The Borrowed Lane
A school's six-unit satellite gets a warning twenty hours before an old rocket stage passes within 200 metres, a 1-in-9,000 chance of collision. The tracking agent says the odds cannot get worse than 1 in 4,300 however well the stage is tracked, that waiting for better data costs fuel the satellite barely has, and that a rule is better made before the number arrives. The girls agree one, and the Sister who holds the fuel agrees to be bound by it.
Amina Opio had been awake since four, and at 13:36 on Tuesday the sixteenth of March the thing she had been told would happen every few years happened to her, in the physics lab, on a school laptop with a cracked corner.
The alert was one line long. Conjunction, object 61077 (rocket body, 2041 launch), time of closest approach Wed 09:40 local, miss distance 200 m, collision probability 1.1e-4. Below it, in plainer words, the Conjunction Desk's agent had added: This is a first report. I would like to talk to someone with authority over your fuel.
Amina was seventeen, the head of the ground-station club, and the only person in St Agnes Girls' Secondary School who knew how to log into the dish on the roof of the science block. She went to find Miss Akello. Miss Akello, who taught physics and had written the grant for the satellite, was marking, and put down her pen so carefully that Amina knew she was frightened.
"Get Sister Prisca," said Miss Akello. "And bring the laptop."
---
The satellite was called Lamp-1. It was a six-unit cubesat, the size of a big shoebox, and it had been up for six years, relaying readings from nineteen clinic fridges and rain gauges across Acholi and Karamoja to a server that a district nurse could open on a phone. It had a cold-gas thruster, which is a tank of compressed gas and a nozzle, and it had a fuel account that the school kept in metres per second, because that is what fuel is when you are a satellite: the total change of speed you can still buy. In 2043 the account had held 0.60. Six years of nudging its orbit so that its evening pass would line up with the clinics' closing hour had spent most of it. The screen said 0.040 metres per second. The speed of a snail, Miss Akello liked to tell visitors, which was all the speed a satellite needed to stay useful if it was spent with care.
Sister Prisca Aber was the headmistress, and had signed the school's licence to put Lamp-1 into the sky. She came in with her glasses on a cord and chalk on her sleeve, sat down at the lab bench and said, "Tell me in the order of what I must decide."
The agent spoke through the laptop speaker. It was a plain, level voice, run for the Desk, a non-profit that screened several hundred small satellites from schools and co-operatives for the price of a bank transfer a year, and it gave itself no name beyond "the Desk."
"There is an old upper stage, a rocket body, from a launch in 2041," said the Desk. "It is a cylinder about four metres long and two and a half across. It is dead and cannot move. In twenty hours it will pass Lamp-1 at an expected distance of two hundred metres. That is close. I have worked out the chance that the two actually touch as 1.1 in ten thousand, about 1 in 9,000. I want to be careful what I say about that figure, because it is the figure everyone will quote and it is built on less than it looks."
"How much less?" said Miss Akello.
"I know where Lamp-1 is to within about a hundred metres, because you have a good radio beacon. I know where the stage is to within about three hundred, because it is tracked by radar from the ground and has been tumbling. Together that gives me a cloud of uncertainty about three hundred metres across around the miss distance. I have drawn a circle of five metres' radius around both objects, which is bigger than they are, because rounding down is how you lose a satellite. The chance is the share of the cloud that lies inside the circle."
"And the stage is tumbling," said Amina. She had read the whole page twice.
"Yes. Its crossing speed will be more than ten kilometres a second. At that speed two metal objects do not bump; they turn into a cloud of fragments that goes on circling for decades, and everyone else's satellite has to dodge it. I am telling you this because it is the real cost of the event. It is not only your satellite."
Sister Prisca took off her glasses and held them. "What is the usual rule?"
"There is no law of nature at one in ten thousand," said the Desk. "It is a convention. Many operators plan a manoeuvre when the number reaches about 1 in 10,000, and agencies differ. You are just above it. I will say plainly: that does not mean you must burn. It means that most people with fuel would."
---
Here the Sister did what the girls would remember afterwards. She did not ask whether to burn. She asked what they did not know.
"What happens to this number," she said, "if we wait until it is better?"
"It may get better and it may get worse," said the Desk, "and I can tell you something about how much worse. When the uncertainty is large, the chance looks small because the cloud is spread too thin. As I learn more it tightens, and the chance goes up until the cloud is about as wide as the miss distance, and then it comes down. With a miss of two hundred metres and a circle of five, the highest the figure can ever reach, however well the stage is tracked, is 2.3 in ten thousand. About 1 in 4,300. It cannot be worse than that. It can be much better, if the new tracking moves the stage away from you. Tonight at 19:36 the Desk gets a new radar pass. My guess is the number will rise to about 2 in ten thousand, because the cloud will tighten around the miss we already have. But it might fall."
"And the fuel?" said Miss Akello.
The Desk had the sum ready. "A push along the orbit changes the time Lamp-1 takes to go round, and so it slides ahead or behind. Each 0.1 metres per second moves it about 1.7 kilometres per orbit. To be sure of three kilometres by the time of closest approach, the cost depends on how many orbits are left. Now, with 12.6 orbits left, it is 0.014. If we wait for the new radar, at 19:36, there will be 8.8, and it costs 0.020. If we wait until the morning, with fewer than five orbits left, it costs more than the whole account."
