Central Field Motion with Variation of Perturbing Acceleration According to the Inverse Square Law in a Reference Frame Associated with the Velocity Vector

Abstract

A problem is considered in which a zero-mass point moves under the attraction of the central body $\mathcal{S}$ and perturbing acceleration ${\mathbf{P}}{\kern 1pt} '$ inversely proportional to the square of the distance to $\mathcal{S}$. Regarding ${\mathbf{P}}{\kern 1pt} ' = {\mathbf{P}}{\text{/}}{{r}^{2}}$, the ${\mathbf{P}}{\kern 1pt} '$ absolute value is small in comparison with the main acceleration caused by the attraction of the central body. The components of the vector ${\mathbf{P}}(\mathfrak{T},\mathfrak{N},W)$ are constant in the reference frame with the origin at $\mathcal{S}$ and the axes directed along the velocity vector, the normal to it in the plane of the osculating orbit, as well as the binormal. For this problem, we previously obtained the mean element equations of motion in the first approximation with respect to a small parameter, the role of which is played by the ratio of the perturbing acceleration to the main acceleration. The solution of the equations averaged over the mean anomaly is proposed. The system is solved for a circular orbit and in the cases when at least one component of the perturbing acceleration vector is zero. The solution for a circular orbit and for $\mathfrak{T} = 0$ is presented in the form of time dependences of the orbital elements and contains either elementary functions or complete elliptic integrals. If the tangential component of the perturbing acceleration is not zero, the time and orbital elements are represented by functions of eccentricity. In these cases, the system is integrated in quadratures leading to non-elementary functions, but they are all expressed as series in powers of eccentricity $e$ converging at $e < 1$. Thus, the solution for $\mathfrak{T} \ne 0$ is obtained in the form of series.

APPENDIX

1.1 SOME FORMULAS FOR ELLIPTIC INTEGRALS

The main part of the paper contains values that are combinations of eccentricity functions and complete elliptic integrals, as well as integrals of these combinations. Below are the coefficients of the power series for these values (Sections A1, A2). In most cases, explicit formulas for the general term of the series are obtained; for some cases, the algorithm for its derivation is given. An arbitrary term of the series is a rational number and is calculated exactly. Tables 1 and 2 show the first five coefficients of the series mentioned in the paper as rational and decimal fractions, which will allow us to evaluate the behavior of the coefficients. In Section A3, the asymptotics of some integrals of combinations of elliptic integrals is derived.

Table 1.   The ${{a}_{{mk}}}$ values as rational (top) and decimal (bottom) fractions

Table 2.   The $a_{k}^{3}$ and $A_{k}^{3}$ values as rational (top) and decimal (bottom) fractions

1.1.1 A1. Linear Combinations of Elliptic Integrals

By the definition of complete elliptic integrals in normal trigonometric form [6],

$${\mathbf{K}}(e) = \int\limits_0^{\pi /2} \,\frac{{dx}}{{h(x,e)}},\quad {\mathbf{E}}(e) = \int\limits_0^{\pi /2} \,h(x,e)dx,$$

where $h(x,e) = \sqrt {1 - {{e}^{2}}{{\sin}^{2}}x}$. Let us write out the expansions in a series of powers of eccentricity for them, as well as for the $\eta$ function [6, 11, 12]:

$$\begin{gathered} \eta = \sqrt {1 - {{e}^{2}}} \;\mathop = \limits^{\text{*}} \;1 - \sum {{{a}_{{1k}}}{{e}^{{2k}}}} , \\ \frac{1}{\eta } = \frac{1}{{\sqrt {1 - {{e}^{2}}} }} = 1 + \sum {{{a}_{{2k}}}{{e}^{{2k}}}} , \\ \end{gathered} $$

(A.1)

$$\frac{2}{\pi }{\mathbf{K}}(e) = 1 + \sum {{{a}_{{3k}}}{{e}^{{2k}}}} ,\;\;\frac{2}{\pi }{\mathbf{E}}(e)\;\mathop = \limits^{\text{*}} \;1 - \sum {{{a}_{{4k}}}{{e}^{{2k}}}} ,$$

(A.2)

where

$$\begin{gathered} {{a}_{{1k}}} = \frac{{(2k - 3){\kern 1pt} !{\kern 1pt} !}}{{(2k)!!}},\quad {{a}_{{2k}}} = \frac{{(2k - 1){\kern 1pt} !{\kern 1pt} !}}{{(2k){\kern 1pt} !{\kern 1pt} !}}, \\ {{a}_{{3k}}} = \mathop {\left[ {\frac{{(2k - 1){\kern 1pt} !{\kern 1pt} !}}{{(2k){\kern 1pt} !{\kern 1pt} !}}} \right]}\nolimits^2 ,\quad {{a}_{{4k}}} = \frac{1}{{2k - 1}}{{a}_{{3k}}}. \\ \end{gathered} $$

Same as in the main part of the paper, unless otherwise indicated, summation is carried out over $k$ from 1 to $\infty$, the radius of convergence of the power series is equal to one, and the asterisk above the equal sign denotes the cases of convergence of the series at $e = 1$.

