MASTER’S THESIS2010:058 CIV

Universitetstryckeriet, Luleå

Karsten Dittrich

The Physics and Evolution of Small Molecular Clouds in Nebulæ - Globulettes as Seeds for Planets?

MASTER OF SCIENCE PROGRAMME Mechanical Engineering

Luleå University of Technology Department of Applied Physics and Mechanical Engineering

Division of Physics

2010:058 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 10/058 - - SE

Hier ist wahrhaftig ein Loch im Himmel!

Sir William Herschel, 1834

(English: Here is truly a hole in the sky!)

Abstract

Globulettes have recently been found in the Rosette Nebula, the Carina Nebula and other nebulæ.They are expected to be seeds of brown dwarfs and free-floating planetary-mass objects. The sizedistribution in the Carina Nebula was found to follow a power-law, and the same power-functionresulted in 880±250 globulettes in total in the Rosette Nebula. Compared to the 145 observedobjects in this nebula, many globulettes are beneath the resolution limit of the Nordic OpticalTelescope, which was used to explore the Rosette Nebula.

A simulation that arranged all these globulettes randomly in the nebula determined that someglobulettes are captured by stars. They are believed to form into one or more planets, orbitingthe star thereafter. The possibility that globulettes result into the formation of planets, orbiting astar, is some 4.75·10−2 per cent. According to this simulation, about 3.35·10−3 per cent of thestars with spectral type A to M host one or more planets that once have been globulettes.

Keywords: free-floating planets - H II regions - ISM: individual: Carina Nebula, Rosette Nebula- ISM: globules, globulettes - planet capturing

i

Acronyms

B68 Barnard 68 (object 68 in Barnard’s catalogue)CN Carina NebulaCTIO Cerro Tololo Inter-American ObservatoryESO European Southern ObservatoryFWHM Full Width at Half MaximumGMC Giant Molecular CloudHST Hubble Space TelescopeIR InfraRedNIR Near InfraRedNOT Norther Optical TelescopeNTT New Technology TelescopeRN Rosette NebulaSPH Smoothed Particle HydrodynamicsSST Spitzer Space TelescopeUV UltraVioletVLT Very Large Telescope

Units and symbolsIn this thesis, cgs units are used. That is, centimeter for length, gram for weight and second fortime.Definitions of other units used to describe distance:

• Angström (Å): 1 Å = 10−8 cm = 10−4 µm.

• Astronomical unit (AU) - the average earth-sun distance: 1 AU = 1.496·1013 cm.

• Parsec (pc) - the distance from which the radius of the Earth’s orbit around the sun takes upan angle of 1′′: 1 pc = 206,265 AU = 3.086·1018 cm. Thus, an angle of 1′′ in the RosetteNebula (at a distance of ≈ 1400 pc), corresponds to a distance of ≈ 1400 AU.

• Light-year (ly) - the distance that light travels in one year:1 ly = 63,240 AU = 0.3066 pc = 9.461·1017 cm.

Definitions of other units used to describe mass:

• The atomic mass unit (u): 1u = 1.6605·10−24 g

• The mass of the Earth (M♁): 1M♁ = 5.974·1027 g.

• The mass of the planet Jupiter (MJ): 1MJ = 1.899·1030 g.

• The solar mass (M�): 1M� = 1048MJ = 1.989·1033 g.

iii

Contents

1 Introduction 11.1 Formation of stars and planets . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Extrasolar planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Contraction of a cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Fragmentation of a molecular cloud . . . . . . . . . . . . . . . . . . . . . . . . 41.5 H II regions and emission nebulæ . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Stellar wind and the motion of the clouds . . . . . . . . . . . . . . . . . . . . . 71.7 Aim and outline of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Observations 92.1 Observations with the Nordic Optical Telescope . . . . . . . . . . . . . . . . . . 92.2 Observations with the Hubble Space Telescope . . . . . . . . . . . . . . . . . . 9

3 Globulettes 113.1 Interstellar cloud structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 What is a globulette? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Physical properties of a globulette . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Mass estimation and density distribution . . . . . . . . . . . . . . . . . . . . . . 163.5 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Size and mass distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.7 Extrapolation fit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Evolution of globulettes 234.1 Possible evolution scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Possibility of a capture as a free-floating planet-like object . . . . . . . . . . . . 264.3 Possibility of a capture as a cloud . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Computing the possibility of a star capturing a globulette . . . . . . . . . . . . . 27

5 Results 295.1 Fitting the Carina Nebula size distribution . . . . . . . . . . . . . . . . . . . . . 295.2 Extrapolating the Rosette Nebula size distribution . . . . . . . . . . . . . . . . . 315.3 Comparison of the two nebulæ . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Probability of globulettes being captured by stars . . . . . . . . . . . . . . . . . 34

6 Discussion 396.1 Size distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

v

Contents

6.2 Evolution of globulettes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Conclusions 45

Acknowledgments 47

Bibliography 51

A Derivation of the virial theorem 53

B Linear fitting with singular value decomposition 55

C The capture mechanism with SPH computation 57

Statutory Declaration 59

vi

List of Figures

1.1 The Rosette Nebula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 The Carina Nebula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Inner Carina Nebula with the HST . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 B68 without star background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 B68 with star background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Different appearances of the globulettes . . . . . . . . . . . . . . . . . . . . . . 153.4 Observed size distribution in RN and CN . . . . . . . . . . . . . . . . . . . . . 203.5 Observed mass distribution in RN and CN . . . . . . . . . . . . . . . . . . . . . 21

4.1 Fragmentation of a globule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 The Wrench trunk and fragmentation of a globulette . . . . . . . . . . . . . . . . 25

5.1 Carina Nebula fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Rosette Nebula fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Evolved and fitted size distribution in RN and CN . . . . . . . . . . . . . . . . . 335.4 Start arrangement of stars and globulettes . . . . . . . . . . . . . . . . . . . . . 355.5 End arrangement of stars and globulettes . . . . . . . . . . . . . . . . . . . . . . 365.6 Captured globulettes by spectral type of the capturing star . . . . . . . . . . . . . 375.7 Percentage of capturing stars for each spectral type . . . . . . . . . . . . . . . . 38

6.1 Twice the capture range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 Double amount of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.3 Time dependence of the captures . . . . . . . . . . . . . . . . . . . . . . . . . . 42

C.1 SPH simulation plots of captured condensations . . . . . . . . . . . . . . . . . . 58

vii

List of Tables

1.1 Percentage of planets around stars of different spectral types . . . . . . . . . . . 3

4.1 Average stellar density in the Milky Way . . . . . . . . . . . . . . . . . . . . . . 28

5.1 Percentage of star hosts for each spectral type . . . . . . . . . . . . . . . . . . . 38

6.1 Comparison of host spectral types . . . . . . . . . . . . . . . . . . . . . . . . . 40

ix

1 Introduction

Earth is “an utterly insignificant little blue-green planet far out in the uncharted backwaters of theunfashionable end of the western spiral arm of the galaxy” (Adams 1979). That is how DouglasAdams described the position of our planet in his novel The hitchhiker’s guide to the galaxy.

The Milky Way galaxy is a spiral galaxy with around 100 billion stars. It is composed of spiralarms, which rotate around the galactic center. If the rotation speed of the stars in a galaxy againstthe distance to the center of the galaxy is measured, there is a big discrepancy with the theoreticalvalue from the law of gravity (Zwicky 1933). The stars in the outer parts are much faster thanexpected. This can be explained, if one assumes that there is more mass in our galaxy than theluminescent one. According to some calculations (Turner 1999b), this would give rise to the ideathat almost 90 per cent of the mass of our galaxy is invisible to us, in short: dark.

This dark matter is mainly composed of nonbaryonic matter. There are many theories for it,but as of yet, all efforts to measure the form of it have failed. Dark objects, as, for instance,“dark stars” (black dwarfs, neutron stars, black holes or objects of mass around or below thehydrogen-burning limit) and interstellar cloud structures (Turner 1999a), were, for a long time,suggested as the missing dark matter. But even all “dark stars” and clouds together amount toonly a small fraction of the measured discrepancy.

Nevertheless, those “dark” objects are very interesting, and, due to some lucky circumstances,one can see interstellar cloud structures in some parts of the galaxy.

1.1 Formation of stars and planets

In order to describe the formation of stars, one should distinguish between low-mass (< 2M�)stars, and high-mass stars. The mechanism for high-mass stars is poorly understood.

Low-mass stars are mostly formed in giant molecular clouds (GMCs) or globules (see Sec-tion 3.1). According to the standard theory, star formation takes place in the core of dense andcold (10−20 K) molecular clouds, due to their gravitational energy.

During star formation, the cloud decreases in size by several orders of magnitude from apc- to a 100 AU-scale. Big clouds, of several hundred or thousand solar masses, fragment intosmaller clouds until the remaining clouds have approximately solar masses. All such clouds havesome initial angular momentum, due to some inhomogeneity in the cloud. Angular momentumis conserved during the contraction process, so the inner parts start to spin relatively fast. Atsome point, the infalling matter has enough transversal velocity to prevent direct accretion to thecentral star. So a disk-like structure, of 1− 2 per cent of the star’s mass, is formed around thestar. Planets are believed to evolve from such “protoplanetary” disks. These disks are assumed tobe the birthplace of planets, asteroids and comets, but the exact formation process is still poorly

1

1 Introduction

understood. The Hubble Space Telescope (HST) imaged numerous protoplanetary disks in theMilky Way. This indicates that solar systems such as ours are rather common in our galaxy.

The basic theory for the formation of planets, as described in Lissauer (1993), is as follows:Dust particles and primitive meteorites in the protoplanetary disks, with sizes of 0.05−100 µm,stick together, collide pairwise and thus form larger bodies, so-called planetesimals. The larger abody is, the faster its surface grows. That results in more headwind, i.e., it is slowed down due tofriction. So those larger bodies spiral inwards, and find their orbits closer to the star. The largestobjects grow rapidly, and become by far the most massive bodies for each accretion zone. Giantgas planets are formed around icy solid cores. These are the most massive bodies in such a stateof the development of a solar system. They catch most of the gas from the surrounding discs.Due to this gas drag, the gaseous planets nowadays consist to 90 per cent (Jupiter) or 77 per cent(Saturn) of hydrogen and helium. Terrestrial planets have a higher proportion of heavier elementsand hence are rocky.

Free-floating planetary objects were recently discovered (Lucas & Roche 2000) in star-richregions, such as the Orion Nebula. Their detection raises the question of their origin. Fogg(1990) gave some ideas on how these free-floating objects may originate. He suggested that theywere either formed like the stars, just out of smaller clouds, or they were formed around stars inprotoplanetary discs and were later ejected by a destructive event. Such an event could be, forinstance, a supernova explosion or a close stellar encounter.

