# Balloon Design for Mars, Venus, and Titan Atmospheres
## 2 Theoretical Model
### Geometric Properties  
While the volume and type of gas play an important role in deciding the amount of mass a balloon can lift, the shape of the balloon has an important role in the aerodynamics and pressure distribution inside the balloon. In this work, we attempted to analyse the behaviour of different shapes for different planet and moon atmospheres. In general, a spherical shape is the preferred shape for any aerobot type as the ratio of volume to surface area (aerobot efficiency) is highest for a sphere shape. In the case of aerobot operation on different planets/moons, this ratio might not be the most important factor as different shapes can complement the mission with varying characteristics, for example, a mission might demand a faster or slower ascent, or an aerobot that can continuously do altitude excursions. When the sphere is deformed by directional scaling, an ellipsoid shape is formed. An ellipsoid is given by the following Cartesian coordinates:  
\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \tag{3}\]  
When an ellipsoid is rotated about its principal axis \(z\), then a spheroid is formed with \(z\) as the symmetry axis and \(a=b\), the equation can be written as:  
\[\frac{x^{2}+y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1 \tag{4}\]  
Here, \(a\) is the equatorial radius of the spheroid and \(c\) is the distance from centre to pole along the axis of symmetry. A spheroid is called an oblate, if \(c<a\) (Figure 1a), and it is called a prolate, if \(c>a\) (Figure 1b), and if \(c=a\), then it is a sphere (Figure 1c). When a prolate shape is rotated by \(90^{\circ}\) around the \(y\)-axis, it is called an airship (Figure 1d). The ratio of \(a\) to \(c\) is defined as fineness ratio \(f\) in this work. For analysing the performance of oblate shape, the fineness is varied between 2 and 9, and for prolate and airship it is varied between 0.1 and 0.8. Different parameters that are used for simulating these particular shapes are: volume (\(\pi a^{2}c\)), diameter (\(2a\), or \(2c\) for airship), surface area \(A\), top area \(A_{xy}\) (projected onto \(x-y\) plane), projected area \(A_{xz}\) (projected onto \(x-z\) plane), height (\(2c\), or \(2a\) for airship), and effective area \(A_{\rm e}\) (\(0.5(A_{xy}+A_{xz})=0.5\pi a(a+c)\) for all shapes). Figure 1 shows the different shapes, and Table 2 gives the area parameters that depend on the shape.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Shape**} & \multirow{2}{*}{**Surface Area \(A\)**} & \multirow{2}{*}{\(\alpha\)} & \multicolumn{2}{c}{**Cross-Sectional Areas**} \\ \cline{3-5}  & & & \(\mathbf{A_{xy}}\) & \(\mathbf{A_{xz}}\) \\ \hline Sphere & \(4\pi a^{2}\) & - & \(\pi a^{2}\) & \(\pi a^{2}\) \\ Prolate & \(2\pi\left[a^{2}+c^{2}\frac{\alpha}{\tan(a)}\right]\) & \(\arccos(1/f)\) & \(\pi a^{2}\) & \(\pi ac\) \\ Oblate & \(2\pi\left[a^{2}+\frac{c^{2}\operatorname{artan}(\sin a)}{\sin(a)}\right]\) & \(\arccos(f)\) & \(\pi a^{2}\) & \(\pi ac\) \\ Airship & \(2\pi\left[a^{2}+c^{2}\frac{\alpha}{\tan(a)}\right]\) & \(\arccos(f)\) & \(\pi ac\) & \(\pi a^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The area parameters of different shapes. The surface area \(A\) is a function of \(a\), \(c\), and \(\alpha\), where \(\alpha\) is a function of \(f\) that depends on the shape.