Icarus 254 (2015) 92–101
Contents lists available at ScienceDirect
Icarus
journal homepage: www.elsevier.com/locate/icarus
Crater-diameter distribution on Comets 9P and 81P and potential
meteoroid streams crossing their orbits
O.V. Ivanova a,⇑, L. Neslušan b, J. Svoreň b, Z. Seman Krišandová b
a
b
Main Astronomical Observatory of NAS of Ukraine, Akademika Zabolotnoho 27, 03680 Kyiv, Ukraine
Astronomical Institute of the Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
a r t i c l e
i n f o
Article history:
Received 17 November 2014
Revised 18 March 2015
Accepted 19 March 2015
Available online 27 March 2015
Keywords:
Comets
Meteors
Cratering
a b s t r a c t
We attempt to answer two questions concerning the impacts of stream meteoroids on the nuclei of
Comets 9P/Tempel 1 and 81P/Wild 2: firstly, how many streams cross the orbits of both comets and,
secondly, what is the index of the differential mass distribution of impactors, s, when we assume that
a prevailing number of the craters on the surfaces of cometary nuclei were created by stream meteoroids? We found that 110 and 129 potential streams originating from comets likely cross the orbits of
9P and 81P, respectively (and 103 potential streams cross the orbit of 1P/Halley, for comparison). If we
consider the more compact streams originating from asteroids, the 9P and 81P pass through such streams
15 664 and 65 368 times. Neither these large numbers of passages imply, however, enough large impactors to excavate the whole observed variety of craters on studied comets. For all craters on 9P and 81P,
s ¼ 2:09 0:01 and s ¼ 2:25 0:03, respectively. The craters on 81P seem to be, however, excavated by
the impactors from four discernible sources. For two numerous enough sources we find s ¼ 5:6 0:2 and
s ¼ 5:2 0:5. The difference between the indices for the set of all craters and the sets of their partial
groups obviously implies an unknown cosmogonic consequence.
Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction
Cometary nuclei contain a volatile material which is released,
and usually causes a cometary activity, when the nucleus
approaches the Sun. Although there is no doubt that the activity
occurs mainly due to the heating of the comet-nucleus surface
by the solar radiation, it sometimes occurs at a large heliocentric
distance, without any correlation with the amount of received
energy from the central star (e.g. Svoreň, 1988). Thus, another
mechanisms of the activity, seen as the sudden minor or larger outbursts of cometary brightness, have to exist and represent a task
for the astronomers to describe them in more detail.
Except for the exothermal chemical reactions, the impacts of
meteorites were suggested as, perhaps, the most serious alternative mechanism that causes the extraordinary cometary activity.
Or, possibly, it can trigger another alternative mechanism (a
chemical reaction). This mechanism does not primarily depend
on the position of the comet with respect to the Sun. (It depends
on the heliocentric distance in the sense that the number density
of potential impactors increases with a decreasing heliocentric
⇑ Corresponding author.
E-mail address: [email protected] (O.V. Ivanova).
http://dx.doi.org/10.1016/j.icarus.2015.03.023
0019-1035/Ó 2015 Elsevier Inc. All rights reserved.
distance because of a smaller volume of the relevant heliocentric
sphere.) Thus, it can be efficient also at large heliocentric distances.
A cometary nucleus can be impacted by projectiles of various
origin. This can be a small asteroidal object, a relatively large fragment of the nucleus of other or the same comet, or a tiny fragment
of such a nucleus, i.e. a meteoroid. Since the spatial density of
meteoroids belonging to an abundant stream is larger than that
of sporadic meteoroids or asteroids, we can deduce that just the
probability of collision of a given comet nucleus with a stream
meteoroid is the largest of all possible kinds of impactors. In the
first part of our work, we map the potential streams crossing the
orbits of selected comets (see below; comet 1P/Halley is also investigated for the sake of comparison). In more detail, we try to
answer the question of how many streams may cross the orbit of
a comet under investigation along its orbit and what the encounter
heliocentric distances and velocities are.
It is worth to note that the stream-meteoroid impacts onto a
cometary nucleus were suggested to explain the cometary activity
several times in the history. Bosler and Roure (1937) suggested
that the passage of comet 2D/Biela through the Leonid meteoroid
stream could have caused its break up. This possibility was properly investigated by Babadzhanov et al. (1991). They confirmed
that the minimum distance between the orbits of Biela and the
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
Leonid stream was very small. They further found that the comet
was close to the main bulk of the Leonid stream in year 1832.
Williams et al. (1993) investigated whether the brightening of
Comet 1P/Halley at large distances was due to collisions with
meteoroids from its own stream. They determined the mean
velocity of collisions between the meteoroids and comet’s nucleus,
which occurred to be about 4.0 km s1, far from negligible. A half of
collisions happened when the comet was close to its perihelion.
Recently, Guliyev et al. (2014) considering 116 outbursting comets
calculated the distribution of their nodes in planes of 68 meteor
showers from the Cook’s catalog. They confirmed that the
outbursts could be caused by collisions of these comets with the
streams.
In addition to a lot of studies of meteor showers occurring after
the collisions of meteoroid streams with the Earth’s atmosphere,
there have also appeared several studies on the meteoroid streams
in the atmosphere of Venus (Beech and Brown, 1995; Beech, 1998;
Christou, 2004, 2010) and Mars (Christou and Beurle, 1999;
Treiman and Treiman, 2000; Ma et al., 2002; Christou, 2010;
Christou et al., 2012). Hughes (2002) compared terrestrial,
Cytherean, and lunar impact cratering. A study providing the
streams at the orbit of comet 29P/Schwassmann-Wachmann 1
was recently published by Neslušan (2014).
