Question: Since the presence of the Rosetta spacecraft near comet 67P allowed for a detailed mass measurement, extremely precise position and velocity determinations, and physcal measurement and imaging of ejecta from the comet as it passed through perihelion, did Rosetta improve on models of non-gravitational effects on comet 67P's orbit?


The calculation of the orbits of comets can be more difficult than those of most asteroids for a number of reasons. Some comets have such highly eccentric orbits that aphelion is too far for the comet to be observed continuously, or the period is so long that only one pass has been observed and a period can not be calculated, or it passes so close to the sun that it's orbit is highly modified. However comet 67P/Churyumov–Gerasimenko currently has a period of only about 6.4 years a perihelion/aphelion of 1.2 AU and 5.7 AU respectively. While it's orbit is within a so called "frost line" it remains further from the sun than the Earth's orbit.

In this answer I plot some data from a recent NASA JPL Horizons ephemeris for comet 67P. The current default solution soln ref.= JPL#K084/25, data arc: 1995-07-03 to 2016-05-30 appears to use Marsden coefficients to model non-gravitational forces on the comet. Brian . Marsden was a British astronomer who contributed greatly to the field of cometary orbits. (See also here and here.) While exact modeling of non-gravitational forces on comets would be extremely complex, he introduced a simple empirical parameterization that provides a framework to discuss the magnitude and potential effects of these forces on the orbits of comets.

Using the following convention: $\hat{\mathbf{e}}_R, \ \hat{\mathbf{e}}_T, \ \hat{\mathbf{e}}_N$ are unit vectors at the location of the comet in the radial, transverse, and normal directions where $\hat{\mathbf{e}}_R$ points away from the sun, $\hat{\mathbf{e}}_N$ is the direction of the angular momentum vector (perpendicular to the orbit plane) and $\hat{\mathbf{e}}_T$ is perpendicular to the first two and approximately in the direction of motion, non-gravitational accelerations can be parameterized using the empirical equations:

$$\mathbf{a}_{NG} = ( A_1\hat{\mathbf{e}}_R \ + \ A_2\hat{\mathbf{e}}_T \ + \ A_3\hat{\mathbf{e}}_N) \ g(r), $$


$$g(r)= 0.111262\left(\frac{r}{2.808}\right)^{-2.15} \left(1+\left(\frac{r}{2.808}\right)^{5.093}\right)^{-4.6142}, $$

and the acceleration coeficients $A_1,A_2,A_3$ commonly have units of $AU / day^2$.

I've reproduced these here to illustrate the basic idea. There are further considerations including a delay term and effects of rotation. However with this parameterization it is possible to discuss and at least get a handle on non-gravitational effects without a detailed physical model. These effects might the Yarkovsky and Poynting-Robertson effects, and of course recoil from material energetically ejected from the comet, especially as it approaches the sun and is heated.

The parameters $A_1,A_2,A_3$ in model can be used to express effects from physical models of comets, but they can also be used as fitting parameters to improve orbital solutions for comets based on observational data.

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above: linear and semi-log plots of $g(r)$ between 1.2 and 5.7 AU.

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above: Example of the non-gravitational parameters used in the most recent JPL Horizons ephemeris for comet 67P. The coefficients have units of $AU/day^2$. For a comparison, the gravitational acceleration at a distance of 1.2 $AU$ is about 0.0041 $m/s^2$ or about 2.1E-04 $AU/day^2$. This suggests that the non-gravitational forces used here have a parts-per-million effect per orbit which will become substantial over a large number of orbits.

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    $\begingroup$ Thanks for this question, far far better than I could have asked it! I was very surprised to learn recently how big these non-gravitational effects can be. One interesting source is lpi.usra.edu/books/CometsII/7009.pdf which says: As the astrometric datasets improve and lengthen and as the modeling of the cometary nongravitational effects becomes more realistic, there remains the strong possibility that some physical characteristics of comets will soon be accurately determined from the orbit-determination process alone. We are already beginning to see signs that this is the case. $\endgroup$ – nealmcb May 16 '17 at 1:30
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    $\begingroup$ Thank you for the comments! Yeomans et al. is also linked here and here. I'm not sure about the "will soon be accurately determined from" part, but certainly continued visits to comets like that of OSIRIS-REx will keep adding to the field. See How will OSIRIS-REx scan and characterise the near-earth asteroid Bennu? $\endgroup$ – uhoh May 16 '17 at 3:05

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