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Transcript
Interstellar comets
How many we should have seen and when we will see them
Tim de Graaff
SN: 5886724
Bachelor thesis Physics: 12 EC
7/5/2012 - 4/7/2012
Universiteit van Amsterdam
Anton Pannekoek institute
Supervisor: Prof. C. Dominik
July 4th 2012
1
Abstract
A discrepancy exists between the number of observed dynamical interstellar comets on
hyperbolic orbits, and the theoretical determined value. This report investigates this
discrepancy by determining how earlier theoretical values were determined, and uses new
estimations to obtain a new value for the expected rate of interstellar comets passing within
2 AU from the sun, where they can be observed. The techniques was largely based up the
work of McGlynn and Chapman (1989) and Sen and Rana (1993). A discrepancy with their
values was found upon re-calculating their results, in which the expected rate of interstellar
comets is lower than originally calculated by the authors named above. Simulations done in
this research have obtained a new value for the expected rate of
interstellar comets per year. This results in a number of 0.8 comets that should have been
observed in the past 170 years, which explains the observed number of zero interstellar
comets on hyperbolic orbits. Using these results, it was determined that we should observe
one ISC, on average, every 200 years. The odds of observing a comet in the next 900 to 1400
years comes quite close to 100%.
Chapter 1
Populair wetenschappelijke samenvatting
Kometen zijn grote bollen van ijs en stof, die zichtbaar worden als ze in de buurt van de zon
komen. Een deel van dat ijs gaat dan sublimeren, waardoor er een regio van gas (een
zogenoemde ‘coma’) ontstaat rondom de komeet. Door de zonnewind van onze ster wordt
een deel van dit gas weggeblazen, waardoor we een staart zien verschijnen aan de komeet.
Kometen zijn heel lang geleden ontstaan, in de tijd dat de planeten gevormd werden. In
die tijd zijn vele kometen in de buurt van de grote gasplaneten gekomen (Jupiter, Saturnus,
Uranus en Neptunus) die de baan van deze kometen kunnen verstoren. Hierdoor kunnen
kometen zelfs het zonnestelsel uitgeslingerd worden.
In dit onderzoek gaan wij er van uit dat andere sterrenstelsels ook op deze manier
kometen verliezen. Hierdoor moet er een grote groep kometen tussen de sterren hangen,
deze noemen we dan ook “interstellaire kometen”. Omdat die groep kometen waarschijnlijk
erg groot is, is het geen gek idee dat er af en toe zo’n komeet door het binnenste van het
zonnestelsel komt vliegen, waar wij hem kunnen observeren. We zien het verschil tussen
een interstellaire komeet, en een komeet die uit ons eigen zonnestelsel komt, door de sterke
hyperbolische baan die de komeet volgt, en de hoge snelheid die interstellaire kometen
hebben ten opzichte van onze eigen kometen.
In de literatuur is te vinden dat we in de afgelopen 170 jaar een aantal van deze kometen
hadden moeten zien, maar we hebben er in werkelijkheid nog nooit een waargenomen. Daar
klopt dus iets niet! In dit project is onderzocht of het vreemd is dat we nog nooit zo’n
komeet gezien hebben.
De aanpak was als volgt: met een simulatie programma is het zonnestelsel nagebootst,
met daarin een grote groep kometen. Vervolgens is er gekeken hoeveel van deze kometen
het zonnestelsel uitgeslingerd zijn, en hoeveel kometen er in de Oort wolk zijn gekomen. De
Oort wolk is een enorme wolk van kometen die zich rondom het zonnestelsel bevind.
2
Ondanks het feit dat deze wolk nog niet is waargenomen, weten we uit onderzoek naar de
kometen die uit dat gebied komen dat er zo’n
kometen in moeten zitten.
Met deze gegevens kunnen we het aantal kometen dat uit het zonnestelsel is geslingerd
in zijn gehele levensloop eenvoudig bepalen. Uit de simulatie bleek namelijk dat er 445 van
de originele 1175 gesimuleerde kometen uit het stelsel gevlogen zijn, terwijl er slechts 9 in
de Oort wolk zijn geplaatst. Dus per Oort wolk komeet zijn er ongeveer 50 kometen
weggevlogen (445/9). Als we dit vermenigvuldigen met het aantal kometen in de Oort wolk
op dit moment, weten we hoeveel kometen er in totaal verloren zijn gegaan, namelijk
.
Met dit gegeven, en een ingewikkelde natuurkundige formule, volgt uit berekeningen
direct het aantal kometen dat we hadden moeten zien in de tijd dat we er al naar zoeken (nu
ruim 170 jaar). Het uiteindelijke aantal dat we hadden moeten zien, was 0.8. Omdat dit
minder dan 1 is, betekent dit resultaat dat het niet vreemd is dat we nog nooit een
interstellaire komeet gezien hebben.
