Anisotropy of partially self-absorbed jets and the jet of Cyg X-1

TL;DR

This paper analyzes the angular dependence of flux in partially self-absorbed jets, deriving formulas and applying results to Cyg X-1 to estimate jet velocity and opening angle, revealing a highly collimated jet structure.

Contribution

It provides analytical and numerical solutions for flux dependence on viewing angle in conical jets, and applies these to observational data of Cyg X-1 to infer jet properties.

Findings

Maximum emission occurs when the jet is viewed from top on-axis.

Jet velocity in Cyg X-1 is estimated to be ≥ 0.8c.

Jet half-opening angle is inferred to be ≤ 1 degree.

Abstract

We study the angular dependence of the flux from partially synchrotron self-absorbed conical jets (proposed by Blandford \& K{\"o}nigl). We consider the jet viewed from either a side or close to on axis, and in the latter case, either from the jet top or bottom. We derive analytical formulae for the flux in each of these cases, and find the exact solution for an arbitrary angle numerically. We find that the maximum of the emission occurs when the jet is viewed from top on-axis, which is contrast to a previous result, which found the maximum at some intermediate angle and null emission on-axis. We then calculate the ratio of the jet-to-counterjet emission for this model, which depends on the viewing angle and the index of power-law electrons. We apply our results to the black-hole binary Cyg X-1. Given the jet-to-counterjet flux ratio of $\gtrsim$50 found observationally and the current estimates of the inclination, we find the jet velocity to be $\gtrsim 0.8c$. We also point out that when the projection effect is taken into account, the radio observations imply the jet half-opening angle of $\lesssim 1^{\rm o}$, a half of the value given before. When combined with the existing estimates of $\Gamma_{\rm j}$, the jet half-opening angle is low, $\ll 1/\Gamma_{\rm j}$, and much lower than values observed in blazars, unless $\Gamma_{\rm j}$ is much higher than currently estimated.