Orbital parameters of asteroids using analytical propagation
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The document discusses generating ephemerides for asteroids using analytical propagation methods. It begins by introducing the objectives, various coordinate systems, and complexities in determining orbital motion. It then covers the equations of motion for two-body motion and conic orbits. The document describes how to generate ephemerides using analytical techniques by calculating orbital element transformations between time steps. Comparisons to JPL ephemerides for Pallas show errors on the order of 10^-1 to 10^-2 in position and velocity components. While computationally simpler, the two-body model results in significant absolute errors, requiring more detailed force models for precision.
Orbital parameters of asteroids using analytical propagation
- 1. Orbital parameters of Asteroids using analytical Propagation Team Members: Chetana D. Lakshmi Narsimhan Lokeswara Rao.N Ramiz Ahmad Ranupriya Didwania Guided by Dr. R.V Ramanan
- 2. Plan of the talk: 1. Objective 2. Introduction to various coordinate system 3. Problems/complexities associated with the parameter calculations. 4. Equations of motion (for two-body motion). 5. Ephemeris generation (related formulas and codes). 6. Conclusions.
- 3. Objective To obtain the orbital parameters of the celestial objects ( asteroids ) at any time with respect to its reference parameters. To study the time evolution of asteroids
- 4. Introduction to the coordinate system Various coordinate systems: 1. Inertial coordinate system (commonly used) Origin - Centre of Earth Principal axis (x-axis) - towards the vernal equinox ( intersection of the Earth Equator and ecliptic plane) from the origin Fundamental Plane - Earth equator 2. The right ascension –declination coordinate system Origin - Centre of Earth Principal axis (x-axis) - towards the vernal equinox ( intersection of the Earth Equator and ecliptic plane) from the origin Fundamental Plane - Earth equator 3. The latitude – longitude coordinate system Origin - Centre of Earth Principal axis (x-axis) - towards the Greenwich meridian from the origin Fundamental Plane - Earth equator
- 5. Complexity in determination of the motion of the body Spacecraft / Celestial body is acted upon by multiple gravity fields (For e.g.. Earth , Sun and Mars for an Earth –Mars Transfer) - 4-body equations of motion must be solved - No closed form solution 2 d R R rE RE rM RM 2 S 3 E 3 3 M 3 3 R OTHERS dt R rE RE rM RM R Others R P lanets R NSG R Drag R SRP - To be solved numerically - Ephemeris (solution) accuracy depends on the Force Model
- 6. Two body Motion and Conic Assumptions The motion of a body is governed by attraction due to a single central body. The mass of the body is negligible compared to that of the central body The bodies are spherically symmetric with the masses concentrated at the center. No forces act on the bodies except for gravitational and centrifugal forces acting along the line of centers If these assumptions hold, it can be shown that conic sections are the only possible paths for orbiting bodies and that the central body must be at a focus of the conic Fundamental equations of motion that describe two-body motion under the assumptions Relative Form where G ( m1 m2 ) Closed form Solution 2 p a (1 e ) - Conic Equation r 1 e cos 1 e cos
- 7. Size and Shape of a Conic a - semi major axis b - semi minor axis r - radial distance ν - true anomaly
- 8. Representation of a point (spacecraft / body) in motion Position and velocity vectors represent a point in motion in space uniquely Z Satellite perigee 0 Vernal i Equator equinox Node a semi major axis ; e Eccentricity i Inclination ; Right ascension of ascending node Argument of perigee; True anomaly True anomaly
- 9. Two-Body Motion : General Description
- 10. Ephemeris Generation Given full characteristics of spacecraft in a conic at time t1 - either state vector (both position and velocity vectors together) or orbital elements Find the characteristics of the spacecraft in the conic at time t2 r , V at t2 r , V at t 1
- 11. Ephemeris generation using analytical techniques (Time evolution)
- 13. Calculating the transformation equations Algorithm: 1. From the calculated value of nu we get the value of parameters using the transformation equations
- 16. Pallas From To 2004-Oct-01 2004-Oct-02 Parameters JPL calculated Error rx (km) -204349767.594200000 -205016214.226665000 666446.632464975 ry (km) 242717229.395200000 242888592.882376000 -171363.487175971 rz (km) -34002778.020140000 -33939477.594100100 -63300.426039897 vx (km/sec) -17.500293070 -17.504129882 0.003836812 vy (km/sec) -13.720366896 -13.730516703 0.010149807 vz (km/sec) 3.945413510 3.940343099 0.005070411 v (km/sec) 22.584840337 22.593095308 -0.008254971 r (km) 319102914.236415000 319653569.959065000 -550655.722650230 alpha (degree) 130.094900000 130.166879000 -0.071979000 delta (degree) -6.117083078 -6.095094145 -0.021988932
- 20. Source of error
- 21. Conclusion: Based on the two body model ephemeris generation was carried out. Even though the % error is of the order of 10-1 -10-2 , , their value in absolute is very high This stress the fact that we need to have a detailed model and do the calculation using them, even though carrying them out is very tedious and takes a lot of time. Precision takes precedence over time!
- 22. References http://ssd.jpl.nasa.gov/horizons.cgi#top Lecture notes on Orbital Dynamics, Dr Ramanan M V, IIST “Orbital mechanics for Engineering Students”, Howard Curtis, Elsevier Aerospace Engineering Series
- 23. Thank You