Using formulas (A.1) and (A.2), we obtain the expansions of the combinations:

$$\begin{gathered} \frac{e}{{\eta (1 + \eta )}} = \frac{e}{2} + \sum {{{a}_{{5k}}}{{e}^{{2k + 1}}}} , \\ \frac{2}{\pi }[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]\;\mathop = \limits^{\text{*}} \;\frac{{{{e}^{2}}}}{2}\left( {1 + \sum {{{a}_{{6k}}}{{e}^{{2k}}}} } \right), \\ \end{gathered} $$

$$\begin{gathered} \frac{2}{{\pi e}}[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]\;\mathop = \limits^{\text{*}} \;\frac{e}{2}\left( {1 + \sum {{{a}_{{6k}}}{{e}^{{2k}}}} } \right), \\ \frac{2}{\pi }[2{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]\;\mathop = \limits^{\text{*}} \;1 + \sum {{{a}_{{7k}}}{{e}^{{2k}}}} , \\ \end{gathered} $$

(A.3)

$$\begin{gathered} \frac{2}{\pi }{{\eta }^{2}}[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]\;\mathop = \limits^{\text{*}} \;\frac{{{{e}^{2}}}}{2}\left( {1 - \sum {{{a}_{{8k}}}{{e}^{{2k}}}} } \right), \\ \frac{2}{\pi }\eta {\mathbf{K}}(e)\;\mathop = \limits^{\text{*}} \;1 - \sum {{{a}_{{9k}}}{{e}^{{2k}}}} . \\ \end{gathered} $$

Here,

$${{a}_{{5k}}} = \left( {k - \frac{1}{4}} \right){{a}_{{1k}}} + \frac{1}{2}\sum\limits_{m = 1}^{k - 1} \,{{a}_{{5m}}}{{a}_{{1,k - m}}},$$

$${{a}_{{6k}}} = \frac{1}{{k + 1}}{{a}_{{3k}}},$$

$${{a}_{{7k}}} = \frac{1}{{{{{(2k - 1)}}^{2}}}}{{a}_{{3k}}},$$

$${{a}_{{8k}}} = \frac{{8k - 1}}{{(k + 1){{{(2k - 1)}}^{2}}}}{{a}_{{3k}}},$$

$${{a}_{{9k}}} = {{a}_{{1k}}} - {{a}_{{3k}}} + \sum\limits_{m = 1}^{k - 1} \,{{a}_{{1m}}}{{a}_{{3,k - m}}}.$$

The numbers ${{a}_{{mk}}}$, $m = 1, \ldots ,8$, are positive. At $k = 1$, ${{a}_{{91}}} = 1{\text{/}}4$, but at $k > 1$, the term $({{a}_{{1k}}} - {{a}_{{3k}}}) < 0$. By analogy with [7], let us prove that ${{a}_{{9k}}}$ for $k > 1$ is positive. Let us set

$$\eta {\mathbf{K}}(e) = \int\limits_0^{\pi /2} \,\sqrt {\varphi (e,x)} dx,$$

(A.4)

where

$$\begin{gathered} \varphi = \frac{{1 - {{e}^{2}}}}{{1 - {{e}^{2}}{{\sin}^{2}}x}} = 1 - {{\varphi }_{1}}, \\ {{\varphi }_{1}} = {{\cos}^{2}}x\sum {{{\sin}^{{2k - 2}}}{\kern 1pt} x\,{{e}^{{2k}}}} . \\ \end{gathered} $$

According to (A.1),

$$\sqrt \varphi = 1 - \sum\limits_{m = 1}^\infty \,{{a}_{{1m}}}\varphi _{1}^{m}.$$