The various effects of the encounters between meteoroid
stream and comet nucleus have also been studied. Matese and
Whitman (1994) investigated a correlation between an episodic
outgassing observed at the new Oort cloud comets, which are most
likely to have experienced large non-gravitational forces, and their
passage through well-known meteor streams. According to these
authors, the impacting grain-sized meteoroids penetrate any surviving mantle and cause a catalytic increase in volatility. If there
exists a trapped gas, either in subsurface pockets or at a crystalline/amorphous-ice interface, then it is much more likely to
erupt and expose adjacent underlying fresh volatiles when the
meteoroids impact the comet, diminishing the tensile strength of
the subsurface crystalline layer.
A temporary dust tail activity of asteroidal-cometary object
7968 Elst-Pizarro (133P/Elst-Pizarro) was also suggested to occur
due to an impact-induced processes generated by multiple collisions with a debris cloud associated with another asteroid (Toth,
2000). Specifically, Toth showed that the most probable candidate
for the parent body of the impactors to generate the temporary
outburst activity is asteroid 427 Galene. But Hsieh et al. (2004)
concluded that the most probable mechanism is the seasonal outgassing activity which lead to low velocity dust emission, which
explains the long-term activity of 133P. However, we pointed in
this paper that the impact events could be important to trigger
cometary activity. Toth (2001) further attempted to explain the
initiation and/or triggering of the breakup of comet C/1999 S4
(LINEAR) nucleus by impact-induced events from possible larger
debris or a debris cloud dispersed around the orbits of known
asteroids. He found that C/1999 S4 crossed the orbits of seven
known asteroids from December 1999 to March 2000. The breakup
of the comet occurred a short time before July 23, 2000.
Beech (2001) investigated the possibility that the activity of
comet 72P/Denning-Fujikawa, over the past 200 years, has been
governed by impacts suffered by the comet as it moves through
the main-belt asteroid region. The outburst activity of the comet
may be impact-modulated in the sense that small-object impacts
might trigger the explosive release of gases trapped in subsurface
cavities.
An investigation of the interaction between the meteoroid
streams and small bodies of the Solar System crossing the corridors
of the streams is difficult because of an impossibility to detect, in a
usual way, the concerning impacts and their consequences.
However, there is an indirect way to gain at least certain
93
information about such impacts, namely via an analysis of craters
on asteroids and cometary nuclei. Concerning the latter, the first
craters were actually observed in situ within the famous missions
VEGA 1, VEGA 2, and Giotto to the Halley comet at its perihelion
passage in 1986. Then, the cometary surfaces covered with craters
were found by the Stardust spacecraft passing the nucleus of
comet 81P/Wild 2 in the beginning of 2004 (Kirk et al., 2005)
and Comet 9P/Tempel 1 in 2005 as the Deep Impact mission
(McFadden et al., 2001; Hampton et al., 2004; A’Hearn and Deep
Impact Project Team, 2005; Thomas et al., 2007). Other detailed
images of the surface of 9P were taken during the repeated visit
of the Stardust spacecraft to its nucleus in 2011 as the StardustNExT mission (Veverka et al., 2012; Schultz et al., 2012; Thomas
et al., 2013).
If a comet nucleus passes through a meteoroid stream, it is
impacted by the meteoroids of this stream repeatedly, once at
every orbital revolution around the Sun. The stream represents a
source of projectiles, which are likely characterized with a unique
mass distribution. Since the encounter, and, therefore, the impact
velocities, are practically the same for all impactors from the
stream, the size distribution of excavated craters can be expected
to correlate with the mass distribution of meteoroid-stream
impactors. Hence, we can gain an implication to the latter by analyzing the size distribution of the craters observed on the surface of
the studied comet nucleus. In the second part of our work, we
assume that an essential number of the craters on the cometary
nuclei were created by the meteoroid-stream impacts and do the
analysis of the distribution of crater diameters observed in situ
on the two comets mentioned above: 9P/Tempel 1 (hereinafter
9P) and 81P/Wild 2 (hereinafter 81P).
Our analysis is completed with a rough estimate of the mean
time between two collisions of 9P nucleus (81P nucleus) with the
meteoroid of given size, which is the member of a typical major
stream.
2. The streams crossing the comet orbits
It is well known that the Earth crosses, during every orbital revolution around the Sun, the orbital corridors of several meteoroid
streams associated with comets or, in a lesser degree, with asteroids. An analogous crossing can be expected at every object in
the Solar System.
Comets 9P and 81P were observed in situ and the distribution of
the sizes of their impact craters is relatively well known. We,
therefore, focus our attention on these two comets. Some information is also given for the famous comet 1P/Halley, revolving around
the Sun in the retrograde orbit.
A stream orbiting the Sun in a corridor situated far from the
Earth’s orbit is hard to be detected. We, however, know that a
majority of comets in orbits approaching the orbit of our planet
have created a meteoroid stream around its orbit, so that we can
suppose the existence of a stream around the orbit of each periodic
comet. An uncertainty due to this simplifying assumption is twofold. On the one hand, there can be some comets without their
streams or the streams are only low numerous. This circumstance
can reduce the number of real streams. On the other hand, some
comets have not been discovered yet. Hence the number of potential parent bodies is underestimated.
Asteroids are known to produce some meteoroid particles in a
much lesser degree than the comets. Nevertheless at least two
asteroids, 3200 Phaethon and 196,256, evidence that the asteroidal
meteoroid streams exist and, thus, these streams cannot be omitted either in our analysis. We note, 3200 Phaethon is the parent
body of major shower Geminids and 196,256 (former designation
94
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
2003 EH1) can be the parent (or one of parents; see Neslušan et al.,
2013) of major shower Quadrantids.
