A `Numbers’ Approach to Astronomical Correlations I: Introduction and Application to galaxy Scaling Relations. (arXiv:1902.08704v1 [astro-ph.GA])
<a href="http://arxiv.org/find/astro-ph/1/au:+Henriksen_R/0/1/0/all/0/1">R. N. Henriksen</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Irwin_J/0/1/0/all/0/1">J. A. Irwin</a>
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.
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.
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