Proper elements for resonant planet-crossing asteroids

Abstract

Proper elements are quasi-integrals of motion of a dynamical system, meaning that they can be considered constant over a certain timespan, and they permit to describe the long-term evolution of the system with a few parameters. Near-Earth objects (NEOs) generally have a large eccentricity, and therefore they can cross the orbits of the planets. Moreover, some of them are known to be currently in a mean-motion resonance with a planet. Thus, the methods previously used for the computation of main-belt asteroid proper elements are not appropriate for such objects. In this paper, we introduce a technique for the computation of proper elements of planet-crossing asteroids that are in a mean-motion resonance with a planet. First, we numerically average the Hamiltonian over the fast angles while keeping all the resonant terms, and we describe how to continue a solution beyond orbit-crossing singularities. Proper elements are then extracted by a frequency analysis of the averaged orbit-crossing solutions. We give proper elements of some known resonant NEOs and provide comparisons with non-resonant models. These examples show that it is necessary to take into account the effect of the resonance for the computation of accurate proper elements.

Crossing singularity

1.1 The minimum orbit intersection distance

Let $(E, \ell ), (E', \ell ') \in \mathbb {R}^6$ be two sets of orbital elements of two Keplerian orbits with a common focus. The components $E, E' \in \mathbb {R}^5$ describe the shape of the orbit, while $\ell , \ell ' \in S^1$ are the mean anomalies. We denote with $\mathcal {E}= (E,E') \in \mathbb {R}^{10}$ the couple of the orbit configurations, and with $V = (\ell ,\ell ') \in \mathbb {T}^2 = S^1 \times S^1$ the parameters along the orbits. We choose a reference frame centered at the common focus and we denote with $\mathcal {X}(E,\ell ), \mathcal {X}'(E', \ell ')$ the Cartesian coordinates of the two bodies. For a given configuration $\mathcal {E}$, we define the Keplerian distance function d as

$$\begin{aligned} d: \mathbb {T}^2 \rightarrow \mathbb {R}, \quad d(\mathcal {E}, V) = |\mathcal {X}-\mathcal {X}'|. \end{aligned}$$

(A.1)

Let $V_h=V_h(\mathcal {E})$ be a local minimum point⁹ of the Keplerian distance function and consider the maps

$$\begin{aligned} \mathcal {E}\mapsto d_h(\mathcal {E}) = d(\mathcal {E}, V_h), \quad \mathcal {E}\mapsto d_{\text {min}}(\mathcal {E}) = \min _{h} d_h(\mathcal {E}, V_h). \end{aligned}$$

(A.2)

A configuration $\mathcal {E}$ is non-degenerate if all the critical points of the Keplerian distance function are non-degenerate. If $\mathcal {E}$ is non-degenerate, then there exists a neighborhood $\mathcal {W}\subseteq \mathbb {R}^{10}$ of $\mathcal {E}$ such that the maps $d_h$, restricted to $\mathcal {W}$, do not have bifurcations.

The functions $d_h$ and $d_{\text {min}}$ are not smooth at crossing configurations, and their derivatives do not exist. However, it is possible to define analytical maps in a neighborhood of a non-degenerate crossing configuration $\mathcal {E}_c$ by choosing an appropriate sign for the maps. We summarize the procedure to deal with the crossing singularity of $d_h$, the procedure for $d_{\text {min}}$ being the same. We consider the points on the two ellipses corresponding to the local minimum points $V_h = (\ell _h, \ell '_h)$ of $d^2$, i.e.,

$$\begin{aligned} \mathcal {X}_h = \mathcal {X}(E, \ell _h), \quad \mathcal {X}'_h = \mathcal {X}'(E',\ell '_h). \end{aligned}$$

(A.3)

We denote with $\tau _h, \tau '_h$ the tangent vectors to the trajectories $E,E'$ at these points, i.e.,

$$\begin{aligned} \tau _h = \frac{\partial \mathcal {X}}{\partial \ell }(E,\ell _h), \quad \tau '_h = \frac{\partial \mathcal {X}'}{\partial \ell '}(E',\ell '_h), \end{aligned}$$

(A.4)

and their cross-product

$$\begin{aligned} \tau ^*_h = \tau _h \times \tau '_h. \end{aligned}$$

(A.5)

