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Numerical models of the early stages of planet formation [Elektronische Ressource] / presented by Anders Johansen

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Numerical models ofthe early stages of planet formationAnders JohansenDissertationsubmitted to theCombined Faculties for the Natural Sciences and for Mathematicsof the Ruperto-Carola University of Heidelberg (Germany)for the degree ofDoctor of Natural Sciencespresented bycand.scient.Anders Johansenborn in Gentofte (Denmark)Oral examination: July 18, 2007Numerical models ofthe early stages of planet formationReferees: Prof. Dr. Thomas HenningProf. Dr. Ralf S. KlessenAbstractThis PhD thesis deals with the early stages of planet formation and the growth from μmdust grains to kilometer-sized planetesimals. Dust grains are diffused by the turbulence intheprotoplanetarydisc. Wemeasurethediffusioncoefficientofmagnetorotationalturbulenceand relate it to the turbulent viscosity. Diffusion is surprisingly as strong as viscosity, eventhoughmostoftheviscositycomesfrommagneticstressesthatdonotdirectlyaffectdiffusion.The ratio between turbulent viscosity and turbulent diffusion (the Schmidt number) is foundtodependstronglyonthestrengthofanimposedverticalmagneticfield. Largefieldstrengthsyield a Schmidt number that is much larger than unity. Larger solid particles, i.e. rocksand boulders, are not only diffused by magnetorotational turbulence, but also experienceconcentrations in transient high pressure regions of the turbulent gas, reaching local densitiestwo orders of magnitude higher than the average.
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Numerical models of
the early stages of planet formation
Anders JohansenDissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg (Germany)
for the degree of
Doctor of Natural Sciences
presented by
cand.scient.
Anders Johansen
born in Gentofte (Denmark)
Oral examination: July 18, 2007Numerical models of
the early stages of planet formation
Referees: Prof. Dr. Thomas Henning
Prof. Dr. Ralf S. KlessenAbstract
This PhD thesis deals with the early stages of planet formation and the growth from μm
dust grains to kilometer-sized planetesimals. Dust grains are diffused by the turbulence in
theprotoplanetarydisc. Wemeasurethediffusioncoefficientofmagnetorotationalturbulence
and relate it to the turbulent viscosity. Diffusion is surprisingly as strong as viscosity, even
thoughmostoftheviscositycomesfrommagneticstressesthatdonotdirectlyaffectdiffusion.
The ratio between turbulent viscosity and turbulent diffusion (the Schmidt number) is found
todependstronglyonthestrengthofanimposedverticalmagneticfield. Largefieldstrengths
yield a Schmidt number that is much larger than unity. Larger solid particles, i.e. rocks
and boulders, are not only diffused by magnetorotational turbulence, but also experience
concentrations in transient high pressure regions of the turbulent gas, reaching local densities
two orders of magnitude higher than the average. Discs that are not susceptible to the
magnetorotational instability can develop turbulence due to the sedimentation of solids. The
radial pressure gradient of the gas, together with a vertical gradient in the solids-to-gas
ratio, leads to a vertical shear in the orbital velocity of the gas, unstable to the Kelvin-
Helmholtz instability. The turbulent state is characterised by a number of dense clumps
of solids that form due to the dependence of the orbital velocity on the local solids-to-gas
ratio, making denser regions plough through less dense regions and scoop up the material
at the full Keplerian speed. Isolating the effect of this streaming instability, by ignoring
vertical stratification, we find that the turbulent state depends strongly on the background
solids-to-gas ratio and on the friction time of the particles. Marginally coupled solids display
huge overdensities and a diffusion coefficient that approaches that of the magnetorotational
turbulence, more tightly coupled solids develop only a very weak non-linear state.