Gegeven dit resultaat is het eenvoudig te berekenen dat we gemiddeld elke 200 jaar een
interstellaire komeet zouden moeten waarnemen. Dit betekent echter niet dat we er een
zullen zien binnen 30 jaar. Wel kunnen we berekenen dat de kans om er een te zien bijna
100% is, gedurende de komende 900 tot 1400 jaar. Het kan dus zijn dat we er morgen een in
de lucht zien verschijnen, maar het kan ook best nog 1000 jaar duren.
Conclusie: gezien de resultaten is het helemaal niet vreemd dat we nog nooit een
interstellaire komeet hebben gezien, en we verwachten gemiddeld 1 zo’n komeet te zien per
200 jaar.
3
Table of contents
1. Populair wetenschappelijke samenvatting
2
2. Introduction
2.1 Comets and belts
2.2 A very odd comet
2.3 Interstellar comets
2.4 Main questions
5
5
5
6
6
3. Theory
3.1 Formation of the Oort Cloud
3.2 Expected rate of extrasolar comets
7
7
8
4. Research and Results
4.1 Assumptions and parameters
4.2 Simulations
4.3 Results
9
9
10
13
5. Conclusions
14
6. Discussion
6.1 On the research
6.2 On the missing interstellar comets
14
14
15
Thank you notes
17
References
17
Appendix 1
18
4
Chapter 2
Introduction
2.1 Comets and belts
In the past few millennia, humans have spotted comets in the skies as fuzzy balls of light
with a tail. Before the invention of binoculars and telescopes, comets seemed to appear out
of nowhere, after which they gradually disappeared from the skies. 2000 years ago, roman
astronomers thought of comets as omens of impending doom. Plagues, wars, failed harvests
and even family feuds were often blamed on comets (Ridpath 1985). Centuries later, with
the use of telescopes, humanity discovered that comets were not messages from the Gods,
but in fact small solar system bodies consisting mostly of ice and dust.
A comet consists of a (relatively small) nucleus ranging from hundreds of meters to
several tens of kilometers across. When the nucleus of ice, dust and small rocks comes close
to the sun, solar radiation evaporates some of the icy volatiles, creating a huge atmosphere
around the nucleus called a coma. The sun’s solar wind and radiation pressure blow parts of
the coma away, resulting in a large tail which points away from the sun.
Comets, while sometimes passing through the inner solar system, are usually confined to
the outer regions of the system. In 1950, Oort postulated a cometary belt beyond 50.000
astronomical units (AU) and up to 150.000 AU, filled with comets that were put in those
large orbits by planetary perturbations and the influence of nearby stars and the galactic tide
(Oort, 1950). This galactic tide, as well as passing stars/gas clouds, can sometimes perturb
Oort Cloud comets such that they are ejected from the system, or pass through the
planetary region of the solar system (Dones et al. 2004) In the planetary region, they either
get ejected from the system by the gas giants or obtain new orbits with a perihelion distance
closer to the sun. Many such comets have been observed when they entered the inner solar
system.
The number of comets with radius r > 1 km in the Oort Cloud has been given several
values in literature, ranging between
and
comets (Levison et al. 2010,
Weissman 1983, Tremaine 1993).
In 1951, Kuiper postulated another belt of comets which is now known as the Kuiper Belt.
The Kuiper extends from roughly 30 to 50 AU and, like the Oort Cloud, consists of comets
which, presumably, were created in the inner solar system in its early history. The belt
consists of approximately 1010 Kuiper Belt objects (KBO) with r > 1 km (Luu & Jewitt, 2002).
2.2 A very odd comet
On may 12 1986 comet hunter Donald Machholz discovered a very unusual short-period
comet, which was later named 96P/Machholz. The comet is special for a number of reasons.
For one, it has a very short orbital period of only 5.2 years and a perihelion distance of just
0.127 AU. The orbit is so close around the sun that, with the proper tools, the comet can be
seen even at its aphelion point. Secondly, the comet has a very high inclination of nearly 60
degrees, because of which it is not considered a Jupiter-family comet (Schleicher 2008).
But what really sets this comet apart from its peers is the composition of the comet.
96P/Machholz seems to be depleted of CN by a factor of 72, in comparison to the average
comet, while the amount of NH seems to be on the high end of the normal comet-range
5
(Schleicher 2008). This low CN-OH ratio gives 96P/Machholz a composition unlike any other
known comet in our solar system, suggesting that it might have originated elsewhere.
There is also a group of comets containing molecules (Methyl- and Hydrogen Cyanide)
that are usually associated with the interstellar medium (ISM), which also suggests that
comets are either interstellar in nature, or that these comets spent some time there before
becoming gravitationally bound to our Oort Cloud (Sekanina, 1975).
