The Population of Centaurs in the Solar System

January 29, 2018

Author: Teemu Granholm

Astronomy Department Unit

University of Oulu

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Contents

1 Observational Constraints of Centaurs 5

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The Origin of Centaurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Rings Around Centaurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Observing the Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Chariklo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.3 Chiron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.4 Rings around TNO Haumea . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.5 Formation of the Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Colors of the Observed Centaurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.1 Differences Between the Populations . . . . . . . . . . . . . . . . . . . . . 20

1.4.2 Possible Explanations for Color Bimodality . . . . . . . . . . . . . . . . . . 20

2 Calculating Centaur Positions and displaying their distribution in the Solar System 23

2.1 Calculation of Centaur Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Programming the Map of the Solar System . . . . . . . . . . . . . . . . . . . . . . . 24

3 Results 27

Appendices 32

A The NAIF SPICE Toolkit 33

A.1 SPICE Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A.1.1 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2

A.1.2 Using the SPICE Toolkit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Bibliography 37

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Abstract

Context. Centaurs are objects with perihelia beyond the orbit of Jupiter and semi-major axes

inside the orbit of Neptune. They show both cometary and asteroid characteristics, have been

observed to have colour bimodality and some of them are suspected to have a ring system around

them. Studying centaurs provides a glimpse to the early stages of the Solar System and to the

formation of the rings, which makes them interesting to study.

Aim. This work aims to gather the current knowledge of the observed bimodality of the colours

of the centaurs and of the ring systems around (10199) Chariklo and (2060) Chiron. The work

provides a map of the outer Solar System, showing the distribution of the observed centaurs. It

is also tested, whether it is useful to use NAIF SPICE-toolkit to plot the locations and orbits of a

large number of centaurs compared to a calculating in terms of orbital elements.

Results. The evidence for (10199) Chariklo having a ring system is compelling. It is also very

likely that (2060) Chiron has a ring system. In addition, the centaurs colour bimodality in B-R

colours is highly significant with probability of 95 %. More observations and theoretical work

is needed in order to find out the cause of the bimodality. It was found, that even though the

NAIF SPICE-toolkit provides more precision to calculating the orbits of centaurs, handling large

amounts of data with it seems inconvenient. If one wants a simple overview of all the observed

centaur locations and orbits, it is time-saving to calculate the coordinates of the centaurs oneself.

On the other hand, the NAIF SPICE-toolkit is very useful when considering only few centaurs,

maintaining a better precision in the calculations.

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1 Observational Constraints of Centaurs

1.1 Background

In 1977 an unusual object,(2060) Chiron (classified as an asteroid), was found in the outer Solar

System. The orbit of this object stretches from the orbit of Saturn to the orbit of Uranus. Later it was

observed, that (2060) Chiron had dust around it. This dust was interpreted to be a cometary coma,

thus getting (2060) Chiron to be classified as both an asteroid and a comet. 15 years later 5 similar

objects were found [1]. This new class of objects was named centaurs. Currently (02.04.2017) 695

centaurs and Scattered-Disc Objects are known [2]. Centaurs are defined by the Minor Planet Center

as objects, which have perihelia beyond the orbit of Jupiter and semi-major axes inside the orbit of

Neptune [2].

Centaurs can transform into Jupiter-family comets due to perturbations. These objects can also

transform from Jupiter-family comets back to centaurs [3]. Jupiter-family comets (JFCs) are short-

period comets with orbital periods shorter than 20 years and low inclinations [4]. Other major comet

groups are Halley-type comets and long-period comets. Long-period comets have orbital periods

larger than 200 years, random orbital inclinations and very high eccentricities [4]. Halley-type comets

have orbital periods of between 20 and 200 years and orbital inclinations and eccentricities between

JFCs and long-period comets [4].

In addition to their cometary characteristics, some centaurs ((10199) Chariklo [5], also possibly

(2060) Chiron [6]) seem to have a system of rings around them. Centaurs are interesting objects to

study on their own, but especially interesting is the fact, that they most likely provide a glimpse of the

early stages of the Solar System and of the origin of the rings.

Studying the Kuiper Belt (a disk of small bodies in the outer Solar System similar to the Asteroid

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Belt) and Scattered Disk (a disk of icy bodies which extends from Kuiper Belt to 100 AU in outer Solar

System) can also reveal interesting new information of the Solar System. For example, a research done

on clustering of orbital elements of a set of Kuiper Belt objects has led to a hypothesis of a planet

(Planet 9) existing well beyond Neptunes orbit that could perturb these objects enough to cause the

clustering [7]. This is an exciting hypothesis, which shows that there are many discoveries yet to

be made in the Solar System and in the Kuiper Belt area, where also centaurs are thought to have

originated.

1.2 The Origin of Centaurs

There is not a complete certainty of the origin of the centaurs yet, but they are thought to have been a

part of the Kuiper belt, that has been then perturbed by Saturn or Uranus. The planetary perturbations

caused by the giant planets during close encounters with the Kuiper Belt objects have changed their

orbits to their current forms. These perturbations happen in the following way. A Kuiper Belt object

can be in an orbital resonance for example with Neptune. Orbital resonance happens, when the

ratio of two or more astronomical object’s orbiting period is stated using integers. This means, that

over certain periods of time, they are in gravitational interaction with each other. In this case, the

gravitational field of Neptune has an effect on the orbit of the object, making it more unstable. With

every interaction the effect of Neptune becomes stronger (hence the term resonance) and finally sends

the object for example on an orbit that takes it close to Uranus [8]. Then Uranus can affect the object

in a similar way, leading the object closer to Jupiter or back towards Neptune. The current orbits

of centaurs are relatively short lived, around 106 to 107 years [9], [3], because they are dynamically

unstable. Dynamically centaurs can develop (because of the resonances) to be JFCs and some of them

can be thrown out of the Solar System or crash into the giant planets. JFCs on the other hand can

transform back to centaurs by the gravitational perturbation of Jupiter, although it is not as likely as a

centaur becoming a JFC [3], [10].