Let us now to deal with the potential cometary meteoroid
streams. As their potential parents, we consider all known comets
with an orbital period, P, up to 1000 years. Further, we take into
account the circumstance that the given comet under study, 9P
or 81P, can collide with a meteoroid of the stream if the orbit of
its parent comet approaches the orbit of the studied comet within
0.15 AU. This limit follows from the most distant approach of a
major-meteor-shower parent body to the Earth’s orbit. Such the
approach is found for the Orionid-corresponding arc of the orbit
of 1P/Halley, which approaches the orbit of the Earth on this arc
as close as 0.155 AU (Neslušan et al., 1998) (we will round the
value of 0.155 AU off to 0.15 AU).
Based on the 16th edition of the Catalogue of Cometary Orbits
(Marsden and Williams, 2005), we selected altogether 360 comets
having P 6 1000 years (the comets under study inclusive) and calculated the closest distances between the post-perihelion as well
as pre-perihelion arc of each of these comets and the orbit of the
studied comet. We found that there are 110 approaches within
the limit of 0.15 AU to the orbits of 87 comets in the case of 9P,
129 such the approaches to the orbits of 102 comets in the case
of 81P, and 103 approaches to the orbits of 97 comets in the case
of 1P. So, approximately one quarter of all known periodic comets,
with their potential streams, approach the orbit of each of our
three studied comets.
The distribution of the heliocentric distances of approaches and,
at the same time, of the encounter velocities for each of the three
studied comets is shown in Fig. 1. The velocity distributions for the
Comets 9P and 81P in the prograde orbits are similar. The encounter velocities with both comets vary from the values of a few kilometers per second up to about 30 km s1 (very few encounter
velocities exceed this upper limit). Comet 1P in the retrograde orbit
can be hit by meteoroids with considerably larger velocities, up to
about 90 km s1. A much larger energy is delivered by the impacts
on its surface in comparison with 9P and 81P.
Considering 518,811 three and more opposition asteroidal
orbits with the appropriate perihelia and aphelia for the
approaches of 9P and 81P to these orbits within 0.15 AU, we found
that there are 130,287 such the approaches of 9P and 429,606
approaches of 81P. We again search for the approaches to the
pre-perihelion and post-perihelion orbital arcs separately, since
the studied comets could approach and actually approached some
orbits twice. The numbers of approached orbits were 97,601 and
262,303 by 9P and 81P, respectively. The orbits of asteroids considered were selected from the database of the Minor Planet Center of
the International Astronomical Union downloaded on February 10,
2015.1 This version of the database contained 677,230 asteroidal
orbits in total.
The Geminid stream however indicates that the asteroidal
streams are more compact than those originating from comets.
(An evidence of the extraordinary compactness of Geminids in
comparison with several other major cometary showers can be
seen, e.g., in the paper by Neslušan et al. (1995), Fig. 11.) Hence,
the threshold distance of the approach of 0.15 AU, used for the
approaches to the cometary parents, is inappropriate for the asteroidal parents. Instead, the limit of 0.02 AU, which is the minimum
approach of 3200 Phaethon to the Earth’s orbit (Neslušan, 2005),
can be assumed to be more realistic. Within this limit, Comet 9P
approached to 11,495 asteroidal orbits 15,664 times and 81P to
36,789 asteroidal orbits 65,368 times.
Among several asteroids approaching the Earth’s orbit, only two
associate a major meteoroid stream. The common feature of the
1
http://www.minorplanetcenter.net/iau/MPCORB.html.
two asteroids, 3200 and 196,256, is their occurrence in the orbit
with a small perihelion distance. Specifically, 3200 Phaethon’s perihelion distance is about 0.14 AU and 196,256’s perihelion distance
decreases below 0.15 AU two times during its 8200-year lasting
libration cycle (Neslušan et al., 2013). This fact indicates that the
meteoroid particles can be released from the asteroidal surface
especially due to a thermal stress of material, intensively heated
by the solar radiation around the perihelion and cooled around
the aphelion. If we consider only the asteroids in orbits with the
perihelion distance lower than 0.15 AU as the active parent bodies
of meteoroid stream, then Comet 9P (81P) approaches only three
(five) times to the orbit of such asteroid within the 0.15 AU limit
and there is no approach within the more appropriate limit of
0.02 AU. Therefore, if the thermal-stress hypothesis is correct,
Comets 9P and 81P (either else) do not likely pass through the real
asteroidal streams often.
3. Mass distribution of impactors – theory
The craters of the sizes observed on the surfaces of Comets 9P
and 81P obviously occurred due to relatively large impactors.
There are much more tiny than large particles in a meteoroid
stream. However, each comet could pass through a stream repeatedly during a long period. In a long term evolution of the cometnucleus surface, a number of impact craters could be created and
these craters accumulated during a long period could be observed
in situ by space probes and further studied.
If we assume that the craters on the cometary nuclei are excavated mainly by the impacts of stream meteoroids, then knowing
the size-distribution of craters we can, in principle, estimate the
mass distribution of impactors. So, we derive the characteristics
of the distribution for the meteoroids in the streams.
We can expect that the mass distributions of individual streams
differ from each other. Moreover, there are many streams crossing
the orbit of each comet under study. To proceed with our estimate
of the impactor-mass distribution, we temporarily adopt another
simplifying assumption that there is only a single stream with
the differential mass distribution that can be described by a single
power-law
n dm ¼ No ms dm;
ð1Þ
where n is the number of meteoroids with masses in the interval
from m to m þ dm; No is a constant, and s is the slope index of the
distribution.
