R-Toroid as a Three-Dimensional Generalization of a Gaussian Ring and Its Application in Astronomy

Abstract

A new analytical model (R-toroid), representing a 3D generalization of the precessing Gaussian ring, is constructed for the study of secular perturbations in celestial mechanics. Our approach is based on triple averaging of the motion of a material point and is reduced to a chain of transformations: 1D Gaussian ring–2D R-ring–3D R-toroid. The figure, structure and gravitational potential of the R-toroid are studied. We obtained the expression for the mutual energy of the R-toroid and the outer Gaussian ring to study the motion of bodies in the gravitational field of the model in two forms (in the integral and in the form of a power-law series). Two equation systems of the secular evolution of osculating orbits (Gaussian rings), in the gravitational field of an R-toroid and in the field of a central precessing star, are derived using the mutual energy. The periods of nodal T_Ω and apsidal T_ω orbital precession were obtained. Examples of three hot Jupiters with a known period of nodal precession are considered. For the exoplanet Kepler-413b, the R-toroid describes the evolution of any orbit with a ≥ 5.48 AU, and for the exoplanet PTFO 8-8695b, the critical value of the semi-major axis turned out to be only a_min ≈ 0.2 AU. The frequency profile of the precession of the test orbit in the field of the star and planet PTFO 8-8695b has been calculated. The minimum value of the period of nodal precession was T_Ω ≈ (26.1 ± 3.0) × 10³ years.

APPENDIX

EXPANSION OF THE EXTERNAL POTENTIAL OF AN R-TOROID IN A SERIES

To derive formula (24), it is advisable to introduce the function

$$F(R,z,x,a,g) = \frac{{K(\bar {k})}}{{\sqrt {{{R}^{2}} + {{z}^{2}} + {{{(x + a)}}^{2}} + 2(x + a)[R\sqrt {1 - {{g}^{2}}(u)} - zg(u)]} }},$$

(A.1)

which we expand in a series in the variables x and u. As a result, we get the series

$$\begin{gathered} F(R,z,x,a,g) \\ = {{\sum\limits_{n,m = 0}^\infty {\left. {\frac{{{{x}^{n}}}}{{n{\kern 1pt} !}}\frac{{{{g}^{m}}}}{{m{\kern 1pt} !}}\frac{{{{\partial }^{{n + m}}}F(R,z,x,a,g)}}{{\partial {{x}^{n}}\partial {{g}^{m}}}}} \right|} }_{{\begin{subarray}{l} x = 0 \\ g = 0 \end{subarray}} }}, \\ \end{gathered} $$

(A.2)

which can be rewritten, taking into account the form of the function (A1), in the form

$$F(R,z,x,a,g) = {{\sum\limits_{n,m = 0}^\infty {\left. {\frac{{{{x}^{n}}}}{{n{\kern 1pt} !}}\frac{{{{g}^{m}}}}{{m{\kern 1pt} !}}\frac{{{{\partial }^{{n + m}}}\bar {F}(R,z,a,g)}}{{\partial {{a}^{n}}\partial {{g}^{m}}}}} \right|} }_{{g = 0}}},$$

(A.3)

where $\bar {F}(R,z,a,g) = F(R,z,x = 0,a,g).$ After substituting the series (A3) in (22), considering (A1) and rearranging the places of summation and integration, the double integral is divided into the sum of the products of two single integrals

$$\begin{gathered} {{\Phi }_{{{\text{out}}}}}(r,\varphi ,\theta ) = \frac{{2GM}}{{{{\pi }^{3}}a}}\sum\limits_{n,m = 0}^\infty {\left[ {\frac{1}{{n{\kern 1pt} !}}\int\limits_{ - ae}^{ae} {\frac{{(x + a){{x}^{n}}dx}}{{\sqrt {{{a}^{2}}{{e}^{2}} - {{x}^{2}}} }}} \mathop {}\limits_{\mathop {}\limits_{_{{}}}^{} }^{} } \right.} \\ \left. {{{{\left. { \times \;\frac{{{{{\sin }}^{m}}i}}{{m{\kern 1pt} !}}\int\limits_{ - \pi /2}^{\pi /2} {{{{\sin }}^{m}}} udu\frac{{{{\partial }^{{n + m}}}\bar {F}(R,z,a,g)}}{{\partial {{a}^{n}}\partial {{g}^{m}}}}} \right|}}_{{g = 0}}}} \right]. \\ \end{gathered} $$

(A.4)

From here, introducing coefficients (25), as a result, we obtain formula (24).