Proper illuminance formula in planetary habitability calculator

[lortordermur]

There is no proper illuminance calculation in the planetary habitability calculator, i.e. the incident light flux per area, weighted for human perception.

The calculator so far simply goes and uses the luminosity reduced by the inverse-square law as “illuminance”:

,

where E(v) = “Illuminance” (Suns), L(1) and L(2) = Star luminosity (Suns) and d = Distance to planet (AU).

Stars feature different wavelength peaks (Wien’s displacement law) and curves correlating with their temperature. To obtain the illuminance, we need to know not only the luminosity and temperature of the star(s) as well as the distance to the planet, but also an estimate of how much of the star’s light is in the visible spectrum. This could obviously be done by integrating over the curve but that would be beyond the scope of a spreadsheet. There must be a simpler way to approximate the proportion of visible light.

[jchylik]

This could obviously be done by integrating over the curve but that would be beyond the scope of a spreadsheet. There must be a simpler way to approximate the proportion of visible light.

You can try a fairly good approximation. The curve showing the energy emitted for each frequency in the spectrum is described by the Radiation Law. And the good news are that the its approximation ( https://en.wikipedia.org/wiki/Wien_approximation )
is analytically integrable.

So we can calculate illuminance for each frequency band (ie band for frequencies between f_a and f_b)
E (f_a, f_b; T) = ((2 L K T)/(c^2 d^2 )) ((h f_a + K T) exp{ - h f_a / (K T)} -(h f_b + K T) exp{ - h f_b / (K T)})

[jchylik]

Reiterating on that, I can fairly easy add a sheet for that, however you have to decide on the sampling:

• selected frequency bands (such as corresponding to ultraviolet, blue, green, ...)
• select the sampling of bands after opening the sheet and generate the list

[lortordermur]

What would be nice would be a calculator (for example "Illuminance" on the "Star" sheet) that could say how much of the star's energy output is in a specific frequency band (as a percentage). Bands would be far-infrared, near-infrared, visible light, UV-A, UV-B etc. The illuminance calculation in the "Planetary habitability" calculator could optionally be updated accordingly, to only take into account visible light.

This is how I can determine it on the fly, however I have not worked on the spreadsheet in quite a while. You could do a fork if you want and add a new calculator, and I pull it in or copy-paste in LibreOffice.

[jchylik]

how much of the star's energy output is in a specific frequency band

That is definitely possible, but there are few more free parameters of a star that determine that.
Still, as the first approximation we can use the black-body emission (i.e. spectral distribution based on temperature only)

could do a fork if you want and add a new calculator, and I pull it in or copy-paste in LibreOffice.

I can try that, with LibreOffice the version might even end up compatible.

[lortordermur]

Nice, I would also test in Google Docs and Excel Online because not all OpenDocument features are 100% compatible between the three. But those are details that can be debugged.

Yes, black-body emission should be sufficient for now.