# Aiming and Stability

## Aiming

Dynamic structures contain recirculating, fast moving mass. Some postulate "open" dynamic structures in vacuum, which are possible in theory, with perfect vacuum, perfect characterizations of perturbing forces, and perfect measurement. The difference between perfection and practical makes long ballistic spans impossible to do.

Imagine a glider airplane without control surfaces, but incredibly accurately characterized and mathematically modelled. We launch it out of a long hanger at airport A, and point it very accurately so it lands just inside the open hanger at airport B, a hundred kilometers away. Assume the spacings between the wingtips and the hangar walls are 1 centimeter. Just how accurate would we have to be?

If there was a perfect vacuum in between, and we launched our glider into a free ballistic, then we must set the azimuth (left right) and elevation (up down) angles on our launcher to an accuracy of 0.01m/100,000m or 1e-7 radians. 1E-7 is a wavelength of green light (500 nm) compared to the circumference of a circle 10 meters (33 feet) in diameter. If this giant protractor was made of stainless steel, we would need to protect it from gravitational sagging, vibration, and other distortions. The coefficient of thermal expansion of stainless steel is 17.3e-6 per Celsius, so we would need to keep the temperature controlled within 0.006 degrees.

In fact, we will probably make extensive use of celestial navigation - we have good maps of the sky, the stars don't move fast, and the sky makes a good protractor. The angular resolution of a telescope mirror (diameter D) is 1.22 λ / D . For an angular resolution of 1e-7 radians and a wavelength of 500 nm, we need a telescope mirror at least 6 meters in diameter to resolve 1 centimeter at 100 kilometer's distance.

But knowing where we are aiming is not enough; we must control our launch angle accurately, and the sum of all the errors in every measurement, structure, and control system adds to the inaccuracy of our targeting. If our launch path is short, with continuous angular corrections on the vertical and horizontal position and velocity of our vehicle, we must estimate the radial velocities from two distance measurements and the time of flight between them. We know the amount of radial acceleration we are applying to the vehicle, and we know Newton's force laws apply, so we know there will be no sudden changes in radial velocity - any large changes will be measurement errors, not sudden changes in velocity. Still, it may be difficult to find the actual radial velocities within the errors.

If our glider is moving slowly, we need to measure small radial velocities to determine launch angle. Fast is better. If we launch our glider at 10,000 meters per second, over a 100 meter track, we will need enormous accelerations - 5e5 m/s^{2}, or 50 thousand gees. Lets assume we have already gotten that velocity somehow, pointed very close to the right direction, and all we are doing is some midcourse correction over our track. This resembles the arrival and correction of a mass in an orbital ring.

Our glider passes over the track in 10 milliseconds; in that space we must control the position to within a fraction of a centimeter, and the tangential velocities to within a fraction of a millimeter per second. If the glider enters with a 1 centimeter spacing error, then we must accelerate it sideways to a large velocity, then decelerate it again, all within 10 milliseconds, to make that 1 centimeter change. In the middle of our track, we are moving sideways at 2 centimeters per 10 milliseconds, or 2 meters per second. We've had only 5 milliseconds to accelerate to that velocity, so the sideways gee forces are 400 meters per second squared, or 40 gees. That results in a large impulsive shock on one end the track segment, and an opposing impulsive shock at the other end. Unless the track segment is very massive, it will start vibrating and rotating. That will be bad news for a subsequent glider.

The physical track is probably too flexible to be the only reference platform. If the 100 meter track spar is half as heavy as the glider, say 500kg, and constructed as a hollow, 2 meter diameter (D) carbon fiber tube, it will have an panel area of 628 square meters and a panel weight of 796 grams per square meter. At 1.57 grams per centimeter (Hexcel IM7 carbon/epoxy prepreg), that is a thickness (t) of 0.507 millimeters. With a modulus E of 163 GPa, the bending moment is π t D^{3} E / 16 or 1.3e8 N-m^{2} .

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