Here's a tikz solution in 3d, with shading and more believable shaped lunes. I expect the code could be shortened with some command definitions, feel free to suggest.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{fadings}
\begin{document}
\begin{center}
\begin{tikzpicture}[scale=3]
\begin{scope}[rotate=-10]
\draw [very thin, opacity=0.5] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=180];
\draw [very thin] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=-180];
\end{scope}
\begin{scope}[rotate=42]
\draw [very thin, opacity=0.5] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=180];
\draw [very thin] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=-180];
\end{scope}
\begin{scope}[rotate=96]
\draw [very thin, opacity=0.5] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=180];
\draw [very thin] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=-180];
\end{scope}
\begin{scope}
\clip [rotate=42] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=-180] -- (-1,-1) -- (1,-1) -- (1,0);
\clip [rotate=96] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=-180] -- (-1,1) -- (1,1) -- (1,0);
\shade [ball color = red, opacity = 0.7] (0,0) circle [radius=1];
\end{scope}
\begin{scope}
\clip [rotate=42] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=-180] -- (-1,1) -- (1,1) -- (1,0);
\clip [rotate=96] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=-180] -- (-1,-1) -- (1,-1) -- (1,0);
\shade [ball color = red, opacity = 0.7] (0,0) circle [radius=1];
\end{scope}
\begin{scope}[transform canvas={rotate=180}, rotate=180]
\clip [rotate=42] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=360];
\clip [rotate=96] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=180] -- (-1,1) -- (1,1) -- (1,0);
\shade [ball color = red!50!black, opacity = 0.35] (0,0) circle [radius=1];
\end{scope}
\begin{scope}[transform canvas={rotate=180}, rotate=180]
\clip [rotate=42] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=180] -- (-1,1) -- (1,1) -- (1,0);
\clip [rotate=96] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=360];
\shade [ball color = red!50!black, opacity = 0.35] (0,0) circle [radius=1];
\end{scope}
\begin{scope}
\clip [rotate=96] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=-180] -- (-1,1) -- (1,1) -- (1,0);
\clip [rotate=42] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=-180] -- (-1,1) -- (1,1) -- (1,0);
\shade [ball color = red!60, opacity = 0.4] (0,0) circle [radius=1];
\end{scope}
\begin{scope}
\clip [rotate=96] (1,0) arc [x radius=1, y radius=0.5, start angle=0, end angle=-180] -- (-1,-1) -- (1,-1) -- (1,0);
\clip [rotate=42] (1,0) arc [x radius=1, y radius=0.25, start angle=0, end angle=-180] -- (-1,-1) -- (1,-1) -- (1,0);
\shade [ball color = red!60, opacity = 0.4] (0,0) circle [radius=1];
\end{scope}
\end{tikzpicture}
\end{center}
\end{document}