# 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).

## 1 Answer

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... Feb 20 '17 at 19:14
• You use the distance to the object in question. For galaxies, there are additional complications due to the expansion of the universe, and they are well explained in that paper by Hogg that I linked in the answer. Feb 21 '17 at 1:58
• I'm returning to this now and I still don't understand your answer. Suppose I'm working with only a theoretical model spectrum, so there is nothing about the "actual distance to the object in question" or Hubble flow or anything. Model spectra (such as from Bruzual & Charlot) typically have luminosity density units (Lsun/AA) vs AA, such that the spectrum tells you the luminosity output normalized to 1 Msun of stars formed. Can we use the fact that abs magnitudes and luminosities assume D=10 parsec, to convert Lsun/AA to erg/s/cm**2/AA, using your F=L/(4piD^2) formula? Thanks! Jan 29 '18 at 19:31
• 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. Jan 29 '18 at 19:53