I'm working with (Nublado) Cloudy, which is essentially a spectral modelling tool for stellar matter. Using the software, I model my own stellar object and obtain its continuum.
According to Hazy, which is Cloudy's official document, the continuum is given in units of "Intensity per octave". Specifically, $4 \pi \nu J_\nu $ [erg $cm^{-2} \ s^{-1}$] units. However, I'm having difficulty making sense of this form of units. What is meant by "Intensity per octave", and why is there an additional factor of $4\pi$ and $\nu$? What exactly does $J_\nu$ refer to here, and how is it different to all the letters astrophysicists use when talking about spectra, like $L_\nu$, $F_\nu$ and $I_\nu$?
What would be the correct approach to integrate this spectra in order to get the luminosity of my object?
Finally, my main goal is to apply the Gaia passband filters to my spectra and integrate in order to get the flux and thus find $(G, G_{bp} - G_{rp})$ coordinates on the HR diagram. At its current form of supposedly $4 \pi \nu J_\nu $ as a function of the wavelength, can this be done directly or do I have to change my continuum in some way.
I have very little to no astrophysics background, so I appreciate your understanding. Thanks.