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1.3 Rings Around Centaurs

Previously it has been thought, that only giant planets (gas and ice giants) could have a ring system.

Recently (observed 2013, published 2014) it was discovered, that there are two rings around the

centaur (10199) Chariklo [5]. This discovery suggested the probability that other centaurs could have

ring systems as well. In 2015 was found evidence, which showed that also (2060) Chiron could have

rings around it [6]. This is a remarkable discovery, because it will give us an opportunity to find

out more about the early stages of the Solar System and about the mechanisms that are affecting the

formation of the ring systems. Next step would be to find out, whether other centaurs have rings

and how common they really are. More observations are also needed to determine what mechanisms

cause the rings to form and how they remain stable.

1.3.1 Observing the Rings

Due to the centaurs being so distant, so dim and the rings being so close to the main body, direct

observation of the ring system is impossible or at least extremely difficult with modern technology.

Hence, the best way to detect the rings is by stellar occultations. In a stellar occultation one observes

the light coming from a star. When the object one is interested in occults the star, i.e it passes in front

of the star, it dims or blocks the light. Usually there is only one dimming caused by the object, but

sometimes there are secondary dimmings, before and after the dimming caused by the main body. If

the dimmings are symmetrical, they can be interpreted to be caused by a ring system. So far the only

centaurs for which stellar occultations have been recorded are (2060) Chiron and (10199) Chariklo

[6]. Both of these objects show secondary dimmings in their occultation profiles [6], [5], which can

be explained by a ring system.

In Fig. 1.1 from [5] the light curve for (10199) Chariklo’s stellar occultation in 2013 is shown.

The secondary dimmings are clearly visible before and after the occultation of the main body (marked

ingress and egress, respectively). The first dip is the occultation of the smaller of Chariklo’s two rings

and the second dip is the occultation of the bigger one (named C2 and C1, respectively). A similar

lightcurve was observed for (2060) Chiron and also interpreted to be a sign of the ring system [6].

Other evidence for the existence of the rings is found from the spectra of these objects. Water-ice

7

Figure 1.1: Image from [5]. Stellar occultation lightcurve for (10199) Chariklo. The secondary

dimmings around the main body are visible.

features have been observed in the spectra of both (2060) Chiron and (10199) Chariklo [6], [5]. These

features later disappeared completely and ultimately came visible again. The previous explanation for

this disappearance in (2060) Chirons case was cometary activity [6]. It was also argued, that a dust

cloud around (2060) Chiron could have dimmed the water-ice spectral line, causing it to disappear.

However, simulations show that even though the water-ice spectral line would be dimmed by the dust

cloud, it should still be detectable [6].

The disappearance of the water-ice band can be more easily explained by the changing opening

angle (aspect angle) of possible rings. This will be discussed in more detail in subsection 1.3.3. Let

us assume that water-ice is present in the rings of (2060) Chiron and (10199) Chariklo and not on the

surfaces of their main bodies. As the pole orientation of these objects changes, the projected surface

of the rings also changes. This makes the rings sometimes completely visible and other times invisible

(when rings are edge-on) to the observer. When the rings are wide open, also the water-ice band will

be most prominent. As the rings change their orientation towards an edge-on view, this spectral band

disappears completely [6]. The observed lightcurve and the disappearance of the water-ice band are

8

the most pressing evidence towards the existence of the rings around both (2060) Chiron and (10199)

Chariklo. This will be discussed in more detail in following subsections and also other evidence to

support the hypothesis of the existence of the rings will be shown. It seems to be widely accepted

that (10199) Chariklo has a ring system, but (2060) Chiron’s case is still open, although the suggested

evidence feels quite compelling.

1.3.2 Chariklo

The first centaur around of which rings were found, was (10199) Chariklo. It is the largest known

centaur and it has a very dark surface [5]. The evidence for the rings comes from stellar occultation

and from photometric and spectroscopic measurements [5], [11]. Chariklo occulted a 12.4 mag star in

2013, which was observed from different sites, four of them in South America. A total of seven sites

observed secondary events during the occultation [5]. All of these events (Fig. 1.1) can be explained

with an azimuthally homogenous (and narrow) ring system. One could argue that cometary jets play

a role in these dimmings, but geometrically that is highly unlikely.

In [5], the researchers managed to fit an ellipse to the measurement, as result from the projection

of the observed symmetry. This fit allows for two possible solutions for the ring axis. The preferred

one is −61.54 ± 0.14, which explains the gradual dimming of Chariklo between 1997 and 2008.

The preferred solution suggests that the rings were edge-on in 2008. This solution also explains the

gradual disappearance of the water-ice features in the spectra during that time period, if we assume

that water-ice is present dominantly in the rings. It has also been observed, that the Chariklo system

is brightening since 2008 when the water-ice should have become again visible in the spectra, which

further suggests that the hypothesis of rings formed by dominantly water-ice is correct [5].

1.3.3 Chiron

(2060) Chiron is also suspected of having a ring system around it. As mentioned in section 1.3.1,

the main evidence comes again from stellar occultations [6]. Originally the secondary dimmings

were explained as comet-like dust jets or symmetric jet-like features [6], [12], [13]. In the light of

the discovery of Chariklo’s rings, these events were reconsidered. Also in 1993, 1994 and 2011

observations, some spectral extinction lines are more compatible with the ring system than with a

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jet [6]. However, Chiron was not reported to be active around the times of those observations (no

outbursts or coma reported [6]), which makes the jets an unlikely explanation. Further argument

against the jets is, that a narrow ejection angle, needed to fit the narrow dips in occultations has not

been observed for cometary jets [6].

The most compelling occultation was observed in 2011. The data was obtained with the NASA 3

m InfraRed Telescope Facility (IRTF) and the 2 m Faulkes telescope [14]. The NASA-IRTF telescope

showed two symmetrical secondary dimmings along with the primary dimming from the main body.

