Systematic ranging and late warning asteroid impacts
We describe systematic ranging, an orbit determination technique especially suitable to assess the near-term Earth impact hazard posed by newly discovered asteroids. For these late warning cases, the time interval covered by the observations is generally short, perhaps a few hours or even less, which leads to severe degeneracies in the orbit estimation process. The systematic ranging approach gets around these degeneracies by performing a raster scan in the poorly-constrained space of topocentric range and range rate, while the plane of sky position and motion are directly tied to the recorded observations. This scan allows us to identify regions corresponding to collision solutions, as well as potential impact times and locations. From the probability distribution of the observation errors, we obtain a probability distribution in the orbital space and then estimate the probability of an Earth impact. We show how this technique is effective for a number of examples, including 2008~TC$_3$ and 2014~AA, the only two asteroids to date discovered prior to impact.
sys_ran_arxiv.tex
%\documentclass[preprint,authoryear,12pt]{elsarticle}

\documentclass[authoryear,preprint,review,12pt]{elsarticle_arxiv}

\usepackage{amssymb,amsmath}
\usepackage{bm}
\journal{Icarus}

\begin{document}

\begin{frontmatter}

  \title{Systematic ranging and late warning asteroid impacts}

  \author[jpl]{D. Farnocchia}
  \ead{Davide.Farnocchia@jpl.nasa.gov}
  \author[jpl]{S.~R. Chesley}
  %\author[ontario]{P.~G. Brown ???}
  %\author[jpl]{P.~W. Chodas ???}
  \author[esa,spacedys,iaps]{M. Micheli}
  \address[jpl]{Jet Propulsion Laboratory, California Institute of
    Technology, Pasadena, CA 91109, USA}
  %\address[ontario]{University of Western Ontario, London, Ontario N6A
  %  3K7, Canada}
  \address[esa]{ESA NEO Coordination Centre, 00044 Frascati (RM), Italy}
  \address[spacedys]{SpaceDyS s.r.l., 56023 Cascina (PI), Italy}
  \address[iaps]{INAF-IAPS, 00133 Roma (RM), Italy}


\begin{abstract}
  We describe systematic ranging, an orbit determination technique
  especially suitable to assess the near-term Earth impact hazard
  posed by newly discovered asteroids. For these late warning cases,
  the time interval covered by the observations is generally short,
  perhaps a few hours or even less, which leads to severe degeneracies
  in the orbit estimation process. The systematic ranging approach
  gets around these degeneracies by performing a raster scan in the
  poorly-constrained space of topocentric range and range rate, while
  the plane of sky position and motion are directly tied to the
  recorded observations. This scan allows us to identify regions
  corresponding to collision solutions, as well as potential impact
  times and locations.  From the probability distribution of the
  observation errors, we obtain a probability distribution in the
  orbital space and then estimate the probability of an Earth
  impact. We show how this technique is effective for a number of
  examples, including 2008~TC$_3$ and 2014~AA, the only two asteroids
  to date discovered prior to impact.
\end{abstract}

\begin{keyword}
  Asteroids \sep Asteroids, dynamics \sep Astrometry \sep Near-Earth
  objects \sep Orbit determination
\end{keyword}

\end{frontmatter}

\section{Introduction}
When an asteroid is first discovered, the few available observations
rarely allow the determination of its orbit. Surveys usually capture
3--5 repeat images of the same area of sky with a typical return time
of 15--30 minutes. Over this time interval, an object in the field of
view moves with respect to background stars at a roughly linear
rate. Thus, the short arc of available data, also called a tracklet
\citep{kubica}, provides an estimate of the angular position and
motion of the object in the plane of sky, which is not enough to
determine a six-parameter orbit \citep{milani05}.

Despite the degeneracies in the orbit estimation process, it is
important to recognize potentially hazardous objects shortly after
discovery. In fact, the only two objects discovered prior to an Earth
impact, namely 2008~TC$_3$ and 2014~AA, were both discovered by
R. Kowalski \citep{mpec_tc3, mpec_aa} of the Catalina Sky Survey
\citep{catalina} only about 20 hours before striking the Earth. On one
hand, 2008~TC$_3$ was quickly recognized as potential impactor and so
it was extensively observed before the impact. The acquired  observations
allowed the estimation of the orbit of 2008~TC$_3$ as well as some
physical characterization \citep{jenniskens_nat, kozubal,
  scheirich}. Using the available data, it was possible to predict the
impact that took place above Sudan on 2008 October
7\footnote{http://neo.jpl.nasa.gov/news/2008tc3.html}
\citep{jenniskens_nat}. On the other hand, 2014~AA was not immediately
recognized as a possible impactor and so only seven astrometric
observations over about 70 minutes were obtained before the object
fell into the Atlantic Ocean on 2014 January
2\footnote{http://www.nasa.gov/jpl/asteroid/first-2014-asteroid-20140102}
\citep{chesley_ieee}.

When the standard differential correction procedure
\citep[e.g.,][]{farnocchia_ast4, orbdet} to find a least-squares orbit
fails, other methods can be used to assess the orbital probability
distribution. In particular, \citet{bayes} show how to put a
probability density on the phase space of the orbital elements by
using Bayesian inversion theory. By using a Monte Carlo approach one
can sample the orbit phase space and thereby derive the probability
distribution from the observation errors corresponding to the sampled
orbit.

The available observations directly constrain the position and motion
of the asteroid in the sky while the distance between the asteroid and
the observer (topocentric range) is poorly constrained. Thus, ranging
methods are the preferred Monte Carlo approach when only a short arc
of observations is available. \citet{stat_ran} describe a method
called statistical ranging that allows one to generate orbital samples
compatible with the observational data. After choosing two
observations, the topocentric ranges at the epochs of the two
observations are randomly sampled and a corresponding orbit
computed. \citet{mcmc} rely on the statistical ranging technique and
use Markov chains to generate an unbiased sequence of orbital samples
distributed according to the probability distribution coming from
Bayesian inversion theory.

\citet{sys_ran_steve} introduces a technique called systematic
ranging\footnote{This technique was actually introduced by Tholen and
  Whiteley in 2002, but the paper was never published.}, which in
contrast to Monte Carlo techniques systematically explores a raster in
the topocentric range and range-rate space. This technique provides a
geometric description of the orbital elements as a function of range
and range rate. Moreover, systematic ranging allows one to
identify regions of the phase space filled with impact solutions and
the corresponding impact times and locations. In this paper we present
a detailed description of systematic ranging and show how to
derive a probability distribution on the range and range-rate space,
which is then mapped to the orbital element space where 
impact probability estimates can be derived.

\section{Systematic ranging}
Systematic ranging relies on the fact that a short arc of observations
yields a direct estimate of the plane of sky position (right ascension
$\alpha$ and declination $\delta$) and motion ($\dot\alpha$ and
$\dot\delta$). These four scalar quantities can be assembled together
in the so-called attributable
${\cal A} = (\alpha, \delta, \dot\alpha, \dot\delta)$
\citep{milani05}. The topocentric range $\rho$ and $\dot\rho$ are only
marginally constrained, if at all. If $\rho$ and $\dot\rho$ were
known, we would have a full description of the asteroid's topocentric
position and velocity in polar coordinates
$(\alpha, \delta, \dot\alpha, \dot\delta, \rho, \dot\rho)$, which can
be easily converted to a Cartesian heliocentric state if the position
and velocity of the observer are known. Note that, to account for
light-time correction, we correct the epoch of the Cartesian state
from that of the attributable by a quantity $\rho/c$, where $c$ is the
speed of light.

To explore the orbit phase space, systematic ranging scans a
suitably dense grid in the $(\rho,\dot\rho)$ space. Such grid needs to
be large enough to contain the reasonably possible orbital configurations. In
particular, the grid must contain the so-called admissible region
\citep{milani04}, i.e., the values of $(\rho, \dot\rho)$ leading to
bounded heliocentric orbits. For each grid point we fix the values of
$\rho = \rho_i$ and $\dot\rho = \dot\rho_j$ and find the best fit
value of the attributable ${\cal A}_{ij}$ that minimizes the cost function:
\begin{equation}\label{e:target}
Q = \bm\nu^T W \bm \nu
\end{equation}
where $\bm\nu$ is the vector of Observed $-$ Computed astrometric
residuals and $W$ is the weight matrix
\citep{farnocchia_ast4}. 

The constrained best-fitting solution can easily be converted to an
orbit, which is in turn propagated to find upcoming Earth
encounters. Moreover, if the observations contain photometric
measurements we also compute the absolute magnitude for
each grid point.

\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{2014aa_ecc}
\includegraphics[width=0.5\textwidth]{2014aa_q}}
\centerline{\includegraphics[width=0.5\textwidth]{2014aa_in}
\includegraphics[width=0.5\textwidth]{2014aa_H}}
%\centerline{\includegraphics[width=0.5\textwidth]{2014aa_dca}
%\includegraphics[width=0.5\textwidth]{2014aa_rms}}
\caption{Eccentricity, perihelion distance, inclination, and absolute
  magnitude as a function of range and range rate for asteroid
  2014~AA. The dashed curve corresponds to parabolic orbits and the
  dash-dotted line to grazing impacts.}\label{f:2014aa}
\end{figure}

Figure~\ref{f:2014aa} shows the application of systematic ranging
to 2014~AA by using the seven astrometric observations obtained by the
Catalina Sky Survey prior to impact. The orbital elements are shown as
a function of $(\rho, \dot\rho)$. The dashed curve encloses bounded
orbits and corresponds to the admissible region of \citet{milani04}.
The dash-dotted line is for grazing impacts: the region on the left of
the curve contains impacting solutions. %The normalized RMS, i.e.,
%$\sqrt{Q/2n}$ where $n$ is the number of astrometric observations,
%indicates what region of the $(\rho,\dot\rho)$ space is more likely
%and strongly suggests that 2014~AA is on a collision course. The
%countour levels of the normalized RMS clearly reveal the nonlinearity
%of the orbit determination process.

\section{Probability distribution on the range \& range rate
  space}\label{s:rrd_dist}
The raster scan presented in the previous section gives a geometric
description of how the orbital configuration depends on the value of
topocentric range and range rate. The next step is to assign a
probability distribution to this space.

