# How to convert theoretical template spectrum from luminosity density to flux density units?

I'm working with galaxy spectral templates (e.g., Bruzual & Charlot 2003) which seem to always come with y-axis units of $L_{\odot}$/A and x-axis units of Angstroms. Thus the y-axis is a luminosity density instead of a flux density. In contrast, observationally, we tend to always work with spectra that have y-axis units of flux density ($F_{\lambda}$ in erg/s/cm$^2$/A or $F_{\nu}$ in erg/s/cm$^2$/Hz). Similarly, spectral energy distributions (SEDs) from photometry tend to have $\lambda F_{\lambda}$ or $\nu F_{\nu}$ such that the y-axis is flux, not flux density.

How do I convert a theoretical template spectrum from units of luminosity density ($L_{\odot}$/A) to flux density (erg/s/cm$^2$/A)?

For context, I want to fit spectral templates to an observed SED. The observed SED is for an object at a redshift $z$, so I think I can either convert the templates to flux density units, or I can convert my observed SED to luminosity density units. I feel like working in flux density units is more natural -- plus I'm not sure if multiplying the observed SED y-axis values by $4\pi D^2$ (D is distance of object) and x-axis (wavelengths) by $1/(1+z)$ would be sufficient (e.g., normalization concerns).

Use this equation: $$F_\nu = \frac{L_\nu}{4\pi D_L^2}.$$ This is the relationship that defines the luminosity distance, $D_L$, in a static Euclidean universe (i.e. not ours). Hogg has a good review on the arXiv of how to handle the relationship in an expanding universe, including when you need to perform something called a "$K$-correction" for spectral densities like the question asks.
• Thanks a lot, but I have one obvious question: for a theoretical template spectrum, what distance do I choose to convert from units of $L_{\odot}/A$ to erg/s/cm$^{-2}$/A? One can imagine having a set of observed spectra (at rest-frame wavelengths) for objects at a variety of redshifts, and a set of template spectra which should just be at some default distance that works for fitting to every observed object. I wonder if just setting D to "1 cm" would work... – quantumflash Feb 20 '17 at 19:14
• Yes, you can use the formula to convert $L_\lambda$ to $F_\lambda$ when $D=10\operatorname{pc}$, there are no cosmological (redshift) concerns then. – Sean Lake Jan 29 '18 at 19:53