The data from the Faulkes telescope shows two very sharp drops of the stellar light, but not the

primary event of the main body. This lightcurve is similar to the one of Chariklo, in that the width

and separation of the two features in both lightcurves are similar [6]. It is natural to think that also

Chiron’s lightcurve could be explained by rings, just like in Chariklo’s case.

Another evidence mentioned before are the water-ice features in the spectra of (2060) Chiron

(and (10199) Chariklo). The water-ice signature was detected in 1997 and 1999, but it completely

disappeared in data from 2001 [6]. This can be explained with a ring system. The rings of Saturn

have water-ice features [6], so it is reasonable to assume that water-ice also forms in the proposed

ring system instead of the main body of (2060) Chiron [6]. If this is the case, the disappearance of

the water-ice band can be explained by the changing aspect angle of the rings. The aspect angle is

defined as the angle that the rotation axis of a body (normal to the rings) makes with the direction to

the observer [6] (Fig. 1.2).

Figure 1.2: Geometry of an observation and definition of the aspect angle.

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When the aspect angle is 90, the rings are edge-on and the observer cannot see them. Similarly,

when the aspect angle is 0 or 180, they are seen fully open from the top or from below. This change

in the aspect angle has been shown to correlate with the disappearance of the water-ice spectra in

(10199) Chariklo’s case, so it could have a similar effect on (2060) Chiron, too.

The composition best matching the reflectance spectra and albedo of (2060) Chiron is an aereal

mixture of 30 % of water-ice and 70 % of refractory material. Assuming that the water-ice is com-

pletely in the rings, would mean that the rings make a minimum of 30 % of the total projected area

of (2060) Chiron’s system [6]. The area of the rings can be up to 50 % of the total system [6]. Based

on spectroscopy constraints, the geometric albedo of the rings is somewhere between 0.14 to 0.17.

It can be also 0.27, if the rings cover 30 % of the total area [6]. Geometric albedo is the ratio of a

body’s brightness seen from the light source (the observer is at the lightsource) to the brightness of a

perfectly diffusing disk, which is the same size and at the same position as the observed body [15].

Fig. 1.3 and Fig. 1.4 show the change of the aspect angle for (2060) Chiron from 1950 to 2050,

using the two possible pole solutions derived by [6]. The geometry (Chiron and Earth location)

was obtained with NAIF SPICE Toolkit (toolkit discussed in appendix A) One can see that for both

configurations the aspect angle was close to 90 in 2001, when the water-ice features disappeared

from the spectra of (2060) Chiron. This would also explain why the water-ice was visible in 1997

and 1999 and predicts an aspect angle of 90 (making the water-ice band to disappear again) to occur

next time around year 2035. The aspect angle can be calculated with the formula

α = π − arccos

(B · A|A|

). (1.1)

Here A is the vector from Earth (observer) to the centaur. B is a unit vector representing the centaur

pole orientation.

We can also plot the evolution of the absolute magnitude of (2060) Chiron and compare that to the

expected aspect angle of the rings. In Fig. 1.5 (same as Fig. 6 in [6]) are shown the changes of absolute

magnitude of (2060) Chiron. One can see, that in the figure there are two pronounced maxima in

absolute magnitude, which correspond to brightness minima. When these results are plotted together

with the change of the aspect angle (Fig. 1.6), it is apparent, that these minima happened at the

moments when the rings are presumed to be edge-on to the observer (aspect angle 90). The first pole

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Aspect angle vs Time

1940 1960 1980 2000 2020 2040 2060Time [Years]

0

50

100

150

200

Asp

ect

an

gle

[d

eg

ree

s]

90 deg

Figure 1.3: The aspect angle derived from the first possible pole orientation (λ = (144 ± 10), β =

(24± 10)) of (2060) Chiron. Figure recreated from [6] using ephemeris data obtained from JPL.

solution shown, in Fig. 1.3, is better matching the changes in the absolute magnitude.

Because the area of the rings is 30 % to 50 % of the whole system, one can understand how

the orientation of the rings affects the overall brightness of the system. When the aspect angle is

close to 90, the rings do not contribute to the brightness at all. The only light observed comes

from the main body. Then, as the aspect angle changes (for (2060) Chiron it ranges from 30 to

150, as seen in Fig. 1.3 and Fig. 1.4) and the rings open more, they start to reflect more light and

make the system brighter, hence the large changes in the observed absolute magnitude. A similar

explanation has also been used to explain (10199) Chariklo’s absolute magnitude changes over time

[6].Besides the ring system, the brightest phase corresponds to aphelion, which is inconsistent with

activity. Some observations remain unexplained, though. The observed absolute magnitude for the

brightness minima cannot be completely reproduced by the model for the ring system presented in [6],

because the two minima are observed to have different values. The model also does not reproduce the

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Aspect angle vs Time

1940 1960 1980 2000 2020 2040 2060Time [Years]

0

50

100

150

Asp

ect

an

gle

[d

eg

ree

s]

90 deg

Figure 1.4: The aspect angle of the second pole orientation (λ = (352 ± 10), β = (37 ± 10)) of

(2060) Chiron. Figure recreated from [6] using ephemeris data obtained from JPL.

brightness of local maxima possibly related to activity outbursts [6]. Such outbursts are unpredictable,

so the simple model cannot reproduce them.

In [6] a possible solution is presented to explain the two different brightness minima. It is sug-

gested, that a decaying cloud of debris or dust (with fading brightness) could explain the differences

in the minima. In principle such cloud could be possible. There are hints of debris or dust around

(2060) Chiron [6] and it has been suggested that it could have an exponentially decaying coma.