%\subsection{Mapping the observation error distribution}
As is common practice, given a set of optical observations we assume
that the observation errors $\bm\nu$ are normally distributed
according to a weight matrix $W$, i.e., their probability density
$f_{err}$ is:
\[
f_{err}(\bm\nu) \propto \exp(-0.5\ \bm\nu^T\,W\,\bm\nu)\ .
\]
It is typical to use a diagonal weight matrix where the individual
weights are $1/\sigma^2$ ($\sigma$ is the uncertainty in the
astrometric positions). Table~\ref{t:weights} shows the data weights
we currently use for the most productive discovery and follow-up
stations. The chosen weights are based on our experience and account
for the fact that discovery observations can sometimes be problematic
because they are not targeted on the object.

%% Tim's advice
%%691 -> 0.7"
%%H21 -> 0.7"
%%W84 -> 0.2"

\begin{table}[t]
\begin{center}
\begin{tabular}{|cc|cc|cc|}
  \hline
  Station & $\sigma_{\alpha,\delta}$ & Station &
                                                 $\sigma_{\alpha,\delta}$
  & Station & $\sigma_{\alpha,\delta}$\\
  \hline 
%! CBM Station-Catalog specific rules
  703 & 1.00$''$ & G96 & 0.50$''$ & F51 & 0.20$''$ \\
  950 & 0.50$''$ & 291 & 0.70$''$ & 691 & 0.70$''$ \\
  568 & 0.15$''$ & H01 & 0.30$''$ & H21 & 0.70$''$ \\
  J04 & 0.40$''$ & W84 &  0.20$''$ & F65, E10 & 0.40$''$\\
  W85, W86, W87 & 0.60$''$ & Q63, Q64, V37 & 0.80$''$ &  K91, K92, K93
  & 0.80$''$\\
\hline
\end{tabular}
\end{center}
\caption{Station-specific weighting rules for the most productive
  discovery and short-term follow-up stations. For all the other
  stations we use weights at 1$''$.}
\label{t:weights}
\end{table}

In principle, it is possible to consider the photometric residuals
together with the astrometric ones. However, photometric measurements
have a much larger uncertainty than that of the astrometry and can be
affected by significant biases \citep{sdss}. Moreover, in case of a
magnitude trend, it is hard to tell if the varying luminosity is due
to a rapidly changing topocentric distance rather than the asteroid's
rotation. Therefore, we do not use the information obtained from the
photometric residuals.

According to Bayesian inversion theory \citep{bayes}, the
posterior probability density function for
$(\rho,\dot\rho)$ is:
\[
f_{post}(\rho,\dot\rho) \propto f_{err}(\bm\nu(\rho, \dot\rho))\,
f_{prior}(\rho,\dot\rho)\ ,
\]
where $f_{prior}$ is a prior distribution on the $(\rho, \dot\rho)$
space.\footnote{It is often convenient to use a logarithmic scale for
  $\rho$ to achieve a better resolution at small topocentric
  distances. In that case the probability density has to be multiplied
  by $\rho$.} The selection of $f_{prior}$ is to some extent arbitrary
and far from trivial \citep{prior}, and yet can significantly affect
the posterior probability distribution, especially for short
observation arcs.

Whatever the choice of $f_{prior}$, we always add a crude constraint
from the population model by setting $f_{prior} = 0$ for hyperbolic
orbits, flagging objects where the astrometric errors favor an unbounded
orbit.

\subsection{Jeffreys' prior}
A mathematically sound choice is Jeffreys' prior \citep{openorb}:
\begin{equation}\label{e:jeffreys}
f_{prior}(\rho, \dot\rho) = \sqrt{\det\left(\frac{\partial
  \bm\nu}{\partial (\rho, \dot\rho)}^T\  W \ \frac{\partial
  \bm\nu}{\partial (\rho, \dot\rho)}\right)}\ .
\end{equation}
Note that the partials are the total derivatives of $\bm \nu$ with
respect to $(\rho, \dot\rho)$, which means that they account for
the fact that the best-fit attributable ${\cal A}$ changes as a
function of $(\rho, \dot\rho)$.

Among other things, Jeffreys' prior secures the invariance of the
probability distribution when changing variables. However, since there
is more sensitivity of the residuals for small topocentric distances,
Jeffreys' prior tends to favor orbital configurations where the object
is close to the observer, e.g., see the examples of 2015~CV and
2015~BU$_{92}$ in Sec.~\ref{s:examples}.

\subsection{Prior based on population model}
As already discussed by \citet{stat_ran}, the prior could be based on
the known asteroid population. Even better, one could use an orbit and
size population model where the known population is extrapolated to
remove survey selection effects and to reach completeness up to a
given apparent magnitude.  One such model is the Pan-STARRS Synthetic
Solar System Model \citep{grav11}, and new population models are
expected in the future \citep{granvik_neo}. From the \citet{grav11}
model we derived a probability density $f_{pop}(q, e, i, H)$ for
perihelion distance $q$, eccentricity $e$, inclination $i$, and
absolute magnitude $H$. A more refined approach would be to remove the
known population from the population model.

Mapping $f_{pop}$ to the best fitting orbits in the
$(\rho,\dot\rho)$ plane is quite complicated. Therefore, to compute such a
prior we iteratively apply Bayesian inversion theory, i.e., the
prior is computed as posterior $f_{post}'$ of another prior
$f_{prior}'$:
\[
f_{prior}(\rho, \dot\rho) = f_{post}' (\rho, \dot\rho) \propto f_{prior}'
(\rho, \dot\rho) f_{pop}(q (\rho, \dot\rho), e (\rho, \dot\rho), i
(\rho, \dot\rho), H (\rho, \dot\rho))\,.
\]
At this point we select $f_{prior}'$ using simple geometric
considerations. By assuming that the Cartesian space of position and
velocity is uniformly filled, we have that $\rho^3$ and $\dot\rho$ are
uniformly distributed. Hence,
$f_{prior}' (\rho, \dot\rho) \propto \rho^2$.
% In other words $f_{pop}$
%is a density to be integrated over the volume element $dV$ in Cartesian
%space:
%\[
%\int f_{pop} dV \propto \int f_{pop} \rho^2
%\]

\subsection{Uniform prior}
A common choice for $f_{prior}$ is a uniform distribution. For
instance, such a prior is used by \citet{stat_ran} and \citet{mcmc}.
Interestingly, it is possible to approximately obtain a prior
distribution from the population model if we only consider the size
distribution $f_{size}(H)$, i.e., $f_{prior} = \rho^2 f_{size}(H) $.

For near-Earth asteroids the cumulative size
distribution is expected to follow a power law
$f_{size}(H) \propto 10^{\eta H}$, with $\eta$ somewhere between 0.35
and 0.47 \citep{bottke02, harris, stuart}. For a given visual
magnitude $V$, we have that $f_{size}(H) \propto \rho^{-5\eta}$
\citep{bowell} and therefore $f_{prior} = \rho^{2 - 5\eta}$. According
to the range of possible values for $\eta$, this prior corresponds to
a power law with exponent between $-0.35$ and $0.25$, which is
compatible with a uniform distribution.

Though a uniform distribution might appear overly simplistic, we have found
that this it does a good job of identifying potential
impactors (e.g., see Sec.~\ref{s:examples}) and distinguishing
interesting objects from more common ones, such as main belt asteroids.

