Osculations in Minor Planet Orbits

In the absence of other objects, the orbit of a minor planet around the Sun would be Keplerian, e.g. for the bound case the minor planet would precisely follow an elliptical path around the Sun. For such a case the orbit can be accurately defined by the six orbital elements: semi-major axis, eccentricity, inclination, longitude of the ascending node, argument (longitude) of the perihelion and the mean anomaly. The orbital elements are constants and can be used to derive directly the object's position and orbital velocity.

Gravitational perturbations by the planets continuously distort an object's orbit, i.e. change the orbital elements. The masses and locations of the perturbing objects are known, therefore the change of the orbital elements can be accurately calculated by the numerical integration of the motions of the major solar system bodies. Thus when the orbital elements are quoted for a minor planet they are given for a particular date (called the epoch of osculation). For the bright minor planet, Niobe(71), the change of the orbital elements over one year is

 Epoch of osculation     =  2005 01 30    2006 01 30 
 Epoch of osculation(MJD)=   53400.5       53765.5   
 Semimajor axis          =  2.75455377    2.75427850 
 Eccentricity            =  0.17609238    0.17624829 
 Inclination             =   23.255867     23.257766 
 Long. of ascending node =  316.118227    316.113724 
 Argument of perihelion  =  267.438638    267.473192 
 Mean anomaly            =  222.942006    301.621193 

While for main-belt minor planets these changes will be very small, the effect of the predicted RA-Dec positions can be typically 600 arcseconds over five years if these perturbations are not applied. Hence, when determining the location of a minor planet the `current' orbital elements need to be used. The Lowell Observatory update the "Epoch of osculation" on their database every 100 days, so the "Epoch of osculation" dates move forward by 100 days, i.e. 2005-Jan-30 (JD=2453400.5), 2005-May-10 (JD=2453500.5), etc.

When using Project Pluto's find_orb the derived orbital elements appear to be quoted for the mean date of the observations hence it is not surprising that the derived values don't precisely agree with the given values from a different "Epoch of osculation".

In order to aid the comparison of the derived orbital elements at a given epoch, we have a short F77 programme (econ) that will calculate the orbital elements for any required date. This programme uses the subroutine SLA_PERTUE from Starlink's positional astronomy SLALIB.

To use econ, create a file with the date and a list of the minor planets required, e.g.

2006 1 30
'Feronia'
'Klytia'

 econ 

 Object:    72   feronia
 --------------------------------------------------  
 Epoch of osculation     =  2005 01 30    2006 01 30 
 Epoch of osculation(MJD)=   53400.5       53765.5   
 Mean anomaly            =  304.773927     50.189207 
 Argument of perihelion  =  102.558110    102.606305 
 Long. of ascending node =  208.140822    208.139700 
 Inclination             =    5.417293      5.417417 
 Eccentricity            =  0.12025772    0.12038333 
 Semimajor axis          =  2.26608230    2.26629427 
 --------------------------------------------------  
  
 Object:    73   klytia
 --------------------------------------------------  
 Epoch of osculation     =  2005 01 30    2006 01 30 
 Epoch of osculation(MJD)=   53400.5       53765.5   
 Mean anomaly            =   58.032542    139.640891 
 Argument of perihelion  =   52.689416     53.747815 
 Long. of ascending node =    7.295432      7.298090 
 Inclination             =    2.377572      2.375957 
 Eccentricity            =  0.04125281    0.04108568 
 Semimajor axis          =  2.66437095    2.66631246 
--------------------------------------------------   

References

"Astronomy on the Personal Computer" by Montenbruck and Pfleger, contains extensive information on orbital calculations. Chapter 5 (4th Edition) is on "Special Perturbations".

"Astronomical Algorithms" by Jean Meeus, contains much useful material on orbital calculations. Chapter 32 (2nd Edition) is on "Positions of the Planets" and includes notes on osculating elements.

James Hilton's article on U.S. Naval Observatory Ephemerides of the Largest Asteroids"

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jrl

2013-Dec-22 23:33:14 UTC