Survey of electron-cesium collision probabilities - Momentum transfer collisions
Electron-cesium atom collision probability for momentum transfer
Dayton, J. A., Jr.↗
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Survey of electron, cesium collision probabilities - Momentum transfer collisions
Thermionic converter electron-cesium atom momentum transfer collision probability, considering scattering cross sections
Dayton, J. A., Jr.↗
NASA Lewis Launch Collision Probability Model Developed and Analyzed
There are nearly 10,000 tracked objects orbiting the earth. These objects encompass manned objects, active and decommissioned satellites, spent rocket bodies, and debris. They range from a few centimeters across to the size of the MIR space station. Anytime a new satellite is launched, the launch vehicle with its payload attached passes through an area of space in which these objects orbit. Although the population density of these objects is low, there always is a small but finite probability of collision between the launch vehicle and one or more of these space objects. Even though the probability of collision is very low, for some payloads even this small risk is unacceptable. To mitigate the small risk of collision associated with launching at an arbitrary time within the daily launch window, NASA performs a prelaunch mission assurance Collision Avoidance Analysis (or COLA). For the COLA of the Cassini spacecraft, the NASA Lewis Research Center conducted an in-house development and analysis of a model for launch collision probability. The model allows a minimum clearance criteria to be used with the COLA analysis to ensure an acceptably low probability of collision. If, for any given liftoff time, the nominal launch vehicle trajectory would pass a space object with less than the minimum required clearance, launch would not be attempted at that time. The model assumes that the nominal positions of the orbiting objects and of the launch vehicle can be predicted as a function of time, and therefore, that any tracked object that comes within close proximity of the launch vehicle can be identified. For any such pair, these nominal positions can be used to calculate a nominal miss distance. The actual miss distances may differ substantially from the nominal miss distance, due, in part, to the statistical uncertainty of the knowledge of the objects positions. The model further assumes that these position uncertainties can be described with position covariance matrices. With these, and some additional simplifying assumptions, a closed-form solution for the probability of collision is obtained. This solution provides clear insights into how each of the independent parameters affects the probability of collision. It shows that for a given maximum probability of collision and prior knowledge of the objects position uncertainties and sizes, only knowledge of the nominal closest approach distance is required to make the launch/no launch decision. The model was completed and used for the mission assurance COLA analysis for the Cassini spacecraft, which was launched on a Titan IV/Centaur rocket on October 15, 1997. Although the model was specifically developed for the Cassini mission, it is clearly applicable for other launches as well. The effect of COLA closures on the launch window is shown. The bar represents the entire 140-min launch window on October 15, 1997; the blackened areas represent the loss of launch opportunities due to the potential for a collision with an orbiting object.
Bollenbacher, Gary↗
Conservative Analytical Collision Probability for Design of Orbital Formations
The literature offers a number of approximations for analytically and/or efficiently computing the probability of collision between two space objects. However, only one of these techniques is a completely analytical approximation that is suitable for use in the preliminary design phase, when it is more important to quickly analyze a large segment of the trade space than it is to precisely compute collision probabilities. Unfortunately, among the types of formations that one might consider, some combine a range of conditions for which this analytical method is less suitable. This work proposes a simple, conservative approximation that produces reasonable upper bounds on the collision probability in such conditions. Although its estimates are much too conservative under other conditions, such conditions are typically well suited for use of the existing method.
Carpenter, J. Russell↗
Conservative Analytical Collision Probabilities for Orbital Formation Flying
The literature offers a number of approximations for analytically and/or efficiently computing the probability of collision between two space objects. However, only one of these techniques is a completely analytical approximation that is suitable for use in the preliminary design phase, when it is more important to quickly analyze a large segment of the trade space than it is to precisely compute collision probabilities. Unfortunately, among the types of formations that one might consider, some combine a range of conditions for which this analytical method is less suitable. This work proposes a simple, conservative approximation that produces reasonable upper bounds on the collision probability in such conditions. Although its estimates are much too conservative under other conditions, such conditions are typically well suited for use of the existing method.
