What is the optional tau parameter in the Logarithmic1D model class?

I’m an undergrad (about to finish my Sophomore year) physics/astrophysics major and recently used Astropy to help me make a very rough model for my data. I ended up needing a logarithmic equation to fit it but I don’t know much of the math involved behind fitting data outside the most basic linear regressions so I’ve mostly just been coasting along using the amazing astropy.modeling package (namely the models and fitting features). However, I’d like to know how to at least get the equation that fitting my data with Logarithmic1D class and the LevMarLSQFitter function use.
Looking at the documentation amplitude makes enough sense, just an external factor… but I don’t understand what tau is or how I should treat it if I say… make a Logarithmic1D class with Amplitude=2 instead of the default 1. Again, my math (and coding knowledge honestly) isn’t the best… or at least it’s not nearly good enough to begin understanding how the fitters work and all that. For now the fitters are magic and that’s okay, all I need is just what tau is and how to turn the returned amplitude and tau values given to me by the fitter into the correct equation for my paper.
So I’ve come to this forum in the hopes that someone who understands the code and/or math better can help me answer these questions. If I need to provide more context or whatever that’s all good, the only thing I can think worth mentioning about my sample is that I wanted it to be quite large, ~1mil samples total as I was interested in the sweeping trends across different regions of the sky. Thanks for any help in advance!

Hi Ataamic, the documentation of the tau parameter in this model class is indeed quite unsatisfactory. While in principle it’s always good to have a basic grasp of the maths you are going to apply to your problem, here you can only infer the meaning from the source code. This shows that it is simply applied as a scale to the x inputs prior to applying the logarithm, while amplitude is scaling the y outputs afterwards. Typically tau is probably to be understood in the sense of an optical depth, at least that’s the first physical interpretation that springs to my mind. Hope this helps in translating this to your problem case.

Hi! That explanation is certainly good enough for me and makes much more sense now. Thanks for the explanation!