Том 63, № 10 (2019)

The averaged semi-analytical motion theory of second order in planetary masses for the four-planet problem has been constructed. The Hamiltonian and equations of motion are given in Jacobi coordinates and written in elements of the second Poincaré system. The eccentric and oblique Poincaré orbital elements are conserved in the equations of motion up to third order. The orbital evolution of the three-planet system HD 39194 and the four-planet systems HD 141399 and HD 160691 (μ Ara) is considered. The numerical integration of the equations of motion was carried out for a set of initial conditions, in which unknown orbital elements and orbital elements that are known from observations with some uncertainty were varied within admissible limits. The ranges of variation of the orbital elements are determined as a function of the initial conditions. The assumption that the observed planetary systems are stable can be used to exclude initial conditions leading to extreme growth in the orbital eccentricities and inclinations. Initial conditions for which the orbital elements remain small over the entire modeling interval are identified. A method that can be used to narrow the range of possible values of the unknown orbital elements and identify most probable values from the point of view of stability is shown.

The correlation between scattering and the dispersion measure of the Crab Nebula pulsar B0531+21, obtained earlier from observations made on the Large Phased Array (LPA) at 111 MHz and at Jodrell Bank Observatory, has been confirmed using new independent measurements at the Pushchino Radio Astronomy Observatory and Jodrell Bank. The scattering varies in the range 10–115 ms, and can be explained by the passage of clouds of plasma in the nebula with sizes 10¹¹–10¹² m, electron densities n_e = 10³–10⁴ cm⁻³, and various degrees of turbulence in front of the pulsar. During a relatively quiescent period in 2002–2007, the turbulence coefficient in the direction of the Crab Nebula pulsar was found to be C²_n = 0.00661 m⁻⁷, and the mean density variations in the Crab Nebula to be Δn_e ~ 0.02 cm⁻³, n_e ~ 2 cm⁻³, so that Δn_e/n_e ~ 0.01. In the comparatively active period 2009–2013, C²_n = 0.0662 m⁻⁷, n_e ~ 2 cm⁻³, Δn_e ~ 0.06 cm⁻³, and Δn_e/n_e ~ 0.03.

A comparative analysis is presented for the physical parameters of gas in various types of molecular clouds, derived from observations and theoretical calculations. Cool, dense fragments of infrared dense clouds (IRDCs)—rare star-forming regions with OH radical emission in the 1720-MHz satellite line, extended green objects (EGOs)— diffuse bipolar outflows discovered by the Spitzer mission at short IR wavelengths, and gas-dust condensations hosting class I and II methanol masers (class I MM and class II MM) are considered. The magnetic fields of these regions are calculated using the empirical criterion of Crutcher (ApJ, 520, 706, 1999) and the equation for numerical modeling of cold turbulent clouds of Ostriker et al. (ApJ, 546, 980, 2001). This modeling is done for three values of the parameter β = (cs/cA)² relating the sound and Alfvén speeds, which describes the effect of the magnetic field: strong (β = 0.01), intermediate (β = 0.1), and weak (β = 1). In 69 EGOs with densities of 10⁴–10^5.6 cm⁻³, derived earlier from observations, and with an acceptable temperature range, from T_k ≈ 30 K to T_k ≈ 100 K, the results of the calculations are most consistent with β = 0.1 (|B̄|= 0.26 mG) and β = 1 (|B̄|= 0.15 mG) for intermediate and weak magnetic fields, respectively. For class I methanol masers, with gas densities not exceeding 10⁶ cm⁻³ and a temperature T_k = 80 K, the most plausible value is β = 1 (|B < 0.4 mG), i.e., the expected effect of the magnetic field is weak. Comparison of the sound and Alfvén speeds suggests that very faint, low-velocity C-type shocks may exist in these objcts, with the magnetic field exerting little control over the chaotic motions of the material. The shocks are continuous, and do not affect the parameters in the maser condensations. The magnetic intensities found for class I MMs and OH (1720 MHz) regions differ only slightly; according to this parameter, which is related to the density of the medium and the kinetic temperature of the gas, they may be observed in either different or the same gas-dust condensations.