A&A 376, 316-324 (2001)
DOI: 10.1051/0004-6361:20010945
A study of the original orbits of ``hyperbolic'' comets
M. Królikowska
Space Research Centre of the Polish Academy of Sciences,
Bartycka 18A, 00-716 Warsaw, Poland
Received 7 March 2001 / Accepted 27 June 2001
Abstract
The problem of hyperbolic incoming orbits for
single-apparition comets is investigated. In this context the
effect of non-gravitational acceleration on cometary dynamics was
analyzed for the sample of 33 "hyperbolic'' comets. The orbital
elements of each cometary orbit were determined by the least
squares procedure based on positional observations. These
osculating orbital elements serve as a basis for the determination
of the incoming orbits. The new method of estimating the
uncertainties of
based on a statistical approach
to the orbit determinations is applied. For sixteen comets it was
possible to determine the non-gravitational effects and original
"non-gravitational'' orbits. We show that for almost all the comets
with detectable non-gravitational effects the incoming barycentric
orbits changed from hyperbolic to elliptical when the
non-gravitational effects were included. However, it was also
ascertained that excess velocity at great distance are not
cancelled by the non-gravitational acceleration itself. We show
that the non-gravitational effects are responsible for small
modifications of the osculating elements (mainly eccentricity),
which are sufficient, however, to obtain the elliptic original
orbits for comets recognized in the literature as hyperbolic.
Key words: solar system - comets - long-period - non-gravitational effects
It is generally accepted that long-period comets (e.g.
comets with periods greater than the traditional lower limit of
200 years) originate from the Oort cloud, from which their stable
orbits have been perturbed into the inner Solar System by the
cumulative effects of the galactic tidal field and the passage of
nearby stars or giant molecular clouds. Such comets possess nearly
parabolic incoming orbits described by positive reciprocals of
semimajor axes, e.g.:
.
However, in the sample of about three hundred long-period comets
with well determined orbits there are about 10% which have
hyperbolic incoming orbits (
). The problem of
the negative tail of the distribution of original binding energies
(proportional to
)
has been discussed in detail
in literature by many authors. First, the quality of astrometric
data, the method of data selection (including the problem of
weighting the individual measurements), and the applied method of
integration may all strongly affect the orbit computation of each
individual "hyperbolic'' comet (even comets with orbits ascribed as
1A - highest quality class). Second, since all the negative
excess energies are small, corresponding to roughly
-10-4 AU-1 in inverse semimajor axis, Marsden et al.
(1973) speculate that neglecting the non-gravitational
effects tends to produce original orbit which appear to be more
hyperbolic than they really are. In particular, it is possible
that most "hyperbolic'' comets with small perihelion distances
originate from the Oort's cloud of comets. The latter suggestion
is strongly supported by the fact that perihelion distances of two
comets with a greatest negative excess of
are
smaller than 0.9 AU.
On the other hand, Yabushita (1991) found that
the maximum excess velocity at great distance due to non-symmetric
outgassing from cometary nuclei is equal to 0.18 kms-1.
Thus, he argues that for at least 15 comets the calculated
original orbits are "highly hyperbolic'' and are not "compatible
with elliptical orbits modified by the non-gravitational
accelerations''. More recently, Bolatto et al. (1995)
concluded that the non-gravitational perturbation in a
near-parabolic comet's energy per orbital revolution is generally
smaller than about 10-4 AU-1. However these two papers
neglect the influence of variations of osculating orbital elements
on
when the non-gravitational effects are
implied from observational data. Present calculations show that
hyperbolic original orbits are produced by hyperbolic osculating
orbits resulting from positional data in the case of pure
gravitational motions.
There are several generally accepted reasons that extrasolar
comets exist. An important question is whether such comets could
be observed if they actually were observed. The modern theories (e.g.
Clube & Napier 1984; Duncan et al. 1987) for the formation of comets in the interstellar
environment predict many more extrasolar comets than bound comets.
