All Figures

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{ms2023f1.eps} \end{figure}$ Figure 1: Basic geometric setup of Öpik's theory. The planet is at the origin, moving in the direction of Y, with the Sun at unit distance on the negative X-axis. $\vec{U}$ is the planetocentric velocity vector of the small body (see text).
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In the text

$\begin{figure} \par\includegraphics[width=4cm,height=40mm,clip]{ms2023f2.eps} \end{figure}$ Figure 2: The deflection of $\vec{U}$ . The vectors $\vec{U}$ and $\vec{U'}$ lie on, respectively, the incoming and outgoing asymptotes of the planetocentric hyperbola; b (the segment PA) is the impact parameter, $\gamma$ is the deflection angle, and the length of the segments AB and BC is c.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{ms2023f3.eps}\end{figure}$ Figure 3: Same as Fig. 1, but with also the post-encounter velocity $\vec{U'}$ .
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{ms2023f4.eps} \end{figure}$ Figure 4: Time variation of the nodal distance and of the MOID with respect to the Earth for 1997 XF₁₁.
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In the text

$\begin{figure} \par\includegraphics[width=17cm,clip]{ms2023f5.eps} \end{figure}$ Figure 5: The outcomes, computed with the extended Öpik theory, of the August 2027 encounter with the Earth of asteroid 1999AN₁₀. Left: final states in the a-e plane (a circle marks the pre-encounter orbit); right: final states in the plane $\Delta t$ (difference in time from closest approach; $\Delta t=0$ for encounter at the MOID) vs. P (post-encounter orbital period). The left panel is equivalent to Fig. 2 of Milani et al. (2000c), the right panel to Fig. 1 of Milani et al. (1999), in order to show how the extended Öpik theory reproduces the important features of the encounter. For the meaning of the inclined lines, see text.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{ms2023f6.eps} \end{figure}$ Figure 6: The dotted lines show the circles, in the b-plane of the August 2027 encounter with the Earth of 1999 AN₁₀, corresponding to the mean motion resonances 7/13 (upper circle), 10/17 (smaller of the two lower circles), 11/19; these resonances lead to returns in 2040, 2044, 2046. The units for the b-plane coordinates $\xi$ and $\zeta$ are Earth radii, and the Earth is shown at the origin with a circle drawn to scale. Note that the radius of the Earth on the b-plane is larger than 1, in this case by about 8%, due to gravitational focusing. The lines on, or close to, each circle, denote b-plane coordinates resulting in encounters within 4 Earth radii at the resonant returns, assuming a linear drift in the MOID by 0.35 Earth radii per year. To guide the eye, a vertical segment is traced at $\xi =5.8$ , corresponding to the actual MOID of about 0.000246 AU.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{ms2023f7.eps} \end{figure}$ Figure 7: Keyholes, in the b-plane of the October 2028 encounter with the Earth of 1997 XF₁₁, for collision at the resonant return in 2040; units for the coordinates are Earth radii. Upper left: keyhole nearer to the Earth, with the latter shown to scale for comparison (the gravitational focusing is here about 30%); upper right: keyhole farther from the Earth; lower left and lower right: enlargements of, respectively, the near and far keyholes shown in the corresponding upper panels. The resonant circle corresponding to the exact resonance is drawn with dots.
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In the text

$\begin{figure} \par\includegraphics[width=12cm]{ms2023f8.eps} \end{figure}$ Figure 8: Keyhole, in the b-plane of the October 2028 encounter with the Earth of 1997 XF₁₁, for collision at the resonant return in 2040; units for the coordinates are Earth radii. The initial conditions have been modified with respect to Fig. 7 in order to force the two keyholes of that figure to coalesce into one. Left: overall view; right: enlargement.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{ms2023f1.eps} \end{figure}$	Figure 1: Basic geometric setup of Öpik's theory. The planet is at the origin, moving in the direction of Y, with the Sun at unit distance on the negative X-axis. $\vec{U}$ is the planetocentric velocity vector of the small body (see text).
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$\begin{figure} \par\includegraphics[width=4cm,height=40mm,clip]{ms2023f2.eps} \end{figure}$	Figure 2: The deflection of $\vec{U}$ . The vectors $\vec{U}$ and $\vec{U'}$ lie on, respectively, the incoming and outgoing asymptotes of the planetocentric hyperbola; b (the segment PA) is the impact parameter, $\gamma$ is the deflection angle, and the length of the segments AB and BC is c.
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$\begin{figure} \par\includegraphics[width=7.5cm,clip]{ms2023f3.eps}\end{figure}$	Figure 3: Same as Fig. 1, but with also the post-encounter velocity $\vec{U'}$ .
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$\begin{figure} \par\includegraphics[width=12cm,clip]{ms2023f4.eps} \end{figure}$	Figure 4: Time variation of the nodal distance and of the MOID with respect to the Earth for 1997 XF₁₁.
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$\begin{figure} \par\includegraphics[width=12cm]{ms2023f8.eps} \end{figure}$	Figure 8: Keyhole, in the b-plane of the October 2028 encounter with the Earth of 1997 XF₁₁, for collision at the resonant return in 2040; units for the coordinates are Earth radii. The initial conditions have been modified with respect to Fig. 7 in order to force the two keyholes of that figure to coalesce into one. Left: overall view; right: enlargement.
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