Homework Assignment #6: Orbital Debris In this assignment you will model the orbital debris collision hazard for spacecraft in Earth orbit. This will require you to program formulas representing the flux (impacts/ unit time/ unit area) of debris, numerically integrate these formulas forward in time, and perform an analysis where different inputs will affect the results (a parametric study). Background The flux of orbital debris on a spacecraft can be represented by a fairly simple formula (the boxed equation below) that depends on two sets of information: Information about the spacecraft such as altitude, inclination, and orientation; and, information about the environment such as particle sizes, growth rates, and the effects of the solar cycle (through changing atmospheric drag). For a little perspective realize that while a .1mm particle may cause serious surface erosion, a 1mm particle can be very damaging. A 3mm particle traveling at 10 km/s has the kinetic energy of a bowling ball thrown at 100 km/hr and a 1cm particle would have the energy of a 180kg safe thrown at the same speed. The U.S. space shuttles have already changed their orbits several times to avoid large debris. There has been pitting of tiles, and the loss of several panes of its multipaned windshield due to impact with a small fleck of paint. Quoting from Kessler, Reynolds, and Phillip, "NASA Technical Memorandum 100 471: Orbital Debris Environment for Spacecraft Designed to Operate in Low Earth Orbit": The natural meteoroid environment has historically been a design consideration for spacecraft. Meteoroids are part of the interplanetary environment and sweep through Earth Orbital space at an average speed of 20 km/s. At any one instant, a total of 200 kg of meteoroid mass is within 2000 km of the Earth's surface. Most of this mass is concentrated in 0.1mm meteoroids. Within this same 2000 km above the Earth's surface, however, are an estimated 3,000,000 kg of manmade orbiting objects. These objects are mostly in high inclination orbits and sweep past one another at an average speed of 10 km/s. Most of the mass is concentrated in approximately 3000 spent rocket stages, inactive payloads and a few active payloads. A smaller amount of mass, approximately 40,000 kg, is in the remaining 4000 objects currently being tracked by U.S. Space Command radars. Most of the objects are the result of more than 90 onorbit satellite fragmentation. Recent ground telescope measurements of orbital debris combined with an analysis of hypervelocity impact pits on the returned surfaces of the Solar Maximum Mission (SMM) satellite indicate a total mass of approximately 2000 kg for orbital debris of 1 cm or smaller and approximately 300 kg for orbital debris smaller than 1 mm. This distribution of mass and relative velocity is sufficient to cause the orbital debris environment to be more hazardous than the meteoroid environment to most spacecraft orbiting below 2000 km altitude. While lowaltitude debris will fall to the Earth due to atmospheric drag, it is quickly replenished by particles higher up and from collisions. It has been proposed that if too much debris gets into orbit, collisions could cause an increasing number of breakups, leading to an exponential growth in the number of particles. Such a catastrophic chainreaction has been referred to as the "Kessler Syndrome". Even with the envisioned growth in launch rate, the growth of orbital debris can be greatly reduced by deorbiting spent rocket stages and satellite at the end of their useful lifetimes. Better design and management can also reduce the likelihood of explosions (often related to propulsion systems) and reduce the amount of debris likely to be generated in a collision. Orbiting robots have also been proposed to scavenge for old satellites so they could be recycled before they disintegrate, perhaps to be used as small source of materials in future space development. Space Debris Flux Modeling
F = flux, in impacts per square meter of surface per year
k = 1 for randomly tumbling surface d = orbital debris diameter in cm t = time in years, t_{1} start year, t_{2} ending year h = altitude in km ()
S = 13month smoothed 10.7 cmwavelength solar flux in 10^{4} Jy; lagged by 1 year from t Use to approximate solar flux for 11 year cycle, where t is year i = inclination in degreesUse 28.5 degrees , typical value for space shuttle where p = the assumed annual growth rate of mass in orbit (p = .05)
= flux enhancement factor, inclinationdependent function that is a ratio of flux on spacecraft to the flux on the spacecraft from the current debris population's average of 60°. N = number of impacts on a spacecraft of surface area A, between time interval t_{1} and t_{2}where A = exposed area in square meters. Use A=100 square meters.
The Assignment 1. Prepare a multicurve plot that illustrates the average number of impacts on a spacecraft (yaxis) vs. altitude in km (xaxis) for particle sizes of 0.1, 0.2, and 0.3 cm. The range of heights should be 100 km to 2,000 km, and assume the 10year time interval: t_{1}=1985, t_{2}=1995. Use the default values listed with the equations, i.e., k=1; random tumbling surface; I = 28.5 degrees inclination, etc. 2. Make at least two other graphs of your choice, in each one varying the effect of altering one variable  change k, S, t_{2}, i () or p. For comparison keep d = {0.1, 0.2, 0.3 cm} or d = 0.1 cm depending on your graph. Explain your choices.
