Abstract
We study the postencounter evolution of fictitious small bodies belonging to the socalled Line of Variations (LoV) in the framework of the analytic theory of close encounters. We show the consequences of the encounter on the local minimum of the distance between the orbit of the planet and that of the small body and get a global picture of the way in which the planetocentric velocity vector is affected by the encounter. The analytical results are compared with those of numerical integrations of the restricted threebody problem.
Introduction
In the framework of their extension of the analytic theory of close encounters originally formulated by Öpik (1976), Valsecchi et al. (2003) introduced the socalled wire approximation, a simple analytic description of the Line of Variations (LoV) in which the orbital uncertainty of a small body encountering a planet is modelled assuming that its orbital parameters a, e, i, \(\Omega \) and \(\omega \) are constant, and all of the uncertainty is in the timing of the encounter, i.e. in the mean anomaly M.
This rather simplified description differs from other more sophisticated modelizations like those described in Milani (1999) and Milani et al. (2005), implemented in the software robots CLOMON2^{Footnote 1} and Sentry,^{Footnote 2} which are in charge of determining whether a given NearEarth asteroid (NEA) can impact the Earth in the coming century. These software robots take into account the uncertainties in all the orbital elements and make accurate propagations in time, including all the known perturbations, while in the wire approximation the only uncertainty taken into account is that on the timing of the encounter, which in turn is computed in a very simplified model. On the other hand, the simple analytical formulation of the wire approximation allows one to capture some important features of the problem.
The outcome of a planetary flyby of a planetcrossing small body strongly depends on its coordinates on the “target plane”, or bplane, of the encounter (Kizner 1961; Greenberg et al. 1988; Valsecchi et al. 2018), i.e. the plane centred on the planet and perpendicular to the planetocentric velocity “at infinity” of the small body. The uncertainty of the postencounter trajectory is a function of the uncertainty in the orbital elements at the time of the encounter and in most cases of interest is dominated by the uncertainty in the time of closest approach. A suitable choice of the target plane coordinates is such that one coordinate represents the local minimum distance between the orbit of the small body and that of the planet, and the other is proportional to the timing of the encounter. In this way, the uncertainty is mostly along a line parallel to one of the coordinate axes and can be modelled by the socalled Line of Variations (LoV). The LoV approach is a crucial ingredient of the Impact Monitoring software developed at the University of Pisa and at the JPL.
In this paper, we show some geometric features of the wire approximation that are deducible from its analytic formulation and that can be useful to interpret the outcomes of impact monitoring computations coming from sophisticated, accurate models of the motion of NEAs on Earthapproaching orbits.
The paper is organized as follows: In Sect. 2 we briefly recall the analytic theory and how the LoV is described by the wire approximation; in Sect. 3 we discuss what happens to the LoV at a close encounter. Afterwards, in Sect. 4 we give the overall geometric picture of how the planetocentric velocity vectors of the points belonging to the LoV are rotated, and in Sect. 5 we summarize the conclusions.
Analytic theory of close encounters
The analytic theory of close encounters has been developed over the years, starting from Öpik (1976), in a sequence of papers (Greenberg et al. 1988; Carusi et al. 1990; Valsecchi et al. 2003; Valsecchi 2006; Valsecchi et al. 2015a, b), to which we refer the reader.
The basic assumptions are that the small body is massless, and the planet moves on a circular orbit about the Sun, similarly to what is assumed in the restricted, circular, threedimensional threebody problem; however, far from the planet, the small body is assumed to move on an unperturbed heliocentric Keplerian orbit, not being subject to the perturbation by the planet. Then, when a close encounter with the planet takes place, the interaction is modelled as an instantaneous transition from the incoming asymptote of the planetocentric hyperbola to the outgoing one, taking place when the small body crosses the bplane (Kizner 1961; Öpik 1976; Greenberg et al. 1988; Carusi et al. 1990).
The computation relies on some intermediate variables, the socalled Öpik variables, constituted by:

the planetocentric velocity vector \(\mathbf {U}\), whose components in a reference frame centred on the planet, with the Xaxis pointing away from the Sun, and the Yaxis in the direction of the planet motion, are given by \(U_x=U\sin \theta \sin \phi \), \(U_y=U\cos \theta \), and \(U_z=U\sin \theta \cos \phi \) (see Fig. 1);

the two bplane coordinates \(\xi \) and \(\zeta \);

the time \(t_b\) at which the bplane is crossed.
