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

15. http://ssd.jpl.nasa.gov/horizons.cgi#top
T0= 2004-Oct-01
T= 2005-Oct-01

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

17. Ephemeris (successive years)

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