Inertial Acceleration of Small Bodies

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    In launching payloads into orbit around Earth the basic problem boils down to putting a large amount of kinetic energy into a small body. Conventionally rocketeers use large quantities of propellant and machinery that must drop off an ascending rocket to achieve that goal. But if a payload can withstand extremely high rates of acceleration, we have available to us another, possibly cheaper, way to go into space.

    So now let's look at one of the dumbest ideas ever conceived for launching payloads into space. How dumb? People will long remember this thing as the astronautical analogue of an early Twentieth-Century attempt to create a heavier-than-air flying machine. Yeah, we've all seen it and had a good laugh over it. It simply consisted of a stripped-down automobile whose motor pumped an oversized Chinese umbrella rapidly up and down. That's about how dumb and laughable this idea is.

    If we want to shoot a payload into space quickly and cheaply, we need to use a slingshot... a really big slingshot. To gain a rough idea of how this thing will work, imagine playing Crack the Whip with a runaway freight train careening up the side of a mountain at TGV speed. Now replace the runaway freight train with a 10,000-tonne double chain moving on a track at 360 kilometers per hour (224 miles per hour) and you thereby transform a horror story into an engineering problem. And we know how to solve those.

    Let's start by looking at the basic physics behind this idea. Consider the crack of a whip, the phenomenon in which the end of a flexible leather cord exceeds the speed of sound. When Indiana Jones cracks his whip he starts by stretching the whip out behind him, then he throws it forward and brings the handle to a halt, giving it a twist as he does so in order to put a half loop into the whip. As each part of the whip goes around the half loop it follows a cycloidal path that brings it to a halt and transfers its kinetic energy into the still-moving part of the whip. That still-moving part thus accelerates due to both the increase in its kinetic energy and its effective loss of mass. Thus, though Indy throws his whip forward at less than one hundred miles per hour, he can make the speed of the tip exceed 730 miles per hour and generate a miniature sonic boom. Now I want to analyze that whipcrack effect in greater depth and see how we might use it to launch payloads into space from Earth's surface.

    Imagine a long chain stretched out straight and moving parallel to its length along a track. That chain contains a certain amount of kinetic energy, which is distributed uniformly among its links (assuming, of course, that the links are all identical to one another). Traveling along its track, the chain comes to a hook that engages the leading link and swings it through a half loop, bringing it to rest against a retaining wall parallel to the track (the wall retains a mass of earth that gives it stability). As the link goes around the half loop its speed goes from the speed V of the chain to zero.

    That deceleration occurs without the intervention of any process that dissipates the link's kinetic energy. We know that the energy can't just vanish: subject to a conservation law, it can only be transferred from one object to another. In this case the energy gets transferred to the part of the chain still moving in a straight line. We can understand how that transfer occurs if we imagine the part of the chain going around the half loop riding on the circumference of a wheel rolling along the retaining wall with its axle moving at a speed of V/2.

    As each link goes around the wheel, its inertial reaction to having its motion changed produces a centrifugal force pulling more or less in the direction of the chain's original motion. By placing two chains back to back we effectively eliminate the lateral component of that force, leaving only the component that acts to accelerate the still-free part of the chain. If we assign a mass M to each link, then we can say that, as each link goes around the semicircle, its momentum along the axis of the system changes from MV to zero. Our imaginary wheel rotates at an angular speed ω, so the time that the link takes to shed its momentum equals π/ω and thus the average force that the link exerts comes out as f=MVω/π. The distance that the link moves as it exerts that force equals the distance that the wheel rolls along the retaining wall in that time interval (πr, in which r represents the radius of the wheel, of the half loop), so the work that the link does equals W=MVrω=MV2/2, in which I used the fact that rω=V/2. Thus, all of the link's original kinetic energy gets transferred to the still-moving part of the chain.

    We will have the chain moving on a long track, which may have gentle curves if the landscape demands them, and accelerates it to 360 kilometers per hour (1/10 kilometer per second). Then we send the chain up the side of a mountain in the direction its payload will fly when it reaches its intended orbit. At the base of the mountain large hooks grab the leading links of the chains and pull them apart in opposite directions, allowing them to come to rest against a wall that parallels the track. As each link goes around a half loop and comes to rest, the centrifugal force acting on it transfers its kinetic energy to the still-moving part of the chain. In the absence of any significant energy-dissipating process, all of the chains' kinetic energy gets transferred to the payload, attached to the end of the chains like a caboose on a freight train. As the last links swing around their half loops, the payload, in its protective shell, gets flung orbitward at ten kilometers per second. During this process the payload will have to endure accelerations as high as five thousand gees.

    Consider in more detail what we need. We need two chains moving, pressed together side by side, along a frictionless track. Each chain ponders 5000 tonnes, but we don't want it to extend over too great a length. If we choose to make each chain 500 meters long (a bit over a quarter mile) and one meter wide, then an average pressure exerted across the bottom of the chain of ten tonnes per square meter (about one atmosphere) will support the chain. If we give the chains' links flat bottoms, then we can support the chains on a track built like an air hockey table.

