations have been reported to the Minor
Planet Center.  The object has been recognized as a Main Belt Object,
and it obtained the provisional designation $2017~$AD$_{17}$. It could
have been the first Gaia discovery, if it hasn't been linked to few
observations from F51 Pan-STARSS submitted earlier (March
  2014).

\begin{figure}[t!]

    \subfloat{%
      \includegraphics[width=0.44\textwidth]{g0T015_AR.pdf}%
    }%
    \hfill%
    \subfloat{%
      \includegraphics[width=0.45\textwidth]{g0T015_AR_zoom.pdf}%
    }%
    \caption{Top panel: grid sample of the $(\rho,\dot{\rho})$ space
      for the Gaia object $g0T015$. Down panel: zoom of the second
      grid.}
    \label{fig:g0T015_AR}
\end{figure}

To be sure that the observed object was the same potentially
  discovered by Gaia, we consider as initial guess the elements
corresponding to the point with the minimum value of $\chi^2$, and we
perform an orbit determination process including the outlier
rejection procedure~\citep{milani:orbdet}.

\begin{center}
  \begin{figure}[h!]
    \includegraphics[width=\columnwidth]{g0T015_all_res.pdf}
    \caption{Residuals in right ascension and declination, in arcsec,
      obtained as a result of the orbit determination applied to the
      whole set of observations: Gaia (red points) and ground based
      (blue points) for the Gaia object $g0T015$ ($2017~$AD$_{17}$).}
    \label{fig:g0T015_res}
  \end{figure}
\end{center}

Figure~\ref{fig:g0T015_res} shows that follow-up observations from the
OHP, and Gaia observations match together, according to the orbit
selected as first guess. The residuals for the OHP (blue points) are
way larger than the ones obtained from Gaia observations (red points),
as expected. The weights used in this case are uniform in right
ascension (RA) and declination (DEC), and correspond to $0.1$
arcsec. The result also shows a high correlation (close to 1) between
RA and DEC, which is typical for Gaia observations due to its scanning
law.

The second Gaia object is $g1j0D7$, again a MBA. It has been observed by
Gaia on 2017, September 2. It has $7$ transits, which cover a time
span of $\sim 22$ hours. The object has follow-up observations from
the Abastuman Observatory (MPC code $119$) the night between September
10 and September 11. It obtained the MPC preliminary designation
$2017$~RW$_{16}$, but then it has been linked to the asteroid
$2006$~UL$_189$ discovered by the Catalina Sky Survey on June
  2005. Figure~\ref{fig:g1j0D7_AR} shows the result of the systematic
ranging when we use only Gaia observations. Again, we choose the point
with the minimum value of the $\chi^2$ as starting point, and we use
it as preliminary point. We then perform a differential corrections
least-squares fit, and we obtain the residuals (see
Fig.~\ref{fig:g1j0D7_res}), as described in the previous case.

\begin{figure}[t!]
    \hspace{-0.3cm}
    \subfloat{%
      \includegraphics[width=0.44\textwidth]{g1jD07_AR.pdf}%
    }%
    \hfill%
    \subfloat{%
      \includegraphics[width=0.45\textwidth]{g1jD07_AR_zoom.pdf}%
    }%
    \caption{Top panel: grid sample of the $(\rho,\dot{\rho})$ space
      for the Gaia object $g1j0D7$. Down panel: zoom of the second
      grid.}
    \label{fig:g1j0D7_AR}
\end{figure}

\begin{center}
  \begin{figure}[h!]
    \includegraphics[width=0.7\columnwidth]{g1j0D7_res.pdf}
    \caption{Residuals in right ascension and declination, in arcsec,
      obtained as a result of the orbit determination applied to the
      whole set of observations: Gaia (red points) and ground based
      (blue points) for the Gaia object $g1j0D7$.}
    \label{fig:g1j0D7_res}
  \end{figure}
\end{center}

\section{Future perspectives}

Gaia alerts runs daily, and they need a large effort to collect
follow-up observations from ground. With the method and the
  software described we have validated the already existing Java code
  written for the alert pipeline. Moreover, our approach can be also
  used to confirm the discoveries by computing an orbit using both
  Gaia and ground-based observations.

