# A Simulation of Contact Graph Routing for Mars-Earth Data Communication
## 2 Model of CGR
### Arrangement of the Contact Graph
For example, we have \(rs_{11}(t)=1\), for \(t\) in \([50,75]\), \([660,695]\), or \([903,915]\), and 0 otherwise. \begin{table}
\begin{tabular}{c c c c} \hline
**Rover** & **Satellite** & **Contact Windows** \\ \hline PR & MRO & 02:34-02.40, 08:13-08:23, 14:44-14:54, 20:50–21:00 \\  & Odyssey & 03:41-03:50, 09:15–25:15, 10:15-22, 21:46–21:58 \\  & TGO & 05:16–05:03, 11:10–12:14, 17:39–17:51, 23:32–23:41, \\  & MAVEN & 00:25–00:34, 05:4–06:03, 11:18–12:18, 16:45–16:55, 21:50–21:58 \\ CR & MRO & 00:46–05:08, 06:39–05:65, 12:31–23:18, 18:48–18:54 \\  & Odyssey & 01:30–01:03, 07:27–03:13, 13:23–13:36, 18:37–18:45 \\  & TGO & 03:34–03:48, 09:34–09:40, 15:19–15:30, 21:22–21:30 \\  & MAVEN & 04:54–05:00, 10:06–10:17, 15:09–15:24, 20:14–20:28, 0:12:9–01:41:00 \\ \hline \end{tabular}
\end{table} 
Table 4: A simulated contact window between a rover and a satellite. \begin{table}
\begin{tabular}{c l c c} \hline
**Satellite** & **Rover** & **Mars Sol Time (MST)** & **Duration** \\
**\& Storage** & **(Minutes)** \\ \hline \multirow{2}{*}{MRO (20 GB)} & PR & \(\sim\)2 AM, \(\sim\)8 AM, \(\sim\)2 PM, \(\sim\)8 PM & \multirow{2}{*}{8 to 12} \\  & CR & \(\sim\)12 AM, \(\sim\)6 AM, \(\sim\)12 PM, \(\sim\)6 PM & & \\ \hline \multirow{2}{*}{Odyssey (8 MB)} & PR & \(\sim\)3 AM, \(\sim\)9 AM, \(\sim\)9 PM, \(\sim\)9 PM & \multirow{2}{*}{9 to 15} \\  & CR & \(\sim\)1 AM, \(\sim\)7 AM, \(\sim\)1 PM, \(\sim\)7 PM & & \\ \hline \multirow{2}{*}{TGO (16 GB)} & PR & \(\sim\)5 AM, \(\sim\)11 AM, \(\sim\)5 PM, \(\sim\)11 PM & \multirow{2}{*}{9 to 15} \\  & CR & \(\sim\)3 AM, \(\sim\)9 AM, \(\sim\)9 PM & \multirow{2}{*}{9 to 15} \\ \hline MAVEN (16 GB) & & Every 4.5 h & & \\ \hline \end{tabular}
\end{table} 
Table 3: Contact windows for Perseerance Rover (FR) and Curiosity Rover (CR) with satellites. For satellite \(j\) and antenna-\(k\), we also define  
\[se_{jk}(t)=\left\{ \begin{array}{ll}1,&\mbox{if satellite $j$ and antenna-$k$ can connect at $t$,}\\ 0,&\mbox{if not.}\end{array} \right. \tag{8}\]  
For instance during 00:00-04:50, the DSN antenna in Spain open at \(t=1\) for time interval 00:00 to 00:01, \(t=2\) for time interval 00:01 to 00:02, until \(t=290\) for time interval 04:49 to 04:50. Due to the very long distance between Mars and Earth, there is a delay time in delivering data. Using the speed of light 299,792 km/s, the shortest travel time, called \(Tt_{short}\), to send a signal (One-Way Light Time) at closest approach (\(\sim\)54.6 million km) is calculated as follows. \[Tt_{short}=\frac{54,600,000\mbox{ km}}{299,792\mbox{ km/s}}=182\mbox{ s}\approx 3.03\mbox{ min}\]  
This travel time needs additional transfer time based on the data size. For example, if 2 Megabits data are transferred and the transfer rate is 256 kilobits per second (kbps) or 0.256 Mbps, the transfer time is  
\[\frac{2\mbox{ Megabits}}{0.256\mbox{ Mbps}}=78,125\mbox{ s}.\]  
This additional time is quite small so we do not add this to the travel time for simplicity. Using the travel time calculation, the average travel time is approximately 12 min. For simplicity, we set the delay time to be the same for all times, but the algorithm facilitates the different delay times in the variable \(dly_{jk}\) for sending a data packet from rover \(j\) to antenna \(k\). A data packet is safely delivered from Mars to Earth if \(rs_{ij}(t)=1\) and \(se_{jk}(t+dly_{jk})=1\). In this paper, we set \(dly_{jk}=12\) min for simplicity.