    We certainly want to make the track robust and as inexpensive as we can. To that end we conceive the track in two parts -- the foundation and the air-cushion plates.

    For the foundation the builders will lay down what looks like an ordinary highway. They'll begin with a low, packed-earth berm with appropriate drainage to prevent erosion. On top of that berm the builders will lay down a sheet of reinforced concrete, like that in modern freeways, though thicker in order to support the more concentrated loads that will pass over them. And on top of the concrete the builders may lay down a layer of something like a combination of rubber and asphalt to act as a cushion between the concrete and the air-cushion plates.

    To make the active part of the track the builders will produce meter-wide steel plates, made flat on one side (the top) and with deep, wide grooves on the opposite side. Where the grooves come closest to the top side the builders will drill holes, perhaps a centimeter apart. Then they weld a flat plate to the bottom of that grooved plate, turning the grooves into tubes, and then weld half pipes to the sides of the plate to create the main channels conveying compressed air into the plate. After attaching the necessary valves the builders then send the plate out to the track layers.

    So where should we put this thing? If we want our payloads to end up on an equatorial orbit, as we must if we want to use our fancy slingshot to support the construction of a space elevator, then we must build it on the Equator. We also need high mountains, so we have to build it in Ecuador, somewhere on the line that passes about forty kilometers north of Quito. We thus have to build this giant slingshot in an area containing some population. Nonetheless, we have to hope that the mountains of Ecuador have some truly remote valleys where lots of people won't get hurt when this thing goes horribly wrong, as it (with a tip o' the hat to the infamous Mr. Murphy) inevitably will. Never mind the sonic booms that will emanate from every launch.

    When the payload leaves the launcher it has to punch through the atmosphere to reach hard vacuum. That punching will take energy out of the payload, so we want to launch the payload onto the steepest possible trajectory to minimize the drag that the payload encounters. On the other hand, we want the payload to reach its orbital altitude with the maximum possible horizontal velocity. If we want to put the payload into a low orbit, then the payload's trajectory must follow a shallow path through the atmosphere. The builders can reconcile those two requirements by enclosing the payload inside a protective shell that acts as a lifting body. Such a body, soaring through the atmosphere, generates a force that shifts the initial steep climb into a more horizontal flight as the payload effectively leaves the atmosphere.

    But that maneuver won't put the payload into a proper orbit. The projectile's perigee will still lie deep inside Earth's atmosphere. In order to maximize the payload, the builders don't want to add extra features, such as rocket motors, to the projectile. So, to raise the projectile's perigee all the way into space, they might borrow an idea first proposed in the late 1970's in association with Gerard O'Neill's space colonization plan: the builders might use an orbiting catcher to snatch up the projectile.

    Originally conceived to catch payloads of material catapulted into space from the moon, the catcher inspired a somewhat different device that uses the same system. It was to consist basically of a net attached by long cables to a set of fast-acting winches: guided by radar, the winches were to move the net to intercept each payload, slow it to a halt, and move it to a collection area. Today that technology, minus the radar and the high-speed projectiles, is used to move television cameras over football fields. We need only adapt that system back to its original purpose.

    Imagine testing such a system by suspending it between two mountain peaks in an isolated area and using it to catch rockets fired at it. If we intend to put the catcher into a circular orbit 300 kilometers (187.5 miles) above Earth's surface, it must move at 7.7 kilometers per second relative to a non-rotating frame associated with Earth. If the lifting body containing our payload shifts its trajectory enough to raise its perigee to sea level, then it will reach its apogee at 300 kilometers with a speed 89.5 meters per second (about 200 miles per hour) less than the circular orbital speed at that altitude. Thus the catcher has to grab a body moving relative to it at a speed slower than the speed, 225 miles per hour, at which airplanes are required to land at Los Angeles International Airport.

    Catching the projectiles and bringing them up to full orbital speed also, by Newton's third law of motion, reduces the catcher's own momentum. Catching enough projectiles would eventually pull the catcher onto a trajectory that enters the atmosphere, so we need to find a way to make up the lost momentum in order to keep the catcher in orbit. The builders will have to equip the catcher with high-specific-impulse electromagnetic rockets and provide those rockets with propellant derived from the projectiles that the catcher captures. To that latter end the shells enclosing the payloads heaved into space will be made up of a combination of sawdust and ice chilled to cryogenic temperatures. Originally invented to be used in the construction of torpedo-proof ships in World War II, this robust, shatterproof material would provide the necessary protection for payloads launched into space and then, when melted down, provide the propellant needed to refuel the catcher's rockets.

    Yes, this super-trebuchet is an incredibly dumb idea. But sometimes dumb ideas actually work. Consider a dumb idea from history. At the beginning of the Nineteenth Century no less an engineering authority than Napoleon Bonaparte declared the utter stupidity of trying to make a ship move against wind and tides by lighting a bonfire under its decks (more or less: he was speaking French at the time). Apparently Robert Fulton was a lot dumber than Napoleon judged, because he came back to America, found investors, and built his idiotic steaming-boat anyway.

    So now I will end this essay with four words that denote another dumb idea, one that may not sound so hilarious in a few years -- the Ecuadorean space program.

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