%%\[
%%f_{post}({\cal A}, {\cal R}) \propto f_{res}(\bm\nu({\cal A}, {\cal
%%  R}))\, f_{prior}({\cal A}, {\cal R})
%%\sqrt{\det(A^T W A)}\ \ ,\ \ A =
%%  \frac{\partial \bm\nu}{\partial \mathbf x}\ .
%%\]
%%In particular, according to \citet{openorb}, we can get the distribution $f_R$
%%on $(\rho, \dot\rho)$ as
%%\begin{equation}\label{eq:f_R}
%%  f_R(\rho,\dot\rho) \propto f_{\bm\nu}\,\sqrt{\det(A^T W A)}\ \ ,\ \ A =
%%  \frac{\partial \bm\nu}{\partial (\rho,\dot\rho)}\ .
%%\end{equation}
%The term $\sqrt{\det(A^T W A)}$ is important as it accounts for the
%partials of the transformation from the orbital element space to that
%of the residuals, similarly to what the Jacobian determinant does in
%the change of variables for integrals.  It is important to mention
%that, since we are analyzing the marginal distribution of
%$(\rho, \dot\rho)$, $A$ is the total derivative of $\bm \nu$ with
%respect to $(\rho, \dot\rho)$, which means that we have to account for
%the fact that the best fit attributable ${\cal A}$ changes as a
%function of $(\rho, \dot\rho)$.  Note that this formulation
%generalizes the standard linear approach, where $A$ is constant and
%therefore $(\rho, \dot\rho)$ are normally distributed around the
%minimum of the cost function $Q$.
%
%It is often convenient to use a logarithmic scale for $\rho$ to
%achieve a better resolution at small topocentric distances. In that
%case $f_{R}$ shall be multiplied by $\rho$.
%
%In absence of a priori information on the probability distribution of
%$({\cal A}, {\cal R})$, a sensible
%choice for the prior distribution is Jeffreys' prior:
%\[
%f_{prior} = \sqrt{\det\left(\frac{\partial
%  \bm\nu}{\partial ({\cal A}, {\cal R})}^T\  W \ \frac{\partial
%  \bm\nu}{\partial ({\cal A}, {\cal R})}\right)}\ .%\ \ ,\ \ A = \frac{\partial
%%  \bm\nu}{\partial ({\cal A}, {\cal R})}\ .
%\]
%Jeffreys' prior accounts for the partials of the transformation from
%the orbital element space to that of the residuals, similarly to what
%the Jacobian determinant does in the change of variables for
%integrals, and therefore $f_{post}$ is invariant when changing to
%other orbital element sets, e.g., Keplerian. Note that this
%formulation generalizes the standard linear approach, where $A$ is
%constant and therefore $\mathbf x$ is normally distributed around the
%minimum of the cost function $Q$.
%
%%%We want the marginal distribution in $(\rho, \dot\rho)$. Thus, after
%%%finding the constrained least squares solution, and the corresponding
%%%residuals, at each grid point we obtain the probability density
%%%function:
%%%\begin{equation}\label{e:prob}
%%%f_{(\rho,\dot\rho)} \propto f_{\bm\nu}\,\sqrt{\det(A^T W A)}\ \ ,\ \ A =
%%%  \frac{\partial \bm\nu}{\partial (\rho,\dot\rho)}\ .
%%%\end{equation}
%%%It is important to mention that $A$ is the total derivative of $\bm \nu$
%%%with respect to $(\rho, \dot\rho)$, which means that we have to
%%%account for the fact that the best fit attributable ${\cal A}$ changes
%%%as a function of $(\rho, \dot\rho)$. To compute $A$ we store the
%%%residual vector $\bm\nu_{ij}$ at each grid point and use a finite
%%%difference approximation:
%%%%\[%begin{equation}
%%%%\left(\frac{\partial\bm\nu}{\partial\rho}\right)_{ij} =
%%%%\frac{\bm\nu(\rho_{i+1}, \dot\rho_{j}) - \bm\nu(\rho_{i-1},
%%%% \dot\rho_{j})}{\rho_{i+1} - \rho_{i-1}}\ \ ,\ \
%%%%\left(\frac{\partial\bm\nu}{\partial\dot\rho}\right)_{ij} =
%%%%\frac{\bm\nu(\rho_i, \dot\rho_{j+1}) - \bm\nu(\rho_{i},
%%%%  \dot\rho_{j-1})}{\dot\rho_{j+1} - \dot\rho_{j-1}}\ .
%%%%\]%end{equation}
%%%\[%begin{equation}
%%%\left(\frac{\partial\bm\nu}{\partial\rho}\right)_{ij} =
%%%\frac{\bm\nu_{(i + 1),j} - \bm\nu_{(i-1),j}}{\rho_{i+1} - \rho_{i-1}}\ \
%%%,\ \ \left(\frac{\partial\bm\nu}{\partial\dot\rho}\right)_{ij} =
%%%\frac{\bm\nu_{i,(j+1)} - \bm\nu_{i,(j-1)}}{\dot\rho_{j+1} - \dot\rho_{j-1}}\ .
%%%\]%end{equation}
%%%Is is often convenient to use a logarithmic scale for $\rho$ to have a
%%%better resolution at small topocentric distances. In that case the
%%%term $\sqrt{\det(A^T W A)}$ has to be multiplied by $\rho$.
%
%
%
%
%\subsection{Prior from asteroid orbit population model}
%When the number of astrometric observations is low and the observed
%arc short, the amount of information contained in the observation
%residuals $\bm\nu$ may not suffice to obtain a realistic
%distribution. In these cases, the prior distribution can be derived
%from an asteroid population model for the orbital elements and the
%absolute magnitude $H$.
%
%For each point of the grid we compute an orbit that can be fully
%described by
%$(\alpha, \delta, \dot\alpha, \dot\delta, \rho, \dot\rho, V)$, where
%$V$ is the apparent magnitude. All these quantities except
%$(\rho, \dot\rho)$ are directly tied to the optical observations.
%From $(\alpha, \delta, \dot\alpha, \dot\delta, \rho, \dot\rho, V)$ we
%can compute the absolute magnitude $H$ and following orbital elements:
%\begin{align*}
%q &  & h &= e\sin(\Omega + \omega)\\
%k &= e\cos(\Omega + \omega) & s &= \sin(i/2)\sin\Omega\\
%u &= \sin(i/2)\cos\Omega & \lambda &= \Omega + \omega + M
%\end{align*}
%where $q$ is the perihelion distance, $e$ the eccentricity $e$, $i$
%the inclination, $\Omega$ the longitude of node, $\omega$ the argument
%of perihelion, and $M$ the mean anomaly. These elements are similar to
%the equinoctial elements \citep{equ} and, unlike Keplerian orbital
%elements, are not singular for zero eccentricity or zero inclination.
%Therefore, we have a map:
%\[
%  ({\cal A}, {\cal R}, V)
%  \Longrightarrow (q, h, k, s, u, \lambda, H).
%\]
%and we indicate with $B$ the partial derivatives matrix of this
%map. Note that these partials can be analytically computed by using
%the chain rule for the following maps:
%\begin{eqnarray*}
%  ({\cal A}, {\cal R}, V) &
%  \Longrightarrow & (\mathbf r, \mathbf v, V)\\
%  (\mathbf r, \mathbf v, V) & \Longrightarrow &  (\mathbf r, \mathbf v, H)\\
%  (\mathbf r, \mathbf v, H) & \Longrightarrow &  (q, h, k, s, u, \lambda, H)\,.
%\end{eqnarray*}
%where $\mathbf r$ and $\mathbf v$ are the Cartesian position and
%velocity of the object.  For each of these maps it is well known how
%to compute the partials \citep[e.g., see][]{milani04, bowell, equ}.
%The prior distribution can be obtained as:
%\[%begin{equation}%eqnarray*}
%  f_{prior} ({\cal A}, {\cal R}, V) = \ f_{pop} (q, h, k, s, u,
%  \lambda, H)\ \left|\det\frac{\partial(q, h, k, s, u, \lambda,
%      H)}{\partial({\cal A}, {\cal R}, V)}\right|\\
%%  B &=&
%%       \frac{\partial(q, h, k, s, u, \lambda, H)}{\partial(\alpha, \delta,
%%        \dot\alpha, \dot\delta, \rho, \dot\rho, V)}.
%\]%end{equation}%eqnarray*}
%The Jacobian determinant factor accounts for the uneven way the
%volumes are mapped from the $({\cal A}, {\cal R}, V)$ space to that of
%the orbital elementsand absolute magnitude, e.g., for smaller values
%of $\rho$ we have smaller volumes in Cartesian space and therefore in
%the orbital element space.
%
%To obtain the probability density function in the orbital element
%space $f_{pop}$ we use the Pan-STARRS Synthetic Solar System model
%\citep{grav11}, which is a realistic model up to a visual
%magnitude $V = 24.5$ and to $H = 25$ for NEAs. The finite number of
%samples in the model produce some limitations that we mitigate with
%the following tweaks:
%\begin{itemize}
%\item The absolute magnitude is assumed to be independent of the
%  orbital elements except $q$:
%\[
%f_{pop} (q, h, k, s, u, \lambda, H) \propto f_{el} (q, h, k, s, u,
%\lambda) f_{mag} (H | q)\ .
%\]
%\item The model is symmetric in $\lambda$, so
%$f_{el} (q, h, k, s, u, \lambda) \propto f_{el} (q, h, k, s, u)$.
%\item To take advantage of the symmetry of the model in $\Omega$ and
%  $\omega$, instead of checking if an object was inside a rectangle
%  $\Delta h \Delta k$ we computed the fraction of the circle of radius
%  $e = \sqrt{h^2 + k^2}$ inside the $\Delta h \Delta k$ rectangle. We
%  applied an analogous approach for $s$, $u$, and $\sin(i/2)$.
%\item To account for Poisson statistics uncertainties, to each orbital
%  element volume $\Delta q\Delta h \Delta k \Delta u \Delta s$ we
%  added a ficticious orbit with $e = \sqrt{h^2 + k^2}$ and
%  $\sin(i/2) = \sqrt{u^2 + s^2}$.
%\item Some very small objects may have $H > 25$ and still be observed
%  at $V < 24.5$ if the are close enough to the observer. Therefore, we
%  extrapolated the number of NEAs for a given $H$ according to
%  \citet{bottke02}:
%  \[
%  N(<H) \propto 10^{0.35 H}\ .
%  \]
%\end{itemize}
%Finally, to avoid making some orbit configurations impossible, we set
%a minimum value for $f_{prior}$ to $\exp(-5^2/2)\max f_{prior}$, which
%can be leveled off  with a $5\sigma$ signal coming from the observation
%residuals.
%
%Since we mostly plan to apply the systematic ranging algorithm to
%newly discovered objects, to account for the fact that
%the object has not already been discovered we can multiply $f_{prior}$
%by a factor $(1 - \epsilon(H, q))$, where $\epsilon(H, q)$ is the discovery
%completeness for given $H$ and $q$.

\section{Impact probability computation}\label{s:imp_prob}
For each grid point, as a result of the constrained least squares fit
to the observations, we obtain not only a best fit value
${\cal A}_{ij}$ of the attributable, but also the corresponding
uncertainty expressed by a covariance matrix $\Gamma_{{\cal
    A}_{ij}}$.
This uncertainty is easily mapped to the orbital element space and
therefore allows us to compute the probability $p_{ij}$ of an Earth
impact for that orbit. Since for a fixed grid point the orbital
uncertainty is small and the horizon for impacts is of the order of
weeks, we compute the impact probability with a linear approach
\citep{milani_ast3, farnocchia_ast4}.

Given $f_{post}$ we can assign each grid point a weight
$w_{ij} \propto f_{post}(\rho_i, \dot\rho_j)$, where
$\sum w_{ij} = 1$. Then, the impact probability can be easily computed
as
\[
P = \sum w_{ij} p_{ij}.
\]

\section{Generation of Monte Carlo orbital samples}
To better visualize the orbital distribution, it is convenient to
generate Monte Carlo orbital samples distributed according to the
probability distribution computed above. The following procedure can
be used to achieve this goal:
\begin{description}
\item[(a)] Randomly draw $\rho_k$ according to a uniform distribution
  between the minimum and maximum values of $\rho$ on the grid;
\item[(b)] Randomly draw $\dot\rho_k$ according to a uniform distribution
  between the minimum and maximum values of $\dot\rho$ on the grid;
\item[(c)] Randomly draw a value $\gamma$ according to a uniform
  distribution between 0 and the maximum of $f_{post}$ on
  the grid;
\item[(d)] Compute a best-fit attributable ${\cal A}_k$ for
  $(\rho, \dot\rho) = (\rho_k, \dot\rho_k)$;
\item[(e)] If $f_{post}(\rho_k, \dot\rho_k) \geq \gamma$ go
  to (f), otherwise start again from (a);
\item[(f)] Add normally distributed noise to ${\cal A}_k$ according to
  the covariance $\Gamma_{{\cal A}_k}$ and add
  $({\cal A}_k, \rho_k, \dot\rho_k)$ to the sequence of orbital
  samples.
\end{description}

\section{Examples}\label{s:examples}
We want to test the algorithm presented in the previous sections on
real cases. On one hand it is important that systematic ranging
provides an early detection of impacting asteroids. On the other hand
we need to avoid generating too many warnings for objects that might
be very far from Earth. Thus, we show examples of impactors as well as
main belt asteroids. Table~\ref{t:examples} summarizes the results for
the examples discussed in more detail in the following subsections.

