Accounting for the Yarkovsky Effect in Reference Frames Associated with the Radius Vector and Velocity Vector

Abstract

The motion of an asteroid in a central gravitational field in the presence of an additional perturbing acceleration due to the Yarkovsky effect was considered. The long-term evolution of the orbit was studied using the analytical solution of the averaged equations of motion in two orbital frames of reference: ${{\mathcal{O}}_{1}}$, associated with the radius vector, and ${{\mathcal{O}}_{2}}$, associated with the velocity vector. The Yarkovsky acceleration components were found as averages over the orbital period based on the thermophysical characteristics and rotation parameters of a small body within the linear thermophysical model of the Yarkovsky force for spherical asteroids. The drifts of the mean anomaly and semimajor axis, as well as the displacement relative to the unperturbed position per 1000 orbital revolutions, were obtained for model asteroids with different orbital eccentricities in both frames of reference. As a result, the drifts of the semimajor axis and the mean anomaly, as well as the displacements, were found to differ by less than 1% at small eccentricities (up to 0.5). When ${{e}_{0}} > 0.5$, the values found in system ${{\mathcal{O}}_{1}}$ are always greater than the same values in system ${{\mathcal{O}}_{2}}$. At ${{e}_{0}} \sim 0.5{-} 0.7$, their difference does not exceed 6%, gradually increasing with the growth of ${{e}_{0}}$. For ${{e}_{0}} > 0.7$, these differences increase exponentially. Thus, when the Yarkovsky effect is modeled with transversal acceleration, the expected values of drifts and displacements for objects with highly elliptical orbits may be overestimated, which may be one of the factors for the low detection of the Yarkovsky effect directly from astrometric observations.

1 INTRODUCTION

At present, humanity has come to recognize the reality of the threat of a collision between the Earth and small bodies of the Solar System. To prevent this threat, it is necessary to identify potentially dangerous objects, determine their orbits, and assess the probability of their collision or close encounter with the Earth. To improve the accuracy of predicting the motion of a body, it is necessary to consider non-gravitational effects, in particular, the Yarkovsky effect. This effect occurs due to the thermal radiation of a rotating body with non-zero thermal inertia, and it causes secular variations in the eccentricity, semimajor axis and, first of all, the mean anomaly, the variation of which increases quadratically with time. Thus, the Yarkovsky effect plays a significant role in the evolution of the orbits of small bodies; therefore, determining the magnitude of this effect and studying its impact on near-Earth asteroids (NEAs) is necessary for calculating their orbits and assessing their potential hazard to the Earth. Also, the Yarkovsky effect is considered one of the reasons for the migration of asteroids to the resonance zones of the Main Belt, after which they may replenish the NEA population. In addition, covering the object with a special substance to change its orbit using the Yarkovsky effect is proposed as one of the ways to manipulate space objects that threaten to collide with the Earth.

To take into account the Yarkovsky effect and study its influence on the long-term evolution of the orbit of a small body, it is necessary to know the values of the components of the perturbing acceleration that occurs due to this effect. At present, the most common method for estimating the Yarkovsky effect is differential orbit correction [1–4]. Since the Yarkovsky effect leads to secular perturbations of the semimajor axis, transversal acceleration is used in the following form

$${{{\mathbf{a}}}_{t}} = {{A}_{2}}{{\left( {\frac{{{{r}_{0}}}}{r}} \right)}^{2}}{\mathbf{\hat {t}}},$$

where $r$ is the heliocentric distance to the asteroid, ${{r}_{0}} = 1$ AU, and ${{A}_{2}}$ is the dynamic parameter determined in the orbital fitting with the orbital elements. Furthermore, the drift of the semimajor axis of the orbit is estimated, and this drift is considered when predicting the motion of the asteroid. However, when the effect is determined by this method, a significant part of it remains unaccounted, which can cause noticeable errors in predicting the motion of a body based on the resulting orbit [5].

A more accurate method, in our opinion, is the calculation of the acceleration components based on some model of the Yarkovsky force. At present, extensive work is underway to determine the thermophysical characteristics of small bodies using observations obtained in ground-based and orbital observatories, for example, determining the shape and rotation parameters of asteroids from their photometry [6, 7], determining the diameter and geometric albedo from the total thermal radiation of bodies found as a result of a four-band thermal infrared all-sky survey carried out by an infrared telescope located in near-Earth orbit (NEOWISE project) [8], or refining the thermal characteristics of the surface during laboratory studies of meteorite and asteroid samples [9], as well as during space missions to asteroids [10–12]. In the future, as the knowledge about the properties of small bodies expands, explicit consideration of the Yarkovsky effect will become more preferable.

As mentioned above, when estimating the drift of the semimajor axis due to the Yarkovsky effect, the perturbing acceleration is usually modeled with a transversal component. However, the tangential acceleration component has a more direct effect on the variation in the object’s velocity and, as a result, on the drifts of the semimajor axis and the mean anomaly. This is not essential for circular orbits and small eccentricities, when the transversal and tangential components almost coincide. However, for highly elliptical orbits, the situation is different. It is also obvious that the magnitude of the variation in the orbital elements should not depend on the choice of the reference frame. Furthermore, we determine the element drifts and the displacement relative to the unperturbed position per 1000 orbital revolutions for model asteroids with thermophysical characteristics similar to the asteroid 101 955 Bennu, but with different orbital eccentricities in two orbital systems and compare them.

For this purpose, let us consider the motion of a zero-mass point $\mathcal{A}$(asteroid) under the influence of attraction to the central body $\mathcal{S}$(Sun) and perturbing acceleration ${\mathbf{P}}{\kern 1pt} '$, which is inversely proportional to the distance to $\mathcal{S}$ squared, i.e., ${\mathbf{P}}{\kern 1pt} ' = {\mathbf{P}}{\text{/}}{{r}^{2}}$, and its value is small compared to the main acceleration ${{\varkappa }^{2}}{\text{/}}{{r}^{2}}$:

$$\max \frac{{{\text{|}}{\mathbf{P}}{\kern 1pt} '{\text{|}}}}{{{{\varkappa }^{2}}{{r}^{{ - 2}}}}} = \max \frac{{{\text{|}}{\mathbf{P}}{\kern 1pt} {\text{|}}}}{{{{\varkappa }^{2}}}} = \mu \ll 1.$$

(1)

Here, ${\mathbf{r}} = \mathcal{S}\mathcal{A}$, $r = {\text{|}}{\mathbf{r}}{\text{|}}$, ${{\varkappa }^{2}}$ is the product of the gravitational constant and mass $\mathcal{S}$, and $\mu$ is a small parameter. Let the acceleration ${\mathbf{P}}{\kern 1pt} '$ be due to the Yarkovsky effect. In this case, condition (1) is satisfied, since for an NEA with a diameter below 1 km, the typical value of the transversal parameter A₂ ~ 10^–15–10^–13 AU/day², and for a distance of 1 AU, $\mu \approx {{10}^{{ - 9}}} \ll 1$. The vector ${\mathbf{P}}$ components are constant and small (on the order of μ) values. For this problem, the authors of [13] obtained evolutionary equations of motion in the mean elements in the first order of smallness in $\mu$ for various frames of reference. In [14, 15], these equations are integrated for two orbital frames of reference: ${{\mathcal{O}}_{1}}$, associated with the radius vector, and ${{\mathcal{O}}_{2}}$, associated with the velocity vector. Section 2 presents particular solutions needed to study the long-term evolution of the semimajor axis and the mean anomaly.