The size of a crater depends on the kinetic energy, W, of the
impactor. The meteoroids of a given stream encounter the studied
comet with essentially the same velocity, therefore the kinetic
energy of various bodies in the stream differs only with respect
to their mass, m. If we denote the encounter velocity of the j-th
stream by v j , the differential energy distribution of impactors from
this stream, which corresponds to the mass distribution (1), can be
given as
nWj dm ¼
1
No v 2j m1s dm:
2
ð2Þ
To relate the energy distribution to the known distribution of
crater diameters on the surface of a given comet under study, we
use the semi-empirical formula relating the diameter, D, with the
energy, W, as (Housen et al., 1979; see also Fernández, 1990)
D ¼ K i W 1=3 ;
ð3Þ
where K i is a constant of proportionality. We note that Fendyke
et al. (2013) recently determined experimentally the relation
between D and W, also in the form D ¼ AW d . Shooting various
high-velocity (1–7 km s1) projectiles into ice, they found
95
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
(a)
encounter velocity [km/s]
50
40
30
20
10
0
1.5
2
3
2.5
3.5
4
4.5
5
heliocentric distance [AU]
35
(b)
encounter velocity [km/s]
30
25
20
15
10
5
0
1.5
2
2.5
3
4
3.5
4.5
5
5.5
heliocentric distance [AU]
Fig. 1. The heliocentric distances of the approaches of potential meteoroid streams, within 0.15 AU, to the orbits of investigated comets, 9P/Tempel 1 (plot a), 81P/Wild 2 (b),
as well as 1P/Halley (c). The distance is shown as a function of the relative velocity of approaching objects at the points of the minimum distance of their orbits. The dashed
curve shows the dependence of the heliocentric velocity of the investigated comet on its heliocentric distance.
d ¼ 0:33 0:03 from their own experimental data and
d ¼ 0:303 0:006 from the older data from experiments performed
by Shrine et al. (2002) and Burchell and Johnson (2005). We see that
the found values of d are practically the same as the power-index of
1/3 found by Housen, which appears in relation (3).
Using relation (3) and summing through all the streams (and
thus abandoning the single-stream assumption), distribution (2)
acquires the following form
nW dD ¼
3N o
s1
2
K 3s6
i
X
v 2s2
D53s dD:
j
ð4Þ
j
Let us further denote the diameter of the smallest crater by Dmin and
the largest crater by Dmax . Integrating Eq. (4) we obtain the
corresponding cumulative distribution
NW ðDÞ ¼
3N o
2
s1
ð6 3sÞ
K 3s6
i
X
v 2s2
j
j
D63s D63s
min
ð5Þ
96
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
90
(c)
encounter velocity [km/s]
80
70
60
50
40
30
20
10
0
1
2
3
4
5
6
heliocentric distance [AU]
Fig. 1 (continued)
for s – 2, where N W ðDÞ is the number of all meteoroids with the
kinetic energy needed to excavate a crater with the diameter equal
to or smaller than D, down to Dmin .
In the fitting of the theoretical distribution to its observed counterpart, it is advantageous to establish the relative size of craters
with respect to the maximum size, Dmax . This is achieved by sube ¼ D=Dmax . If N all is the number of all impactors (that crestitution D
ated all observed craters), then Eq. (5) can be re-written into a
relative form
e 63s D
e 63s
min
e ¼ Nall D
NW ð DÞ
e 63s
1D
ð6Þ
min
for s – 2.
In Section 2, we demonstrated that the number of streams
crossed by each of the studied comets is relatively large.
However, the largest craters could be excavated only by the relatively large impactors and we can ask if there is enough large
objects in the streams to create all observed craters. At the end
of this section, we describe the way to give a rough estimate of
the impact probability for the impactors of various size.
Since we do not know the specific size- or mass-distribution of
the stream or streams creating the craters on the surfaces of studied comets, we use the overall mass distribution of interplanetary
material impacting the Earth, which was compiled and published
by Ceplecha (1992). In more detail, we fit a linear function,
log 10 ðNÞ ¼ Alog 10 ðm½kgÞ þ B, to the cumulative number of all
objects, N, down to the mass m (Ceplecha’s Fig. 1). The fit of the
interval from 1012 kg to 106 kg yields A 0:845. The
corresponding differential distribution is given by relation (1) with
s ¼ 1 A and constant N o gauged in the way described below.
Let us to establish a ‘‘typical major stream’’. We assume, it corresponds to a major meteor shower in the Earth’s atmosphere with
the hourly zenithal rate equal to 50. From a given observational
station, there can be seen all meteors, which pass through a circular atmospheric area with the radius of Ra 100 km and the center
above the station. The mean period of activity of a major shower at
the Earth can be assumed to be, roughly, 10 days, whereby the
studied comet can be expected to pass through the typical stream
about the same time. Hence, there are N a ¼ 50 24 10 12; 000
meteors passing through the area during a single passage of the
Earth through the typical stream. Since the mean radius of comet,
Rc , is smaller than Ra , the number of meteors colliding with the
cometary nucleus, N 3 , is obtained after the reduction of N a by
the factor of R2c =R2a .
According to Ceplecha (1992), the typical mass of the prevailing
number of visually observed meteors is about 1 g. Using this information, we can gauge constant N o in the distribution (1) as
A1
N o ¼ N 3 =m3
, where m3 ¼ 103 kg. Then, using (1), we can estimate the number of the objects of given mass colliding with the
nucleus of given comet per time unit, when the cometary nucleus
passes through the typical stream. Or, we can calculate the mean
time, ht 2 i, between two collisions of the object of given mass with
the nucleus. Specifically, the mean time ht2 i could be calculated as
the reciprocal value of the number per unit time if the nucleus of
studied comet permanently passed through the stream. Because
the nucleus is assumed to pass through our typical stream only
during 10 days, then time ht 2 i can be expected, in reality, longer
than the simple reciprocal value about the factor P=ð10mÞ, where
m is the number of passages of the nucleus through a stream in
its complete orbit around the Sun and P is its orbital period given
in days.