\begin{table}[t]
\small
\begin{center}
\begin{tabular}{|lccc|ccc|}
  \hline
  Object & Number & Arc & Weights & \multicolumn{3}{c|}{Impact probability}\\
 & of obs & length & & Jeffreys & Uniform & Population\\
  \hline 
  2008~TC$_3$ & 4 & 43 min & 0.5$''$ & $6.3\times 10^{-1}$ & $4.4\times 10^{-2}$
 & $2.4\times 10^{-2}$\\
  2008~TC$_3$ & 4 & 43 min & 1.0$''$ & $6.7\times 10^{-1}$ & $3.7\times 10^{-3}$
 & $3.5\times 10^{-3}$\\
  2008~TC$_3$ & 7 & 99 min & 0.5$''$ & $1.0$ & $1.0$
 & $1.0$\\
  2008~TC$_3$ & 7 & 99 min & 1.0$''$ & $9.7\times 10^{-1}$ & $9.8\times 10^{-1}$
 & $9.6\times 10^{-1}$\\
  \hline
  2014~AA & 3 & 28 min & 0.5$''$ & $8.1\times 10^{-1}$ & $2.9\times 10^{-2}$
 & $3.1\times 10^{-2}$\\
  2014~AA & 3 & 28 min & 1.0$''$ & $8.4\times 10^{-1}$ & $3.3\times 10^{-3}$
 & $8.4\times 10^{-3}$\\
  2014~AA & 7 & 69 min & 0.5$''$ & $1.0$ & $1.0$
 & $1.0$\\
  2014~AA & 7 & 69 min & 1.0$''$ & $9.9\times 10^{-1}$ & $9.6\times 10^{-1}$
 & $9.0\times 10^{-1}$\\
  \hline
  Hayabusa & 3 & 7 min & 1.0$''$ & $9.4\times 10^{-1}$ & $2.7\times 10^{-3}$
 & $2.0\times 10^{-3}$\\
  Hayabusa & 6 & 34 min & 1.0$''$ & $2.4\times 10^{-1}$ & $1.7\times 10^{-1}$
 & $2.2\times 10^{-2}$\\
  \hline
  Hayabusa 2 & 3 & 43 min & 0.2$''$ & $2.8\times 10^{-1}$ & $3.6\times 10^{-1}$
 & $4.2\times 10^{-1}$\\
  Hayabusa 2 & 3 & 43 min & 0.5$''$ & $3.6\times 10^{-1}$ & $3.3\times 10^{-2}$
 & $7.2\times 10^{-2}$\\
  Hayabusa 2 & 6 & 26 h & 0.2$''$ & $2.4\times 10^{-13}$ & $2.3\times 10^{-13}$
 & $2.2\times 10^{-13}$\\
  Hayabusa 2 & 6 & 26 h & 0.5$''$ & $1.9\times 10^{-3}$ & $1.9\times 10^{-3}$
 & $1.8\times 10^{-3}$\\
  \hline
  2004~AS$_1$ & 4 & 71 min & 1.0$''$ & $2.2\times 10^{-1}$ & $1.2\times 10^{-1}$
 & $7.8\times 10^{-2}$\\
  2004~AS$_1$ & 4 & 71 min & 1.5$''$ & $1.2\times 10^{-1}$ & $1.3\times 10^{-2}$
 & $1.1\times 10^{-2}$\\
  \hline
  2004~FU$_{162}$ & 4 & 44 min & 1.0$''$ & $3.6\times 10^{-1}$ & $7.3\times 10^{-3}$
 & $8.7\times 10^{-4}$\\
  2004~FU$_{162}$ & 4 & 44 min & 1.5$''$ & $5.0\times 10^{-1}$ & $1.3\times 10^{-2}$
 & $1.0\times 10^{-3}$\\
  \hline
  2015~CV & 4 & 34 min & 0.5$''$ & $8.5\times 10^{-3}$ & $4.7\times 10^{-6}$
 & $9.9\times 10^{-7}$\\
  2015~CV & 4 & 34 min & 1.0$''$ & $1.7\times 10^{-1}$ & $1.5\times 10^{-4}$
 & $4.6\times 10^{-5}$\\
  \hline
  2015~BU$_{92}$ & 3 & 36 min & 0.5$''$ & $1.7\times 10^{-1}$ & $7.2\times 10^{-5}$
 & $3.5\times 10^{-10}$\\
  2015~BU$_{92}$ & 3 & 36 min & 1.0$''$ & $4.8\times 10^{-1}$ & $4.0\times 10^{-4}$
 & $8.5\times 10^{-8}$\\
  \hline
\end{tabular}
\end{center}
\caption{Impact probabilities for the examples considered in
  Sec.~\ref{s:examples} for different choices of $f_{prior}$,
  different number of tracklets, and different data weights.}
\label{t:examples}
\end{table}

\subsection{2008~TC$_3$}
\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{2008tc3_4_rms}
\includegraphics[width=0.5\textwidth]{2008tc3_7_rms}}
\caption{Normalized RMS as a function of $(\rho, \dot\rho)$ for
  2008~TC$_3$. Left panel is for the first tracklet (four observations
  over 43 min), right panel is for the first two tracklets (total of
  seven observations over 99 min). Astrometric weights are at
  0.5$''$. The dashed curve corresponds to parabolic heliocentric
  orbits and the dash-dotted line to grazing
  impacts.}\label{f:tc3_rms}
\end{figure} 
As already discussed in the introduction, 2008~TC$_3$ was the first
asteroid discovered prior to impact. This object was first spotted and
promptly followed up by R.~A. Kowalski at the Mt. Lemmon
observatory. In the 20 hours between discovery and impact hundreds of
astrometric observations were reported to the Minor Planet
Center. These observations allowed the calculation of a least squares
orbit and the prediction of the impact. How would systematic
ranging have performed with the first observations of 2008~TC$_3$?

The left panel of Fig.~\ref{f:tc3_rms} shows the normalized RMS, i.e.,
the square root of the target function $Q$ in Eq.~\eqref{e:target}, as
a function of $(\rho, \dot\rho)$ by using the first tracklet of four
observations obtained by Kowalski. The nonlinearity of the orbit
determination problem is evident from the level curves of the RMS,
which clearly suggest that 2008~TC$_3$ could be on an impact
trajectory. As a matter of fact, the corresponding impact probability,
computed as described in Sec.~\ref{s:imp_prob}, is already significant
with the first four observations (Table~\ref{t:examples}). With our
defaults data weights at 0.5$''$, the impact probability is few
percents when using a uniform prior or one based on the population
model, and $\sim$60\% when using Jeffreys' prior. With more conservative
weights at 1$''$ Jeffreys' prior leads to a $\sim$70\% impact
probability, while the uniform and population prior result in a
probability of few in a thousand.

%4\%. Therefore,
%systematic ranging would have been able to recognize that 2008~TC$_3$
%had significant chances of striking the Earth starting from the first
%batch of observations.

%\begin{figure}
%\centerline{\includegraphics[width=8 cm]{ip_ev}}
%\caption{Impact probability as a function of the number of
%  observations for 2008~TC$_3$ (solid), 2014~AA (dashed), Hayabusa
%  (dash-dotted), and Hayabusa 2 (dotted).}\label{f:imp_ip_ev}
%\end{figure}

With the addition of Kowalski's second tracklet, three additional
observations, the orbit becomes much better constrained by the
observation residuals (see right panel of Fig.~\ref{f:tc3_rms}). The
impact is now almost certain, especially with 0.5$''$ weights.
%with a probability of 99.99\%.
%Figure~\ref{f:imp_ip_ev} shows the evolution of the impact probability
%as a function of the number of observations. It is worth noticing that
%with the first three observations the probability of impact would have
%been $2.3 \times 10^{-3}$, which suggests that objects with an initial
%impact probability of at least few in a thousand deserve attention and
%should promptly be followed up.

\begin{figure}
\centerline{\includegraphics[width=10 cm]{tc3_loc}}
\caption{Potential impact locations for 2008 TC$_3$ based on the first 
  two tracklets of data.}\label{f:tc3_loc}
\end{figure}

By using the first two tracklets of data and adopting a uniform prior, we
also generated Monte Carlo samples to find potential impact locations,
as shown in Fig.~\ref{f:tc3_loc}. As expectable, the region of
possible impact location contains the Sudan desert where 2008~TC$_3$
actually ended its journey \citep{jenniskens_nat}.

\subsection{2014~AA}
Though 2014~AA was not extensively followed up after discovery, there
are several similarities with 2008~TC$_3$. In particular, 2014~AA
was also discovered about 20 hours before impact by R.~A. Kowalski at the
Mt. Lemmon observatory.

\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{2014aa_3_rms}
\includegraphics[width=0.5\textwidth]{2014aa_7_rms}}
\caption{Normalized RMS as a function of $(\rho, \dot\rho)$ for
  2014~AA. Left panel is for the first tracklet (three observations
  over 28 min), right panel is for the first two tracklets (total of
  seven observations over 69 min). Astrometric weights are at
  0.5$''$. The dashed curve corresponds to parabolic heliocentric
  orbits and the dash-dotted line to grazing impacts.}\label{f:aa_rms}
\end{figure}

Figure~\ref{f:aa_rms} shows the normalized RMS for 2014~AA as a
function of $(\rho, \dot\rho)$ by using the first tracklet of data
(three observations) or the full available dataset (seven
observations). Again, the RMS curves clearly indicate that an impact
is possible. Systematic ranging immediately recognizes the potential
impact from the first tracklet and make the impact almost certain when
the second tracklet is added. Table~\ref{t:examples} reports the
computed impact probabilities, which are similar to those computed for
2008~TC$_3$,.

%with an impact probability of 3\%. The addition
%of the second tracklet unequivocally confirms the impact,
%with a probability of 99.998\%.

