The integral expression for the gravitational potential of a homogeneous circular torus composed of infinitely thin rings is obtained. Approximate expressions for the torus potential in the outer and inner regions are found. In the outer region, the torus potential is shown to be approximately equal to that of an infinitely thin ring of the same mass; it is valid up to the surface of the torus. It is shown in a first approximation that the inner potential of the torus (inside a torus body) is a quadratic function of the coordinates. A method of sewing together the inner and outer potentials is proposed. This method provides a continuous approximate solution for the potential and its derivatives, working throughout the region.

We investigate cosmological perturbations for nonlinear f( R) models within the cosmic screening approach. Matter is considered both in the form of a set of discrete point-like massive bodies and in...

The emission from dark ages halos in the lines of transitions between the lowest rotational levels of hydrogen and hydrogen deuteride molecules is analyzed. It is assumed that molecules are excited by the cosmic microwave background (CMB) and collisions with hydrogen atoms. The physical parameters of halos and the number density of molecules are precalculated assuming that halos are homogeneous top-hat spheres formed from the cosmological density perturbations in the four-component universe with post-Planck cosmological parameters. The differential brightness temperatures and differential spectral fluxes in the rotational lines of H₂–HD molecules are computed for two phenomena: thermal luminescence and resonant scattering of CMB radiation. The results show that the expected maximal values of differential brightness temperature of warm halos (T _K ∼ 200–800 K) are at the level of nanokelvins, are comparable for both phenomena, and are below the sensitivity of modern submillimeter radio telescopes. For hot halos (T _K ∼ 2000–5000 K) the thermal emission of H₂-ortho molecules dominates and the differential brightness temperatures are predicted to be of a few microkelvins at the frequencies 300–600 GHz, which could be detectable with next-generation telescopes.

In this paper, a comparison of the impact of major sudden stratospheric warmings (SSWs) in the Arctic in February 2018 (SSW1) and January 2019 (SSW2) on the mid-latitude mesosphere is given. The mesospheric carbon monoxide (CO) and zonal wind in these two major SSW events were observed at altitudes of 70–85 km using a microwave radiometer (MWR) at Kharkiv, Ukraine (50.0°N, 36.3°E). Data from ERA-Interim and MERRA-2 reanalyses and Aura Microwave Limb Sounder measurements were also used. It is shown that: (i) The differences between SSW1 and SSW2, in terms of local variability in zonal wind, temperature, and CO in the stratosphere and mesosphere, were clearly defined by the polar vortex (westerly in cyclonic circulation) and mid-latitude anticyclone (easterly) migrating over the MWR station, therefore; (ii) mesospheric intrusions of CO-rich air into the stratosphere over the Kharkiv region occurred only occasionally, (iii) the larger zonal wave 1–3 amplitudes before SSW1 were followed by weaker polar vortex recovery than that after SSW2, (iv) the strong vortex recovery after SSW2 was supported by earlier event timing (midwinter) favoring vortex cooling due to low solar irradiance and enhanced zonal circulation, and (v) vortex strengthening after SSW2 was accompanied by wave 1–3 amplification in March 2019, which was absent after SSW1. Finally, the influence of the large-scale circulation structures formed in individual major SSW events on the locally recorded characteristics of the atmosphere is discussed.

The planetary wave activity in the stratosphere–mesosphere during the Arctic major Sudden Stratospheric Warming (SSW) in February 2018 is discussed on the basis of microwave radiometer (MWR) measurements of carbon monoxide (CO) above Kharkiv, Ukraine (50.0° N, 36.3° E) and the Aura Microwave Limb Sounder (MLS) measurements of CO, temperature and geopotential heights. From the MLS data, eastward and westward migrations of wave 1/wave 2 spectral components were differentiated, to which less attention was paid in previous studies. Abrupt changes in zonal wave spectra occurred with the zonal wind reversal near 10 February 2018. Eastward wave 1 and wave 2 were observed before the SSW onset and disappeared during the SSW event, when westward wave 1 became dominant. Wavelet power spectra of mesospheric CO variations showed statistically significant periods of 20–30 days using both MWR and MLS data. Although westward wave 1 in the mesosphere dominated with the onset of the SSW 2018, it developed independently of stratospheric dynamics. Since the propagation of upward planetary waves was limited in the easterly zonal flow in the stratosphere during SSW, forced planetary waves in the mid-latitude mesosphere may exist due to the instability of the zonal flow.

We analyze the thermal emission and resonant scattering of CMB radiation from the dark ages halos in the rotational lines of the helium hydride ion (${\mathrm{HeH}}^{+}$), one of the first chemical compounds in the Universe. Evaluating the optical depth, thermal, and resonant brightness temperatures and spectral fluxes of dark ages halos is based on computing the cross sections and rate coefficients of excitation/deexcitation of the lowest five rotational states of ${\mathrm{HeH}}^{+}$ by inelastic collisions with atomic hydrogen. It was shown that in the Dark Ages the collisional excitation/deexcitation by atoms of neutral hydrogen and electrons are competitive, whereas in the denser regions, e.g., in the halos, the contribution of collisions with atomic hydrogen is dominant. We demonstrate the peaklike time-dependence of halo luminosities in ${\mathrm{HeH}}^{+}$ lines.