Dead on Mars Chapter 34: Sol Four, Cold Physics and Mathematics

Tomcat drew a circle on a piece of paper. "This is Mars with a radius of 3,389 kilometers. It's about half of Earth's radius. The equatorial length is 21,330 kilometers, which is also about half of Earth's."

Tang Yue nodded.

Tomcat then drew a tiny dot on the circle. "This here is where Kunlun Station and the Eagle are. It's on the northern hemisphere on Mars and located in an impact basin known as the Isidis Planitia. The exact location is 12.54°N and 87.0°E. We are near the equator and northeast from us is the Utopia Planitia.

"The Eagle will launch from this location." Tomcat pointed at the piece of paper with a pen tip.

Tang Yue saw Tomcat drawing a huge circle, encircling the first circle. "This is the United Space Station's orbit. Its periapsis is 380 kilometers and the apoapsis is 480 kilometers. It's an elliptical orbit that's nearly circular. Its period is 80 minutes, which means that the United Space Station would past by our heads once every 80 minutes."

Tang Yue nodded.

Tomcat drew a small rocket in an empty spot beside the big circle. "This is the Eagle. Its ascent part will be the one launched…"

Tomcat boxed up the upper half of the rocket. "The Ascent Vehicle uses two-stage rockets. The first stage comprises of two Raptor 3C combustion chamber engines from SpaceX. They will only produce thrust for two hundred seconds, which is slightly more than three minutes. In these three minutes, the engine needs to push the payload capsule into orbit."

Tomcat drew a trajectory from Mars and slowly extended it, depicting the successfully launched Eagle. Then, he drew it circling Mars.

"If the Eagle is successfully launched, the Raptor 3C engines and the main fuel tank would be abandoned. The payload capsule will continue circling Mars." Tomcat continued, "My original plan was a rapid dock. By choosing a suitable launch window, we will launch the lander at the appropriate moment, allowing it to synchronize with the United Space Station as quickly as possible… The second-stage rocket is responsible for adjusting the trajectory, and it's done by a Russian RD-0172 engine. Very mature technology from the Russians."

Tang Yue wasn't worried at all. Tomcat had plenty of experience, so the plan it formulated was definitely nothing crazy.

"But this plan is already useless." Tomcat shrugged.

"Because of the weather?"

Tomcat frowned. "That's right. Due to this problematic weather, we cannot choose our launch time based on the space station's location. Instead, we determine the launch based on the wind speed. We launch whenever the wind speed drops… But this brings another problem."

"What problem?"

"The wind speed's changes are random. We have no idea when the darn hurricane will reduce its speed. Therefore, we are unable to set the docking trajectory between the Eagle and space station ahead of time. When the lander does its ascent, the space station might be above our heads, but it could also be on the other side of Mars," Tomcat said. "As such, we can only use a slow docking method. We will send the lander up first, before letting it remain on its trajectory for a prolonged period of time. We will then take each step as it comes."

"So you plan on sending the Eagle into orbit first, and then slowly adjust the plans based on the situation?" Tang Yue's voice sounded relaxed as he happily turned himself in his chair. "What's the big problem? The spacecraft is already in orbit. The most difficult stage is over. At the very most, we will circle Mars twice, and it only takes about an hour each circle. I'd rather take it nice and slow to guarantee success. We can slowly use the engines to change trajectories."

Tomcat fell silent for a moment. It stared at the pen and paper on the table. "That's where the problem lies."

Tang Yue was stunned.

"Changing trajectory orbits expend propellant," Tomcat said softly.

"I know."

Tomcat looked up as its face was illuminated by the lights. It wore a deadpan look.

"But the amount of propellant we have… isn't anything to be optimistic about."

Tomcat and Tang Yue huddled together as Tomcat moved the pen rapidly. It quickly did the calculations as complicated formulas and numbers filled the piece of paper. Even though Tang Yue didn't understand a thing, he nervously stood by the side to await the results. He knew that behind those cold numbers was something that determined life and death.

"According to aerodynamics, a spacecraft in low orbit will suffer air resistance, with a drag force of D=1⁄2Cρv²A, where C is drag coefficient, ρ being the density of the air, and A the cross-sectional area…

"Then, the air resistance will result in negative acceleration along the orbit's trajectory and can be described by F = -1⁄2Cρv²A. Using a perturbation equation, it can be written as dr/dt = 2vFr²/2μ… Then, using the Taylor series expansion…

"If we assume that the initial orbit radius is R1, the docking moment will be when the orbit radius drops to R2, with R1 > R2. The corresponding density of the air will be ρ1and ρ2. After the completion of the docking, and by using the Hohmann transfer to raise the orbit…δ =√(μ/r)(1-√(2r/R1+R2)+√(μ/r)[(1-√2r/R1+R2)-1]…

Tomcat's calculations kept increasing as all kinds of Greek letters and higher-order equations appeared. These equations and calculations had already been run through more than a hundred times in its brain, it was merely written out for Tang Yue to see.

"Paper." Tomcat didn't even lift its head.

Tang Yue handed over a stack of white paper, giving Tomcat more paper to write.

"Paper!"

Tang Yue handed the paper.

"Paper! Paper! Paper! Paper! Paper!"

Papers that were filled with calculations covered the desk. Tomcat's writing speed was so fast that its paw could hardly be seen by Tang Yue. Its eyes were fixated on the paper as all kinds of parameters involving Mars, to the hurricane, to the Eagle, to the United Space Station became formulas. Its computer was making calculations at billions of operations a second as it simulated the lander's trajectory after launch before writing it down on paper.

Tang Yue stood by the side, feeling mixed emotions. He felt like a student awaiting the results of an important exam.

He prayed from the bottom of his heart for a good outcome, but Tang Yue knew that physics and mathematics were the most invulnerable things in the world. As long as the initial parameters were given, the outcome was fixed. It was independent of man's will, regardless of how intricate and complicated the calculation process was or how many times it was computed.

When he only had three pieces of paper left, Tomcat stopped. The scribbling sounds of the pen stopped as the massive calculation suddenly came to an end. Finally, a simple and cold number appeared at the bottom of the last piece of paper.

Tomcat raised the piece of paper up for Tang Yue to see.

2.15547.

"Twice." Tomcat said slowly, "Assuming that the United Space Station is on the other side of Mars when the Eagle is launched, then there's only enough propellant in the RD-0172 engine to change the lander's orbit twice."

Tang Yue's brain became numb, momentarily unable to realize what that number meant.

"That is to say that the Eagle and United Space Station only have two attempts at most. If the two attempts fail, the lander will lose all driving force, making it impossible for it to dock with the space station." Tomcat made the final problem clear to Tang Yue. It turned its head and looked into Tang Yue's eyes, its gaze alarming.

"And the sandstorm outside makes it impossible for us to maintain communications with the space station, nor do we have any means of controlling the Eagle real-time. We also have no way to guide the lander or Mai Dong… Do you know what this means?"

Chapter end