These comets encourage the idea of a comet group populating the interstellar medium.
2.3 Interstellar comets
The group of comets that holds the main focus of this research is the group of dynamical
comets on hyperbolic orbits. These comets have eccentricities larger than unity and are thus
unbound to any star. From here on out, the term interstellar comets (ISC’s) will represent
this particular group of comets. Of course, comets perturbed from the Oort Cloud can enter
the inner solar system on hyperbolic orbits as well, but their velocity will be substantially
lower than interstellar comets, allowing us to distinguish them from one another unless their
velocities are less than 1 km/s (McGlynn and Chapman, 1989).
We know that the planets in the outer solar system can give comets a boost when they
enter the Hill radius of the gas giants. If they are given a high enough dose of extra energy,
the comets can be ejected into interstellar space. In our solar system, the planets Jupiter and
Saturn are found to be most effective in ejecting comets into the ISM, rather than placing
them in the Oort Cloud (Dones et al, 2004).
The current study of exoplanets reveals more and more planets of Jupiter mass (or
greater) almost on a weekly basis. In fact, over half of the extrasolar planets found up till July
2nd 2012 had masses greater than Jupiter (www.exoplanet.eu/). Since a Jupiter or Saturn mass
planet is needed for the ejection of comets, it can be assumed that the major planets of
other star systems will likewise eject comets into ISM, resulting in a rather large group of
ISC’s.
There is a theory that a large part of our current Oort Cloud was captured from nearby
stars when the sun was still in its birth cluster. As much as 90% of our Oort Cloud could have
been captured during the first few million years after the sun was born (Levison et al. 2010).
Because the relative velocities between the stars in the sun’s birth cluster were low, comets
could easily become gravitationally bound to our star.
Their results imply that the number of Oort Cloud comets that were put in their orbits by
the major planets has been greatly exaggerated. But it also implies that other star systems
are indeed producing, and losing, cometary systems of their own.
As the sun moves in its orbit around the centre of the galaxy with a velocity of 16.5 km/s
with respect to the local standard of rest (LSR), a number of those ISC’s should pass through
the solar system per century (McGlynn and Chapman, 1989). Observations of these comets
however, have remained non-existent up to this day, which raises the question: where are
the interstellar comets?
2.4 Main questions
According to theories posed by McGlynn and Chapman (1989) we should be seeing several
ISC’s per century, but in our almost 170 years of observations (Sen and Rana, 1993 state it
was 150 years in at the time of their article), no such comets were spotted. The main goal for
6
this research will be to calculate the expected rate of ISC’s and determine whether this value
is consistent with the observed value of zero. Also, the time in which we shall see our first
extrasolar comet, based on calculations in this research, will be calculated.
But why would one want to study ISC’s in the first place? In their article on Kuiper Belt
Objects, Luu and Jewitt (2002) describe comets as relics from the early ages of the solar
system. They claim that the Kuiper Belt itself is a remnant of the solar nebula, from which
the planets and comets were created, filled with primordial fossils that have escaped
collisions with each other and were not ejected from the system. Studying extrasolar comets
might provide us with a window into the conditions under which planetary systems around
other stars were formed. This makes ISC’s a prime target for further investigation.
The two main questions of this research will thus be: How many interstellar comets
should we have seen in the past 170 years? And when will we see an interstellar comet?
In the next chapter, the theory behind the formation of the Oort Cloud and the calculations
behind the expected number of ISC’s will be explained, followed by a chapter giving
explanation about the actual research and its results. In the following chapter, conclusions
are drawn and the final chapter of this thesis contains the discussion.
Chapter 3
Theory
3.1 Formation of the Oort Cloud
When the planets in the solar system had fully formed, it is believed that many small solid
bodies remained in the planetary regions. Most of these so called ‘planetisimals’ in the
region of the giant planets were likely to contain volatiles such as (water)ice. In the early
years of the solar system, many such comets would have been perturbed by the major
planets and as a result, they were ejected from the solar system. Most of those comets
would have entered planet-crossing orbits within the first 10 million years or so (Dones et al.
2004).
While a lot of comets received energies high enough to let them escape the solar system,
not all of the planet-crossing comets were ejected. Some comets did not gain enough energy
to be lost to the ISM but instead were perturbed onto orbits with very large semi-major
axes. Their perihelion distances however, remained within the planetary orbits. This means
that, although the comet will spend most of its time far outside the range of Jupiter and
Saturn, it will always return back to the planetary region where it can still be ejected from
the solar system.
