Orbital Mechanics

Copyright © Karl Dahlke, 2022

Recall our girl on the swing. She is moving back and forth in an easy arc. You come along and give her a push at the bottom of her swing, then dart out of the way. As your hands leave her back, she is still at the bottom of her arc, but she is moving faster than she was before. She has more kinetic energy than she had before. This extra energy carries her higher up in the arc. She is swinging higher, and loving it.

As she comes back down, she will return to the exact point where you pushed her, at the bottom of the arc, but she will be moving faster, thanks to your push. In fact she will pass through that same point, at the bottom of the arc, again and again, but with more speed. However, she reaches new heights, as she swings up in front, and in back.

This intuition can be helpful as we start to understand orbital mechanics. Imagine a satellite that is in a perfect circular orbit around the equator. It is 100 miles above the earth, all the way around. As it passes over Brazil, you give it a push, like the girl in the swing. Not with your hand of course, maybe a thruster fires. So you push it in some fashion, due east, in the direction it is already travelling, just as you pushed the girl, in the direction she was moving. The satellite is now moving faster than it was before, let's say a lot faster. It has more energy, and thus, it can go higher in its orbit. The orbit has changed from a circle to an ellipse. The satellite flies up and away from the earth, and when it reaches the far side, over the Indian Ocean, it is 1,000 miles above the earth. However, like the girl in the swing, it will come back down to the same point where you gave it a push. It is going faster, yes, but it is still 100 miles above Brazil. Without air to slow it down, the next orbit is just like the last; it flies up with its extra energy, high above the Indian Ocean, then back around to South America in a crack-the-whip motion, then back up again, and so on.

I am skating past an important detail here. The earth turns under the orbit of the satellite. The lowest point of the orbit, called perigee, is over Brazil, but South America moves to the east as the earth turns. After two or three orbits, perigee is now over the Pacific Ocean, off the coast of Ecuadore. A person standing on the earth might think the orbit is shifting, but the orbit is fixed in space, as the earth turns underneath it.

Let's say we want the satellite to be 1,000 miles above the earth, all the way around. We want to "circularize" the orbit, but at a higher altitude. How should we apply thrust to achieve this?

Just to keep things simple, pretend for a moment like the earth isn't turning. The highest point of the orbit, called apogee, is always over the Indian Ocean, and perigee is over Brazil. Fire the thruster at the highest point, at apogee, pushing the satellite due east. After this push, the satellite will return to this point again and again, but it is moving faster than it was before. The extra push represents extra energy, and thus the satellite does not fall back down to earth, or at least not as far. It does not descend to 100 miles above Brazil. Maybe perigee is now 600 miles above Brazil. Maybe Perigee is 1,000 miles above Brazil, thus forming a perfect circle around the earth. Maybe, if you gave it too much thrust, there is more than enough energy, and the satellite flies up, 1,350 miles above Brazil. Maybe it is an ellipse extending high above the western hemisphere. Well we're pretty good at orbital mechanics, having done it since 1957, so we gave it just enough push to circularize the orbit. Our communications satellite is 1,000 miles above the earth, all the way around.

When a space craft ferries cargo or people to the International Space Station, it performs these same maneuvers. The launch gives the craft just enough energy to reach the height of the Space Station, 254 miles above the earth, but it is an elliptical orbit. The astronauts coast up to the top of the orbit, enjoying 15 minutes of weightlessness. If they don't do anything, their craft will fall back to earth, and crash before it makes even one trip around. Thus, at the top of the arc, they need a push to circularize the orbit. Engines fire, in a maneuver that NASA calls a burn. The craft receives a push from behind, and the orbit becomes a circle. The top of the orbit doesn't change, but the extra speed pushes the bottom of the orbit, on the far side of the earth, up, so that the spacecraft is 254 miles above the earth, all the way around. Over the next two days, it creeps up to the station, and delivers its cargo, or passengers.

At this point we spot a technical error in Star Trek, The Next Generation, Dija Q. This is where Q appears naked, having been turned into a mortal human being. Setting this distraction aside for the moment, the crew is given the task of changing the orbit of the moon, which is dangerously close to its host planet. We're not sure how it got so low, but there it is. The tides are destroying coastal cities, and if the moon keeps grazing the top of the atmosphere, it might slow down enough to crash into the planet and kill everyone on the surface. The Enterprise fires up its warp engines, and pushes on the moon at perigee, i.e. the lowest point in its orbit. This fact is mentioned more than once. The writers believed this would push the moon up and away from the planet, but it won't. The extra speed raises the far side of the orbit, stretching it into a long ellipse, but the moon will return to this point again and again, with additional speed. The tides are just as high, and it is still grazing the atmosphere, as it did before. The Enterprise should push the moon at apogee, at the highest point of its orbit. This raises the entire orbit, including perigee, away from the planet, thence the moon isn't close enough to create destructive tides or earthquakes.

Now that you've read this chapter, you know more about orbital mechanics than the script writers of Star Trek.

Spoiler alert: at the end of the show, Q gets his powers back, and in an atypical gesture of gratitude and kindness, he puts the moon back up to its proper orbit, and the planet is saved.

Shoot Yourself in the Back

Disclaimer: you should never play or experiment with guns or firearms of any kind, nor use them to harm anyone, nor for any destructive purpose. This chapter is actually about orbital mechanics, not guns.

What if you could shoot a gun straight and true, so that the bullet traveled directly forward from your chest at high velocity, went all the way around the earth, and hit you in the back? That might be the only way to shoot yourself in the back.