From this, we obtain that the ${{\varphi }_{{1k}}}$ values in the expansion

$$\sqrt \varphi = 1 - \sum {{\varphi }_{{1k}}}{{e}^{{2k}}}$$

are polynomials in ${{\sin}^{2}}{\kern 1pt} x,\;{{\cos}^{2}}{\kern 1pt} x$ with positive coefficients. In particular, ${{\varphi }_{{11}}} = (1{\text{/}}2){{\cos}^{2}}{\kern 1pt} x$, ${{\varphi }_{{12}}} =$ $(1{\text{/}}8){{\cos}^{4}}{\kern 1pt} x + (1{\text{/}}2){{\cos}^{2}}{\kern 1pt} x{{\sin}^{2}}{\kern 1pt} x$, i.e., the integrand in (A.4) can be represented as

$$1 - \frac{1}{2}{{\cos}^{2}}{\kern 1pt} x\,{{e}^{2}} - \frac{1}{8}{{\cos}^{2}}{\kern 1pt} x(1 + 3{{\sin}^{2}}{\kern 1pt} x){{e}^{4}} - \sum\limits_{k = 3}^\infty {\kern 1pt} {{\varphi }_{{1k}}}{{e}^{{2k}}}.$$

Integration will give ${{a}_{{91}}} = 1{\text{/}}4$, ${{a}_{{92}}} = 7{\text{/}}64$, ${{a}_{{9k}}} > 0$ at $k > 1$.

The left-hand side of the last relation (A.3) tends to zero at $e \to 1$. From this and the positivity of ${{a}_{{9k}}}$, it follows that the series on the right-hand side converge at $e = 1$ [13].

1.1.2 A2. More Complex Combinations of Elliptic Integrals

The recurrence for the general expansion term

$$\frac{{2[2{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}}{{\pi {{\eta }^{2}}}} = 1 + \sum {{{a}_{{10,k}}}{{e}^{{2k}}}} $$

(A.5)

is derived from the relation

$$(1 - {{e}^{2}})\left( {1 + \sum {{{a}_{{10,k}}}{{e}^{{2k}}}} } \right) = 1 + \sum {{{a}_{{7k}}}{{e}^{{2k}}}} ,$$

which follows from (A.3) and (A.5). From here,

$${{a}_{{10,k}}} = {{a}_{{10,k - 1}}} + {{a}_{{7k}}}\;\;\;{\text{at}}\;\;\;{{a}_{{10,1}}} = \frac{5}{4}.$$

(A.6)

From (A.6) it follows that ${{a}_{{10,k}}}$ is positive, but at $e = 1$ series (A.5) diverges to infinity.

Similarly, we obtain the general term of the expansion

$$\frac{{{\mathbf{K}}(e)}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} = \frac{1}{{{{e}^{2}}}} + \frac{1}{8} + \sum {{{a}_{{11,k}}}{{e}^{{2k}}}} ,$$

(A.7)

$$\begin{gathered} \left( {\frac{1}{{{{e}^{2}}}} + \frac{1}{8} + \sum {{{a}_{{11,k}}}{{e}^{{2k}}}} } \right){{e}^{2}}\left( {1 + \sum {{{a}_{{6k}}}{{e}^{{2k}}}} } \right) \\ = 1 + \sum {{{a}_{{3k}}}{{e}^{{2k}}}} , \\ \end{gathered} $$

$${{a}_{{11,k}}} = {{a}_{{3,k + 1}}} - {{a}_{{6,k + 1}}} - \frac{1}{8}{{a}_{{6k}}} - \sum\limits_{m = 1}^{k - 1} \,{{a}_{{11,m}}}{{a}_{{6,k - m}}},$$

or

$${{a}_{{11,k}}} = \frac{{k(8k + 7)}}{{8(k + 2)}}{{a}_{{6k}}} - \sum\limits_{m = 1}^{k - 1} \,{{a}_{{11,m}}}{{a}_{{6,k - m}}},$$

where

$${{a}_{{11,1}}} = {{a}_{{3,2}}} - {{a}_{{6,2}}} - \frac{1}{8}{{a}_{{61}}} = \frac{5}{8}{{a}_{{61}}} = \frac{5}{{64}}.$$

Coefficients ${{a}_{{11,k}}}$ are positive, since, according to (A.2) and (A.3), the series expansions of the numerator and denominator of the left-hand side of (A.7) have positive coefficients.

