Quantum tunneling makes a Reissner–Nordstrom black hole turn into a Kerr–Newman black hole under angular momentum conservation

Hawking radiation can be viewed as a process of particle quantum tunneling near a black hole horizon. When a particle with angular momentum tunnels across the event horizon of a Reissner–Nordstrom black hole, the black hole will change into a Kerr–Newman one. In previous papers, axisymmetric black hole has been studied only when a keeps constant. Changing from Reissner–Nordstrom to Kerr–Newman should be a simple case when a varies. After this, more general radiation including changed a in Kerr or Kerr–Newman spacetimes can be studied in the future. The emission rate of the massless particles with angular momentum is calculated, and the result is consistent with an underlying unitary theory.      

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”. The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating, extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as long as they are rightly identified as the formulae for the inertial angular velocity).     

Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth

We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes de référence spatio-temporels. l’Observatoire de Paris, pp 74–81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251–283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117–180, 1995), but considering the contribution of the convective terms. Specifically, we derive analytical formulas that determine the elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron. 105:261–274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded the motion of the tide raising bodies with respect to a non-rotating reference system when compared with the Earth rotational motion. We demonstrate that, from a numerical perspective, the convective part provides the principal contribution to the variation of the directional angles and of length of day. In the case of the x and y components in the Tisserand system, the convective contribution is of the same order of magnitude as the direct one. Finally, we show that the approximation employed in Kubo (Ibid.) leads to significant numerical differences at the level of a hundred micro-arcsecond.     

Continuity and stability of families of figure eight orbits with finite angular momentum

Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron. 78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given numerically, and some examples are given of nearby stable orbits which bifurcate from these families.     

Lense-thirring precession of radio pulsars

It is shown that if neutron stars contain dynamically de-coupled components, most plausibly discrete superfluid zones, then it is possible for the spin axes of these components o become slightly misaligned with respect to the crustal spin following a series of glitches. The crust will then undergo Lense-Thirring precession about the total angular moment with a period of ∼ 6–7.5P (assuming a crustal superfluid) and ∼ 3–6P (if the core superfluid is not tightly coupled to the crust). The precise precessional period is diagnostic of the mass distribution within each component. The implications of recent observational inferences concerning glitching pulsars are discussed. The conditions necessary for precession to be observable are analysed phenomenologically and a search of pulse-timing data for evidence of a Lense-Thirring modulation within the period range ∼ 3–8P is proposed.     

Astrometric CCD observations of visual double stars at the Pulkovo Observatory

We present the results of 3489 astrometric observations for 361 visual double stars performed in 2003–2007 with the 26-inch refractor of the Pulkovo Observatory. The angular separations between the components (ρ) and the position angles (θ) are given. The errors in these quantities are, on average, 0″.009 for ρ and 0°.40/ρ for θ, where ρ is the separation in arcseconds.     

Carbon recombination lines from the Galactic plane at 34.5 & 328 MHz

We present the results of a search for carbon recombination lines in the Galaxy at 34.5 MHz (C575α) made using the dipole array at Gauribidanur near Bangalore. Observations made towards 32 directions resulted in detections of lines, in absorption at nine positions. Followup observations at 328 MHz (C272α) using the Ooty Radio Telescope detected these lines in emission. A VLA D-array observation of one of the positions at 330 MHz yielded no detection implying a lower limit of 10′ for the angular size of the line forming region. The longitude-velocity distribution of the observed carbon lines indicate that the line forming regions are located mainly between 4 kpc and 7 kpc from the Galactic centre. Combining our results with published carbon recombination line data near 76 MHz (Erickson, McConnell & Anantharamaiah 1995), we obtain constraintson the physical parameters of the line forming regions. We find thatif the angular size of the line forming regions is ≥ 4°, then the range of parameters that fit the data are:T _e =20–40 K,n _e ∼ 0.1–0.3 cm⁻³ and pathlengths ∼ 0.07–0.9 pc which may correspond to thin photodissociated regions around molecular clouds. On the other hand, if the line forming regions are ∼ 2° in extent, then warmer gas (T _e ∼ 60–300 K) with lower electron densities (n _e ∼ 0.03–0.05 cm⁻³) extending over several tens of parsecs along the line of sight and possibly associated with atomic HI gas can fit the data. Based on the range of derived parameters, we suggest that the carbon line regions are most likely associated with photo-dissociation regions.     