So far the rings are the best candidate to explain the observation of (2060) Chiron. Especially the

symmetry of the sharp occultation features is difficult to explain with cometary jets. (2060) Chiron

was also not generally active during the time [6], which makes it difficult to see how such jets could

have formed at such large distance from the Sun. Jets can’t explain the sharp minima in (2060)

Chiron’s long term brightness either [6]. (2060) Chiron’s rings would be similar in nature to (10199)

Chariklo’s rings, which lends further support to the notion of (2060) Chiron having a ring system [6].

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Figure 1.5: The absolute magnitude vs time. Fig. 6 from [6].

14

Figure 1.6: The change of the sinus of the aspect angle. Fig. 5 from [6].

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1.3.4 Rings around TNO Haumea

Rings were observed around dwarf planet Haumea in 2017. Haumea is one of the largest known

TNOs, although still smaller than Pluto or Eris [27]. The occultation of Haumea was observed and the

results suggest that there is a 70–km-wide ring of material surrounding the main body, approximately

1000 km away from the surface [27]. More research is needed in order to determine the brightness of

the ring [27].

Changes in the magnitude of Haumea have been observed. These are similar changes as in the

cases with Chiron and Chariklo, where the rings are changing from edge-on orientation to a face-on

orientation. However, these results require formation mechanisms that can be applied to object that

is much larger than centaurs and is farther in the Solar System. It also has only one ring compared to

the two rings that Chiron and Chariklo have. [27]

One can speculate that in the light of this recent discovery, rings might not be uncommon in the

outer Solar System and there might be more discoveries to be made.

1.3.5 Formation of the Rings

In [5] four hypotheses have been presented for the formation of (10199) Chariklos rings (they can be

generalized for other centaurs as well). All of these rely on a debris disk, where the biggest particles

are acting as shepherd satellites for the smaller ones, thus confining the ring system into its current

shape. It is suggested that these shepherds are kilometre-sized satellites that are confining the rings

[5]. The presence of shepherd satellites has been suggested, because the rings are so narrow. They are

either extremely young or confined by shepherds, although so far there have not been any observations

of such satellites around (10199) Chariklo [5]. Shepherd satellites have also been suggested to confine

the rings of (2060) Chiron. This would explain the sharp edges of the rings and the presence of the

gap in the rings of both (2060) Chiron and (10199) Chariklo [6].

The first hypothesis is that an object collided with the centaur and ejected material from its surface

[5], [6] that has then become a ring. Alternatively, the colliding object might have destroyed a satellite

around the centaur or got itself destroyed in the collision with the centaur [5]. From these options

most likely seems to be the scenario where a satellite around centaur is destroyed in a collision with

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another object, because it is more likely for the debris from the collision to accumulate as a ring

compared to the formation of the ring from the ejected debris. The problem with the ejection from the

centaur’s surface is, that the debris needs to escape from the surface, but must stay inside its Roche

sphere to form a ring. This could be possible under certain conditions, but it is not as likely as the

scenario of a destroyed satellite.

The second hypothesis is, that a debris disk has formed by a rotational disruption of the main

body or by cometary activity [5]. Cometary activity forming the rings does not seem very plausible,

because again there would appear the problem mentioned in the first hypothesis with dust needing to

escape from the surface and staying inside the Roche sphere. Also it seems unlikely, that cometary

activity would happen at so great orbital distances. One more reason for it not being very plausible

comes from the fact, that even though (2060) Chiron has shown some cometary activity, activity has

not been observed for (10199) Chariklo [5].

The third scenario is, that two satellites around the centaur have collided and the ring has been

formed from their remains [5]. This scenario is the most problematic, because there has not been sug-

gested any mechanism that would cause such collision between two satellites. The satellites shouldn’t

collide, because in order to orbit the primary, they need to be in relatively stable orbits. Therefore,

they should only collide if some external force perturbs them, which is unlikely to happen due to large

distances between the objects.

The fourth hypothesis is, that a retrograde satellite could have migrated inwards and been dis-

rupted by tidal forces, leaving enough debris for the rings to be formed [5]. The satellite needs to be

retrograde or inside a synchronous orbit to be able to do the migrating. The synchronous orbit is the

distance where the Kepler period is equal to the spin period of central body. Inside synchronous orbit

the spin period is smaller than the Kepler period. Outside the synchronous orbit, the satellite needs to

be retrograde in order to move inwards, while a prograde satellite would eventually migrate outwards.

This is because of tidal interactions with the main body, where the satellite raises a tidal bulge on the

main body. Since the main body is rotating, the bulge is not directly under the satellite. Instead, it is

shifted towards the direction of rotation. If the bulge is leading the satellite, it accelerates the satellite.

If the bulge is trailing, it will decelerate the satellite. For a retrograde satellite, the interaction always

tends to reduce its semi-major axis. This hypothesis could be a likely option, if the material of the

17

satellite is loose enough and the mass of the primary is high enough to cause strong enough tidal

forces to disrupt the satellite.

Altogether, the second and the fourth hypothesis seem to be the most plausible ones. One could

argue for example, that a centaur has encountered a rotational disruption and some of its material

has formed a cloud of debris around the main body. The debris would then naturally form a ring.

This same principle could be also applied to the earlier scenario with a satellite of a centaur getting

destroyed in a collision. Also tidal disruption of a retrograde satellite seems like a plausible option,

because it does not depend on the debris getting ejected into the orbit around a centaur, thus having a

higher chance to happen.

A modeling of the spectrum of the rings of (10199) Chariklo shows, that they consist of a mix-

ture of bright and dark material, similar to (10199) Chariklo’s surface material [11]. This could be

explained by a disruption of an object, which has formed in the same region as (10199) Chariklo, but

with a mixture of ices, silicates and carbonaceus materials, that have been darkened by high-energy

irradiation [11]. The similarity between the material of the rings and the main body could be inter-

preted either by satellites forming around the main body in the same region, or by the formation as a

binary system.