Then, when the accuracy for the short term will improve, we will also
be able to run the systematic ranging on less than three transits, given
the correct error model to the observations. Here it is worth noting
that the concept of short arc strongly depends on the concept of
curvature (see Sections~\ref{sec:cobweb} and \ref{sec:results}), which
is related not only to the time interval, but also to the accuracy of
the observations themselves.

The score computation represents a key feature in the Gaia frame (as
already pointed out in Sec.~\ref{sec:Gaia_ST}), but it will also be
very useful in future applications, like the ESA Euclid mission. Euclid
is an ESA mission with the aim of mapping the geometry of the dark
universe down to V$_{AB}$=24.5, with a launch scheduled for
  2020. While conducting its primary goal survey, Euclid will also
observe asteroids during his whole lifetime, and a Solar System
Working Group has been created within the Euclid consortium.

Euclid will observe at solar elongation $\sim 91^{\circ}$, and each
SSO will be imaged $16$ times (more precisely, $95\%$ of them $12$
times, and $50-60 \%$ of them $16$ times) over $67$
minutes~\citep{carry_euclid}. With a Hubble-like angular resolution,
astrometry will be quite good, at least at the same level of the best
observatories on ground (like Mauna Kea), or as the short term
accuracy of the Gaia alerts as it is now. The estimate accuracy is
around $100$ mas. While the southern sky will be repeatedly covered to
the same depth by LSST~\citep{lsst}, only Euclid will systematically
cover high declinations, with a strong potential for discoveries. For
each, having the score will be crucial to select the asteroid for
which trigger or not follow-up observations.

\section{Conclusions}
\label{sec:conclusions}

One of the main issue in the impact hazard assessment for imminent
impactors is given by the computation of the impact probability. The
main results of this article are a new algorithm to propagate the
probability density function from the space of the astrometric
residuals to the Manifold Of Variation, a geometric device to sample
the set of possible orbits, available even after a very short observed
arc.  In previous works, this computation was supported with an a
priori number density of asteroids. Our computation is complete,
rigorous, and uses no a priori hypothesis.

Does this new algorithm solve the problem of assessing the risk of
imminent impacts from a freshly discovered asteroid, with observations
limited to 1--2 tracklets? By using the AR and one of
our grid sampling, we have shown how to approximate a probability
integral on the portion of the MOV leading to an imminent impact, if
it is found. However, to accept this integral as Impact Probability we
need to check three conditions.

First, the probability density on the space of residuals needs to be
based upon a probabilistic model of the astrometric errors, taking
into account the past performances of the observatories. Second, the
observations used in the computation must be ``typical'' of the
observatory: even the best astronomical program produces a
comparatively small subset of ``faulty'' observations, with errors
much larger than the usual ones.  Third, we should assume that the
small sample of observations has statistical properties, such as mean
and standard deviation (STD), close to the ones of the full
distribution.

The first hypothesis is reasonable, in that a lot of work has been
developed in the last 20 years to produce astrometric error models for
asteroid observations (see \cite{carpino:outrej}, \cite{cbm10},
\cite{baer:err_mod}, \cite{farnocchia:fcct}). These models are not
perfect, but they represent an increasingly reliable source of
statistical information. The second hypothesis is not trivial: the
current format for asteroid observations does not contain sufficient
metadata to discriminate the ``weak observations'' from the good
ones. The full adoption of the new Astrometric Data Exchange Standard
(ADES\footnote{It is available at
  \url{http://minorplanetcenter.net/iau/info/IAU2015_ADES.pdf}.}),
approved by the IAU in 2015, will provide information such as SNR,
timing uncertainty, and so on, allowing to adapt the weighting of the
individual observations. The example of $P10vxCt$ shows how just one
lower quality observation can completely spoil the orbit results,
generating a false impact alarm. This can be avoided either with
remeasuring by the observer or by the orbit computer, provided such
down-weighting is supported by the metadata.

The third hypothesis is the most troublesome. Assuming that the
probability density of an astrometric error model is a perfect
statistical description, then by the law of large numbers a large
enough sample of $N$ observations shall have approximately the same
statistical properties of the model, with the differences going to
zero for $N\to +\infty$ (law of large numbers). Unfortunately,
$N=3,\ 4,\ 5$ is not large enough for the law of large numbers to
apply. For instance, a tracklet with $N=3$ observations can have all
the observations in one coordinate with errors $>2.5$ STD: this
statistical fluke would be very rare, occurring in a little more than
$1$ tracklet over $1$ million. Still, if a large asteroid survey
submits to the MPC more than one million tracklets per year, such a
fluke may occur about once a year, whereas the discovery of imminent
impactors is currently more rare (2 in 10 years). Detection of a rare
astronomical event cannot be a priori discriminated from rare
statistical events.