Reconstruction of the Heliospheric Current Sheet Tilts Using Sunspot Numbers

We have investigated the correlation between the relative sunspot number and tilt of the heliospheric current sheet (HCS) in solar cycles 21–23. Strong and highly significant positive correlation (r > 0.8, P < 0.001) was found for corresponding data in the time interval from May 1976 through December 2004. Cross-correlation analysis does not reveal any time shift between the data sets. Reconstructed values of the HCS tilt, for the time interval before 1976, are found using sunspot numbers. To take different amplitude of solar cycles into account they were then normalized to zero in the minima of the solar activity and to average in solar cycles 21–23 maximal calculated HCS tilt in the maxima. These normalized reconstructed HCS data are compared with the angular positions of the brightest coronal streamers observed during total solar eclipses in 1870–2002, and their agreement is better for the minima of the solar activity than for the maxima.     

Motion Around The Triangular Equilibrium Points Of The Restricted Three-Body Problem Under Angular Velocity Variation

We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The symmetric periodic orbits emanating from the collinear Lagrangian point L ₃, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2

Radio synthesis imaging of anisotropic angular broadening in the solar wind

We present Very Large Array observations at wavelengths of 2, 3.5, 6, and 20 cm, of angular broadening of radio sources due to the solar wind in the region 2–16 solar radii. Angular broadening is anisotropic with axial ratios in the range 2–16. Larger axial ratios are observed preferentially at smaller solar distances. Assuming that anisotropy is due to scattering blobs elongated along magnetic field lines, the distribution of position angles of the elliptically broadened images indicates that the field lines are non-radial even at the largest heliocentric distances observed here. At 5R⊙, the major axis scattering angle is ∼ 0.7" atλ= 6 cm and it varies with heliocentric distance asR ^-1.6. The level of turbulence, characterized by the wave structure function at a scale of 10 km along the major axis, normalized toλ = 20 cm, has a value 20 ± 7 at 5R⊙and varies with heliocentric distance asR ^-3. Comparison with earlier resu lts suggest that the level of turbulence is higher during solar maximum. Assuming a power-law spectrum of electron density fluctuations, the fitted spectral exponents have values in the range 2.8–3.4 for scale sizes between 2–35 km. The data suggests temporal fluctuations (of up to 10%) in the spectral exponent on a time scale of a few tens of minutes. The observed structure functions at different solar distances do not show any evidence for an inner scale; the upper limits are l k m at 2R⊙ and 4 km at 13R⊙. These upper limits are in conflict with earlier determinations and may suggest a reduced inner scale during solar maximum.     

Nonaxisymmetric instabilities of self-gravitating disks. I toroids

We perform an extensive linear investigation of the nonaxisymmetric disk modes referred to in the literature as P, I, and J modes in self-gravitating polytropic toroids with power law angular velocity distributions. For selected models, we also follow the development of instability from the linear regime through the quasi-linear regime to deep into the nonlinear regime. We consider modes with azimuthal dependence e ^imφ , where m is an integer and φ is the azimuthal angle. We find that instability sets in through m=2 barlike I modes at T/|W|∼0.16–0.18 depending upon the chosen angular velocity law where T is the rotational kinetic energy and W is the gravitational energy of the toroid. Instability in the barlike I mode peaks in strength around T/|W|=0.22–0.23 after which it weakens, eventually stabilizing around T/|W|∼0.25–0.26. One-armed modes (m=1 modes) become unstable just after instability in the m=2I modes sets in; instability in m=1 modes sets in at T/|W|∼0.19. They dominate the barlike I modes in toroids with T/|W|≳0.25. However, almost immediately after the m=1 mode overtakes the barlike I mode, higher-m J modes appear. J modes with m=2, 3, and 4 become unstable for T/|W|≳0.25–0.26, 0.23–0.25, and 0.25–0.26, respectively. m≥3J modes dominate the m=1 mode in toroids with T/|W|≳0.27. As T/|W| increases further, nonaxisymmetric instability sets in through higher and higher m modes. We find quantitative agreement between the early nonlinear behavior of the tested unstable toroids and our linear results. Quasi-linear modeling suggests that a gravitational self-interaction torque which arises early in the nonlinear regime saturates growth of the mode and leads to significant transport of mass and angular momentum. Neither I mode nor J mode instabilities produce prompt fission in toroids.     