The same galactic tide that perturbs comets in the Oort Cloud, and thus sends them back
into the planetary region, could also have been used as a means to create the Oort Cloud in
the first place. Long period comets spend most of their time in places where the
perturbations from the gas giants is negligible next to influences from outside the solar
system (Rickman, 1976). Galactic tide, a passing star, a brown dwarf, a gas cloud and even
dark matter can lift the perihelion of the comets out of the planetary region (Dybczyński
2002). These passing masses will increase the perihelion distance far more than they will
increase the overall size of the orbit (Dones et al. 2004). If the star lifts the perihelion
enough, the comet could obtain an Oort Cloud orbit, safely away from planetary
7
perturbations. This mechanism sets the range of the Oort Cloud between 10.000 (Tremaine
1993) and 100.000 AU (Dones et al. 2004).
3.2 Expected rate of extrasolar comets
In this section, the calculations for the expected number of interstellar comets will be
discussed. These calculations follow the works of McGlynn and Chapman’s 1989 paper on
the nondetection of extrasolar comets. A more abundant explanation can be found there.
The first thing that needs to be estimated is the total number of comets that roam free in
between the stars. If we assume that all other stars eject as many comets as the solar system
does, the density of comets in the interstellar medium will simply be the density of stars,
multiplied by the number of comets lost per star.
is used by McGlynn
and Chapman.
Another phenomenon that needs to be taken into account is the process of gravitational
focusing (figure 1). Imagine a sphere of radius r around the sun. We want to count all the
comets that go through this sphere. Some of the comets that are just outside of the sphere
can be pulled into it due to the sun’s strong gravitation. This effect is called gravitational
focusing.
r
Figure 1: The blue circle of radius r around the sun represents the area in which we want to observe comets. The five lines
represent incoming comets. On the lift, in the situation without gravitational focusing, only 3 out of 5 comets will be observed. In
the left figure, gravitational focusing is taken into account, resulting in all five comets to be observable.
As such, we shall see all comets with an initial impact parameter p, given by:
The total number of comets that we shall be able to see, per unit time, will then be given by
the formula:
The distribution of comet velocities,
, is approximated by integrating an isotropic
threedimensional Gaussian with a centroid at velocity δ (= 16.5 km/s) with respect to the
sun, and a one dimensional dispersion σ (=21 km/s). We assume that comets are ejected
from their systems with relatively low peculiar velocities.
8
The axes of velocity are chosen in such a way that the Local Standard of Rest is on the
axis. This integral is taken over a shell of constant velocity (McGlynn & Chapman, 1989). The
minus sign in the exponential term is absent in the McGlynn and Chapman’s article, yet is
present in a subsequent study by Sen and Rana (1993). The author assumes it to be a typo in
the original paper.
Finally,
can be determined if the number of comets lost to ISM per comet that is
placed in the Oort Cloud is known. After this value is determined from simulations, it can be
multiplied by the current number of Oort cloud comets to find the number of comets
ejected from the solar system.
Chapter 4
Research and results
In this chapter, the simulation done on comets in the solar system will be explained. The
assumptions made and values given to the parameters of the simulations and consequent
calculations will be addressed. In the final section of this chapter, the results will be
tabulated.
4.1 Assumptions and parameters
In an attempt to make a new estimate in the number of comets ejected by the solar system,
a simulation was made with a program called “Swifter”. This program allows the user to
generate a star and a surrounding planetary system. During a chosen integration time,
gravitational computations determine the orbits of the planets in the simulation.
A group of test particles, in this case representing comets, can be put into the system to
find what effects the planets and the star have on this group. The test particles do not
interact with one another, to keep calculation times low, and also did not influence the
orbits of the major planets, as the comet mass is negligible compared to the planets.
For the simulations used in this research, the solar system containing only the sun and the
four gas giants (Jupiter, Saturn, Uranus and Neptune) was used. A group of test particles was
introduced to the system at specific orbits, and allowed to run for a variable amount of time.
The module ‘’Symba’’ was used to accurately calculate the event in which a test particle
came inside of the Hill radius of one of the planets. The Hill radius is an area around each
planet in which the gravitational force on a particle, exerted by the planet, is stronger than
the gravitational influence of the sun. Whenever a comet would enter a Hill radius, the
Symba module would lower the time step used in the integrations to accurately calculate the
course of the comet after the interaction with the planet in question.
A certain set of parameters was given to Swifter’s simulation runs. For one, the total
simulation time was set to be 107 years, as it was within this time span that the solar system
supposedly lost most of its comets to either the ISM, or the Oort Cloud (Dones et al, 2004).
The time steps the integration process used was set to be one year per step. This was
deemed sufficient, as the smallest orbital period (that of Jupiter) is nearly 12 years.