Carpenter, J. Russell↗
Asteroidal collision probabilities
Several past calculations of collision probabilities between pairs of bodies on independent orbits have yielded inconsistent results. We review the methodologies and identify their various problems. Greenberg's (1982) collision probability formalism (now with a corrected symmetry assumption) is equivalent to Wetherill's (1967) approach, except that it includes a way to avoid singularities near apsides. That method shows that the procedure by Namiki and Binzel (1991) was accurate for those cases where singularities did not arise.
Bottke, William F., Jr.↗
Collision probability at low altitudes resulting from elliptical orbits
The probability of collision between a spacecraft and another object is calculated for various altitude and orbit conditions, and factors affecting the probability are discussed. It is shown that a collision can only occur when the spacecraft is located at an altitude which is between the perigee and apogee altitudes of the object and that the probability per unit time is largest when the orbit of the object is nearly circular. However, at low altitudes, the atmospheric drag causes changes with time of the perigee and the apogee, such that circular orbits have a much shorter lifetime than many of the elliptical orbits. Thus, when the collision probability is integrated over the lifetime of the orbiting object, some elliptical orbits are found to have much higher total collision probability than circular orbits. Rocket bodies used to boost payloads from low earth orbit to geosynchronous orbit are an example of objects in these elliptical orbits.
Kessler, Donald J.↗
Time Dependence of Collision Probabilities During Satellite Conjunctions
The NASA Conjunction Assessment Risk Analysis (CARA) team has recently implemented updated software to calculate the probability of collision (P (sub c)) for Earth-orbiting satellites. The algorithm can employ complex dynamical models for orbital motion, and account for the effects of non-linear trajectories as well as both position and velocity uncertainties. This “3D P (sub c)” method entails computing a 3-dimensional numerical integral for each estimated probability. Our analysis indicates that the 3D method provides several new insights over the traditional “2D P (sub c)” method, even when approximating the orbital motion using the relatively simple Keplerian two-body dynamical model. First, the formulation provides the means to estimate variations in the time derivative of the collision probability, or the probability rate, R (sub c). For close-proximity satellites, such as those orbiting in formations or clusters, R (sub c) variations can show multiple peaks that repeat or blend with one another, providing insight into the ongoing temporal distribution of risk. For single, isolated conjunctions, R (sub c) analysis provides the means to identify and bound the times of peak collision risk. Additionally, analysis of multiple actual archived conjunctions demonstrates that the commonly used “2D P (sub c)” approximation can occasionally provide inaccurate estimates. These include cases in which the 2D method yields negligibly small probabilities (e.g., P (sub c)) is greater than 10 (sup -10)), but the 3D estimates are sufficiently large to prompt increased monitoring or collision mitigation (e.g., P (sub c) is greater than or equal to 10 (sup -5)). Finally, the archive analysis indicates that a relatively efficient calculation can be used to identify which conjunctions will have negligibly small probabilities. This small-P (sub c) screening test can significantly speed the overall risk analysis computation for large numbers of conjunctions.
Hall, Doyle T.↗
A Simple, Velocity Dependent, Collision Probability Algorithm for Small Combined Hard Body Radii Close Approach Events
Over the last few years progress has been made in the velocity dependent collision probability problem. Recent progress in this problem has been approached via integration of the probability flux into the surface of the combined hard body object. The algorithm presented uses the surface flux approach and is designed to compute the collision probability rate between two objects given their time dependent states and state error covariance matrices. With regards to the computation of the collision probability (Pc) rate, two major differences exist between the development of this algorithm and the usual Pc algorithm. First, the shape of the at-risk volume is assumed to be a cube rather than a sphere. The size of the cube is chosen so that it circumscribes the usual hard body sphere chosen for the spherical Pc problem. This will result in half the length of a side of the cube being equal to the hard body radius (HBR) of the sphere. Second, it is assumed that the HBR of the cube is much smaller than the smallest combined position uncertainty (σ(sub min) > 5*HBR) at each time point of evaluation. This is necessary as the actual collision probability calculation is based on a first order, small variable expansion in the position components of the Gaussian probability density function. This leads to the collision probability rate being to second order in the combined HBR. The collision probability rate is in a concise, closed form containing exponential and error functions.