However, the estimates of the expected rate for detection of
extrasolar comets are uncertain. McGlynn & Chapman (1989)
show that the rate of hyperbolic comets passing with perihelion
q<2 AU is 0.6 per year which implies that a total of about six
extrasolar comets should have been detected over the last 150
years. More recently, Sen & Rana (1993) argued that the
density of stars in the solar neighbourhood was overestimated by
McGlynn and Chapman. Using a value of 0.014 
pc-3Sen & Rana found that the expected number of detectable
interstellar comets per century is less than one. Hughes
(1991) gives an even smaller number of expected comets on
hyperbolic orbits. He argues that hyperbolic comets occur at a
rate of 0.00225 yr-1, which yields one such comet on average
every 450 yrs. It seems clear that at most only a few known
"hyperbolic'' comets could really have interstellar origins.
Nevertheless, using modern techniques for comet searches, the
detection rate of extrasolar comets will increase dramatically in
the future. Presently it is possible to weed out false members from the sample of
"hyperbolic'' comets.
In this paper the non-gravitational motions of individual
"hyperbolic'' comets from the Catalogue of Cometary Orbits (Marsden
& Williams 1997; hereafter MW Catalogue) are examined. The
aim of the present study is to prove that incoming orbits of
"hyperbolic'' comets for which it was possible to determine the
non-gravitational effects are almost all elliptical. Next, we show
that using objective statistical criteria for the remaining
"hyperbolic'' comets, excess energies derived for pure
gravitational motion are systematically smaller than those given
in the MW Catalogue. The new method of
uncertainty estimates are also used.
In their catalogue Marsden & Williams (1997)
listed 307 comets with periods longer than 200 yrs and with
well-determined orbits, e.g. comets belonging to quality classes 1 or 2 originally defined by Marsden et al. (1978). In this
sample, 29 comets have negative reciprocals of original semimajor
axes:
.
Furthermore, 16 long-period comets
with determined non-gravitational effects and unknown
(in the MW Catalogue) were taken into account. It turned
out, however, that only six of these 16 comets had negative
.
Thus, our sample consists of 35 candidates that
have hyperbolic incoming orbits. Since the original observations
were not available for four comets of the sample (1849 G2 Goujon;
1959 O1 Bester-Hoffmeister; 1895 W1 Perrine; 1911 S3 Beljawsky)
the present analysis includes 31 comets observed before 1997 (see
Tables 1 and 2). This sample was appended by two "hyperbolic''
comets discovered after 1997.
The orbital elements of each comet were recomputed on
the basis of archive observations available at the Minor Planet
Center (Cambridge, USA), with some additional data selected from
the literature. For comets observed during the period of 1900-1950
the data collected in Warsaw in cooperation with Slovakian
astronomers at the Astronomical Institute from Bratislava and
Tatranska Lomnica were also taken into account. The observations
used for the orbit determination in the present work are more
complete than the data used in the MW Catalogue (see Col. 5 of
Tables 1 and 2) except for two comets: 1898 V1 (Chase) and 1955 O1
(Honda). The observations for each comet were selected according
to the objective criteria elaborated by Bielicki & Sitarski
(1991).
It has been shown that barycentric elements of the orbit
determined at a distance above 150-200 AU change insignificantly
(Todorovic-Juchniewicz 1981). For this reason each comet is
followed from its position at a given epoch (see Table 3)
backwards (original orbit) and forwards (future orbit) until the
comet reaches a distance of 250 AU from the Sun. The barycentric
orbital parameters of incoming comets (before planetary
perturbations) and outgoing comets (whose orbits have been
perturbed) are called "original'' and "future'' quantities,
respectively. In the numerical calculations the equations of
motion are integrated in barycentric coordinates using recurrent
power series method (Sitarski 1989) taking into account the
perturbations by all nine planets. The respective values of
original, osculating and future reciprocals of semimajor axis are
given in Cols. 2-4 of Tables 1 and 2. The first line for each
comet gives respective values of reciprocals of semimajor axis
taken from the MW Catalogue.
The most hyperbolic original orbit in the sample was Comet Sato
1975 X1 (
in units of 10-6 AU-1used in this paper) with q=0.86, and Comet Honda 1955 O1
(
)
with q=0.89 (see Table 1). In the
next section we show that both these comets have strongly positive
values of
if the non-gravitational solutions are
considered.
