CMMSE: Study of a new symmetric anomaly in the elliptic, hyperbolic and parabolic Keplerian motion

In the present work, we define a new anomaly, $\Psi$, termed semifocal anomaly. It is determined by the mean between the true anomaly, $f$, and the antifocal anomaly, $f^{\prime}$; Fukushima defined $f^{\prime}$ as the angle between the periapsis and the secondary around the empty focus. In this first part of the paper, we take an approach to the study of the semifocal anomaly in the hyperbolic motion and in the limit case correspoding to the parabolic movement. From here we find a relation beetween the semifocal anomaly and the true anomaly that holds independently of the movement type. We focus on the study of the two-body problem when this new anomaly is used as the temporal variable.\\ In the second part, we show the use of this anomaly —combined with numerical integration methods— to improve integration errors in one revolution. Finally, we analyze the errors committed in the integration process —depending on several values of the eccentricity— for the elliptic, parabolic and hyperbolic cases in the apsidal region.

Brouwer’s satellite solution redux

Brouwer’s solution to the artificial satellite problem is revisited. We show that the complete Hamiltonian reduction is rather achieved in the plain Poincaré’s style, through a single canonical transformation, than using a sequence of partial reductions based on von Zeipel’s alternative for dealing with perturbed degenerate Hamiltonian systems. Beyond the theoretical interest of the new approach as regards the complete reduction of perturbed Keplerian motion, we also show that a solution based on a single set of corrections may yield computational benefits in the implementation of an analytic orbit propagator.

Analytic Inversion of Closed form Solutions of the Satellite’s J2 Problem

This report provides some closed form solutions -and their inversion- to a satellite’s bounded motion on the equatorial plane of a spheroidal attractor (planet) considering the J2 spherical zonal harmonic. The equatorial track of satellite motion- assuming the co-latitude φ fixed at π/2- is investigated: the relevant time laws and trajectories are evaluated as combinations of elliptic integrals of first, second, third kind and Jacobi elliptic functions. The new feature of this report is: from the inverse t = t(c) we get the period T of some functions c(t) of mechanical interest and then we construct the relevant c(t) expansion in Fourier series, in such a way performing the inversion. Such approach-which led to new formulations for time laws of a J2 problem- is benchmarked by applying it to the basic case of keplerian motion, finding again the classic results through our different analytic path.

Estimating supermassive black hole masses in active galactic nuclei using polarization of broad Mg ii, H α, and H β lines

ABSTRACT For type-1 active galactic nuclei (AGNs) for which the equatorial scattering is the dominant broad-line polarization mechanism, it is possible to measure the supermassive black hole (SMBH) mass by tracing the Keplerian motion across the polarization plane position angle φ. So far, this method has been used for 30 objects but only for H α emission line. We explore the possibilities of this method for determining SMBH masses using polarization in broad emission lines by applying it for the first time to Mg ii λ2798 Å spectral line. We use three-dimensional (3-D) Monte Carlo radiative transfer code stokes for simultaneous modelling of equatorial scattering of H α, H β, and Mg ii lines. We included vertical inflows and outflows in the Mg ii broad-line region (BLR). We find that polarization states of H α and H β lines are almost identical and SMBH-mass estimates differ by 7 per cent. For Mg ii line, we find that φ exhibits an additional ‘plateau’ with a constant φ, which deviates than the profiles expected for pure Keplerian motion. SMBH-mass estimates using Mg ii line are higher by up to 35 per cent than those obtained from H α and H β lines. Our model shows that for vertical inflows and outflows in the BLR that are higher or comparable to the Keplerian velocity, this method can be applied as a first approximation for obtaining SMBH mass.

Nonsingular recursion formulas for third-body perturbations in mean vectorial elements

The description of the long-term dynamics of highly elliptic orbits under third-body perturbations may require an expansion of the disturbing function in series of the semi-major axes ratio up to higher orders. To avoid dealing with long series in trigonometric functions, we refer the motion to the apsidal frame and efficiently remove the short-period effects of this expansion in vectorial form up to an arbitrary order. We then provide the variation equations of the two fundamental vectors of the Keplerian motion by analogous vectorial recurrences, which are free from singularities and take a compact form useful for the numerical propagation of the flow in mean elements.