## 3 Passive Thermal Control Methods
### Passive Radiator  
Radiators used on the spacecraft are passive radiating elements. In space, there is no medium present to convect or conduct away the waste heat from the spacecraft components to deep space, and it has to be lost only through radiative heat transfer. Radiator panels are specially designed for this purpose, and they come in various configurations such as single active face (Body-mount radiators, BMR) and double active face (deployable radiators, DR) [28; 29]. Radiator design and its mounting location on the spacecraft body greatly influences the performances of the radiating panel. Most importantly, radiators are mounted in a location where external fluxes are minimized. Body-mount radiators are designed as an integral part of the spacecraft (S/C) or can be designed separately and mounted to the external surface of the S/C structural body [30]. Secondly, optical properties of the radiators are considered important as the radiating power is dependent on surface coating. Radiating power (\(Q_{rad}\)) of the radiator is defined by the Stefan-Boltzmann equation as in (8). \[Q_{rad}=A_{Rad}e\sigma F_{y}(T_{rad}^{4}-T_{s}^{4}) \tag{8}\]  
Figure 4: Thermal strap conductance. where, \(A_{Rad}\)--Radiator surface area, \(\epsilon\)--emissivity of the radiator surface, \(\sigma\)--[PERSON] constant (5.6703/74419 \(\times\) 10\({}^{-8}\) W/m\({}^{2}\).K\({}^{4}\)), \(F_{v}\)--view factor and it will normally be close to unity and it can be lower if the respective surface is partially occulted by other spacecraft components, \(T_{rad}\)--Temperature of the radiator surface and \(T_{s}\)--deep space temperature (assumed 3 K). \(Q_{rad}\) is radiating power of the radiator and it is strongly dependent on the surface temperature. Thermal equilibrium condition is defined as the total incoming heat (\(Q_{in}\)) equal to heat leaving (\(Q_{out}\)) the system and it is given by Equation (9). The following governing equations are used for calculating the radiator area for any given environmental conditions [11; 31]. \[Q_{in}=Q_{out} \tag{9}\]  
On comparing,  
\[q_{s}\alpha A_{s}\cos(\theta)=e\sigma T^{4}A \tag{10}\]  
Steady state radiating surface temperature is computed by,  
\[T=\left[\frac{q_{s}\alpha\cos(\theta)}{e\sigma}\right]^{\frac{1}{3}} \tag{11}\]  
At LEO, a satellite's thermal equilibrium condition is given by Equation (12),  
\[Q_{Internal}+Q_{Solar}+Q_{Albolo}+Q_{Infrared}=Q_{rad} \tag{12}\]  
Radiator sizing depends on the total external heat load which is a sum of the solar flux (\(Q_{Solar}\)), planet albedo (\(Q_{Albolo}\)), planet IR flux (\(Q_{Infrared}\)), and internal heat load (\(Q_{Internal}\)). Equation (13) shows the heat balance between the radiator and the space environment in steady state. \[(A_{s}q_{s}+A_{A}q_{A})\alpha+A_{E}q_{E}\epsilon+Q_{internal}=A\sigma eF_{v}T_{ rad}^{4} \tag{13}\]  
Equation (13) can be further simplified as follows,  
\[Q_{external}+Q_{internal}=A\sigma eF_{v}T_{rad}^{4} \tag{14}\]  
Alternatively,  
\[q_{absorbed}\cdot A+Q_{internal}=A\sigma eF_{v}T_{rad}^{4} \tag{15}\]  
where, \(Q_{external}\)--Total external flux and it is a sum of \(Q_{solar}\), \(Q_{Albolo}\), and \(Q_{Infrared}\). \(A_{s}\), \(A_{A}\) and \(A_{E}\) are the projected areas receiving, respectively, solar, albedo and planetary radiation. \(F_{v}\) is the view factor from the radiator to the space environment (assumed perfect view into space, \(F_{v}=1\)) and \(A_{E}\) is the projected area of the earth. \(q_{s}\), \(q_{a}\), and \(q_{E}\) are solar flux, albedo flux, and earth IR flux, respectively. \(\dot{q}_{absorbed}\) is the total flux or heat absorbed from solar, albedo and infrared radiation per unit area. Radiator area is calculated using Equation (16) [32],  
\[A_{rad}=\frac{Q_{internal}}{e\sigma F_{v}T_{rad}^{4}-\dot{q}_{absorbed}} \tag{16}\]  
And the surface temperature of the radiator is calculated using Equation (17).