We demonstrate that in the framework of standard general relativity polytropic spheres with properly fixed polytropic index n and relativistic parameter σ, giving ratio of the central pressure pc to the central energy density ρc, can contain region of trapped null geodesics. Such trapping polytropes can exist for n > 2.138 and they are generally much more extended and massive than the observed neutron stars. We show that in the n-σ parameter space the region of allowed trapping increase with polytropic index for interval of physical interest 2.138 < n < 4. Space extension of the region of trapped null geodesics increases with both increasing n and σ > 0.677 from the allowed region. In order to relate the trapping phenomenon to astrophysically relevant situations, we restrict validity of the polytropic configurations to their extension rextr corresponding to the gravitational mass M ∼ 2M of the most massive observed neutron stars. Then for the central density ρc ∼ 10 15 g cm −3 the trapped regions are outside rextr for all values of 2.138 < n < 4, for the central density ρc ∼ 5 × 10 15 g cm −3 the whole trapped regions are located inside of rextr for 2.138 < n < 3.1, while for ρc ∼ 10 16 g cm −3 the whole trapped regions are inside of rextr for all values of 2.138 < n < 4, guaranteeing astrophysically plausible trapping for all considered polytropes. The region of trapped null geodesics is located closely to the polytrope centre and could have relevant influence on cooling of such polytropes or for binding of gravitational waves in their interior.

Abstract. We study behaviour of gravitational waves in the recently introduced general relativistic polytropic spheres containing a region of trapped null geodesics extended around radius of the stable null circular geodesic that can exist for the polytropic index N > 2.138 and the relativistic parameter, giving ratio of the central pressure p c to the central energy density ρ c , higher than σ = 0.677. In the trapping zones of such polytropes, the effective potential of the axial gravitational wave perturbations resembles those related to the ultracompact uniform density objects, giving thus similar long-lived axial gravitational modes. These longlived linear perturbations are related to the stable circular null geodesic and due to additional non-linear phenomena could lead to conversion of the trapping zone to a black hole. We give in the eikonal limit examples of the long-lived gravitational modes, their oscillatory frequencies and slow damping rates, for the trapping zones of the polytropes with N ∈ (2.138, 4). However, in the trapping polytropes the long-lived damped modes exist only for very large values of the multipole number ℓ > 50, while for smaller values of ℓ the numerical calculations indicate existence of fast growing unstable axial gravitational modes. We demonstrate that for polytropes with N ≥ 3.78, the trapping region is by many orders smaller than extension of the polytrope, and the mass contained in the trapping zone is about 10 −3 of the total mass of the polytrope. Therefore, the gravitational instability of such trapping zones could serve as a model explaining creation of central supermassive black holes in galactic halos or galaxy clusters.

Plasma modification of powder has recently attracted much interest because of new prospects of the interfacial properties supervision. There was mostly used the low-pressure plasma modification in fluidized bed, but for industrial-scale application the employment of mechanical stirring also appears to be very promising at the moment. It was shown that application of plasma modification methods led to substantial process time reduction. In case of the polyethylene, parts sintered from the plasma-modified polyethylene powder were characterized with high surface tension, which allowed e.g. direct painting or adhesive bonding without any additional pre-treatment. Plasma modification also significantly enhanced the adhesion of the polymer to the substrate. Adhesion and wettability measurements were carried out to reveal surface modification of plasma treatment of HDPE powders. This paper describes results of the polyethylene powder plasma treatment in microwave low-pressure plasma and discusses possibilities of its application.