Let us consider two reference systems with the origin at $\mathcal{S}$: ${{\mathcal{O}}_{1}}$ with the axes oriented along the radius vector, transversal (perpendicular to the radius vector in the plane of the osculating orbit along the direction of motion), and binormal (along the area vector), and ${{\mathcal{O}}_{2}}$ with the axes along the velocity vector, normal to it in the plane of the osculating orbit, and the binormal. Let vector ${\mathbf{P}}$ have components ${{P}_{r}},\;{{P}_{t}},\;{{P}_{n}}$ in system ${{\mathcal{O}}_{1}}$ and ${{P}_{\mathfrak{T}}},\;{{P}_{\mathfrak{N}}},\;{{P}_{n}}$ in ${{\mathcal{O}}_{2}}$. We denote the values of the vector ${\mathbf{P}}$ components averaged over the orbital period as follows: $S = {{\bar {P}}_{r}}$, $T = {{\bar {P}}_{t}}$, $\mathfrak{T} = {{\bar {P}}_{\mathfrak{T}}}$, $\mathfrak{N} = {{\bar {P}}_{\mathfrak{N}}}$, $W = {{\bar {P}}_{n}}$ and call them the radial, transversal, tangential, normal, and binormal parameters, respectively. In [16], the corresponding expressions are derived for the parameters $S$, $T$, $W$ based on the formulas for the components of the Yarkovsky acceleration in the projection onto the system ${{\mathcal{O}}_{1}}$ axes obtained in the linear thermophysical model of the Yarkovsky force for spherical asteroids [17] and published in [18]. In this paper, the tangential $\mathfrak{T}$ and normal $\mathfrak{N}$ parameters are derived (see Section 3).

In Section 4, the drifts of the semimajor axis and the mean anomaly, as well as the displacement relative to the unperturbed position due to the Yarkovsky effect, are found and compared for model asteroids in two orbital reference systems.

2 EQUATIONS OF MOTION

As shown in [16], the binormal component P_n averaged over the orbital period is zero (W = 0), so we use particular analytical solutions.

For reference frame ${{\mathcal{O}}_{1}}$ [14]:

$$t = \frac{{{{\varkappa }^{2}}}}{{{{n}_{0}}T}}{{\left( {\frac{{{{\eta }_{0}}}}{{1 - {{\eta }_{0}}}}} \right)}^{3}}\left( {2\ln \frac{\eta }{{{{\eta }_{0}}}} + \frac{1}{\eta } - \eta - \frac{1}{{{{\eta }_{0}}}} + {{\eta }_{0}}} \right),$$

$$\begin{gathered} a = {{a}_{0}}{{\left[ {\frac{{{{\eta }_{0}}\left( {1 - \eta } \right)}}{{\eta \left( {1 - {{\eta }_{0}}} \right)}}} \right]}^{2}}, \\ i = {{i}_{0}},\quad \Omega = {{\Omega }_{0}},\quad \omega = {{\omega }_{0}}, \\ \end{gathered} $$

(2)

$$M = {{M}_{0}} + \frac{{{{\varkappa }^{2}} - 2{\kern 1pt} S}}{T}\left( {\eta + \ln \frac{{1 - \eta }}{{1 - {{\eta }_{0}}}} - {{\eta }_{0}}} \right)$$

and for ${{\mathcal{O}}_{2}}$ [15]:

$$\begin{gathered} t = \frac{{\pi \eta _{0}^{3}{{\varkappa }^{2}}}}{{4{{n}_{0}}\mathfrak{T}}}\int\limits_{{{e}_{0}}}^e \frac{e}{{{{\eta }^{3}}[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} \\ \times \;\left( {\exp \int\limits_{{{e}_{0}}}^e \frac{{3x{\mathbf{K}}(x){\kern 1pt} dx}}{{2[{\mathbf{E}}(x) - (1 - {{x}^{2}}){\mathbf{K}}(x)]}}} \right){\kern 1pt} de, \\ \end{gathered} $$

$$a = {{a}_{0}}{{\left( {\frac{{{{\eta }_{0}}}}{\eta }} \right)}^{2}}\exp \left[ {\int\limits_{{{e}_{0}}}^e \frac{{e{\mathbf{K}}(e){\kern 1pt} de}}{{[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}}} \right],$$

$$i = {{i}_{0}},\quad \Omega = {{\Omega }_{0}},$$

(3)

$$\omega = {{\omega }_{0}} + \int\limits_{{{e}_{0}}}^e \left\{ {\frac{{e{\mathbf{K}}(e)}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}}\frac{\mathfrak{N}}{\mathfrak{T}}} \right\}{\kern 1pt} de,$$

$$\begin{gathered} M = {{M}_{0}} + \int\limits_{{{e}_{0}}}^e \left\{ {\frac{{\pi {{\varkappa }^{2}}e}}{{4[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]\mathfrak{T}}}} \right. \\ \left. {\, + \frac{{e\eta {\mathbf{K}}(e)}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}}\frac{\mathfrak{N}}{\mathfrak{T}}} \right\}{\kern 1pt} de. \\ \end{gathered} $$

Here and below, the subscript $0$ marks the values of the variables at the initial epoch $t = 0$, $e$ is the eccentricity, $a$ is the semimajor axis, $n = \varkappa {{a}^{{ - 3/2}}}$ is the mean motion, $\eta = \sqrt {1 - {{e}^{2}}}$, $i$ is the imclination, $\Omega$ is the longitude of the ascending node, $\omega$ is the argument of the pericenter, and $M$ is the mean anomaly. The standard notations for complete elliptic integrals in normal trigonometric form are also used [19]:

$$\begin{gathered} {\mathbf{K}}(e) = \int\limits_0^{\pi /2} \frac{{dx}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} , \\ {\mathbf{E}}(e) = \int\limits_0^{\pi /2} \sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} {\kern 1pt} dx. \\ \end{gathered} $$

(4)

Solution (3) contains definite integrals from combinations of complete elliptic integrals, which can be found by numerical methods.

The first expression in systems (2) and (3) is a kinematic equation, which can be used to find the time during which a given change in eccentricity occurs, and, vice versa, by solving the kinematic equation $t(e) = \Delta t$ with respect to $e$, we can find the change in eccentricity over time $\Delta t$.

The definition domain of solutions of (2) and (3) is given in [14, 15]. The representations of these solutions are also provided there in the form of series expansions in powers of eccentricity. According to [16], in the case of solution (2), for reference frame ${{\mathcal{O}}_{1}}$ at $e \leqslant 0.8$, it is necessary to use power-law series, so we present them here:

$$\begin{gathered} t = \frac{{{{\varkappa }^{2}}}}{{{{n}_{0}}T}}{{\left[ {\sum\limits_{k = 0}^\infty \frac{{(2k + 1)!!}}{{(2k + 2)!!}}e_{0}^{{2k}}} \right]}^{{ - 3}}} \\ \times \;\sum\limits_{k = 0}^\infty \left( {\frac{{(2k + 3)!!}}{{(2k + 4)!!}} - \frac{1}{{k + 3}}} \right)\left( {\frac{{{{e}^{6}}}}{{e_{0}^{6}}}{{e}^{{2k}}} - e_{0}^{{2k}}} \right), \\ \end{gathered} $$

$$\begin{gathered} a = {{a}_{0}}{{\left( {\frac{e}{{{{e}_{0}}}}} \right)}^{4}}{{\left[ {\sum\limits_{k = 0}^\infty \frac{{(2k + 1)!!}}{{(2k + 2)!!}}e_{0}^{{2k}}} \right]}^{{ - 2}}} \\ \times \;{{\left[ {\sum\limits_{k = 0}^\infty \frac{{(2k + 1)!!}}{{(2k + 2)!!}}{{e}^{{2k}}}} \right]}^{2}}, \\ \end{gathered} $$

$$\begin{gathered} M = {{M}_{0}} + \frac{{{{\varkappa }^{2}} - 2{\kern 1pt} S}}{T} \\ \times \;\left[ {2\ln \left( {\frac{e}{{{{e}_{0}}}}} \right) - \sum\limits_{k = 1}^\infty \frac{{(2k - 2)!}}{{{{2}^{{2k}}}{{{(k!)}}^{2}}}}({{e}^{{2k}}} - e_{0}^{{2k}})} \right]. \\ \end{gathered} $$

