# Spacecraft and Asteroid Thermal Image Generation for Proximity Navigation and Detection Scenarios
## 2 Thermal Modeling
### Radiation Modeling  
There are two types of thermal cameras: photon counters and microbolometers [12]. The presented research work focuses on the latter one, being more versatile and more robust with respect to the former ones. Furthermore, they have a wide space heritage. For more details, the reader can refer to [13; 14; 15]. Microbolometer pixels are sensitive to the heat flux that hits the pixel; thus, the goal is to compute the heat flux per unit area emitted by each mesh face in the frequency spectrum of the thermal camera. For microbolometers, it is the Long Wavelength Infrared (LWIR) spectrum from \(8\,\upmu\)m to \(14\,\upmu\)m; therefore, the heat flux emitted in this range by each face over a hemisphere [16], i.e., _emissive power_ [W m\({}^{-2}\)], is given by [13]:  
\[q_{out}(T)=F(T)=\pi\int_{0}^{\infty}\varepsilon(\lambda)\,B(\lambda,T)\,R( \lambda)\,d\lambda\,, \tag{1}\]  
where \(\varepsilon(\lambda)\) is the emissivity, \(B(\lambda,T)\) is the Max Planck equation, and \(R(\lambda)\) is a function limited between 0 and 1. It represents the detection efficiency of the bolometer in the LWIR range, the transmittance of the band-pass filter, and the transmittance of the germanium lens. An example of the shape of this function is reported in [13]. In order to unify the notation with ([16], Section 10.1), the emissive power can be rewritten in terms of _intensity of radiation_, \(i\) [W m\({}^{-2}\,\)sr\({}^{-1}\)]:  
\[q_{out}(T)=\pi\int_{0}^{\infty}\varepsilon(\lambda)\,B(\lambda,T)\,R(\lambda) \,d\lambda=\pi\,i\,. \tag{2}\]  
The next step is to compute the portion of heat flux emitted by an infinitesimal part of a generic mesh face and intercepted by an infinitesimal part of a generic camera pixel, hence introducing the view factor between the mesh face and the pixel [16]. An infinitesimal area of a mesh face \(dA_{f}\) emits heat flux in all directions, distributing it over a hemisphere with a total amount described by Equation (2). An infinitesimal area of the pixel of a camera \(dA_{p}\) intercepts a portion of that hemispheric heat flux, and based on the solid angle \(d\omega_{p}\), it subtends with respect to \(dA_{f}\). Indeed, the heat flux that leaves \(dA_{f}\) within \(d\omega_{p}\) stays within \(d\omega_{p}\) as it travels towards \(dA_{p}\). If \(dA_{p}\) is placed at a distance \(r=|\mathbf{r}_{fp}|\) with an angle \(\beta_{fp}\) with respect to the normal of \(dA_{f}\) and it is normal to the radial direction \(\mathbf{r}_{fp}=\mathbf{r}_{fp}/|\mathbf{r}_{fp}|\), then \(dA_{p}\) will see \(dA_{f}\) as having an area of \(\cos\beta_{fp}\,dA_{f}\). At the same time, if \(dA_{p}\) is not normal to the radial direction \(\mathbf{\hat{r}}_{fp}\), the actual area \(dA_{f}\) will see \(\cos\beta_{pf}\,dA_{p}\), where \(\beta_{pf}\) is the angle between the radial direction \(\mathbf{\hat{r}}_{fp}\) and the normal of \(dA_{p}\), hence the approximated projection (assuming \(dA_{p}\) infinitesimal) of \(dA_{p}\) onto the hemisphere of radius \(r\). A visualization of the angles is reported in Figure 1.  
Figure 1: Infinitesimal part of the mesh face area projected on the direction of the center of the infinitesimal part of the pixel area and vice versa. Using such a description, the heat flux emitted by \(dA_{f}\) and intercepted by \(dA_{p}\) can be written as:  
\[dQ_{fp}=i\,d\omega_{p}\cos\beta_{fp}\,dA_{f}\,. \tag{3}\]  
Using the definition of a solid angle, \(d\omega_{p}\) can be written as the portion of the area of the sphere over the radius of the sphere squared. The portion of the area of the sphere to be used is the projection of \(dA_{p}\) onto the sphere, and hence \(\cos\beta_{pf}\,dA_{p}\), pictured in Figure 2. Thus, Equation (3) can be written as:  
\[dQ_{fp}=i\,\frac{\cos\beta_{pf}\,dA_{p}}{|\mathbf{r}_{fp}|^{2}}\cos\beta_{fp} \,dA_{f}\,, \tag{4}\]  
where \(\mathbf{r}_{fp}\) is the position vector that goes from \(dA_{f}\) towards \(dA_{p}\), hence the position of \(dA_{p}\) with respect to \(dA_{f}\).