To Catch Up, Slow Down
Here is a puzzle that stumped two NASA astronauts in 1965.
You are flying a spacecraft in a circle around Earth. A space station flies the same circle, a short way ahead of you. Your engine can push you forward or backward.
Which way should you fire your engine to catch the station?
- Push forward, to speed up
- Push backward, to slow down
- Either one works, as long as you aim at the station
Show answer
B. Push backward, to slow down
Slow down is right. On a road, braking leaves you further behind. In orbit, braking is how you catch up, and speeding up makes you fall behind. If you picked "speed up", you are in good company.
Try to dock
Try to dock with the station: tap Speed up first and watch the gap.
Stuck? A plan that docks
Tap Slow down 4 times. A ring marks the spot where you fired.
Wait one lap. When your arrow reaches the ring again, the game switches to slow motion.
Tap Speed up 4 times while your arrow is on the ring.
To save fuel, tap Slow down only once and wait four laps.
Speed up, and the station pulls away. Why would a push toward the station carry you away from the station? The crew of Gemini 4 ran into that question for real.
What Gemini 4 saw
June 3, 1965. Gemini 4 had just separated from the top stage of its own rocket. On the first lap around Earth, the commander, Jim McDivitt, tried to fly back to that empty rocket stage.
McDivitt aimed the capsule at the rocket stage and fired the thrusters. Each push seemed to change the gap in the wrong direction. After using almost half of the thruster fuel, the crew gave up the chase.
The rocket stage was venting leftover fuel, and the capsule had no radar to judge distance. But the strangest part was the physics: in orbit, a push toward a target does not carry you to the target.
Why would a push forward lift you up? Start with a simpler question: what keeps a spacecraft up at all?
Falling around Earth
A spacecraft in orbit is falling. Gravity pulls the spacecraft down the whole time, but the spacecraft also moves sideways, very fast. Earth's round surface curves away underneath as fast as the spacecraft falls, so the spacecraft never lands. That endless fall is an orbit.
Picture a tower 6,371 km tall, as high as the docking game, and throw a ball sideways from the top. Drag the throw speed to find the one speed that makes a circle, then go faster and watch the far side rise.
Too slow, and the ball hits the ground. Too fast, and the far side of the path rises into an oval. Whatever the speed, the ball comes back to the throw point. So your speed at one point sets your height on the other side of Earth.
You are circling Earth and fire your engine forward for a moment. Where does your path change the most?
- Right where you fired: you jump up at once
- Half a lap later, on the far side of Earth
- Nowhere: you go faster but stay on the same circle
Show answer
B. Half a lap later, on the far side of Earth
The far side rises. At the moment of the push, your height has not changed yet. Half a lap later, you reach the high point of your new oval.
So a forward push lifts you. But the Gemini 4 capsule also fell behind. Why would a higher path be a slower one?
Higher is slower
First, climbing costs speed. A ball thrown straight up slows down as the ball rises. A spacecraft climbing onto a higher path slows down the same way.
Second, a higher circle is a slower circle. Far from Earth, gravity is weaker, so a gentler speed is enough to bend the path into a circle. A slower speed on a longer circle adds up to a much longer lap.
| Height above the ground (km) | Speed (km/s) |
|---|---|
| 0 | 7.91 |
| 4000 | 6.2 |
| 8000 | 5.27 |
| 12000 | 4.66 |
| 16000 | 4.22 |
| 20000 | 3.89 |
| 24000 | 3.62 |
| 28000 | 3.41 |
| 32000 | 3.22 |
| 36000 | 3.07 |
One lap around Earth, at real heights (rounded):
- Space station, 400 km up: 1.5 hours
- The docking game, 6,371 km up: 4 hours
- GPS satellites, 20,200 km up: 12 hours
- Geostationary satellites, 35,800 km up: 24 hours, one lap per turn of Earth, so these satellites hang over one spot
- The Moon, 380,000 km away: 27 days
A forward push puts you on a higher, slower, longer lap, and the station gains on you every lap. So what happens if you push backward?
Why braking wins
A backward push slows you down. Too slow to hold your height, you fall inward, and you speed up as you fall, like a dropped ball.
Your path becomes a smaller oval with a shorter lap. The spot where you braked is the highest point of the smaller oval. Everywhere else on the smaller oval, you are lower and faster than the station.
You arrive back at the spot where you braked. At that moment, who is moving faster: you or the station?
- You are faster
- The station is faster
- You both move at the same speed
Show answer
B. The station is faster
The station is faster. The spot where you braked is the highest, slowest point of your smaller oval: you gained all your ground lower down. So the last push is a speed-up, to match the station.