We also give a solution for a circular orbit, which is preferable to use instead of (2), (3) for $e \approx 0$:

$$a = {{a}_{0}}{{\left( {1 + \frac{t}{{{{t}_{1}}}}} \right)}^{{2/3}}},\quad i = {{i}_{0}},\quad \Omega = {{\Omega }_{0}},\quad \omega = {{\omega }_{0}},$$

$$\lambda = {{\lambda }_{0}} + {{n}_{0}}{{t}_{1}}\left( {1 + \frac{{2\mathfrak{N}}}{{{{\varkappa }^{2}}}}} \right)\ln \left( {1 + \frac{t}{{{{t}_{1}}}}} \right)\quad {\text{at}}\quad {{t}_{1}} = \frac{{{{\varkappa }^{2}}}}{{3\mathfrak{T}{{n}_{0}}}},$$

where $\lambda = \Omega + \omega + M$ is the mean longitude. This solution is also valid in reference frame ${{\mathcal{O}}_{1}}$ with replacement $\mathfrak{T} \to T$, $\mathfrak{N} \to - S$.

3 YARKOVSKY ACCELERATION MODEL

Within the linear thermophysical model of the Yarkovsky acceleration for spherical asteroids [17], the radial, transversal, and binormal components of this acceleration in the frame of reference ${{\mathcal{O}}_{1}}$ have the following form [18, Eq. (12)]:

$$\begin{gathered} P_{r}^{'} = \frac{{{{P}_{r}}}}{{{{r}^{2}}}} = \frac{{4\alpha \Phi }}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}\sin ({{\delta }_{{R_{s}^{'}}}} + \lambda )\sin \lambda {{\sin }^{2}}\gamma \\ \, + {{E}_{{R_{d}^{'}}}}\cos {{\delta }_{{R_{d}^{'}}}}[{{\cos }^{2}}\lambda + {{\sin }^{2}}\lambda {{\cos }^{2}}\gamma ]\} , \\ \end{gathered} $$

$$\begin{gathered} P_{t}^{'} = \frac{{{{P}_{t}}}}{{{{r}^{2}}}} = \frac{{4\alpha \Phi }}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}\sin ({{\delta }_{{R_{s}^{'}}}} + \lambda )\cos \lambda {{\sin }^{2}}\gamma \\ - \;{{E}_{{R_{d}^{'}}}}[\cos {{\delta }_{{R_{d}^{'}}}}\sin \lambda \cos \lambda {{\sin }^{2}}\gamma + \sin {{\delta }_{{R_{d}^{'}}}}\cos \gamma ]\} , \\ \end{gathered} $$

(5)

$$\begin{gathered} P_{n}^{'} = \frac{{{{P}_{n}}}}{{{{r}^{2}}}} = \frac{{4\alpha \Phi }}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}\sin ({{\delta }_{{R_{s}^{'}}}} + \lambda )\sin \gamma \cos \gamma \\ - \;{{E}_{{R_{d}^{'}}}}[\cos {{\delta }_{{R_{d}^{'}}}}\sin \lambda \sin \gamma \cos \gamma - \sin {{\delta }_{{R_{d}^{'}}}}\cos \lambda \sin \gamma ]\} , \\ \end{gathered} $$

where the subscript $s$ corresponds to the seasonal Yarkovsky effect, and $d$ is the daily effect. In (5), $\gamma$ is the obliquity of the asteroid’s spin axis with respect to the normal to its orbital plane, $\lambda = {{\omega }_{{{\text{rev}}}}}(t - {{t}_{0}})$ is the mean longitude, ${{\omega }_{{{\text{rev}}}}}$ is the mean motion, $t$ is the time, ${{t}_{0}}$ is the initial time point, $\alpha = 1 - A$ is the optical absorption coefficient, $A$ is the Bond albedo,

$$\begin{gathered} \Phi = \frac{{{{\mathcal{E}}_{ \star }}\pi {{R}^{2}}}}{{mc}},\quad \chi = \frac{{{{\Theta }_{s}}}}{{\sqrt 2 R_{s}^{'}}},\quad {{\Theta }_{s}} = \frac{{\Gamma \sqrt {{{\omega }_{{{\text{rev}}}}}} }}{{\epsilon \sigma T_{ \star }^{3}}}, \\ \Gamma = \sqrt {K\rho C} ,\quad {{T}_{ \star }} = {{\left( {\frac{{\alpha {{\mathcal{E}}_{ \star }}}}{{\epsilon \sigma }}} \right)}^{{1/4}}}, \\ \end{gathered} $$

$\Gamma$ is the thermal inertia of the surface, ${{T}_{ \star }}$ is the temperature of the subsolar point, ${{\mathcal{E}}_{ \star }} = {{L}_{ \odot }}{\text{/}}(4\pi {{a}^{2}})$ is the solar radiation flux at the heliocentric distance $a$, ${{L}_{ \odot }} = 3.86 \times {{10}^{{26}}}$ W is the luminosity of the Sun, $c = 299{\kern 1pt} {\kern 1pt} 792{\kern 1pt} {\kern 1pt} 458$ m/s is the speed of light, σ = $5.670374419 \times {{10}^{{ - 8}}}$ W m^–2 K^–4 is the Stefan–Boltzmann constant, $m$, $R$, $\rho$, $\epsilon$, $K$, $C$ are the mass, radius, bulk density, thermal emissivity, thermal conductivity, and specific heat capacity of the asteroid, respectively. Furthermore,

$$\begin{gathered} R_{s}^{'} = \frac{R}{{{{l}_{s}}}},\quad {{l}_{s}} = \frac{\Gamma }{{\rho C\sqrt {{{\omega }_{{{\text{rev}}}}}} }},\quad {{\omega }_{{{\text{rev}}}}} = \frac{{2\pi }}{{{{P}_{{{\text{rev}}}}}}}, \\ R_{d}^{'} = \frac{R}{{{{l}_{d}}}},\quad {{l}_{d}} = {{l}_{s}}\sqrt {\frac{{{{\omega }_{{{\text{rev}}}}}}}{{{{\omega }_{{{\text{rot}}}}}}}} ,\quad {{\omega }_{{{\text{rot}}}}} = \frac{{2\pi }}{{{{P}_{{{\text{rot}}}}}}}, \\ \end{gathered} $$

where ${{P}_{{{\text{rev}}}}}$ is the period of revolution of the asteroid around the Sun, and ${{P}_{{{\text{rot}}}}}$ is the period of its rotation around the axis. In addition, the amplitude ${{E}_{{R'}}} =$ $E(\sqrt 2 R{\kern 1pt} ')$ and phase ${{\delta }_{{R'}}} = \delta (\sqrt 2 R{\kern 1pt} ')$ are defined, like in [17], by the relations

$$\begin{gathered} {{E}_{{R{\kern 1pt} '}}}\exp (i{{\delta }_{{R{\kern 1pt} '}}}) = \frac{{A(x) + iB(x)}}{{C(x) + iD(x)}}, \\ {{E}_{{R{\kern 1pt} '}}}\exp ( - i{{\delta }_{{R{\kern 1pt} '}}}) = \frac{{A(x) - iB(x)}}{{C(x) - iD(x)}}, \\ \end{gathered} $$