The tests on real cases discussed in this paper, and many more
from the NEOCP, convinced us that our algorithm
computes a reliable impact probability when the impact actually
occurs. Nevertheless, we cannot show that our algorithm is immune from
``false'' alarms. They are not false in the sense of a wrong
computation, or even worse a malicious disinformation, they are
statistical flukes which cannot be avoided because of lack of
information (hypothesis~2) and the need to use statistics on a small
sample (hypothesis~3). The question is what should be done to mitigate
the damage by these false alarms, given that we cannot avoid
disseminating them: otherwise, how could we disseminate the alarm in
the true case?

The only answer is to have a follow-up chain which does not waste
resources: the discoverers could themselves either remeasure or
follow-up on the short term, like 1 hour after discovery, the cases
announced as possible impactors.  Other telescopes should be available
to perform follow-up, to avoid improper use of survey telescopes for a
less demanding task. The ideal solution should be the availability of
a \textit{Wide Survey}, capable of covering the entire dark sky every
night and of detecting, e.g., an asteroid with absolute
magnitude $H=28$ at $0.03$ au distance (near opposition). Then the
same asteroid would be recovered by the survey the next day, before
the impact, and without the need for auxiliary follow-up.

\section*{acknowledgements}
This work has made use of data from the European Space Agency (ESA)
mission Gaia (\url{http://www.cosmos.esa.int/gaia}), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
\url{http://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for
the DPAC has been provided by national institutions, in particular the
institutions participating in the Gaia Multilateral Agreement.

Part of this work is based on observations made at Observatoire de
Haute Provence (CNRS), France. The authors would also like to
acknowledge all the observers involved in the detection of the
asteroids discovered by Gaia: for g0T015 they are E.~Saquet,
M.~Dennefeld, D.~Gravallon, and for g1j0D7 they are R.~Y.~Inasaridze,
V.~Ayvazian, T.~Mdzinarishvili, Y.~Krugly.

F.~Spoto acknowledges support by the CNES post--doctoral program, and
A.~Del Vigna acknowledges support by the company SpaceDyS.

redpajama_set_name: RedPajamaArXiv


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Many of the most unpleasant aspects of urban life are caused by cars, for both driver and the city itself. Large sweltering expanses of tarmac in cities contribute heavily to the urban heat island effect, whilst cars also become unbearably hot in summer sitting in these urban deserts.
Solar Tree is unique among anti-auto pollution designs in that it empathizes with the anthropomorphized automobile, but particularly with the electric car, providing a place for overworked, spent cars to regain their lost torque, to relax beneath the trees, to gain back their juice in an optimal stress-free environment.
DCF reveals the first model of the E_TREE to the Minister of Energy of China and Dutch Minster of Economic Affairs on the Day Of Sustainability.


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The Higgins Firm, PLLC, was established to be a full service law firm in Tennessee for the general public. It has been divided into four interwoven divisions to help serve its clients more efficiently and to capitalize on each lawyer's strengths: Civil Litigation, Workers Compensation, Employment Law and Probate. The three principal partners oversee these individual sections.
Jim Higgins
Jim is a 4th generation Nashvillian, and proud of it. He also comes from a long line of law. His father has been on the bench as a judge for the past 30+ years. Oddly enough, Jim's career in law began defending insurance companies. During his time there, he learned the tactics insurance companies use to try and prevent people from getting their fair share in settlements. However, those days are long gone. For the past several years he has exclusively represented the injured, and he's good at it. Mr. Higgins is a proud member of the Million Dollar Advocates Forum, which is an organization limited to attorneys who have won million and multi-million dollar verdicts and settlements for their clients. He graduated from the University of Tennessee at Knoxville and earned his Juris Doctor at the University of Memphis. He litigates serious injury cases all over Tennessee and throughout the country. He is even licensed in three states: Tennessee, Kentucky, and Georgia.
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Rick is a principal partner for the Serious Injury and Wrongful Death Litigation and the Probate and Estate Planning sections of the firm. He is primarily in charge of the firm's nursing home abuse and neglect and medical malpractice division. He has litigated cases across the southeast and regularly assists other law firms with these complex cases. He earned his undergraduate degree from the University of Tennessee and his Juris Doctor from DePaul University College of Law in Chicago, Illinois. In his free time, Rick is a spectacular rock climber and sometimes even takes trips to climb around the country.
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Mr. Miller received his B.A. from Vanderbilt University in Nashville, TN and his J.D. from Wake Forest University School of Law in Winston-Salem, NC. While in law school, Ben met his wife, Elizabeth, who is also a lawyer. In his free time he enjoys playing tennis.
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My daughter is attending American Academy of Art and one of the student groups she likes to go to is "character design." they come up with a list of potential character ideas and then have to create a character based on the name. This last week they had one called "octopus dragon," which piqued my interest and so I decided to try my hand at creating the creature.
Technical pencil on sketchbook paper.