From (A.7), we obtain

$$\begin{gathered} \sigma \int\limits_{{{e}_{0}}}^e \,\frac{{e{\mathbf{K}}(e)de}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} \\ = \;\ln\mathop {\left( {\frac{e}{{{{e}_{0}}}}} \right)}\nolimits^\sigma + \sum {\frac{{\sigma {{a}_{{12,k}}}}}{{2k}}{{e}^{{2k}}}} - \sigma {{C}_{1}}, \\ \end{gathered} $$

(A.8)

where

$$\begin{gathered} {{C}_{1}} = \sum {\frac{{{{a}_{{12,k}}}}}{{2k}}e_{0}^{{2k}}} , \\ {{a}_{{12,1}}} = \frac{1}{8}, \\ {{a}_{{12,k}}} = {{a}_{{11,k - 1}}}\quad {\text{at}}\quad k > 1. \\ \end{gathered} $$

(A.9)

Coefficients $a_{k}^{\sigma }$ of the expansion

$$\begin{gathered} \exp\int\limits_{{{e}_{0}}}^e \,\frac{{\sigma e{\mathbf{K}}(e)de}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} \\ = \exp( - \sigma {{C}_{1}})\mathop {\left( {\frac{e}{{{{e}_{0}}}}} \right)}\nolimits^\sigma \left( {1 + \sum {a_{k}^{\sigma }{{e}^{{2k}}}} } \right) \\ \end{gathered} $$

(A.10)

can be obtained by means of computer algebra using the classical exponent series:

$$\exp\left( {\sum {\frac{{\sigma {{a}_{{12,k}}}}}{{2k}}{{e}^{{2k}}}} } \right) = 1 + \sum\limits_{n = 1}^\infty \,\frac{{{{{\left( {\sum\limits_{k = 1}^\infty {\tfrac{{\sigma {{a}_{{12,k}}}}}{{2k}}{{e}^{{2k}}}} } \right)}}^{n}}}}{{n{\kern 1pt} !}}.$$

The positivity of ${{a}_{{11,k}}}$ implies that $a_{k}^{\sigma } > 0$.

Considering (A.10), we obtain the expansion coefficients

$$\begin{gathered} \exp\left( { - \int\limits_{{{e}_{0}}}^e \,\frac{{\sigma e{\mathbf{K}}(e)de}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}}} \right) \\ = \exp(\sigma {{C}_{1}})\mathop {\left( {\frac{{{{e}_{0}}}}{e}} \right)}\nolimits^\sigma \left( {1 - \sum {A_{k}^{\sigma }{{e}^{{2k}}}} } \right), \\ \end{gathered} $$

(A.11)

where

$$A_{k}^{\sigma } = a_{k}^{\sigma } - \sum\limits_{m = 1}^{k - 1} \,A_{m}^{\sigma }a_{{k - m}}^{\sigma },\quad A_{1}^{\sigma } = \frac{\sigma }{{16}}.$$

Coefficients ${{a}_{{mk}}}$, $a_{k}^{\sigma }$, and $A_{k}^{\sigma }$ at $k = 1, \ldots ,5$, $\sigma = 3$ are given in Tables 1 and 2. Each line contains the exact value in the form of a rational fraction at the top and the approximate value in the form of a decimal fraction at the bottom.

1.1.3 A3. Asymptotics of Integrals of Elliptic Integral Combinations

The asymptotic behavior of elliptic integrals at $e \to 1$, $\eta \to 0$ is known [6, 11, 12, 14]:

$${\mathbf{E}}(e) = 1 - \frac{1}{2}{{\eta }^{2}}\left( {\ln\frac{\eta }{4} + \frac{1}{2}} \right) - \ldots ,$$

$${\mathbf{K}}(e) = - \ln\frac{\eta }{4} - \frac{1}{4}{{\eta }^{2}}\left( {\ln\frac{\eta }{4} + 1} \right) - \ldots .$$

For their combinations, we obtain

$$\begin{gathered} \frac{1}{{{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)}} = 1 - \frac{1}{2}{{\eta }^{2}}\left( {\ln\frac{\eta }{4} - \frac{1}{2}} \right) + \ldots , \\ \frac{{{\mathbf{K}}(e)}}{{{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)}} \\ = \; - {\kern 1pt} \ln\frac{\eta }{4} - {{\eta }^{2}}\left( {\frac{1}{4} + \frac{1}{2}\ln\frac{\eta }{4} - \frac{1}{2}\mathop {\left( {\ln\frac{\eta }{4}} \right)}\nolimits^2 } \right) + \ldots \;. \\ \end{gathered} $$

(A.12)

Here and below, the dots denote infinitesimals of higher order with respect to $\eta$.

