Correction: Bifurcation of frozen orbits in a gravity field with zonal harmonics

Correction to: Celestial Mechanics and Dynamical Astronomy (2022) 134:49

https://doi.org/10.1007/s10569-022-10103-6

Under the heading 4.2 The J4-problem, the formula following the paragraph “After introducing it in the Hamiltonian, …” should read as follows:

$$\begin{aligned} g(Z,a) &= \frac{{ - 5\rho ^{2} + 2Z + 1}}{{\sqrt2 \left( {\rho ^{2} + 2Z + 1}\right)^{{\frac{5}{2}}} }} -\frac{{3\lambda }}{{16\sqrt 2 \left({\rho ^{2} + 2Z + 1}\right)^{{\frac{{11}}{2}}} }}(40Z^{3} + \left({ - 84\rho ^{2} +20} \right)Z^{2} \hfill \\ &\quad + \left( {-74\rho ^{4} - 44\rho ^{2} - 10} \right)Z - 11\rho ^{6} +273\rho^{4} - \rho ^{2} - 5 \hfill \\ & \quad +4\sqrt {2\rho ^{2} + 4Z + 2} \left( { - 5\rho ^{2} + 2Z + 1} \right)^{2} )+\frac{{3\lambda j_{4} }}{{16\sqrt 2 \left( {\rho ^{2} + 2Z +1}\right)^{{\frac{{11}}{2}}} }}\\&\quad( - 72Z^{3} + 12\left( {51\rho ^{2} + 1} \right)Z^{2} +3\left( {- 58\rho ^{4} - 156\rho ^{2} + 22} \right)Z - 249\rho ^{6}\hfill \\&\quad + 743\rho ^{4} - 387\rho ^{2} +21), \hfill \\ f(Z,a) &= - \frac{3}{2} \lambda \frac{{\left( { -29 \rho ^{2}+ 2 Z + 1} \right)}}{{\sqrt 2 \left( {\rho ^{2} + 2 Z+ 1}\right)^{{\frac{{11}}{2}}} }} + \frac{{15}}{2}\lambda j_{4}\frac{{\left( { - 13\rho ^{2} + 2Z + 1} \right)}}{{\sqrt 2 \left({\rho ^{2} + 2Z + 1} \right)^{{\frac{{11}}{2}}} }}, \hfill \\\hfill \\ \end{aligned}$$

Under the heading 4.2.3 Existence of the equilibrium points of type E₊  and E₋ , the formula following the paragraph “Concerning the equilibrium points of type E₋, we have…”should read as follows:

$$ \begin{gathered} \hat{A}_{ - } = 80G^{4} + \lambda \left( {(595G^{2} - 1035)j_{4} - 175G^{2} - 96G + 189} \right), \hfill \\ \hat{B}_{ - } = \frac{{\hat{A}_{ - }^{2} - 5\lambda \hat{C}_{ - } \hat{D}_{ - } }}{{16}}, \hfill \\ \hat{C}_{ - } = (315G^{2} - 539)j_{4} - 63G^{2} - 72G - 11, \hfill \\ \hat{D}_{ - } = 32G^{4} + \lambda \left( {(95G^{2} - 175)j_{4} - 35G^{2} - 24G + 49} \right). \hfill \\ \end{gathered} $$

Under the heading 4.2.4 About the existence of ${{\bar{\boldsymbol{E}}}_{1}} \;{{\textbf{and}}}\;{\bar{\boldsymbol{E}}}_{2}$, the formula following the paragraph “If existing, the equilibrium points. E₁ and E₂ have coordinates…” should read as follows:

$$ \begin{aligned}\bar{X} =&{\text{ }} - \frac{{\rho ^{2} }}{{\lambda \left( {4375j_{4}^{5} -5375j_{4}^{4} + 2550j_{4}^{3} - 590j_{4}^{2} + 67j_{4} - 3}\right)}}\\&\left( - 2000\rho ^{4} (j_{4} - 1)(7j_{4} - 3)^{3} + \lambda \big((5j_{4} - 1)(55125j_{4}^{4} \rho ^{2} - 28700j_{4}^{3} \rho ^{2}- 14875j_{4}^{4} \vphantom{\sqrt {\frac{{7j_{4} - 3}}{{5j_{4} - 1}}} } \right.\\&- 23130j_{4}^{2} \rho ^{2} + 3350j_{4}^{3} +17460j_{4} \rho ^{2} + 7000j_{4}^{2} - 2835\rho ^{2} - 2950j_{4}+ 307)\\&\left. + 48\sqrt {\frac{{7j_{4} - 3}}{{5j_{4} - 1}}} \sqrt 5 |\rho |(125j_{4}^{4} - 250j_{4}^{3} + 160j_{4}^{2} - 38j_{4} + 3)\big)\right),\end{aligned} $$