As a consequence of an encounter, the direction of \(\mathbf {U}\) changes but its modulus U does not; explicit expressions linking the preencounter to the postencounter orbital parameters, making use of Öpik variables, are given in Carusi et al. (1990), Valsecchi et al. (2003) and Valsecchi (2006).
The bplane
As was already mentioned, the bplane of an encounter is the plane containing the planet and perpendicular to the planetocentric unperturbed velocity \(\mathbf {U}\). The vector from the planet to the point in which \( \mathbf {U}\) crosses the plane is \(\mathbf {b}\), and the coordinates on the bplane are \(\xi \) and \(\zeta \). As defined in Valsecchi et al. (2003) and Valsecchi (2006), \(\xi =\xi (a,e,i,\omega ,\mathop {\mathrm {sgn}}{f_b})\) is the local MOID (minimum orbital intersection distance), and \(\zeta =\zeta (a,e,i,\Omega ,\omega ,f_b,\lambda _p)\) is related to the timing of the encounter. In these expressions, a, e, i, \(\Omega \) and \(\omega \) are the semimajor axis, eccentricity, inclination, longitude of node and argument of perihelion of the preencounter orbit of the small body, and \(f_b\) and \(\lambda _p\) are the true anomaly of the small body and the longitude of the planet at the crossing of the bplane.
The wire approximation
In the wire approximation, we consider the encounter of a stream of small bodies spaced in mean anomaly (that is, in true anomaly and therefore in \(\zeta \)), all on the same orbit, with given local MOID \(\xi _0\). Then, as previously mentioned, U does not change as a result of the close encounter and \(t_b\) does not concern us here.
The encounter changes the angles \(\theta \) and \(\phi \) into \(\theta '\) and \(\phi '\). Moreover, we can consider that for each pair \(\theta ',\phi '\) we can define an “outgoing” bplane, normal to postencounter velocity vector \(\mathbf {U}'\),^{Footnote 3} that is crossed by the small body at coordinates \(\xi '\) and \(\zeta '\). We call this plane the “postencounter bplane”.
Valsecchi et al. (2003) give the relevant equations to compute the postencounter quantities of the small bodies along the wire:
with c given by:
where m is the mass of the planet in units of that of the Sun. Particularly noteworthy is the expression for \(\xi '\), which gives the postencounter local MOID; we discuss its implications in Sect. 3.
Postencounter local MOID along the wire
On the postencounter bplane, the size of the postencounter impact parameter \(b'\)must be the same as that of the preencounter one b, due to the conservation of the planetocentric orbital angular momentum; thus, the postencounter local MOID \(\xi '\) is bounded:
Moreover, since \(\theta \) and \(\theta '\) take values between \(0^\circ \) and \(180^\circ \) (Carusi et al. 1990), \(\xi \) and \(\xi '\) have the same sign. (We remind the reader that \(\xi \) and \(\xi '\) are coordinates on different planes.)
Let us now discuss the variation in size of the local MOID due to the encounter, for a wire that has \(\xi =\xi _0\). Equations (1) and (2) show that, the larger the value of \(\zeta \), the closer \(\theta '\) will be to \(\theta \), and thus, the closer \(\xi '\) will be to \(\xi _0\).
For smaller values of \(\zeta \), there must be a minimum and a maximum value of \(\sin \theta '\) that correspond, respectively, to the maximum and minimum values of \(\xi '\). To find them, let us consider the derivative of \(\sin \theta '\) with respect to \(\zeta \):
The zeroes of \(\partial \sin \theta '/\partial \zeta \) include those of \(\partial \cos \theta '/\partial \zeta \), as well as the values of \(\zeta \) such that \(\cos \theta '=0\). As regards the zeroes of \(\partial \cos \theta '/\partial \zeta \), these can be found by zeroing the numerator of this derivative, since its denominator cannot be negative:
Concerning the values of \(\zeta \) such that \(\cos \theta '=0\), they are given by the intersections of the straight line \(\xi =\xi _0\) with the circle having coordinates of the centre (0, D) and radius R, given by Valsecchi et al. (2000, 2003):
The equation of the circle is:
Its intersections with the straight line \(\xi =\xi _0\) are the roots of the equation:
that are:
Summarizing, the zeroes of \(\partial \sin \theta '/\partial \zeta \) are:
Note that for
there is no intersection of the circle corresponding to \(\cos \theta '=0\) with the straight line \(\xi =\xi _0\), so that there are no real values for the roots \(\zeta _{1,2}\).
