When we last left Felinian catronaut Major Thomas Fluffison, he was contemplating the world of Catalonia from his crippled capsule. Unknown to him, a small orange kitten agent of Catstroph had sabotaged his stage 1 engine causing it to fail during launch. To achieve orbit, Major Tom had been forced to ignite his stage 2 engine early. His desperate plan worked, but at a terrible cost: he’d used too much fuel and was now trapped in orbit. Despite his dire situation, he had no regrets. Being the first cat to see his world from space had been worth it.
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However, he couldn’t escape the fact that he only had forty-eight hours of oxygen left. Undeterred, he dutifully transmitted all requested data, but he had no intention of just accepting his fate. Major Tom didn’t tell ground control, but when his air supply was nearly gone, he’d attempt reentry with whatever trajectory gave him the best chance for success, no matter how small. He was a catronaut after all, and if he had to go it would be at the controls of his rocket.
Every cat on Catalonia was now glued to their television, anxiously awaiting news about the brave catronaut’s fate. As good wishes from throughout Catalonia poured into Felinia, the leaders of Catstroph fumed as their plan to humiliate their enemy had spectacularly backfired.
The small orange kitten (code name Agent Kitten) who’d caused Major Tom’s predicament didn’t just slip away. No, cheeky little kitten that he was, he snuck back into the launch facility, donned a lab coat, and joined the scientists trying to save Major Tom (you’d think the scientists would’ve noticed him, but that whole absent-minded professor thing affects cats too). Agent Kitten quickly learned that Felinia’s scientists saw no chance of success.
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Alarmed that his sabotage would turn Major Tom into a martyr for Felinia, Agent Kitten knew something had to be done. Slipping under a nearby couch, he contacted his superiors. Catstroph would have to save Major Tom if they were to turn this situation to their advantage.
Catstroph’s leaders hastily called a press conference and offered the use of their country’s rocket: the one they’d hoped would beat Felinia to space. Although still unable to carry passengers, it could get much needed fuel to the stranded Felinian ship. Felinia’s leaders had no choice but to accept their enemies help or they’d be blamed for Major Tom’s demise.
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Felinia’s best rocket scientists were dispatched to Catstroph to help design a mechanism that would allow the fuel transfer to Major Tom’s ship. Even working around the clock with every available resource, Major Tom was down to five hours of oxygen when the Catstrophian rocket was ready.
The timing of the launch was critical. The rocket’s orbit had to be close enough to Major Tom’s orbit so he could reach it with his limited fuel supply. Every television on Catalonia was tuned in as Catstroph’s rocket left the ground. Fifteen minutes later it was in orbit, and everyone waited breathlessly as its final orbital position was calculated. Had they gotten close enough?
A cheer went up in ground control as the final numbers came in. The fuel ship was in an orbit thirty km higher than Major Tom’s: within the range of his fuel supply. Major Tom anxiously waited at the controls of his ship for the final go ahead to rendezvous with Catstroph’s ship.
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Igniting the capsule’s engine, Major Tom was pushed back against his chair as his ship accelerated forward and began moving into a higher orbit. A minute later, he shut down the engine. Forty-five minutes later, he fired the engine once again for one minute. Scanning his radar screen, Major Tom saw a blip about three hundred meters ahead of him.
“I have contact with the fuel ship,” he reported. “Using maneuvering thrusters to approach.”
The distinct sound of applause could be heard coming over the radio in response. Over the next half hour, Major Tom slowly approached the other ship until he was flying along beside it. Now came the hard part: docking. There’d been no time to test the system on the ground so there was no way of knowing if it would even work.
“I'm go for docking,” he finally announced sensing the tension on the other side of the radio.
He fired his thrusters and inched towards the other ship, trying to line up his fuel inlet with the fuel dispenser on the other ship. It took several tries, and after each he anxiously looked over at the fuel gauge. Almost out of fuel, he let out a whoop of relief when the two ships finally connected.
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Activating the fuel transfer system he waited, but his heart sank as nothing happened for several minutes. He was about to report failure when with a whir the pumps kicked in and his fuel level began to increase.
“Fuel transfer in progress,” he called out excitedly.
Every cat on Catalonia let out a simultaneous cheer as Major Tom’s words were transmitted worldwide. Within fifteen minutes there was enough fuel for a safe reentry. Undocking from the other ship, Major Tom maneuvered his ship to a safe distance and heaved a sigh of relief. He was ready to go home, but we’ll save that part for the next exciting installment of Rocket Cat.