(6)

where $i = \sqrt { - 1}$, $x = \sqrt 2 R{\kern 1pt} '$, and auxiliary functions

$$A(x) = - (x + 2) - {{e}^{x}}[(x - 2)\cos x - x\sin x],$$

$$B(x) = - x - {{e}^{x}}[x\cos x + (x - 2)\sin x],$$

$$\begin{gathered} C(x) = A(x) + \frac{\chi }{{1 + \chi }}\left( {3(x + 2)\mathop {}\limits_{} } \right. \\ + \;\left. {{{e}^{x}}[3(x - 2)\cos x + x(x - 3)\sin x]} \right), \\ \end{gathered} $$

$$\begin{gathered} D(x) = B(x) + \frac{\chi }{{1 + \chi }}\left( {\mathop {x(x + 3)}\limits_{} } \right. \\ \left. { - \;\mathop {{{e}^{x}}[x(x - 3)\cos x - 3(x - 2)\sin x]}\limits_{} } \right). \\ \end{gathered} $$

Taking into account (6), we obtain

$$\begin{gathered} {{E}_{{R{\kern 1pt} '}}}\cos {{\delta }_{{R{\kern 1pt} '}}} = \frac{{A(x)C(x) + B(x)D(x)}}{{C{{{(x)}}^{2}} + D{{{(x)}}^{2}}}}, \\ {{E}_{{R{\kern 1pt} '}}}\sin {{\delta }_{{R{\kern 1pt} '}}} = \frac{{B(x)C(x) - A(x)D(x)}}{{C{{{(x)}}^{2}} + D{{{(x)}}^{2}}}}. \\ \end{gathered} $$

(7)

The linear thermophysical model [17] was developed under simplifying assumptions of a circular orbit around the Sun and a spherical shape of the asteroid. For an elliptical orbit, the coefficient $\Phi$ will depend on the heliocentric distance $r$ as $\Phi = ({{\Phi }_{0}}r_{0}^{2}){\text{/}}{{r}^{2}}$, where ${{r}_{0}} = 1$ AU, and ${{\Phi }_{0}}$ is $\Phi$ calculated for the distance of 1 AU. This inverse proportionality to the square of the distance is already included in our equations of motion (2) and (3) in accordance with the problem posed in the Introduction. Furthermore, we replace the mean longitude $\lambda$, which describes the position of the body in orbit, with the mean anomaly $M$ and average the expressions for the tangential ${{P}_{\mathfrak{T}}}$ and normal ${{P}_{\mathfrak{N}}}$ components of vector P with respect to the mean anomaly over the orbital period. At the same time, we take into account that

$$\begin{gathered} {{P}_{\mathfrak{T}}} = {{P}_{r}}\sin f + {{P}_{t}}\cos f, \\ {{P}_{\mathfrak{N}}} = - {{P}_{r}}\cos f + {{P}_{t}}\sin f, \\ \end{gathered} $$

(8)

where ${{P}_{r}}$ and ${{P}_{t}}$ are the radial and transversal components of vector P, $f$ is the angle by which the velocity vector must be rotated to coincide with the transversal (Fig. 1),

$$\cos f = \frac{\eta }{{\sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}E} }}{\kern 1pt} ,\quad \sin f = \frac{{e\sin E}}{{\sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}E} }}{\kern 1pt} ,$$

(9)

$E$ is the eccentric anomaly, and $M = E - e\sin E$.

Fig. 1.

The angle of rotation $f$ of the velocity vector ${\mathbf{v}}$ until it coincides with the transversal. The axes of system ${{\mathcal{O}}_{1}}$ are radial ${{x}_{1}}$ and transversal ${{y}_{1}}$. The axes of system ${{\mathcal{O}}_{2}}$ are tangential ${{x}_{2}}$ and normal ${{y}_{2}}$. The binormal axes ${{z}_{1}}$ and ${{z}_{2}}$ are oriented toward us orthogonally to the orbital plane. The $\theta$ angle is the true anomaly.

While averaging, we assume that the orientation of the asteroid’s axis of rotation and the periods ${{P}_{{{\text{rev}}}}}$ and ${{P}_{{{\text{rot}}}}}$ do not change with time.

Considering in (5) that $\Phi = ({{\Phi }_{0}}r_{0}^{2}){\text{/}}{{r}^{2}}$, we write the expressions for ${{P}_{r}}$ and ${{P}_{t}}$ in the form

$$\begin{gathered} {{P}_{r}} = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}{{\sin }^{2}}\gamma {\kern 1pt} {\kern 1pt} \text{[}\sin {{\delta }_{{R_{s}^{'}}}}\sin 2M \\ \, + \cos {{\delta }_{{R_{s}^{'}}}}(1 - \cos 2M)] \\ \end{gathered} $$

$$ + \;{{E}_{{R_{d}^{'}}}}\cos {{\delta }_{{R_{d}^{'}}}}[1 + \cos 2M + (1 - \cos 2M){{\cos }^{2}}\gamma ]\} ,$$

(10)

$$\begin{gathered} {{P}_{t}} = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}{{\sin }^{2}}\gamma {\kern 1pt} {\kern 1pt} \text{[}\sin {{\delta }_{{R_{s}^{'}}}}(1 + \cos 2M) \\ + \;\cos {{\delta }_{{R_{s}^{'}}}}\sin 2M] \\ - \;{{E}_{{R_{d}^{'}}}}[\cos {{\delta }_{{R_{d}^{'}}}}\sin 2M{{\sin }^{2}}\gamma + 2\sin {{\delta }_{{R_{d}^{'}}}}\cos \gamma ]\} . \\ \end{gathered} $$

Let us substitute (10) into (8) and combine similar terms:

$$\begin{gathered} {{P}_{\mathfrak{T}}} = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}{{\sin }^{2}}\gamma {\kern 1pt} {\kern 1pt} \text{[}\sin {{\delta }_{{R_{s}^{'}}}}(\cos f \\ + \;\cos 2M\cos f + \sin 2M\sin f) \\ \end{gathered} $$

$$\begin{gathered} + \;\cos {{\delta }_{{R_{s}^{'}}}}(\sin f - \cos 2M\sin f \\ + \;\sin 2M\cos f)] \\ + \;{{E}_{{R_{d}^{'}}}}[\cos {{\delta }_{{R_{d}^{'}}}}([1 + \mathop {\cos }\nolimits^2 \gamma ]\sin f \\ \end{gathered} $$

(11)

$$\begin{gathered} + \;{{\sin }^{2}}\gamma {\kern 1pt} {\kern 1pt} \text{[}\cos 2M\sin f - \sin 2M\cos f]) \\ - \;2\sin {{\delta }_{{R_{d}^{'}}}}\cos \gamma \cos f]\} , \\ \end{gathered} $$

$$\begin{gathered} {{P}_{\mathfrak{N}}} = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}{{\sin }^{2}}\gamma {\kern 1pt} {\kern 1pt} \text{[}\sin {{\delta }_{{R_{s}^{'}}}}(\cos 2M\sin f \\ - \;\sin 2M\cos f + \sin f) \\ \end{gathered} $$

$$\begin{gathered} + \;\cos {{\delta }_{{R_{s}^{'}}}}(\sin 2M\sin f + \cos 2M\cos f \\ - \;\cos f)] - {{E}_{{R_{d}^{'}}}}[\cos {{\delta }_{{R_{d}^{'}}}}([1 + {{\cos }^{2}}\gamma ]\cos f \\ \end{gathered} $$

