# Stars that don't fit $L=4\pi R^{2}\sigma T^{4}$?

When calculating luminosity/radius/temperature for several stars, I've noticed that while the majority of stars conform to the luminosity relationship $$L=4\pi R^{2}\sigma T^{4}$$, there are a few that don't. Take 8 Ursae Minoris. As far as I can tell it has a luminosity 56 times that of the sun, a radius of 8.5 solar radii and an effective temperature of about 5000 K. When I substitute those numbers in they don't match. The math isn't wrong so I can only assume either the data isn't reliable, or maybe there is a certain class of stars that don't conform to that relation. I don't know enough astronomy to determine which is correct, but I suspect obtaining a reliable luminosity value for some stars could be the problem.

Does anyone know the answer to this riddle?

• Using the Stefan-Boltzmann law, I get $L\approx64L_{\odot}$, which isn't bad - only about a 15% difference from the measured value. Is that around what you got? Keep in mind that the Stefan-Boltzmann law only holds for black bodies, and while stars are close to black bodies, they aren't perfect black bodies - hence the deviations in various cases. – HDE 226868 Jun 26 '19 at 23:03
• @HDE226868 oops, I wrote 5600 K for the surface temperature rather than 5000 K, which is what my data says. I only get about $L\approx 40 L_{⊙}$ in that case. – TeeJay Jun 26 '19 at 23:09
• Ah, I see. Would you mind listing the sources of those values, just for completeness? Thanks. – HDE 226868 Jun 27 '19 at 4:41
• Well, there's always the simple "star thermal behavior isn't a pure black-body" but I suspect that only applies to real oddballs like neutron stars. – Carl Witthoft Jun 27 '19 at 14:37
• The equation applies to stars which are well approximated as being spherical, uniform-temperature blackbodies. There are various possible circumstances where this is not a good approximation but it doesn't look like these would apply to 8 UMi. Likely this mismatch is caused by collating different sources, which is why it would be good to know where your values are coming from. – antispinwards Jun 27 '19 at 20:35