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

<p>I assume therefore that you are talking about <em>absolute</em> magnitudes in those filter band passes.</p>

<p>Luminosity can also <em>not</em> 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.</p>

<p>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.</p>

<p><span class="math-container">$$ L_{\rm band} = \int f_{\nu} S_{\nu} d\nu$$</span></p>

<p>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.</p>

<p>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.</p>

<p>Page 100 of <a href="http://ads.harvard.edu/cgi-bin/bbrowse?book=hsaa&amp;chap=2&amp;page=0100" rel="nofollow noreferrer">Zombeck (1992)</a> gives the flux <em>density</em> zeropoints in UBVRIJHK corresponding to magnitude 0.0. So take your magnitude and scale the zeropoint flux accordingly (by <span class="math-container">$10^{-0.4m}$</span>) and then multiply the flux density by the effective bandpass width. Finally to convert an observed flux to a luminosity, multiply by <span class="math-container">$4\pi d^2$</span>, where <span class="math-container">$d=10$</span> pc, if you are dealing with absolute magnitudes. </p>

<p>By "PARSEC" I assume you are talking about the stellar evolution models produced by the <a href="http://stev.oapd.inaf.it/cgi-bin/cmd" rel="nofollow noreferrer">Padova group</a>. In which case a basic output of the model is the stellar luminosity. Any magnitudes that are provided by the models <em>are calculated afterwards</em> by using a particular set of <a href="https://en.wikipedia.org/wiki/Bolometric_correction" rel="nofollow noreferrer">bolometric corrections</a> (the relationship between absolute bolometric magniude and absolute magnitude in a particular passband).</p>

<p>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.</p>

CC BY-SA 4.0
2018-10-25T13:55:17.403Z
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!
KarenRei
CC BY-SA 4.0
2018-10-25T14:15:50.173Z
@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). http://stev.oapd.inaf.it/tmp/output59540467643.dat  It has logL as a column.
ProfRob
CC BY-SA 4.0
2018-10-25T17:39:39.580Z
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: http://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.
KarenRei
CC BY-SA 4.0
2018-10-25T19:02:47.347Z
Sample file like the ones I'm parsing: http://stev.oapd.inaf.it/tmp/output498688187523.dat
KarenRei
CC BY-SA 4.0
2018-10-25T19:26:32.370Z
@KarenRei So the 5th column is log (Luminosity).
ProfRob
CC BY-SA 4.0
2018-10-25T20:04:38.817Z
Yes, but not broken down into spectral bands.  Spectral band magnitudes do me no good; I need watts.
KarenRei
CC BY-SA 4.0
2018-10-25T20:18:20.937Z
@KarenRei The magnitudes are derived from **the** bolometric luminosity that is converted to a bolometric magnitude and then a bolometric correction is added to get the magnitude in a given band
ProfRob
CC BY-SA 4.0
2018-10-25T20:30:42.960Z
See the edit I've just done. This gives you an approximate route. You cannot do anything else without the spectrum of the star and the detailed filter bandpasses.
ProfRob
CC BY-SA 4.0
2018-10-25T21:11:48.077Z
Very helpful! If I've got this, then it looks like there's one detail you left out - the formula I get is: W = 10^(-0.4 * mag + log10(fv0)) * Δλ0 * 1e-6 m/μm * m². The last term should be the surface area of the star, since these zero points are per unit area.  Correct?  If so I'll go ahead and accept your answer and offer you my thanks  :)
KarenRei
CC BY-SA 4.0
2018-10-25T21:13:45.170Z
Wait... ugh... I don't see any stellar radii in the dataset....
KarenRei
CC BY-SA 4.0
2018-10-25T21:15:28.283Z
You know, maybe I'm just better off assuming a blackbody and using that to generate spectral components...
KarenRei
CC BY-SA 4.0
2018-10-25T21:32:38.840Z
No.The zero points are the flux received at earth. You still need to multiply by $4\pi d^2$.
ProfRob
CC BY-SA 4.0
2018-10-25T21:36:55.777Z
Also don't mix definitions. You have to use $f_{\lambda}$ with $\Delta \lambda$. And of course if you did need it (you don't), stellar radius is defined by L and the effective temperature.
ProfRob
CC BY-SA 4.0
2018-10-25T08:15:15.040Z
2018-10-25T21:34:55.100Z
2018-10-25T21:34:55.100Z
ProfRob
<p>No comment on PARSEC data, but here is how to do bolometric luminosity from UBVRIJHK magnitudes:<br>
 1. Map filter to its effective wavelength (e.g., U = 3650 A).<br>
 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.<br>
 3. Integrate {specific flux (i.e., in erg/s/A/cm^2) * effective wavelength (i.e., in A)}. Normally, trapezoid rule is enough.<br>
 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.<br>
 5. L = flux * area. If you assume spherical symmetry, area = <span class="math-container">$4 \pi r^2$</span>, where r = luminosity distance in this case. Note that you get the units right.<br>
 6. If you assume BB-SED, an easier way to get the bolometric luminosity, but less preferred in most cases, is to simply do <span class="math-container">$L = 4 \pi r^2 \sigma T^4$</span>.</p>