Orbital velocity on earth is 17,500 mph, and no gun is that powerful. Military railguns have reached 5,370 mph, which is pretty impressive, but still not fast enough to achieve orbit. And then there's that pesky air, as thick as syrup at hypersonic speeds. In just a couple miles the projectile is subsonic. So that's just not going to work. If you are determined to shoot yourself in the back, then you better go to the moon.

  1. There's no air on the moon, but that's a bigger problem for you than for your gun. You could use one of the aforementioned rail guns, but those are large and bulky, and they don't have the satisfying feel of a rifle in your arms. The White Album reminds us, "happiness is a warm gun". So let's stay with traditional firearms. Fortunately for us, gun powder (and its chemical cousins) includes its own oxidizer, e.g. potassium nitrate. Air is not required to drive the chemical reaction behind the bullet. In fact, burning in air would be too slow for an effective explosion. The powder must include both fuel and oxidizer intermixed. Thus a gun, perhaps with minor modifications, should work on the moon.

  2. How fast does a bullet have to travel to achieve lunar orbit? The moon has 27.3% of the earth's diameter, and 1.23% of its mass. Following the universal law of gravitation, gravity on the moon is 0.0123 / 0.2732 ɡ, or one sixth the gravity of earth. A 186 pound man only weighs 31 pounds on the moon. So gravitational acceleration is reduced by a factor of 6, and circular acceleration, given by the formula v2/r, must balance. Start with 17,500 mph, orbital velocity here on earth, and multiply v by the square root of 27.3% to compensate for the moon's smaller radius. Then multiply by the square root of 1/6 to adjust for the lower gravity. The new velocity is 21.3% of 17,500 mph, or 3,720 mph. This is at the high end of modern rifles. The .220 Swift realizes a muzzle velocity of 2,660 mph, 71% of what we need, and that is a street weapon, easily acquired by the general public. It is used to hunt small game, such as rabbit, from a great distance, whence accuracy is vital. There's no room for a bullet to drop on the way to its target. An experienced hunter can hit a groundhog at 375 yards. With this in mind, I will assume that the military possesses, or could build, a portable rifle with the necessary power.

  3. You are standing on the moon with an adequate gun. The bullet will trace a circle around the moon, and hit you in the back at a speed of 3,720 mph, in just under two hours. If you want to avoid bodily injury, you can simply step out of the way. Of course the bullet will continue to circle the moon until it hits something or someone.

    But there's a catch. What if your aim isn't level and true? What if the gun is tilted down just a fraction of a degree? The bullet will crash into the lunar surface within a couple hundred miles. The circle that is the bullet's trajectory tips down and touches the moon. Similarly, if the barrel is tipped up just a fraction of a degree, the circle tips back and touches the moon behind you. The bullet hits the ground 200 miles behind you and never reaches your location. So the gun has to be impossibly level.

    The solution is to use a gun with just a bit more power. This pushes the circular orbit into an ellipse. You stand at the low point of the ellipse, just a couple meters off the ground, while the high point of the ellipse, on the opposite side of the moon, is 100 miles up. You only need a modest increase in muzzle velocity to produce the desired ellipse, 3,850 mph should do the trick. Now you have some breathing room. The gun doesn't have to be spot-on level, and the bullet will even clear a lunar mountain range over the horizon that you hadn't anticipated. You don't have to be at the highest elevation along the bullet's trajectory, but neither can you stand in a crater, nor even a shallow depression. You should be at least at the top of a hill, the highest point within your ken.

  4. The next problem : the moon moves. Specifically, it turns on its axis once every 27.3 days. The bullet leaves the gun and traces a circle around the moon, or an ellipse if you prefer, but in just a few seconds you are no longer standing under its path. You have moved to the east. If you are near the equator, and you aim north, then you are moving perpendicular to the circle at 4.6 meters per second. You will be kilometers away by the time the bullet returns.

    One way around this is to stand at the equator and aim due east. The circle runs around the equator, and you're still standing on the equator when the bullet returns. But there are two problems with this plan. As mentioned above, the circle has to be an ellipse, and you were standing at its lowest point. After two hours you have moved forward along the path of the ellipse, and it is higher off the ground. The bullet sails harmlessly over your head.

    The second problem is one of precision. Just as the gun cannot be perfectly level, so it cannot aim perfectly east. An arc second to the north or south will tilt the ellipse, and once you and the ellipse have diverged, the bullet will pass you by.

    As it turns out, the only location that might be feasible, is the north pole. Fortunately, the north pole has some accommodating mountains, so this might work out. In contrast, the south polar region has many craters, and the precise south pole could be in permanent darkness.

  5. If the moon were a perfect sphere of uniform density, and if you knew the precise location of the north pole, and if your aim was true and level, perhaps guided by some technology, and if the gun had enough power to put the bullet in orbit, we might declare "Mission accomplished!" The bullet will return to you in 2 hours, and hit you in the back. But the moon is not a perfect sphere, and is not uniform in its composition. As the bullet flies down from the north pole headed south, it might pass a mountain on its right, or perhaps an unusually dense region of rock beneath the moon's surface. Either way the trajectory is bent ever so slightly. The ellipse tilts, and when the bullet returns, it passes a few meters to your left. Apollo 11 had to compensate for these gravitational perturbations, as it flew, close to the ground, in search of a safe landing site. Even from 60 miles up, the effect is noticeable after a dozen orbits. In an ultralow orbit, meters high at its lowest point, the ellipse is probably bent after just one transit around the moon. It only has to move by a meter to miss you.

    Is there a longitude that runs perfectly between the mountains, wherein the ellipse would not be appreciably disturbed after one orbit? I doubt it, and if there was, it would once again require precise aim, neither an arc second to the left nor the right.

After all that work, I'm afraid it is not possible to shoot yourself in the back, even on the moon. That's ok; we all have better things to do in space.