Dubbed the "Big Easy" and "most unique city in America", New Orleans is an enchanting alchemy of soulful acoustics, hoodoo folklore, eccentric old world cookin' and outlandish traditions. New Orleans is the largest city in Louisiana and home to an estimate population of over 400,000 people. Attractions include Jackson Square, Bourbon Street and St. Louis Cathedral.
Allow yourself to surrender to the rugged voodoo rhythms and jazzy decadence of the Big Easy with cheap flights to New Orleans. The city is home to Louis Armstrong New Orleans International Airport (MSY), which is a 25-minute drive from the city.
No city does it better than New Orleans. Masquerade balls, decadent parties, extravagant parades and flamboyant feather-clad revellers continue this wild and colourful tradition that has endured since 1699.
Embark on a tour into the mysterious bayou and discover why its mystical murky depths have inspired songs and legends. Kayak through cypress groves or brave a night swamp tour.
New Orleans is the birthplace of jazz and home to the iconic French Quarter, a hub of juke joints, dive bars and blues clubs. Bourbon Street is where your night begins and your wildest fantasies come true.
New Orleans experiences a humid subtropical climate, with hot summers and mild winters. Temperatures range from 32 degrees Celsius in summer to 8 degrees Celsius in winter. The city experiences significant rainfall throughout the year. The best time to visit New Orleans is from February to May when the weather is balmy and major festivals such as Mardi Gras and the New Orleans Jazz & Heritage Festival are in full swing. December and January are great months to visit If you want to avoid the boisterous celebrations and high rates. Notable events include the Creole Tomato Festival (June), Essence Festival (July) and Voodoo Music + Art Experience (October). For cheap flights to New Orleans, the best time to visit is during summer (June to August). These months coincide with hurricane season.
New Orleans is a colourful quilt of sensational eccentricities. It is a city that inspires indulgence and celebrates life like no other place on earth.
The feverish soul of New Orleans is everywhere. It is in the sweet tang that hangs in the air and jazzy riffs that drift out of Frenchmen Street. It is etched into the faces of the locals and entwined in the shaggy roots of the bayou. It dances on the sidewalk and hunches over saxophones in smoky blues bars.
You will feel it prickle your skin as you wander down Bourbon Street and stroll through the bustling walkways of Jackson Square. You will see it hanging off the lacy Victorian façades in the French Quarter and historic tomb of the 'Voodoo Queen' Marie Laveau.
It will invite you to join in the wild revelry of Carnival and explore obscure attractions such as the Lafayette Cemetery, New Orleans' Historic Voodoo Museum and Old Absinthe House. And it will convince you to chase the velvety rhythms of jazz through the city streets on a streetcar.
New Orleans invites you to give in to its hoodoo magic and let the good times roll!
Compare several airlines and book your cheap flights to New Orleans online at Travelstart!
Although New Orleans is a fairly safe city, it is a sprawling city, so crime does happen. However, if you use common sense and take certain precautions, you should be able to avoid troublesome situations. Travel insurance to cover theft and medical expenses is recommended.
Smash and grab robberies and pick-pocketing does happen, so valuables should be left in the hotel safe. You should also avoid flashy displays of wealth and carrying large sums of cash. Avoid venturing out of the busy, well-lit areas at night and solo forays into isolated areas (particularly cemeteries). Use registered taxis to get around at night.
New Orleans is at risk for tropical storms. Follow news channels for updates on approaching storms and familiarise yourself with emergency protocols in the event of flooding and hurricanes.