Our
values differ substantially from those
published by Marsden and Williams in their catalogue. These
discrepancies in
are caused![[*]](/icons/foot_motif.gif)
by an enlargement of the observational data
and by adoption of the objective data selection procedure
(Bielicki & Sitarski 1991). The influence of data
selection is seen for the first and third comet of Table 2 where
the more restrictive selection procedure results in less negative
values of
for these comets. Generally, the
present
values are less negative than those
given in the MW Catalogue (see Tables 1 and 2). Out of seventeen
comets listed in Table 2, roughly half (nine comets) turned out to
have positive
values and none had a more
negative value of
.
Four comets with more
negative values of
are given in Table 1, but for
all of them the non-gravitational effects change the original
orbits from hyperbolical to elliptical (see the next section).
The most positive value of
belongs to
Comet 1940 S1 Okabayashi-Honda (Table 2) whose orbit was calculated
on the basis of 40 positional observations spanning a three months
period. This orbit was poorly determined (the worst of whole
sample), and significantly different from that given in the MW
catalogue which was mainly caused by the three last observations
of January 3rd, 1940. According to Kresak (1992) estimates of
internal errors in 1/a for orbits of quality classes 1A-2B,
this comet does not belong even to the lowest quality class 2B.
We have estimated the uncertainties of
values.
This was made by means of the following numerical simulations. The
orbital elements (hereafter standard, or nominal elements) were
determined by an iterative, least squares process from equations
based on the positional observations. This "best solution'' orbit
is one of the potential orbits allowed by observations of the
comet. Since we do not know the true cometary orbit, we will
construct a set of randomly selected orbits which all fit well
with the observations used for the orbit determination. The orbit
selection procedure was adopted from Sitarski (1998) where
the entire method was described in details. According to this
procedure, we randomly selected a set of 500 orbits belonging to
the same celestial body, e.g., a series of orbits in which the
comet "could'' move. A random number generator with Gaussian
distribution was applied to select random values of the right
sides of the elimination equations described in Sect. 3 of
Sitarski's paper (1998). Then new values of orbital
elements were calculated from the elimination equations. Each set
of six orbital elements describing the chosen orbit differs from
the nominal values, but they always fit the observations with an
accuracy defined by residuals calculated according to Sitarski's
method. The mean rms residuals of randomly selected orbits
are given in the last column of Tables 1 and 2. Next, for each
random orbit we calculated the original orbit. Thus, for each
comet we obtained the distribution of original reciprocals of
semimajor axes (Figs. 1 and 2). The normal distribution was then
fitted to the histogram of original
.
Examples of the Gaussian function fitting to the
obtained
distributions are given in Fig. 1 for
Comet 1997 P2 (Spacewatch), and in Fig. 2 for three other comets.
In this way we calculated the mean value of
and
its standard deviation 
.
The goodness of fit was tested by
a 
method and fitting by normal distributions was very
good in all cases. The respective values of
with their 's are given in Tables 1 and 2.
One finds that the planetary perturbations also contribute to the
calculated uncertainties. However, for all the comets from the
sample the uncertainties in original and future orbits are similar
to uncertainties in randomly selected orbits and reflect the
internal errors of orbital elements derived from astrometric data.
This is illustrated in Fig. 1 for Comet 1997 P2 (Spacewatch) which
underwent the strongest energy perturbation (among the group of
analyzed objects) during its passage through the planetary system.
Before its discovery this comet passed close to Jupiter (at a
distance of about 0.65 AU) and in the middle of 2000 it passed
2.7 AU from Saturn losing
(see Table 2), mostly due to these two
close encounters. Thus, this comet suffers planetary perturbations
of almost a factor of 4 greater than the typical (mean) value of
energy perturbations of long-period comets (about
;
Yabushita 1979). For all 500 randomly selected orbits
we obtained, the calculated uncertainties of
and
are equal to 36 and 21, respectively. This means
that the observational uncertainties usually do not affect our
estimates of planetary perturbations and do not contribute
substantially to the uncertainties of calculated
and
.
Therefore, uncertainties of
given in Tables 1 and 2 are closely connected with the
orbital quality classes 1A-2B.