(12)

$$\begin{gathered} + \;{{\sin }^{2}}\gamma {\kern 1pt} {\kern 1pt} \text{[}\cos 2M\cos f + \sin 2M\sin f]) \\ \, + 2\sin {{\delta }_{{R_{d}^{'}}}}\cos \gamma \sin f]\} . \\ \end{gathered} $$

Let us carry out the averaging procedure of expressions (11) and (12):

$$\begin{gathered} \mathfrak{T} = {{{\bar {P}}}_{\mathfrak{T}}} = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {{P}_{\mathfrak{T}}}(M)dM \\ = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {{P}_{\mathfrak{T}}}(E)(1 - e\cos E)dE, \\ \end{gathered} $$

$$\begin{gathered} \mathfrak{N} = {{{\bar {P}}}_{\mathfrak{N}}} = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {{P}_{\mathfrak{N}}}(M)dM \\ = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {{P}_{\mathfrak{N}}}(E)(1 - e\cos E)dE, \\ \end{gathered} $$

and, considering the results given in Appendix A (Eqs. (A14)), we obtain the tangential and normal parameters:

$$\mathfrak{T} = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9\pi (1 + \chi )}}\left\{ {\mathop {{{E}_{{R_{s}^{'}}}}\sin {{\delta }_{{R_{s}^{'}}}}{{{\sin }}^{2}}\gamma {\kern 1pt} {\kern 1pt} }\limits_{}^{} \left( {\mathop {2\eta {\mathbf{K}}(e)}\limits_{_{{_{{_{{_{{}}}}}}}}} } \right.} \right.$$

$$\begin{gathered} - \;\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}[2{{e}^{2}}({{I}_{{k + 1}}} - {{I}_{{k + 2}}}) + \eta (2{{I}_{{k + 1}}} - {{I}_{k}})] \\ + \;\left. {\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}(2{{I}_{{k + 2}}} - {{I}_{{k + 1}}} + 2\eta ({{I}_{{k + 2}}} - {{I}_{{k + 1}}}))} \right) \\ \end{gathered} $$

(13)

$$ - \;\left. {\mathop {4\eta {\mathbf{K}}(e){{E}_{{R_{d}^{'}}}}\sin {{\delta }_{{R_{d}^{'}}}}\cos \gamma }\limits_{}^{} } \right\},$$

$$\begin{gathered} \mathfrak{N} = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9\pi (1 + \chi )}}\left\{ {{{E}_{{R_{s}^{'}}}}\cos {{\delta }_{{R_{s}^{'}}}}{{{\sin }}^{2}}\gamma \left( {\mathop { - 2\eta {\mathbf{K}}(e)}\limits_{}^{} } \right.} \right. \\ + \;\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}[\eta ({{I}_{k}} - 2{{I}_{{k + 1}}}) - 2{{e}^{2}}({{I}_{{k + 1}}} - {{I}_{{k + 2}}})] \\ \end{gathered} $$

$$\begin{gathered} \left. { - \;\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}[{{I}_{{k + 1}}} - 2{{I}_{{k + 2}}} + 2\eta ({{I}_{{k + 1}}} - {{I}_{{k + 2}}})]} \right) \\ \, - {{E}_{{R_{d}^{'}}}}\cos {{\delta }_{{R_{d}^{'}}}}\left[ {\mathop {2\eta {\mathbf{K}}(e)(1 + {{{\cos }}^{2}}\gamma )}\limits_{}^{} } \right. \\ \end{gathered} $$

(14)

$$\begin{gathered} + \;{{\sin }^{2}}\gamma \sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}[\eta ({{I}_{k}} - 2{{I}_{{k + 1}}}) - 2{{e}^{2}}({{I}_{{k + 1}}} - {{I}_{{k + 2}}})] \\ \left. {\left. { - \;{{{\sin }}^{2}}\gamma \sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}[2\eta ({{I}_{{k + 1}}} - {{I}_{{k + 2}}}) + {{I}_{{k + 1}}} - 2{{I}_{{k + 2}}}]} \right]} \right\}, \\ \end{gathered} $$

where ${{E}_{{R{\kern 1pt} '}}}\cos {{\delta }_{{R{\kern 1pt} '}}}$ and ${{E}_{{R{\kern 1pt} '}}}\sin {{\delta }_{{R{\kern 1pt} '}}}$ are defined by formulas (7), and ${{I}_{k}}$ is expressed by the recursive formula:

$${{I}_{k}} = \left\{ \begin{gathered} 2{\mathbf{K}}(e)\quad {\text{at}}\quad k = 0 \hfill \\ \frac{2}{{{{e}^{2}}}}[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]\quad {\text{at}}\quad k = 1 \hfill \\ \frac{{(2{{e}^{2}} - 1)(2k - 2)}}{{{{e}^{2}}(2k - 1)}}{{I}_{{k - 1}}} + \frac{{(1 - {{e}^{2}})(2k - 3)}}{{{{e}^{2}}(2k - 1)}}{{I}_{{k - 2}}} \hfill \\ {\text{at}}\quad k > 1, \hfill \\ \end{gathered} \right.$$

coefficients ${{g}_{{1k}}}$ and ${{g}_{{2k}}}$ are given in Appendix A (Eqs. (A7)), and their values for $k = 0{-} 15$ are given in Table 4 in Appendix A.

The expressions for the radial, transversal, and binormal parameters were obtained in [16]. Let us write them in the following form:

$$\begin{gathered} S = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}\cos {{\delta }_{{R_{s}^{'}}}}{{\sin }^{2}}\gamma \\ + \;{{E}_{{R_{d}^{'}}}}\cos {{\delta }_{{R_{d}^{'}}}}(1 + {{\cos }^{2}}\gamma )\} , \\ T = \frac{{2\alpha {{\Phi }_{0}}r_{0}^{2}}}{{9(1 + \chi )}}\{ {{E}_{{R_{s}^{'}}}}\sin {{\delta }_{{R_{s}^{'}}}}{{\sin }^{2}}\gamma \\ - \;2{{E}_{{R_{d}^{'}}}}\sin {{\delta }_{{R_{d}^{'}}}}\cos \gamma \} ,\quad W = 0. \\ \end{gathered} $$

(15)

Comparing (13) and (14) with (15), we see that the tangential and normal parameters differ from the transversal and radial parameters in multiplying the daily and seasonal components by eccentricity-dependent coefficients. Taking into account the expression (A15) in the Appendix, we obtain $- \mathfrak{N} = S$, $\mathfrak{T} = T$ for $e = 0$, i.e., the trihedron ($S,\;T,\;W$) coincides with the trihedron ($- \mathfrak{N},\;\mathfrak{T},\;W$) for circular orbits, as it should be. At $e = 1$, the elliptical orbit degenerates into rectilinear motion and the procedure of averaging over the orbital period loses its meaning; therefore, formulas (13)–(15) are inapplicable at $e = 1$.