The values of \(\xi '\) corresponding to \(\zeta _\pm \) are:
while those of \(\xi '\) corresponding to \(\zeta _{1,2}\), when present, are:
To see how the above expressions work in practice, let us apply the analytical theory to the encounter of 2012 TC\(_4\) with the Earth that has taken place on 12 October 2017. The geocentric velocity of 2012 TC\(_4\) is \(U=0.235\), relatively low for a NEA, making this case rather challenging for the analytic theory that works best for encounters in which the planetocentric velocity is high.
Figure 2 shows the bplane relative to this encounter; in it, the black circle centred in the origin is the gravitational cross section of the Earth, and the unit adopted for the axes is the physical radius of our planet \(r_\oplus \). In the case of 2012 TC\(_4\), the effective radius of the Earth, due to gravitational focussing, is \(1.89r_\oplus \). The black dots represent LoV points for two different values of \(\xi \), namely \(\xi _0=4\) and \(\xi _0=2\) Earth radii; the red dots show the postencounter values \(\xi ',\zeta '\) corresponding to each pair \(\xi _0, \zeta \), for \(\xi _0=4\) Earth radii, and the blue dots do the same for \(\xi _0=2\) Earth radii. As already said, to each pair \(\xi _0, \zeta \) corresponds a postencounter pair \(\xi ', \zeta '\) defined on a different postencounter bplane; here, however, we plot them on the preencounter bplane in order to show how the LoV is deformed as a consequence of the close encounter. It is noteworthy how the variation of the local MOID can be, at least in this case, comparable to the radius of the Earth.
Figure 3 is similar to Fig. 2, but also shows the relevant bplane loci (Valsecchi et al. 2018) whose intersections with the LoVs give origin to specific values of \(\xi '\). These loci are:

the condition \(\partial \cos \theta '/\partial \zeta =0\), shown by the cyan hyperbola;

the condition \(\theta '=\theta \) (Valsecchi et al. 2000), implying \(a'=a\) and \(\xi '=\xi \), shown by the black horizontal straight line;

the condition \(\cos \theta '=0\) (in this particular case giving \(a'=1.05\) au), corresponding to \(\xi '=\xi \sin \theta \), shown by the green circle.
Let us examine the LoV with \(\xi _0=4\) Earth radii, going from positive \(\zeta \) values towards negative ones. For large positive values of \(\zeta \), as already said, \(\theta '\) tends to \(\theta \), so the variation of \(\xi \) is small.
Going towards \(\zeta =0\), the LoV crosses the hyperbola for \(\zeta =\zeta _+\): This corresponds to the maximum of \(\cos \theta '\), i.e. to the minimum of \(\sin \theta '\), and thus to the maximum of \(\xi '\). The values of \(\theta '\), \(\phi '\) and \(\xi '\) are given by:
The next locus encountered by the \(\xi _0=4\) Earth radii LoV is the horizontal straight line corresponding to \(\cos \theta '=\cos \theta \). In this case, the local MOID is unchanged, \(\xi '=\xi _0\).
Finally, the LoV encounters the other branch of the hyperbola, in \(\zeta _\); here, the values of \(\theta '\), \(\phi '\) and \(\xi '\) are given by:
A noteworthy feature is that \(\phi '_+=\phi '_\), implying that the extrema of the postencounter values of \(\theta \) lie on the same meridian. The difference \(\Delta _{\mathrm{max}}\theta '\) between them is:
we will return on this issue in Sect. 4.
Coming now to the \(\xi _0=2\) Earth radii LoV, the crossings of the hyperbola and of the straight line are as before. However, this LoV crosses also the green circle corresponding to \(\cos \theta '=0\); actually, one of these crossings happens also to take place at the border of the cross section of the Earth. Anyway, at these two crossings we have:
Numerical check
To test the validity of the theoretical predictions about the variation of the local MOID along the LoV, we proceeded as in Section 4 of Valsecchi et al. (2018). That is, we integrated the equations of the restricted, circular, threedimensional threebody problem using the RA15 integrator (Everhart 1985) with initial conditions corresponding to the 12 October 2017 encounter of 2012 TC\(_4\).
