Mathematical Analysis of Perturbed Orbits Special and General
![Mathematical Analysis of Perturbed Orbits (Special and General Perturbations) Mathematical Analysis of Perturbed Orbits (Special and General Perturbations)](https://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-1.jpg)
Mathematical Analysis of Perturbed Orbits (Special and General Perturbations)
![Perturbed Orbits Ø A perturbation is a deviation from some normal or expected motion. Perturbed Orbits Ø A perturbation is a deviation from some normal or expected motion.](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-2.jpg)
Perturbed Orbits Ø A perturbation is a deviation from some normal or expected motion. Ø In case of a satellite, the actual path will vary from theoretical two-body path due to perturbations caused by various forces and other bodies as well. Ø These perturbations have a significant impact, when studying the long term behavior of satellite orbits. Ø Major impact on interplanetary missions.
![Special Perturbations Approach Ø Need to solve the relative equation of motion with perturbing Special Perturbations Approach Ø Need to solve the relative equation of motion with perturbing](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-3.jpg)
Special Perturbations Approach Ø Need to solve the relative equation of motion with perturbing forces. Ø Special perturbations methods generate just one special trajectory for a definite satellite, given its unique initial conditions. Ø A determination of the orbit is made by means of a numerical step-by-step process. (positions and velocities at later times by numerical integration) Ø Cowell’s method and Encke’s method.
![Cowell’s Method Ø Simplest and most widely used method for the computation of perturbed Cowell’s Method Ø Simplest and most widely used method for the computation of perturbed](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-4.jpg)
Cowell’s Method Ø Simplest and most widely used method for the computation of perturbed satellite orbits. Ø The application of Cowell’s method is simply to write the equations of motion of object (of a two bodies problem), with the perturbations, and then to integrate them step-by-step numerically. Ø Thus we have a second order differential equation of a form
![Cowell’s Method Cowell’s Method](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-5.jpg)
Cowell’s Method
![Encke’s Method Ø In this method, the difference between the primary acceleration and all Encke’s Method Ø In this method, the difference between the primary acceleration and all](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-6.jpg)
Encke’s Method Ø In this method, the difference between the primary acceleration and all perturbing accelerations is integrated. Ø Encke’s method uses a reference two-body orbit whose initial position and velocity equal that of the orbit with perturbations. Ø Thus all the calculations and position, velocity would be with respect to the reference trajectory (Osculating orbit).
![Encke’s Method Encke’s Method](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-7.jpg)
Encke’s Method
![Encke’s Method For true orbit At epoch, , For reference orbit Encke’s Method For true orbit At epoch, , For reference orbit](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-8.jpg)
Encke’s Method For true orbit At epoch, , For reference orbit
![Encke Manipulations A more attractive form: So finally, Encke Manipulations A more attractive form: So finally,](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-9.jpg)
Encke Manipulations A more attractive form: So finally,
![Encke’s Method – Summary Ø At time = 0, Ø The actual orbit position Encke’s Method – Summary Ø At time = 0, Ø The actual orbit position](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-10.jpg)
Encke’s Method – Summary Ø At time = 0, Ø The actual orbit position and velocity vectors can be computed through Ø Orbit rectification For too large. A new reference orbit must be selected such that at the particular time of rectification ,
![Cowell’s vs Encke’s method Cowell’s vs Encke’s method](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-11.jpg)
Cowell’s vs Encke’s method
![Reference(s) (1) Roger R. Bate, Donald D. Mueller & Jerry E. White, Fundamentals of Reference(s) (1) Roger R. Bate, Donald D. Mueller & Jerry E. White, Fundamentals of](http://slidetodoc.com/presentation_image_h/2f911b0340db6717e1b9e4ef8ad623d3/image-12.jpg)
Reference(s) (1) Roger R. Bate, Donald D. Mueller & Jerry E. White, Fundamentals of Astrodynamics, Dover Publications Inc. , New York, 1971. (2) Vladimir A. Chobotov, Orbital Mechanics, Third Edition, American Institute of Aeronautics and Astronautics Inc. , 2002. (3) John L. Junkins, “Integration of Perturbed Motion”, Teaching presentation notes, University at Buffalo, The State University of New York, 2006.
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