If the new star has a mass lower than 0.08M�, no hydrogen fusion can start. This means thestar appears very dark, which is why it was called a “black” dwarf in Kumar (1962). Later Tarternamed these objects brown dwarfs in her Ph.D. thesis. Actually brown dwarfs are red, but theterm red dwarf was already used, when Nakajima et al. (1995) first observed a brown dwarf.Today we know of the existence of many brown dwarfs, e.g., in the Pleiades Cluster (Martínet al. 2000).

1.2 Extrasolar planets

“Just fourteen years ago the Solar System represented the only known planetary system in theGalaxy, . . . Since then, 320 planets have been discovered orbiting 276 individual stars.” (John-son 2009). Extrasolar planets (or shorter, exoplanets) were mostly discovered using Dopplertechniques or photometric transit surveys. Some more have been detected by gravitational mi-crolensing, and a few were directly imaged. These techniques are described in various articlesand textbooks, e.g., in Lunine et al. (2009). Today, the “planet count” is at 429 planets around362 stars (NASA Planet Quest 2010). At this web address, a list of all exoplanets can be found.

According to Johnson et al. (2007), stars of different masses have different possibilities ofbeing host for a planet. Johnson et al. plotted the percentage of stars with planets against thestellar mass. The main quantities from this plot are summarized in Table 1.1. This table showsthat most stars with planets are sun-like, while the relative number is highest for larger stars.This table includes neither the biggest, nor the smallest stars. The reason, put simply, is that noexoplanets have been found around those stars yet.

In our solar system we find the heavy gas planets in the outer parts, whereas the terrestrial,

2

1.3 Contraction of a cloud

lighter planets are in the inner parts. However, there are observations also of massive planetsvery close to some stars. They form a distinct class of exoplanets, called “hot Jupiters”. Theseplanets have masses of around 1MJ and orbit very close to their stars. They have some remarkableproperties, such as their low density. Due to the latter, they would “float on water.” Also, theyhave to be migrated in this low orbit, as they cannot have been formed that close to a star. “HotJupiters” are easier to observe as they transit the star more frequently. Also, those planets are verybig, due to their mass and low density. Although most “hot Jupiters” have a mass of around 1MJ ,some with the mass of 21M♁ were detected (Léger et al. 2009). However, there is no evidencethat there is a lower mass limit for those planets.

Table 1.1 – The percentage of stars with detectable planets as a function of stellar mass, from Johnsonet al. (2007, Fig. 6). The number of hosts (NHosts), i.e., stars with planets, and the number of studiedstars (NStars) can be found in the same figure. The spectral classification of the stars was taken fromJohnson (2009). The uncertainty limits are from Poisson statistics.

Spectral type Stellar mass[M�

]NStars NHosts % stars with planets

M4 - K7 0.1 - 0.7 169 3 1.8 ± 1.2K5 - F8 0.7 - 1.3 803 34 4.13 ± 0.67F5 - A5 1.3 - 1.9 101 9 8.80 ± 2.27

1.3 Contraction of a cloud

Every massive object affects its surroundings by gravity. So molecular clouds, which consistof tiny particles such as molecules of hydrogen and helium, and dust particles can contract dueto gravity. But for such diffuse objects, one has to take the movement of the particles intoconsideration.

The virial theorem (derived in Appendix A) applicable for clouds is

2(Kint−Kext)+W +Emag =12

d2

dt2 I, (1.1)

where the total kinetic energy inside the cloud can be described by

Kint =∫V

(32

Pth +12

ρv2)

dV ≡ 32

PV . (1.2)

Here Pth is the local thermal pressure, v the local turbulent velocity, V the volume of the cloudand P is expressed by the pressure of the ideal gas equation:

P =NkBT

V=

MµuV

kBT . (1.3)

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1 Introduction

The symbols have the same meanings as explained above.The external energy Kext is a term concerning, for example, the outer pressure from the sur-

rounding environment. For a spherical cloud of uniform density, mass M and radius R, thegravitational energy W becomes

Wsphere =−3GM2

5R. (1.4)

The magnetic energy Emag can be described by the magnetic field B. The total magnetic energyfor a sphere of radius R can be expressed by the magnetic energy density B2/8π times the volumeof the sphere:

Emag =B2

8π·4π

3R3 =

B2R3

6. (1.5)

On the right-hand side of Equation (1.1), one finds the second derivative of the moment ofinertia I. Generally, the moment of inertia can be written as

I =∫

r2 dm. (1.6)

The rate of change of the size and shape of the gas system is described by I in the virial theorem.If one neglects the magnetic energy and the outer pressure, then for static clouds, i.e., I ∼= 0,

Equation (1.1) simplifies to

2Kint +W = 0. (1.7)

When the kinetic energy is balanced by the gravitational energy, the cloud is referred to bein virial equilibrium. Thus, if a cloud changes in shape and size, Equation (1.7) becomes aninequality. If I < 0, the ratio 2Kint

−W < 1 and the cloud becomes smaller, i.e., contracts. For theopposing case I > 0, the ratio 2Kint

−W > 1, and the cloud dissipates into space.

1.4 Fragmentation of a molecular cloud

Usually GMCs and globules have masses of several hundred M� (compare Section 3.1). Butthere are no stars that are so massive. Thus these clouds have to fragment before they collapse toa star. The size distribution of such fragments is important for this analysis.

The clouds split up into smaller structures with a mass distribution that follows a power law.This power law was determined with the help of smoothed particle hydrodynamic (SPH) sim-ulations. They lead to a mass distribution following dN/dM ∝ Mν with ν . −2 for turbulentmodels (Klessen 2001). Without turbulence, the exponent becomes rather ν ≈ −1.5 (Klessen2001; Klessen et al. 1998).

If asteroids collide with each other, they form many small objects and a relatively small numberof larger objects. The size distribution of such objects follows a power law (Capaccioni et al.

4

1.5 H II regions and emission nebulæ

1986). The size distribution of molecular clouds shall also follow a power law. There are somevalues for the exponent in Ormel et al. (2009), yet they are for microscopic grains. Hence thesize distribution also of the globulettes is assumed to follow a power law.

1.5 H II regions and emission nebulæ

H II regions are regions with ionized hydrogen. For all elements, the following description isvalid: Elements marked with “I” are neutral, those marked with “II” are ionized once, and so on.There are a lot of H II regions in the Milky Way, and some of these bright nebulæ are visible tothe naked eye. Nevertheless, they were first noticed in the 17th century, after the advent of thetelescope.

H II regions can be very large - one single O-type star may ionize a region of up to 100 pc,while a B type star still ionizes a few parsecs. However, most H II regions are ionized by agroup of O- and B-type stars - the brightest members of a stellar cluster. O- and B-type stars arevery massive and luminescent, with very strong stellar winds (a few times 1000 kms−1), as wellas with powerful and energetic radiation. Due to their high loss of mass, their lifetime is verylimited, so O-type stars leave the main sequence after 3−6 million years. The radiation pressureand the stellar wind condense the surrounding gas, creating a thin, but dense, shell in the innerpart of the local cloud. The cloud expands outwards with accelerated speed. The region near thecluster is filled with warm plasma and appears as a “bubble” in the cloud. Typically H II regionshave a particle number density of 10− 100 cm−3, consisting mainly of hydrogen (73 per cent)and helium (35 per cent). Heavier elements are much rarer, and make up the remaining 2 per cent.

H II regions usually appear reddish. It is easily explained why they have this appearance.The high-energetic photons from the UV radiation from the O- and B-type stars knock out theelectrons of hydrogen. In the region of the warm plasma, the protons and electrons are too fast torecombine. In the bright regions, it is cold enough for them to recombine, so the electrons cascadedown to the lowest energy level. During this they emit light at fixed wavelengths. This givesan emission line-spectrum, wherefrom we know the constituents of the nebulæ. In the visiblespectrum, by far the strongest emission line is the Hα -line with λ = 6563 Å (= 0.6563 µm).Light with this wavelength is emitted by the electrons going from the third to the second excitedlevel of hydrogen.

The closest of such nebulæ is the Orion Nebula, at a distance of 414 pc (Menten et al. 2007).This nebula has been the object of numerous studies, due to its closeness. There is an over-whelming amount of evidence that this nebula is a star-birth region (Hillenbrand 1997). O‘dellet al. (1993) have found protoplanetary discs (proplyds) in the Orion Nebula. Another very fa-mous H II region is the Rosette Nebula (Figure 1.1), which is one of the two main areas of thiswork. It is situated around the cluster NGC 2244, with several O- and B-type stars within. Themost accurate distance to NGC 2244 is derived to be 1.39± 0.10 kpc (Hensberge et al. 2000).It spans some 30.6 pc in diameter and has an age of a few Myr (Viner et al. 1979). It is, as theOrion Nebula, a good region to study star formation and the interaction between the H II regionand the surrounding molecular cloud. The Rosette Nebula (RN) shows many interesting struc-tures, such as the rotating elephant trunks (Schneps et al. 1980; Gahm et al. 2006) and many

5

1 Introduction

globulettes (Grenman 2006; Gahm et al. 2007). The elephant trunks have an estimated age of(2.6−5.8) ·105 yr (Schneps et al. 1980).

Figure 1.1 – The Rosette Nebula, a large emission nebula some 1400 pc away. The hole in themiddle was blown free by the strong stellar winds of the open cluster NGC 2244, seen greenish inthis representation. In the bright region, several filamentary structures and other dark markings can bespotted. This false-colour image uses three emission lines: Hα for red, [O III] for green and [S II] forblue. Image source: http://www.noao.edu/image_gallery/images/d2/ngc2237.jpg.

6

1.6 Stellar wind and the motion of the clouds

Probably the most impressive H II region is the Carina Nebula (Figure 1.2). It hosts also thebiggest star found in our galaxy so far. The star (η Carinae) has a mass of 100 solar masses(Davidson & Humphreys 1997) and is 5 ·106 times as bright as our sun. This star is currentlyclassified as a luminous blue variable binary star that has brightness variations. For instance, itappeared as the second brightest star at the night sky in 1843. Compared to the large distance ofapproximately 2.3 kpc (Smith et al. 2003), this shows the amazing power of this star. The CarinaNebula hosts several clusters and spans about 80 pc across, that is about seven times the sizeof the Orion Nebula. In this nebula, several objects were found that might be proplyds (Smithet al. 2003) or globulettes (Grenman & Gahm 2009/10). In this analysis it is assumed that theseobjects are globulettes.

