Chapter 5: I

“I’m just a spaceflight enthusiast,” Lin Ran said calmly.

“A spaceflight enthusiast?” Haines mused to himself.

Haines began to suspect the man was sent by China, but that made no sense—how could China have produced someone who could write out all three J3, J4, and J5 terms?

He himself didn’t even know J4 and J5.

“If you say functionality outweighs precision now, then why do you think I need to derive the Newtonian gravitational perturbation correction formula?” Haines felt the initiative slipping away; he wanted to regain control.

Lin Ran smiled: “Because you want to go to the Moon.”

If it weren’t for his graduation project choosing Apollo lunar landing, if it weren’t for the darkness he’d endured and trained through in the void, Lin Ran could never have so clearly reconstructed the fragments of his memory and presented them before this man as the key to rooting himself in 1960s America.

“Your formula decomposes Earth’s irregularities using spherical harmonics, starting from the second term.

The second term is the gravitational perturbation caused by the equatorial bulge.

The third term is the north-south asymmetry caused by Earth’s slightly pear-like shape.

The fourth term is higher-order subtle irregularities.

The fifth term is the key—the very foundation of your calculations.”

Lin Ran paused. Haines’s hand holding his Americano trembled; coffee dripped onto the table, unnoticed. His voice shook: “What is the fifth term?”

“The fifth term is the gravitational pull of the Sun and Moon. Satellites themselves need not consider these, but if you intend to land on the Moon, you must account for the influence of solar and lunar gravity.

You must calculate the fifth term.”

Haines rose, stepped beside him, placed both hands on Lin Ran’s shoulders, leaned close, and stared into his eyes:

“Randolph, who are you?”

Seeing the white engineer’s horrified expression, Lin Ran felt the thrill of a ghost from another time, tearing apart a fixed history.

Had he not known the two timelines could not interfere, he would never have dared this.

Yet even calculating the fifth term ahead of time wasn’t a major breakthrough for the lunar program—it wouldn’t significantly accelerate overall progress.

“Me? Didn’t I already tell you I’m just a spaceflight enthusiast? Isn’t it normal to derive these?”

Lin Ran brushed Haines’s hands off his shoulders, feeling the surprising strength in the young man’s arms—strength that clashed violently with his appearance—deepening Haines’s sense that this young man was anything but ordinary.

“Pen and paper,” Lin Ran said.

Haines dashed out of the private room, terrified he’d return and find the miraculous Chinese-American youth gone.

He returned breathless, clutching pen and paper—and even bought an extra metal basin.

“Here,” Haines handed the pen and paper to Lin Ran, relieved to see he was still there.

“Earth’s non-spherical gravitational perturbations aren’t some profound theory.

Newton mentioned them in his Mathematical Principles of Natural Philosophy: Earth’s rotation generates centrifugal force, causing an equatorial bulge and forming an oblate spheroid. Newton estimated the difference between equatorial and polar radii at 27 kilometers—I wouldn’t be surprised if your deviation angle is around three degrees.

Also, two years ago, Professor Desmond Kincheloe detailed in his paper ‘The Effect of the Earth’s Oblateness on the Orbit of a Near Satellite’ the long-term perturbative effects of the J2 term on near-Earth satellite orbits, providing quantitative formulas for perigee precession and orbital plane rotation.

From Kincheloe’s J2 term, deriving J3, J4, and J5 is perfectly normal, isn’t it?”

As Lin Ran spoke, his metal pen tip scribbled furiously across the stationery, filling over ten pages with dense derivations—until the room fell silent.

Haines took the derivations, thinking: “Where is this normal?”

Though Ebenezer Haines was unknown in spaceflight history, he was unquestionably a top-tier genius—a man who could hand-calculate corrections for spacecraft orbital inclination due to Earth’s oblateness.

His hand calculations had errors under 5%; his steps even became NASA’s standard template for similar manual computations.

“Here, the fourth term accounts for the crustal density difference between the Hawaiian seafloor and the Mariana Trench.”

“The J5 term corresponds to the Coriolis-tidal coupling effect.”

“I forgot to mention—if you pursue even higher precision, the J3 term can also incorporate gravitational anomalies caused by polar geological irregularities.”

Haines stared, spellbound; each explanation struck him like a revelation.

If this man were sent by another country, wouldn’t they send more? Haines thought.

No country would ever send a genius of this caliber to do menial work.

In his view, the J4 and J5 terms this man derived were far more meticulous than his own. The original lunar landing tolerance was 100 kilometers; now, with this, orbital and landing errors would shrink to within 20 kilometers.

Yet Haines wasn’t satisfied. He set the papers down and asked:

“Randolph, why does our theoretical calculation say Explorer I’s perigee precession should be 1 to 1.5 degrees per orbit, but the actual value is 3 degrees?”

Lin Ran tapped his finger on the black walnut table in Fibonacci rhythm.

“Haines, you missed a condition. Even pure theoretical calculation wouldn’t yield 1 to 1.5 degrees per orbit. The fact that it does means your calculation considered only the pure J2 term.

But if you include the fifth term and the coupling effect of atmospheric drag, the theoretical result matches reality closely.”

Lin Ran’s fingertips tapped faster and faster on the table; as his words ceased, so did his fingers—his final strike delivered thirteen taps within one second.

Haines felt only the tapping and Lin Ran’s voice—two threads weaving into his ears like fireworks exploding.

“No! Impossible!

Atmospheric drag coupling was only observed after Explorer I’s launch.

The complexity of interaction between non-spherical perturbations and atmospheric drag was only theoretically confirmed within NASA this year, establishing the mathematical foundation for future orbital control algorithm improvements.

Atmospheric drag coupling is not mentioned in any published paper.

This is NASA’s latest breakthrough in orbital mechanics! No!”

Haines snatched up the half-empty iced Americano and drained it. “Randolph, no spaceflight enthusiast could possibly know this.”