The Swifter program discards comets from calculations if they venture outside of an inner
and outer radius, specifiable by the user. The inner boundary for comets in the solar system
was taken to be 0.1 AU, as any closer ranges would result in (nearly) all volatiles being lost
from the comet, thus making it unobservable. The outer boundary, which would correspond
9
with the edge of the solar system, was calculated to be 114,000 AU, or 1,8 light years. This is
the estimated distance at which the gravitational dominance of the Sun ends and the
dominance of Alpha Centauri, nearly 4.4 light years away, begins. This estimation was made
based upon the comparison of gravitational forces on a comet of random mass somewhere
between the two stars.
This 114,000 AU is also taken to be the boundary of the Oort Cloud, as any comets
obtaining higher orbits would be gravitationally unbound to the solar system, and thus no
longer a part of the Oort Cloud. The inner boundary of the Oort Cloud is taken from
literature to be 10,000 AU (Dones et al. 2004, Tremaine 1993, to name a few).
In the next section, the simulations will be fully explained. But several more assumptions
need to be made in order to calculate the number of interstellar comets we should have
seen in the past century. For one thing, the density of stars needs to be determined in order
to make the final calculations. Where McGlynn and Chapman took the density of stars in the
solar neighborhood, 72 systems within 6.5 pc from the sun, resulting in a density of 0.1Msun
pc-3, it can be argued that not all star systems will produce interstellar comets. Binary, triple
and quadruple systems will not have Oort Clouds like ours, and will probably not be able to
place comets into interstellar space. It can therefore be argued that these systems should
not be taken into account, ultimately leading to a stellar density (of single stars) of 0,014M sun
pc-3 (Sen and Rana, 1993). This is the value that will be adopted in the rest of this research
paper.
The probability of observing a comet at a certain distance around the sun poses a bit of a
problem, as no such estimation based on current surveys and telescopes were found. The
best estimate to date, it seems, comes from E. Everhart (1967) and was also used in the
calculations done by McGlynn and Chapman. In this research, the same values will be
adopted. They state that the probability of observing a comet that comes within 2 AU of the
sun is equal to 7%. This value was determined by dividing the observed number of 238
comets within this region by the theoretical amount of comets that passed through the inner
solar system (within 2 AU) of 3568 comets. The author of this thesis was unable to find a
more recent value for the probability of observation, so the 7% value will be used.
The only variable that remains is the number of comets ejected by the solar system,
which will be calculated in the simulations.
4.2 Simulations
With this simulation, the fraction of comets that are lost to the ISM per comet that is
currently in the Oort Cloud is determined. In order to do this, a population of comets was
introduced to the solar system, consisting of the Sun and the four major gas planets. As
explained earlier, the integration time was 107 years.
Due to the large amounts of time required to integrate a population of even just 1000
comets simultaneously for 107 years, a different approach was used. The solar system was
divided into eight fractions, each of which would receive a group of comets in separate
simulation runs. The comets are always evenly divided within a given interval. The comets
have zero eccentricity, so they all start out in circular orbits. As such, their initial semi-major
axis is also their perihelion point. To save calculation time, the chosen population would be
25 comets per 1 AU interval, giving a grand total of 1175 test particles (TP) for the region
between the giant gas planets, and part of the area beyond (3-50 AU).
10
Due to the relatively short period in which this research was able to take place, larger
populations of comets or longer integration times were not an option. The results of the
swifter program are given in table 1.
Interval (AU)
3-7 (Jupiter)
7-8
8-12 (Saturn)
12-18
18-22 (Uranus)
22-28
28-32 (Neptune)
32-50
Totals
# TP
# TP
lost*
100
85
25
19
100
82
150
106
100
50
150
53
100
30
450
20
1175 445
# TP on
planets
6
0
8
3
2
1
2
1
23
#TP too close
to the sun
7
4
3
6
3
3
1
1
28
# Oort Cloud
candidates
1
0
2
2
1
2
0
1
9
# TP that
survived
1
2
5
33
44
91
67
427
670
Table 1: Swifter simulation results. TP stands for “test particles”.
* This is the total number of comets lost from the system (i.e. that ventured beyond 114,000 AU) and the number of
comets that were on orbits that would result in their ejection from the system, at the end of the simulation.
A grand total of 9 comets are possible candidates to become Oort Cloud comets. 8 of
these comets were placed in such high orbits by Jupiter and one by Saturn, which is in good
agreement with literature, which states that comets near Uranus and Neptune will spiral
inward and be either ejected from the system, or placed in the Oort Cloud by Jupiter and
Saturn (Dones et al, 2004).
The term Oort Cloud “candidates” comes from the uncertainty whether these comets will
actually become Oort Cloud comets. As can be seen in Figure 2 comets that obtain orbits
that take them into the Oort cloud will eventually return to the planetary region where their
perihelion point lies.
Figure 2: This comet was perturbed into an orbit that took it within the Oort Cloud.
9
Little before 2.5*10 days after the integration started, it returned to its perihelion
point near Jupiter, where it was perturbed onto
11 an even higher orbit.
