Abstract:
We analytically and numerically investigate the possibility that a still undiscovered body X, moving along an unbound hyperbolic path from outside the solar system, may penetrate its inner regions in the next few years posing a threat to the Earth. By conservatively using as initial position of X the lower bounds on the present‐day distance of X dynamically inferred from the gravitational perturbations induced by it on the orbital motions of the planets of the solar system, both the analyses show that, in order to reach the Earth’s orbit in the next 2 yr, X should move at a highly unrealistic speed , whatever its mass is. For example, by assuming for it a solar ( M ) or brown dwarf mass ( ), now at not less than kau (1 kau=1000 astronomical units), v would be of the order of and of the speed of light c, respectively. By assuming larger present‐day distances for X, on the basis of the lacking of direct observational evidences of electromagnetic origin for it, its speed would be even higher. Instead, the fastest solitary massive objects known so far, like hypervelocity stars (HVSs) and supernova remnants (SRs), travel at , having acquired so huge velocities in some of the most violent astrophysical phenomena like interactions with supermassive galactic black holes and supernova explosions. It turns out that the orbit of the Earth would not be macroscopically altered by a close (0.2 au) passage of such an ultrafast body X in the next 2 yr. On the contrary, our planet would be hurled into the space if a Sun‐sized body X would encounter it by moving at . On the other hand, this would imply that such a X should be now at just 20-30 au, contrary to all direct observational and indirect dynamical evidences.

Abstract:
We summarize some critical issues pertaining the tests of the general relativistic Lense-Thirring effect performed by I. Ciufolini and coworkers in the gravitational field of the Earth with the geodetic satellites LAGEOS and LAGEOS II tracked with the Satellite Laser Ranging technique.

Abstract:
The impact of possible a-priori “imprinting” effects of general relativity itself on recent attempts to measure the general relativistic Lense-Thirring effect with the LAGEOS satellites orbiting the Earth and the terrestrial geopotential models from the dedicated mission GRACE is investigated. It is analytically shown that general relativity, not explicitly solved for in the GRACE-based models, may “imprint” their even zonal harmonic coeffi-cients of low degrees at a non-negligible level, given the present-day accuracy in recovering them. This trans-lates into a bias of the LAGEOS-based relativistic tests as large as the Lense-Thirring effect itself. Further analyses should include general relativity itself in the GRACE data processing by explicitly solving for it.

Abstract:
I calculate the classical effects induced by an isotropic mass loss of a body on the orbital motion of a test particle around it; the present analysis is also valid for a variation of the Newtonian constant of gravitation. I perturbatively obtain negative secular rates for the osculating semimajor axis a, the eccentricity e and the mean anomaly , while the argument of pericenter ω does not undergo secular precession, like the longitude of the ascending node Ω and the inclination I. The anomalistic period is different from the Keplerian one, being larger than it. The true orbit, instead, expands, as shown by a numerical integration of the equations of motion in Cartesian coordinates; in fact, this is in agreement with the seemingly counter-intuitive decreasing of a and e because they only refer to the osculating Keplerian ellipses which approximate the trajectory at each instant. By assuming for the Sun it turns out that the Earth's perihelion position is displaced outward by 1.3 cm along the fixed line of apsides after each revolution. By applying our results to the phase in which the radius of the Sun, already moved to the Red Giant Branch of the Hertzsprung-Russell Diagram, will become as large as 1.20 AU in about 1 Myr, I find that the Earth's perihelion position on the fixed line of the apsides will increase by AU (for ); other researchers point towards an increase of AU. Mercury will be destroyed already at the end of the Main Sequence, while Venus should be engulfed in the initial phase of the Red Giant Branch phase; the orbits of the outer planets will increase by AU. Simultaneous long-term numerical integrations of the equations of motion of all the major bodies of the solar system, with the inclusion of a mass-loss term in the dynamical force models as well, are required to check if the mutual N-body interactions may substantially change the picture analytically outlined here, especially in the Red Giant Branch phase in which Mercury and Venus may be removed from the integration.

The high velocities observed in supernovae require a relativistic treatment for the equation of motion in the presence of gradients in the density of the interstellar medium. The adopted theory is that of the thin layer approximation. The chosen medium is auto-gravitating with respect to an equatorial plane. The differential equation which governs the relativistic conservation of momentum is solved numerically and by recursion. The asymmetric field of relativistic velocities as well the time dilation is plotted at the age of 1 yr for SN 1987A.

Abstract:
The conservation of the energy flux in turbulent jets which propagate in the intergalactic medium (IGM) allows deducing the law of motion in the classical and relativistic cases. Three types of IGM are considered: constant density, hyperbolic and inverse power law decrease of density. An analytical law for the evolution of the magnetic field along the radio-jets is deduced using a linear relation between the magnetic pressure and the rest density. Astrophysical applications are made to the centerline intensity of synchrotron emission in NGC315 and to the magnetic field of 3C273.

Abstract:
We present an analytical solution for the luminosity distance in spatially flat cosmology with pressureless matter and the cosmological constant. The complex analytical solution is made of a real part and a negligible imaginary part. The real part of the luminosity distance allows finding the two parameters H_{0} and Ω_{M} . A simple expression for the distance modulus for SNs of type Ia is reported in the framework of the mini-max approximation.

Abstract:
A superbubble which advances in a symmetric Navarro-Frenk-White density profile or in an auto-gravitating density profile generates a thick shell with a radius that can reach 10 kpc. The application of the symmetric and asymmetric image theory to this thick 3D shell produces a ring in the 2D map of intensity and a characteristic “U” shape in the case of 1D cut of the intensity. A comparison of such a ring originating from a superbubble is made with the Einstein’s ring. A Taylor approximation of order 10 for the angular diameter distance is derived in order to deal with high values of the redshift.

Abstract:
A classical model based on a power law assumption for the radius-time relationship in the expansion of a supernova (SN) allows to derive an analytical expression for the flow of mechanical kinetic energy and the time duration of gamma-ray burst (GRB). A random process based on the ratio of two truncated lognormal distributions for luminosity and luminosity distance allows to derive the statistical distribution for time duration of GRBs. The high velocities involved in the first phase of expansion of a SN require a relativistic treatment. The circumstellar medium is assumed to follow a density profile of Plummer type with eta = 6. A series solution for the relativistic flow of kinetic energy allows to derive in a numerical way the duration time for GRBs. Here we analyze two cosmologies: the standard cosmology and the plasma cosmology.

Abstract:
The luminosity function (LF) for stars is here fitted by a Schechter function and by a Gamma probability density function. The dependence of the number of stars on the distance, both in the low and high luminosity regions, requires the inclusion of a lower and upper boundary in the Schechter and Gamma LFs. Three astrophysical applications for stars are provided: deduction of the parameters at low distances, behavior of the average absolute magnitude with distance, and the location of the photometric maximum as a function of the selected flux. The use of the truncated LFs allows modeling the Malmquist bias.