According to [5], (10199) Chariklos orbit was perturbed by Uranus less than 10 Myr ago. This

perturbation caused (10199) Chariklo to migrate inwards from the Kuiper Belt. Such an event could

only disturb the ring system if Uranus is closer than about five Uranus radii distance from (10199)

Chariklo [5]. Such close encounter is not very likely, so it appears to be a likely possibility, that

the rings were formed in the trans-Neptunian region and managed to survive the change of (10199)

Chariklo’s orbit [5]. This could be true for other centaurs as well, if it turns out that they also have

rings, and one may suspect that rings systems exist around Kuiper Belt objects.

1.4 Colors of the Observed Centaurs

Compared to other objects in the Solar System, centaurs have a peculiar characteristic: they are

divided into two colour populations (B-R colour measurement of centaurs reflectance spectra. B-R

is acquired by using filters sensitive to blue (B) and red (R) light) [3]. The populations are either

grey (icy, cometary-like objects) or red (organic, asteroid-like objects). This is interesting, because

18

this kind of physical properties are not seen among any other objects in the Solar System [3]. Color

variation may arise from solar and cosmic radiation that ionizes, and thus transforms, the surfaces

of these objects to show more red colours [3]. On the other hand, the collision times in Kuiper Belt

are smaller than the age of the Kuiper Belt (according to a widely accepted Nice model, KB was

formed when a 2 : 1 resonance of Jupiter and Saturn scattered planetesimals from the Solar System

and started Late Heavy Bombardment [16]). Thus, collisions may play a role, especially with smaller

objects, because the distances in the KB are so large. These collisions could break the initially red

surface and reveal the underlaying grey, icy material from inside the objects. Therefore the objects

in both groups should show red and grey colours and also different combinations of the two. It is

important to note, however, that V-R and R-I colour surveys have not shown a two colour population.

So far the survey has been done for 24 centaurs [3]. Fig. 1.7 shows a picture of Pluto, a Kuiper Belt

object, that illustrates the red and grey colours.

In [3], statistical tests have been applied with the result that the bimodality is highly significant at

probability of 95 %.

Figure 1.7: Image from [17]. Four images were combined with colour data to create this coloured

view of Pluto. Here red and gray types of surface are clearly visible.

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1.4.1 Differences Between the Populations

By plotting the semi-major axis (a), aphelion distance (Q), perihelion distance (q), inclination angle

(i), eccentricity (e) and absolute visual magnitude (H) it is found that the red centaurs have slightly

larger semi-major axes than the gray ones (median ared = 19.8 AU and median agray = 17.8 AU).

Similarly, the aphelion distances and the eccentricities seem to be larger for red centaurs than for the

gray ones (median Qred = 29.1 AU and median Qgray = 21.8 AU). It appears, that almost all red

centaurs have smaller inclination angles than the gray ones (the probability for the inclination angle

distribution to be the same for both populations is only 10 %). The distribution of absolute magnitudes

and perihelion distances are virtually identical for both populations. [3]

A comparison between the observations of the Spitzer Space Telescope and ground based optical

photometry with the B-R colours of 15 centaurs, indicates that the gray centaurs have lower albedos

than the red ones. According to the Wilcoxon rank sum test, the probability for the albedo distribution

of the two populations being the same is only 1 %, which makes the difference of the albedos of red

and gray centaurs statistically significant. [3]

1.4.2 Possible Explanations for Color Bimodality

There are two possible explanations for the observed colour bimodality: evolutionary and primordial

[3]. More data and theoretical work is needed in order to say which one is more plausible.

Evolutionary Explanation

This explanation assumes that the subsurface material of KBOs remains intact while the surface ma-

terial becomes redder and darker as a result of solar radiation [3]. Another assumption is that random

collisions happen occasionally. These collisions then break part of the red surfaces, revealing the in-

tact and more volatile subsurface material [3]. At Kuiper belt distances, the temperatures are too low

for sublimation of most of the molecular ices (25 K to 50 K, only very volatile ices can sublimate at

these temperatures [18]). When KBOs orbits transfer to centaur orbits, they are taken closer towards

the Sun, so that the sublimation processes may begin [3].

Because the region occupied by centaurs is not so densely populated, the process of radiation red-

20

dening and sublimation dominate the evolution of the centaurs while collisions become unimportant

[3].

In this picture, the red surface protects the volatile material from sublimation and therefore appears

as the observed reddish colour. The ones that underwent recent collisions in the Kuiper Belt have

some of the volatile material exposed to the solar heating and the material begins to sublimate. This

in turn breaks more of the red surface and reveals more gray subsurface. Another possibility is, that

the sublimating material coats the surfaces of the centaurs with gray debris. This process could be

fast and therefore explain why there are no intermediate coloured centaurs observed. [3]

This could be tested by observing KBOs and seeing if the intermediate coloured KBOs show any

B-R colour variations while rotating [3]. Due to the collisions being random, there should be some

visible variation of red and gray colours (similar to Pluto in Fig. 1.7) at the surface of the KBOs.

Primordial Explanation

This hypothesis assumes that the centaurs preserve their original primordial colours [3]. Maybe from

distances slightly smaller than 40 AU to over 40 AU from the Sun, methane (CH4) condensed as pure

CH4 instead of a H2O-rich clathrate (at distances much smaller than 40 AU methane usually does

condense as H2O-rich clathrate) [3]. A pure CH4-ice crust could serve as a base for a transformation

into red organic compounds [3]. This could mean that objects forming at heliocentric distances less

than 40 AU have more grayish colours and those formed at distances larger than 40 AU are more red-

dish in colour. CH4-ice bands have been observed in the spectra of (134340) Pluto, Triton, (136199)

Eris and (136472) 2005 FY9 [3]. Also CH3OH could supply enough carbon and hydrogen for the

reddening to happen by radiation [3]. This radiation reddening would happen only at larger distances,

because methanol (CH3OH) ice is highly volatile and would evaporate at smaller orbital distances.