Figure 1.2 – The Carina Nebula here as a mosaic of images collected with the 1.5-m Danish telescopeat ESO’s La Silla Observatory. The nebula, 7500 ly away, spans some 260 ly across and is as big asseven Orion Nebulæ. The brightest star in this image is what can be the heaviest star in our MilkyWay galaxy, called η Carinae. Image source: http://www.eso.org/public/archives/images/large/etamosaicnm2.jpg.

1.6 Stellar wind and the motion of the clouds

All stars lose mass over time. The mass loss M is measured in M�/yr. It can be so severe thata star throws out 10−3 solar masses per year. Our sun loses 10−14 M�/yr, so its mass loss does

7

1 Introduction

not affect its development. Stars of the spectral type O and B have very strong stellar winds, withvelocities up to 2000 kms−1 (Weaver et al. 1977). These stars have a mass loss M = 10−6 M�/yr(Snow & Morton 1976).

For the RN, the velocity of the stellar wind is ∼ 2000 kms−1 (Conti 1978). However, thestellar winds from the central cluster of the RN are stopped at the shock front. But, due to thisstop, the warm gas outside the inner bubble expands. This expansion moves the molecular cloudsagainst the stars, of which the latter are many orders of magnitude smaller than the globulettes insurface area.

The globules (and with them the globulettes) in the RN move outwards with the velocity of23 kms−1 (Schneps et al. 1980). However, Schneps et al. used 1600 pc as the distance to theRN, so this value has to be corrected to ∼ 20 kms−1.

1.7 Aim and outline of the workIn this work, the statistics of the globulettes in the Rosette Nebula (RN) (Grenman 2006) and theCarina Nebula (CN) (Grenman & Gahm 2009/10) will be compared in order to estimate the totalnumber of globulettes in the RN.

Therefore, the size distribution of the globulettes in the CN, which goes to much smallersizes, will be fitted to a power-function. This curve will be used to fit and extrapolate the sizedistribution of the globulettes in the RN.

Thereafter, a simulation is used to estimate how many of these globulettes may be captured bya star. This program also distinguishes between the spectral types of the stars.

The outline of this work is as follows:

• Chapter 2 gives an overview of the observations of the studied nebulæ.

• Chapter 3 describes interstellar cloud structures, gives details about the physical parame-ters of the globulettes and how they are computed, along with some statistical calculations.

• Chapter 4 examines the evolution of the globulettes and gives an outline of how they mayevolve further on.

• Chapter 5 presents the results.

• Chapter 6 contains a discussion about the fate of the globulettes, and age discrepancybetween the CN and the RN. Also some suggestions for future work are given.

• Chapter 7 gives the conclusions.

• Appendix A shows the derivation of the virial theorem.

• Appendix B explains how one can fit linear graphs with singular value decomposition.

• Appendix C summarizes the capture mechanism from Oxley & Woolfson (2000).

8

2 Observations

2.1 Observations with the Nordic Optical Telescope

Ten H II regions were observed during 5 nights in December 1999 and 5 nights in Novemberand December in 2000. The observations were carried out on behalf of Gahm and Kristen withthe Nordic Optical Telescope (NOT) on the island of La Palma, Spain. NOT’s main mirror has adiameter of 2.56 meter.

The Rosette Nebula was observed during one night in 2000 with the ALFOSC camera. Thiscamera has a field of view of 6.4′× 6.4′. The used filters were centered at 6563 Å, the wave-length of the Hα radiation. The exposure time was typically 1800 s, while the resolution underbest conditions was about 0.4′′. The CCD detector has a resolution of 0.188′′ per pixel. In theobservation in 2000, a 180 Å FWHM filter was used. This filter allowed the inclusion of thewavelengths of the [N II]-emission lines at 6548.1 and 6583.6 Å. From the stars in the field,angular resolution was measured to be in the range of 0.7− 1.3′′. Hence, the smallest objectsresolved by the NOT in the Rosette Nebula span 980−1820 AU across, depending on the fieldwhere the objects are found. It has to be noted, that not the whole nebula was covered by theobservation and thus some smaller objects in other areas can have escaped detection.

The images were corrected for instrumental effects and cosmic ray excitations. Bias, darkcurrent and flat field correction were applied, as well as corrections for the sky background asextracted from sky field exposures. The night during the observations was dark, except for somemoonlight in parts of the nights of December 2 and 3.

2.2 Observations with the Hubble Space Telescope

The Carina Nebula was observed with the Wide-Field Channel of the Hubble Space Telescope’s(HST’s) Advanced Camera for Surveys. “Because of the large area of the Carina Nebula, it wasimpractical to make a single contiguous map of the entire region with HST . . . ” (Smith et al.2010). Thus, the nebula was mapped with two joint programs (GO-10241 and GO-10475) inCycles 13 and 14 of the HST. During Cycle 13, a large (13.5′× 27′) mosaic image was made.In this first survey, the brightest inner parts (the clusters Tr14 and Tr16, the Keyhole Nebulaand η Carinae) were given priority. An HST image of the inner Carina Nebula can be seen inFigure 2.1.

In all observations of this survey, the F658N filter was used. This filter transmits Hα and [NII] at λ = 6583.6 Å. The total exposure time was 1000 s at each position (500 s per individualexposure). The pixel resolution of the Wide-Field Channel in the HST is 0.05′′ per pixel, which

9

2 Observations

is also the angular resolution, as there are no atmospheric turbulences. Thus the smallest objectsin the Carina Nebula that are resolved by the HST span ∼ 115 AU across.

All images were reduced and combined using the PYRAF routine MULTIDRIZZLE, withmanually determined pixel shifts using the pipeline-calibrated, flat-fielded individual exposures.This routine not only combines the exposures, it also generates static bad-pixel masks, correctsfor image distortions and removes cosmic rays using the CR-SPLIT pairs. During this investiga-tion, 98 per cent of the Carina Nebula was imaged with the HST. The remaining gaps were filledwith images obtained with the MOSAC camera on the 4 m Blanco telescope at Cerro TololoInter-American observatory (CTIO).

Figure 2.1 – This image of the inner Carina Nebula was made from a large Hα mosaicfor the intensity scale using the Advanced Camera for Surveys at the Hubble Space Tele-scope (HST). The ground-based narrow-band images obtained with the MOSAIC camera onthe 4 m Blanco telescope at CTIO was used for color coding. Blue is [O III], green isHα and red is [S II]. North is to the upper right. This image can be found in full res-olution (24,000× 12,000 pixel) at http://hubblesite.org/gallery/wall_murals/download/2-carina-color/full/carina_80x40full.jpg.

10

3 Globulettes

Globulettes were first investigated in Grenman (2006). In her licentiate thesis all observed prop-erties of the globulettes were determined. The size of the globulettes was measured, while fittingan ellipse in the dark pattern of each globulette. For non-spherical globulettes the orientationwas determined. The mass was calculated due to their extinction values. Assuming ellipsoidicbodies, their volume, mass density and particle density were evaluated. Also, their tendency tocollapse or dissipate into space was computed with the help of the virial theorem.

There are several evidences (Grenman 2006; Gahm et al. 2007) that globulettes are not smallerglobules, or a “tail” of normal globules. They form a separated group in mass and size distribu-tions. They also have a much higher particle density than “tails” of globules usually have. Finally,they have another appearance than globules. Globulettes form more roundish or teardrop shapesrather than complex structures.

Grenman included two H II regions in her investigation (2006), and in Gahm et al. (2007)all 10 regions of the observation with the NOT (see Section 2.1) were included. A total of 173clouds were found in these surveys, and 145 of them lie in the Rosette Nebula (RN). In order tomake the statistics more reliable, only the number of objects from the RN are used in this work.

3.1 Interstellar cloud structures

What Herschel in 1834 has described as “holes in the sky” (Houghton 1942) is nowadays wellknown as dark nebulæ. The Barnard Catalogue (Barnard 1919) was a first list of “182 darkmarkings”. One very distinct cloud from Barnard’s catalogue is Barnard 68, seen in Figure 3.1.As its distance to the sun is just 500 light-years (ly), it is not surprising that not a single star is inour field of view to this dark nebula. This opaque cloud blocks the light of over 3700 Milky Waystars (Alves 2001), due to its short distance to us. They are visible at longer wavelengths, as themolecular clouds are transparent for infrared (IR) light. Images, like Figure 3.2, prove that theseclouds are not real holes in the sky.

In optical wavelengths, the internal structure of the clouds is hard to study. For such investiga-tions, CO-emission lines are used. These studies show that the dark clouds consist of molecules,mainly hydrogen. After these observations, dark clouds were named molecular clouds.

Molecular clouds differ very much in size and mass, so they are classified accordingly. Thereare tiny structures with masses in the range of a Jupiter mass (MJ). But there are also huge cloudswith more than 100,000 solar masses (M�). Sizes range from some 100 astronomical units (AU)to 100 parsecs (pc). Star formation takes place in gravitationally unstable and massive clouds (seeSection 1.1).

11

3 Globulettes

Figure 3.1 – B68 seen by the Very Large Telescope (VLT) of the European Southern Observatory(ESO) and reproduced from http://www.eso.org/public/archives/images/large/eso0102a.jpg. This cloud appears very distinct against a star-rich background of the Milky Way.The molecular gas absorbs almost all visible light from the background stars. B68 is 0.5 light-yearsin diameter, and is composed of a gas of about two solar masses (M�). This cloud might collapse dueto its own gravity, and form one or more stars, possibly with planets. See Figure 3.2 for backgroundstars.

12

3.1 Interstellar cloud structures

Figure 3.2 – This view of the dark cloud B68 is a false-color composite based on a visible (hererendered as blue), a near-infrared (green) and an infrared (red) image. Since the light from starsbehind the cloud is only visible at the longest (infrared) wavelengths, they appear red. This imageis composed by measurements from the Very Large Telescope (VLT) and New Technology Tele-scope (NTT), and is reproduced from http://www.eso.org/public/archives/images/large/eso0102b.jpg. See Figure 3.1 for the VLT image.

13

3 Globulettes

The largest molecular clouds build the class of the Giant Molecular Clouds (GMCs). GMCshave material with the mass of 104M�. Their lifetime is calculated to be some 107 years, thedensity is about 102 cm−3, and the temperature inside is 10−100 K.

