We review the statistical properties of the main populations of radio sources, as emerging from radio and millimeter sky surveys. Recent determinations of local luminosity functions are presented and compared with earlier estimates still in widespread use. A number of unresolved issues are discussed. These include: the (possibly luminosity-dependent) decline of source space densities at high redshifts; the possible dichotomies between evolutionary properties of low- versus high-luminosity and of flat- versus steep-spectrum AGN-powered radio sources; and the nature of sources accounting for the upturn of source counts at sub-milli-Jansky (mJy) levels. It is shown that straightforward extrapolations of evolutionary models, accounting for both the far-IR counts and redshift distributions of star-forming galaxies, match the radio source counts at flux-density levels of tens of μJy remarkably well. We consider the statistical properties of rare but physically very interesting classes of sources, such as GHz Peak Spectrum and ADAF/ADIOS sources, and radio afterglows of γ-ray bursts. We also discuss the exploitation of large-area radio surveys to investigate large-scale structure through studies of clustering and the Integrated Sachs–Wolfe effect. Finally, we briefly describe the potential of the new and forthcoming generations of radio telescopes. A compendium of source counts at different frequencies is given in Supplementary Material.

Radio and Millimeter Continuum Surveys and their Astrophysical Implications

This paper deals with the numerical investigations of the locations of the five equilibrium points by taking into consideration the effect of oblateness of the more massive primary for some systems of astronomical interest. This note is further concerned with the periodic solutions of the linearized equations of motion around the five equilibrium points. Interesting differences in the trends of the angular frequencies of these motions have been noticed.

Stationary solutions and their characteristic exponents in the restricted three-body problem when the more massive primary is an oblate spheroid

This paper deals with the stationary solutions of the planar restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The collinear equilibria have conditional retrograde elliptical periodic orbits around them in the linear sense, while the triangular points have long- or short-periodic retrograde elliptical orbits for the mass parameter 0 ≤ μ < μcrit, the critical mass parameter, which decreases with the increase in oblateness and radiation force. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case μ = μcrit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness.

The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid

We discuss the existence, location, and stability of the collinear equilibrium points of a generalized Hill problem with radiation of the primary (the Sun) and oblateness of the secondary (the planet), and present some remarkable fractals created as basins of attraction of Newton’s method applied for their computation in several cases of the parameters.

Collinear equilibrium points of Hill’s problem with radiation and oblateness and their fractal basins of attraction

In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J2 and J4. After that, we show that the triangular points are stable for 0<μ<μc and unstable when $\mu_{c}\leq \mu\leq \frac{1}{2}$, where μc is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μc using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J4 for some planets systems as in Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems.

https://www.tpbin.com/Uploads/Subjects/39bc02d2-48ed-4fe0-8a58-45605d133c56.pdf

Existence and stability of triangular points in the restricted three-body problem with numerical applications

A note on the stability of the triangular points of equilibrium in the restricted three-body problem.

In this paper location of the collinear libration points is investigated numerically, by taking the oblateness of the primaries into consideration, for 19 systems of astronomical interest. It is found that in some of the systems the shifts are significant. These equilibria are shown to be unstable in general, though the existence of conditional, infinitesimal (linearized) periodic orbits around them can be established, in the usual way. It is shown that the eccentricity and synodic period of these orbits are functions of oblateness too. Numerical study, in this connection, with the above systems, revealed that the orbits around the libration point, which is farthest from the primary whose oblateness effect is included, exhibit a different trend from those around the other two points.

Collinear equilibria and their characteristic exponents in the restricted three-body problem when the primaries are oblate spheroids

Non-linear stability of the libration point L
4 of the restricted three-body problem is studied when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion, Moser's conditions are utilised in this study by employing the iterative scheme of Henrard for transforming the Hamiltonian to the Birkhoff's normal form with the help of double D'Alembert's series. It is found that L
4 is stable for all mass ratios in the range of linear stability except for the three mass ratios: 
$$\begin{gathered} \mu _{c1} = 0.0242{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.1790{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c2} = 0.0135{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0993{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c3} = 0.0109{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0294{\text{ }}...{\text{ }}A_1 . \hfill \\ \end{gathered} $$

Ram Krishan Sharma

Papers

Stationary solutions and their characteristic exponents in the restricted three-body problem when the more massive primary is an oblate spheroid

The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid

A note on the stability of the triangular points of equilibrium in the restricted three-body problem.

Collinear equilibria and their characteristic exponents in the restricted three-body problem when the primaries are oblate spheroids

Effect of oblateness on the non-linear stability of L 4 in the restricted three-body problem