A dynamical simulation predicts that Neptune was migrating outward in the early Solar System

and has scattered objects from 25 AU heliocentric distances to orbits of present-day SDOs ( objects

with large orbital eccentricity and perihelia greater than 30 AU), high-inclination classical KBOs and

high-inclination Plutinos (a trans-Neptunian object in a present day 2 : 3 mean-motion resonance

with Neptune) [19]. Low-inclination classical KBOs were not perturbed much, because they stayed

far enough from Neptune to be affected. The scattering changed the original orbits of these low-

21

inclination classical KBOs, so one cannot use the present-day orbits to deduce their original ones.

However, if the ideas and the dynamical simulation are correct, SDOs, high-inclination classical

KBOs and high-inclination Plutinos formed at distances less than 40 AU and should have grayer

surface colours. On the other hand, low-inclination classical KBOs should have formed at distances

larger than 40 AU and should be showing redder surface colours (so far, observations seem to support

this). [3]

This idea could then explain the bimodality of the centaurs in the following way. The orbits of

centaurs are relatively short-lived (as mentioned in 1.2), so the centaur colour distribution depends

on recent scattering events due to Neptune. As Neptune migrated outwards, it scattered SDOs, high-

inclination classical KBOs, high-inclination Plutinos and grey centaurs originally at 25 AU from the

Sun to their current orbits. Neptune could today be far enough from the Sun to be able to scatter also

the redder, low-inclination KBOs into centaur orbits. We must remember, ( 1.4.1) that redder centaurs

show lower inclination than the grayer ones. This suggests that the different colour populations could

have been born in different regions. [3]

These ideas are only hypotheses to explain the colour bimodality of centaurs and should not yet be

taken as a solid fact. More research needs to be done in order to prove or to disprove these scenarios.

22

2 Calculating Centaur Positions and display-

ing their distribution in the Solar System

In order to visualize the positions of currently known centaurs, a text file containing the orbital el-

ements of centaurs and Scattered-Disc Objects was downloaded from [20]. A file containing 562

objects (centaurs and SDOs) has been used in the program, but a more recent file is now available

from the Minor Planet Center with 612 objects. The calculations needed for the program are de-

scribed in 2.1 and the actual program is discussed in 2.2. In one part of the program the NAIF SPICE

toolkit was used. The toolkit is discussed in more detail in appendix A. The results achieved with the

program are described in detail in section 3.

2.1 Calculation of Centaur Positions

In order to plot a map of centaurs in the Solar System, we transform the orbital elements to the x,

y and z coordinates of the positions in an inertial frame. In order to do this, one can solve first the

eccentric anomaly numerically from Kepler’s equation (2.1) for given eccentricity and mean anomaly

of the object

M = E − e sin(E). (2.1)

Here M is the mean anomaly of the object in radians, E is the eccentric anomaly and e is the eccen-

tricity of the object. Eccentric anomaly cannot be derived analytically from Kepler’s equation, but it

23

can be solved numerically by iteration. This is done by setting (derived from Eq. 2.54 from [21]):

E = M + ψ, (2.2)

where ψi+1 = e sin(M + ψi) and ψ0 = 0.

In order to calculate the x, y and z coordinates, one needs to calculate the true anomaly and the

radius of the object’s orbit. The true anomaly is (derived from Eq. 2.43 from [21])

f = arccos

(cos(E)− e

1− e cos(E)

)(2.3)

and the radius is obtained from

r = a(1− e cos(E)). (2.4)

for given semi-major axis a. Next, one can calculate the x, y and z coordinates in the following way

(Eq. 2.122 from [21])

x = r(cos(Ω) cos(ω + f)− sin(Ω) sin(ω + f) cos(I))

y = r(sin(Ω) cos(ω + f) + cos(Ω) sin(ω + f) cos(I))

z = r(sin(ω + f) sin(I))

.

Here ω is the argument of pericentre, I is inclination and Ω is longitude of ascending node. These

are also obtained from the data file in the angular J2000.0 frame of reference.

2.2 Programming the Map of the Solar System

The programs (called Input_program.pro and Spice_centaurs.pro) were written in the IDL program-

ming language. Spice_centaurs.pro utilizes the NAIF SPICE-toolkit (appendix A) in order to calculate

the orbits of (2060) Chiron, (10199) Chariklo and Pholus.

To start the program, a function to read the data file containing the orbital elements of the centaurs

and SDOs was needed. The data obtained from the file was perihelion, aphelion, visual magnitude,

epoch of mean anomaly, mean anomaly, argument of perihelion, longitude of ascending node, incli-

24

nation, eccentricity and semi-major axis. Not every piece of data was needed, but getting everything

at once allows one to add other applications to the program in later time.

A subroutine is needed to solve Kepler’s equation (2.1).

The third function is the most important one. It calculates the centaur’s coordinates. As an input,

it takes semi-major axis, inclination, eccentricity, longitude of ascending node, argument of pericentre

and mean anomaly. In order to calculate the radius and true anomaly of the object, one needs to know

the eccentric anomaly. This is achieved by first calling the function to solve for the eccentric anomaly

from the data. Then one can use equations 2.3 and 2.4 to solve the radius and true anomaly. When

these are known, one can finally calculate the coordinates for the centaur at hand.

The main program uses the aforementioned functions to process the data and to plot the results.

The main program also calls the program Spice_centaurs, to plot the orbits of aforementioned cen-

taurs, over which the other results are overplotted to illustrate how the orbits of these centaurs relate

to those of gas planets and to the locations of other centaurs.

Spice_centaurs.pro comprises of 2 helping functions, a program to draw the plots (Plot_draw) and

the main program. The first function generates the timeperiod used in handling the data.

The second function defines the length of each month and checks for leap years during the defined

100 year time period.

