<p>The page you quote is guilty of a gross simplification (or error). The luminosity ratio <em>cannot</em> be estimated from the difference in visual magnitudes for stars of different spectral type. You need to use the difference in bolometric magnitude, which indicates the luminosity integrated over all wavelengths. This will involve making <a href="https://en.m.wikipedia.org/wiki/Bolometric_correction" rel="nofollow noreferrer">bolometric corrections</a> to the visual absolute magnitudes.</p>

<p>A star like Proxima Cen radiates much of its light outside the visible band and has a much different and larger bolometric correction than the Sun or Sirius. This accounts for your problem.</p>

<p>Example:</p>

<p>The correct calculation uses a table of bolometric corrections vs temperature. This rapidly changes for M-dwarfs in the visual band (you would be more precise using K magnitude), but is about 2.6 mag for a star at 3100 K (see <a href="http://ads.harvard.edu/cgi-bin/bbrowse?book=hsaa&amp;chap=2&amp;page=0069" rel="nofollow noreferrer">Zombeck 1992</a>). That is, the bolometric magnitude is smaller than $M_V$ by 2.6 mag. The equivalent correction for the Sun is only 0.1 mag.
$$ M_{\rm bol}= M_V - BC$$</p>

<p>Thus there is a 2.5 mag <em>less</em> difference in the <em>bolometric</em> magnitudes in your final equation than the difference in absolute visual magnitudes, and I get a radius for Barnard's star of $0.23\ R_{\odot}$.</p>

<p>The probable reason for the discrepancy between this and the radius quoted in wikipedia (actually it is the radius determined by an interferometric angular diameter measurement plus the parallax by Lane et al. 2001; a more recent measurement by <a href="https://arxiv.org/abs/1208.2431" rel="nofollow noreferrer">Boyajian et al. 2012</a> gives $0.187\pm 0.001\ R_{\odot}$) is that (a) the assumed temperature is too low or (b) that the bolometric correction scale for V-band magnitudes is not very well defined for very cool stars. Both (a) and (b) are possible, especially as Barnard' star probably has a significantly subsolar metallicity.</p>

CC BY-SA 3.0
2017-04-15T08:33:08.910Z
2017-07-16T17:16:58.427Z
2017-07-16T17:16:58.427Z
ProfRob
<p>In order to calculate this, you can use the Stefan-Boltzmann law to calculate the star's surface flux and its absolute magnitude to get the luminosity.  Once you know the surface flux and luminosity, you can find the radius of the star.</p>

<p>Stefan-Boltzmann Law:
$$
F=\sigma T^4
$$</p>

<p>Absolute Magnitude - Luminosity calculation:
$$
\frac{L}{L_\odot} = 10^{0.4(M_\odot - M)}
$$
where $L_\odot$ is solar luminosity and $M_\odot$ is the absolute magnitude of the sun.</p>

<p>Next, you use the formula for intensity and solve for distance:
$$
r = \sqrt{\frac{L}{4\pi F}}
$$
Compiling all this into one equation,</p>

<p>$$
r = \sqrt{\frac{L_\odot * 10^{0.4(M\odot - M)}}{4\pi \sigma T^4}}
$$</p>

<p>As long as you use SI units for $L_\odot$, $\sigma$, and $T$, you'll get $r$ in meters.</p>

CC BY-SA 3.0
2017-04-12T02:33:53.073Z
You're missing an important caveat: bolometric. I'm not sure, but I think that the distinction between band-pass (approximately spectral) fluxes and luminosities and bolometric (total) quantities is probably what's causing problems here.
Sean Lake
CC BY-SA 3.0
2017-04-12T13:29:30.490Z
Yes I played around with Stefan-Boltzman, but the trick explained both in the link I put in the original equation, and this link: http://www.astronomynotes.com/starprop/s4.htm (at the bottom), we can get the ratio of the two stars' luminosity if you have their absolute magnitudes from L1/L2 = 2.512^(M2 - M1). I wonder if the issue is indeed to do with bolometric magnitude than anything else. I suspect Barnard's star has much more going on in the R and I magnitudes...
Awoogamuffin
CC BY-SA 3.0
2017-04-12T18:33:35.640Z
Did you double check on your units?
Phiteros
CC BY-SA 3.0
2017-04-15T08:34:44.937Z
This just makes the same mistake as the linked page that the OP gives.
ProfRob
CC BY-SA 3.0
2017-04-11T22:10:00.580Z
2017-04-11T22:10:00.580Z
Phiteros
<p>In an attempt to visualise nearby stars, I've tried to get the star radius (measured in solar radii) using the equation provided on the following site:</p>

<p><a href="http://cas.sdss.org/dr4/en/proj/advanced/hr/radius1.asp" rel="nofollow noreferrer">http://cas.sdss.org/dr4/en/proj/advanced/hr/radius1.asp</a></p>

<p>Now though the equation seems to work fine for a star like Sirius, when I plug in the values for absolute magnitude and temperature for Barnard's star (according to wikipedia, 13.21 and 3134 K respectively) I get a radius of 0.0722. The radius provided by wikipedia is more than twice that value, at 0.196.</p>

<p>There are many potential culprits. Could it be floating point errors in the php code? The equation looks like this by the way:</p>

<p>$$\text{radius = pow(5800/3134, 2)} \times \text{pow(pow(2.512, (4.83 - 13.21)), 0.5})$$</p>

<p>Another possibility is that the equation is too simplistic, and there are other variables I need to take into account.</p>

<p>Any insights would be greatly appreciated!</p>

<p>Michael</p>

CC BY-SA 3.0
2017-04-11T21:57:35.067Z
I think you would want to use the Stefan-Boltzmann law and calculate flux from absolute magnitude. I'm on mobile now, so I can't work it out in full.
Phiteros
CC BY-SA 3.0
2017-09-14T18:49:13.940Z
"Any insights would be greatly appreciated!". hmm...
ProfRob
<star><temperature><absolute-magnitude>
Calculating star radius from temperature and absolute magnitude - what am I missing?
Awoogamuffin
# Calculating star radius from temperature and absolute magnitude - what am I missing?