Note 1. The radial $S$, transversal $T$, and binormal $W$parameters are analogues of non-gravitational parameters ${{A}_{1}}$, ${{A}_{2}}$, and ${{A}_{3}}$ [20]. They are related through the expressions ${{A}_{1}} = S{\text{/}}r_{0}^{2}$, ${{A}_{2}} = T{\text{/}}r_{0}^{2}$, ${{A}_{3}} = W{\text{/}}r_{0}^{2}$, where ${{r}_{0}} = 1$ AU.

Note 2. Expressions (13)–(15) allow us to estimate the magnitude of the Yarkovsky effect, but they require knowledge of such characteristics of the body as diameter, bulk density, rotation rate, obliquity of the spin axis to the orbital plane, Bond albedo, thermal inertia of the surface, specific heat capacity, and thermal emissivity.

4 EVOLUTION OF THE ORBITS OF MODEL ASTEROIDS

Let us consider model objects with different orbital eccentricities from 0 to 0.99, while all other orbital and thermophysical characteristics correspond to the asteroid 101 955 Bennu (Table 1). We find the orbit-averaged values of the vector P components, the drifts of the elements, and the displacement relative to the unperturbed position in two frames of reference. The element drifts and displacement are marked with the subscript “1” if they are defined in reference frame ${{\mathcal{O}}_{1}}$, and “2” in reference frame ${{\mathcal{O}}_{2}}$.

Table 1.   Orbital elements, thermophysical characteristics, and rotation parameters of asteroid 101 955 Bennu

For comparison, we will also calculate the drift of the mean anomaly and the displacement using the estimation formulas given in [21]:

$$\Delta M\simeq 0.01'' {{\dot{a}}_{4}}{{({{\Delta }_{10}}t)}^{2}} a_{\text{AU}}^{-5/2},$$

(16)

$$\Delta \rho \simeq 7{{\dot {a}}_{4}}{{({{\Delta }_{{10}}}t)}^{2}}a_{{{\text{AU}}}}^{{ - 3/2}}\;{\text{km}},$$

(17)

where ΔM is the variation in the mean anomaly in arcseconds; $\Delta \rho$ is the displacement in kilometers; ${{\dot {a}}_{4}}$  is  the drift of the semimajor axis due to the Yarkovsky effect in units of 10^–4 AU/Myr (the values are   given in the eight column of Table 2); ${{\Delta }_{{10}}}t = 119.5479063961725$ is the time period for which the estimation is made in tens of years; and ${{a}_{{{\text{AU}}}}}$ is the semimajor axis in astronomical units (Table 1). Estimates (16) and (17) were obtained in reference frame ${{\mathcal{O}}_{1}}$ neglecting the fourth-order terms in eccentricity.

Table 2.   Tangential $\mathfrak{T}$ and normal $\mathfrak{N}$ parameters, variations in the mean anomaly $d{{M}_{1}}$, $d{{M}_{2}}$ and semimajor axis $d{{a}_{1}}$, $d{{a}_{2}}$ for 1000 revolutions around the Sun ($\approx {\kern 1pt} 1195.48$ years) depending on the initial eccentricity ${{e}_{0}}$

Table 2 shows the tangential $\mathfrak{T}$ and normal $\mathfrak{N}$ parameters for different values of the initial eccentricity ${{e}_{0}}$. According to (15), the radial $S$ and transversal $T$ parameters do not depend on the eccentricity, $S = 9.91079 \times {{10}^{{ - 14}}}$ AU³/day², and $T = - 5.10168$ × 10^–14 AU³/day² for all ${{e}_{0}}$. Figure 2 illustrates the difference of $T$ from $\mathfrak{T}$ in percentage for ${{e}_{0}} > 0.7$: at ${{e}_{0}} \in (0.7,0.9)$ $T$ exceeds $\mathfrak{T}$ by 19 to 58%, and in the interval ${{e}_{0}} \in (0.9,\;0.99)$ this difference is from 58 to 230%. The difference of $S$ from $\mathfrak{N}$ behaves similarly.

Fig. 2.

The difference between the transversal parameter $T$ and the tangential parameter $\mathfrak{T}$in percentage at ${{e}_{0}} > 0.7$.

Using the values of the acceleration components and formulas (2), (3), we find the variations in the mean anomaly $d{{M}_{1}}$, $d{{M}_{2}}$ and semimajor axis $d{{a}_{1}}$, $d{{a}_{2}}$ for 1000 revolutions around the Sun ($\approx {\kern 1pt} 1195.48$ years) at different initial eccentricities ${{e}_{0}}$ (see Table 2). The differences of $d{{M}_{1}}$ from $d{{M}_{2}}$ and $d{{a}_{1}}$ from $d{{a}_{2}}$ are less than $1\%$ for small eccentricities (up to 0.5). At ${{e}_{0}}$ from 0.5 to 0.7, their difference does not exceed $6\%$, gradually increasing with ${{e}_{0}}$. At ${{e}_{0}} > 0.7$, these differences grow exponentially from 6 to 160%. The last column of Table 2 shows the variation in the mean anomaly calculated by formula (16). The comparison of columns $d{{M}_{1}}$ and $\Delta M$ gives good agreement between these values (the discrepancy does not exceed $3\%$).

Using the known formulas of celestial mechanics [24], we determine the displacement $d$ of a small body relative to the unperturbed position, which will occur due to the Yarkovsky effect. For this purpose, we calculate the rectangular coordinates of the body from the unperturbed and perturbed orbital elements and then find the distance between these positions. Table 3 shows displacements ${{d}_{1}}$ and ${{d}_{2}}$ for 1000 orbital revolutions ($\approx {\kern 1pt} 1195.48$ years) at various ${{e}_{0}}$, and Fig. 3 illustrates the difference of ${{d}_{1}}$ from ${{d}_{2}}$ in percentage at ${{e}_{0}} > 0.7$. With an increase in the initial eccentricity from 0.7 to 0.9, the excess of displacement ${{d}_{1}}$ over ${{d}_{2}}$ changes from 6 to 30%, and at ${{e}_{0}}$ from 0.9 to 0.99, the difference is from 30 to 127%.

Table 3.   Displacements ${{d}_{1}}$, ${{d}_{2}}$ of a small body relative to the unperturbed position for 1000 orbital revolutions ($\approx {\kern 1pt} 1195.48$ years) depending on the initial eccentricity ${{e}_{0}}$

Fig. 3.

The difference of displacement ${{d}_{1}}$ from displacement ${{d}_{2}}$ in percentage at ${{e}_{0}} > 0.7$.

The calculation of displacement ${{d}_{2}}$ also considers the variation in the argument of pericenter $d{{\omega }_{2}}$; however, it is small (less than an arc second) over the considered period of time, and depends little on the eccentricity of the orbit. The variations in the eccentricity $d{{e}_{1}}$, $d{{e}_{2}}$ are small as well. If they are not considered, displacements ${{d}_{1}}$, ${{d}_{2}}$ will change by no more than 0.01%. Table 3 shows the differences ${{d}_{1}} - d_{1}^{'}$ and ${{d}_{2}} - d_{2}^{'}$, where $d_{1}^{'}$ and $d_{2}^{'}$ are the displacements calculated without taking into account $d{{e}_{1}}$ and $d{{e}_{2}}$. If the variation in the mean anomaly is also neglected, the displacement relative to the undisturbed position only due to a variation in the semimajor axis will be insignificant (columns $d_{1}^{{''}}$ and $d_{2}^{{''}}$ in Table 3). Also for comparison, the fifth column of Table 3 shows displacement $\Delta \rho$ calculated by formula (17). For ${{e}_{0}} > 0.7$ the estimate $\Delta \rho$ exceeds ${{d}_{1}}$ by a factor of $1.65{-} 3.2$ and ${{d}_{2}}$ by a factor of $1.75{-} 7.3$, which indicates that the displacement estimate calculated by formula (17) can be significantly overestimated. This may explain the low detection of the Yarkovsky effect directly from astrometric observations: in the JPL database of small bodies, the parameter ${{A}_{2}}$ among NEAs less than 5 km in diameter was determined for 45 out of 588 objects with orbital eccentricities up to 0.5, and only for 18 out of 640 objects with $e > 0.5$, although the estimated value of the displacement at such eccentricities is greater, all other conditions being equal.