We define also $\Delta = \mathcal {X}-\mathcal {X}', \Delta _h = \mathcal {X}_h-\mathcal {X}'_h$. The vector $\Delta _h$ joins the points attaining a local minimum value of $d^2$, hence $|\Delta _h|=d_h$. From the definition of critical points of $d^2$, both vectors $\tau _h, \tau '_h$ are orthogonal to $\Delta _h$, therefore $\tau ^*_h$ and $\Delta _h$ are parallel. Denoting with $\widehat{\tau }^*_h,\widehat{\Delta }_h$ the corresponding unit vectors, the distance with sign

$$\begin{aligned} \tilde{d}_h = \big ( \widehat{\tau }^*_h \cdot \widehat{\Delta }_h \big ) d_h, \end{aligned}$$

(A.6)

is an analytic function in a neighborhood of a crossing configuration, provided that $\tau _h$ and $\tau '_h$ are not parallel, situation happening only when the trajectories are tangent at the crossing point (Gronchi and Tommei 2007). The derivatives of $\tilde{d}_h$ with respect to the component $\mathcal {E}_k, \, k=1,\dots ,10$ of $\mathcal {E}$ are given by

$$\begin{aligned} \frac{\partial \tilde{d}_h}{\partial \mathcal {E}_k} = \widehat{\tau }^*_h \cdot \frac{\partial \Delta }{\partial \mathcal {E}_k}(\mathcal {E},V_h). \end{aligned}$$

(A.7)

1.2 Extraction of the singularity

Denote by $\mathcal {E}_c$ a non-degenerate crossing configuration with only one crossing point. We choose the index h such that $d_h(\mathcal {E}_c) = 0$. For each $\mathcal {E}$ in a neighborhood of $\mathcal {E}_c$, we consider the Taylor development of $V\mapsto d^2(\mathcal{E},V) = |\mathcal {X}-\mathcal {X}'|^2$, in a neighborhood of the local minimum point $V_h=V_h(\mathcal {E})$, i.e.,

$$\begin{aligned} d^2(\mathcal{E},V) = d_h^2(\mathcal{E}) + \frac{1}{2}(V-V_h)\cdot H_h(\mathcal{E})(V-V_h) + \mathcal{R}^{(h)}(\mathcal{E},V)\,, \end{aligned}$$

(A.8)

where

$$\begin{aligned} H_h(\mathcal{E}) = \frac{\partial ^2 d^2}{\partial V^2}(\mathcal {E},V_h(\mathcal {E})), \end{aligned}$$

(A.9)

is the Hessian matrix of $d^2$ at $V_h=(\ell _h,\ell _h')$, and $\mathcal{R}^{(h)}$ is the Taylor remainder. We introduce the approximated distance

$$\begin{aligned} \delta _h = \sqrt{d_h^2 + (V-V_h)\cdot \mathcal{A}_h(V-V_h)}\,, \end{aligned}$$

(A.10)

where

$$\begin{aligned} \mathcal {A}_h = \frac{1}{2}H_h = \left[ \begin{array}{cc} |\tau _h|^2+ \displaystyle \frac{\partial ^2\mathcal {X}}{\partial \ell ^2}(E,\ell _h)\cdot \Delta _h &{}-\tau _h\cdot \tau _h' \\ &{} \\ -\tau _h\cdot \tau _h' &{}|\tau _h'|^2 -\displaystyle \frac{\partial ^2\mathcal {X}'}{\partial \ell '^2}(E',\ell _h')\cdot \Delta _h \\ \end{array} \right] , \end{aligned}$$

(A.11)

and

$$\begin{aligned} \Delta _h = \Delta _h(\mathcal {E})\,, \quad \tau _h = \frac{\partial {\mathcal X}}{\partial \ell }(E,\ell _h)\,, \quad \tau _h' = \frac{\partial {\mathcal X}'}{\partial \ell '}(E',\ell _h')\ . \end{aligned}$$

(A.12)

If the matrix $\mathcal {A}_h$ is non-degenerate, then it is positive definite since $V_h$ is a minimum point, and this property holds in a suitably chosen neighborhood $\mathcal {W}$ of $\mathcal {E}_c$. The matrix $\mathcal {A}_h$ is degenerate at the crossing configuration if and only if the tangent vectors $\tau _h,\tau _h'$ are parallel; therefore, in the following, we always assume that the crossing is not tangent.