Frisbee, Joe↗
An Upper Bound on Orbital Debris Collision Probability When Only One Object has Position Uncertainty Information
Upper bounds on high speed satellite collision probability, P (sub c), have been investigated. Previous methods assume an individual position error covariance matrix is available for each object. The two matrices being combined into a single, relative position error covariance matrix. Components of the combined error covariance are then varied to obtain a maximum P (sub c). If error covariance information for only one of the two objects was available, either some default shape has been used or nothing could be done. An alternative is presented that uses the known covariance information along with a critical value of the missing covariance to obtain an approximate but useful P (sub c) upper bound. There are various avenues along which an upper bound on the high speed satellite collision probability has been pursued. Typically, for the collision plane representation of the high speed collision probability problem, the predicted miss position in the collision plane is assumed fixed. Then the shape (aspect ratio of ellipse), the size (scaling of standard deviations) or the orientation (rotation of ellipse principal axes) of the combined position error ellipse is varied to obtain a maximum P (sub c). Regardless as to the exact details of the approach, previously presented methods all assume that an individual position error covariance matrix is available for each object and the two are combined into a single, relative position error covariance matrix. This combined position error covariance matrix is then modified according to the chosen scheme to arrive at a maximum P (sub c). But what if error covariance information for one of the two objects is not available? When error covariance information for one of the objects is not available the analyst has commonly defaulted to the situation in which only the relative miss position and velocity are known without any corresponding state error covariance information. The various usual methods of finding a maximum P (sub c) do no good because the analyst defaults to no knowledge of the combined, relative position error covariance matrix. It is reasonable to think, given an assumption of no covariance information, an analyst might still attempt to determine the error covariance matrix that results in an upper bound on the P (sub c). Without some guidance on limits to the shape, size and orientation of the unknown covariance matrix, the limiting case is a degenerate ellipse lying along the relative miss vector in the collision plane. Unless the miss position is exceptionally large or the at-risk object is exceptionally small, this method results in a maximum P (sub c) too large to be of practical use. For example, assuming that the miss distance is equal to the current ISS alert volume along-track (+ or -) distance of 25 kilometers and that the at-risk area has a 70 meter radius. The maximum (degenerate ellipse) P (sub c) is about 0.00136. At 40 kilometers, the maximum P (sub c) would be 0.00085 which is still almost an order of magnitude larger than the ISS maneuver threshold of 0.0001. In fact, a miss distance of almost 340 kilometers is necessary to reduce the maximum P (sub c) associated with this degenerate ellipse to the ISS maneuver threshold value. Such a result is frequently of no practical value to the analyst. Some improvement may be made with respect to this problem by realizing that while the position error covariance matrix of one of the objects (usually the debris object) may not be known the position error covariance matrix of the other object (usually the asset) is almost always available. Making use of the position error covariance information for the one object provides an improvement in finding a maximum P (sub c) which, in some cases, may offer real utility. The equations to be used are presented and their use discussed.
Frisbee, Joseph H., Jr.↗
High Fidelity Collision Probabilities Estimated Using Brute Force Monte Carlo Simulations
The NASA Conjunction Assessment Risk Analysis team has implemented new software to estimate the probability of collision (P (sub c)) for Earth-orbiting satellites. The algorithm employs a brute force Monte Carlo (BFMC) method which differs from most other methods because it uses orbital states and covariances propagated from their orbit determination epoch times using the full set of Astrodynamics Support Workstation higher order theory models, including the High Accuracy Satellite Drag Model. This paper de-scribes the BFMC algorithm, presents comparisons of BFMC P (sub c) estimates to those calculated using other methods, and discusses the implications for conjunction risk assessment.
Probabilities↗
A new technique for predicting geosynchronous satellite collision probability
A new technique has been developed to predict the probability of an expired geosynchronous satellite colliding with an active satellite. This new technique employs deterministic methods for modeling the motion of satellites and applies statistical techniques to estimate the collision probability. The collision probability is used to estimate the expected time between collisions based on realistic distributions of expired and active satellites. The primary advantage of this new technique is that realistic distributions can be used in the prediction process instead of uniform distributions as has been used in previous techniques. The expected time between collisions based on a current NORAD database is estimated to be in the hundreds of years.