text:
By Dejiki on January 8, 2021 in Attractions, Theme Parks, USS Park Updates
Universal Studios Singapore welcomes the Year of the Ox with its Usher In The Fun park programme.
Usher In The Fun at Universal Studios Singapore
From now till 28 February 2021, USS will be holding its Chinese New Year themed event. During this period, the park will be open 4 days a week, from Thursday to Sunday. An exception would be the Monday (15 Feb) right after the Chinese New Year public holidays (12-13 Feb).
As street shows have not resumed at USS, this year's programme is like a lighter version of the 2020 Chinese New Year event, which was already cut down from the last "full-fat" event in 2019.
Festive Meet and Greet for the New Year
Snap photos with Winnie and Woody Woodpecker (all decked out with lucky red costumes) at the new (or niu) stage at Hollywood zone!
You can also meet Gru and the Minions in their usual outfits – but with one of them wearing a Ox Head.
Po (Kung Fu Panda) is making a special return. He was retired from the park's line-up back in 2015.
Even though Singapore is now in Phase 3 of Re-opening, the safe distancing rule of photography still apply. You have to stand at a marker on the floor away from the characters, and also you will need to take the photo yourself – either a selfie or with a tripod.
Good news: More characters may appear together now, as seen over at the New York Public Library stage.
The Majestic Dragon Dance will not be taking its flight around the park, but you can still take photos with the dancers, who will appear along with Gru and the Minions or the friends from Sesame Street such as Elmo, Grover, Abby Cadabby, Cookie Monster, and Oscar the Grouch.
You can also collect free stickers during these 'Majestic Dragon & Friends' Meet and Greet experiences.
Megatron and Optimus Prime will be speaking some lines in Mandarin during their Meet and Greet this season.
The Mandarin-speaking Megatron has been a staple at Universal Studios Hollywood's Lunar New Year programming. The big difference is that Megatron there speaks 100% Mandarin during their event.
Other characters from Madgascar and Far Far Away zones will also be donning lucky outfits as part of the Chinese New Year event. Check out how they looked like from the 2019 event coverage.
January 2021 park updates
Festive decor around the park is quite similar to last year's. Apart from the ones at the park entrance, you'll find refreshed store display windows at the Universal Studios Store. These are all simple but "you can snap a photo with it".
Minions take focus again at the grand store window.
There's also Gudetama – the next yellow/ gold-coloured thing.
Along with the return of the Po Meet and Greet: merchandise from Kung Fu Panda film franchise.
The Minion Zodiac collection of goods that debuted last year is also back.
More attractions reopen
Shrek 4D Adventure and Lights, Camera, Action! have reopened in December 2020 – the park's busiest season last year. These show attractions were only operating on select days with limited capacity to comply with the prevailing safe management measures.
During the period when they were open, the RWS website had the following information for the attractions.
Only open on Public Holidays and select days. Please check the park map outside Universal Studios Singapore or enquire with a team member upon arrival for availability.
This has been removed and replaced with information that the attractions are "temporarily suspended until further notice".
It is likely that these show attractions have become seasonal and will only be open on days when the park has reached or is near its maximum capacity. Having more attractions open on those days could help control the crowds in other areas of the park.
SafeEntry Check-In QR codes can be seen around the queue line for Shrek 4D Adventure. This is also required for entry at the various indoor restaurants in the park.
Some restaurants have extra barriers installed indoors for safe distancing.
In Dec 2020, the park transformed the outdoor seating area beside Loui's NY Pizza Parlor into a charming al fresco dining area.
Do not be alarmed! This is not the queue for Battlestar Galactica. Rather, people are in line to redeem a free snack included with their park admission as part of a seasonal promotion. Last year, the park gave out face masks during its Halloween and Christmas events.
Universal Studios Singapore expansion
Madagascar: A Crate Adventure is still running, which could mean a lot of things… but the only thing for sure is that the park's expansion plans will be further delayed. The good news is the 'Nintendo Direct' video on Japan's Super Nintendo World has reaffirmed that Singapore will eventually have its own. Though no further details were provided, it would not be an exact copy of Japan's Super Nintendo World, though the Mario Kart: Bowser's Challenge ride is expected to be built in Singapore as the star attraction.
USS Annual and Season Passes: Sales/ Renewal still suspended
I'm sure some people are looking forward to be passholders again. While most attractions in Singapore have resumed their membership programmes, the sale and renewal of Annual/ Season Pass at USS are still suspended until further notice.
In the interim, if you plan to visit the park a few times this year, the park's new promotion might be attractive: 3 Visits for the Price of 2. You can either buy a $132 ticket direct from RWS that can be used for 3 visits (by 30 Sep 2021), or upgrade a one-day ticket before leaving the park on the day of your visit, if eligible. Check with the park crew near the park entrance (at Guest Services or Membership Lobby) for more details. Tickets purchased from official partners using SingapoRediscover vouchers may be eligible for upgrade as well.
Buying USS Tickets with SingapoRediscover vouchers
If you plan to use SingapoRediscover vouchers to buy tickets to Universal Studios Singapore, feel free to use my Klook link!
NOTE: Reservations Required for Tickets purchased from Official Partners
For all tickets purchased from official ticketing partners, making a reservation prior to visiting the park is required. Please go to the RWS Attractions Reservations website to do so.
Only tickets purchased directly from RWS do not require reservations.
attractions, chinese new year, event, photos:a7, resorts world sentosa, universal studios singapore, uss


