LETTER
PASJ: Publ. Astron. Soc. Japan 59, L7–L10, 2007 April 25
c 2007. Astronomical Society of Japan.
New Approach to Study Non-Gravitational Motion of a
Comet Normal to the Orbital Plane
Naoya M IURA
Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902
[email protected]
Masateru I SHIGURO
School of Physics and Astronomy, Seoul National University, San 56-1, Shillim-dong, Kwanak-gu, Seoul 151-742, Korea
[email protected]
Yuki S ARUGAKU
Department of Earth and Planetary Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902
[email protected]
and
Munetaka Ueno
Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902
[email protected]
(Received 2006 November 29; accepted 2007 January 8)
Abstract
An edge-on view of the 22P/Kopff image showed that the nucleus was not on the locus of the maximum
brightness of its dust cloud, which is likely to be a neck-line structure composed of large dust particles ejected 409
days before the observation. Assuming that the non-gravitational recoil force normal to the orbital plane generated
a displacement of the nucleus, we demonstrate a modeled image similar to that of the observation. The obtained
normal component of the non-gravitational acceleration is consistent with the predicted value extrapolated by data
obtained during 1958–1983.
Key words: comets: individual (22P/Kopff) — meteors, meteoroids
1. Introduction
Different from the motion of other bodies in the solar
system, the motion of comets is affected by the so-called
non-gravitational forces. It is a rocket-like force by the
outgassing of material from the nucleus during perihelion
passage. Marsden et al. (1973) introduced three component
Style II non-gravitational parameters, A1 , A2 , and A3 , which
are the radial, transverse, and normal acceleration components.
The transverse component parameter, A2 , provides information
about perturbations of the mean motion and the eccentricity.
The parameter A2 is almost always well determined for
periodic comets, because it is sensitive to the time of the orbital
elements, and is confirmed by several observations. Because
A1 can be determined by perturbations of the longitude
of perihelion, it is a poorly determined quantity. The
least well-determined parameter is A3 , which measures the
perturbations of the orbital inclination and of the line of nodes,
because the orbital-plane orientations in space are generally
small (Yoemans et al. 2005).
The non-gravitational motion of a short-periodic comet,
22P/Kopff, has been well-investigated because of its long-term
apparition. Yeomans (1974) found secular changes of the
quasi-regular appearance in the parameter A2 of 22P/Kopff.
Using the values of A2 , Sekanina (1984) studied the
precessional pattern of the spin axis along with some related
physical parameter of the nucleus. Rickman et al. (1987)
collected an enormous amount of astrometric data observed
between 1958 and 1983, and found a regular change with time
for A1 , A2 , and A3 . At least three apparitions are generally
applied to determine these non-gravitational parameters.
The cometary dust trails (Ishiguro et al. 2002) and neck-line
structures (Kimura & Liu 1977) record the history of the
position of comets ranging in age from tens of days to
centuries. A nearly edge-on view of 22P/Kopff showed
an intriguing dust structure extending parallel to the comet
orbit, but slightly away from the orbital plane (Ishiguro et al.
2007, hereafter Paper I). In section 3, we discuss the reason
why the nucleus was not on the extension of the elongated
dust structure.
By introducing the normal component
of non-gravitational motion with the dust ejection model
presented in Paper I, we consider how a non-gravitational
force develops the gap between the nucleus position and the
line-of-dust structure. Our approach is unique in that we apply
a non-gravitational force to the neck-line image, which should
record the passed position of the nucleus.
2. 22P/Kopff Image by CFHT/MegaCam Observation
An optical image of 22P/Kopff was obtained on 2003 July
31 at the 3.6-m CFHT, Mauna Kea, Hawaii. The prime-focus
camera MegaCam was applied to cover the extended dust
cloud. Detailed descriptions of the observation and a data
analysis are presented in Paper I. The observation was
scheduled for when the Earth crossed the orbit of 22P/Kopff;
that is, the viewing angle with respect to the comet’s orbital
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N. Miura et al.
[Vol. 59,
Fig. 2. Position of maximum brightness of the dust structure. The
nucleus of 22P/Kopff was at the point of origin. The horizontal and
vertical coordinates are right ascension and declination relative to the
position of the nucleus. The solid line is a linear fitting of the position
of the maximum brightness.