There are some smaller molecular clouds, of which the smallest ones are called globules. Theyhave a mass of a few solar masses up to 103M�. They are much denser, with some 104 particlesper cubic centimeter, and also colder, with temperatures around 10 K. They are thought to bea site of star formation (Bok 1977). Yun & Clemens (1990) proved this assumption with thedetection of dense cores in many clouds. The globules are divided into two classes, namely thecometary globules and the Bok globules. Cometary globules have a cometary shape. Small,isolated clouds with a dense core, called the “head”, and a long tail with various structuresare some major characteristic features for them. Some have bright rims, usually at the headof the globule. The tails point away from the most luminescent nearby stars. This shows theinteraction between the ultraviolet (UV) radiation, and the strong stellar winds of O- or B-typestars and the molecular clouds. They are located in emission nebulæ such as the Gum Nebula(Hawarden & Brand 1976; Sandqvist 1976) or the RN (Herbig 1974), and have sizes that rangefrom 0.05 to 1 pc, densities of some 104− 105 cm−3, and temperatures around 10 K. Bokglobules are named after Bart J. Bok, who first named them globules (Bok & Reilly 1947). Theyare relatively isolated, can be seen in optical spectral regions, and are sharply outlined againsta background of stars. Bok globules can be larger than cometary globules (0.2− 2 pc), havemasses of 5−500M�, and are shown (Yun & Clemens 1990) to be sites of star formation. Coreswithin the globules typically consist of material with 1−5M� and have radii of 0.05 pc, givinga density of 106 cm−3.

The smallest distinct class of molecular clouds are the globulettes. The mass distribution ofthe globulettes reach from the lightest objects to some with 0.1M� (≈ 100MJ). They were firstdiscussed in Grenman (2006), and might be the site of planet or low-mass brown dwarf formation(Gahm et al. 2007). These objects are the focus of my project work, and will now be describedin more detail.

3.2 What is a globulette?

A globulette is a “tiny” molecular cloud with a mass smaller than 0.1M�. They can be distin-guished in massive globulettes within the mass range of 30− 100MJ

(≈ 0.1M�

)and low-mass

globulettes with masses lower than 30MJ . Usually, globulettes measure some kAU in size andas such they are much bigger than the solar system. But due to their distance they appear as tinydark spots in the observational limits with nowadays telescopes. They occur in many differentshapes, as shown in Figure 3.3.

To give a definition for the size of a globulette, Grenman (2006) described the contour as theline where the pixel intensity dropped to 85 per cent intensity of the interpolated background.With this description the contour line is defined and, as the globulettes have an elliptic appeare-ance, they are fitted to ellipses, giving semi-major and semi-minor axes. This is done because allcomputations are performed much easier if one has an ellipse, rather than an undefined, almostelliptic shape. Because we do not have any possibility to determine the real three-dimensional

14

3.2 What is a globulette?

Figure 3.3 – Illustration of the variety of the globulettes in the Rosette Nebula and the Heart Nebula(IC 1805). All images are taken with the NOT in Hα light. They are marked with numbers of the datatables in Grenman (2006). Each panel is 16′′×16′′, except the second-lowest row, which has the size5.5′′×5.5′′. North is up and east to the left. This image was reproduced from Grenman (2006) withpermission of the author.

15

3 Globulettes

shape yet, an ellipsoidic body with two semi-minor axes is assumed. This gives the potential ofcalculating the volume and the densities.

3.3 Physical properties of a globulette

The contour of a globulette is defined as the border, where the pixel intensity is 85 per cent ofthe interpolated background. The semi-minor axis β as well as the semi-major axis α are givenby an ellipse that is fitted to this contour. The size of the globulette is then defined by

R =√

αβ . (3.1)

This provides one comparable value for all globulettes. Assuming the whole globulette is anellipsoid with two semi-minor axes, as in Gahm et al. (2007), the volume can be calculated as

V =4π

3αβ

2 =4π

3R2

β . (3.2)

This assumption leads rather to a lower bound on the volume for such a globulette. They mightbe more elongated in the line of sight. Also, if the ellipsoid is inclined, the measured axesare smaller than the real axes and thus the volume is estimated to be too small. In Schnepset al. (1980), an inclination of 45◦ is assumed for the calculations, and this can also be done forthe observed globulettes, as they have some connections to the globules. However, there is noevidence that they cannot have any other inclination.

The orientation, given through the position angle, was calculated for good elliptic globulettes,where the relation between the axes is α/β > 1.5. Most globulettes point inward to the centralcluster, showing a strong relation to the stellar wind and the extreme radiation of the latter.

3.4 Mass estimation and density distribution

The mass of each globulette is estimated through measurements of the extinction rate. Therelation between the extinction rate and the column particle density of hydrogen is described byBohlin et al. (1978). It is expressed as

N (H2) = 9.4·1020AV[cm−2mag−1] , (3.3)

where AV is the extinction (absorption plus scattering) for the photometric V-band, i.e. the visibleband. The extinction Aλ for a wavelength λ is defined by

Aλ =−2.5log(

II0

), (3.4)

where I is the light-intensity measured in a certain area inside the globulette, and where I0 refersto the nebular background. However, the observations were carried out in the wavelength of

16

3.4 Mass estimation and density distribution

6563 Å (Hα ). Therefore the relationship between the extinction at λ = 6563 Å and the V-band(centered at λ = 5500 Å) is needed. It was found, with the help of some interpolation of thetabulated values in Savage & Mathis (1979), to be

AV = 1.2Aα (see Grenman (2006) for further description). (3.5)

In order to get the column mass density, the mass of the hydrogen molecule has to be taken intoaccount. Furthermore, one has to take into consideration that hydrogen makes up only 73 per centof the total mass. So, the relationship between the column particle density and the column massdensity is

N = 2·1.67·10−24 ·1

0.73N (H2) = 4.58·10−24N (H2)

[ gcm2

]. (3.6)

Hence, inserting Equations (3.6) and (3.5) in Equation (3.3) gives a direct relationship betweenthe measured extinction and the column mass density:

N = 5.2·10−3Aα

[g

cm2mag

]. (3.7)

With this relation, one can calculate the mass per pixel in each globulette. This pixel mass isthen used to compute the total mass of the globulette by simply multiplying this mass with thenumber of pixels inside its contour. But, we do not know whether the measured extinction isaltered by foreground emission in the H II region. Therefore Grenman (2006) considered twoextreme cases. In the first case, she assumed that the globulette is in front of the nebula in theline of sight, and hence has no foreground emission at all. That leads to a minimal mass. Thesecond case pretends that the globulette is in the background of the nebula. Thus in this casethe light measured from the darkest pixel (i.e., the highest extinction) is caused by foregroundHα emission by 95 per cent. The value of 95 per cent is chosen for “technical” reasons to avoiddivision through zero, which would lead to infinite mass. The latter gives a maximal mass. Forcalculation, the arithmetic mean of these cases is used, denoted in my work by M. Comparisonwith objects from Gonzalez-Alfonso & Cernicharo (1994) gave good agreement.

By computing the volume with Equation (3.2), the mass density can be derived, simply fol-lowing the relation ρ = M/V . The particle density then is computed as

n =ρ

µ u, (3.8)

where µ = 2.4 is the mean molecular weight for molecular clouds, and u is the atomic mass unit.

Density distribution evaluations in Grenman (2006) show that the globulettes in the RosetteNebula are likely to have uniform densities. The density seems to flatten with respect to distancethan expected from a uniform distribution in the very outmost parts of the globulettes. This mightbe a sign of outgassing due to interaction with the plasma environment.

17

3 Globulettes

3.5 The virial theorem

In order to determine whether a globulette is about to dissipate into space, collapse due to itsown gravity, or not change its appearance, one may apply the virial theorem (see Section 1.3).For the sake of simplicity and calculability, a spherical cloud is assumed, with a homogeneousgas temperature and density without turbulence or rotational energy. For the first investigationthe simple form of Equation (1.1),

2Kint +W =12

d2

dt2 I (3.9)

is used. Therefore, one uses Equation (1.3), inserting the derived values for the mass and the vol-ume. The temperature is assumed to be 10 K for round globulettes, whereas teardrop globuletteshave T = 15 K (Gonzalez-Alfonso & Cernicharo 1994). With this, the kinetic energy as well asthe gravitational energy can be calculated, using the mass and size for each globulette.

The ratio

2Kint

−W

< 1 the object will contract= 1 the object will not change in size> 1 the object will dissipate into space

(3.10)

checks the future of the globulettes. For all globulettes in Grenman (2006), the ratio lies between5 and 120, indicating that all objects would expand and disrupt.

However, Equation (3.9) is a much simplified version of the whole virial theorem (Equa-tion 1.1). Thus more terms have to be taken into consideration. Internal energy, like turbulenceor magnetic fields, would make this situation even more pronounced. Turbulence inside the glob-ulettes is so far neglected, as there are no signs of any internal activity that would show someturbulence. Magnetic fields are assumed to be negligible, although there is no observational in-formation about that. Another disrupting force is photo-evaporation, and this was discussed inKuutmann (2007). Due to his computations, the photo-evaporation of the globulettes is severeand dominates all other terms, reducing the lifetime to 5 ·104 years, compared to the calculatedlifetime of (2−4) ·105 years in Gahm et al. (2007). However, Kuutmann states in the discussionthat some values (e.g., the UV photon flux) might be overestimated. Additionally, no bow shock(i.e., the boundary between the stellar wind of the central cluster and the H II region) is includedin his work, although it would stop the stellar wind. Thus, just the innermost globulettes of thenebula may experience severe disruption due to photo-evaporation.

Furthermore, the external pressure from the environment, i.e., the thermal (turbulent and gaspressure) and radiation pressures, is expected to be strong. The thermal pressure for a stableobject is derived from the general virial theorem (Equation 1.1) to be

Pout =2Kint +W

3V. (3.11)

The radiation pressure just affects the side of the cloud facing the central star cluster. As there

18

3.6 Size and mass distributions

is no concrete information about the surface towards the central cluster this pressure was notincluded in the analysis. The calculated values for the globulettes, using Equation (3.11), are ingood agreement with the estimations made by Gahm et al. (2006). Now a new ratio determinesthe future of the globulettes:

2Kint

−W +3PoutV, (3.12)

with the same cases as in Equation (3.10).

This new relation points out that most globulettes will either contract, or are close to virialequilibrium. For the ones that are about to contract, the free fall time can be calculated, usingEquation (4.1) in the next Chapter.

3.6 Size and mass distributions

There is not an unrestricted resolution in our observation. In the Carina Nebula (CN) the res-olution limit is 115 AU. Yet the distribution falls off for objects smaller than ∼ 300 AU. Thisshows that not all objects at the resolution limit can be detected. It seems logical to choosethe maximum in the bigger graph in Figure 3.4 for the minimum of the fit range, as the power-function is described best with these bins. As maximum it is sensible to chose the fifth bin, asall bins beyond this one do not follow the power-law very well. It has specific to be noted, thatthe observed globulettes in both nebulæ were extracted from some regions. Not the whole neb-ulæ were covered by the surveys. The CN was fully covered by the observations, but the data Igot from Grenman & Gahm (2009/10) are just from selected regions. Hence, there can be someincompleteness in the distributions.