4. Mass distribution of impactors on 9P and 81P
4.1. Single power-law
The cratering on the surfaces of Comets 9P and 81P was studied
by Thomas et al. (2013) and Kirk et al. (2005), respectively. Based
on these studies, we used the lists of the crater diameters (Kirk,
2014; Thomas, 2014) that are summarized in Table 1. Using the
method of least squares, we fit distribution (6) to the corresponding cumulative distribution constructed for each of the Comets 9P
and 81P. The constructed cumulative distributions (full-circles)
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
Table 1
The lists of crater relative diameters of Comets 9P/Tempel 1 and 81P/Wild 2 measured
by Thomas (2014) and Kirk (2014), respectively.
No.
D ½Dmax No.
D ½Dmax Craters on 9P
1
2
3
4
5
6
0.032
0.045
0.062
0.088
0.134
0.184
7
8
9
10
11
0.257
0.380
0.499
0.734
1.000
Craters on 81P
1
2
3
4
5
6
7
8
9
10
11
12
0.121
0.145
0.191
0.220
0.243
0.249
0.249
0.254
0.260
0.266
0.272
0.283
13
14
15
16
17
18
19
20
21
22
23
0.295
0.341
0.410
0.422
0.457
0.474
0.503
0.699
0.821
0.884
1.000
together with their power-law fits (dotted curves) are shown in
Fig. 2. We found that s ¼ 2:04 0:01 for the craters on Comet 9P
and s ¼ 2:14 0:03 for those on comet 81P.
The maximum crater diameter on the surface of the 9P nucleus
is 821 m and that on the 81P nucleus 1730 m. The impactors
excavating these craters had to be large. Considering value
K i ¼ ð0:0223 0:0032Þ m J0.33 found by Fendyke et al. (2013),
the masses of these impactors, calculated according to relation
(3), were as high as 7:6 104 and 7:3 105 kg for 9P and 81P,
respectively, even if we consider the maximum reasonable
encounter speed (see Section 2) of 30 km s1.
If the mass distribution of impactors is found in the way
described at the end of Section 3, then time ht 2 i as the function
of mass for Comets 9P and 81P and for m ¼ 100 (a typical number
of passages through the cometary streams) is that shown in Fig. 3.
If we considered only the meteoroid streams originating from
comets, a significant statistical probability of the impactors creating the maximum craters occurs only at the timescale exceeding
the common survival of a comet in short-period orbit. So far, the
timescale exceeds the age of the Solar System. If the streams originating from asteroids are included, time ht2 i in Fig. 3 must be
reduced about two to three orders of magnitude, approximately,
because m giving the passages through the asteroidal streams is
about two to three orders of magnitude larger than that for cometary streams. The largest impactors would then collide with 9P and
81P in a timescale comparable to the age of the Solar System, but
both comets have not resided in their short-period orbits more
than several million or, at maximum, several ten million years.
Thus, there is still a deficit of large impactors.
This fact evokes the question of what was the actual source of
the impactors modeling the surfaces of the studied comets?
There are several possible reasons to explain the found deficit of
large impactors. Fendyke et al. (2013) derived the parameters K i
and s using the relatively small projectiles and, hence, low energies, up to about 240 J. However, the energy needed to excavate
the largest crater on 81P was about 6:6 1014 J. It is possible that
the Fendyke et al.’s values of K i and s cannot be used in the extreme
extrapolation. The mass of the impactor depends on the third
power of K i , therefore the mass of the largest on-81P impactor
would decrease to only about 680 kg if we assumed that the value
of K i is one order of magnitude underestimated for the case of
high-energetic impacts.
97
The used mass distribution by Ceplecha (1992) was derived for
many groups of objects that impacted the Earth and Moon. It is
possible that the distribution is different in the region between
the orbits of Mars and Jupiter, where the orbits 9P and 81P are situated, and this can be the second possible reason for the relatively
large number of large craters on 9P and 81P. Maybe, a numerous
streams consisting of large boulders, created at the collisions of
asteroids or occurring after a disruptions of rubble-pile objects,
exist in this region.
A difference between the Ceplecha’s mass-distribution law and
the above-found mass distribution of impactors seems to eliminate
this possibility, however. While s ¼ 1 A ¼ 1:845 according to our
fit of the Ceplecha’s law, the masses of impactors are distributed by
the law with a higher value of s, implying even a smaller relative
abundance of more massive objects in comparison with the
Ceplecha’s distribution. And, we will derive, in Section 4.2, even a
much higher value of s for the multiple sources of objects impacting comet 81P.
Or, if neither of two above-mentioned reasons is relevant, we
are forced to accept that the craters are the surface features from
the era, when the cometary nuclei resided in the scattered disk
beyond Neptune, which is regarded as the primary source of
short-period comets. The craters would then imply an existence
of a numerous population of meter-sized bodies in the scattered
disk.
At the moment, we have not a strong enough argument to support any hypothesis. A further research to solve the puzzle of the
too numerous cratering observed in situ on the surfaces of 9P
and 81P will be desirable.
4.2. Multi-source impacts on 81P
In Fig. 2, we can see a different smoothness of the observed distributions of 9P and 81P. While the distribution of 9P almost perfectly follows the single power-law, the distribution of 81P is not
so smooth. We can speculate that the craters of the first comet
were formed by the impacts originating from a single source, i.e.
from a single stream or from sporadic background. On the other
hand, the craters of the second comet seem to be created by the
impacts from four sources, most probably four dominant streams,
each with its own mass distribution.
In more detail, the four smallest craters in the lower plot of
Fig. 2 constitute a group which can be associated with the impactors from the first source, another 11 craters (6 craters) are the
group associated with the impactors from the second (third)
source, and three largest craters are the group associated with
the impactors from the fourth source. The numbers of craters in
the first and last groups are too small, therefore we further do a
numerical analysis only for the second (A, hereinafter) and third
(B) group. It is not clear if the last (11th) crater of group A belongs
to this group or should rather be associated with group B, as its
first crater. Because of this circumstance, we consider it in both
groups at the same time (it is the last crater of group A and the first
crater of group B).
