# Balloon Design for Mars, Venus, and Titan Atmospheres
## 2 Theoretical Model
### Buoyancy  
Aerobots work on the principle of buoyancy or Archimedes principle, where the buoyant force is equal to the weight of the displaced fluid. In the case of a balloon, the buoyancy is complicated by the presence of a lighter compressible lifting gas, and so the principle can be described as: the net buoyant force is equal to the weight difference between the displaced air and the lifting gas. This net buoyant force, in the case of a balloon also called gross inflation \(GI\), can be written as  
\[GI=g\cdot V\cdot(\rho_{\rm a}-\rho_{\rm g}), \tag{5}\]  
where \(g\) is the planetary acceleration of gravity, \(V\) is the volume of the balloon, and the terms \(g\rho_{\rm a}V\) and and \(g\rho_{\rm g}V\) are the weight of the displaced air, i.e., the gravitational force acting on the displaced air (equal to the total buoyant force as per Archimedes principle), and the weight of the gas inside the balloon, respectively. The balloon starts ascending if initially \(GI\) is larger than the gross weight \(M_{\rm gross}g\), where \(M_{\rm gross}\) is the gross mass, i.e., the total mass \(M_{\rm t}\) without the mass of the lifting gas (\(M_{\rm gross}=M_{\rm t}-m_{\rm g}\)). While ascending, the lifting gas is cooling adiabatically due to its expanding volume. This adiabatic cooling rate is usually stronger than the atmospheric lapse rate; thus the weight of the displaced atmospheric gas is decreasing. Consequently, both the net buoyant force and the gross inflation are decreasing. The increasing volume reaches the maximum volume \(V_{\rm design}\) at some point and the volume expansion comes to a stop. While ascending beyond this point, in the case of ZP balloons, the lifting gas starts venting, and, in the case of SP balloons, the internal gas pressure remains constant, leading to a larger than ambient pressure inside the balloon. In both cases specific buoyancy is further reduced, until, at some particular altitude, neutral buoyancy is achieved, i.e., the gross weight of the balloon system equals \(GI\) (or total weight equals the total buoyant force \(g\)\(\rho_{\rm a}V\)). This particular altitude is called float altitude and once the balloon reaches this altitude, it floats horizontally in the direction of winds. The maximum design volume may be calculated from the neutral buoyancy condition at the float altitude as:  
Figure 1: Oblate, prolate, sphere, and airship shapes of balloons. \[V_{\rm design}=\frac{M_{\rm t}}{\rho_{\rm a}}\quad\text{ or} \tag{6a}\] \[V_{\rm design}=\frac{M_{\rm gross}}{\rho_{\rm a}-\rho_{\rm g}} \tag{6b}\]  
with \(\rho_{\rm a}\) at atmospheric temperature and pressure and \(\rho_{\rm g}\) at atmospheric temperature and gas pressure (inside the balloon) at the float altitude. Equations (6a) and (6b) are equivalent as the mass of lifting gas required for neutral buoyancy (at the float altitude) is now:  
\[m_{\rm g}=V_{\rm design}\cdot\rho_{\rm g} \tag{7}\]  
However, to guarantee stable ascent to the desired float altitude, a free lift (\(F\)) is added, i.e., a lifting capability that is in addition to the required neutral buoyancy. Hence, the amount of lifting gas to be filled can be calculated from Equations (6) and (7) as:  
\[m_{\rm g}=\frac{(1+F)\cdot M_{\rm gross}\cdot\rho_{\rm g}}{(\rho_{\rm a}-\rho_{ \rm g})} \tag{8}\]  
Note, that neither \(V_{\rm design}\) nor \(m_{\rm g}\) can be calculated directly due to interdependencies. For example, the gross mass, which needs to be known to calculate \(V_{\rm design}\) (Equation (6)) depends on \(V_{\rm design}\) itself. The size of the balloon, i.e., \(V_{\rm design}\) determines the balloon envelope or film mass \(m_{\rm f}\), which is included in the gross mass. Thus Equation (6) must be used in a recursive method to calculate \(V_{\rm design}\). Then \(m_{\rm g}\) can be calculated using Equation (8). In this paper, the method of inflating the balloon is not considered, which means that any additional mass required for that, such as the mass of gas tanks, is assumed to be included in the payload mass.