By trial and error, we found the preencounter values of \(\omega \) corresponding to \(\xi _0=4\) and \(\xi _0=2\) Earth radii, and then we integrated sets of initial conditions equally spaced in \(\zeta \), thus reproducing the two LoVs of interest. Then, we determined the postencounter values \(\xi ', \zeta '\) and plotted them in Fig. 4 that has to be compared with Fig. 2.
The theoretical behaviour of the LoV is very well confirmed by the numerical integrations.
Rotation of \(\mathbf {U}\) into \(\mathbf {U}'\) along the wire
The conservation of U implies that the preencounter and postencounter velocity vectors \(\mathbf {U}\) and \(\mathbf {U}'\) span a sphere in X–Y–Z space, the \(U\!\)sphere, of radius U centred in the origin and on which the angles \(\theta \) and \(\phi \) define a system of parallels and meridians: on the \(U\!\)sphere \(\theta \) is the colatitude measured from the Yaxis (the direction of motion of the planet), and \(\phi \) is the longitude, counted from the ZY plane.
Let us examine the path followed by the tip of \(\mathbf {U}'\) on the \(U\!\)sphere for a given value of \(\xi _0\). The angle \(\gamma \) between the vectors \(\mathbf {U}\) and \(\mathbf {U}'\) is given by:
this implies that, for \(\zeta \rightarrow \pm \infty \), \(\gamma \rightarrow 0\). On the other hand, \(\gamma _{\mathrm{max}}\), the maximum value of \(\gamma \), is reached for \(\zeta =0\) and is given by:
A comparison of (24) and (26) shows that \(\Delta _{\mathrm{max}}\theta '\) and \(\gamma _{\mathrm{max}}\) are equal; this suggests the possibility that the path followed by the tip of \(\mathbf {U}'\) on the \(U\!\)sphere for a given value of \(\xi _0\) might be a circle.
To check whether this is true, let us consider meridian \(\phi _P=\phi '_+=\phi '_\). On it lie the points \(P_+\), of coordinates \((X_+, Y_+, Z_+)\), and \(P_\), of coordinates \((X_, Y_, Z_)\), where
and
These two points correspond to the tips of the postencounter velocity vector for, respectively, \(\zeta =\zeta _+\) and \(\zeta =\zeta _\); halfway between them, on the same meridian, we now consider point P, whose colatitude \(\theta _P\) is halfway between \(\theta '_+\) and \(\theta '_\), so that:
From Eqs. (14), (16), (17) and (19), we can compute \(\theta _P, \phi _P\) as functions \(c, \theta , \phi , \xi _0\):
In the X–Y–Z frame, the coordinates of P are:
On the other hand, the postencounter values \(\theta ', \phi '\) for a generic initial condition “on the wire”, of coordinates \(\xi _0, \zeta \), can be computed using Eqs. (1)–(4). The corresponding point on the \(U\!\)sphere will have coordinates:
From Eqs. (1)–(4), we rewrite the above expressions as follows:
The square of distance \(D_P\) from P to a generic point on the \(U\!\)sphere corresponding to an initial condition “on the wire” is:
substituting the expressions as functions of \(U, \theta , \phi , \xi _0, \zeta \), one obtains:
Thus, the postencounter values of \(\theta '\) and \(\phi '\) accessible to a small body encountering the planet “on the wire” define the circle resulting from the intersection of the cone of aperture \(\gamma _{\mathrm{max}}\), centred in the centre of the Usphere, and the sphere itself. The pole of the spherical cap delimited by the circle is point P.
For \(\xi _0^2\gg {c}^2\), i.e. when the MOID is relatively large, and very close encounters are not possible, \(\theta _P, \phi _P\) become:
that is, P tends towards the tip of \(\mathbf {U}\). On the other hand, for \(\xi _0^2\ll {c}^2\), \(\theta _P, \phi _P\) become:
Moreover, it is clear that the smaller becomes \(\xi _0\) relative to c, the larger becomes the circle spanned by the initial conditions “on the wire” until, for \(\xi _0=0\), it becomes the great circle corresponding to the \(\phi \)meridian. For \(\xi _0\ne 0\), the circle is tangent to the \(\phi \)meridian in the point of spherical coordinates \(\theta , \phi \).