%Figure~\ref{f:imp_ip_ev} shows the impact probability for 2014~AA for
%increasing number of observations. Remarkably, the impact probability
%with three observations is higher than that of 2008~TC$_3$, which is
%due to the larger rate of motion of 2014~AA. Moreover, the addition of
%just the fourth observation already makes the observed arc long enough
%to have clearly identify 2014~AA as an impactor (99.97\% probability).

\begin{figure}
\centerline{\includegraphics[width=10 cm]{aa_loc}}
\caption{Potential impact locations for 2014~AA based on all available
  data, i.e., seven observations.}\label{f:aa_loc}
\end{figure}
The impact locations (see Fig.~\ref{f:aa_loc}) found by using the
seven available observations and adopting a uniform prior are
statistically consistent with the detection by the infrasound
component of the International Monitoring System (IMS) operated by the
Comprehensive Nuclear Test Ban Treaty Organization (CTBTO), which
places the impact location in the Atlantic Ocean at a latitude of
about 15$^\circ$ N and a longitude of about 43$^\circ$ W
\citep{chesley_ieee,brown_acm}.

\subsection{Hayabusa \& Hayabusa 2}
Because of the low number of actual impactors we can use as a
validation, we tested our algorithm on Hayabusa and Hayabusa 2
\citep{hayabuse}.

\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{hayabusa_3_rms}
  \includegraphics[width=0.5\textwidth]{hayabusa_6_rms}}
\caption{Normalized RMS as a function of $(\rho, \dot\rho)$ for
  Hayabusa. Left panel is for the first tracklet (three observations
  over 7 min), right panel is for the first two tracklets (total of
  seven observations over 34 min). Astrometric weights are at
  1.0$''$. The dashed curve corresponds to parabolic heliocentric
  orbits and the dash-dotted line to grazing impacts.}\label{f:hayabusa_rms}
\end{figure}

The reentry of Hayabusa took place on 2010 June
13. \citet{hayabusa_obs} report ground-based observations acquired
during the reentry phase, which ended with a landing in the South
Australia outback. We used these observations for applying systematic
ranging technique to Hayabusa. Figure~\ref{f:hayabusa_rms} shows the
normalized RMS as a function of $(\rho, \dot\rho)$ by using the first
tracklet of data (three observations) or the full available dataset
(six observations). Table~\ref{t:examples} gives the impact
probability as a function of the number of observations. With a
uniform prior the initial impact probability is $2.7\times 10^{-3}$
and increases to 17\% with the second tracklet. With the
population-based prior the initial impact probability is
$2.0\times 10^{-3}$ and increases to 2\% with the second tracklet. On
the other hand, with the use of Jefrreys' prior we immediately have a
94\% impact probability that decreases to 24\% with the second
tracklet. This counterintuitive behavior is due to the presence of two
local RMS minima with the first tracklet, one of which clearly
corresponds to an impact and is ruled out by the second tracklet.

Hayabusa 2 was launched on 2014 December
3.\footnote{http://global.jaxa.jp/press/2014/12/20141203\_h2af26.html}
After launch Hayabusa 2 was initially put on a low Earth parking
orbit, until a maneuver moved it onto a ballistic trajectory leaving
the Earth. On its way out, on 2014 December 6, Hayabusa 2 happend to
be in the field of view of the Pan-STARRS PS1 survey \citep{ps1},
which collected three observations of the spacecraft over 43 min. The
corresponding astrometry was submitted to the Minor Planet Center,
which, since the observations were not recognized as belonging to the
Hayabusa 2 spacecraft, put the object on the NEO Confirmation Page
\footnote{http://www.minorplanetcenter.net/iau/NEO/ToConfirm.html}
with the temporary designation P10gJfF. The following day, 2014
December 7, Pan-STARRS obtained three additional observations from
which it became clear that the object was Hayabusa
2. Table~\ref{t:examples} shows how systematic ranging recognizes
Hayabusa 2 as potential impactor from the first three observations
with an impact probability probability of few to ten percents, depending on the data
weights and the prior used. Then, the subsequent observations better
constrain the orbit and lower the impact probability since the
ballistic trajectory does not correspond to an impact but to the
post-injection escape orbit.

\subsection{2004~AS$_1$}
Asteroid 2004~AS$_1$ was discovered by LINEAR \citep{linear} on 2004
January 13 \citep{mpec_as1}. The four discovery observations span 71
min and show an
unusually high curvature thus suggesting that 2004~AS$_1$ was close to
the Earth and that an impact was possible. Follow-up observations
later showed that the object was much farther than originally thought,
thus ruling out any impact, and that the observed curvature was due to
large astrometric errors in the discovery observations.

\citet{virt_as1} employed the statistical ranging technique
\citep{stat_ran} to analyze the 2004~AS$_1$ case. They found an 18\%
impact probability with 1.0$''$ weights and concluded that
such a large impact probability was mostly due to the
poor quality of the discovery night observations.

The left panel of Fig.~\ref{f:04_rms} shows the application of
systematic ranging to 2004~AS$_1$. The astrometric observations are
weighted at 1.0$''$. The RMS favors an impact solution while the true
solution (marked with a cross) correspond to larger astrometric
errors. The impact probabilities computed for the different choices of
the prior distribution are of the order of 10\%, see
Table~\ref{t:examples}. Decreasing the weights at 1.5$''$ gives
significantly lower probabilities for the uniform and population-based
priors, while with Jefrreys' prior we still have a 12\% impact
probability.

\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{2004as1_rms}
\includegraphics[width=0.5\textwidth]{2004fu162_rms}}
\caption{Normalized RMS as a function of $(\rho, \dot\rho)$ for
  2004~AS$_1$ (left panel) and 2004~FU$_{162}$ (right
  panel). Astrometric weights are at 1.0$''$.  The dashed curve
  corresponds to parabolic heliocentric orbits and the dash-dotted
  line to grazing impacts. For 2004~AS$_1$ the cross marks the true
  solution.}\label{f:04_rms}
\end{figure}

As shown by Table~\ref{t:pval}, we computed the $p$-value of the true
solution, i.e., the fraction of less likely orbital solutions, for the
different choices of the prior and different data weights. Jeffreys'
prior produces very small $p$-values thus indicating that the true
solution is deemed as extremely unlikely by the computed probability
distribution. For the uniform distribution we have 3\% $p$-value with
1$''$ weights, which is due to the large errors in the astrometric
observations. As a matter of fact, relaxing the weights increases the
$p$-value to 22\%. On the other hand, the population-based prior compensates
for the large astrometric errors since impacting solutions with
low RMS are less likely in the population model than non-impacting solutions
with a higher RMS. The $p$-values are 13.6\% and 38.7\% depending on
the data weights.

\begin{table}[t]
%\small
\begin{center}
\begin{tabular}{|lc|ccc|}
  \hline
  Object & Weights & \multicolumn{3}{c|}{$p$-value}\\
         & & Jeffreys & Uniform & Population\\
  \hline 
  2004~AS$_1$ & 1.0$''$ & $3.1\times 10^{-5}$ & $2.6\times 10^{-2}$
 & $1.4\times 10^{-1}$\\
  2004~AS$_1$ & 1.5$''$ & $4.0\times 10^{-4}$ & $2.2\times 10^{-1}$
 & $3.9\times 10^{-1}$\\
  \hline
  2015~CV & 0.5$''$ & $5.0\times 10^{-5}$ & $4.9\times 10^{-1}$
 & $4.2\times 10^{-1}$\\
  2015~CV & 1.0$''$ & $1.3\times 10^{-5}$ & $4.9\times 10^{-1}$
 & $4.2\times 10^{-1}$\\
  \hline
  2015~BU$_{92}$ & 0.5$''$ & $1.1\times 10^{-5}$ & $9.1\times 10^{-1}$
 & $8.8\times 10^{-1}$\\
  2015~BU$_{92}$ & 1.0$''$ & $2.3\times 10^{-6}$ & $9.1\times 10^{-1}$
 & $8.9\times 10^{-1}$\\
  \hline
\end{tabular}
\end{center}
\caption{$p$-values for some of the examples considered in
  Sec.~\ref{s:examples} for different choices of $f_{prior}$ and different data weights.}
\label{t:pval}
\end{table}

The conclusion fpr 2004~AS$_1$ is that the discovery tracklet,
weighted at 1$''$, yields impact probabilities $\sim10\%$ even as the
trueth is compatible with the data at the 3--14\% level. This outcome
is a manifestation of the large astrometric errors in the data.

\subsection{2004~FU$_{162}$}
Asteroid 2004~FU$_{162}$ was also discovered by LINEAR on 2004 March
31 \citep{mpec_fu162}.  \citet{sys_ran_steve} first analyzed this
object and estimated that there was a 0.3\% probability of an impact and
that the object was likely to pass within 10 Earth radii.

The right panel of Fig.~\ref{f:04_rms} shows our implementation of
systematic ranging, while Table~\ref{t:examples} reports the impact
probabilities for different weights and prior distributions. Again,
Jeffreys' prior results in very high impact probabilities while the
population-based prior gives the lowest impact chances. For 1$''$
weights the impact probability is 0.7\% -- 0.9\% with the uniform and
the population based priors. An Earth encounter within 10 Earth radii
is assured with all three priors, thus confirming the findings of
\citet{sys_ran_steve}.

\subsection{2015~CV \& 2015~BU$_{92}$}
\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{2015cv_rms}
\includegraphics[width=0.5\textwidth]{2015bu92_rms}}
\caption{Normalized RMS as a function of $(\rho, \dot\rho)$ for
  2015~CV (left panel) and 2015~BU$_{92}$ (right panel). Astrometric
  weights are at 0.5$''$. The dashed curve corresponds to parabolic
  heliocentric orbits and the dash-dotted line to grazing
  impacts. Crosses mark the true solutions.}\label{f:non_imp_rms}
\end{figure}
The previous examples show how systematic ranging is capable of
quickly identifying actual and potential impactors. However, there is
the risk of having significant impact probabilities ($10^{-3}$ or
higher) for a large number of objects, which would make it difficult
to distinguish interesting objects from less remarkable ones. Here we
show the example of two main belt objects that were recently posted on
the NEO Confirmation Page when the initial observations were
compatible with an NEO orbit: 2015~CV and 2015~BU$_{92}$.

2015~CV was discovered by the Catalina Sky survey, Mt. Lemmon station,
on 2015 February 10, while 2015~BU$_{92}$ was discovered by Spacewatch
\citep{spacewatch}, Kitt Peak observatory, on 2015 January 16
\citep{mpec_bu92}. Figure~\ref{f:non_imp_rms} shows the application of
systematic ranging to these two asteroids.  For both objects the
region with reasonable astrometric errors extends from impacting
solutions to more than 1 au from the observer.