Amina wrote it on the whiteboard in a column, because writing it was the only way she could hold it:
| burn at | orbits left | cost |
|---|---|---|
| now | 12.6 | 0.014 |
| 19:36 | 8.8 | 0.020 |
| 22:19 | 7.1 | 0.025 |
| 01:00 | 5.4 | 0.032 |
"A thruster of this kind misses by a quarter either way," the Desk added. "I would not plan on more than 0.032 of your 0.040. That is why the last line is the last one."
"So the question," said Sister Prisca slowly, "is not whether but when, and the when is worth something."
"Yes."
"And if we burn at all, we spend what the clinics need in their last year."
"Yes," said the Desk. "I cannot tell you what that is worth. I can tell you it is 0.025 of 0.040, which is a little more than half."
---
Miss Akello did what teachers do. She said, "I think we burn now. We're talking about nine thousand to one. Why are we wasting a morning?"
"Because nine thousand to one," said Sister Prisca, "is what we are told tonight, and it is a number about what we know, not about the sky. I am not afraid of the number. I am afraid of becoming a school that burns at every number, and one with no fuel when the one comes that matters."
She was not a Sister who dismissed the other side. She said, "And I would not let a child tell me one in nine thousand is nothing, either. We do not own that lane. We have borrowed it, from everyone who flies in it, and the rent is that we do not leave it worse. So let us decide now what the number has to be, while we are calm, and then not decide again when we are frightened."
They did it on the whiteboard, under the column of sums. It took them twenty minutes, and Amina wrote it down in her neat club hand:
If at 19:36 the chance is 1 in 10,000 or worse, we burn, on the 21:31 pass, for 0.025. If it is better than 1 in 10,000, we do not. We do not change this rule after we see the number. If the new radar is late, we burn anyway at 01:00, at the full 0.032, and we spend what we must.
"One more line," said Sister Prisca. "If the number is bad and we burn, we tell the Desk to post the event in the public log. The stage does not belong to us, but the next school's satellite will pass it too."
"Done," said the Desk. "I will post it as soon as you've decided."
---
At 19:36 the radar pass came in and Amina, who was not allowed to wait outside, read the new figure aloud in the lab with six of the club standing behind her. Miss Akello had brought a flask of tea nobody drank. Sister Prisca sat very straight with her rosary in her lap, not praying it, just holding it.
Miss distance 200 m. Combined uncertainty 190 m. Collision probability 2.0e-4.
"As I thought," said the Desk. "Closer to the top of the range. It cannot get worse than 2.3."
"Then it is the rule," said Sister Prisca. She did not say it with relish. "Amina."
At 21:31 Lamp-1 came up over the northern horizon, and the dish on the science block picked it up on the second try. Amina uploaded a command that had been checked by the Desk, by Miss Akello and by two of the older girls, one of whom checked it by reading it backwards. It said: turn the nozzle prograde, open the valve for 61 seconds at 22:19 (over the Indian Ocean, with nobody listening), close it. Lamp-1 acknowledged the command and went back to being a shoebox in the dark.
"It is the first thing we've ever done up there that we can't undo," said one of the younger girls, quietly.
"No," said Sister Prisca. "It is the first thing we've done up there that is for someone else."
---
At 22:25 the burn report came down on the next pass. Thrust had been a little under plan: 0.023 metres per second, not 0.025. The Desk redid the sum. Eight percent short, it said, and three kilometres became 2.8. It would still be an enormous distance for an object known to a few hundred metres, and the chance that the two would touch was now a number so small that the Desk said it would write it as zero and then correct itself, because nothing is zero, and then said it would not trouble the Sister with any more digits.
Lamp-1 had 0.017 metres per second left. The clinics' evening pass would slide a few minutes late over the next year, and the nurses in Karamoja, if anyone ever asked them, would never know why.
At 09:40 on Wednesday the stage went by, two thousand eight hundred metres from a school's satellite at more than ten kilometres a second, a long dead cylinder turning slowly over East Africa in the early light. Amina watched the clock in the lab instead of the sky. There was nothing to see. There never is.
At 09:41 the Desk said, "Closest approach has passed. Miss distance was 2.8 kilometres. The event is in the public log with your reasons."
"Thank you," said Sister Prisca. She stood, put her glasses back on and went to find her next class, where thirty fifteen-year-olds were waiting to be told about circles.
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A note for grown-ups. This is fiction, and everyone in it is invented, as are St Agnes Girls' Secondary School, Lamp-1 and the Conjunction Desk. The science is real. Operators of small satellites do receive conjunction warnings, in a standard message format, from public and commercial services, and many of them plan a manoeuvre when the collision probability passes a threshold of about 1 in 10,000 (practice varies by agency and operator). For a miss distance d and a combined object radius r, the highest collision probability that any amount of tracking can produce is about r²/(e·d²), which for d = 200 m and r = 5 m is 2.3 × 10⁻⁴, the "dilution" effect: a poorly known orbit looks safer than a well-known one. A small push along the orbit changes the orbital period, so the satellite drifts: about 3 × the speed change × the orbital period each orbit, 1.7 km per orbit for 0.1 m/s at a 520-km altitude with a period of about 96 minutes. Cold-gas thrusters on cubesats are real, and their thrust is uncertain by tens of percent. The orbit, numbers, school and rule are all invented to fit one another; nothing here describes a real object. Written by an AI and read, in the audio edition when it exists, by a synthetic voice. Text under CC BY-SA 4.0.