The Astronomical Journal, 142:81 (4pp), 2011 September C 2011.
doi:10.1088/0004-6256/142/3/81
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
NONGRAVITATIONAL FORCES ON COMETS: AN EXTENSION OF THE STANDARD MODEL K. Aksnes1 and E. Mysen2 1
Institute of Theoretical Astrophysics, University of Oslo, Norway;
[email protected] 2 Norwegian Mapping Authority, Hønefoss, Norway Received 2011 May 20; accepted 2011 July 7; published 2011 August 11
ABSTRACT The accuracy of comet orbit computations is limited by uncertain knowledge of the recoil force due to outgassing from the nuclei. The standard model assumes an exponential dependence of the force on distance from the Sun. This variable force times constants A1 , A2 , and A3 represents the radial, transverse, and normal components of the net force. Orbit solutions show that the As often vary considerably over a few apparitions of the comets. In this paper, we allow for time variations of the As, and we show that for several comets this improves the orbit accuracy considerably. Key words: celestial mechanics – comets: individual (19P, 22P, 24P, 46P, 64P, 67P, 71P) the force models based on a comet’s state of rotation give more physical and dynamical insight than does the standard model, their parameters tend, in our experience, to be more unstable and less well determined than is the case for the parameters A1 , A2 and A3 . This may be due to more than one gas jet emanating from the nucleus and/or a chaotic state of rotation. This is supported by the close-up views from space probes of several gas jets on the comets 1P/Halley in 1986 and 19P/Borrelly in 2002. Some researchers even question the importance of the visible jets as a source of thrust. They point to simulations showing that the jets may consist of dust particles in shocked regions of the expanding gas (see, e.g., Mysen et al. 2010, and references therein), and that these jets may appear over both active and inactive regions of the nucleus (Crifo et al. 2002). The clue to understanding the nongravitational forces may therefore be a better modeling of the sublimation processes on the nucleus leading to an improved function g(r). For instance, the current law g(r) can be derived from the assumptions that the comet nucleus is spherical, isothermal, and that the sublimation of water ice takes place at a uniform surface (Rickman 1992). However, observations show that comet nuclei are nonspherical (1P/Halley, 19P/Borrelly, 81P/Wild 2, 9P/Tempel 1), not isothermal (A’Hearn et al. 2005), and that most of the sublimation does not take place at the surface (Sunshine et al. 2006) in general. As a first step, we therefore decided to look into the possibility of improving the standard model by allowing for time variations of the As. That is the object of the following sections.
1. INTRODUCTION Outgassing mainly from water ice in comet nuclei in the vicinity of the Sun produces recoil forces that affect the orbits of the comets. These so-called nongravitational forces are difficult to estimate and they therefore limit the accuracy of orbit computations. Marsden (1968, 1969) pioneered the investigation of nongravitational forces by introducing them directly into the equations of motion through terms depending exponentially on time and distance from the Sun. His model (known as Style I) for the nongravitational parameters was later improved by Marsden et al. (1973) in what has since become the standard model (Style II). This semi-empirical model, based on the Sun’s insolation and the volatility of water ice, uses a force function g(r) that depends exponentially on the heliocentric distance. The comet’s acceleration is decomposed into radial A1 g(r), transverse A2 g(r), and sometimes normal A3 g(r) components. Here A1 , A2 , and A3 are constants solved for along with orbital elements, in a fit to observed comet positions. The rationale behind this model is that solar heating of the nucleus produces a gas jet which, due to thermal inertia, is rotated away from the Sun, so that the force is not entirely radial. For some comets the orbit fit has been improved by introducing an asymmetry in the outgassing around the perihelion by replacing g(r(t)) by g(r(t−τ )), where τ is a time shift solved for together with the other parameters in a least-squares solution (Yeomans & Chodas 1989). Sekanina (1981) and others have attempted to formulate the jet force in terms of the orientation of the comet and the direction of the Sun. The parameters A1 , A2 , A3 are replaced by four parameters A, η, I, φ, where A = (A21 + A22 + A23 )1/2 , η is a rotation lag angle, I is the obliquity of the comet’s spin axis, and φ is the cometo-centric longitude of the Sun at perihelion. Assuming that the spin axis of a non-spherical nucleus precesses, Sekanina (1984) has derived formulae for the variation of I and φ. For a spherical nucleus, Bielicki & Sitarski (1991) solved for linear precession rates dI/dt and dφ/dt in addition to A, η, I, φ, and the six orbital elements of the comet 64P/Swift-Gehrels. They succeeded in linking four apparitions of observations spanning about 100 yr with a mean residual of 1.94 arcsec. A much more detailed review of the large amount of work devoted to the analysis of nongravitational forces on comets has been offered by Yeomans et al. (2004). In general, none of the force models discussed above give satisfactory orbit fits over more than three apparitions. Although
2. EXTENSION OF THE STANDARD MODEL We consider the heliocentric motion of a comet perturbed by the eight planets, Pluto, and the Moon, and suffering a nongravitational acceleration: r r r˙ − r˙ r h δ r¨ = g(r) A1 + A2 + A3 , (1) r h h where h = r × r˙ and g(r) = α(r/r0 )−m {1 + (r/r0 )n }−k .