Time (days) 
The comet in Figure 2 was perturbed by Jupiter and placed on an orbit well within the
Oort Cloud. When it returned to its perihelion point near Jupiter, it was perturbed again, but
this time on an even higher orbit. During a next encounter with Jupiter, the energy of the
comet might even be increased enough for it to be ejected from the system completely. If
the galactic tide can lift its perihelion point before this happens, this Oort Cloud candidate
can become an actual Oort Cloud comet. See Appendix 1 for more plots of Oort Cloud
candidates and their eccentricities. To get the most optimistic value for
and thus for
the expected rate of ISC’s, we shall assume that all Oort Cloud candidates become Oort
Cloud comets
Conclusions on simulations in the article of Dones et al. (2004) state that the fraction of Oort
Cloud comets lies between 5% and 7,6% as opposed to the 0.77% found in this simulation.
The difference may be due to other initial orbits (4-40 AU as opposed to 3-50 AU), although
that would only bring up the percentage found here up to 1%. The main difference however,
might be due to the much shorter integration time and the fact that not all comets have
been either ejected or put into Oort Cloud orbits by the end of the simulation (Figure 3).
Figure 3: The number of comets that remained in the solar system as time progressed for 10 million years
Another noticeable fact in Table 1 is that the number of objects in the region of 32-50 AU
has remained fairly stable over the first 107 years (nearly 95% of the original population
remains). This region of space is currently known as the Kuiper Belt, and as such it is to be
expected that a large number of comets survived there.
Based on the results found in this simulation, the desired fraction of comets lost to the ISM
per comet that ends up in the Oort Cloud, would be roughly 49. As stated in the
introduction, the number of current Oort Cloud comets is thought to be somewhere
between
and
comets. To get the most optimistic final result for
observational ISC’s, the value of
comets shall be used, resulting in an outflow of
12
comets to the ISM, which is similar to the value adopted by McGlynn and
Chapman (1014 comets).
Another way to calculate the number of ejected comets is to make use of the Kuiper Belt.
For the 427 surviving KBO’s, a total of 445 comets were lost to the ISM, meaning that per
KBO little over one comet is ejected from the system. Adopting the total number of KBO’s
given by Luu and Jewitt (2002) of 1010 comets, one quickly finds that
comets are
lost to ISM, which is significantly lower than the amount found using the Oort Cloud.
Given that this result is a factor of 104 lower than the number ejected comets used by
McGlynn and Chapman (1989), on whose work this paper is based, it would seem that this
value is not adequate for determining the number of comets in the ISM. Instead, one might
say that the current amount of comets in the Kuiper Belt is surprisingly low. Theories of
migrating planets, which scatter comets and other small bodies away as they migrate
through the solar system might be the reason behind the low abundance of comets in the
Kuiper Belt.
4.3 Results
All that remains now is to use the formulae given in section 3.2 to calculate the expected
rate of ISC’s we should have seen in the past century. Combining the formulae, the number
of comets that pass through the inner solar system is given by:
In which δ = 16.5 km/s, σ = 21 km/s and r = 2 AU. For r = 2 we can adopt the probability of
actually detecting a comet to be 7%, as stated earlier (McGlynn & Chapman, 1989).
Multiplying
by this number will give us the number of comets that should have been
observed each year for the past century. The results can be found in Table 2.
# comets lost per star
# comets observed each year
# comets in past century
1014
1014
Table 2: The number of ISC's that should have been seen in the past century, given certain values of ρcomets
In Table 3 the numbers will be scaled up to encompass the past 170 years that we have been
observing, however there is one particular observation the author would like to make,
considering the numbers in Table 2.
While the first row is filled with the values calculated according to the number of lost
comets found in the simulations, the second row contains the values calculated by Sen and
Rana in 1993. However, when re-calculating their expected rate of ISC’s per century of 0.56
comets, the author of this paper came up short by a factor of 2.8, which can be seen in the
third row of the table. To be more precise, the results are a factor of factor
lower that
the results given in their article. When re-calculating the values of McGlynn and Chapman,
the author also comes up a factor of
lower than expected.
This discrepancy suggests that the
in the integral above was, for some reason, not
taken into the original calculations by McGlynn and Chapman. In the later study by Sen and
13
Rana, the results from the previous paper were scaled with new estimations for
without re-calculating the integral. As a result, the number of comets we should have
observed according to their papers, should be a factor
lower, resulting in a value much
closer to the observed population of zero interstellar comets. This missing factor was taken
into account in the first row of Table 2.
# comets lost per star
# comets observed each year
# comets in past 170 years
1014
Table 3: The number of comets we should have observed in the past 170 years
In Table 3 the number of comets we would expect to have seen in the past 170 years (the
number of years in which we have been observing) are still below one, which corresponds to
the observed number of zero ISC’s. The first row in this table contains the results based on
the simulations of this paper, and the second row contains the values found in the article of
Sen and Rana (1993). If the odds of observing a comet were 100% instead of 7%, we should
have seen roughly 12.1 ISC’s.
