scholarly journals Analytical solutions of radiative transfer equations in accretion discs with finite optical depth

2020 ◽  
Vol 496 (2) ◽  
pp. 1655-1666
Author(s):  
M Samadi ◽  
F Habibi ◽  
S Abbassi

ABSTRACT The main purpose of this paper is to obtain analytical solutions for radiative transfer equations related to the vertical structure of accretion discs with finite optical depth. In the non-grey atmosphere, we employ the optical-depth dependent Eddington factor to define the relationship between the mean intensity and radiation stress tensor. Analytical solutions are achieved for two cases: (i) radiative equilibrium, and (ii) a disc with uniform internal heating and both cases are assumed to be in local thermodynamical equilibrium (LTE), too. These solutions enable us to study probable role of scattering and disc optical depth on the emergent intensity and other radiative quantities. Our results show that for the first case, the surface value of mean intensity with constant Eddington factor is three times larger than that with variable factor. Moreover, scattering has no role in the vertical radiative structure of discs with the assumptions of the first case. On the other hand, for the second case, we encounter reductions in all radiative quantities as the photon destruction probability decreases (which is equivalent to increasing scattering). Furthermore, for both cases with total optical depth less than unity, the outward intensity towards the polar direction becomes less than that from the edges of disc which is contrary to limb-darkening. At the end, we apply our results to find the spectrum from accretion systems, based on two dynamical models. Consequently, we can see that how the total optical depth varies with frequency and causes remarkable changes on the emergent spectra.

2020 ◽  
Vol 12 (8) ◽  
pp. 1290 ◽  
Author(s):  
Xu Ma ◽  
Tiejun Wang ◽  
Lei Lu

In modeling the canopy reflectance of row-planted crops, neglecting horizontal radiative transfer may lead to an inaccurate representation of vegetation energy balance and further cause uncertainty in the simulation of canopy reflectance at larger viewing zenith angles. To reduce this systematic deviation, here we refined the four-stream radiative transfer equations by considering horizontal radiation through the lateral “walls”, considered the radiative transfer between rows, then proposed a modified four-stream (MFS) radiative transfer model using single and multiple scattering. We validated the MFS model using both computer simulations and in situ measurements, and found that the MFS model can be used to simulate crop canopy reflectance at different growth stages with an accuracy comparable to the computer simulations (RMSE < 0.002 in the red band, RMSE < 0.019 in NIR band). Moreover, the MFS model can be successfully used to simulate the reflectance of continuous (RMSE = 0.012) and row crop canopies (RMSE < 0.023), and therefore addressed the large viewing zenith angle problems in the previous row model based on four-stream radiative transfer equations. Our results demonstrate that horizontal radiation is an important factor that needs to be considered in modeling the canopy reflectance of row-planted crops. Hence, the refined four-stream radiative transfer model is applicable to the real world.


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