As shown by Table~\ref{t:examples} and Table~\ref{t:pval}, in both
cases Jeffreys' prior provides significant impact probabilities and
extremely low $p$-values for the true solutions. In fact, Jeffreys'
prior is proportional to the partials of the astrometric residuals and
the residuals do not change much for large topocentric distances while
they are very sensitive when the object is close to the
observer. Therefore, Jeffreys' prior tends to favor small topocentric
distances even if the astrometric errors are larger. On the other hand
the uniform and the population-based priors give much more reasonable
probabilities and $p$-values.

%\section{Discussion}

\section{Discussion and future work}
Small asteroids are expected to strike the Earth on a regular
basis. For instance, according to \citet{impact_frequency} impacts
with 10 m sized objects take place every few years. Since such small
asteroids are likely to be observed, if at all, only shortly before
impact, it is important to be capable of an early recognition of
hazardous objects.

The shortness of the observation arcs typically available for these
objects leads to critical degeneracies in the orbit estimation process
and limits the applicability of the standard impact monitoring systems
\citep{milani_im}. The systematic ranging technique presented in this
paper relies on Bayesian inversion theory, which overcomes these
degeneracies and results in a posterior distribution on the orbit phase space.

A critical point is the choice of a prior distribution for the poorly
constrained space of topocentric range and range rate. This choice is
far from trivial and indeed is somewhat arbitrary. We analyzed three
options: Jeffreys' prior, a uniform prior, and a prior distribution
based on the asteroid population model.

Jeffreys' prior does a good job at assigning high impact probabilities
to actual impactors. However, this prior produces high impact
probabilities also for less remarkable objects such as main belt
asteroids. The reason for this behavior is that Jeffreys' prior favors
regions in the phase space that are closer to the observer. As a
matter of fact, Jeffreys' prior is proportional to the partials of the
residuals with respect to topocentric range and range rate, which are
large for small topocentric distances and vanish at large
distances. Moreover, in the non-impacting examples we analyzed we
found that the true solution was often deemed as extremely unlikely by
the posterior distribution obtained starting from Jeffreys'
prior. Therefore, we avoid the use of Jeffreys' prior for systematic
ranging.

The prior based on the asteroid orbit and size distribution represents
the highest fidelity and most complex choice. However, it is worth
pointing out that a population model is necessarily limited by the
finite number of samples and is therefore affected by Poisson
statistics errors. In particular, small impactors might have peculiar
orbits that are deemed unlikely by the model, which could be too
much of a risk when the main goal is to identify potential impactors.


A uniform distribution represents a compromise between simplicity and
reliability. Though the corresponding posterior distribution and
impact probabilities are not as rigorous as the one resulting from the
population-based prior, we found that a uniform prior is effective at
flagging potential impactors and discriminating between hazardous
asteroids and more distant ones, which is the most important task to
be achieved by systematic ranging. Another advantage of the uniform
prior is that it has little prejudice on the actual orbit.

To complement the standard impact monitoring performed by JPL's
Sentry\footnote{http://neo.jpl.nasa.gov/risk} and the University of
Pisa's NEODyS,\footnote{http://newton.dm.unipi.it/neodys} we have
implemented an automated system to continually analyze the objects on
the Minor Planet Center's NEO Confirmation Page with systematic
ranging. This system quickly computes an impact probability to
identify potential impactors and notifies us of interesting
results. In such cases, we have communicated with discoverers and
other observers to solicitate a quality control of the reported astrometry
and for acquiring follow-up observations, which are necessary to
decrease the orbital uncertainties and conclusively determine if the
impact is going to take place.  The examples of 2008~TC$_3$ and
2014~AA, i.e., the only two asteroids discovered before striking the
Earth, suggest that objects with an impact probability above $10^{-3}$
should be prioritized for prompt follow-up. In recent experience this
happens a few to several times a month.

Among the possible additional applications of the systematic ranging
technique presented in this paper, one could be the identification
potential radar targets or mission-accessible targets, e.g., for the
NASA's Asteroid Redirect Mission \citep{arm}.

\section*{Acknowledgments}
We thank T.~B. Spahr for his advice on the astrometric weights of
Table~\ref{t:weights}.

Part of this research was conducted at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with NASA.

Copyright 2015 California Institute of Technology.
%Government sponsorship acknowledged.