In system ${{\mathcal{O}}_{2}}$, parameters $\mathfrak{T}$ and $\mathfrak{N}$ gradually decrease with increasing ${{e}_{0}}$, but despite this decrease, the drifts of the elements and the displacement relative to the unperturbed position increase. In system ${{\mathcal{O}}_{1}}$, the growth of the element drifts and displacements with ${{e}_{0}}$ is much greater than in ${{\mathcal{O}}_{2}}$ for the same T and S for all e₀. Thus, when the Yarkovsky effect is modeled by radial and transversal acceleration, the estimates of element drifts and displacement relative to the unperturbed position for objects with highly elliptical orbits may turn out to be overestimated. For such objects, it is necessary to develop special methods for taking into account the Yarkovsky effect.

5 CONCLUSIONS

Expressions for the mean values of the tangential and normal components of the Yarkovsky acceleration over the orbital period have been derived based on the rotation parameters and thermophysical characteristics of the asteroid in the linear thermophysical model of the Yarkovsky force for spherical asteroids. At $e \to 0$, $\mathfrak{T} \to T$, and $\mathfrak{N} \to - S$, as it should be. For an elliptical orbit, the tangential and normal parameters are always smaller in absolute value than the transversal and radial parameters, and their difference increases with orbital eccentricity. The drifts of the semimajor axis and the mean anomaly, as well as the displacements relative to the unperturbed position calculated for two orbital reference systems are almost identical at small eccentricities (${{e}_{0}} < 0.5$). When ${{e}_{0}} > 0.5$, the values found in system ${{\mathcal{O}}_{1}}$ are always greater than the corresponding values in system ${{\mathcal{O}}_{2}}$. At ${{e}_{0}}$ from 0.5 to 0.7, the difference is moderate ($< {\kern 1pt} 6\%$), but it gradually increases, and at $e > 0.7$, the differences in the drifts and displacement grow exponentially. Thus, when the Yarkovsky effect is modeled by radial and transversal acceleration, the estimates of element drifts and displacement relative to the unperturbed position for objects with highly elliptical orbits may be overestimated. For such objects, it is necessary to develop special methods for taking into account the Yarkovsky effect.

APPENDIX A

1.1 MEAN VALUES OF THE FUNCTIONS OF THE ECCENTRIC ANOMALY

The main part of the paper involves the mean values of the functions $\sin f$, $\cos f$, $\sin 2M\sin f$, $\sin 2M\cos f$, $\cos 2M\sin f$, and $\cos 2M\cos f$. All of them can be expressed explicitly as analytic $2\pi$-periodic functions of the eccentric anomaly depending on the parameter $e$, $0 \leqslant e < 1$. The functions sin f, $\sin 2M\cos$ f, and $\cos 2M\sin f$are odd in $E$, because, according to (9),

$$\sin f = \frac{{e\sin E}}{{\sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}E} }} = - \frac{{e\sin ( - E)}}{{\sqrt {1 - {{e}^{2}}{{{\cos }}^{2}}( - E)} }}{\kern 1pt} ,$$

$$\begin{gathered} \sin 2M = \sin [2(E - e\sin E)] \\ \, = - \sin [2( - E - e\sin ( - E))]{\kern 1pt} . \\ \end{gathered} $$

The mean value of the odd function is zero [24]. Therefore, we restrict ourselves to even functions for which the mean value is

$$\mathcal{E}w = \frac{1}{\pi }\int\limits_0^\pi \,w{\kern 1pt} dM = \frac{1}{\pi }\int\limits_0^\pi \,w(1 - e\cos E){\kern 1pt} dE.$$

(A1)

The following properties may be useful:

$$\begin{gathered} \int\limits_0^\pi \,w(\cos x){\kern 1pt} dx = 0, \\ {\text{if}}\quad w(\cos x) = - w( - \cos x), \\ \end{gathered} $$

(A2)

$$\begin{gathered} \int\limits_0^\pi \,w(\cos x){\kern 1pt} dx = 2\int\limits_0^{\pi /2} \,w(\cos x){\kern 1pt} dx, \\ {\text{if}}\quad w(\cos x) = w( - \cos x). \\ \end{gathered} $$

(A3)

Let us find the mean values of the functions we need. Taking into account (9), (A1), (A2), (A3), and (4),

$$\begin{gathered} \frac{1}{\pi }\int\limits_0^\pi \cos fdM = \frac{1}{\pi }\int\limits_0^\pi \frac{\eta }{{\sqrt {1 - {{e}^{2}}\mathop {\cos }\nolimits^2 E} }}{\kern 1pt} dE \\ - \;\frac{1}{\pi }\int\limits_0^\pi \frac{{e\eta \cos E}}{{\sqrt {1 - {{e}^{2}}\mathop {\cos }\nolimits^2 E} }}{\kern 1pt} dE = \frac{2}{\pi }\int\limits_0^{\pi /2} \frac{\eta }{{\sqrt {1 - {{e}^{2}}\mathop {\cos }\nolimits^2 E} }}{\kern 1pt} dE \\ \, = \frac{2}{\pi }\int\limits_{\pi /2}^\pi \frac{\eta }{{\sqrt {1 - {{e}^{2}}\mathop {\sin }\nolimits^2 x} }}{\kern 1pt} dx = \frac{{2\eta }}{\pi }{\mathbf{K}}(e), \\ \end{gathered} $$

(A4)

where the new variable $x = E + \frac{\pi }{2}$. We express $\cos 2M$ and $\sin 2M$ through $x$ and represent them as

$$\begin{gathered} \cos 2M = \cos {\kern 1pt} {\kern 1pt} \text{[}2(E - e\sin E)] \\ = - \cos (2x + 2e\cos x) \\ = - \cos 2x\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}{{\cos }^{{2k}}}x \\ + \;\sin 2x\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 1}}}{{\cos }^{{2k + 1}}}x, \\ \end{gathered} $$

(A5)

$$\begin{gathered} \sin 2M = \sin [2(E - e\sin E)] \\ = - \sin (2x + 2e\cos x) \\ = - \sin 2x\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}{{\cos }^{{2k}}}x \\ - \;\cos 2x\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 1}}}{{\cos }^{{2k + 1}}}x, \\ \end{gathered} $$

(A6)

where

$${{g}_{{1k}}} = ( - {{1)}^{k}}\frac{{{{2}^{{2k}}}}}{{(2k)!}}{\kern 1pt} ,\quad {{g}_{{2k}}} = ( - {{1)}^{k}}\frac{{{{2}^{{2k + 1}}}}}{{(2k + 1)!}}{\kern 1pt} .$$

(A7)

Expansions (A5) and (A6) were obtained using standard formulas for adding angles and expanding trigonometric functions in power-law series [19]. The values of the coefficients ${{g}_{{1k}}},\;{{g}_{{2k}}}$for $k = 0{-} 15$ are given in Table 4 in the form of rational and decimal fractions, which will allow us to judge their behavior. The sum symbols in (A5) and (A6) have an infinite upper limit, but since the coefficients rapidly decrease with the growth of $k$, in practice it is sufficient to use 15–20 terms. Expansions into a series of functions $\cos (2e\cos x)$ and $\sin (2e\cos x)$ converge for any values of $2e\cos x$: $- \infty < 2e\cos x < \infty$, hence they converge for all $e \in [0,1]$.