To extract the singularity at an orbit crossing, we split the integral as

$$\begin{aligned} \int _{\mathbb {T}^2}^{}\frac{1}{d}\, \text {d}\ell \text {d}\ell ' = \int _{\mathbb {T}^2}^{}\bigg (\frac{1}{d}-\frac{1}{\delta _h} \bigg ) \text {d}\ell \text {d}\ell ' + \int _{\mathbb {T}^2}^{}\frac{1}{\delta _h}\text {d}\ell \text {d}\ell '. \end{aligned}$$

(A.13)

Let us set $\mathcal {S} = \{ \mathcal {E}\in \mathcal {W}: \, d_h(\mathcal {E})=0 \}$, and denote with $y_k \in \{ \Sigma , U, V, \sigma , u, v \}$ one of the coordinates. The derivatives of the first term in the right-hand side of Eq. (A.13) are integrable, and the map

$$\begin{aligned} \mathcal {W}\setminus \mathcal {S} \ni \mathcal {E}\mapsto \int _{\mathbb {T}^2}^{}\frac{\partial }{\partial y_k}\bigg (\frac{1}{d}-\frac{1}{\delta _h}\bigg ) \text {d}\ell \text {d}\ell ', \end{aligned}$$

(A.14)

can be extended continuously to the whole set $\mathcal {W}$. To compute the derivatives in Eq. (A.14), we can use

$$\begin{aligned} \frac{\partial }{\partial y_k}\bigg (\frac{1}{\delta _h}\bigg ) = -\frac{1}{2\delta _h^3}\frac{\partial \delta _h^2}{\partial y_k}. \end{aligned}$$

(A.15)

From Eq. (A.8), we obtain the derivatives of the approximated distance as

$$\begin{aligned} \frac{\partial \delta _h^2}{\partial y_k} = \frac{\partial d_h^2}{\partial y_k} - 2 \frac{\partial V_h}{\partial y_k}\cdot \mathcal {A}_h(V-V_h) + (V-V_h)\cdot \frac{\partial \mathcal {A}_h}{\partial y_k}(V-V_h). \end{aligned}$$

(A.16)

The derivatives of $V_h$ are computed by differentiating the relation

$$\begin{aligned} \frac{\partial }{\partial y_k}d_h^2(\mathcal {E}, V_h(\mathcal {E})) = 0, \end{aligned}$$

(A.17)

which holds since $(\mathcal {E}, V_h(\mathcal {E}))$ is a stationary point of $d^2$. Hence,

$$\begin{aligned} \frac{\partial V_h}{\partial y_k}(\mathcal {E}) = - [H_h(\mathcal {E})]^{-1}\frac{\partial }{\partial y_k} \nabla _V d^2(\mathcal {E},V_h(\mathcal {E})). \end{aligned}$$

(A.18)

Note that the derivatives of 1/d are obtained with standard computations, and they can be expressed through the derivatives of the position of the asteroid with respect to the Delaunay variables. On the other hand, the average over $\mathbb {T}^2$ of the derivatives of $1/\delta _h$ is non-convergent integrals for $\mathcal {E}\in \mathcal {S}$ and contains the main part of the singularity of the vector field.

1.3 Integration of $1/\delta _h$ and its derivatives

Let $(\mathcal {E}_c, V_h(\mathcal {E}_c))$ be a crossing configuration. We consider the transformations

$$\begin{aligned} \mathcal{T}_h(V) = V + V_h, \quad \mathcal{L}_h(V) = \sqrt{\mathcal{A}_h}\ V, \end{aligned}$$

(A.19)

where $\sqrt{\mathcal {A}_h}$ is defined as the unique positive definite matrix such that $(\sqrt{\mathcal {A}_h})^2 = \mathcal{A}_h$. With these constraints, the entries $a_{ij}$ of $\sqrt{\mathcal {A}_h}$ are

$$\begin{aligned} a_{11} = \frac{\alpha +A_{11}}{\sqrt{2\alpha + A_{11}+A_{22}}}, \quad a_{22} = \frac{\alpha +A_{22}}{\sqrt{2\alpha + A_{11}+A_{22}}}, \quad a_{12} = \frac{A_{12}}{\sqrt{2\alpha + A_{11}+A_{22}}}, \end{aligned}$$