\begin{thebibliography}{}
\bibliographystyle{elsarticle-harv}
\bibitem[Abell et al.(2014)]{arm} Abell, P., Gates, M., Johnson, L.,
  Chodas, P., Brophy, J., Mazanek, D., Muirhead, B.\ 2014.\ NASA's
  Asteroid Redirect Mission: Overview and Status.\ 40th COSPAR
  Scientific Assembly 40, 17.
\bibitem[Blythe et al.(2004a)]{mpec_as1} Blythe, M, Shelly, F.,
  Bezpalko, M., et al. 2004.\ 2004~AS1.\ Minor Planet Electronic Circulars
  2004-A56.
\bibitem[Blythe et al.(2004b)]{mpec_fu162} Blythe, M, Shelly, F.,
  Bezpalko, M., et al. 2004.\ 2004~FU162.\ Minor Planet Electronic Circulars
  2004-Q22.
\bibitem[Bottke et al.(2002)]{bottke02} Bottke, W.~F., Morbidelli,
  A., Jedicke, R., Petit, J.-M., Levison, H.~F., Michel, P., Metcalfe,
  T.~S.\ 2002.\ Debiased Orbital and Absolute Magnitude Distribution
  of the Near-Earth Objects.\ Icarus 156, 399-433.
\bibitem[Bowell et al.(1989)]{bowell} Bowell, E., Hapke, B., Domingue,
  D., Lumme, K., Peltoniemi, J., Harris, A.~W.\ 1989.\ Application of
  photometric models to asteroids.\ Asteroids II 524-556.
\bibitem[Bressi and Tubbiolo(2015)]{mpec_bu92} Bressi, T.~H.,
  Tubbiolo, A.~F., 2015.\ 2015~BU92.\ Minor Planet Electronic
  Circulars 2015-B95.
%\bibitem[Broucke and Cefola(1972)]{equ} Broucke, R.~A., Cefola, P.~J.\
%  1972.\ On the Equinoctial Orbit Elements.\ Celestial Mechanics 5,
%  303-310.
\bibitem[Brown et al.(2002)]{impact_frequency} Brown, P., Spalding,
  R.~E., ReVelle, D.~O., Tagliaferri, E., Worden, S.~P.\ 2002.\ The
  flux of small near-Earth objects colliding with the Earth.\ Nature
  420, 294-296.
\bibitem[Brown(2014)]{brown_acm} Brown, P.\ 2014.\ Using the
  international monitoring system of seismic, infrasound, and
  hydroacoustic sensors for global airburst detection.\ Asteroids,
  Comets, Meteors 2014 63.
%\bibitem[Chambers and PS1 Science Consortium(2015)]{ps1} Chambers,
%  K.~C., PS1 Science Consortium 2015.\ Results from the Pan-STARRS1
%  Sky Surveys.\ American Astronomical Society Meeting Abstracts 225,
%  \#332.01.
\bibitem[Chesley(2005)]{sys_ran_steve} Chesley, S.~R.\ 2005.\
  Very short arc orbit determination: the case of asteroid 2004
  FU$_{162}$.\ IAU Colloq.~197: Dynamics of Populations of
  Planetary Systems 255-258.
%\bibitem[Chesley et al.(2014)]{chesley_dps14} Chesley, S.~R.,
%  Farnocchia, D., Brown, P., Chodas, P.~W.\ 2014.\ Orbit Estimation
%  for Late Warning Asteroid Impacts: The Case of 2014 AA.\
%  AAS/Division for Planetary Sciences Meeting Abstracts 46, \#403.03.
\bibitem[Chesley et al.(2015)]{chesley_ieee} Chesley, S.~R.,
  Farnocchia, D., Brown, P., Chodas, P.~W.\ 2015.\ Orbit Estimation
  for Late Warning Asteroid Impacts: The Case of 2014 AA.\
  IEEE Aerospace Conference, \#2.1202.
%\bibitem[Farnocchia et al.(2015a)]{debiasing} Farnocchia, D., Chesley,
%  S.~R., Chamberlin, A.~B., Tholen, D.~J.\ 2015.\ Star catalog
%  position and proper motion corrections in asteroid astrometry.\
%  Icarus 245, 94-111.
\bibitem[Farnocchia et al.(2015)]{farnocchia_ast4} Farnocchia, D.,
  Chesley, S.~R., Milani, A., Gronchi, G.~F., Chodas, P.~W.\ 2015.\
  Orbits, long-term predictions, impact monitoring.\ Asteroids IV, In
  press.
\bibitem[Granvik et al.(2009)]{openorb} Granvik, M., Virtanen, J.,
  Oszkiewicz, D., Muinonen, K.\ 2009.\ OpenOrb: Open-source asteroid
  orbit computation software including statistical ranging.\
  Meteoritics and Planetary Science 44, 1853-1861.
\bibitem[Granvik et al.(2014)]{granvik_neo} Granvik, M., Morbidelli,
  A., Jecicke, R., et al.\ 2014.\ Unbiased dynamical and physical
  characteristics of the near-Earth-object population.\ Asteroids,
  Comets, Meteors 2014 180.
\bibitem[Grav et al.(2011)]{grav11} Grav, T., Jedicke, R., Denneau,
  L., Chesley, S., Holman, M.~J., Spahr, T.~B.\ 2011.\ The Pan-STARRS
  Synthetic Solar System Model: A Tool for Testing and Efficiency
  Determination of the Moving Object Processing System.\ Publications
  of the Astronomical Society of the Pacific 123, 423-447.
\bibitem[Harris(2002)]{harris} Harris, A.~W.\ 2002.\ A New Estimate of
  the Population of Small NEAs.\ Bulletin of the American Astronomical
  Society 34, 835.
\bibitem[Jaynes(1968)]{prior} Janyes, E.~T. 1968. Priori Probabilities. IEEE
  Transactions On Systems Science and Cybernetics 4, 227-241.
%\bibitem[Jedicke et al.(2015)]{jedicke_ast4} Jedicke, R., Granvik, M.,
%  Micheli, M., Ryan, E., Spahr, T., Yeomans, D.~K.\ (2015).\ Surveys,
%  Astrometric Follow-Up, and Population Statistics.\ Asteroids IV, In
%  press.
\bibitem[Jenniskens et al.(2009)]{jenniskens_nat} Jenniskens, P.,
  Shaddad, M.~H., Numan, D., et al. 2009.\ The impact and recovery of
  asteroid 2008 TC$_{3}$.\ Nature 458, 485-488.
%\bibitem[Jenniskens and Shaddad(2010)]{jenniskens_mps} Jenniskens, P.,
%  Shaddad, M.~H.\ 2010.\ 2008 TC$_{3}$: The small asteroid with an
%  impact.\ Meteoritics and Planetary Science 45, 1553-1556.
\bibitem[Juri{\'c} et al.(2002)]{sdss} Juri{\'c}, M., Ivezi\'c,
  \v{Z}., Lupton, R.~H., et al. 2002.\ Comparison of Positions and
  Magnitudes of Asteroids Observed in the Sloan Digital Sky Survey
  with Those Predicted for Known Asteroids.\ The Astronomical Journal
  124, 1776-1787.
\bibitem[Kawaguchi et al.(2008)]{hayabuse} Kawaguchi, J., Fujiwara,
  A., Uesugi, T.\ 2008.\ Hayabusa---Its technology and science
  accomplishment summary and Hayabusa-2.\ Acta Astronautica 62,
  639-647.
\bibitem[Kowalski et al.(2008)]{mpec_tc3} Kowalski, R.~A., Beshore,
  E.~C., Boattini, A., et
  al. 2008.\ 2008~TC3.\ Minor Planet Electronic Circulars 2008-T50.
\bibitem[Kowalski et al.(2014)]{mpec_aa} Kowalski, R.~A., Boattini,
  A., Christesen, E.~J., et al. 2014.\ 2014~AA.\ Minor Planet
  Electronic Circulars 2014-A02.
\bibitem[Kozubal et al.(2011)]{kozubal} Kozubal, M.~J., Gasdia, F.~W.,
  Dantowitz, R.~F., Scheirich, P., Harris, A.~W.\ 2011.\ Photometric
  observations of Earth-impacting asteroid 2008~TC$_{3}$.\ Meteoritics
  and Planetary Science 46, 534-542.
\bibitem[Kubica et al.(2007)]{kubica} Kubica, J., Denneau, L., Grav,
  T., Heasley, J., Jedicke, R., Masiero, J., Milani, A., Moore, A.,
  Tholen, D., Wainscoat, R.~J.\ 2007.\ Efficient intra- and
  inter-night linking of asteroid detections using kd-trees.\ Icarus
  189, 151-168.
\bibitem[Larson et al.(1998)]{catalina} Larson, S., Brownlee, J.,
  Hergenrother, C., Spahr, T.\ 1998.\ The Catalina Sky Survey for
  NEOs.\ Bulletin of the American Astronomical Society 30, 1037.
\bibitem[McMillan(2000)]{spacewatch} McMillan, R.~S.\ 2000.\
  Spacewatch Survey of the Solar System.\ NASA STI/Recon Technical
  Report N 1, 18998.
\bibitem[Milani et al.(2002)]{milani_ast3} Milani, A., Chesley, 
S.~R., Chodas, P.~W., Valsecchi, G.~B.\ 2002.\ Asteroid Close Approaches: 
Analysis and Potential Impact Detection.\ Asteroids III 55-69.
\bibitem[Milani et al.(2004)]{milani04} Milani, A., Gronchi, G.~F.,
  Vitturi, M.~D., Kne{\v z}evi{\'c}, Z.\ 2004.\ Orbit determination
  with very short arcs. I admissible regions.\ Celestial Mechanics and
  Dynamical Astronomy 90, 57-85.
\bibitem[Milani et al.(2005)]{milani_im} Milani, A., Chesley, S.~R.,
  Sansaturio, M.~E., Tommei, G., Valsecchi, G.~B.\ 2005.\ Nonlinear
  impact monitoring: line of variation searches for impactors.\ Icarus
  173, 362-384.
\bibitem[Milani and Kne{\v z}evi{\'c}(2005)]{milani05}
  Milani, A., Kne{\v z}evi{\'c}, Z.\ 2005.\ From Astrometry to
  Celestial Mechanics: Orbit Determination with Very Short Arcs.\
  Celestial Mechanics and Dynamical Astronomy 92, 1-18.
\bibitem[Milani and Gronchi(2010)]{orbdet} Milani, A.,
  Gronchi, G.~F.\ 2010.\ Theory of Orbit Determination. Cambridge
  University Press, Cambridge, UK, 2010.
\bibitem[Muinonen and Bowell(1993)]{bayes} Muinonen, K.,
  Bowell, E.\ 1993.\ Asteroid orbit determination using Bayesian
  probabilities.\ Icarus 104, 255-279.
\bibitem[Oszkiewicz et al.(2009)]{mcmc} Oszkiewicz, D., Muinonen, K.,
  Virtanen, J., Granvik, M.\ 2009.\ Asteroid orbital ranging using
  Markov-Chain Monte Carlo.\ Meteoritics and Planetary Science 44,
  1897-1904.
%\bibitem[Shaddad et al.(2010)]{shaddad} Shaddad, M.~H., Jenniskens,
%  P., Numan, D., et al. 2010.\
%  The recovery of asteroid 2008 TC$_{3}$.\ Meteoritics and Planetary
%  Science 45, 1557-1589.
\bibitem[Scheirich et al.(2010)]{scheirich} Scheirich, P., et
  al. 2010.\ The shape and rotation of asteroid 2008 TC$_{3}$.\
  Meteoritics and Planetary Science 45, 1804-1811.
\bibitem[Stokes et al.(2000)]{linear} Stokes, G.~H.,
  Evans, J.~B., Viggh, H.~E.~M., Shelly, F.~C., Pearce, E.~C.\ 2000.\
  Lincoln Near-Earth Asteroid Program (LINEAR).\ Icarus 148, 21-28.
\bibitem[Stuart and Binzel(2004)]{stuart} Stuart, J.~S., 
Binzel, R.~P.\ 2004.\ Bias-corrected population, size distribution, and 
impact hazard for the near-Earth objects.\ Icarus 170, 295-311.
\bibitem[Virtanen et al.(2001)]{stat_ran} Virtanen, J.,
  Muinonen, K., Bowell, E.\ 2001.\ Statistical Ranging of Asteroid
  Orbits.\ Icarus 154, 412-431.
\bibitem[Virtanen and Muinonen(2006)]{virt_as1} Virtanen,
  J., Muinonen, K.\ 2006.\ Time evolution of orbital uncertainties for
  the impactor candidate 2004 AS$_{1}$.\ Icarus 184, 289-301.
\bibitem[Wainscoat et al.(2014)]{ps1} Wainscoat, R.,
  Veres, P., Bolin, B., Denneau, L., Jedicke, R., Chastel, S.,
  Micheli, M.\ 2014.\ The Pan-STARRS search for near-Earth objects.\
  Asteroids, Comets, Meteors 2014 574.
\bibitem[Yamaguchi et al.(2011)]{hayabusa_obs} Yamaguchi, T., 
  Yoshikawa, M., Yagi, M., Tholen, D.~J.\ 2011.\ Entry Dispersion Analysis 
  for the HAYABUSA Spacecraft using Ground-Based Optical Observation.\ 
  Publications of the Astronomical Society of Japan 63, 979-985.
\end{thebibliography}