3. Interpretation of the Gap
Fig. 1. r 0 -band (wavelength 0.63 m) images of 22P/Kopff on 2003
July 31. These images are the standard orientation in the sky, that is,
North is up and East is to the left. The brightness limits and the contrast
were modulate in order to see the extended dust structure (a, upper)
and the proximity of the nucleus and its cone-shaped jet (b, lower),
respectively. The cross in figure 1a means the position of the nucleus,
and the solid line in figure 1b denotes the extension of the neck-line.
The box in figure 1a is the corresponding area of figure 1b.
plane was 2:ı 1. The obtained images are shown in figure 1.
The nucleus was in the bright coma, but the position was
remarkably on the north-side (figure 1a). The scaling and the
contrast were modulated in order to look at the proximity of the
nucleus (figure 1b) and the extended dust structure (figure 1a),
respectively. It was found that the faint southward jet was
embedded in a spherical coma. Moreover, a faint beard-like
structure extended to the southwest. From the image, it seems
that the position of the nucleus shifted in the counter direction
of the cone-shaped jet.
We applied 44 boxcar smoothing in order to reduce the
pixel-to-pixel noise, and plotted the maximum brightness of the
dust structure (figure 2). The good seeing condition (0:00 6–0:00 8)
as well as a well-designed observational condition (i.e. edge-on
view) enabled us to detect a subtle gap between the position
of the dust particles and the nucleus. We derived the distance
between the locus of the maximum brightness of the neck-line
and the nucleus by a linear fitting, and found that the nucleus
was shifted 3:008˙0:00 5 to the north.
Dust tubes lying along the comet orbits are called “dust
trails”. Neck-line structures are brightness enhancements by
particles ejected at a point (first node) away in true anomaly
from the observed point (second node) (Kimura & Liu 1977).
The initial dust shells, ejected from the nucleus spherically,
became ellipsoidal in shape by collapsing after the perihelion
passage on the orbital plane of the parent comet at the second
node. This behavior is understandable if we think that each
dust particle must cross the orbital plane of the parent comet
at the nodes of its orbit. As a result, it looks like a narrow
extended structure (neck-line structure). In Paper I, the narrow
extended dust structure presented in figure 1 is most likely
the neck-line; that is to say, the dust particles responsible for
the scattered light were emitted 409 days before the time of
the observation. The estimated particle size in the extended
structure is between 1 mm and 1 cm. What is significant in
this argument is that neck-line structure records the position of
the nucleus at a passed specific day. The particles distributed
on the ellipsoid must collapse on the “the past orbital plane of
nucleus”. Because our data were taken from an edge-on view
after perihelion passage, the vertical displacement was easily
detected.
Since planetary perturbations affect the orbital change of
both the nucleus and dust particles, it is hard to explain a
displacement by the planetary perturbations, because both of
these dust particles and the nucleus moved on the similar
trajectory. The Poynting–Robertson effect and the solar wind
drag force act on the orbits of the dust particles (Mukai 1985).
It is, however, unlikely that the dust cloud was nudged out by
the Poynting–Robertson effect or the solar wind drag, because
they act parallel to the plane of the dust orbits. In addition,
the solar-wind drag is negligible for large particles in the
22P/Kopff’s neck-line structure (Mukai & Yamamoto 1982).
Accordingly, it is likely that the normal component of the
non-gravitational jet accelerations might move the orbits of the
22P/Kopff nucleus toward the north with respect to the orbital
plane.
To test the hypotheses mentioned above, we carried out
a numerical simulation of the positions of the nucleus and
the dust particles. The motion of the nucleus is affected
New Approach to Study Non-Gravitational Motion of Comet Normal to the Orbital Plane
by not only the solar gravity, but also the non-gravitational
acceleration, whereas the motion of the dust particles,
which were ejected with non-zero velocity, are affected by
the solar gravity field reduced by the radiation pressure.
We applied non-gravitational parameters of the nucleus
A1 = 5:676160473407425 1010 AU d2 and A2 =
1:05620534539366 109 AU d2 .1 A3 is not available
from the web-site. Here, we supposed that A3 is variable,
ranging from 1 108 to +1 108 AU d2 at 1 109
AU d2 intervals. A dimensionless empirical function, g.r/,
was employed in order to produce the heliocentric dependence
of the acceleration. A dust emission model (Paper I) from the
22P/Kopff nucleus, which was obtained by fitting the dust trail
observed in 2002 May, was applied, i.e.,
h r iu2
h
;
(1)
Vej = V0 ˇ u1
AU
L9
Fig. 3. Modeled image of the dust structure with non-gravitational
acceleration. Here, we assumed that all dust particles were ejected from
a cone-shape jet symmetrically with respect to the Sun–comet axis. The
position of the nucleus is indicated by the cross.