(A.20)

where $\alpha = \sqrt{\det \mathcal {A}_h}$, and $A_{ij}$ are the entries of $\mathcal {A}_h$. Using these transformations to change the coordinates in the integral, we get

$$\begin{aligned} \int _{\mathbb {T}^2}^{}\frac{1}{\delta _h}\text {d}\ell \text {d}\ell ' = \int _{\mathcal{T}_h(\mathbb {T}^2)}\frac{1}{\delta _h} \text {d}V = \frac{1}{{\sqrt{\det \mathcal {A}_h}}} \int _{\mathcal{L}_h(\mathbb {T}^2)}\frac{1}{\sqrt{d_h^2 + |W|^2}}\text {d}W \end{aligned}$$

(A.21)

where

$$\begin{aligned} W = \mathcal{L}_h\circ \mathcal{T}_h^{-1}(V) = \mathcal{L}_h(V - V_h). \end{aligned}$$

(A.22)

Let us consider the points $P_1 \equiv (\pi ,\pi ), P_2 \equiv (-\pi ,\pi ), P_3 \equiv (-\pi ,-\pi ), P_4 \equiv (\pi ,-\pi )$ and their images $Q_j \equiv (x_j,y_j), j=1,\dots ,4$ through $\mathcal{L}_h$, so that

$$\begin{aligned} (x_1, y_1)&= \pi (a_{11}+a_{12}, a_{12}+a_{22}), \quad (x_2, y_2) = \pi ( -a_{11}+a_{12}, -a_{12}+a_{22}), \end{aligned}$$

(A.23)

$$\begin{aligned} (x_3, y_3)&= -(x_1,y_1), \quad (x_4, y_4) = -(x_2,y_2). \end{aligned}$$

(A.24)

Set $P_5=P_1$ and, for $j=1,\ldots ,4$, let $\mathscr {R}_j$ be the straight line passing through the points $P_j, P_{j+1}$, i.e.,

$$\begin{aligned} \xi _j(y-y_j) = \eta _j(x-x_j), \end{aligned}$$

(A.25)

where $\xi _j = x_{j+1}-x_j, \ \eta _j = y_{j+1}-y_j.$ Introducing polar coordinates $(\rho ,\theta )$ such that $W = (\rho \cos \theta , \rho \sin \theta )$, we can write these lines in polar form

$$\begin{aligned} \mathscr {R}_j = \bigl \{\bigl (r_j(\theta )\cos \theta ,r_j(\theta )\sin (\theta )\bigr ): \theta \in (\bar{\theta }_j,\bar{\theta }_j+\pi )\bigr \} \end{aligned}$$

(A.26)

with

$$\begin{aligned} r_j(\theta ) = \frac{\xi _j y_j - \eta _jx_j}{\xi _j\sin \theta -\eta _j\cos \theta } \end{aligned}$$

(A.27)

and

$$\begin{aligned} \bar{\theta }_j = \left\{ \begin{array}{ll} \arctan ({\eta _j}/{\xi _j}), \qquad &{}\xi _j\ne 0, \\ \pi /2, &{}\xi _j= 0. \\ \end{array} \right. \end{aligned}$$

(A.28)

Note that

$$\begin{aligned} (\xi _1,\eta _1)&= -2\pi (a_{11}, a_{12}),\qquad (\xi _2,\eta _2) = -2\pi (a_{12}, a_{22}), \end{aligned}$$

(A.29)

$$\begin{aligned} (\xi _3,\eta _3)&= -(\xi _1,\eta _1), \qquad (\xi _4,\eta _4) = -(\xi _2,\eta _2), \end{aligned}$$

(A.30)

so that, for each $j=1,\ldots ,4$,

$$\begin{aligned} \xi _j y_j - \eta _j x_j = -2\pi ^2\sqrt{\det \mathcal{A}_h} \end{aligned}$$

(A.31)

and

$$\begin{aligned} r_1(\theta )&= \frac{\pi \sqrt{\det \mathcal{A}_h} }{a_{11}\sin \theta -a_{12}\cos \theta }, \qquad r_2(\theta ) = \frac{\pi \sqrt{\det \mathcal{A}_h}}{a_{12}\sin \theta -a_{22}\cos \theta }, \end{aligned}$$

(A.32)

$$\begin{aligned} r_3(\theta )&= -r_1(\theta ), \qquad r_4(\theta ) = -r_2(\theta ). \end{aligned}$$

