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# Orbital parameters of asteroids using analytical propagation

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### Orbital parameters of asteroids using analytical propagation

1. 1. Orbital parameters of Asteroids using analytical PropagationTeam Members:Chetana D.Lakshmi NarsimhanLokeswara Rao.NRamiz AhmadRanupriya Didwania Guided by Dr. R.V Ramanan
2. 2. Plan of the talk:1. Objective2. Introduction to various coordinate system3. 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. 3. ObjectiveTo obtain the orbital parameters of the celestialobjects ( asteroids ) at any time with respect toits reference parameters.To study the time evolution of asteroids
4. 4. Introduction to the coordinate systemVarious coordinate systems:1. Inertial coordinate system (commonly used)Origin - Centre of EarthPrincipal axis (x-axis) - towards the vernal equinox ( intersection of the Earth Equator and ecliptic plane) from the originFundamental Plane - Earth equator2. The right ascension –declination coordinate systemOrigin - Centre of EarthPrincipal axis (x-axis) - towards the vernal equinox ( intersection of the Earth Equator and ecliptic plane) from the originFundamental Plane - Earth equator3. The latitude – longitude coordinate systemOrigin - Centre of EarthPrincipal axis (x-axis) - towards the Greenwich meridian from the originFundamental Plane - Earth equator
5. 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       2d R R rE RE rM RM   2 S 3 E 3 3 M 3 3 R OTHERSdt 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. 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. 7. Size and Shape of a Conica - semi major axisb - semi minor axisr - radial distanceν - true anomaly
8. 8. Representation of a point (spacecraft / body) in motion Position and velocity vectors represent a point in motion in space uniquely Z Satellite perigee 0Vernal i Equatorequinox Node a semi major axis ; e Eccentricity i Inclination ; Right ascension of ascending node Argument of perigee; True anomaly True anomaly
9. 9. Two-Body Motion : General Description
10. 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. 11. Ephemeris generation using analytical techniques(Time evolution)
12. 12. Calculating the transformation equationsAlgorithm:1. From the calculated value of nu we get the value of parametersusing the transformation equations
13. 13. http://ssd.jpl.nasa.gov/horizons.cgi#topT0= 2004-Oct-01T= 2005-Oct-01
14. 14. Pallas From To 2004-Oct-01 2004-Oct-02Parameters JPL calculated Errorrx (km) -204349767.594200000 -205016214.226665000 666446.632464975ry (km) 242717229.395200000 242888592.882376000 -171363.487175971rz (km) -34002778.020140000 -33939477.594100100 -63300.426039897vx (km/sec) -17.500293070 -17.504129882 0.003836812vy (km/sec) -13.720366896 -13.730516703 0.010149807vz (km/sec) 3.945413510 3.940343099 0.005070411v (km/sec) 22.584840337 22.593095308 -0.008254971r (km) 319102914.236415000 319653569.959065000 -550655.722650230alpha (degree) 130.094900000 130.166879000 -0.071979000delta (degree) -6.117083078 -6.095094145 -0.021988932
15. 15. Ephemeris (successive years)
16. 16. Source of error
17. 17. 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 highThis stress the fact that we need to have a detailed model and do the calculation usingthem, even though carrying them out is very tedious and takes a lot of time.Precision takes precedence over time!
18. 18. Referenceshttp://ssd.jpl.nasa.gov/horizons.cgi#topLecture notes on Orbital Dynamics, Dr Ramanan M V, IIST“Orbital mechanics for Engineering Students”, Howard Curtis, Elsevier AerospaceEngineering Series
19. 19. Thank You
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