\end{document}

1504.00025.extracted_bib
[Abell et al.(2014)]{arm Abell, P., Gates, M., Johnson, L., Chodas, P., Brophy, J., Mazanek, D., Muirhead, B.\ 2014.\ NASA's Asteroid Redirect Mission: Overview and Status.\ 40th COSPAR Scientific Assembly 40, 17.
1504.00025.extracted_bib
[Blythe et al.(2004a)]{mpec_as1 Blythe, M, Shelly, F., Bezpalko, M., et al. 2004.\ 2004~AS1.\ Minor Planet Electronic Circulars 2004-A56.
1504.00025.extracted_bib
[Blythe et al.(2004b)]{mpec_fu162 Blythe, M, Shelly, F., Bezpalko, M., et al. 2004.\ 2004~FU162.\ Minor Planet Electronic Circulars 2004-Q22.
1504.00025.extracted_bib
[Bottke et al.(2002)]{bottke02 Bottke, W.~F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H.~F., Michel, P., Metcalfe, T.~S.\ 2002.\ Debiased Orbital and Absolute Magnitude Distribution of the Near-Earth Objects.\ Icarus 156, 399-433.
1504.00025.extracted_bib
[Bowell et al.(1989)]{bowell Bowell, E., Hapke, B., Domingue, D., Lumme, K., Peltoniemi, J., Harris, A.~W.\ 1989.\ Application of photometric models to asteroids.\ Asteroids II 524-556.
1504.00025.extracted_bib
[Bressi and Tubbiolo(2015)]{mpec_bu92 Bressi, T.~H., Tubbiolo, A.~F., 2015.\ 2015~BU92.\ Minor Planet Electronic Circulars 2015-B95. %
1504.00025.extracted_bib
[Broucke and Cefola(1972)]{equ Broucke, R.~A., Cefola, P.~J.\ % 1972.\ On the Equinoctial Orbit Elements.\ Celestial Mechanics 5, % 303-310.
1504.00025.extracted_bib
[Brown et al.(2002)]{impact_frequency Brown, P., Spalding, R.~E., ReVelle, D.~O., Tagliaferri, E., Worden, S.~P.\ 2002.\ The flux of small near-Earth objects colliding with the Earth.\ Nature 420, 294-296.
1504.00025.extracted_bib
[Brown(2014)]{brown_acm Brown, P.\ 2014.\ Using the international monitoring system of seismic, infrasound, and hydroacoustic sensors for global airburst detection.\ Asteroids, Comets, Meteors 2014 63. %
1504.00025.extracted_bib
[Chambers and PS1 Science Consortium(2015)]{ps1 Chambers, % K.~C., PS1 Science Consortium 2015.\ Results from the Pan-STARRS1 % Sky Surveys.\ American Astronomical Society Meeting Abstracts 225, % \#332.01.
1504.00025.extracted_bib
[Chesley(2005)]{sys_ran_steve Chesley, S.~R.\ 2005.\ Very short arc orbit determination: the case of asteroid 2004 FU$_{162$.\ IAU Colloq.~197: Dynamics of Populations of Planetary Systems 255-258. %
1504.00025.extracted_bib
[Chesley et al.(2014)]{chesley_dps14 Chesley, S.~R., % Farnocchia, D., Brown, P., Chodas, P.~W.\ 2014.\ Orbit Estimation % for Late Warning Asteroid Impacts: The Case of 2014 AA.\ % AAS/Division for Planetary Sciences Meeting Abstracts 46, \#403.03.
1504.00025.extracted_bib
[Chesley et al.(2015)]{chesley_ieee Chesley, S.~R., Farnocchia, D., Brown, P., Chodas, P.~W.\ 2015.\ Orbit Estimation for Late Warning Asteroid Impacts: The Case of 2014 AA.\ IEEE Aerospace Conference, \#2.1202. %
1504.00025.extracted_bib
[Farnocchia et al.(2015a)]{debiasing Farnocchia, D., Chesley, % S.~R., Chamberlin, A.~B., Tholen, D.~J.\ 2015.\ Star catalog % position and proper motion corrections in asteroid astrometry.\ % Icarus 245, 94-111.
1504.00025.extracted_bib
[Farnocchia et al.(2015)]{farnocchia_ast4 Farnocchia, D., Chesley, S.~R., Milani, A., Gronchi, G.~F., Chodas, P.~W.\ 2015.\ Orbits, long-term predictions, impact monitoring.\ Asteroids IV, In press.
1504.00025.extracted_bib
[Granvik et al.(2009)]{openorb Granvik, M., Virtanen, J., Oszkiewicz, D., Muinonen, K.\ 2009.\ OpenOrb: Open-source asteroid orbit computation software including statistical ranging.\ Meteoritics and Planetary Science 44, 1853-1861.
1504.00025.extracted_bib
[Granvik et al.(2014)]{granvik_neo Granvik, M., Morbidelli, A., Jecicke, R., et al.\ 2014.\ Unbiased dynamical and physical characteristics of the near-Earth-object population.\ Asteroids, Comets, Meteors 2014 180.
1504.00025.extracted_bib
[Grav et al.(2011)]{grav11 Grav, T., Jedicke, R., Denneau, L., Chesley, S., Holman, M.~J., Spahr, T.~B.\ 2011.\ The Pan-STARRS Synthetic Solar System Model: A Tool for Testing and Efficiency Determination of the Moving Object Processing System.\ Publications of the Astronomical Society of the Pacific 123, 423-447.
1504.00025.extracted_bib
[Harris(2002)]{harris Harris, A.~W.\ 2002.\ A New Estimate of the Population of Small NEAs.\ Bulletin of the American Astronomical Society 34, 835.
1504.00025.extracted_bib
[Jaynes(1968)]{prior Janyes, E.~T. 1968. Priori Probabilities. IEEE Transactions On Systems Science and Cybernetics 4, 227-241. %
1504.00025.extracted_bib
[Jedicke et al.(2015)]{jedicke_ast4 Jedicke, R., Granvik, M., % Micheli, M., Ryan, E., Spahr, T., Yeomans, D.~K.\ (2015).\ Surveys, % Astrometric Follow-Up, and Population Statistics.\ Asteroids IV, In % press.
1504.00025.extracted_bib
[Jenniskens et al.(2009)]{jenniskens_nat Jenniskens, P., Shaddad, M.~H., Numan, D., et al. 2009.\ The impact and recovery of asteroid 2008 TC$_{3$.\ Nature 458, 485-488. %
1504.00025.extracted_bib
[Jenniskens and Shaddad(2010)]{jenniskens_mps Jenniskens, P., % Shaddad, M.~H.\ 2010.\ 2008 TC$_{3$: The small asteroid with an % impact.\ Meteoritics and Planetary Science 45, 1553-1556.
1504.00025.extracted_bib
[Juri{\'c et al.(2002)]{sdss Juri{\'c, M., Ivezi\'c, \v{Z., Lupton, R.~H., et al. 2002.\ Comparison of Positions and Magnitudes of Asteroids Observed in the Sloan Digital Sky Survey with Those Predicted for Known Asteroids.\ The Astronomical Journal 124, 1776-1787.
1504.00025.extracted_bib
[Kawaguchi et al.(2008)]{hayabuse Kawaguchi, J., Fujiwara, A., Uesugi, T.\ 2008.\ Hayabusa---Its technology and science accomplishment summary and Hayabusa-2.\ Acta Astronautica 62, 639-647.
1504.00025.extracted_bib
[Kowalski et al.(2008)]{mpec_tc3 Kowalski, R.~A., Beshore, E.~C., Boattini, A., et al. 2008.\ 2008~TC3.\ Minor Planet Electronic Circulars 2008-T50.
1504.00025.extracted_bib
[Kowalski et al.(2014)]{mpec_aa Kowalski, R.~A., Boattini, A., Christesen, E.~J., et al. 2014.\ 2014~AA.\ Minor Planet Electronic Circulars 2014-A02.
1504.00025.extracted_bib
[Kozubal et al.(2011)]{kozubal Kozubal, M.~J., Gasdia, F.~W., Dantowitz, R.~F., Scheirich, P., Harris, A.~W.\ 2011.\ Photometric observations of Earth-impacting asteroid 2008~TC$_{3$.\ Meteoritics and Planetary Science 46, 534-542.
1504.00025.extracted_bib
[Kubica et al.(2007)]{kubica Kubica, J., Denneau, L., Grav, T., Heasley, J., Jedicke, R., Masiero, J., Milani, A., Moore, A., Tholen, D., Wainscoat, R.~J.\ 2007.\ Efficient intra- and inter-night linking of asteroid detections using kd-trees.\ Icarus 189, 151-168.
1504.00025.extracted_bib
[Larson et al.(1998)]{catalina Larson, S., Brownlee, J., Hergenrother, C., Spahr, T.\ 1998.\ The Catalina Sky Survey for NEOs.\ Bulletin of the American Astronomical Society 30, 1037.
1504.00025.extracted_bib
[McMillan(2000)]{spacewatch McMillan, R.~S.\ 2000.\ Spacewatch Survey of the Solar System.\ NASA STI/Recon Technical Report N 1, 18998.
1504.00025.extracted_bib
[Milani et al.(2002)]{milani_ast3 Milani, A., Chesley, S.~R., Chodas, P.~W., Valsecchi, G.~B.\ 2002.\ Asteroid Close Approaches: Analysis and Potential Impact Detection.\ Asteroids III 55-69.
1504.00025.extracted_bib
[Milani et al.(2004)]{milani04 Milani, A., Gronchi, G.~F., Vitturi, M.~D., Kne{\v zevi{\'c, Z.\ 2004.\ Orbit determination with very short arcs. I admissible regions.\ Celestial Mechanics and Dynamical Astronomy 90, 57-85.
1504.00025.extracted_bib
[Milani et al.(2005)]{milani_im Milani, A., Chesley, S.~R., Sansaturio, M.~E., Tommei, G., Valsecchi, G.~B.\ 2005.\ Nonlinear impact monitoring: line of variation searches for impactors.\ Icarus 173, 362-384.
1504.00025.extracted_bib
[Milani and Kne{\v zevi{\'c(2005)]{milani05 Milani, A., Kne{\v zevi{\'c, Z.\ 2005.\ From Astrometry to Celestial Mechanics: Orbit Determination with Very Short Arcs.\ Celestial Mechanics and Dynamical Astronomy 92, 1-18.
1504.00025.extracted_bib
[Milani and Gronchi(2010)]{orbdet Milani, A., Gronchi, G.~F.\ 2010.\ Theory of Orbit Determination. Cambridge University Press, Cambridge, UK, 2010.
1504.00025.extracted_bib
[Muinonen and Bowell(1993)]{bayes Muinonen, K., Bowell, E.\ 1993.\ Asteroid orbit determination using Bayesian probabilities.\ Icarus 104, 255-279.
1504.00025.extracted_bib
[Oszkiewicz et al.(2009)]{mcmc Oszkiewicz, D., Muinonen, K., Virtanen, J., Granvik, M.\ 2009.\ Asteroid orbital ranging using Markov-Chain Monte Carlo.\ Meteoritics and Planetary Science 44, 1897-1904. %
1504.00025.extracted_bib
[Shaddad et al.(2010)]{shaddad Shaddad, M.~H., Jenniskens, % P., Numan, D., et al. 2010.\ % The recovery of asteroid 2008 TC$_{3$.\ Meteoritics and Planetary % Science 45, 1557-1589.
1504.00025.extracted_bib
[Scheirich et al.(2010)]{scheirich Scheirich, P., et al. 2010.\ The shape and rotation of asteroid 2008 TC$_{3$.\ Meteoritics and Planetary Science 45, 1804-1811.
1504.00025.extracted_bib
[Stokes et al.(2000)]{linear Stokes, G.~H., Evans, J.~B., Viggh, H.~E.~M., Shelly, F.~C., Pearce, E.~C.\ 2000.\ Lincoln Near-Earth Asteroid Program (LINEAR).\ Icarus 148, 21-28.
1504.00025.extracted_bib
[Stuart and Binzel(2004)]{stuart Stuart, J.~S., Binzel, R.~P.\ 2004.\ Bias-corrected population, size distribution, and impact hazard for the near-Earth objects.\ Icarus 170, 295-311.
1504.00025.extracted_bib
[Virtanen et al.(2001)]{stat_ran Virtanen, J., Muinonen, K., Bowell, E.\ 2001.\ Statistical Ranging of Asteroid Orbits.\ Icarus 154, 412-431.
1504.00025.extracted_bib
[Virtanen and Muinonen(2006)]{virt_as1 Virtanen, J., Muinonen, K.\ 2006.\ Time evolution of orbital uncertainties for the impactor candidate 2004 AS$_{1$.\ Icarus 184, 289-301.
1504.00025.extracted_bib
[Wainscoat et al.(2014)]{ps1 Wainscoat, R., Veres, P., Bolin, B., Denneau, L., Jedicke, R., Chastel, S., Micheli, M.\ 2014.\ The Pan-STARRS search for near-Earth objects.\ Asteroids, Comets, Meteors 2014 574.
1504.00025.extracted_bib
[Yamaguchi et al.(2011)]{hayabusa_obs Yamaguchi, T., Yoshikawa, M., Yagi, M., Tholen, D.~J.\ 2011.\ Entry Dispersion Analysis for the HAYABUSA Spacecraft using Ground-Based Optical Observation.\ Publications of the Astronomical Society of Japan 63, 979-985.