Table 4.   Values ${{g}_{{1k}}},\;{{g}_{{2k}}}$ as rational (top) and decimal (bottom) fractions

Taking into account (9), (A1), and (A5), as well as the commutativity of the integration and summation operations, we find

$$\begin{gathered} \frac{1}{\pi }\int\limits_0^\pi {\cos 2M\cos fdM} \\ = - \frac{\eta }{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\cos 2x{{{\cos }}^{{2k}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx \\ \end{gathered} $$

$$ + \;\frac{\eta }{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 1}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\sin 2x{{{\cos }}^{{2k + 1}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx$$

(A8)

$$\begin{gathered} + \;\frac{\eta }{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k + 1}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\sin x\cos 2x{{{\cos }}^{{2k}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx \\ - \;\frac{\eta }{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\sin x\sin 2x{{{\cos }}^{{2k + 1}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} {\kern 1pt} dx. \\ \end{gathered} $$

By passing back to $E$, it is easy to show that, by virtue of (A2), the second and third terms in expression (A8) are zero. After small trigonometric transformations, we bring (A8) to the form

$$\begin{gathered} \frac{1}{\pi }\int\limits_0^\pi \cos 2M\cos fdM \\ = \frac{\eta }{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\mathop {\cos }\nolimits^{2k} x - 2{{{\cos }}^{{2k + 2}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx \\ - \;\frac{{2\eta }}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\mathop {\cos }\nolimits^{2k + 2} x - {{{\cos }}^{{2k + 4}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx. \\ \end{gathered} $$

(A9)

Similarly, we obtain

$$\begin{gathered} \frac{1}{\pi }\int\limits_0^\pi \sin 2M\sin fdM \\ = \frac{1}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k + 1}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\sin 2x{{{\cos }}^{{2k + 1}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}dx \\ \end{gathered} $$

$$ - \;\frac{1}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k + 2}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\sin x\sin 2x{{{\cos }}^{{2k + 1}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}dx$$

$$ + \;\frac{1}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\cos 2x{{{\cos }}^{{2k + 2}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}dx$$

$$ - \;\frac{1}{\pi }\sum\limits_{k = 1}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 3}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{\sin x\cos 2x{{{\cos }}^{{2k + 2}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}dx,$$

where the first and last terms are zero. As a result,

$$\begin{gathered} \frac{1}{\pi }\int\limits_0^\pi \sin 2M\sin fdM \\ = - \frac{2}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k + 2}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{(\mathop {\cos }\nolimits^{2k + 2} x - {{{\cos }}^{{2k + 4}}}x)}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}dx \\ - \;\frac{1}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}\int\limits_{\pi /2}^{3\pi /2} \frac{{(\mathop {\cos }\nolimits^{2k + 2} x - 2{{{\cos }}^{{2k + 4}}}x)}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}dx. \\ \end{gathered} $$

(A10)

Thus, finding the necessary mean values is reduced to the integrals of the form

$${{I}_{k}} = \int\limits_{\pi /2}^{3\pi /2} \frac{{{{{\cos }}^{{2k}}}x}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx\quad {\text{at}}\quad k \geqslant 0.$$

(A11)

In [19], the following formula is given:

$$\begin{gathered} \int \frac{{{{{\cos }}^{n}}x}}{{\sqrt {1 - {{k}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx = \frac{{{{{\cos }}^{{n - 3}}}x}}{{(n - 1){{k}^{2}}}}\sin x\sqrt {1 - {{k}^{2}}{{{\sin }}^{2}}x} \\ + \;\frac{{n - 2}}{{n - 1}}\frac{{2{{k}^{2}} - 1}}{{{{k}^{2}}}}\int \frac{{{{{\cos }}^{{n - 2}}}x}}{{\sqrt {1 - {{k}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx \\ + \;\frac{{n - 3}}{{n - 1}}\frac{{1 - {{k}^{2}}}}{{{{k}^{2}}}}\int \frac{{{{{\cos }}^{{n - 4}}}x}}{{\sqrt {1 - {{k}^{2}}{{{\sin }}^{2}}x} }}{\kern 1pt} dx, \\ \end{gathered} $$

with the help of which we obtain the recursive relation for calculating (A11):

$${{I}_{k}} = \frac{{2k - 2}}{{2k - 1}}\frac{{2{{e}^{2}} - 1}}{{{{e}^{2}}}}{{I}_{{k - 1}}} + \frac{{2k - 3}}{{2k - 1}}\frac{{1 - {{e}^{2}}}}{{{{e}^{2}}}}{{I}_{{k - 2}}}{\kern 1pt} .$$

(A12)

The first terms of the ${{I}_{k}}$ sequence are easy to find using the reference book [19]:

$$\begin{gathered} {{I}_{0}} = \int\limits_{\pi /2}^{3\pi /2} \frac{{dx}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }} = 2{\mathbf{K}}(e), \\ {{I}_{1}} = \int\limits_{\pi /2}^{3\pi /2} \frac{{{{{\cos }}^{2}}xdx}}{{\sqrt {1 - {{e}^{2}}{{{\sin }}^{2}}x} }} = \frac{2}{{{{e}^{2}}}}[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]{\kern 1pt} {\kern 1pt} . \\ \end{gathered} $$

(A13)

Taking into account (A4), (A9), (A10), and (A11), we write the final expressions for the desired mean v-alues:

$$\frac{1}{\pi }\int\limits_0^\pi \cos fdM = \frac{{2\eta }}{\pi }{\mathbf{K}}(e),$$

$$\begin{gathered} \frac{1}{\pi }\int\limits_0^\pi \cos 2M\cos fdM = \frac{\eta }{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k}}}({{I}_{k}} - 2{{I}_{{k + 1}}}) \\ - \;\frac{{2\eta }}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}({{I}_{{k + 1}}} - {{I}_{{k + 2}}}), \\ \end{gathered} $$

(A14)

$$\begin{gathered} \frac{1}{\pi }\int\limits_0^\pi \sin 2M\sin fdM = - \frac{2}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{1k}}}{{e}^{{2k + 2}}}({{I}_{{k + 1}}} - {{I}_{{k + 2}}}) \\ - \;\frac{1}{\pi }\sum\limits_{k = 0}^\infty \,{{g}_{{2k}}}{{e}^{{2k + 2}}}({{I}_{{k + 1}}} - 2{{I}_{{k + 2}}}). \\ \end{gathered} $$

Let us consider the behavior of the integrals (A14) for $e = 0$. The last two expressions on the right contain the uncertainty of the form $0:0$ due to (A12) and (A13). Therefore, we set $e = 0$ in the original integrals and calculate them:

$$\frac{1}{\pi }\int\limits_0^\pi \cos fdM = \frac{1}{\pi }\int\limits_0^\pi \,dE = 1,$$

$$\frac{1}{\pi }\int\limits_0^\pi \cos 2M\cos fdM = \frac{1}{\pi }\int\limits_0^\pi \cos 2E{\kern 1pt} dE = 0,$$

(A15)

$$\frac{1}{\pi }\int\limits_0^\pi \sin 2M\sin fdM = 0.$$