Calculating flux

I think a slightly more helpful quantity to calculate is the flux received by the planet - the power per unit area from the star. The mean flux on Earth is the solar constant, $F_e=1.36\times {10}^3{\text{ W m}}^{-2}$. If a planet orbits a star of luminosity $L_{\ast }$ at a distance $r_p$, the flux received is $$F_p=\frac{L_{\ast }}{4\pi r_p^2}$$ The ratio of the planet's flux $F_p$ to the solar constant $F_e$ is $$\frac{F_p}{F_e}=\frac{L_{\ast }}{L_{\odot }}{\left(\frac{r_p}{1\text{ AU}}\right)}^{-2}$$ For an M4V dwarf, I'd expect $L_{\ast }\approx 0.006L_{\odot }$. Plugging this in, we get $$F_p=1.1\times {10}^{-3}{F}_e=1.50{\text{ W m}}^{-2}$$ for an M4V star.

Why should we use flux instead of lux?

• It's easier to calculate. And when I say easier, I mean much easier. Lux is a lot more complicated.
• It's great for calculating things like the effective temperature of a planet.
• It's much more commonly used in this sort of scenario.

I used stellar models by Eric Mamajek to find the star's luminosity - he gives a $\log L_{\ast }/L_{\odot }=-2.2$, so $L_{\ast }\approx 0.006L_{\odot }$. It's worth noting that those models list an M4V star as having $T_{eff}\approx 3200\text{ K}$ and $R\approx 0.258R_{\odot }$ - so this is based on those figures. Using your numbers (which are for a smaller, cooler star) and the Stefan-Boltzmann law, I get $L_{\ast }\approx 0.00186L_{\odot }$, leading to a flux of $F_p\approx 0.48{\text{ W m}}^{-2}$. I believe your values are slightly off for the given spectral type.

I've written a Python program to calculate the flux received on a planet based on a given spectral type, using those models. It should be a quick shortcut for you in the future.

Astronomical lux

It's claimed that you can convert between a star's apparent magnitude in the V-band ($m_V$) and its illuminance ($I_V$) - the value I think you're looking for. Wikipedia's formula is $$I_V={10}^{(-14.18-m_V)/2.5}$$ which matches values from the National Park Service. The absolute magnitude of an M4V star is - according to the same stellar models - $M_V\approx 12.80$. Apparent magnitude can be calculated from absolute magnitude by $$m_V=M_V+5\log \left(\frac{r_p}{10\text{ parsecs}}\right)$$ Putting this together for our case yields $m_V\approx -16.96$, and, finally, we get $I_V\approx 12.7\text{ lux}$.

posted over 6 years ago

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