(Asked by: username_0 on 2017-04-11)

In an attempt to visualise nearby stars, I've tried to get the star radius (measured in solar radii) using the equation provided on the following site:

<http://cas.sdss.org/dr4/en/proj/advanced/hr/radius1.asp>

Now though the equation seems to work fine for a star like Sirius, when I plug in the values for absolute magnitude and temperature for Barnard's star (according to wikipedia, 13.21 and 3134 K respectively) I get a radius of 0.0722. The radius provided by wikipedia is more than twice that value, at 0.196.

There are many potential culprits. Could it be floating point errors in the php code? The equation looks like this by the way:

$$\text{radius = pow(5800/3134, 2)} \times \text{pow(pow(2.512, (4.83 - 13.21)), 0.5})$$

Another possibility is that the equation is too simplistic, and there are other variables I need to take into account.

Any insights would be greatly appreciated!

Michael

**Tags:** star, temperature, absolute-magnitude

> **Comments:**
> **username_1**: I think you would want to use the Stefan-Boltzmann law and calculate flux from absolute magnitude. I'm on mobile now, so I can't work it out in full.  
> **username_2**: "Any insights would be greatly appreciated!". hmm...  

---

## 2 answers

### Answer by username_2 on 2017-04-15. Score: 3

The page you quote is guilty of a gross simplification (or error). The luminosity ratio *cannot* be estimated from the difference in visual magnitudes for stars of different spectral type. You need to use the difference in bolometric magnitude, which indicates the luminosity integrated over all wavelengths. This will involve making [bolometric corrections](https://en.m.wikipedia.org/wiki/Bolometric_correction) to the visual absolute magnitudes.

A star like Proxima Cen radiates much of its light outside the visible band and has a much different and larger bolometric correction than the Sun or Sirius. This accounts for your problem.

Example:

The correct calculation uses a table of bolometric corrections vs temperature. This rapidly changes for M-dwarfs in the visual band (you would be more precise using K magnitude), but is about 2.6 mag for a star at 3100 K (see [Zombeck 1992](http://ads.harvard.edu/cgi-bin/bbrowse?book=hsaa&chap=2&page=0069)). That is, the bolometric magnitude is smaller than $M\_V$ by 2.6 mag. The equivalent correction for the Sun is only 0.1 mag.
$$ M\_{\rm bol}= M\_V - BC$$

Thus there is a 2.5 mag *less* difference in the *bolometric* magnitudes in your final equation than the difference in absolute visual magnitudes, and I get a radius for Barnard's star of $0.23\ R\_{\odot}$.

The probable reason for the discrepancy between this and the radius quoted in wikipedia (actually it is the radius determined by an interferometric angular diameter measurement plus the parallax by Lane et al. 2001; a more recent measurement by [Boyajian et al. 2012](https://arxiv.org/abs/1208.2431) gives $0.187\pm 0.001\ R\_{\odot}$) is that (a) the assumed temperature is too low or (b) that the bolometric correction scale for V-band magnitudes is not very well defined for very cool stars. Both (a) and (b) are possible, especially as Barnard' star probably has a significantly subsolar metallicity.


---

### Answer by username_1 on 2017-04-11. Score: -1

In order to calculate this, you can use the Stefan-Boltzmann law to calculate the star's surface flux and its absolute magnitude to get the luminosity. Once you know the surface flux and luminosity, you can find the radius of the star.

Stefan-Boltzmann Law:
$$
F=\sigma T^4
$$

Absolute Magnitude - Luminosity calculation:
$$
\frac{L}{L\_\odot} = 10^{0.4(M\_\odot - M)}
$$
where $L\_\odot$ is solar luminosity and $M\_\odot$ is the absolute magnitude of the sun.

Next, you use the formula for intensity and solve for distance:
$$
r = \sqrt{\frac{L}{4\pi F}}
$$
Compiling all this into one equation,

$$
r = \sqrt{\frac{L\_\odot \* 10^{0.4(M\odot - M)}}{4\pi \sigma T^4}}
$$

As long as you use SI units for $L\_\odot$, $\sigma$, and $T$, you'll get $r$ in meters.

> **Comments:**
> **username_3**: You're missing an important caveat: bolometric. I'm not sure, but I think that the distinction between band-pass (approximately spectral) fluxes and luminosities and bolometric (total) quantities is probably what's causing problems here.  
> **username_0**: Yes I played around with Stefan-Boltzman, but the trick explained both in the link I put in the original equation, and this link: http://www.astronomynotes.com/starprop/s4.htm (at the bottom), we can get the ratio of the two stars' luminosity if you have their absolute magnitudes from L1/L2 = 2.512^(M2 - M1). I wonder if the issue is indeed to do with bolometric magnitude than anything else. I suspect Barnard's star has much more going on in the R and I magnitudes...  
> **username_1**: Did you double check on your units?  
> **username_2**: This just makes the same mistake as the linked page that the OP gives.  

---