The odds of not seeing an ISC in a 170 year period can easily be determined:
Which also confirms that it is not so strange that we have not seen an interstellar comet in
the past 170 years of observing. Dividing one by the number of comets we would expect to
observe each year, it is found that one comet should, on average, be observed every 200.6
years.
That does not necessarily mean, however, that we shall observe an ISC in the next 30
years or so. What can be determined however, is that the odds of observing one such comet
over the next 920 years is 99%, and even 99.9% in the next 1382 years, assuming that the
probability of detecting a comet within 2 AU of the sun does not improve at all in the nearfuture.
Chapter 5
Conclusions
Based on the numbers found in this research, and on those found in earlier papers, the nondetection of interstellar comets is to be expected. Including the Kuiper Belt into the
calculations, one finds that this belt is surprisingly empty. The origin of this problem may lie
in the migration of planets through the solar system.
Using the number of comets that should be observed each year, one can easily determine
the time span in which the total number of observed comets equals one. We should see one
ISC, on average, every 200 years. The odds of seeing one ISC in the next 900 to 1400 years is
quite close to 100%, but there will always remain a chance of missing them when they enter
the system.
If the probability of observing a comet within 2 AU from the sun was set to 100%, we
should have seen at least a dozen interstellar comets by now. As such, further investigation
on the current observation probability is vital for an accurate calculation of the expected
rate of observed interstellar comets.
One last conclusion to be drawn is that the values calculated in the papers of McGlynn and
Chapman, and Sen and Rana were incorrect, due to a missing factor in their calculations.
14
Chapter 6
Discussion
6.1 On the research
First off, the integration times for the simulations was very short. The outflow of comets
from the solar system did not seem to be stabilizing near the end of the run (see Figure 3)
which means that a lot of the comets that remained in the system could still have been
ejected or put in the Oort Cloud, given more time. Unfortunately, due to the short time
available for this Bachelor Project, longer integration times were not an option.
Secondly, on working with Swifter. The program in itself worked like a charm, but in
hindsight was perhaps not the wisest choice for this research, as the program cannot
simulate the galactic tide needed to place comets in the Oort Cloud.
Finally, a thorough investigation of the effects of different time steps was not performed,
as it would take up a lot of the limited time, and thus was perhaps not its most efficient
value. The number of comets placed in the system was a bit on the low side in order to safe
time. A higher comet count in each of the regions might have given more accurate results, as
the program would more often be prompted to lower the time steps of integration when a
comet comes within a planet’s Hill radius, which unfortunately also cranks up the integration
times.
Further studies with Swifter might want to consider determining the optimal value for the
time steps and the number of comets placed in the system, to accurately describe the
evolution of the solar system. Also, it might be possible to place a second star in the
neighborhood of the Sun and thus determine if comets actually make it to the Oort Cloud.
The author has attempted such a study, but was (in the limited available time) unable to find
the optimum distance and mass of this second star, so that the planets in the solar system
would remain unperturbed, but the comets themselves would be influenced when they
were far away from the planetary system.
6.2 On the missing interstellar comets
There are several points of discussion that can be raised on the topic of the missing
interstellar comets. The points mentioned below can change the outcome of the calculations
above rather drastically, and should be further investigated in future studies.
(1) A lot of comets on the southern hemisphere are not spotted, partly due to the fact
that there are far less observers, amateurs and professionals, surveying the skies.
Everhart (1967) gave an example in which 203 comets that are invisible to the naked
eye were found on the northern hemisphere, while only 10 such comets were found
in the south, between 1840 and 1919. Also, of the 337 unexpected long-period
comets studied, 47 were discovered in the north that were not visible in the south,
while only 8 such comets were found in the south. Unequal study of the skies around
us will give rise to a lot of unobserved comets.
(2) The number of assumed Oort Cloud and/or Kuiper Belt objects is incorrect. This
number relates directly to the number of comets hurled into interstellar space. If the
amount of comets in the Oort Cloud increases by a factor 10, then so does the
number of ISC’s we should see each year.
15
(3) Solar systems like ours are rare (McGlynn and Chapman, 1989). Meaning that only a
small portion of stars form planets and Oort Clouds, which create the outflow of
comets. In the last couple of years, hundreds of exoplanets (planets orbiting other
stars) have been found, more than half of which are more massive than Jupiter.
Therefore, there is no reason to assume our system is unique in producing
interstellar comets. However, it is quite possible that a fraction of planet-bearing
stars failed to create planets of sufficient mass to eject comets into the ISM. Too few
such systems are known to date to make any sort of estimate in the calculations
done above, but it is an interesting idea to keep in mind for the decades to come.