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Mesut Ozil, Sead Kolasinac Survive Horror Attack by Knife-Wielding Carjackers on London Streets, Arsenal Duo Escapes Unhurt | WATCH VIDEO
Arsenal footballers Sead Kolasinac and Mesut Ozil fought off knife-wielding car-jackers in a terrifying ordeal in north London which was captured on video.
Updated: July 26, 2019 2:45 PM IST
By Aditya Chauhan Email
Edited by Aditya Chauhan Email
Arsenal footballers - Mesut Ozil, Sead Kolasinac attacked by knife-wielding carjackers on London street, duo escape unhurt_Tweet
Arsenal's star footballers Mesut Ozil and Sead Kolasinac were chased for a mile on the London streets before surviving a horror attack by 'knife-wielding car-jackers' in broad daylight on Thursday. Driving his black Mercedes G-Class through leafy Childs Hill, the star Arsenal midfielder and his teammate were ambushed by the robbers. Kolasinac, in particular, showed courage and can be seen coming out of the car and fought with the armed robbers with bare hands.Also Read - Liverpool Re-Open Training Ground After Shutdown Due to Covid-19
As Kolasinac chased the attackers, Ozil stopped at a nearby restaurant to file the complaint with the policemen. In the video which has gone viral on social media, the Premier League club's Bosnian defender Kolasinac is seen leaping from the vehicle to confront the masked aggressors, who had pulled up alongside the car on mopeds in the Golders Green district of the capital. Also Read - ISL 2021-22 Today Match Report: Bartholomew Ogbeche Scores Brace as Hyderabad FC Thrash Odisha FC 6-1
WATCH HERE: Also Read - Premier League Today Results: Manchester City, Arsenal, Tottenham Hotspur Register Big Wins
Mesut Ozil's car got jumped, Kolasinac jumped out and backed the beef. #YaGunnersYa pic.twitter.com/CxsJeiCvxw
— James (@smhjaames) July 25, 2019
Both attackers appeared to brandish knives at the 26-year-old Kolasinac. "We have been in contact with both players and they are fine," said a spokesman for Arsenal. Police confirmed that they had responded to reports of an attempted robbery.
"It was reported that suspects on motorbikes had attempted to rob a man who was driving a car," a spokesman for the Met Police was quoted by AFP.
"The driver, along with his passenger, managed to get away unharmed and travelled to a restaurant in Golders Green, where they were spoken to by officers.
"There have been no arrests. Enquiries continue."
Kolasinac and German midfielder Ozil are not the first footballers to be targeted on London's streets. In 2016, Andy Carroll, a former West Ham player, was threatened at gunpoint on his way home from training.
For breaking news and live news updates, like us on Facebook or follow us on Twitter and Instagram. Read more on Latest Sports News on India.com.
arsenalArsenal FCKnife attack LondonLatest Football Newsmesut ozilSead Kolasinac
Published Date: July 26, 2019 2:44 PM IST
Updated Date: July 26, 2019 2:45 PM IST


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