Correlation Analyses Between the Characteristic Times of Gradual Solar Energetic Particle Events and the Properties of Associated Coronal Mass Ejections

It is generally believed that gradual solar energetic particles (SEPs) are accelerated by shocks associated with coronal mass ejections (CMEs). Using an ice-cream cone model, the radial speed and angular width of 95 CMEs associated with SEP events during 1998 – 2002 are calculated from SOHO/LASCO observations. Then, we investigate the relationships between the kinematic properties of these CMEs and the characteristic times of the intensity-time profile of their accompanied SEP events observed at 1 AU. These characteristic times of SEP are i) the onset time from the accompanying CME eruption at the Sun to the SEP arrival at 1 AU, ii) the rise time from the SEP onset to the time when the SEP intensity is one-half of peak intensity, and iii) the duration over which the SEP intensity is within a factor of two of the peak intensity. It is found that the onset time has neither significant correlation with the radial speed nor with the angular width of the accompanying CME. For events that are poorly connected to the Earth, the SEP rise time and duration have no significant correlation with the radial speed and angular width of the associated CMEs. However, for events that are magnetically well connected to the Earth, the SEP rise time and duration have significantly positive correlations with the radial speed and angular width of the associated CMEs. This indicates that a CME event with wider angular width and higher speed may more easily drive a strong and wide shock near to the Earth-connected interplanetary magnetic field lines, may trap and accelerate particles for a longer time, and may lead to longer rise time and duration of the ensuing SEP event.     

A statistical study of Galactic SNRs using the PMN survey

The Parkes–MIT–NRAO (PMN) radio survey has been used to generate a quasi all-sky study of Galactic Supernova Remnants (SNRs) at a common frequency of 4.85 GHz (λ=6 cm). We present flux densities estimated for the sample of 110 Southern Galactic SNRs (up to δ=−65°) observed with the Parkes 64-m radio telescope and an additional sample of 54 from the Northern PMN (up to δ=+64°) survey undertaken with the Green Bank 43-m (20 SNRs) and 91-m (34 SNRs) radio telescopes. Out of this total sample of 164 selected SNRs (representing 71% of the currently 231 known SNRs in the Green catalogue) we consider 138 to provide reliable estimates of flux density and surface brightness distribution. This sub-sample represents those SNRs which fall within carefully chosen selection criteria which minimises the effects of the known problems in establishing reliable fluxes from the PMN survey data. Our selection criteria are based on a judicious restriction of source angular size and telescope beam together with careful evaluation of fluxes on a case by case basis. Direct comparison of our new fluxes with independent literature values gives excellent overall agreement. This gives confidence in the newly derived PMN fluxes when the selection criteria are respected. We find a sharp drop off in the flux densities for Galactic SNRs beyond 4 Jy and then a fairly flat distribution from 5 to 9 Jy, a slight decline and a further flat distribution from 9 to 20 Jy though the numbers of SNR in each Jy bin are low. We also re-visit the contentious Σ–D (radio surface brightness–SNRs diameter) relation to determine a new power law index for a sub-sample of shell type SNRs which yields β=−2.2±0.6. This new evaluation of the Σ–D relation, applied to the restricted sample, provides new distance estimates and their Galactic scale height distribution. We find a peak in the SNR distribution between 7–11 kpc with most restricted to ±100 pc Galactic scale height.     

The GraF instrument for Imaging Spectroscopy with the Adaptive Optics

The GraF instrument using a Fabry-Perot interferometer cross-dispersed with a grating was one of the first integral-field and long-slit spectrographs built for and used with an adaptive optics system. We describe its concept, design, optimal observational procedures and the measured performances. The instrument was used in 1997–2001 at the ESO3.6 m telescope equipped with ADONIS adaptive optics and SHARPII+camera. The operating spectral range was 1.2–2.5 μm. We used the spectral resolution from 500 to 10 000 combined with the angular resolution of 0.1″–0.2″. The quality of GraF data is illustrated by the integral field spectroscopy of the complex0.9″ × 0.9″ central region of η Car in the1.7 μm spectral range at the limit of spectral and angular resolutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

New CCD photometry for the extreme lower mass-ratio binary AW Ursae Majoris

This paper presents charge-couple device (CCD) photometric observations for the eclipsing binary AW UMa. The V-band light curve in 2007 was analyzed using the 2003 version of the Wilson–Devinney code. It is confirmed that AW UMa is a total eclipsing binary with a higher degree of contact f=80.2% and a lower mass ratio of q=0.076. From the (O−C) curve, the orbital period shows a continuous period decrease at a rate of dP/dt=−2.05×10⁻⁷ d yr⁻¹. The long-term period decrease suggested that AW UMa is undergoing the mass transfer from the primary component to the secondary one, accompanied by angular momentum loss due to mass outflow L ₂. Weak evidence indicates that there exists a cyclic variation with a period of 17.6 yr and a small amplitude of A=0. ^d 0019, which may be attributed to the light-time effect via the third body. If the existence of an additional body is true, it may remove a great amount of angular momentum from the central system. For this kind of contact binary, as the orbital period decreases, the shrinking of the inner and outer critical Roche lobes will cause the contact degree f to increase. Finally, this kind of binary will merge into a single rapid-rotation star.     

