# Analytical calculation of black hole spin using deformation of the shadow

## TL;DR

This paper derives simple analytical formulas to estimate black hole spin from shadow deformation measurements, enabling easier observational analysis without full shadow modeling.

## Contribution

It introduces compact analytical expressions linking black hole spin to shadow oblateness, simplifying the extraction process from observational data.

## Key findings

- Analytical formulas for black hole spin from shadow oblateness.
- Approximate methods for nearly extreme black holes.
- Spin estimation without full shadow curve modeling.

## Abstract

We succeed to find compact analytical expressions which allow to easily extract the black hole spin from observations of its shadow, without need to construct or model the entire curve of the shadow. The deformation of Kerr black hole shadow can be characterised in a simple way by oblateness (the ratio of the horizontal and vertical angular diameters which are supposed to be measured by an observer). The deformation is significant in case the black hole is nearly extreme and observer is not so far from the equatorial plane. In this approximation, we present: (i) the spin lower limit via oblateness, (ii) the spin via oblateness and viewing angle, in case the latter is known from other observations.