(4) We are currently in a comet shower. McGlynn and Chapman (1989) argue that their
estimates are made with the assumption that the present comet flux from the Oort
Cloud represents a long-term average. If the flux is currently higher than usual, the
lack of ISC’s is no longer a problem. McGlynn and Chapman dismiss this idea as
unlikely, but Valtonen et al. (1992) (and references therein) point out that this idea is
not very farfetched. At the present time, there seems to be a maximum in the types
of geological and paleontological events that have a relation with comets, and that
we are at a maximum in galactic tides, which could cause a comet shower.
However, as McGlynn and Chapman already pointed out, if we were truly in a comet
shower, the appearing comets should not be randomly orientated, but come from a
certain part of space, indicating where a star or gas cloud has passed that perturbed
the comets. This is not seen in observations. Obviously, further research into this
general idea is needed.
(5) The assumption made by Sen and Rana (1993) that binary, triple and quadruple
systems do not form Oort Clouds and thus do not eject comets into the ISM might be
false. Reasons for their statement were not given in their article, although it stands to
reason that a comet cloud exactly like ours would be impossible in a binary system if
planets like Jupiter cannot form in such environments. However, comets might be
able to form there, and be ejected from the system by interactions with the two
stars, instead of one star and a massive planet.
However, the planet Kepler-16b, found in 2011, might counter this point brought up
by Sen and Rana. The planet, of about one third of Jupiter’s mass, is an exoplanet
found in a binary system. If any planet can form in this environment, it stands to
reason that more massive planets should be able to form as well, and thus be able to
inject the ISM with a steady stream of comets. If so, the expected rate of ISC’s would
go up.
16
Thank you notes
The author would like to thank Professor Carsten Dominik for his support and supervision in
this project. Also, without his scripts for calculating the integral for
(section 3.2) the
missing
factor in previous calculations would not have been discovered.
The author would also like to thank Msc. Sebastiaan Krijt for his help with setting up the
Swifter program and providing additional scripts to make my life easier and speed up the
research process.
References
Dones L, Weissman P.R, Levison H.F and Duncan M.J 2004 – Oort Cloud formation and
dynamics. ASP conference series, vol.323, 2004.
Dybczyński P.A. 2002 – Simulating observable comets. A&A 396, 283-292.
Everhart E. 1967 – Comet discoveries and observational selection. AJ 72 volume 6.
Levison H.F, Duncan M.J, Brasser R, Kaufmann D.E 2010 – Capture of the sun’s Oort Cloud
from stars in its birth cluster. Science 329, 187 (2010).
Luu J.X, Jewitt D.C. 2002 – Kuiper Belt objects: relics from the accretion disk of the sun.
Annu. Rev. Astron. Astrophys. 2002. 40:63-101.
McGlynn T.A, Chapman R.D. 1989 – On the detection of extrasolar comets.
AJ 346: L105-L108, 1989 November 15
Oort, J.H. 1950 – The structure of the cloud of comets surrounding the solar system, and a
hypothesis concerning its origin.
Rickman, H. 1976 – Stellar perturbations of orbits of long-period comets and their
significance for cometary capture. Vol 27, No. 2.
Ridpath, I 1985 – an extract from his book ‘A comet called Halley’ found on
http://www.ianridpath.com/halley/halley1.htm (June 22 2012).
Sekanina Z. 1975 – A probability of encounter with interstellar comets and the likelihood of
their existence. Icarus 27, 123-133 (1976)
Sen, A.K, Rana N.C. 1993 – On the missing interstellar comets.
Tremaine, S. 1993 – The distribution of comets around stars. ASP conference series, vol. 36
1993.
Valtonen M.J, Zheng J. and Mikkola S. 1992 – Origin of Oort Cloud comets in the interstellar
space. Celestial Mechanics and Dynamical Astronomy 54: 37-48.
Weissman P.R. 1983 – The mass of the Oort Cloud. A&A 118, 90
Websites:
Exoplanet catalogue – www.exoplanet.eu (as seen on July 2nd 2012)
17
Appendix 1
Figure 4: Above: semi-major axes of a particle lost to the system. Below: the eccentricity of that
particle. The eccentricity became well above one and the semi-major axis below zero, both
mean that the comet has been lost from the system, or is on its way out.
18
Figure 5: Above: the semi-major axis of an Oort Cloud candidate. Below: its eccentricity
remains less than or equal to one, meaning that it remains bound to the system.
19
Figure 6: Above: the semi-major axis of an Oort Cloud candidate. Below: its eccentricity
remains less than or equal to one, meaning that it remains bound to the system. The
eccentricity slowly builds up to unity, as opposed to the particle in figure 5 where it
happened very quick.
20