Plot_draw handles the data and plots it. We use the Ecliptic J2000 as a reference frame centered in

the Sun. When the reference frame has been chosen, one needs to get the data from the spice kernels

(more info about kernels in appendix A). Here we have gotten the vector from the Sun to the chosen

centaur to get the coordinates for drawing the whole orbit. Here is shown the code for (2060) Chiron

as an example:

for it=0L,nt-1 do begin

cspice_spkez,2002060L,ettab(it), $

refer, abcorr, 10, state, none

xschi(it)=state[0] ;Position components, km

yschi(it)=state[1]

25

zschi(it)=state[2]

endfor

In the "for-loop" the SPICE command "cspice_spkez" is used. This command gets the position

data from the kernels. The number "2002060L" is the NAIF ID code for (2060) Chiron (which can

be found here along other centaurs: [22]). The "L" at the end of the id is making sure that IDL

handles the id properly, as "long" number. Therefore it is not part of the original NAIF ID, but should

be added for the program to work properly. Next input is "ettab(it)" which gives the scalar double

precision ephemeris time (contains a time period of 1950–2050 with 50 000 time steps) as seconds

past J2000 Barycentric Dynamical Time system (TDB). This relates the object’s position to the time

at the observers location. Keyword "refer" defines the reference frame (Ecliptic J2000). Keyword

"abcorr" defines possible aberration corrections. Here "abcorr" was defined to be "none", because no

aberration correction needs to be used in calculation of the position data. The next number is the NAIF

ID for Sun that indicates the position of the observer. Second to last input ("state") is the variable in

which the needed data will be returned. The data is saved as a double precision Cartesian 6-vector

representing the position in km and velocity in km/s of the target body relative to the observer. The

first three components are the x, y and z components of the targets position and last three components

are the corresponding velocity vectors. The last command has been set to "none". It is the scalar

double precision one-way light time between the observer and target in seconds.

Next the x, y and z coordinates are extracted from the needed data. After getting the needed

coordinates for the wanted objects, the coordinates have been transformed from kilometers to astro-

nomical units (AU) to make it possible to overplot and compare the orbits to the ones calculated with

Input_program.pro. Lastly the orbits are plotted into one picture.

First the main program loads the Spice kernels. The time period for the kernels is defined and the

used variables are made global. Finally the program calls the Plot_draw program to draw the plot.

26

3 Results

In this section, we discuss the results of the program presented in sections 2.1 and 2.2. The goal of

the program was to illustrate the distribution of the centaurs in the Solar System. It was also tested,

whether it is useful to use the NAIF SPICE-toolkit to directly extract the positions of centaurs, or if

it is better to do the calculations by oneself, which introduces some loss of precision. The figures

presented below show the locations of the centaurs along with the orbits of the giant planets.

Top view of the Solar System

-100 -50 0 50 100Orbit X-coordinate

-100

-50

0

50

100

Orb

it Y

-co

ord

ina

te

Figure 3.1: Distribution of Centaurs in the Solar System (black dots). The orbits of the giant planets

are also shown (yellow – Jupiter, green – Saturn, red – Uranus, light blue – Neptune).

The instantaneous points of centaurs in the Solar System are shown in Fig. 3.1. The maximal

27

distance from the Sun has been chosen to be 100 AU in order to filter out some of the Scattered Disk

Objects (SDOs). This is done, because the Minor Planet Center has combined the information of

centaur orbits with the information of SDO orbits. The objects further away than 100 AU should

consist only of SDOs.

Most of the centaurs are found either close to Neptune’s orbit or inside of it. Most of the SDOs

should be far away from Neptunes orbit. The most dense concentration of centaurs seem to be between

Jupiters and Uranus’ orbits. One can see from Fig. 3.4 that (2060) Chiron’s (pink) and (10199)

Chariklo’s (dark blue) orbits are confined inside Uranus’ orbit. Also part of Pholus’ (orange) orbit

lies well inside Uranus’ orbit, all the way to the orbit of Saturn.

Top view of the Solar System

-40 -20 0 20 40Orbit X-coordinate

-40

-20

0

20

40

Orb

it Y

-co

ord

ina

te

Figure 3.2: Same as Fig. 3.1 but limited to 40 AU from the Sun.

From the snapshot in Fig. 3.2 can be seen, that there are some centaurs inside Jupiter’s orbit.

That means, that the centaurs cross Jupiter’s, orbits. Some of these centaurs transform into Jupiter-

family comets (short-period comets with orbital periods shorter than 20 years and low inclinations).

As mentioned in section 1.1, the Jupiter family comets can also transfer to centaur orbits again.

28

Chiron, Chariklo and Pholus Orbits with known Centaurs

-100 -50 0 50 100X (AU)

-100

-50

0

50

100

Y (

AU

)

Figure 3.3: Outer Solar System with the orbit of (2060) Chiron (pink), (5145) Pholus (orange) and

(10199) Chariklo (dark blue).

In Fig. 3.3 and Fig. 3.4 are shown orbits of (2060) Chiron, (10199) Chariklo and (5145) Pholus

orbits on top of the map of the Solar System. Individual orbits have been created with the SPICE-

toolkit in order to get more precise results. This seems to be a good way to create specific orbits and

investigate them closer. It also gives higher precision and additional data (for example the velocities

of the objects, which one could also get from the elements) that one can use if needed.

If one wants to create all the centaur orbits, this is not the best way to do so if one does not

want to spend a lot of time manually downloading and configuring the kernels one by one. The

actual data handling can be automated easily, but managing the kernels in the first place seems to be

difficult to make automatic. One would need to create a program that reads the centaur names from a

list, accesses the terminal (in our case Linux terminal), inputs the centaur names in the right format,

downloads the kernels and transforms them into the correct binary format to be used. If one wants to

use larger data sets to get a quick grasp of the centaur (or asteroid etc.) orbits, it is faster to calculate

29

Chiron, Chariklo and Pholus Orbits with known Centaurs

-40 -20 0 20 40X (AU)

-40

-20

0

20

40

Y (

AU

)

Figure 3.4: Zoomed image of the outer Solar System with (2060) Chiron (pink), (5145) Pholus (or-

ange) and (10199) Chariklo (dark blue) orbits added (radius 40 AU).

the positions from the elements provided by MPC and only use SPICE for the most interesting objects

to get a better precision on the orbits and other additional data one might want.