Objects in the Rosette Nebula (RN) with sizes beneath 980 AU are invisible, according tothe minimal angular resolution mentioned in Section 2.1. In some areas the angular resolutionis 1.3′′, corresponding to 1820 AU as the smallest object that can be seen. Observations withthe HST show that there can be much smaller globulettes found in the CN and, based on thatfact, also smaller ones in the RN are expected. As explained above, it is reasonable to chose themaximum of the distribution. Hence the minimum of the fit range is chosen to be 2.25 kAU asthe smallest reliable size.

The size distribution of the globulettes in both nebulæ can be described by a power-function:

f (R) ∝ R−α . (3.13)

This behavior is expected according to the origin of the globulettes. As discussed in Section 1.4,the size distribution of the fragments of a cloud follows a power-law. In Section 4.1 the glob-ulettes’ origin is shown to be due to fragmentation.

19

3 Globulettes

Figure 3.4 – This size distribution shows the calculated sizes (see Equation (3.1)) in both the Rosetteand the Carina Nebulæ versus the observed number of globulettes for each size bin with two differentbin ranges. The height is normalized due to different numbers of globulettes in each nebula. Theerror bar (square-root of the number, as they are counted values) to the left corresponds to the Carinadata, while the one to the right corresponds to the Rosette data. The histogram of the globulettes inthe Carina Nebula has a slope (excluding the first bin) that can be described by a power-function.The resolution limit for the Carina Nebula is 0.115 kAU. The histogram of the globulettes in theRosette Nebula seems to have a similar power-law up to the bin for sizes between 2.25 kAU and2.5 kAU, while it decreases for smaller sizes. The resolution limit is at 1.82 kAU in the worst case(see Section 2.1).

In Figure 3.5, the mass distributions of both nebulæ are mapped. As the mass is approximatelyproportional to the radius squared, this distribution is more spread for the Rosette Nebula. Also,for the Carina Nebula, the mass distribution is more spread (upper-right graph in Figure 3.5), butit still has a shape that can be explained with a power-function.

It follows that the CN distribution can be fitted with a power-law according to the fact that thefragmentation of clouds, as discussed in Section 1.4, follows a power-law.

20

3.7 Extrapolation fit functions

Figure 3.5 – This mass distribution shows the calculated masses of the globulettes, in both the Rosetteand the Carina Nebulæ versus the observed number of globulettes for each mass bin. Despite the factthat there are less globulettes in the Rosette Nebula, it is easily noticed that the observed globulettesin the Carina Nebula are much lighter than the ones in the Rosette Nebula. The smaller graph in theupper right shows that also in the Carina Nebula not all small globulettes could be observed. For bothnebulæ the square-root of each bin height was taken for the error bar.

3.7 Extrapolation fit functions

As M ∝ R2, it is reasonable to use the size distributions of the nebulæ for the extrapolations.Since the CN distributions can be fitted by a power-function, a power-law fit like

fCN(R) = ACN ·RBCN (3.14)

can be applied to the graph. Easier to fit is the logarithm of this function, which is

log fCN(R) = log(ACN ·RBCN

)= logACN +BCN · logR. (3.15)

21

3 Globulettes

This is a linear function, and hence it can be fitted by a singular value decomposition algorithm(see Appendix B). Thus the fit function for the CN is

yCN(x) = log fCN(R) = aCN +bCN ·x, (3.16)

with aCN = logACN , bCN = BCN and x = logR.For the power-function in the RN, the function

fRN(R) = ARN ·RBRN (3.17)

is assumed, as the globulettes in the RN have the same way of origination as the ones in the CN.As the power-law in the RN is not that pronounced, the same slope for its fit function can beassumed. So the RN fit function is

yRN(x) = log [ fRN(R)] = aRN +bCN ·x, (3.18)

with aRN = logARN and bCN = BRN .

22

4 Evolution of globulettes

Of major interest regarding the globulettes is their evolution. As these clouds have just a fewJupiter masses, they may evolve into free-floating planetary objects or low-mass brown dwarfs(Gahm et al. 2007). So it is necessary that the future fate of these objects be investigated. In thiswork, the possibility that a captured globulette will evolve into a planet around a star is estimated.

The origin of the globulettes was discussed (Grenman 2006; Gahm et al. 2007), and thereseems to be no other explanation than the fragmentation of globules, or that a parent molecu-lar cloud disperses, leaving free cores that appear as globulettes. Filamentary connections (seeFigure 4.1) and connected globulettes (Figure 4.2) are evidences for the fragmentation theory.

Since some free-floating planetary objects in the Orion Nebula were discovered recently (Lu-cas & Roche 2000), the question of their origin is raised. Earlier, Fogg (1990) gave some ideas asto how such planets could be formed. He suggested that they can either form like a star - out ofsmaller clouds of less mass - or that they are developed around stars in protoplanetary discs. Thelatter case would require a catastrophic event, such as a supernova explosion or a close stellarencounter, for the planet to become free-floating. The first explanation would fit very well withthe observed globulettes in several H II regions, as discussed by Gahm et al. (2007).

On the other hand, it is also very interesting how the globulettes might evolve in the future.Is it possible that the resulting free-floating planetary object will be captured by a star? Maysome parts of the cloud, once close enough to the gravitational field of a star, evolve into planetsorbiting this star? These questions will be discussed in the following section.

4.1 Possible evolution scenarios

Depending on its virial ratio (Equation 3.12), the investitgated globulettes will collapse, remainstable (i.e., will not change in size) or dissipate into space. The latter is a rather boring case,as this would not result in anything we can observe or investigate. Clouds in virial equilibriumwill neither collapse nor dissipate into space, according to the virial theorem (Equation 1.1).These globulettes may remain stable for a long time and will just propagate through the space.Collapsing globulettes close to virial equilibrium is very interesting, as they will be stable for along time. On the other hand, globulettes that collapse too fast may evolve into a free-floatingplanetary object or brown dwarf without encountering any star before it collapses.

23

4 Evolution of globulettes

Figure 4.1 – A structure in the Rosette Nebula with connected globulettes marked by their numbers.The dashed lines show possible “routes” of some of the globulettes. The globulettes 38 and 39 areclose to the large globule, which suggests that they once broke off from it. Globulette 41 has still a tailthat ends in the globule, while the globulettes 42 and 43 lost their tails, probably due to evaporation.The image spans 118′′× 121′′ and was reproduced from Grenman (2006) with permission of theauthor. North is up and east to the left.

24

4.1 Possible evolution scenarios

Figure 4.2 – The globule, known as the “Wrench trunk” in the Rosette Nebula, is shown in the leftpanel in a color-coded image. It primarily consists of thin threads, which are twisted and connectedto jaws at the massive head, seen at the bottom of that image. This image spans 140′′× 306′′. Inthe upper-right panel globulette 44 is shown, having an unusual shape. This image spans 18′′×18′′.The intensity profile in the lower-right panel shows that this object contains three cores. They mayform individual globulettes in the future. This is a very good example for the fragmentation of theglobulettes out of bigger structures. All these images were reproduced from Grenman (2006) withpermission of the author. North is up and east to the left.

A collapsing globulette will form an object with the mass of the former globulette, i.e., objectswith masses of some Jupiter masses. In order to start hydrogen fusion a star needs 0.075M�,which is equal to 80MJ . Thus stars lighter than 0.075M� form brown dwarfs. Brown dwarfsalso are sites of nuclear fusion, but there deuterium and lithium (for those with masses above

25

4 Evolution of globulettes

65MJ) is fused. The mass limit to start deuterium-burning is 13MJ . All objects lighter than 13MJhave planetary masses. Thus most collapsing globulettes will evolve into planetary-like objects.More information about the distinction between planets and brown dwarfs can be found in Cole(2000).

A question that arises is whether those objects can somehow support life. Of course mostof them will be gas giants and as such not capable of supporting any known life form. But onthe other hand, life on Earth would never have evolved as we know it, if there would not bethe comet-magnet Jupiter. This gas giant attracts lots of debris that is flying around in our solarsystem, and Earth would much more often be the site of meteorite strikes without this shelter.Also, a moon orbiting these exoplanets may support life (like Europa around Jupiter might harborlife). However, in order to support life the planet must orbit a star. Otherwise neither a planet,nor a moon, will ever gain enough energy to heat the surface up to temperatures that are livable.

Consequently, the possibility of a free-floating planetary object to be captured has to be es-timated. Another interesting theory was presented by Oxley & Woolfson (2000): A star maycapture parts of a cloud and these parts then evolve into orbiting planets. This theory is moreclosely described in Section 4.3.

4.2 Possibility of a capture as a free-floating planet-likeobject

This possibility can be assumed to be zero. There is almost no chance that a developed free-floating planet-like object is captured by a star. For this it needs to be slowed down once itapproaches the star. A possible scenario is a three-body problem such as in the case of Triton,Neptune’s moon, with a strange orbit around the planet, as described in Agnor & Hamilton(2006). If a globulette will evolve into a binary system this scenario is possible. However, thereis no way to determine the possibility for a binary system developed out of a globulette with thepresent observational data. The resolution of nowadays instruments is just not good enough.

4.3 Possibility of a capture as a cloud

By smoothed-particle-hydrodynamics (SPH) modeling (Oxley & Woolfson 2000) it is shown thata cloud, close enough to a star, can result into some planets orbiting the star. Oxley & Woolfsonincluded three different initial setups in their paper. They found always some condensations intheir models, resulting in 3−5 captured planets of a few Jupiter masses. This is possible due tothe friction of the cloud’s parts. They slow some parts of the cloud down, which are then capturedand orbit the star further on, while the other parts carry the remaining momentum. These otherparts also condense, escaping the star or dissipating into space. This capture mechanism isdescribed in more detail in Appendix C.

Once a condensation is captured, it has an orbit of typically a few 1000 AU. According toOxley & Woolfson (2000) they migrate inwards due the friction of the gas around the star.

26

4.4 Computing the possibility of a star capturing a globulette

In their analysis, Oxley & Woolfson used distances of 2700 and 1600 AU. As at 2700 AU,the captured cloud results in some condensations, it can be assumed that any cloud, as near as2700 AU, will result into some objects orbiting the star. The authors say that the mechanism isstable against changing of parameters. Thus a cloud approaching a star at a distance of 2700 AUcan be claimed to be captured by the star.

The initial globules in Oxley & Woolfson (2000) have a mass of (0.7−1.3) ·1033 g, corre-sponding to 0.35 − 0.65M�. The captured condensations have a mass of around 10MJ . If wewant to have condensations of at least 1M♁, the cloud must have a mass of some 53M♁. Thisvalue was just assumed due to comparison with the condensations in Oxley & Woolfson (2000)and the desired minimal mass of a captured planet.

