# A `Numbers' Approach to Astronomical Correlations I: Introduction and Application to galaxy Scaling Relations

## TL;DR

This paper introduces a systematic method based on dimensional analysis to study correlations in astronomical systems, applying it to galaxy scaling relations like Tully-Fisher, revealing specific power-law dependencies.

## Contribution

It presents a novel approach using Buckingham's theorem to analyze correlations via dimensionless numbers, applied here to galaxy properties and scaling relations.

## Key findings

- Confirmed the Tully-Fisher relation: luminosity proportional to rotation velocity to the fourth power.
- Identified the Baryonic Tully-Fisher relation: baryonic mass proportional to rotation velocity cubed.
- Suggested different causal origins for the two scaling relations.

## Abstract

We propose a new systematic method of studying correlations between parameters that describe an astronomical (or any) physical system. We recall that behind Dimensionless scaling laws in complex, self-interacting physical objects lies a rigorous theorem of Dimensional analysis, known widely as the Buckingham theorem. Once a {\it catalogue} of properties and forces that define an object or physical system is established, the theorem allows one to select a complete set of Dimensionless quantities or {\it Numbers} on which structure must depend. The internal structure takes the form of a functionally defined manifold in the space of these Numbers. Simple and familiar examples are discussed by way of introduction. Correlations in properties of astronomical objects can be sought either through the constancy of these Numbers or between pairs of the Numbers. In either case, within errors, the functional dependences take on an absolute numerical character. As our principal application, we study a well defined sample of galaxies in order to reveal the implied Tully Fisher and Baryonic Tully Fisher relations. We find that $L\,\propto\,v_{rot}^4$ for the former and $M_b\,\propto\,v_{rot}^3$ for the latter, suggesting that these relations may have different causal origins.