(2)
Here r is the comet’s heliocentric radius vector and g(r) is a function that shows that the nongravitational force drops 1
The Astronomical Journal, 142:81 (4pp), 2011 September
Aksnes & Mysen
Table 1 Nongravitational Parameters (the As are in units of 10−8 AU day−2 ) Comet 19P-3210 19P-3210 19P-3210 19P-3210 22P-6410 22P-6410 22P-6410 24P-7601 24P-5101 24P-5101 24P-5101 46P-5486 46P-8508 46P-8508 46P-8508 46P-8508 64P-7301 64P-7310 64P-7310 64P-7310 64P-7310 67P-6910 67P-6910 67P-6910 71P-9506 71P-9511 71P-9511 71P-9511
A1
A2
A3
β × 105 (day−1 )
τ (days)
σ
Obs./Rej.
+0.153 ± 0.001 +0.158 ± 0.001 +0.195 ± 0.002 +0.174 ± 0.002 +0.187 ± 0.016 +0.187 ± 0.016 +0.165 ± 0.004 +0.109 ± 0.009 +0.669 ± 0.018 +0.667 ± 0.018 +0.376 ± 0.004 +0.543 ± 0.040 +0.335 ± 0.004 +0.337 ± 0.004 +0.313 ± 0.002 +0.336 ± 0.004 +0.275 ± 0.025 +0.433 ± 0.021 +0.433 ± 0.022 +0.136 ± 0.013 +0.166 ± 0.014 +0.080 ± 0.001 +0.079 ± 0.001 +0.078 ± 0.002 +1.313 ± 0.019 +1.167 ± 0.050 +0.872 ± 0.039 +0.671 ± 0.011
−0.038 ± 0.000 −0.038 ± 0.000 −0.037 ± 0.000 −0.019 ± 0.001 −0.100 ± 0.000 −0.100 ± 0.000 −0.090 ± 0.000 −0.059 ± 0.000 −0.054 ± 0.000 −0.055 ± 0.000 −0.064 ± 0.000 −0.093 ± 0.003 −0.140 ± 0.000 −0.140 ± 0.000 −0.136 ± 0.000 −0.130 ± 0.001 +0.026 ± 0.000 +0.036 ± 0.000 +0.036 ± 0.000 +0.041 ± 0.000 +0.007 ± 0.005 +0.010 ± 0.000 +0.010 ± 0.000 +0.010 ± 0.000 −0.380 ± 0.001 −0.277 ± 0.001 −0.196 ± 0.003 −0.071 ± 0.003
... +0.021 ± 0.002 +0.010 ± 0.002 +0.004 ± 0.002 ... −0.038 ± 0.009 −0.039 ± 0.003 ... ... +0.147 ± 0.018 +0.218 ± 0.004 ... ... +0.079 ± 0.005 +0.078 ± 0.003 +0.077 ± 0.003 ... ... −0.000 ± 0.011 +0.024 ± 0.005 +0.014 ± 0.005 ... +0.025 ± 0.001 +0.025 ± 0.001 −0.490 ± 0.007 −0.573 ± 0.026 −0.406 ± 0.020 −0.301 ± 0.005
... ... −0.396 ± 0.014 −0.369 ± 0.013 ... ... −3.284 ± 0.010 ... ... ... +1.927 ± 0.013 ... ... ... −3.270 ± 0.022 −3.327 ± 0.023 ... ... ... +5.829 ± 0.228 +6.194 ± 0.236 ... ... +0.110 ± 0.055 ... ... −36.89 ± 1.05 −47.88 ± 0.34
... ... ... −27.65 ± 1.03 ... ... ... ... ... ... ... ... ... ... ... −3.11 ± 0.18 ... ... ... ... 47.09 ± 4.49 ... ... ... ... ... ... −51.82 ± 0.8