4.4 Computing the possibility of a star capturing aglobulette

Calculating the real situation is impossible with the current information. There is no informationabout where the smaller, not observable, globulettes are situated. Hence, simulations with ran-dom arrangements have to be made. In such a simulation, a number of stars and globulettes arearranged randomly in space. Then the computer program checks how many globulettes are closeenough to a star.

Additionally, such a program can calculate the movement of the globulettes against the starsdue to the stellar wind of the central cluster. The stars are much less affected to the stellar windand the radiation pressure of the central cluster than the globulettes. This is due to the muchsmaller surface of the stars. See Section 1.6 for further explanation. For this reason the motionof the stars can be neglected compared to the motion of the globulettes.

To calculate the number of stars for the simulation, the stellar density of this area is needed.The clusters with the high stellar densities are inside the nebulæ, where no globulettes can survivethe strong radiation of the luminescent cluster stars. The globulettes have to be placed outside theplasma bubble, where there is normal galactic stellar density. So no more stars than the averagenumber of stars per cubic parsec can be assumed for the region the globulettes pass by. Theaverage stellar density in the galactic plane is given in Table 4.1.

Stars are not all the same. Their masses differ, and thus their forces of attraction do as well.So the simulation must also consider different distances for stars of various spectral types. Asa B-type star has much more mass than a brown dwarf it has a wider range of attraction forglobulettes. For this reason their capture range is different for each spectral type.

By simply applying Kepler’s third law, one can derive the free-fall time for clouds to be:

τ f f =14

√3π

2Gρ0, (4.1)

where G is the gravitational constant and ρ0 is the mean density of the cloud.

27

4 Evolution of globulettes

Table 4.1 – The average decadic logarithm of the stellar density of the Milky Way. The density of allstars in the Milky Way is approximately 0.13 stars/pc3 (Chabrier 2001). The stellar density of browndwarfs is estimated by Chabrier (2002). Other stellar densities were extracted from Allen (1973).

Spectral type log(number density) Average massin log

(stars/pc3) in M/M�

O-type -7.6 40B-type -4 17A-type -3.3 3.2F-type -2.6 1.7G-type -2.2 1.1K-type -2.2 0.8M-type -0.96 0.5Giants -3.2 4.3White dwarfs -2.3 1.26Brown dwarfs -1 0.05total -0.89

For initially uniform density clouds, the collapse time is also uniform throughout the cloud.Equation (4.1) becomes incorrect for stellar formation due to inner forces, such as hydrostaticpressure or the pressure due to the nuclear fusion in the star. However, for this investigation itis enough to assume hydrostatic equilibrium throughout the globulettes, as shown in Grenman(2006).

The simulation should run over the time a globulette is about to collapse. The average free falltime can be calculated, using the densities tabulated in Gahm et al. (2007) and Equation (4.1) tobe some 3·105 yr. On the other hand, some globulettes have a free fall time of over 5 ·105 yr. Inorder to take into account all globulettes, the simulation should run for 500,000 years.

28

5 Results

For all fitting issues I used the program Igor Pro in Version 6.1.2.1 from WaveMetrics (http://www.wavemetrics.com/). Igor provides the singular value decomposition algorithmfor linear and polynomial fits. The data fits were applied with Equations (3.16 & 3.18) to thelogarithmic representation of the data in Figure 3.4. For the fitting, absolute values were used.As mentioned earlier both nebulæ were just surveyed for globulettes in some regions. Thus theremay be some incompleteness in the histograms.

The capture simulation was carried out with a C++ program, I have written during writing thiswork. It considers all facts from Section 4.4, and a brief description is given in Section 5.4. Thefull program is available upon request.

The simulation plots in Figures 5.4 and 5.5 were done with the program 3D Grapher in Version1.21 from RomanLab software (http://www.romanlab.com/3dg/).

5.1 Fitting the Carina Nebula size distribution

In Figure 5.1 the fit for the Carina Nebula (CN) is plotted. In the histogram, just four bins (i.e.,from 0.25 to 1.25 kAU) follow a power law. Those four points were used to plot the logarithmicvalues, using Equation (3.16). The fit gave the coefficients, with 95 per cent confidence,

aCN = 1.332±0.083bCN =−1.49 ±0.35. (5.1)

Thus the parameters for the power-function are

ACN = 21.49±1.21BCN =−1.49±0.35. (5.2)

It has to be noted that the absolute numbers of the globulettes were used for fitting issues.The fit quality can be determined with a χ2 test. For the fit parameters in Equation (5.2) it is

χ2 = 0.25. For the 2 degrees of freedom (d f ), this gives a probability of the fit of

P(χ

2 = 0.25,d f = 2)

= 88 per cent, (5.3)

showing that the fit was good for these 4 points.

29

5 Results

Figure 5.1 – The size distribution of both nebulæ together with the logarithmic data of the CarinaNebula. The fit from Equation (3.16) is applied to the logarithmic data. The fit range was from bin2 to 5, as it represents the best agreement for the power law. The parameters for the power-functionwere then calculated, using the coefficient values from the linear fit. The power-function fits well forthe second to the fifth bin, while it not fits for the other ones.

In order to get an estimate for the total number of globulettes in the CN, one has to integrateEquation (3.14):

NCN =1

0.25 kAU

Rmax∫R0

[ fRN (R)] dR

= 4Rmax∫R0

(ACN ·R−BCN

)[kAU−1]

30

5.2 Extrapolating the Rosette Nebula size distribution

= 4[

R ·(

ACN ·RBCN

BCN +1

)]Rmax

R0

= 4Rmax

(ACN · (Rmax)

BCN

BCN +1

)−4R0

(ACN · (R0)

BCN

BCN +1

). (5.4)

As the value for Rmax, I use the upper limit of the graph (i.e., 10 kAU). One problem withthis integration is that the number of globulettes now highly depends on the quantity R0. AsBCN <−1, an integration up to size zero would diverge. Thus a minimal size for globulettes hasto be determined. As discussed in Section 4.3 a globulette of at least 53 Earth masses (M♁) wouldresult in planets of at least 1M♁ according to Oxley & Woolfson (2000). This mass corresponds todifferent sizes depending on the mass density of the globulette. Yet a good approximation mightbe R0 = 100 AU for the minimal size, when comparing the masses and sizes of the smallestglobulettes in the CN.

Including the coefficients from Equation (5.1), this gives NCN = 484. The propagation ofuncertainty gives an uncertainty for NCN of uNCN = 137. Hence the total number of globuletteswith sizes between 100 AU and 10 kAU is

NCN = 484±137. (5.5)

5.2 Extrapolating the Rosette Nebula size distribution

For the RN data I used Equation (3.18) with bCN =−1.49. Fitting the logarithmic data from the10th to the 21st bin leads the program to give the following coefficient value with 95 per centconfidence,

aRN = (1.595±0.089) . (5.6)

The fit quality was determined to be χ2 = 8.1, leading to P(8.1,12) = 78 per cent. Thus this fitis in good agreement with the observed sizes.

In the RN, the total number of globulettes with sizes between 100 AU and 10 kAU was deter-mined in a similar way to Equation (5.4). The result is

NRN = 880±250. (5.7)

5.3 Comparison of the two nebulæ

The extrapolation seems to lead to a much larger number of globulettes in the RN than in the CN.This discrepancy is quite severe. But there is a good explanation that can also give these numbersreliability: The CN structure is older than the RN. While the RN globules have an estimated ageof (2.6−5.8) ·105 yr (Schneps et al. 1980), the CN region has an age of some 106 yr (Tapia et al.2003).

31

5 Results

Figure 5.2 – Here the size distribution of both nebulæ together with the logarithmic data of theRosette Nebula is shown. Equation (3.18) is applied to fit the logarithmic data. The fit range wasfrom bin 10 to 21. They were chosen as they represent the best straight line in the logarithmic data.The power-law parameters were calculated using the coefficient values from the linear fit. It fits wellfor the data beyond 2.25 kAU. For comparison, the Carina data fit is shown with a dashed line. Notethe different logarithmic scale in comparison with Figure 5.1.

Just taking into account gravitational force, I “evolved” the globulettes of the RN over atimescale of several 100,000 years. The calculation just decreases the radius of the investi-gated globulettes, as they would decrease in free fall. Thus the distribution of these globulettesmoves towards smaller sizes. The evolution is described by

R(t) = Robs−GM

2R2obs

· t2, (5.8)

where R(t) expresses the evolved size, Robs stands for the size when the observations were done,G is the gravitational constant, M is the mass and t gives the evolution time.

32

5.3 Comparison of the two nebulæ

Figure 5.3 – Shown here is the distribution of the evolved globulettes of the Rosette Nebula during350,000 years. For comparison, the graph of the observed data from Figure 5.2 is shown in lightgray. After the evolution time the fits of both nebulæ are comparable. This shows the reliability ofthe extrapolation.

This evolved distribution was then treated as the observed one in Section 5.2. This assessmentgives comparable numbers of globulettes in the RN after some 300 kyr. For this evolution time,the computed number of globulettes results in 419± 120, whereas after 350 kyr this numbershrinks to 316± 92 globulettes in the RN. This means that most of the globulettes will havecollapsed after 350 kyr. For large evolution times, the size distribution is too flat to give good fitfunctions. Figure 5.3 shows the evolved size distribution after 350 kyr.

Of course Equation (5.8) does not include all forces on a globulette. For instance, it does nottake into account the inner pressure, the outer pressure, the radiation pressure or any turbulence.However, it shows that the calculated globulette number of the RN can be much higher than theone in the CN due to the difference in age.

33

5 Results

5.4 Probability of globulettes being captured by stars

For this task I have programmed a simulation that uses a C++ code, while taking all points ofSection 4.4 into consideration.

The simulation first places the stars, according to their stellar density, randomly in the spacethat will be passed by the moving globulettes. Afterwards the simulation places the 880 glob-ulettes, as calculated above, randomly in the space that is covered by the RN. The RN shape isgenerally considered to be spherical, although it may equally well be cylindrical (Townsley et al.2003). Hence for this simulation I assumed the shape of the RN to be completely spherical. Inthe next simulation step, the program calculates the distances between all globulettes and all starsin order to determine whether there is already a globulette in the surroundings of a star.

Then the globulettes move with a velocity of 20 kms−1. Thereafter again the program checksfor globulettes that are close enough to a star to be captured. The last two steps are looped untilthe time of the investigation is over. In the last step the program ejects the parameters of eachcaptured globulette. This includes the distance to the star that has captured the globulette, thetime when the globulette was captured, starting with zero time at nowadays observations fromEarth, the type of the star that has captured the globulette and the difference of both radial valuesto the center of the nebula (i.e., rradial =

∣∣rposition o f star− rposition o f globulette∣∣). The latter is used

to estimate whether the globulette was about to move directly into the star, or just pass by at theedge of the sphere around the star.