where V0 = 150 m s1 is the reference ejection velocity of the
ˇ = 1 particles at a heliocentric distance of rh = 1 AU; u1
(= 0.5) and u2 (= 0.5) are power indices of the ˇ and rh
dependence of the ejection velocity. ˇ is the ratio of the solar
radiation pressure to the gravitational attraction. By definition,
ˇ=
KQpr
;
a
(2)
where K = 5:7 105 g cm2 . Assuming spherical particles
with a radius of a cm and a mass density of = 1 g cm3 ,
we can compute the ˇ values using the Mie code. Here,
we supposed Qpr = 1 and a 1 m. We assumed that
the dust particles were ejected sunward from a cone-shape jet
symmetrically with respect to the Sun–comet axis with a half
opening angle of w = 45ı . Assuming that the comet activity
was symmetric with respect to the perihelion, the production
rate of the particles emitted within a size range of a da=2
and a + da=2 at time t can be expressed by
rh .t/ k
a q
N.aI t/ da dt = N0
da dt
(3)
AU
m
in the size range of amin = 57 m and amax = 11 mm,
respectively. We fixed k = 3 and q = 3:25.
4. Results and Discussion
The resultant image is shown in figure 3. We can reproduce
the 3:00 8 gap between the position of the nucleus and the locus
of the maximum brightness of the neck-line. The obtained
A3 is +6 109 AU d2 , which implies that the nucleus
was accelerated toward the north. The estimated A3 over
the perihelion passage of 2002/2003 is consistent with the
value of +5 109 , extrapolated from observations during
1958–1983 (Rickman et al. 1987). The result suggests that the
pole orientation of 22P/Kopff might have been continuously
changing during the last half-century by precession (Sekanina
1984). The observed spherical coma, which might be caused
by ubiquitous mini-jets, was also duplicated by our simulation.
However, the prominent southward jet embedded in the coma
was not reproduced by our model, because we supposed
1
hhttp://ssd.jpl.nasa.gov/i.
Fig. 4. Modeled image similar to figure 3, but dust particles were
emitted from the southward jet. The solid line denotes the position
of the neck-line structure relative to the nucleus.
a cone-shape jet symmetrically with respect to the Sun–comet
axis. We carried out a simulation in which dust particles
were ejected southward normal to the orbital plane. Here,
we assumed w = 45ı , amin = 57 m and amax = 11 mm,
V0 = 150 m s1 . It was found that the neck-line structure
cannot be reproduced by the southward jet alone. Instead, the
beard-like structure extended to the southwest was obtained by
the southward jet as in figure 4.
From these results we can infer that: (1) the neck-line (which
records the past position of the nucleus) and the spherical coma
resulted from the cone-shape jet symmetrically with respect to
the Sun–comet axis, and (2) the southward jet produced the
beard-like structure.
5. Summary
In this paper, we considered an unusual image of the
22P/Kopff dust cloud; that is, the nucleus is 3:00 8 away from the
locus of the peak position of the neck-line structure. A feasible
scenario is that the inclination of the nucleus had been changed
by non-gravitational acceleration normal to the orbital plane.
LETTER
No. 2]
L10
N. Miura et al.
Based on this assumption, we estimated a non-gravitational
parameter, A3 , over the perihelion passage of 2002/2003. The
deduced A3 is consistent with the value extrapolated from
observations during 1958–1983. The result suggests that the
pole orientation of 22P/Kopff might have been continuously
changing during the last half-century.
The non-gravitational acceleration of comets has generally
been examined based on astrometric observations of nuclei.
Our new idea would open a new possibility for studying the
motion of comets, because the neck-line structures record the
passed orbital parameters of the parent bodies.
We thank Ingrid Mann for her critical reading of the
manuscript and helpful comments. CFHT/MegaCam data
were obtained by a joint project of CFHT and CEA/DAPNIA,
at the Canada-France-Hawaii Telescope (CFHT), which is
operated by the National Research Council (NRC) of Canada,
the Institut National des Science de l’Univers of the Centre
National de la Rechearch Scientifique (CNRS) of France,
and the University of Hawaii. This research was supported
by Ministry of Education, Culture, Sports, Science and
Technology, Grant-in-Aid for Scientific Research on Priority
Areas, “Development of Extra-solar Planetary Science”.
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