When searching for SPICE kernels that provide the newest information of centaurs and SDOs, a

kernel for 2015 RR245 was found. This is a newly discovered dwarf planet listed together with the

centaurs and SDOs in the file used in creating the program Input_program.pro. Part of 2015 RR245’s

orbit is shown in Fig. 3.5. It is not complete, because the kernel is only covering a period of 100

years. 2015 RR245 is estimated to take 700 years to orbit the Sun once and it has been observed only

for one year [23]. Therefore its orbit is not yet well constrained and the hundred year period is only

showing part of its orbit.

30

Chiron, Chariklo and Pholus Orbits with known Centaurs

-100 -50 0 50 100X (AU)

-100

-50

0

50

100

Y (

AU

)

Figure 3.5: Zoomed image of the outer Solar System with (2060) Chiron, Pholus, (10199) Chariklo

and 2015 RR245 orbits added.

31

Appendices

32

A The NAIF SPICE Toolkit

The NAIF SPICE toolkit is an important part of an astronomers arsenal in creating simulations, han-

dling datasets and calculating positions of celestial bodies (and spacecraft) in the Solar System.

A.1 SPICE Concept

SPICE is a system that assists NASA to plan space missions and to interpret data from scientific

observations. It can be used to develop mission concepts and to analyse post-mission data. It can also

correlate data sets from different instruments with data from other instruments either on the same or

another spacecraft.

A.1.1 Kernels

The SPICE datasets are called kernels. They are composed of navigation information that has been

made available for easy use by scientists and engineers. Kernels are produced from accurate sources

(usually from mission operations center). They are normally accompanied by metadata that provides

descriptive information needed by users.

The kernel files consist of:

S- Spacecraft ephemeris as a function of time. (SPK-file)

P- Location of any target body (Planets, comets, asteroids etc.) as a function of time (SPK). P

kernels also include certain physical, dynamical and cartographic constants for the target bodies (for

example size and shape, orientation of the spin axis and prime meridian, in a PCK-file).

I- Instrument description kernel. It contains the descriptive data of a particular scientific instru-

ment (field-of-view size, shape and orientation parameters, in an IK-file).

33

C- Pointing kernel that contains a transformation (C-matrix) which provides orientation for a

spacecraft bus or a spacecraft structure on which the scientific instruments are attached.

E- Events kernel. It summarizes planned and unanticipated mission activities. The file set consists

of science plans, sequences and notes, but the kernel is not used very often (EK-file).

Aforementioned letters are contained in the SPICE acronym, since they are the most important

ones. However, there are some other important kernels that did not make it to the acronym.

For instance, a frame kernel (FK) contains specifications of the assortment of reference frames

used by flight projects. It also includes mounting alignment information for instruments, antennas

and possible other structures.

SCLK and LSK stand for spacecraft clock and leap seconds kernels. They are used in converting

time tags between various time measurement systems.

A.1.2 Using the SPICE Toolkit

The SPICE toolkit can be easily integrated into a users own program for an easy access of a wide

variety of data. However, one needs to do some initial setting up to get the toolkit to work properly

on ones operating system. The toolkit is available for C, FORTRAN, IDL and MATLAB. Here is

detailed information for setting it up on IDL and Linux, but the main principles should apply also on

other languages and operating systems, although specific details might differ a bit.

For IDL, the toolkit is available for a variety of operating systems. It includes the most common

ones (OSX, Linux and Windows) and Solaris. One can download the toolkit here [24].

In order to get the toolkit to work on Linux one must download a package called ICY. This can

be downloaded from the aforementioned website after choosing the operating system and whether

one needs a 32bit or 64bit version of the toolkit. ICY contains all the toolkit components in one

compressed file. There is also available an installation script for the package and specific information

on how to install ICY on a computer.

After succesfully installing ICY, one wants to get the specific kernels to work with. The easiest

way to do this is to use the terminal in Linux. In order to download the kernels straight through the

terminal, one needs to download a script called "smb_spk". The newest version of it is available here

[25].

34

smb_spk [-t|-b|-1|-2] [small-body] [start] [stop] [e-mail] file_name

As an example, here is the command used to get the kernels for (2060) Chiron used in Spice_centaurs.pro:

smb_spk -b "Chiron;" 1950-Jan-1 2050-JAN-1 email chiron19502050.bsp

This command gets a binary file named chiron19502050 from a time period from 1950 to 2050. It

also saves the email address for NAIF servers, which they can use to contact the user if need be.

When one has the kernels, one needs to create a text file where all the kernel file paths are defined.

Here is an example of that:

\begindata

PATH_VALUES = ( ’/home/granholm/Bachelors/Kernels2’ )

PATH_SYMBOLS = ( ’path’ )

KERNELS_TO_LOAD = (

’$path/LSK/naif0011.tls’ ,

’$path/PCK/gm_de431.tpc’ ,

’$path/PCK/PCK00010.TPC’ ,

’$path/SPK/chiron19502050.bsp’ ,

’$path/SPK/chariklo19502050.bsp’ ,

’$path/SPK/pholus19502050.bsp’ ,

’$path/SPK/2015RR245.bsp’ ,

’$path/SPK/2015HX10.bsp’ ,

’$path/SPK/2016LS.bsp’ ,

’$path/SPK/UN524.bsp’ ,

’$path/SPK/DE405.BSP’

)

This is the content of kernels2.txt that was used to configure the kernels in Spice_centaurs.pro

(not all of the kernels were used in the end). One can just copy the format of this example and change

the path names to fit the file path of the files. When that has been done, the kernels can be called in

IDL simply with the command:

35

spicedir=/path/to/the/directory

kernelfile=spicedir+’/Kernels2/kernels2.txt’

cspice_furnsh,kernelfile

One can find useful SPICE commands to be used in the programs from here [26].

36

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