We again construct, separately for group A and group B, the
unity-normalized cumulative distribution of crater diameters, in
the relative unit of the actual largest crater, and fit the powerlaw (6). The fitted distribution is illustrated with the dotted curves
in Fig. 4, the upper plot for group A and the lower one for group B.
The indices of the distributions are found to be equal to
s ¼ 4:9 0:1 and s ¼ 4:6 0:5 for group A and group B, respectively. These values are more than twice of that (s ¼ 2:14) found
for the set of all measured craters of 81P.
It seems that the mass distributions in various meteoroid
streams are self-similar, with a similarity index having a value of
4.5 to 5.0, but different in the absolute scale. The index
98
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
unit-normalized cumulative number
1
9P
(a)
0.8
0.6
0.4
0.2
0
observed
non-corrected
corrected
0
unit-normalized cumulative number
1
0.1
0.2
0.3
0.4 0.5 0.6
D [D max ]
0.7
0.8
0.9
1
81P
(b)
0.8
0.6
0.4
0.2
0
observed
non-corrected
corrected
0
0.2
0.4
0.6
D [D max ]
0.8
1
Fig. 2. The cumulative distribution, normalized to unity, of the crater diameters of two investigated comets, 9P/Tempel 1 (upper plot) and 81P/Wild 2 (lower plot). The
diameters are given as multiples of that of the largest crater.
decreases down to the value of 2.0, when the particles of several
streams are combined. The reason for the self-similarity and scale
difference can originate in the formation of the stream parent bodies (cometary nuclei) in the once existing proto-planetary disk. A
relation between the index values and individual processes of the
small-body formation are still unknown.
5. Crater erosion effect
In the previous section we found the index of the mass distribution of all meteoroids hitting a given cometary nucleus. This
distribution can, however, differ from the distribution corresponding to the actually observed crater-diameter distribution due to an
erosion of older craters by younger impacts. In what follows, we
will try to estimate the influence of this effect on the distribution.
If a smaller crater is created in an area of the surface where a
larger crater is situated, then both the craters can later be
observed. However, if a larger crater is created in the area occupied
by a smaller crater, then the latter is destroyed. Later, we can
observe only the larger of both craters.
On every cometary nucleus, there are more smaller than larger
craters and we can assume that half of the craters smaller than a
99
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
12
10
log10(<t2> [yr])
8
6
4
2
0
-2
9P
81P
-4
-4
-3
-2
-1
0
1
log10(m [kg])
2
3
4
5
Fig. 3. The dependence of mean time interval, ht 2 i, between two impacts of the objects of given mass, m, on the surfaces of Comets 9P/Tempel 1 and 81P/Wild 2 assuming
that the comets were permanently in their current orbits.
considered crater are, statistically, created before it and the other
half after it. Hence, the observed number of craters (craters after
the correction for the discussed effect) having diameters in the
range from D to D þ dD obviously is
S dD ¼
p
4
D2 n dD ¼
3p N o
2
sþ1
K 3s6
i
X
v j2s2 D73s dD
ð7Þ
j
where SðDÞ is the net area of all craters with diameters from Dmin to
D; Stot is the net area of all observed craters, and n is the original,
non-corrected number of impacts creating the craters with diameters from D to D þ dD.
The cumulative distribution, from diameter Dmin to D,
corresponding to the differential area-distribution (7), reads
SðDÞ ¼
3p N o
2
sþ1
ð8 3sÞ
K 3s6
i
X
v 2s2
j
D83s D83s
min
ð8Þ
j
for s – 8=3. Since Stot ¼ SðDmax Þ, we can use relation (7) to calculate
the latter and with the help of it, as well as the already established
e ¼ D=Dmax , the corrected differential distribution can
substitution D
be given as
e¼
nW2 d D
63s X
3No
Dmax
v 2s2
j
e min Þ K i
2s ð1 D
j
h
i
e min Þ D
e 53s þ D
e 136s d D:
e
ð1 2 D
ð9Þ
The corrected cumulative distribution, N corr ðDÞ, can be obtained
after the integration of the last distribution
e ¼
Ncorr ð DÞ
63s X
3No
Dmax
v 2s2
j
e min Þ K i
2s ð1 D
j
"
#
e min 1 2D
1
63s
63s
146s
146s
e
e
e
e
D
D
D min þ
D min
14 6s
6 3s
ð10Þ
for s – 2 and s – 7=3. We can equivalently express this distribution
in a relative form, with the help of the total observed number of craters N obs ¼ N corr (1). In this case, one finds
h
e ¼ N obs ð14 6sÞ 1 2 D
e 83s D
e 63s
e 63s D
Ncorr ð DÞ
min
min
i
e 146s D
e 146s
þð6 3sÞ D
min
h
i1
e 83s 1 D
e 63s þ ð6 3sÞ 1 D
e 146s
ð14 6sÞ 1 2 D
min
min
min
ð11Þ
for s – 2 and s – 7=3.
Fitting distribution (10) to its counterpart constructed on the
basis of data in Table 1 for each studied comet, we found
scorr ¼ 2:09 0:01 for Comet 9P and scorr ¼ 2:25 0:03 for all craters of 81P. If the fitting is done for the partial groups, A and B
(see Section 4.2), of 81P, then we obtain scorr;A ¼ 5:6 0:2 and
scorr;B ¼ 5:2 0:5. The fittings are also illustrated in Figs. 2 and 4
(solid curves).