For completeness, we now give expressions for the coordinates of the centre and for the radius \(R_C\) of the circle. The centre C lies on the axis of the cone, at a distance \(U\sin (\gamma _{\mathrm{max}}/2)\) from the origin of axes, while the radius \(R_C\) is equal to \(U\cos (\gamma _{\mathrm{max}}/2)\). Since
the coordinates of the C are:
and the radius \(R_C\) is:
As an application of the above considerations, let us consider the already mentioned recent encounter of 2012 TC\(_4\) with the Earth. The relevant quantities in this case are:
where \(r_\oplus \) is the radius of the Earth.
Figure 5 helps to visualize the situation. It shows \(\mathbf {U}\) in the X–Y–Z frame, as well as the Usphere, spanned by \(\mathbf {U}'\) for all possible values of \(\xi , \zeta \); the red circle is the intersection of the Usphere with the X–Y plane, and the angles \(\theta , \phi \) are indicated. The green circle is spanned by \(\mathbf {U}'\) for all possible values of \(\zeta \) and \(\xi =2.38\) Earth radii.
Figure 6 shows the situation for a value of \(\xi _0=c\); this value is still negative, but closer to 0, and for it \(\gamma _{\mathrm{max}}\) amounts to \(90^\circ \). Note that we are showing the behaviour of \(\mathbf {U}'\) also for deflections that would imply a perigee of the real asteroid smaller than the radius of the Earth, in order to give the overall view of the geometry involved. Obviously, in a realistic computation, parts of the green circle would be forbidden, due to the impact.
Finally, Fig. 7 shows what happens when \(\xi _0\) changes sign. In the case shown, \(\xi _0=3.65\) terrestrial radii, so that \(\gamma _{\mathrm{max}}=38^\circ \!\!.9\). As already said, for \(\xi _0=0\) the green circle becomes a great circle; afterwards, the green circle starts to shrink on the other side, as \( \xi _0\) starts to increase after having passed through 0.
Conclusions
We discussed how a close encounter, in which the local MOID is well determined and the timing is somewhat uncertain, can be modelled with the wire approximation, in which the LoV on the bplane is described by \(\xi =\xi _0\) and \(\zeta \) takes any value within the uncertainty range.
Explicit expressions can then be given that describe the behaviour of the LoV after the encounter. In particular, these expressions allow one to describe the variation of the local MOID that in some cases can be of the order of the radius of the Earth and thus have consequences for the possibility of subsequent impacts.
Numerical integrations in the restricted, circular threedimensional threebody problem confirm that the theoretical results on the variation of the local MOID are satisfactorily accurate.
Moreover, the theory allows us to give the overall geometrical description of how the planetocentric velocity vector is deflected at the encounter, as a function of the MOID of the orbits described by the LoV in the wire approximation. In fact, for a LoV of given \(\xi _0\), the postencounter values of \(\theta '\) and \(\phi '\) lead to a circle resulting from the intersection of the cone of aperture \(\gamma _{\mathrm{max}}=\gamma _{\mathrm{max}}(\xi _0, c)\), centred in the centre of the sphere spanned by \(\mathbf {U}'\), and the sphere itself.
Comparisons of these results with those that can be obtained in realistic situations, for real asteroids possibly impacting the Earth, will be the subject of future work.
Notes
 1.
 2.
 3.
The components of \(\mathbf {U}'\) in the X–Y–Z reference frame are (\(U\sin \theta '\sin \phi ',~U\cos \theta ',~U\sin \theta '\cos \phi '\)).
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Acknowledgements
We are grateful to D. Farnocchia for his very useful comments.
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Valsecchi, G.B., Del Vigna, A. & Ceccaroni, M. The evolution of the Line of Variations at close encounters: an analytic approach. Celest Mech Dyn Astr 131, 47 (2019). https://doi.org/10.1007/s1056901999242
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Keywords
 Close encounter
 Perturbation
 Asteroid
 Comet