Orbital Mechanics for Pony Princesses
The north-eastern side of Canterlot Mountain was a special place for any pony who loved the stars. Being shielded from the city by the rocky promontory and overlooking arid uninhabited land meant that light pollution was virtually non-existent. Upon flying or climbing up to a ledge on the cliff, an adventurous traveller could enjoy an amazing view across the high plateau and rocky desert in the rain shadow of the rock. It was impressive by daylight, but it was during a dark moonless night that the cloudless sky became truly awe inspiring, when you could admire the full glory of the heavens. A glittering dome, broken on one side by the black silhouette of the mountain, with the glowing haze of the Milky Way, the bright spots of the nearby planets and brighter stars, and an uncountable number of fainter specks formed by distant stars, star-clusters, nebulae and galaxies.
Princess Luna and Princess Twilight Sparkle had been lying side by side on a blanket laid on the ground for several hours, staring up at the plethora of luminous dots. They were both familiar with all the visible constellations, and viewed the patterns of Cepheus, Cygnus, Pegasus and others as old friends. The light gathering power of their huge pony eyes let then see far more distant stars than less optically-gifted species. Eventually Twilight felt the need to break the silence.
“Thank you for sharing this with me Luna,” she said softly to the fellow alicorn, and rolled over to gently nuzzle her head.
“You're welcome,” replied the Princess of the Night. “I usually come here alone. But it's also good to come with a friend who appreciates the night sky.”
They had been through a long day of royal business in Canterlot with the other princesses. At the end of which Celestia, with heavy eyes, had lowered the sun and withdrawn to bed. Cadance had departed for the Crystal Empire, mumbling something about marital duties, leaving Twilight and Luna standing together on the castle parapets. Both felt restless, but not in the mood for Canterlot night-life.
At Luna's suggestion, they had flown out of the city and around the summit. On the far side they found a spot, in the shadow of the rock, to camp for the night, and settled down to watch the darkening sky, warmed by a dry wind blowing down the lee slope of the mountain.
“I should raise the moon now,” said Luna. “Although it seems a shame to destroy the darkness and mask the stars. Let's leave it a little longer.” She paused for a moment and then looked at the younger princess with a playful grin. “Would you like to try it?”
“What!” cried Twilight. “Raise the moon! Me?” She had never admitted to either Celestia or Luna just how scared she had felt when she had briefly been responsible for both the sun and moon during Tirek's rampage.
Luna laughed. “Well, maybe the moon is a rather large rock for a beginner. But there's no reason why you shouldn't launch your own micro-satellite. Nothing to it. It's just like skimming stones. Here—” she picked up a small boulder, illuminating its outline with her magical aura, and tossed it over to Twilight, who caught it with her own telekinesis. “Throw that in the sky as hard as you can.”
Unsure where this was leading, Twilight lifted the rock, feeling the shape of the space around it with her mind, and warping it to support the stone. She then formed this into a wave, and let it ripple forwards, carrying the rock, so that it flew far across the mountain landscape before crashing to the ground a few miles away amid other geological debris. Luna laughed again.
“You need to push it a bit higher and faster than that to reach orbit,” she said. “Give it a strong sustained upward thrust for a few minutes, then release.”
She tossed another small chunk of limestone to Twilight, who caught it, and focussed her magic not just to balance gravity, as done in everyday levitation, but to accelerate the rock upwards, several times as fast as it would fall down. She felt it moving faster and faster, high into the night sky. After two or three minutes she released it and sat back down on the ground, looking up. No longer illuminated by her magic, it was hard to see the rock. She stared upwards for many minutes trying to make it out. Was it still moving? Would it fall back down? What had happened to it? Luna waited patiently beside her without comment. Suddenly there was a bright streak of light above their heads, which came and disappeared in less than a second.
“Congratulations Twilight. You just made your first shooting star.”
Twilight's face broke into a smile. “I did that? How?”
“You threw the rock up in the air. It reached a maximum height, probably about a thousand kilometres, then fell back to earth accelerating under gravity. It entered the atmosphere moving at several kilometres per second, and burnt up just before it hit the ground.”
“Wow!”
“That's what happens when you let it move through the air too fast. Now, to get a stable orbit, you need to get it into the thermosphere, moving parallel to the surface. Launch it at a shallow angle.”
On her third attempt, Twilight pushed her projectile forwards and upwards at a slight angle. She felt the rock accelerate, impeded by air resistance and pulled downwards by gravity. But once it was above most of the atmosphere she could feel it fly free. She bent its path around until parallel to the surface, and only stopped pushing once it was at the horizon. She then let out a breath and stared forwards tracking the rapidly dimming speck of light.
