Calculating star radius from temperature and absolute magnitude - what am I missing?

Question

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

Answer 1

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 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). 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 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 2

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.