Mccormick, B.↗
Derivation of the collision probability between orbiting objects The lifetimes of Jupiter's outer moons
A general form is derived for Opik's equations relating to the probability of collision between two orbiting objects to their orbital elements, and used to determine the collisional lifetime of the eight outer moons of Jupiter. The derivation is based on a concept of spatial density, or average number of objects found in a unit volume, and results in a set of equations that are easily applied to a variety of orbital collision problems. When applied to the outer satellites, which are all in irregular orbits, the equations predict a relatively long collisional lifetime for the four retrograde moons (about 270 billon years on the average) and a shorter time for the four posigrade moons (0.9 billion years). This short time is suggestive of a past collision history, and may account for the orbiting dust detected by Pioneers 10 and 11.
Kessler, D. J.↗
New method for estimating low-earth-orbit collision probabilities
An unconventional but general method is described for estimating the probability of collision between an earth-orbiting spacecraft and orbital debris. This method uses a Monte Caralo simulation of the orbital motion of the target spacecraft and each discrete debris object to generate an empirical set of distances, each distance representing the separation between the spacecraft and the nearest debris object at random times. Using concepts from the asymptotic theory of extreme order statistics, an analytical density function is fitted to this set of minimum distances. From this function, it is possible to generate realistic collision estimates for the spacecraft.
Vedder, John D.↗
Remediating Non-Positive Definite State Covariances for Collision Probability Estimation
The NASA Conjunction Assessment Risk Analysis team estimates the probability of collision (Pc) for a set of Earth-orbiting satellites. The Pc estimation software processes satellite position+velocity states and their associated covariance matri-ces. On occasion, the software encounters non-positive definite (NPD) state co-variances, which can adversely affect or prevent the Pc estimation process. Inter-polation inaccuracies appear to account for the majority of such covariances, alt-hough other mechanisms contribute also. This paper investigates the origin of NPD state covariance matrices, three different methods for remediating these co-variances when and if necessary, and the associated effects on the Pc estimation process.
Hall, Doyle T.↗
Non-Parametric Collision Probability for Low-Velocity Encounters
An implicit, but not necessarily obvious, assumption in all of the current techniques for assessing satellite collision probability is that the relative position uncertainty is perfectly correlated in time. If there is any mis-modeling of the dynamics in the propagation of the relative position error covariance matrix, time-wise de-correlation of the uncertainty will increase the probability of collision over a given time interval. The paper gives some examples that illustrate this point. This paper argues that, for the present, Monte Carlo analysis is the best available tool for handling low-velocity encounters, and suggests some techniques for addressing the issues just described. One proposal is for the use of a non-parametric technique that is widely used in actuarial and medical studies. The other suggestion is that accurate process noise models be used in the Monte Carlo trials to which the non-parametric estimate is applied. A further contribution of this paper is a description of how the time-wise decorrelation of uncertainty increases the probability of collision.
Carpenter, J. Russell↗
A STATISTICAL-MECHANICAL ESTIMATE OF ORBITAL COLLISION PROBABILITIES
Statistical-mechanical estimate of orbital collision probabilities
ORBIT↗
Implementation of a Collision Probability Prediction Technique for Constellation Maneuver Planning
On March 22, 2006, the Space Technology 5 (ST5) constellation spacecraft were successfully delivered to orbit by a Pegasus XI, launch vehicle. An unexpected relative motion experienced by the constellation after orbit insertion brought about a problem. Soon after launch the observed relative position of the inert rocket body was between the leading and the middle spacecraft within the constellation. The successful planning and execution of an orbit maneuver that would create a fly-by of the rocket body was required to establish the.formation. This maneuver would create a close approach that needed to conform to predefined collision probability requirements. On April 21, 2006, the ST5 "155" spacecraft performed a large orbit maneuver and successfully passed the inert Pegasus 3rd Stage Rocket Body on April 30, 2006 15:20 UTC at a distance of 2.55 km with a Probability of Collision of less than 1.0E-06. This paper will outline the technique that was implemented to establish the safe planning and execution of the fly-by maneuver. The method makes use of Gaussian distribution models of state covariance to determine underlying probabilities of collision that arise under low velocity encounters. Specific numerical examples used for this analysis are discussed in detail. The mechanics of this technique are explained to foster deeper understanding of the concepts presented and to improve existing processes for use in future constellation maneuver planning.
Concha, Marco a.↗