The kinematical mechanism for the perturbation of the rotational axis in the rotation of the elastic Earth

Kubo (Celest Mech Dyn Astron 110:143–168, 2011) investigated the kinematical structure of the perturbation in the rotation of the elastic Earth due to the deformation caused by the outer bodies. In that paper, while the mechanism for the perturbation of the figure axis was made clear, that for the rotational axis was not shown explicitly. In the present study, following the same method, the structure of the perturbation of the rotational axis is investigated. This perturbation consists of the direct perturbation and the convective perturbation. First the direct perturbation is shown to be (A − C)/A times as large as that of the figure axis, coinciding with the analytical expressions obtained in preceding studies by other authors. As for the convective perturbation, which appears only in the perturbation of the rotational axis but not in that of the figure axis, it is shown to be (A − C)/A times the angular separation between the original figure axis and the induced figure axis produced by the elastic deformation, A and C being the principal moments of inertia of the Earth. If the perturbing bodies are motionless, the conclusion of Kubo (Celest Mech Dyn Astron 105:261–274, 2009) holds strictly, i.e. the sum of the direct and the convective perturbations of the rotational axis coincides with the perturbation of the figure axis.     

Report on the dynamical evolution of an axially symmetric quasar model

The role of the angular momentum in the regular or chaotic character of motion in an axially symmetric quasar model is examined. It is found that, for a given value of the critical angular momentumL _zc , there are two values of the mass of the nucleusM _n for which transition from regular to chaotic motion occurs. The [L _zc – M _n ] relationship shows a linear dependence for the time independent model and an exponential dependence for the evolving model. Both cases are explained using theoretical arguments together with some numerical evidence. The evolution of the orbits is studied, as mass is transported from the disk to the nucleus. The results are compared with the outcomes derived for galactic models with massive nuclei.     

Relaxation time of the superfluid core of a neutron star

The superconducting proton condensate in the “npe” phase of a neutron star is considered. It is shown to be a type II superconductor in the outer layer of the “npe” phase and a type I superconductor in the inner layer. Relaxation times are found for elastic scattering of normal relativistic electrons from the magnetic fields of proton vortex clusters in the case of a type II superconductor and elastic scattering from the magnetic field at the center of a neutron vortex in the case of a superconductor of the first kind. The dynamical relaxation times obtained for the angular velocity of the pulsar PSR 0833—45 vary, as a function of the density of the layers taking part in the relaxation process, within a fairly wide range: from several hours to l0⁹ years. This means that the characteristic times of variation of pulsar angular velocity may be observed to lie in the indicated time range. Translated from Astrofizika, Vol. 40, No. 4, op. 497–506, October–December, 1997.     

Spectral and spatial fluctuations of cosmic microwave background radiation. RATAN-600 radio telescope observations with a Fourier spectral analyzer

The “circumpolar survey” technique allowed us to achieve a rather high antenna temperature sensitivity of 0.5 mK at 6 cm on a fixed antenna with a relatively narrow frequency resolution band (600 kHz) and an angular resolution of 45″ × 7′. Starting from 2007 we use a Fourier spectral analyzer with a survey band 10 times broader than that of previous observations. This upgrade allowed us to rapidly perform repeated test observations, which confirmed the power-law nature and the frequency structure of spatial spectra in two strips in the vicinity of the celestial pole in the 4′–16′ interval of angular periods. These observations also showed that the Galactic CH molecular line that was by chance captured in our survey strip (Λ-doubling of rotational transition) has nothing to do with the emission spectra we obtained. The angular pattern of emission in two main lines of Λ-doubling (4847.84 and 4870.12 MHz) is absolutely different, whereas there is practically no maser amplification in the CH line to explain this fact. Our observations also confirm the H110 α line (4874.157 MHz), which forms in the tenuous ionized Galactic hydrogen at such relatively high latitudes. Other features of the spatial spectra also appear to be quite real, although their interpretation remains unclear and requires further studies.     

Orbit determination with the two-body integrals. II

The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or by radar observations. We write polynomial equations for this problem, which can be solved using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. This paper continues the research started in Gronchi et al. (Celest. Mech. Dyn. Astron. 107(3):299–318, 2010), where the angular momentum and the energy integrals were used. The use of a suitable component of the Laplace–Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.