“I think you overshot that one a bit,” said Luna. “Too fast. It look like it's on a hyperbola and will escape forever. Or it may establish a highly eccentric orbit and come back eventually. Unless it hits the moon.”
“Sorry,” said Twilight.
“It's no problem—the moon is hit by many meteorites every day. The trick is to get the speed just right. Too slow, and it falls back to the ground. Too fast, and it gets into a hyperbolic or highly elliptical orbit.
“I'm not sure I can do this,” said Twilight, feeling discouraged.
“Let's try it together.”
Luna picked up two rocks—larger than the previous one—and tossed one of them to Twilight.
“I'll give them a polish, to make them nice and shiny.” She levitated a cloud of sand around the rocks, blasting them until the surface was as smooth and reflective as a mirror. “Celestia's probably parked the sun not far below the horizon, so they will be well-lit once up high. Now let's go!”
Together the two princesses launched their rocks into the air. Twilight mimicked Luna's motions to set the angle and speed such they moved in unison. They both accelerated upwards, and at an angle, bending around into a horizontal path. Then, when Luna judged the speed was right, they released their rocks and watched them move forwards and down beneath the horizon. Two bright stars—the light from Celestia's sun reflected by the mirrored surface.
“I think we did it,” said Luna. “Mission accomplished.”
“Yes!” said Twilight grinning and hopping up in the air on all four hooves. “What will happen to them now?”
“They will orbit around the earth, and come back overhead in one or two hours.”
“How do you know?”
“We set them moving at a fixed speed. As they move, their path will be bent by gravity, so they follow a circle all around the planet. You're good at algebra—I bet you can work out the orbital period.”
They lay back down on the blanket. Twilight took out a notebook and, by the light of her horn, scribbled calculations with a pencil. Meanwhile Luna passed the time idly braiding her friend's tail—separating out the indigo, pink and purple hairs, plaiting them together, and weaving in a little of her own magic sparkle.
Once Twilight had completed the calculations. She then took a pocket watch out of her saddlebag and set it on the ground. She turned to Luna with the glowing face of a school filly who had completed an assignment.
“They should reappear an hour before dawn.”
They settled down to wait, lying on their backs, staring back up at the sky.
“Tell me how you raise the moon,” asked Twilight.
“In much the same way—just on a larger scale. The moon is just a huge rock orbiting the earth with a period of just over twenty-seven days. We see it rise and fall as our planet spins on its axis.”
“So why do you need to move it at all? If the orbit is stable, can't you leave it alone?”
“If only... Unfortunately those Canterlot bureaucrats have other ideas.” Luna rolled her eyes.
“What?” Twilight stared at her with a curious look.
“For some silly reason—I never understood why—it's just what the officials tell me—the ponies of Equestria think the moon should rise every evening, just after sunset. By its natural cycle, moonrise would be an hour or so later each day, sometimes when the sun is still in the sky. But instead I have to sync it to the solar rhythm, or else I get complaints. And to make things even more complicated they don't understand the proper order of the lunar phases. Usually they want a full moon, to keep the streets well-lit. But then on some special occasions they want a crescent, because they think it looks artistic. At the last Summer Sun Celebration, they wanted me to lower a full moon below the horizon, just as Celestia would raise the sun at almost the same azimuth. That took quite a bit of behind-the-scenes magic by my sister and I, and some large mirrors, but we did it.”
“I didn't know it was so complicated.” Twilight thought back to the celebration. At the time she had been so focussed on her own role, that she hadn't noticed the apparently unphysical stage lighting.
“It's not easy being a princess.”
Twilight nodded.
“There's a particularly annoying retired civil servant in Trottingham,” continued Luna. “Who has nothing better to do with his time than write letters to me every time moonrise is late. I bet he's drafting one at this moment. But tonight I don't care.”
They were both wide awake, and knew there was no chance of getting any sleep on this night. They were content to spend the time watching the stars. Eventually the awaited time arrived.
“Any minute now...”
Two bright stars rose above the rock behind them and moved quickly, in unison, tracing an arc across the sky. After a minute or so they had crossed half the sky and fell into the shadow of the planet, no longer illuminated, and disappeared from sight.
Twilight continued staring at the sky with a big grin. “What will happen to them now?”
“They will keep on orbiting. As they are at a low altitude, they will likely drift into the atmosphere and burn up in a few weeks or so.”
Twilight threw her hooves around Luna's neck in an impromptu hug, which her teacher returned with a happy smile.
Meanwhile, on the other side of the mountain, Princess Celestia, having got out of bed, washed her face, and brushed her mane, walked out onto her balcony, ready to raise the sun and start the new day. She looked out at the dark pre-dawn sky, saw the two new stars moving together, and smiled softly to herself, having guessed the pair of ponies who were responsible.