Under the heading 4.3 The J₂-problem with relativistic terms, the formula following the paragraph “We study now the zonal problem containing both the…” should read as follows:

$$\begin{aligned}{\mathcal{K}}_{c} = & -\frac{{\mu ^{2} }}{{2L^{2} }} + \frac{{\mu ^{4} J_{2} R_{P}^{2}(G^{2} - 3H^{2} )}}{{4G^{5} L^{3} }} + \frac{{\mu ^{4} }}{{c^{2}L^{4} G}}\left( {5G - 8L} \right) + \frac{{3\mu ^{6} J_{2}^{2}R_{P}^{4} }}{{128L^{5} G^{{11}} }} \Big[ - 5G^{6} - 4G^{5}L{\text{ }}\\& + 24G^{3} H^{2} L - 36GH^{4} L - 35H^{4} L^{2} +G^{4} (18H^{2} + 5L^{2} ) \\&- 5G^{2} (H^{4} + 2H^{2} L^{2} ) +2(G^{2} - 15H^{2} )(G^{2} - L^{2} )(G^{2} - H^{2} )\cos 2g \Big] \\&+\frac{{\mu ^{6} J_{2} R_{P}^{2} }}{{8c^{2} L^{5} G^{7} }} \Big[(G^{2} -3H^{2} )(6L^{2} - 5G^{2} ) - 6(G^{2} - 3H^{2} )\\&(4G^{2} - 3GL -5L^{2} ) - 9(L^{2} - G^{2} )(G^{2} - H^{2} )\cos 2g \Big]. \end{aligned}$$

Under the heading 4.3.3 About the existence of the equilibrium points of type E₊ and E₋ , the formula following the paragraph “We obtain $\tilde{S}_{ + }$(G; a) = 0 for ${\rho}^2=\tilde{{\rho}} ^{{2}}_{{E}+{1},{2},}$ with…” should read as follows:

$$\begin{aligned} \tilde{\rho }^2_{E_{+_{{1,2}}}}&= {} \frac{G^2}{5\lambda }\frac{\tilde{A}_+\pm 4\sqrt{\tilde{B}_+}}{\tilde{C}_+}, \\ \tilde{A}_+&= {} -(-1040G^4 j_C+864G^3 j_C+1848G^2 j_C+49G^2-96G-99)\lambda -80G^4, \\ \tilde{B}_+&= {} \frac{\tilde{A}_+^2 +5\lambda \tilde{C}_+\tilde{D}_+}{16}, \qquad \tilde{C}_+ = -45G^2+72G+143, \\ \tilde{D}_+&= {} (-320G^4j_C+384G^3j_C+720G^2j_C-15G^2-24G+21)\lambda -128G^6j_C+32G^4 .\end{aligned}$$

Under the heading 4.3.5 About the stability of the equilibrium points of type E₊ and E₋ and of $\overline{\user2{E}} _{1} \, and{\text{ }}\overline{\user2{E}}_{2}$, the formula following the paragraph “Let us consider value of j_C sufficiently….” should read as follows:

$$ \begin{aligned} \frac{{d^{2} \tilde{X}}}{{dZ^{2} }} \pm\,&\frac{{d^{2} \hat{X}}}{{dZ^{2} }}\sim {\text{ }}16\sqrt { - 80j_{C}\rho ^{2} + 1} \bigg( - 144000j_{C}^{3} \rho ^{6} +28400j_{C}^{2} \rho ^{4} - 880j_{C} \rho ^{2} + 7{\text{ }} \\&+\sqrt { - 80j_{C} \rho ^{2} + 1} (10000j_{C}^{2} \rho ^{4} -600j_{C} \rho ^{2} + 7) \bigg).{\text{ }}\end{aligned}$$

$$\begin{aligned} \frac{{d^{2}\tilde{X}}}{{dZ^{2} }}\pm \,& \frac{{d^{2} \hat{X}}}{{dZ^{2} }}{\text{}} \sim 16\sqrt { - 80j_{C} \rho ^{2} + 1} \bigg( 144000j_{C}^{3}\rho ^{6} - 28400j_{C}^{2} \rho ^{4} + 880j_{C} \rho ^{2} -7{\text{ }} \\&+ \sqrt { - 80j_{C} \rho ^{2} + 1} (10000j_{C}^{2}\rho ^{4} - 600j_{C} \rho ^{2} + 7) \bigg)\end{aligned}$$