How to obtain luminosity distance from a light curve

Question

I have some light curve data of a SN Ia from which I want to find the luminosity distance $D_L$.

How can I do it mathematically?

I'll then try to implement the answer using Python.

Answer 1

You want to basically assume that the supernova is a "standard candle," i.e., if we know with confidence how bright it $\it{should}$ be, then we can estimate the distance after measuring it's luminosity.

In a simple, crude approach, one can just use the good 'ole luminosity formula: assuming the source radiates with constant luminosity, its brightness will decrease with the inverse square law, and you invert the formula to get the luminosity distance, $$ D_{\rm L} = \sqrt{\frac{L}{4\pi F}}$$ where $L$ is the luminosity and $F$ is the flux.

Here's a GUI you may or may not find useful (ie see item 13 here).

Lots of helpful info here, and they also have an applet that you can download and play around with which uses the distance modulus (as answered here).

Note this is only a (crude) approximation, since it assumes zero extinction due to dust and doesn't account for cosmology. If this is for a research project, you'll likely need to do something more sophisticated and should consult your advisor and the literature, i.e., see this nice python notebook, or this python module.

I hope this helps!