We can see only a tiny difference between the non-corrected
and corresponding corrected distributions at both cometary nuclei
when all craters are considered. The similarity is also apparent
comparing the numerical values of the corresponding indices s
and scorr and, especially, their corresponding determination errors
implying the same quality of corrected and non-corrected fits, in
the case of all craters.
Since the effect of erosion we deal with destructs small craters
at a higher rate than larger craters, it must more influence the distribution with a higher s-value. This is also apparent from the
increase of scorr in comparison to s of the non-corrected distribution
of partial groups A and B of 81P craters. This increase is significantly larger (cf. 5.6 vs. 4.9 for group A and 5.2 vs. 4.6 for group
B) than that for the set of all 81P craters (cf. 2.25 vs. 2.14).
Anyway, the effect of the erosion crater by crater does not seem
to be a dominant agent re-shaping the cometary surface. This conclusion is consistent with the observation of smooth terrains on the
100
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
unit-normalized cumulative number
1
81P - source A
(a)
0.8
0.6
0.4
0.2
0
0.55
observed
non-corrected
corrected
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
D [Dmax]
unit-normalized cumulative number
1
81P - source B
(b)
0.8
0.6
0.4
0.2
0
0.55
observed
non-corrected
corrected
0.6
0.65
0.7
0.75 0.8
D [Dmax]
0.85
0.9
0.95
1
Fig. 4. The cumulative distribution, normalized to unity, of the crater diameters of comet 81P. Two distributions of craters excavated by the impacts assumed to originate in
two well-defined sources, A and B (see Section 3), are shown. The diameters are given as multiples of that of the largest crater.
surface of Comet 9P (e.g., Thomas et al., 2013) and with re-shaping
the cometary surfaces at outgassing. Thus, these surfaces are largely different, in this aspect, from the surfaces of, e.g., the Moon
or large inactive asteroids, where several generations of craters
covering the whole surface exist simultaneously. The cometary
activity itself evidently changes the cometary surfaces more than
cratering.
6. Summary
We fitted the power law to the cumulative distribution of crater
diameters of two comets, 9P/Tempel 1 and 81P/Wild 2, which were
observed in situ within the Stardust space missions. The surface of
9P appears to be covered with the craters excavated by the impactors whose masses are well distributed according to a single power
law. If the erosion of the older craters by younger impactors is
taken into account, the index of the differential distribution
corresponding to this power law equals 2:09 0:01. The effect of
the erosion is not significant at 9P nucleus, since the index is very
similar, equal to 2:04 0:01, if we ignore the erosion.
For the sake of comparison, we also fitted the single power law
to the distribution of diameters of all measured craters on the surface of comet 81P. The index of the corresponding differential distribution is 2:25 0:03 (if the effect of crater erosion is not taken
O.V. Ivanova et al. / Icarus 254 (2015) 92–101
into account, the index is, again, very similar to the last value; it
equals 2:14 0:03). Hence, we see that the indices for both studied
cometary surfaces, when considered as a whole, do not differ from
each other significantly.
However, the distribution of crater diameters on the surface of
81P appears to be better approximable by an analytical behavior,
when we divide the craters into four groups and fit the single
power law separately to the diameter distribution of the craters
within a given group. Two of the four groups have enough craters
at least for rough statistics. The indices of corresponding differential distributions are scorr;A ¼ 5:6 0:2 and scorr;B ¼ 5:2 0:5
for groups A and B, respectively. These indices are considerably
higher than those for the set of all craters. Likely, this fact points
out an unknown cosmogonic feature in formation of the parent
bodies of meteoroid streams, from which the impactors originated.
Thus, the increase of the index for the partial groups of craters
should be taken into account in the scenarios of the formation of
small bodies in the Solar System.
Considering the known mass distribution of small interplanetary bodies, we found that there are not enough large objects, even
in the meteoroid streams, to explain the observed number of craters on the surfaces of in situ-observed Comets 9P and 81P. Thus,
the source of appropriate impactors is unknown and a further
research is needed to solve this problem.
Acknowledgements
The authors are extremely thankful to Dr. Randolph L. Kirk and
Dr. Peter C. Thomas for providing them with the exact data characterizing the craters on the studied comets. They also acknowledge
support from VEGA – the Slovak Grant Agency for Science (Grants
No. 2/0031/14 and 2/0032/14) and the implementation of the project SAIA – the Slovak Academic Information Agency.
References
A’Hearn, M.F., Deep Impact Project Team, 2005. The Deep Impact project. Highlights
of Astronomy 13, 746.
Babadzhanov, P.B. et al., 1991. The Leonids, Comet Biela and Biela’s associated
meteoroid stream. Month. Not. Roy. Astron. Soc. 253, 69–74.
Beech, M., 1998. Venus-intercepting meteoroid streams. Month. Not. Roy. Astron.
Soc. 294, 259–263.
Beech, M., 2001. Comet 72P/Denning-Fujikawa: Down but not necessarily out.
Month. Not. Roy. Astron. Soc. 327, 1201–1207.
Beech, M., Brown, P., 1995. On the visibility of bright venusian fireballs from Earth.
Earth Moon Planets 68, 171–179.
Bosler, J., Roure, H., 1937. La Comète de Biéla et l’essaim des Léonides. J. Observ. 20,
105–111.
Burchell, M.J., Johnson, E., 2005. Impact craters on small icy bodies such as icy
satellites and comet nuclei. Month. Not. Roy. Astron. Soc. 360, 769–781.
Ceplecha, Z., 1992. Influx of interplanetary bodies onto Earth. Astron. Astrophys.
263, 361–366.
Christou, A.A., 2004. Prospects for meteor shower activity in the venusian
atmosphere. Icarus 168, 23–33.
Christou, A.A., 2010. Annual meteor showers at Venus and Mars: Lessons from the
Earth. Month. Not. Roy. Astron. Soc. 402, 2759–2770.
