From: wgw@vax.oxford.ac.uk
Newsgroups: sci.astro
Subject: Comet Nongravs
Date: 17 Nov 92 14:30:46 GMT
Organization: Oxford University VAX 6620

Comet Nongravitational Paramters:

In response to a number of requests for further info on how to model the
nongravitational (outgassing) effects on a comet's orbit the following
may be useful.


Putting     _R_  as unit vector pointing away from the sun
            _T_  as unit vector in the plane of the comet's orbit
                 pointing in the direction of motion
            _N_  as unit vector normal to the orbital plane


  We can write the nongravitational acceleration as (Marsden, Sekanina &
Yeomans, 1973),

                A1 * g(r) _R_  +  A2 * g(r) _T_  +  A3 * g(r) _N_

where A1, A2 and A3  can depend on time if need be.  The A's are given 
for a time unit of 10000 days (so, if like me, you tend to work in days
the values have to be multiplied by 10^-8 when used in an integration
routine).

Since the effect of the nuclear outgassing is parameterized by using a
function, g(r), that depends on heliocentric distance, r, this formalism
is symmetric about the perihelion point.  Many cometary light-curves do
not peek at perihelion, so this method has limitations  (see Yeomans &
Chodas  and  Festou et al. for discussion of how to get round this).

It is usual to assume that A3 = 0 and to drop this term completely since
what it adds in on the pre-perihelion leg, it removes on the post-perihelion
leg of the orbit and so it cannot be easily solved for from the observations.

The effect on the time of perihelion passage is amost all due to A2 in this
parameterization.

The form of g(r) is based on the expected vaporization of water-ice due to
solar heating and is,

 
      g(r)  =  0.111262 * ( rr0 ^ -2.15 ) * ( 1 + rr0 ^ 5.093 ) ^ -4.6142

  where  rr0 =  r / 2.808    for r in astronomical units.


The IAU Catalogue of Cometary Orbits 1992  (Marsden & Williams) gives 
derived values of A1 and A2 on p.73-75  (( it also gives results derived
using a different g(r), noted as "style I", which was used prior to the
Marsden, Sekanina & Yeomans paper which defined the "style II" above).
From this catalogue, some representative values are,

      P/Halley                    A1 = 0.04   A2 = 0.0155
      P/Brooks 2                  A1 = 0.56   A2 = -0.1381
      P/Encke (discovery)         A1 = -2.17  A2 = -0.0361
              ( recently)         A1 =  0     A2 = -0.0023
      P/Schwassmann-Wachmann 2    A1 = 1.31   A2 = -0.3095
    


Technical references:

B.G.Marsden, Z.Sekanina & D.K.Yeomans  Astron.J. vol. 78, p.211-225
(1973) -- this paper defines the "style II" version of parameterizing
the nongravs and tabulates results for several comets.

Brian Marsden  p.184-190 in The Evolution of the Small Bodies of the 
Solar System (ed. Fulchignoni & Kresak, 1987).

D.K.Yeomans & P.W.Chodas  Astron.J. vol 98, p.1083-1093 (1989)
deals with the problem of g(r) being symetrical about perihelion.

M.Festou, H.Rickman & L.Kamel  Nature vol. 345, p.235-238 (1990)
deals with relating the nongravs to the observed light-curve.


Useful reference :

A good discussion can be found in Introduction to Comets by Brandt &
Chapman p.69-75.  Their recent book (Rendezvous in Space) also has a
small amount on this topic.




Hope this helps,

                  Graeme Waddington     wgw @ vax.ox.ac.uk