![\begin{figure}
\par\includegraphics[width=11.8cm,clip]{ms1250f1.eps}\end{figure}](/articles/aa/full/2001/34/aa1250/Timg65.gif) |
Figure 1:
Distributions of reciprocals of semimajor axes for Comet
1997 P2 (Spacewatch). The left panel shows distribution of
heliocentric
for 500 dynamical clones of the
osculating orbit (from the normal distribution given by black
dots) which were then integrated backwards and forwards to the
distance of 250 AU from the Sun. In this way the distributions of
and
for barycentric orbits
were derived and these are shown in the middle and in the right
panels, respectively. The Gaussian functions fitted to the
and
distributions are
represented by open circles. |
| Open with DEXTER |
![\begin{figure}
\par\includegraphics[width=11.3cm,clip]{ms1250f2.eps}
\end{figure}](/articles/aa/full/2001/34/aa1250/Timg66.gif) |
Figure 2:
Distribution of original reciprocals of semimajor axes
(in units of 10-6 AU-1) derived from
integrations of 500 dynamical clones of the osculating orbit of
three comets. In each panel, the left histogram corresponds to the
pure gravitational solution and the right one - to the
non-gravitational solution. The Gaussian functions fitted to the
distributions are shown with squares; the mean
values of Gaussian distributions
 and their uncertainties are given in the tenth column of Table 1. |
| Open with DEXTER |
Traditionally, the term "non-gravitational effect'' has
been reserved for the outgassing acceleration of the comet nucleus
caused by asymmetric sublimation of cometary volatiles. The method
to determine non-gravitational effects on the orbital motion of a
comet was proposed by Marsden et al. (1973). In the commonly
accepted formalism the three orbital components of the
non-gravitational acceleration acting on a comet are symmetric
relative to perihelion and can be written as:
 |
(1) |
where
F1,F2,F3 represent the radial, transverse and
normal components of the non-gravitational acceleration,
respectively. The analytical function g(r) simulates the ice
sublimation rate as a function of the heliocentric distance rand was based on an empirical water sublimation curve given by
Delsemme & Miller (1971). The non-gravitational effects
play an important role in the vast majority of short-period
comets. At least three consecutive apparitions are needed to
precisely determine the independent non-gravitational parameters
A1, A2, A3 within an investigated time interval. This means
that for one-apparition comets these effects will be poorly
determined.
However, a greater difficulty was pointed out by
Yabushita (1991, 1996). He showed that the
non-gravitational accelerations of some nearly parabolic comets
could be inconsistent with the above physical model of the comet
nucleus if only H2O sublimation was assumed (represented by the
function g(r)). In particular, he argued that for the Comet 1989 Q1 the recoil due to outgassing of H2O is too small (almost by
a factor of 10) to be compatible with the observed value of the
non-gravitational acceleration unless the nuclear density is as
low as 0.1 g cm-3 and nucleus diameter is smaller than 4 km.
There are strong reasons to suppose that the sublimation from
nearly parabolic comets may be controlled by CO or N2 molecules
rather than the less volatile H2O molecules. It seems
especially true for comets with perihelia distances outside of
3 AU, where carbon monoxide production significantly exceeds
production of H2O. Moreover, coming "fresh'' from the Oort cloud
these comets should exhibit a much higher content of the most
volatile ices (e.g. CO, CO2) that are substantially or
completely depleted during its first perihelion passage.
Thus, Yabushita proposed a convenient form for the dependence of
acceleration on the heliocentric distance based on the assumption
that the sublimating molecules are CO instead H2O:
 |
(2) |
These two formulae for non-gravitational accelerations (Eqs. (1) and (2)) are considered in the present investigation.
For sixteen comets from the sample it was possible to obtain the
non-gravitational solutions due to outgassing of H2O volatile
molecules from the cometary surface. For five of them only the
radial component of the non-gravitational force was determined;
for the remaining comets the radial, A1, and transverse, A2,
components were determined with reasonable accuracies (and in two
cases also the normal, A3, component). Starting from such
"non-gravitational'' solutions the original "non-gravitational''
orbits could be calculated.
Results of the new determinations of the
,
taking
into account the non-gravitational effects, are presented for 16
comets in Table 1. The models given in the square brackets
represent non-gravitational parameters determined with poor
accuracy. Distributions of original reciprocals of semimajor axes
for two comets (1975 X1 Sato and 1955 O1 Honda) with the most
hyperbolic original orbits in the MW Catalogue, and for a comet
with small perihelion distance
(1956 R1
Arend-Roland) are shown in Fig. 2.