In Figure 5.4 the start arrangement of one simulation is plotted. This particular simulationlead to one capture. It occurred after 17,850 years, at a distance of r = 2965 AU, and with aradial distance of rradial = 2858 AU to a G-type star. With the radial distance I can calculate howclose this particular globulette encounters the star. Hence I can just use Pythagors’s theorem tocalculate the closest encounter:

rmin =√

r2− r2radial , (5.9)

which leads to a minimal distance of rmin = 787.6 AU. The end arrangement of the same simu-lation is seen in Figure 5.5.

I have carried out 500 runs of this simulation to assure statistical reliability. It occurred thatmost captured globulettes are captured by B-type stars (i.e., 70 occasions), followed by M-typestars with 50 occasions. A category plot is shown in Figure 5.6. In 500 cycles with each 880globulettes at the start I get totally 209.0 ± 14.5 captured. This gives a capture probability of

Pall =209.0±14.5

500·880= (4.75±0.33) ·10−4. (5.10)

34

5.4 Probability of globulettes being captured by stars

Figure 5.4 – Here a random arrangement of stars and globulettes plotted three-dimensionally is re-produced in two dimensions. The time of this plot is 0, i.e., this is the initial arrangement. The stars,as well as the globulettes, appear to be uniformly distributed. The outermost globulettes are situatedat the border of the Rosette Nebula, shown as a blue line. The globulettes are black, while the starsare yellow. The one globulette that is captured during this simulation is marked with a red point. Thestar that captures this globulette is marked green. The upper axis points in the z direction, the axisto the lower left points in the x direction and the one to the lower right points in the y direction. Theaxes range from −520 kAU to +520 kAU.

35

5 Results

Figure 5.5 – This plot shows the arrangement of Figure 5.4 after the development of 500 kyr. Duringthis run of the simulation, one globulette, indicated by the red point, was captured by a star (greenpoint, but not visible as it is covered by the red point). The innermost globulettes are now outside theRosette Nebula, seen as a blue line (this line corresponds to a spherical shape). All other globulettesare black, stars are yellow. In this simulation just the space that the globulettes pass by is considered.Thus, there are no stars outside the outermost possible position of a globulette. The x, y and z axesare to the lower left, the lower right and up, respectively, with a range from−520 kAU to +520 kAU.

However, a capture around a B-type star would most likely lead to the disruption of the glob-ulette, as the stellar winds of these type of stars are very strong. Giant stars and white dwarfs

36

5.4 Probability of globulettes being captured by stars

will not support life on their planets. Brown dwarfs are much too cold to heat a planet. Only A-,F-, G-, K- and M-type stars are expected to be able to support life. These AFGKM-type starscaptured a total of 100 ± 10 globulettes in the simulation. Hence, the capture probability for aglobulette to be caught by either one of these stars is

PAFGKM =100±10500·880

= (2.27±0.28) ·10−4. (5.11)

Figure 5.6 – This category plot shows the result of 500 cycles of the simulation. In total 209.0 ± 14.5globulettes were captured in these cycles. This corresponds to 0.418 ± 0.029 captures per simula-tion. Most globulettes were captured by B-type stars. AFGKM-type stars captured in total 100±10globulettes. The errors correspond to the square-root of the number of captured globulettes for eachspectral type over the total number of captured globulettes (i.e., ∆ =

√n/N).

Each spectral type has a different stellar density, as shown in Table 4.1. For instance, B-type stars are rare, with just 5 out of 10,741 in each simulation cycle. However, they seemto be responsible for most captures of globulettes. Surely this is due to their mass and hencetheir capture range. Table 5.1 shows how many stars are possible hosts for planets due to thecalculation, while the same data is plotted in Figure 5.7.

37

5 Results

Table 5.1 – The number of computed hosts, initial stars, and the resulting percentage of hosts in 500runs for each spectral type.

Spectral type Number of hosts Number of stars Percentage of hostsin 500 simulations in each simulation ± Poisson uncertainty

O-type 0 0 -B-type 70 5 2.80 ± 0.34A-type 11 23 0.096 ± 0.029F-type 11 117 0.0188 ± 0.0057G-type 18 295 0.0122 ± 0.0029K-type 10 304 0.0066 ± 0.0021M-type 50 5063 0.00198 ± 0.00028Giants 18 30 0.120 ± 0.029White dwarfs 21 234 0.0179 ± 0.0040Brown dwarfs 0 4670 0 ± 0total 209 10741 0.00389 ± 0.00027main sequence stars 170 5807 0.00586 ± 0.00045

Figure 5.7 – The percentage of capturing stars for each spectral type. As B-type stars are rare, and yethave most captures, they have the highest percentage. Note the logarithmic scale of the percentage.The error bars are calculated with the same method using the errors from Figure 5.6.

38

6 Discussion

6.1 Size distributions

The number of the globulettes in both nebulæ dependens very much on the quantity R0 in Equa-tion (5.4). For instance, R0 = 50 AU would lead to NRN = 1280± 620, or R0 = 500 AU wouldgive NRN = 347± 56. Thus, this quantity has to be determined with, for example, smoothedparticle hydrodynamic (SPH) simulations like in Oxley & Woolfson (2000). For the SPH simu-lations, the mass and size of the globulettes have to be included, in order to determine what kindof condensations would form.

6.2 Evolution of globulettes

The simulation program I worked out is still fairly simple. I just assume that a globulette iscaptured as soon as it is close enough to a star. However, in Oxley & Woolfson (2000), theclouds (protostars in their vocabulary) are of the same size as the globulettes, whereas their massis much higher. This means that their clouds are denser, which gives more friction. On theother hand, it might be that a globulette is captured further out than the 2700 AU used in theirinvestigation. An n times bigger radius for capturing would lead to an n2 times bigger number ofcaptured globulettes. Results of 100 runs with twice the capture radius are shown in Figure 6.1.Further, Oxley & Woolfson start their simulations with clouds in hyperbolic trajectory, whereasthe globulettes in this investigation move along straight lines.

For the mass estimation in Table 4.1, I took the values from Allen (1973, page 254), whichare tabulated for the corresponding spectral type of the star. Yet, I can think of a range of massesfor each spectral type, also randomly distributed for each star, with respect to the function ofoccurrence.

Also, this simulation does not take the gravitational field into account, which would force theglobulettes not to move straight outwards, but rather in a “zig-zag” pattern line. The gravitationalfield of each star would disturb a globulette in its movement, and also pull at the cloud.

Another point to discuss is the stellar density. As I could not find any explicit numbers for thestellar density in and around the RN, I had to assume the average stellar density in the galacticplane. But, as this nebula is a star-birth region, it can be assumed that there are, or will be, morestars than in an average galactic region. Anyway, an n times larger number of stars would lead ton times more occasions of a star capturing a globulette. Figure 6.2 shows the comparison of 100cycles with the double amount of stars and the ones with the normal amount.

39

6 Discussion

Figure 6.1 – The result of 100 runs with twice the capture range, compared to Figure 5.6. In total188.0±13.7 globulettes were captured in the cycles with the double capture range. The errors corre-spond to the square-root of the values, as they are counted. In the upper right graph, the number ofcaptures for the simulations with twice the capture range was divided by 4, in order to compare theresults from runs with the different parameters.

Table 6.1 – This table compares Tables 1.1 and 5.1. All hosts of the spectral types were taken intoaccount for the computed numbers, thus some were counted twice. The tendency of the observedpercentages could be reproduced.

Spectral type % stars with planets % stars with planetsobserved simulated

M4 - K7 1.8 ± 1.2 0.00877 ± 0.00097K5 - F8 4.13 ± 0.67 0.0634 ± 0.0062F5 - A5 8.80 ± 2.27 0.221 ± 0.032

40

6.2 Evolution of globulettes

Figure 6.2 – This plot shows the comparison between 100 cycles of the double amount of stars andthe 500 cycles of the normal amount of stars. In total 91.0± 9.5 globulettes were captured in thesimulations with more stars. The errors correspond to the square root of the values. The inset graphshows the number of captures for the runs with twice the stellar density divided by 2, for bettercomparison.

One more point is the contradiction that after 350 thousand years there will be just 315±92 globulettes in the RN, while the program still runs with all 880 globulettes (diminished bythe captured ones) until the end of 500 kyr. One way to bring the contraction into effect is touse the evolution function (Equation (5.8)) for each calculation step. Then some globulettesare deleted randomly to get the correct number of globulettes for the time. But, as alreadymentioned, Equation (5.8) does not include all forces, and so the timescale will most likelyincrease. However, there were just 67 out of 209 occasions (i.e., approximately a third) that aglobulette was captured after 250 kyr. This shows that most captures occur in the first half of thesimulation. This might be a coincidence, but there is no evidence that it is not. Figure 6.3 showsa histogram of the captures over time.

41

6 Discussion

Figure 6.3 – This histogram shows the number of captures with respect to time. There is a tendencyfor most captures to occur early in the simulation.

Comparing Tables 1.1 and 5.1, it is shown that the number of captured globulettes cannot beresponsible for the observed extrasolar planets of the “hot Jupiter” type. The observed percent-ages are at least an order of magnitude higher than the computed ones. Yet, the tendency fora higher percentage of the brighter stars to host planets, could be reproduced. A comparison isshown in Table 6.1.

6.3 Future research

Further observations of the globulettes are needed. One suggestion is to observe the RN with tele-scopes that have a higher resolution, e.g., the Hubble Space Telescope (HST). This would proveor disprove the extrapolated number of globulettes in the RN. Also, a more precise measurementof the velocities of the globulettes would provide more information for the further development.For these explorations, radio telescopes are used to observe the 12CO and 13CO lines. However,

42

6.3 Future research

the globulettes are too small to be resolved by a single radio dish. Thus, interferometric tech-niques are desired. If in future observation smaller globulettes are found, their properties have tobe determined, as well as their tendency to collapse.

There were recently discovered many free-floating planetary objects in the Orion Nebula (Lu-cas & Roche 2000). One plausible reason why they have been discovered only in the OrionNebula so far is that it is much closer to us than the RN or the CN. A search for free-floatingplanetary objects in both nebulæ would be very interesting. Near infrared (NIR) imaging in theOrion Nebula (Kristensen et al. 2003) resulted in the finding of some globulette-like structuresclose to the central O-type star cluster of the nebula. Kristensen et al. speculate that free float-ing planetary objects are formed from these objects. NIR observations in the RN showed thatthe globulettes around the Wrench emit H2 radiation at 2.12 µm. This emission is most likelyexcited by fluorescence from UV light from the central O stars (Gahm 2010).