0.97 0.95 0.91 0.83 3.98 3.97 1.12 0.86 5.08 4.97 1.13 2.00 1.50 1.44 0.84 0.83 0.92 2.02 2.03 0.93 0.91 0.71 0.71 0.71 0.84 3.59 2.72 0.90
2187/326 2187/326 2187/326 2187/326 2897/418 2897/418 2897/418 628/178 694/155 694/155 694/155 32/3 1828/42 1828/42 1828/42 1828/42 62/30 249/19 249/19 249/19 249/19 2616/236 2616/236 2616/236 496/56 604/64 604/64 604/64
observation (e.g., 3210 means from 1932 to 2010). The next to last column in Table 1 gives the rms residual (σ ) in arcseconds; the last column shows the number of observations used and the number rejected in the solution. A rejection criterion of about three times the rms residual is used. Many observations could also be excluded because their residuals much exceed those of neighboring observations. 19P/Borrelly. For this comet the A1 , A2 solution is nearly as good as the A1 , A2 , A3 solution although Chesley et al. (2001) found that inclusion of A3 significantly improved the prediction ephemeris for the close flyby of this comet by the Deep Space spacecraft in 2001. Our solution A1 = 0.153, A2 = −0.038, σ = 0.97 for the period 1932 September 29 to 2010 February 11 is comparable to the solution A1 = 0.120, A2 = −0.038, σ = 1.00 obtained by Nakano (1998) for the subinterval 1980–1997. Only very small improvements were obtained by also solving for A3 , β, and τ . This did not change A1 much which is remarkably stable over 11 apparitions of the comet. This suggests that the outgassing from the comet has changed little during these apparitions. 22P/Kopff. Also for this comet the A1 , A2 and A1 , A2 , A3 solutions have almost identical residuals, but inclusion of β gives a dramatic improvement in σ from nearly 4 to a little over 1 arcsec for the observations from 1964 May 8 to 2010 February 5. With β = −3.3 × 10−5 and A1 = 0.165 at the epoch 2003 January 1, A1 decreases from 0.241 at the beginning to 0.073 at the end of the 46 yr time span of the observations. This is in reasonable accord with the A1 values listed for several subintervals in Catalogue of Cometary Orbits 2008 (Marsden & Williams 2008). The corresponding variations of A2 and A3 are small. An attempt to also include τ in the solution failed because of poor convergence.
