**Publication date:** Feb 2017

**Abstract:**

We demonstrate that in the framework of standard general relativity,

polytropic spheres with properly fixed polytropic index n and

relativistic parameter σ , giving a ratio of the central pressure

p_{c} to the central energy density ρ_{c} , can contain

a 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 increases with the polytropic

index for intervals 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 the validity of the polytropic configurations to their

extension r_{extr} corresponding to the gravitational mass M ̃2

M_{☉} of the most massive observed neutron stars. Then, for the

central density ρ_{c}̃1 0^{15} g cm^{-3} , the

trapped regions are outside r_{extr} for all values of 2.138

<n <4 ; for the central density ρ_{c}̃5 ×1 0^{15}

g cm^{-3} , the whole trapped regions are located inside

r_{extr} for 2.138 <n <3.1 ; while for ρ_{c}̃1

0^{16} g cm^{-3} , the whole trapped regions are inside

r_{extr} for all values of 2.138 <n <4 , guaranteeing

astrophysically plausible trapping for all considered polytropes. The

region of trapped null geodesics is located close to the polytrope

center and could have a relevant influence on the cooling of such

polytropes or binding of gravitational waves in their interior.

**Authors:**

Novotný, Jan; Hladík, Jan; Stuchlík, Zdeněk;