Trapped, Two-Armed, Nearly Vertical Oscillations in Disks with Toroidal Magnetic Fields

TL;DR

This study investigates how toroidal magnetic fields influence the trapping and frequencies of nearly vertical two-armed oscillations in relativistic disks, revealing that magnetic strength affects trapping regions and oscillation frequencies.

Contribution

It provides a detailed analysis of the effects of toroidal magnetic fields on wave trapping in relativistic disks, highlighting the dependence of oscillation frequencies and trapping regions on magnetic field strength.

Findings

Oscillations are trapped in the inner disk region.

Frequencies decrease as magnetic field strength increases.

Trapped regions are narrow for fundamental and first-overtone modes, wider for higher overtones.

Abstract

We have examined trapping of two-armed ($m=2$) nearly vertical oscillations (vertical p-mode) in vertically isothermal ($c_{\rm s}=$ const.) relativistic disks with toroidal magnetic fields. The magnetic fields are stratified so that the Alfv{\'e}n speed, $c_{\rm A}$, is constant in the vertical direction. The ratio of $c_{\rm A}^2/c_{\rm s}^2$ in the vertical direction is taken as a parameter examining the effects of magnetic fields on wave trapping. We find that the two-armed nearly vertical oscillations are trapped in the inner region of disks and their frequencies decrease with increase of $c_{\rm A}^2/c_{\rm s}^2$. The trapped regions of the fundamental ($n=1$) and the first-overtone ($n=2$) are narrow (less than the length of the Schwarzschild radius, $r_{\rm g}$) and their frequencies are relatively high (on the order of the angular frequency of disk rotation in the inner region). On contrast to this, the second-overtone ($n=3$) are trapped in a wide region (a few times $r_{\rm g}$), and their frequencies are low and tend to zero in the limit of $c_{\rm A}^2/c_{\rm s}^2=2.0$.