<p>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.</p>

CC BY-SA 4.0
2018-10-25T08:49:09.147Z
3.828E26 is the defined "solar luminosity".
ProfRob
CC BY-SA 4.0
2018-10-25T04:26:44.823Z
2018-10-25T17:54:40.640Z
2018-10-25T17:54:40.640Z
Kornpob Bhirombhakdi
<p>So, I know that the standard for conversion from bolometric magnitude to luminosity established in 2015 equates to:</p>

<pre><code>L = 10^(-0.4 * magnitude) * 3.0128e28
</code></pre>

<p>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:</p>

<pre><code>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
</code></pre>

<p>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.</p>

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

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

<p>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....</p>

<p>Tau1m: ??? Usually 0.00, but sometimes something small like "0.06"</p>

<p>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.</p>

<units><absolute-magnitude><parsec>
Conversion of UBVRIJHK magnitudes to luminosity?
KarenRei
# Conversion of UBVRIJHK magnitudes to luminosity?

(Asked by: username_0 on 2018-10-25)

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.

**Tags:** units, absolute-magnitude, parsec

---

## 2 answers

### Answer by username_1 on 2018-10-25. Score: 1 | Accepted answer

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)](http://ads.harvard.edu/cgi-bin/bbrowse?book=hsaa&chap=2&page=0100) 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](http://stev.oapd.inaf.it/cgi-bin/cmd). 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](https://en.wikipedia.org/wiki/Bolometric_correction) (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.

> **Comments:**
> **username_0**: 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!  
> **username_1**: @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). http://stev.oapd.inaf.it/tmp/output59540467643.dat  It has logL as a column.  
> **username_0**: 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: http://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.  
> **username_0**: Sample file like the ones I'm parsing: http://stev.oapd.inaf.it/tmp/output498688187523.dat  
> **username_1**: @KarenRei So the 5th column is log (Luminosity).  
> **username_0**: Yes, but not broken down into spectral bands.  Spectral band magnitudes do me no good; I need watts.  
> **username_1**: @KarenRei The magnitudes are derived from **the** bolometric luminosity that is converted to a bolometric magnitude and then a bolometric correction is added to get the magnitude in a given band  
> **username_1**: See the edit I've just done. This gives you an approximate route. You cannot do anything else without the spectrum of the star and the detailed filter bandpasses.  
> **username_0**: Very helpful! If I've got this, then it looks like there's one detail you left out - the formula I get is: W = 10^(-0.4 * mag + log10(fv0)) * Δλ0 * 1e-6 m/μm * m². The last term should be the surface area of the star, since these zero points are per unit area.  Correct?  If so I'll go ahead and accept your answer and offer you my thanks  :)  
> **username_0**: Wait... ugh... I don't see any stellar radii in the dataset....  
> **username_0**: You know, maybe I'm just better off assuming a blackbody and using that to generate spectral components...  
> **username_1**: No.The zero points are the flux received at earth. You still need to multiply by $4\pi d^2$.  
> **username_1**: Also don't mix definitions. You have to use $f_{\lambda}$ with $\Delta \lambda$. And of course if you did need it (you don't), stellar radius is defined by L and the effective temperature.  

---

### Answer by username_2 on 2018-10-25. Score: 0

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.

> **Comments:**
> **username_1**: 3.828E26 is the defined "solar luminosity".  

---