off rapidly beyond the distance r0 . For the force due to the vaporization of water ice on a rapidly rotating nucleus with Bond albedo 0.1, the constants are given the values r0 = 2.808 AU, m = 2.15, n = 5.093, k = 4.6142, and α = 0.111262. This value of α normalizes g(r) to 1 AU day−2 at the distance 1 AU. This is the standard model as introduced by Marsden et al. (1973). In our extension of the standard model, we in Equation (1) make the replacement g(r) → g{r(t − τ )}(1 + βt),
(3)
where a time shift τ has been introduced to account for a possible asymmetry of the nongravitational force around the perihelion, while β absorbs any linear change in A1 , A2 , and A3 . The time t is here reckoned from the epoch of the osculating orbital elements. In a numerical integration of the comet’s equations of motion and of a set of variational equations (Aksnes & Grav 2005), we solve for the six orbital elements and some or all of A1 , A2 , A3 , β,τ by a least-squares fit to observed positions of the comet. 3. RESULTS For investigation with our extended model, we have selected seven short-period comets that are noticeably influenced by nongravitational forces and for which fairly long series of observations are available from the IAU Minor Planet Center. Table 1 summarizes different solutions for the nongravitational parameters, while Table 2 gives the orbital elements for the best solution (there are very small differences between the orbital elements of the different solutions for the same comet). The comets are identified with their catalog numbers followed by the last two digits of the first year and the last year of 2
The Astronomical Journal, 142:81 (4pp), 2011 September
Aksnes & Mysen Table 2 Orbital Elements
Comet 19P 22P 24P 46P 64P 67P 71P
Epoch
P (yr)
T
ω
Ω
I
q
E
2008-08-02.0 TT 2003-01-01.0 TT 2001-05-11.0 TT 2003-12-27.0 TT 2000-04-06.0 TT 2003-12-27.0 TT 2006-05-25.0 TT
6.853 6.457 8.247 5.440 9.180 6.562 5.518
2008-07-22.3434 2002-12-12.0901 2001-05-02.6575 2002-08-26.6788 2000-04-21.7980 2002-08-18.2869 2006-06-06.8001
353.3790 162.7629 57.8698 356.3781 92.4131 11.4091 208.7516
75.4446 120.9187 79.8335 82.1669 306.1406 50.9291 59.6521
30.3243 4.7189 11.7518 11.7384 8.4375 7.1241 9.4897
1.3547838 1.5836307 1.2050041 1.0586238 1.3388532 1.2906450 1.5621388
0.6244813 0.5433013 0.7048010 0.6577529 0.6946153 0.6317517 0.4997604
solution based on 2616 observations from six apparitions 1969 September 11 to 2010 July 7 gave A1 = 0.080 ± 0.001, A2 = 0.010 ± 0.000, σ = 0.71. These values hardly changed when more parameters were added to the solution, in agreement with the very small value of 0.11 × 10−5 of β. 71P/Clark. We first made a solution with observations in the interval 1995 August 14 to 2006 December 31, obtaining A1 = 1.313, A2 = −0.380, A3 = −0.490, σ = 0.84. This is very close to a solution obtained by Nakano (2008) for the same interval. For the expanded observation interval 1995 May 3 to 2011 March 2, the fit got much worse with σ = 3.59. This was reduced to 2.72 arcsec by solving also for β = −36.89. A large further reduction to 0.90 arcsec resulted when adding also τ to the solution, but then the three As, and presumably also the outgassing, decreased markedly during the three apparitions considered, since β = −47.88. The large negative value of −51.8 days for τ implies that maximum outgassing occurred almost a month after perihelion. Through the seven examples above we have documented the ability of our extension of the standard model to represent the nongravitational forces within the time spans of the observations used in the orbital solutions. This approach gives information about not only the size of the nongravitational forces but also how they vary with time. It is also of interest to investigate to what extent the model can predict future comet positions. For this purpose we have selected the comets 19P, 22P, 24P, and 67P which have been observed during many apparitions. We have for these four comets obtained orbital solutions as in Table 1 but excluding the last apparitions which cover observation arcs of respectively 2.5, 2, 0.6, and 2 yr length. The solutions are then used to “predict” the positions in these arcs. For 19P and 67P the standard model is almost as good as the extended model while for 22P and 24P the extended model gives large improvements. Not surprisingly, for 19P and 67P the differences between observed and predicted positions are comparable to the rms residuals of the solutions (∼0.90 arcsec) and the inclusion of the parameters β and τ gives only marginal improvements. For the comets 22P and 24P, on the other hand, the inclusion of β substantially improves the solutions which predict the positions of the last apparition with errors increasing with time from 2 to 40 and 1 to 8 arcsec, respectively, compared with 4 to 170 and 45 to 80 arcsec when β is excluded. The larger range of errors for 22P reflects the fact that they cover a three times longer interval than for 24P.
