# Conversion of UBVRIJHK magnitudes to luminosity?

So, I know that the standard for conversion from bolometric magnitude to luminosity established in 2015 equates to:

L = 10^(-0.4 * magnitude) * 3.0128e28


Easy enough. But what does one do about luminosities in specific bands? I've seen some information that makes it look like it should be something like:

U: 10^(0.4*(5.61 - U-band magnitude)) * 3.828e26
B: 10^(0.4*(5.48 - B-band magnitude)) * 3.828e26
V: 10^(0.4*(4.83 - V-band magnitude)) * 3.828e26
R: 10^(0.4*(4.42 - R-band magnitude)) * 3.828e26
I: 10^(0.4*(4.08 - I-band magnitude)) * 3.828e26
J: 10^(0.4*(3.64 - J-band magnitude)) * 3.828e26
H: 10^(0.4*(3.32 - H-band magnitude)) * 3.828e26
K: 10^(0.4*(3.28 - K-band magnitude)) * 3.828e26


But I'm not sure about these, because I've seen varying figures from varying sources for the magnitude of the sun in various bands. I'm trying to extract the luminosity values from PARSEC data.

Speaking of PARSEC data, the documentation is rather lacking, so any clue about the following fields?

McoreTP: Sounds like it should be something about core temperature, but the value is usually either 0, or something small like "0.546"

Mloss: Presumably mass loss, but there's no units! But I can probably work out the units just by comparing how star masses change over time....

Tau1m: ??? Usually 0.00, but sometimes something small like "0.06"

Cexcess: ??? Generally "-1.000", but sometimes things like "7.356". Note that there's an entirely separate field for Xc, which is clearly carbon mass fraction.

Luminosities cannot be calculated from magnitudes without knowing the distance to the source (and perhaps something about extinction).

I assume therefore that you are talking about absolute magnitudes in those filter band passes.

Luminosity can also not be arrived at from a single absolute magnitude measurement in the way you suggest in your post, since for a source of a given luminosity, how much of that luminosity emerges in a specific band is entirely dependent on the spectrum of the source.

OK, but now I see what you want - you want to divide up the bolometric luminosity of the Sun between the individual bands using the absolute magnitude of the Sun in those bands. But the luminosity doesn't divide up neatly like that, partly because the bands overlap and partly because the bandpasses are not top-hat functions so it's difficult to see what a "luminosity in the U-band" actually means other than an integration of the intrinsic stellar spectrum (which you don't have) over the (normalised) bandpass sensitivity profile.

$$L_{\rm band} = \int f_{\nu} S_{\nu} d\nu$$

The formulae you quote contain the absolute magnitudes of the Sun in those bands. That is the first number in the brackets. If you subtract the bolometric magnitude from this, then that gives you (minus) the bolometric correction. The problem is that the UBV etc. magnitude system and the bolometric magnitude system do not share the same zeropoint. The bolometric correction in the V band is about zero, so your formulae would have the entire luminosity of the Sun emerging in the V band. The zeropoints of the UBV etc. system are set by defining an A0 star (like Vega) to have an absolute magnitude of zero in all pass bands.

What you can do is convert the magnitudes into some sort of mean flux density over the bandpass and then just multiply this by the "effective width" of the filter. This is very approximate and essentially assumes the spectrum is flat across the filter bandpass.

Page 100 of Zombeck (1992) gives the flux density zeropoints in UBVRIJHK corresponding to magnitude 0.0. So take your magnitude and scale the zeropoint flux accordingly (by $$10^{-0.4m}$$) and then multiply the flux density by the effective bandpass width. Finally to convert an observed flux to a luminosity, multiply by $$4\pi d^2$$, where $$d=10$$ pc, if you are dealing with absolute magnitudes.

By "PARSEC" I assume you are talking about the stellar evolution models produced by the Padova group. In which case a basic output of the model is the stellar luminosity. Any magnitudes that are provided by the models are calculated afterwards by using a particular set of bolometric corrections (the relationship between absolute bolometric magniude and absolute magnitude in a particular passband).

As regards the specific columns in the tables, you will likely have to go to the original papers that describe the models. I cannot comment further without knowing exactly which models they are or where you got them.

• I (perhaps mistakenly) thought it was pretty clear that I was talking about A) absolute, not apparent magnitudes, and that B) I already have have the absolute UBVRIJHK magnitudes for a given star (not the sun), but want its luminosity in those bands. My apologies if there was any confusion. Re, PARSEC: the isochrone outputs from the model are presented as magnitudes, not luminosities. I need luminosities. Thanks for your help! Oct 25 '18 at 13:55
• @KarenRei Can you say where you got the models from. Stellar models deal with luminosities and effective temperatures. Magnitudes can only be calculated afterwards using bolometric corrections. Here is one I generated at the web address in my answer (only available for 2 hours). stev.oapd.inaf.it/tmp/output59540467643.dat It has logL as a column. Oct 25 '18 at 14:15
• Thanks for your reply Rob. Unfortunately your link is a 404 - think I'm an hour too late. I'm parsing the output of the web-based stellar isochrone dataset generator here: stev.oapd.inaf.it/cgi-bin/cmd - using PARSEC v1.2S + COLIBRI PR16 data to generate isochrones of constant metallicity (one for each of a wide range of metallicities) in order to incorporate stellar evolution into another model. You can see that they provide Umag through Kmag. Clearly they generated those figures from luminosities, but I need to reverse that and get luminosities back. Oct 25 '18 at 17:39
• Sample file like the ones I'm parsing: stev.oapd.inaf.it/tmp/output498688187523.dat Oct 25 '18 at 19:02
• @KarenRei So the 5th column is log (Luminosity). Oct 25 '18 at 19:26

No comment on PARSEC data, but here is how to do bolometric luminosity from UBVRIJHK magnitudes:
1. Map filter to its effective wavelength (e.g., U = 3650 A).
2. Convert magnitudes to specific fluxes + extinction correction and other. Make sure that magnitudes, which are expressed in Vega, AB, or other reference systems, are converted accordingly. Extinction correction and other corrections may or may not what you want to do depending on what you want to get in the end.
3. Integrate {specific flux (i.e., in erg/s/A/cm^2) * effective wavelength (i.e., in A)}. Normally, trapezoid rule is enough.
4. Consider how to deal with wavelength outside the observed bandpass (i.e., wavelength shorter than U, and longer than K). Normally, people have the edges to be zero and quote the luminosity as "pseudo." If you really want "bolometric," you can assume blackbody spectral energy distribution (BB-SED) to reconstruct the edges. But, this depends on whether the BB-SED is suitable for your object. If this is the case, then you fit the observation to BB function to get temperature and scale factor. Then, bolometric flux = flux calculated in step 3 + correction from the edges estimated by the BB-SED.
5. L = flux * area. If you assume spherical symmetry, area = $$4 \pi r^2$$, where r = luminosity distance in this case. Note that you get the units right.
6. If you assume BB-SED, an easier way to get the bolometric luminosity, but less preferred in most cases, is to simply do $$L = 4 \pi r^2 \sigma T^4$$.

According to your formula up there, U: 10^(0.4*(5.61 - bolometric magnitude)), this is the 2nd step with magnitude system reference to Vega system (I guess). '* 3.828e26' is probably the area.

• 3.828E26 is the defined "solar luminosity". Oct 25 '18 at 8:49