Brake, go once around, speed up: that is the whole trick. But how hard should you brake?
How hard to brake?
The station starts 36° ahead, a tenth of a lap. Pick a brake and press Fly one lap: try to meet the station just as you get back to the start.
A small brake gains a few degrees each lap. A big brake gains more, but your oval dips toward Earth, and too big a brake hits the ground. Real crews brake gently and wait several laps.
Now the station is behind you, not ahead. Which push comes first?
- Speed up
- Slow down
- No push: wait for the station to catch up
Show answer
A. Speed up
Speed up first. The push lifts you onto a longer lap, so the station gains on you. When the station has caught up, slow down at the same spot. Waiting does nothing: on the same circle, you and the station keep the same speed forever.
Six months after Gemini 4, two Gemini capsules put this trick to the test. Did the trick work?
How Gemini 6A did it
Yes. On December 15, 1965, Gemini 7 was already in orbit. Gemini 6A launched behind Gemini 7, onto a lower, faster orbit, and gained on Gemini 7 every lap. About six hours after launch, Gemini 6A climbed up alongside.
The two capsules flew together for more than five hours, at times about 30 cm (one foot) apart. That flight was the first crewed meeting in orbit, called a rendezvous.
Buzz Aldrin had written his doctoral thesis at MIT on exactly this problem, and fellow astronauts nicknamed him "Dr. Rendezvous". Today, some crews reach the International Space Station about 3 hours after launch.
Every step so far rests on one fact: higher orbits are slower. Can a few lines of algebra prove that fact?
The math
Yes. Call your distance from Earth's centre \(r\), your speed \(v\), your mass \(m\), and Earth's mass \(M\). Moving in a circle takes a pull toward the centre of \(mv^2/r\), and gravity supplies exactly that pull:
Solve for the speed \(v\). Then divide the length of one lap, \(2\pi r\), by that speed to get the lap time \(T\):
A bigger \(r\) gives a smaller \(v\) and a bigger \(T\): higher is slower. After a brake, your path is an oval, called an ellipse. The ellipse's lap time \(T'\) follows the same formula, with \(r\) swapped for \(a\), half the long width of the oval. Each lap, you gain this angle on the station:
In the docking game, \(T\) = 3.98 h. A brake of 200 m/s gives \(T'\) = 3.59 h, a gain of about 35° per lap: nearly the whole 36° gap.
Full derivation
- Circle speed. Gravity is the centripetal force: \(\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}\). Cancel \(m\) and one \(r\): \(v^2 = \dfrac{GM}{r}\), so \(v = \sqrt{GM/r}\).
- Lap time. One lap is \(2\pi r\) long, so \(T = \dfrac{2\pi r}{v} = 2\pi\sqrt{\dfrac{r^3}{GM}}\). Square both sides: \(T^2 \propto r^3\), which is Kepler's third law.
- After the brake. For any orbit, energy per kilogram fixes the semi-major axis \(a\): \(\dfrac{v^2}{2} - \dfrac{GM}{r} = -\dfrac{GM}{2a}\). A brake lowers \(v\) at the same \(r\), so the left side drops, and \(a\) shrinks.
- The new lap. Kepler's third law holds for ellipses with \(a\) in place of \(r\): \(T' = 2\pi\sqrt{a^3/GM}\), shorter than \(T\).
- Small brakes. For a small brake \(\Delta v\), step 3 gives \(\Delta a/a \approx 2\,\Delta v/v\), and step 4 gives \(\Delta T/T \approx \tfrac32\,\Delta a/a\). Together: \(T'/T \approx 1 - 3\,\Delta v/v\). A brake of 1% of your speed shortens your lap by about 3%.
Now you can explain
- Docking: brake, go once around the smaller, faster oval, then speed up at the same spot.
- Gemini 4: a push toward the rocket stage lifted the capsule onto a higher, slower lap, so the rocket stage pulled away.
- The thrown ball: the ball falls the whole time, but Earth curves away just as fast.
- The one-lap brake: about 200 m/s closes 36° in one lap. Gentler brakes take more laps.
- Higher is slower: \(v = \sqrt{GM/r}\), so a bigger \(r\) means a smaller \(v\).
For your class
Ask Phy
- If braking drops me lower, why am I slower than the station when I get back to the spot where I braked?
- Why did the Gemini 4 rocket stage seem to drift away when the crew flew toward it?
- How do crews reach the ISS in about 3 hours today: how many burns, and when?