We derive alternate and new closed-form analytic solutions for the non-equatorial eccentric bound trajectories, , around a Kerr black hole by using the transformation . The application of the solutions is straightforward and numerically fast. We obtain and implement translation relations between the energy and angular momentum of the particle, ( E , L ), and eccentricity and inverse-latus rectum, ( e , ), for a given spin, a , and Carter’s constant, Q , to write the trajectory completely in the ( e , , a , Q ) parameter space. The bound orbit conditions are obtained and implemented to select the allowed combination of parameters ( e , , a , Q ). We also derive specialized formulae for equatorial, spherical and separatrix orbits. A study of the non-equatorial analog of the previously studied equatorial separatrix orbits is carried out where a homoclinic orbit asymptotes to an energetically bound spherical orbit. Such orbits simultaneously represent an eccentric orbit and an unstable spherical orbit, both of which share the same E and L values. We present exact expressions for e and as functions of the radius of the corresponding unstable spherical orbit, r _s , a , and Q , and their trajectories, for ( ) separatrix orbits; they are shown to reduce to the equatorial case. These formulae have applications to study the gravitational waveforms from extreme-mass ratio inspirals (EMRIs) using adiabatic progression of a sequence of Kerr geodesics, besides relativistic precession and phase space explorations. We obtain closed-form expressions of the fundamental frequencies of non-equatorial eccentric trajectories that are equivalent to the previously obtained quadrature forms and also numerically match with the equivalent formulae previously derived. We sketch non-equatorial eccentric, separatrix, zoom-whirl, and spherical orbits, and discuss their astrophysical applications.

We report results from a one-week multiwavelength campaign to monitor the BL Lacertae object (BL Lac) S5 0716+714 (on 2009 December 9–16). Nine groun

The blazar 3C 454.3 exhibited a strong flare seen in γ-rays, X-rays and optical/near-infrared bands during 2009 December 3–12. Emission in the V and J bands rose more gradually than did the γ-rays and soft X-rays, though all peaked at nearly the same time. Optical polarization measurements showed dramatic changes during the flare, with a strong anticorrelation between optical flux and degree of polarization (which rose from ∼3 to ∼20 per cent) during the declining phase of the flare. The flare was accompanied by large rapid swings in polarization angle of ∼170°. This combination of behaviours appears to be unique. We have cm-band radio data during the same period but they show no correlation with variations at higher frequencies. Such peculiar behaviour may be explained using jet models incorporating fully relativistic effects with a dominant source region moving along a helical path or by a shock-in-jet model incorporating three-dimensional radiation transfer if there is a dominant helical magnetic field. We find that spectral energy distributions at different times during the flare can be fit using modified one-zone models where only the magnetic field strength and particle break frequencies and normalizations need change. An optical spectrum taken at nearly the same time provides an estimate for the central black hole mass of ∼2.3 × 10⁹ M_⊙. We also consider two weaker flares seen during the ∼200 d span over which multiband data are available. In one of them, the V and J bands appear to lead the γ-ray and X-ray bands by a few days; in the other, all variations are simultaneous.