So how exactly did Major Tom rendezvous with the Catstrophian ship? Why with physics of course. The first thing that had to be done was launch the rescue ship so its orbit would end up near Major Tom’s. Most of us have played darts (or some variant of it) and it’s not too difficult, just aim and throw. Now imagine that the dart board is on a train going past you at 50 mph. Not so easy anymore, is it?
In Part 1 of Rocket Cat we calculated Major Tom’s orbital velocity to be over 17000 mph. While the Catstrophian rocket heads for space, Major Tom continues his journey around the planet. The Catstrophian rocket’s launch had to be timed so it would reach space just as Major Tom’s ship arrives. If you’ve ever watched a rocket launch, you may have heard the term “launch window.” You must launch within this time range if you want to meet up with an object in orbit.
Of course just launching at the right time doesn’t guarantee you’ll end up it the exact orbit you want. That’s because of all the other variables (such as the thrust generated by the rocket engine and the aerodynamic loads on the rocket) involved. That the Catstrophian rocket ended up in an orbit 30 km higher than Major Tom’s is actually quite good.
So all Major Tom should have to do is aim at the other ship and fire his engines and he’ll catch up to it, right? No, catching up with something in orbit is a bit more complicated than that. To simplify it, let’s think of Major Tom’s capsule as a train on a circular track.
There he goes, round and round seemingly forever (must get boring). Wait! What’s that over there? Why it’s a larger circular track surrounding the one the train is on. I’ll bet it’d be a lot more fun to ride on. Oh how to get there? Of course the only way to get from the inner to the outer track is if there’s a transfer track between them.
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That’s exactly how moving from one orbit to another works. However, instead of tracks, it’s the capsule’s velocity that determines its orbit. As you may recall from Rocket Cat Part I, we found that velocity is a function of the orbital radius. Changing the capsule’s velocity will change its orbit. Thus, when Major Tom fired his engine the first time, he put himself into what’s called a Hohmann transfer orbit (named after the German scientist Walter Hohmann who developed the concept in 1925). As the name implies, it’s an elliptical (oval) orbit which takes you from one circular orbit to another.
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To get to the transfer orbit, Major Tom had to adjust his ship’s speed. But by how much? Well it’s the difference between the velocities required for the elliptical transfer orbit and the original circular orbit.
In Part 1 of Rocket Cat we calculated the orbital velocity for a circular orbit. The orbital velocity for an elliptical orbit is more complicated because it changes as you move around the planet. But what’s a complicated equation between friends. So without further ado we present to you the elliptical orbit velocity equation.
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So now that we have the velocity equations for circular and elliptical orbits we can calculate how much of a velocity change Major Tom needed from his first engine firing. Now who wants some nice complicated equations? No one? Sorry Math Cat, maybe tomorrow you'll get to solve some equations.
Time to get out the calculator. Plugging in all the numbers (What numbers? Well the numbers for all the stuff in the equation we didn't show of course.), we get a velocity change of 8.95 m/s (20 mph). That’s not very much is it? Well 30 km isn’t much of an orbit change.
In our story Major Tom had to fire his engine a second time. Without the second engine firing he’d stay in the transfer orbit and end up right back where he started. To stay in the Catstrophian’s ship orbit, he had to adjust his velocity. It’s like using a switching post to change to a different railroad line.
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And why was the second engine firing 45 minutes later? That’s because Major Tom had to travel half an orbit to reach the Catstrophian ship’s orbit. Since it takes 90 minutes for a complete orbit (there’s an equation for that too, but I’ll resist the temptation to include it), half that is 45 minutes.
So the change in velocity for this engine firing is the velocity difference between the Catstrophian ship’s orbit and the transfer orbit. And so once again we call upon that wizard of mathematical manipulation, Math Cat, to calculate the required velocity change. Sorry, we did promise to cut back on the math.
Based on Math Cat’s calculations which aren't here (and she’s never wrong), we get a velocity change of 8.94 meters/second (again 20 mph) which really isn’t all that different from the first engine firing. Again, we’re dealing with relatively small orbit changes, so Major Tom didn’t need to change his velocity very much .
Once in the same orbit as the other ship, only small thruster firings (too small to affect his orbit) were needed to move Major Tom towards the other ship. From there it was just a matter of Major Tom having nerves of steel to dock with the other ship and replenish his fuel. Now he’d best quickly prepare for reentry as his air supply is nearly gone. But for us, that will have to wait until the next spine-tingling installment of Rocket Cat.
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