The emissivity of Boltzmann's law

[leilei zhu]

Dear DART support team,
I entered a spectral curve of a surface feature into DART to simulate its broadband radiance between 8–13.5 µm, so I selected Boltzmann. However, the emissivity shown in simulation.properties.txt differs by about 0.07 from the value I obtain when integrating the curve myself. I have checked the DART manual but still do not understand how Boltzmann computes this emissivity.
Could you give me some hints?
Thank you for your time and help.
Best regards,
Leilei Zhu

[Yingjie Wang]

Dear Leilei,

The Boltzmann law (exitance = emissivity * sigma * T^4, https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law#:~:text=For%20an%20ideal%20absorber%2Femitter,the%20black%20body's%20temperature%2C%20T%3A&text=%CF%83%20%3D%205.670374419...) gives exitance (W/m2) integrated for the whole spectrum and for all hemispherical directions.

The emissivity is an input for Boltzmann law, not an output. If you define a boradband in DART, the band-mean emissivity will be computed according to your input spectral emissivity. 

If you want to compute the broadband radiance for a given spectral interval and a given direction. You should use the Planck's law instead. DART provides Broadband tools for computing broadband radiance taking into account the spectral response function of sensor. 

Hope it clarifies the problem,

Best,

Yingjie

[leilei zhu]

Dear DART support team,
My ultimate goal is to calculate the broadband emissivity. I input a spectral curve for simulation because I noticed in your help documentation that the emissivity calculated using Boltzmann's law is displayed as shown in the figure below. I'm very curious to know whether you compute the broadband emissivity by dividing it into very small intervals and performing a weighted sum, or whether you use integration to obtain the emissivity for this range. Is it correct that I need to simulate the band range across many small intervals and use the Planck method for simulation? Could you provide some reliable advice?
Another issue I've noticed is that when DART identifies spectral curve ranges, it encounters errors in calculating emissivity values for bands with excessively high decimal places.
Best regards,
Leilei Zhu