Using relation (A.12) and considering that $ede = - \eta d\eta$, let us proceed to the variable $\eta$ and integrate the following expression:

$$\begin{gathered} \int\limits_{{{e}_{0}}}^e \,\frac{{\sigma e{\mathbf{K}}(e)de}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} \\ = \,\int\limits_{{{\eta }_{0}}}^\eta \frac{{\sigma \eta }}{2}\left( {\ln\frac{\eta }{4}\, + \,{{\eta }^{2}}\left( {\frac{1}{4}\, + \,\frac{1}{2}\ln\frac{\eta }{4}\, - \,\frac{1}{2}\mathop {\left( {\ln\frac{\eta }{4}} \right)}\nolimits^2 } \right)\, + \, \ldots } \right)d\eta \\ \, = \frac{\sigma }{4}{{\eta }^{2}}\left( {\ln\frac{\eta }{4} - \frac{1}{2}} \right) + \sigma {{C}_{2}} + \ldots , \\ \end{gathered} $$

(A.13)

where ${{C}_{2}} = - \tfrac{1}{4}\eta _{0}^{2}\left( {\ln\tfrac{{{{\eta }_{0}}}}{4} - \tfrac{1}{2}} \right)$. From here,

$$\begin{gathered} \exp\int\limits_{{{e}_{0}}}^e \,\frac{{\sigma e{\mathbf{K}}(e)de}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} \\ = \exp(\sigma {{C}_{2}})\left( {1 - \frac{\sigma }{8}{{\eta }^{2}} + \frac{\sigma }{4}{{\eta }^{2}}\ln\frac{\eta }{4}} \right) + \ldots \;. \\ \end{gathered} $$

Further, the asymptotics of the integrals used in formulas (14), (20), and (24) are obtained:

$$\begin{gathered} \int\limits_{{{e}_{0}}}^e \,\frac{e}{{2{{\eta }^{3}}[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} \\ \times \;\left( {\exp\int\limits_{{{e}_{0}}}^e \,\frac{{3e{\mathbf{K}}(e)de}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}}} \right)de \\ \end{gathered} $$

$$\begin{gathered} = \; - {\kern 1pt} \int\limits_{{{\eta }_{0}}}^\eta \,\frac{1}{{2{{\eta }^{2}}}}\left[ {1 - \frac{1}{2}{{\eta }^{2}}\left( {\ln\frac{\eta }{4} - \frac{1}{2}} \right) + \ldots } \right] \\ \times \;\left[ {\exp(3{{C}_{2}})\left( {1 - \frac{3}{8}{{\eta }^{2}} + \frac{3}{4}{{\eta }^{2}}\ln\frac{\eta }{4}} \right) + \ldots } \right]d\eta \\ \\ \end{gathered} $$

(A.14)

$$\begin{gathered} = \int\limits_{{{\eta }_{0}}}^\eta \,\left[ {\exp(3{{C}_{2}})\left( {\frac{1}{{16}} - \frac{1}{{2{{\eta }^{2}}}} - \frac{1}{8}\ln\frac{\eta }{4}} \right) + \ldots } \right]d\eta \\ = \exp(3{{C}_{2}})\left( {\frac{1}{{2\eta }} + \frac{\eta }{{16}}\left( {3 - 2\ln\frac{\eta }{4}} \right)} \right) - C_{2}^{*} + \ldots , \\ \end{gathered} $$

where

$$C_{2}^{*} = \exp(3{{C}_{2}})\left( {\frac{1}{{2{{\eta }_{0}}}} + \frac{{{{\eta }_{0}}}}{{16}}\left( {3 - 2\ln\frac{{{{\eta }_{0}}}}{4}} \right)} \right),$$

$$\int\limits_{{{e}_{0}}}^e \,\frac{{ede}}{{{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)}}$$

(A.15)

$$ = - \int\limits_{{{\eta }_{0}}}^\eta \,\left( {\eta - \frac{1}{2}{{\eta }^{3}}\left( {\ln\frac{\eta }{4} - \frac{1}{2}} \right) + \ldots } \right)d\eta = - \frac{{{{\eta }^{2}}}}{2} + \frac{{\eta _{0}^{2}}}{2} + \ldots ,$$

and considering (A.13),

$$\begin{gathered} \int\limits_{{{e}_{0}}}^e \,\frac{{e\eta {\mathbf{K}}(e)de}}{{2[{\mathbf{E}}(e) - {{\eta }^{2}}{\mathbf{K}}(e)]}} = \int\limits_{{{\eta }_{0}}}^\eta \,\frac{{{{\eta }^{2}}}}{2}\left( {\ln\frac{\eta }{4} + \ldots } \right)d\eta \\ = \frac{{{{\eta }^{3}}}}{6}\left( {\ln\frac{\eta }{4} - \frac{1}{3}} \right) - C_{2}^{{**}} + \ldots , \\ \end{gathered} $$

(A.16)

where

$$C_{2}^{{**}} = \frac{{\eta _{0}^{3}}}{6}\left( {\ln\frac{{{{\eta }_{0}}}}{4} - \frac{1}{3}} \right).$$