Christou, A.A. et al., 2012. Orbital observations of meteors in the martian
atmosphere using the SPOSH camera. Planet. Space Sci. 60, 229–235.
101
Christou, A.A., Beurle, K., 1999. Meteoroid streams at Mars: Possibilities and
implications. Planet. Space Sci. 47, 1475–1485.
Fendyke, S., Price, M.C., Burchell, M.J., 2013. Hydrocode modelling of hypervelocity
impacts on ice. Adv. Space Res. 52, 705–714.
Fernández, J.A., 1990. Collisions of comets with meteoroids. In: Lagerkvist, C.I.,
Rickman, H., Lindblad, B.A. (Eds.), Asteroids, Comets, Meteors III. pp. 309–312.
Guliyev, A.S., Kokhirova, G.I., Poladova, U.D., 2014. Comet outbursts and the meteor
showers. Meteoroids 2013, 263–265.
Hampton, D. et al., 2004. The NASA Deep Impact mission instrument suite and
opportunities for ground-based collaborative observations. In: American
Astronomical Society Meeting Abstracts #204, Bulletin of the American
Astronomical Society, vol. 36, pp. 675–675.
Housen, K.R. et al., 1979. Asteroidal regoliths. Icarus 39, 317–351.
Hsieh, H.H., Jewitt, D.C., Fernández, Y.R., 2004. The strange case of 133P/ElstPizarro: A comet among the asteroids. Astron. J. 127, 2997–3017.
Hughes, D.W., 2002. A comparison between terrestrial, Cytherean and lunar impact
cratering records. Month. Not. Roy. Astron. Soc. 334, 713–720.
Kirk, R.L., 2014. private comm.
Kirk, R.L. et al., 2005. Topography of the 81/P Wild 2 nucleus derived from Stardust
stereoimages. In: Mackwell, S., Stansbery, E. (Eds.), 36th Annual Lunar and
Planetary Science Conference, Lunar and Planetary Science Conference, vol. 36,
p. 2244.
Ma, Y. et al., 2002. The velocity distribution of periodic comets and the meteor
shower on Mars. Astron. Astrophys. 394, 311–316.
Marsden, B., Williams, G., 2005. Catalogue of Cometary Orbits, XVIth ed. IAU Central
Bureau for Astronomical Telegrams/Minor Planet Center, Smithsonian
Astrophysical Observatory, Cambridge, MA (USA).
Matese, J.J., Whitman, P.G., 1994. Meteoroid streams as probes of the subsurface
regions of comets. Icarus 109, 258–266.
McFadden, L.A. et al., 2001. The Deep Impact discovery mission: Update. In: AAS/
Division for Planetary Sciences Meeting Abstracts #33, Bulletin of the American
Astronomical Society, vol. 33, pp. 1148–1148.
Neslušan, L., 2005. The potential meteoroid streams crossing the orbits of terrestrial
planets. Contrib. Astron. Observ. Skalnate Pleso 35, 163–179.
Neslušan, L., 2014. The meteoroid streams crossing the frequently outbursting
comet 29P/Schwassmann-Wachmann. Planet. Space Sci. 101, 162–169.
Neslušan, L., Svoreň, J., Porubčan, V., 1995. A procedure of selection of meteors from
major streams for determination of mean orbits. Earth Moon Planets 68, 427–
433.
Neslušan, L., Svoreň, J., Porubčan, V., 1998. A computer program for calculation of a
theoretical meteor-stream radiant. Astron. Astrophys. 331, 411–413.
Neslušan, L., Hajduková, M., Jakubík, M., 2013. Meteor-shower complex of asteroid
2003 EH1 compared with that of Comet 96P/Machholz. Astron. Astrophys. 560.
A47, 10 pages.
Schultz, P.H., Hermalyn, B., Veverka, J., 2012. The Deep Impact crater as seen from
the Stardust-NExT mission. In: Lunar and Planetary Science Conference, Lunar
and Planetary Science Conference, vol. 43, p. 2440.
Shrine, N.R.G., Burchell, M.J., Grey, I.D.S., 2002. Velocity scaling of impact craters in
water ice over the range 1–7.3 km s1. Icarus 155, 475–485.
Svoreň, J., 1988. Analysis of observations of long-period comets at maximum
heliocentric distances. Contrib. Astron. Observ. Skalnaté Pleso 17, 7–20.
Thomas, P.C. et al., 2007. The shape, topography, and geology of Tempel 1 from
Deep Impact observations. Icarus 191, 51–62.
Thomas, P. et al., 2013. The nucleus of Comet 9P/Tempel 1: Shape and geology from
two flybys. Icarus 222, 453–466.
Thomas, P.C., 2014. private comm.
Toth, I., 2000. Impact-generated activity period of the asteroid 7968 Elst-Pizarro in
1996: Identification of the asteroid 427 Galene as the most probable parent
body of the impactors. Astron. Astrophys. 360, 375–380.
Toth, I., 2001. Impact-triggered breakup of comet C/1999 S4 (LINEAR):
Identification of the closest intersecting orbits of other small bodies with its
orbit. Astron. Astrophys. 368, L25–L28.
Treiman, A.H., Treiman, J.S., 2000. Cometary dust streams at Mars: Preliminary
predictions from meteor streams at Earth and from periodic comets. J. Geophys.
Res. 105, 24571–24582.
Veverka, J. et al., 2012. A Second Ecounter with 9P/Tempel 1: Overview of the
Stardust-NExT results. LPI Contrib. 1667, 6171.
Williams, I.P. et al., 1993. Collisions between the nucleus of Comet Halley and dust
from its own meteoroid stream. Month. Not. Roy. Astron. Soc. 260, 43–48.