(A.33)

With these changes of coordinates, Eq. (A.21) becomes

$$\begin{aligned} \int _{\mathcal{T}_h(\mathbb {T}^2)}\frac{1}{\delta _h}\text {d}\ell \text {d}\ell ' = \frac{1}{\sqrt{\det \mathcal{A}_h}}\left( \sum _{j=1}^4\int _{\theta _j}^{\theta _{j+1}}\sqrt{d_h^2 + r_j^2(\theta )}\text {d}\theta - 2\pi d_h\right) \end{aligned}$$

(A.34)

with

$$\begin{aligned} \cos \theta _j = \frac{x_j}{\sqrt{x_j^2+y_j^2}}, \qquad \sin \theta _j = \frac{y_j}{\sqrt{x_j^2+y_j^2}}, \end{aligned}$$

(A.35)

and

$$\begin{aligned} \theta _1<\theta _2<\theta _3<\theta _4<\theta _5 = 2\pi +\theta _1. \end{aligned}$$

(A.36)

The integrals in Eq. (A.34) are bounded; hence, they are differentiable functions of the elements. On the contrary, the term $-2\pi d_h/\sqrt{\det \mathcal{A}_h}$ is not differentiable at $\mathcal {E}= \mathcal {E}_c\in \mathcal {S}$, and the loss of regularity is due only to this term. The derivatives of Eq. (A.34) with respect to $y_k \in \{ \Sigma , U, V, \sigma , u,v \}$ can be computed by exchanging the integral sign and the derivative, i.e.,

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial y_k}\int _{\mathbb {T}^2}\frac{1}{\delta _h}\text {d}\ell \text {d}\ell '&= \left( \frac{\partial }{\partial y_k}\frac{1}{\sqrt{\det \mathcal{A}_h}}\right) \left( \sum _{j=1}^4\int _{\theta _j}^{\theta _{j+1}}\sqrt{d_h^2 + r_j^2(\theta )}\text {d}\theta - 2\pi d_h\right) \\&\quad + \frac{1}{\sqrt{\det \mathcal{A}_h}}\left( \sum _{j=1}^4\int _{\theta _j}^{\theta _{j+1}} \frac{d_h\frac{\partial d_h}{\partial y_k}+ r_j(\theta )\frac{\partial r_j}{\partial y_k}(\theta ) }{\sqrt{d_h^2 + r_j^2(\theta )}}\text {d}\theta - 2\pi \frac{\partial d_h}{\partial y_k}\right) . \end{aligned} \end{aligned}$$

(A.37)

The term $-2\pi d_h/\sqrt{\det {\mathcal {A}_h}}$ is not differentiable at the orbit crossing; however, the derivatives admit two analytic extensions $(\frac{\partial \mathcal {K}_{\text {sec}}}{\partial y_k})_h^{\pm }$ on $\mathcal {W}^+ = \mathcal {W}\cap \{\tilde{d}_h > 0\}$ and $\mathcal {W}^- = \mathcal {W}\cap \{\tilde{d}_h < 0\}$, where $\mathcal{W}$ is a neighborhood of the crossing configuration $\mathcal{E}_c$ where $\tilde{d}_h$ is defined, and $\mathcal{A}_h$ is non-degenerate (Gronchi and Tardioli 2013). Moreover, the jump in the derivatives passing from $\mathcal {W}^+$ to $\mathcal {W}^-$ is given by

$$\begin{aligned} \begin{aligned} \text {Diff}_h\bigg ( \frac{\partial \mathcal {K}_{\text {sec}}}{\partial y_k} \bigg )&:= \bigg ( \frac{\partial \mathcal {K}_{\text {sec}}}{\partial y_k}\bigg )_h^- - \bigg ( \frac{\partial \mathcal {K}_{\text {sec}}}{\partial y_k}\bigg )_h^+ \\&= \frac{1}{\pi }\bigg [ \frac{\partial }{\partial y_k}\bigg ( \frac{1}{\sqrt{\det \mathcal {A}_h}}\bigg )\tilde{d}_h + \frac{1}{\sqrt{\det \mathcal {A}_h}}\frac{\partial \tilde{d}_h}{\partial y_k}\bigg ]. \end{aligned} \end{aligned}$$

(A.38)