One can see (Table 1) that in almost all cases, the original
orbits changed from hyperbolic to elliptical when the
non-gravitational effects represented by A1, A2 (and A3)
parameters were included. This means that the
become positive for the "non-gravitational'' orbits. The most
interesting solution was obtained for Comet 1975 X1 Sato, the most
"hyperbolic'' comet from the MW Catalogue. For this comet only A1 could be determined with a reasonably small error. However, it seems that A1 plays an important role in
the orbit determination for this comet, changing the osculating
orbit from open to closed, which - as a consequence - generates
an elliptic original orbit, too. The analogous tendency is evident
for the next three solutions with A1 alone (M1986P1n1,
M1971E1n1, and M1991Y1n); only in the case of Comet 1996 N1 is the
modification insufficient to change the original
orbit from open to close curve.
Only two negative values of "non-gravitational''
are present in Table 1. The first is quoted above, Comet
Brewington 1996 N1, for which it was possible to
obtain only A1 and is presented only to show that the non-gravitational
solution does not help in this case. In the considered group of
comets potentially exposed to the action of the non-gravitational
effects this one seems to be the best candidate for an
interstellar object. The second exception was Comet
1996 E1 NEAT. For this best quality osculating orbit (1A class in
the MW Catalogue) we derived a slightly hyperbolic original orbit
in the pure gravitational solution. Next, we considered two
"non-gravitational'' solutions: with two (A1 and A2) and with
all three non-gravitational parameters (Model M1996E1n1 and Model
M1996E1n2, respectively). In the first case the original orbit
still seems to be hyperbolic, but with a slightly smaller absolute
value of
.
For the second non-gravitational
solution the original orbit became elliptical, but its
value is very small. This second solution
indicates that if non-gravitational effects really play an
important role in the motion of Comet 1996E1 (which is suggested
by the substantial decrease of the residuals), the observational
data are insufficient to determine all three non-gravitational
parameters, e.g. the full "non-gravitational'' solution. The
analogous calculations for the remaining comets from Table 1 were
performed, and in all cases where the value of A3 parameter was
determinable, osculating and original orbits were more elliptical
than those given in Table 1, but like in the Comet 1996E1 case,
A3 determinations were very poor. Only the full
"non-gravitational'' solution for two comets (1998 P1 Williams,
1968 N1 Honda) seem to be realistic, and for this reason are given
in Table 1.
The present calculations confirm the Marsden et al. (1978)
hypothesis that original orbits of comets of small and moderate
q appear to be negative due to disregard of the
non-gravitational forces during calculations. It is important to
stress that influence of the non-gravitational effects on
is caused by different orbital elements (in
particular eccentricity) derived for the non-gravitational
solutions and gravitational solutions on the basis of
observational data. The case of Comet 1975 X1 Sato is an adequate
example (Table 3). The non-gravitational perturbations in the
comet's energy per orbital revolution are significantly smaller
than
AU-1 for all the considered comets,
which is in agreement with previous results (e.g. Yabushita
1991; Weissman 1979; Bolatto et al. 1995).
Finally, we examined how the function f(r) (Eq. (2))
approximates the non-gravitational accelerations of all the
sixteen comets with detectable non-gravitational effects.
Generally, it turned out that the "non-gravitational'' orbits for
comets sublimating CO volatile molecules from their surfaces can
be fit to observational data with the same residuals as orbits for
comets undergoing the H2O sublimation, however the
A2*/A1* ratios substantially changed in some cases. Comet
1946 U1 is a good example: in the model 1946U1n the A2parameter dominated over A1 almost by a factor of two, whereas
in the model 1946U1c both components (A1* and A2*) are
comparable. The orbital elements and values of
are also fairly similar. Therefore, in Table 4 the
non-gravitational determinations of
are
presented for some representative cases only. One can see that for
only two comets (models 1998P1 c and 1993A1 c), the
non-gravitational parameters based on CO sublimation are a better
fit to observations than those based on standard H2O
sublimation.
The main conclusions can be summarized as follows:
1. Dividing the present sample of the long-period comets which
hypothetically are of interstellar origin in two subgroups
according to criterion of q<2.5 AU and q>2.5 AU, we have 27
members (82%) of the former group and only 6 (18%) of the
latter. This reflects the trend visible in the full sample of
long-period comets with well determined orbits where about 80% of
comets have q<2.5.