The number of stars in the clusters of the RN is well investigated, but not stellar density inand around the RN. Hence, further investigation of the stellar density in this region would beappropriate. Also it would be useful with a three-dimensional map of all stars and globulettes inthis region, but this is rather utopistic nowadays.

For further simulations it will be a good idea to include the size and mass of the globulettes,as well as gravitational forces of the stars, so the contraction can be computed and the path ofthe globulettes would not be straight anymore. The deflection angle for a globulette passinga star can be used to estimate the effect of gravitational forces. The model, used in Oxley &Woolfson (2000), should also be changed to a scenario where the globulettes are, in order to getbetter estimations for a distance when a globulette might be captured. The SPH computationswere carried out with clouds more dense than the globulettes. It was assumed that they also arecaptured, but the stability of the program against changing the initial parameters of the cloudtowards those of a globulette has to be confirmed.

A similar simulation as the one carried out for the RN could also be done for the CarinaNebula.

43

7 Conclusions

This project work investigates the possibility of the globulettes, found in the Rosette Nebula, tobe captured by a star and then form an orbiting planet around this star. In order to determine this,the total number of globulettes (including smaller ones below the resolution limit) is estimatedand used for the capture simulation. This simulation arranges 880 globulettes and 10,741 starsrandomly in the space of the Rosette Nebula and moves them according to the observed motion ofthe cloud structures. In conclusion, very few globulettes were captured during 500 simulations.In total 39.0 ± 6.3 of 209.0 ± 14.5 captured globulettes are orbiting stars similar to the sun(i.e., F-, G- or K-type stars).

The main conclusions of this work are summarized as follows:

• The extrapolation of the size distribution of the globulettes in the Carina Nebula results ina power-function distribution, proportional to R−1.49.

• Assuming the same power-law in the Rosette Nebula, the size distribution of the globulettesin this nebula was extrapolated. This resulted in 880 ± 250 globulettes in the RosetteNebula in total. Most of them are under the resolution limit of the NOT, which was usedto observe the globulettes there. This figure is higher than the corresponding one for theCarina Nebula, but it is consistent when one looks at the age discrepancy between bothnebulæ.

• In total 500 cycles of the simulation program showed that globulettes in the Rosette Nebulaare captured by stars with a possibility of (4.75 ± 0.33) ·10−4. Then 70.0 ± 8.4 ofthe 209.0 ± 14.5 globulettes were captured by B-type stars, while 100 ± 10 of them werecaptured around main sequence stars with spectral types between A and M.

• The theory that captured globulettes are responsible for the observed exoplanets, or at leastof the “hot Jupiter” type, has to be rejected. Yet they might be responsible for some ofthem.

45

Acknowledgments

First of all I want to thank my supervisor Sverker Fredriksson, who suggested my work andhelped me in many questions. I would also like to express my gratitude to Gösta Gahm and TiiaGrenman for providing their data and answering questions. As well, I would like to thank my co-supervisor Roland Waldi for critical and very helpful advices. Furthermore, I would like to thankNils Almqvist, Erik Elfgren and Hans Weber for being patient with my sometimes annoyingquestions, and helping me in many small, but nevertheless important, issues.

My gratitude goes to all my friends in Luleå, especially to Ilya Dobryden, three Swedish guys(Léon Löwered, Andreas Öberg and Ted Sjöberg), who helped me very much during my firsttwo weeks in Luleå, and to Brenton Earl, who is behind a lot of commas in this work. Also, myfriends in Germany and throughout the world deserve thanks for always being constructive whilereading some parts of this work. Furthermore, I want to thank my parents for always supportingme, not only in financial issues.

Additionally, I want to thank the Erasmus program for granting a scholarship that providedfinancial support for my time in Luleå.

I also want to thank whoever came up with LATEX, for making up such a nice program forwriting such a work.

Last, but not least, I want to thank my love Friederike Meurer for always supporting me incountless ways.

47

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51

A Derivation of the virial theorem

The virial theorem analyses the stability of a gas system. It was first described by Clausius(1870) as “a mechanical theorem applicable to heat.” This theorem can be used to determinethe gravitational stability of gaseous objects. In various textbooks, e.g., Carroll & Ostlie (1996)the derivation of the virial theorem can be found. A short guideline is given by starting with thequantity

Q≡∑i

~pi ·~ri, (A.1)

where ~pi and~ri are the linear momentum and position vectors for particle i. The time derivate ofQ is then

dQdt

= ∑i

(d~pi

dt·~ri +~pi ·

d~ri

dt

). (A.2)

The left-hand side of Equation (A.2) is just

dQdt

=ddt ∑

imi

d~ri

dt·~ri =

ddt ∑

i

12

ddt

(mir2

i)

=12

d2Idt2 , (A.3)

where I = ∑i mir2i is the moment of inertia of the system of particles (see Equation (1.6)). The

second term on the right-hand side of Equation (A.2) becomes

∑i

~pi ·d~ri

dt= ∑

imi~vi ·~vi = 2∑

i

12

miv2i = 2K, (A.4)

and is hence twice the total kinetic energy of the system. It can be shown that the first term at theright-hand side of Equation (A.2) is

∑i

d~pi

dt·~ri = ∑

i

~Fi ·~ri = U , (A.5)

where U is the total potential energy of the system. Here the last step uses the virial theoremfrom Clausius (1870).

53

Appendix

Now Equation (A.2) can be expressed with

12

d2Idt2 = 2K +U , (A.6)

which becomes Equation (1.1) if the total kinetic energy is K = Kint −Kext . The total potentialenergy can be described by U = W +Emag.

54

B Linear fitting with singular valuedecomposition

A general linear function of the form

y(x) = a ·x+b (B.1)

is desired for a set of data. This set is composed of the values xi and yi for i = 1 . . .m. A linear fitis done by solving the linear system of equations

A ·~x =~y, (B.2)

where the (m×2) matrix A and the vectors~x and~y are composed as followed:

A =

1 x1...

...1 xm

~x =(

ba

)~y =

y1...

ym

. (B.3)

The solution of this system of equations is

~x = A+ ·~y, (B.4)

where A+ is the pseudoinverse of the matrix A. It can be determined with the help of singularvalue decomposition.

Every (m×n) matrix A with complex values and rank r can be decomposed with

A = UΣV †, (B.5)

where U is a unitary (m×m) matrix, V † = V T is the adjoint of V , a unitary (n×n) matrix and a(m×n) matrix Σ. The diagonal values of the matrix Σ are the singular values σi. Without loss ofgenerality, σ1 ≥ . . .≥ σr > 0. Then Equation (B.2) becomes

UΣV T ·~x =~y, (B.6)

and the solution in Equation (B.4) becomes

~x = V Σ+UT ·~y. (B.7)

55

Appendix

Here Σ+ is the pseudoinverse of Σ, which is formed by replacing every nonzero entry by itsreciprocal and transposing the resulting matrix (Moler 2006).

In order to determine the values for U , Σ and V , the eigenvalue problem for AT A has to besolved, resulting in the eigenvalues λi = σ2

i , which form Σ and the eigenvectors~vi. The latter canbe formed to the matrix

V = (~v1, . . . ,~vn) . (B.8)

The matrix

U = (~u1, . . . ,~um) (B.9)

is computed by reordering Equation (B.5) into the relationship

AV = UΣ, (B.10)

and solving for the ~ui (Höllig 2007).

56

C The capture mechanism with SPHcomputation

The mechanism described by Oxley & Woolfson (2000) is computed with a smoothed particlehydrodynamics (SPH) simulation. They use a diffuse protostar, i.e., a molecular cloud, in closedistance to a star. The used protostars have a mixture of 70 per cent hydrogen to 30 per centhelium and a mass of approximately 1M�. The 1− 2 per cent of dust are neglected in theirstudies. The star, in each simulation, has the mass and luminosity of the sun.

The equation of state includes various parameters. The variation of the specific heat withchange of temperature was considered. Also, the variation of the temperature itself is takeninto consideration. Finally, the variation of the mean molecular mass with change of the internalenergy is included in the equation of state. Molecular hydrogen dissociates during the simulation.This was given attention by monitoring the pressure and the temperature.

Oxley & Woolfson (2000) simulated 3 different initial arrangements of star and protostar. Asmentioned above, the star does not change in mass or luminosity. On the contrary, the propertiesof the protostar and those between the protostar and the star were changed in the followingranges:

• Radius of the protostar: 800−1300 AU.

• Mass of the protostar: (0.7−1.3) ·1033 g.

• Temperature of the protostar: 15−30 K.

• Initial orbit periastron: 600−800 AU.

• Initial eccentricity: 0.9−1.1.

• Initial star-protostar distance: 1600−2700 AU.

The number of SPH particles had a constant value (5946) in each simulation. Figure C.1 showsone of these simulations.

The mechanism seems to be robust, due to the fact that it operates at a large range of distancescales. Additionally, it does not depend on the fine tuning of the parameters. “Indeed, presentexperience indicates that any passage of a still-diffuse protostar with a periastron distance lessthan the radius of the protostar and with an orbit that is elliptical, or even just hyperbolic (e.g.,[eccentricity] e = 1.1), will give rise to a capture event” (Oxley & Woolfson 2000).

57

Appendix

Figure C.1 – The panel to the left shows the initial arrangement between a star with solar charac-teristics and a protostar with mass 0.5M�, radius 1000 AU and temperature 30 K. The protostarconfiguration after 17,000 years is also shown in the left panel. The right panel then shows the fila-ment after 17,000 years. It has broken up into a number of condensations of which the three indicatedare captured. These plots were reproduced from http://www-users.york.ac.uk/~mmw1/starsbrowndwarfsandplanets.pdf (Oxley & Woolfson 2000).

The resulting protoplanets then have high semi-major axes and eccentricities. They migrateinwards, due to the resistance of the medium

D = 2πρ(r)G2Mp

W 2 ln

(1+

s2W 4

G2M2p

)(C.1)

Oxley & Woolfson (2000). Here ρ(r) is the density of the medium, G is the gravitational con-stant, W the relative speed of the protoplanet with respect to the medium, and Mp is the proto-planet mass. Finally,

s = r(

Mp

2M�

) 13

(C.2)

is the radius of the sphere of influence of the planet at a distance r from the star. Equation (C.1)offers many adjustable parameters. Thus, “. . . it is possible to mimic almost any observed com-bination of semi-major axis and eccentricity” (Oxley & Woolfson 2000).

58

Statutory Declaration

I declare that I have authored this project work independently, that I have not used other than thedeclared sources / resources, and that I have explicitly marked all material that has been quotedeither literally or by content from the used sources.

Luleå, March 2010

59