24P/Schaumasse. The solution in Table 1 for A1 and A2 for the period 1976 December 27 to 2001 July 23 is almost identical to a solution obtained by Nakano (2006), with a very satisfactory σ = 0.86. When increasing the interval to six apparitions from1951 September 30 to 2001 July 23, σ increased to about 5 arcsec. Inclusion of A3 improved the fit only marginally, but after adding β also the residual was reduced to 1.13 arcsec. The last solution decreased the value of A1 and increased the value of A2 very appreciably while their uncertainties were reduced by a factor of three or four. 46P/Wirtanen. Since this was the original target for ESA’s Rosetta mission, much effort has been devoted to investigating the comet which was discovered in 1948 and has been observed during 11 apparitions until 2008. Marsden & Williams (2008) concluded that it was impossible to link all the apparitions up to 2007 with constant values of A1 , A2 , and A3 . Even using the seven parameters A, η, I0 , φ 0 , dφ/dt, dI/dt, τ of the rotating nucleus model, Krolikowska & Sitarski (1996) got rms residuals as large as 2.4 arcsec for the 1948–1991 observations. While two-parameter solutions for the observation intervals 1954–1986 and 1985–2008 gave rms residuals of respectively 2.0 and 1.5 arcsec (Table 1), the entire interval 1954–2008 could not be satisfied to less than 5.6 arcsec, even with a five-parameter solution. This seems to suggest a rather abrupt change in the outgassing from the comet. In four solutions for the interval 1985 November 13 to 2008 June 23 with 2–5 parameters, there is a very marked drop from 1.50 to 0.83 arcsec in σ when including β in the solution but the addition of τ had practically no effect. Our values of A1 , A2 , A3 , τ agree fairly well with the respective values 0.370 ± 0.006, −0.131 ± 0.001, 0.080 ± 0.003, and −4.17 ± 0.6 obtained by Mastrodemos (2008) with σ = 0.8 by fitting to 1815 observations from 1995 to 2008. With β = −3.3 × 10−5 and A1 = 0.313 at the epoch 2003 December 27, A1 varies from 0.381 in 1985 to 0.296 in 2008. 64P/Swift-Gehrels. A solution for A1 and A2 , based on 62 observations from 1973 February 8 to 2001 February 26, is almost identical to a solution by Nakano (2006). Solutions for two or three of the As for four apparitions from 1973 February 8 to 2010 May 4 gave almost identical values with rms residuals of about 2 arcsec, A3 being practically 0 in the latter case. However, when adding β to the solution, the residual was reduced to 0.93 arcsec. A1 decreased from 0.433 to 0.136, A2 changed very little, while A3 increased slightly. Further addition of τ had very little effect. We thus regard the four-parameter solution as the most realistic one. Since β came out with a relatively large positive value of 5.83 × 10−5 , this indicates an increase in the comet’s activity during the interval, A1 being increased from 0.057 to 0.165. 67P/Churyumov-Gerasimenko. The Rosetta spacecraft is set to rendezvous with this comet in 2014. The comet was discovered in 1969 and shows a very stable behavior as far as the nongravitational effects are concerned. A two-parameter
4. CONCLUSIONS Very substantial decreases of the rms residuals were obtained when allowing for linear changes in A1 , A2 , and A3 during three to six apparitions of the comets 22P/Kopff, 24P/Schaumasse, 46P/Wirtanen, 64P/Swift-Gehrels, and 71P/Clark. This clearly demonstrates that for active comets much improved orbit fits 3
The Astronomical Journal, 142:81 (4pp), 2011 September
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may be obtained for several apparitions by introducing such time variations. For more quiescent comets like 19P/Borrelly and 67P/Churyumov-Gerasimenko, the standard model including just A1 and A2 or A1 , A2 , and A3 is adequate. Despite the improvements obtained here in modeling the nongravitational forces, it may be worthwhile to take a new look at the simplified function g(r) to see if it can be modified to better represent the outgassing forces on comets. For instance, it may be that if the improved g(r) studied in this paper is applied to observations, the Ai parameters could more easily be resolved in terms of the nucleus’ rotation state and shape. As previously mentioned, parameterizations of the force model in terms of the state, shape, and the old g(r) have been successfully applied, but they tend in our experience to be somewhat unstable.
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