2. The values of original semimajor axis (and values of orbital
elements) are susceptible to selection criteria. For this reason
the objective criteria described by Bielicki & Sitarski
(1991) are recommended. Using such criteria we found that
10 comets from the entire sample of 33 hyperbolic comets (30%)
have positive
values (see Tables 1 and 2). We
also showed that for the remaining "hyperbolic'' comets, excess
energies derived for pure gravitational motion are systematically
smaller than those given in the MW Catalogue, with the exception
of four cases. All four comets with more negative values of
change their original orbits from hyperbolic to
elliptic (see point 3) when non-gravitational effects are taken
into account.
3. For 16 comets from the sample, non-gravitational solutions were
obtained. In most cases the incoming barycentric orbits changed
from hyperbolic to elliptic when the non-gravitational effects
were included. There are only two exceptions to this rule. For
both of them the tendency of
to decrease is
evident, but not enough to modify the shape of their orbit from an
open to closed curve (see Table 1). This result is in formal
disagreement with Yabushita (1991) who drew a conclusion
that for all comets which have an original orbit with excess
hyperbolic velocity (calculated under the assumption of pure
gravitational motion) larger than 0.18 kms-1 it is
impossible to explain its hyperbolicity in terms of
non-gravitational effects. However, he neglected the fact that
osculating orbital elements are changed when the
non-gravitational effects are applied to observational data.
Including non-gravitational effects into the orbit determination,
we obtain osculating elements somewhat different from those
determined with an assumption of gravitational motion alone. In
particular, the pure gravitational osculating orbits are more
hyperbolic than in the non-gravitational cases, which have a
direct influence on the shape of their original orbits. The
present analysis reveals that these small modifications of the
osculating elements (mainly eccentricity), are sufficient to
obtain elliptic original orbits for comets which previously -
under the assumption of pure gravitational motion - appeared to
be "hyperbolic''. It was also shown that all non-gravitational
parameters, even A1 alone, play an important role in orbit
determination from positional observations.
4. Determining the cometary orbit from observational data in the
pure gravitational case and in the non-gravitational case, and
then integrating both orbits backwards in time we obtained
differences between original reciprocal semimajor axes
of up to 2637 (in units of 10-6 AU-1,
see Table 1). This means that non-gravitational orbits could
significantly affect the Oort peak, i.e. concentration of
long-period comets in the range of
AU-1 which was previously missed in literature (e.g.
Weissman 1985; Bolatto et al. 1995). All the
authors considered the non-gravitational perturbation in comet's
energy per orbital revolution (in fact significantly smaller than
AU-1), whereas the problem lies in
different non-gravitational osculating orbits resulting from the
observational data.
5. It is difficult to judge which relation between the
heliocentric distance and the acceleration (Eqs. (1) and (2))
better characterizes the non-gravitational trends in the
astrometric data of one-apparition comets. To do this, additional
photometric and spectroscopic data at great distances from the sun
are needed. The "non-gravitational'' orbits for comets sublimating
CO volatile molecules from their surfaces are fit to astrometric
data with almost the same residuals as orbits for comets
undergoing H2O sublimation; only in two cases did the
non-gravitational parameters based on CO sublimation give a
slightly better fit to observations than those based on standard
H2O sublimation.
6. Present analysis reduced the sample of 33 putative "hyperbolic''
comets to only 10 objects with negative original reciprocals.
Acknowledgements
The author is deeply indebted to Professor Grzegorz Sitarski for
helpful discussions on various aspects of this investigations.
This work was supported by the Polish Committee of Scientific
Research (the KBN grant 2.P03D.002.18).
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Copyright ESO 2001
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4.372
.1642
)
)
![\begin{figure}
\par\includegraphics[width=11.8cm,clip]{ms1250f1.eps}\end{figure}](/articles/aa/full/2001/34/aa1250/img65.gif)
![\begin{figure}
\par\includegraphics[width=11.3cm,clip]{ms1250f2.eps}
\end{figure}](/articles/aa/full/2001/34/aa1250/img66.gif)
![\begin{displaymath}g(r) = \alpha \left( r/r_{\rm0} \right) ^{-2.15} \left[ 1 +\left( r/r_{\rm0}
\right) ^{5.093} \right] ^{-4.6142},
\end{displaymath}](/articles/aa/full/2001/34/aa1250/img68.gif)
O sublimation
[AU]
)