iZusammenfassung
Diese Doktorarbeit behandelt die ersten Stufen der Planetenentstehung und des Wachstums
von μm-grossen Staubk¨ornern zu km-grossen Planetesimalen. Staubk¨orner werden von der
Turbulenz durch die protoplanetare Scheibe diffundiert. Wir bestimmen den Diffusionsko-
effizienten von magnetorotationeller Turbulenz und seinen Zusammenhang mit der turbu-
lenten Viskosit¨at. Die Diffusion ist ub¨ erraschenderweise so stark wie die Viskosit¨at, ob-
wohl der Grossteil der Viskosit¨at aus der magnetischen Spannung entsteht, die die Diffusion
nicht direkt beeinflußt. Das Verhaltni¨ s von turbulenter Viskosit¨at und turbulenter Diffusion
(die Schmidt-Zahl) h¨angt empfindlich von der St¨arke des u¨berlagerten vertikalen Magnet-
feldes ab. Große Feldst¨arken ergeben Schmidt-Zahlen, die viel gr¨oßer als eins sind. Gr¨oßere
Festk¨orperteilchen wie etwa Steine und Brocken werden nicht nur durch die magnetorota-
tionelle Turbulenz diffundiert, sondern auch in kurzlebigen Hochdruckgebieten des turbulen-
¨tenGaseskonzentriertunderreichenlokaleUberdichtenvonzweiGr¨oßenordnungenu¨berdem
Mittelwert. Scheiben, in denen die magnetorotationelle Instabilit¨at nicht auftritt, entwickeln
TurbulenzaufgrundderSedimentationvonFestk¨orpern. DerradialeDruckgradientdesGases
fu¨hrt zusammen mit einem vertikalen Gradienten im Massenverhaltni¨ s Gas-zu-Festk¨orper zu
einervertikalenScherunginderKreisgeschwindigkeitdesGases,unddadurchzueinerKelvin-
Helmholtz-Instabilit¨at. DerturbulenteZustandistdurcheineAnzahldichterPartikelklumpen
gekennzeichnet, die durch die Abh¨angigkeit der Kreisgeschwindkeit vom lokalen Staub-zu-
Gas-Verh¨altnis entstehen, wodurch sich dichtere Klumpen mit Kepler-Geschwindigkeit durch
weniger dichte Gebiete pflugen.¨ Bei Isolation dieses Strom¨ ungsinstabilit¨atseffektes durch
Vernachlas¨ sigung der vertikalen Schichtung zeigt sich, dass der turbulente Zustand stark
vom globalen Staub-zu-Gas-Verhaltn¨ is und von der Reibungszeit der feste Partikel abh¨angt.
¨Marginal gekoppelte Partikel zeigen große Uberdichten und einen Diffusionskoeffizienten ver-
gleichbar dem bei magnetorotationeller Turbulenz. Kleinere Partikel entwickeln nur einen
sehr schwach nicht-linearen Zustand.
iiPreface
This thesis work was performed at the Max-Planck-Institut fur¨ Astronomie in the period
between August 2004 and May 2007 under the supervision of Dr. Hubert Klahr and Prof. Dr.
Thomas Henning.
The thesis itself consists of six papers published in refereed journals:
1• “Dust diffusion in protoplanetary discs by magnetorotational turbulence” (2005)
Johansen A., & Klahr H.
The Astrophysical Journal, vol. 634, p. 1353-1371
• “Gravoturbulent formation of planetesimals” (2006)
Johansen A., Klahr H., & Henning Th.
The Astrophysical Journal, vol. 636, p. 1121-1134
• “Dust sedimentation and self-sustained Kelvin-Helmholtz turbulence in protoplanetary
disc mid-planes” (2006)
Johansen A., Henning Th., & Klahr H.
The Astrophysical Journal, vol. 643, p. 1219-1232
• “Turbulent diffusion in protoplanetary discs: The effect of an imposed magnetic field”
(2006)
Johansen A., Klahr H., & Mee A. J.
Monthly Notices of the Royal Astronomical Society, vol. 370, p. L71-L75
• “Protoplanetary disc turbulence driven by the streaming instability: Linear evolution
and numerical methods” (2007)
Youdin A., & Johansen A.
The Astrophysical Journal, in press
• “Protoplanetary disc turbulence driven by the streaming instability: Non-linear satu-
ration and particle concentration” (2007)
Johansen A., & Youdin A.
The Astrophysical Journal, in press
1
This paper was awarded the “Patzer Prize” for best refereed paper by a PhD student at the MPIA in
2005.
iiiI was not the first author of Youdin & Johansen (2007), but this paper came about as a split
of a longer paper, so I was heavily involved with the writing, and all code improvements and
computer simulations for the paper were done by me.
I have devoted a chapter of the thesis to each of the abovementioned projects (Chapters 2–
7), in order by which they were finished, so that the evolution of theoretical concepts and
computational methods appears in its natural sequence. Thus the two chapters on the dif-
fusion properties of magnetorotational turbulence (Chapters 2 and 5) are not consecutive.
The scientific chapters are meant to be self-contained each with an individual introduction
and conclusion. Chapters are cross referenced internally either by chapter number or (more
commonly) by author (year) publication id.
All computer simulations in this thesis were done with the Pencil Code. The code, including
improvements done for the thesis work, is publicly available under a GNU open source license
and can be downloaded at http://www.nordita.dk/data/brandenb/pencil-code/.
Notation and terminology
The individual papers have been edited for fitting together in the thesis. Certain repetitions
have been deleted and language has been made uniform, especially on terminology of solids: I
decided to use the word “solids” rather than “dust” to generally describe condensed material
in discs. Thus the terminology in the first four chapters has been changed from what appears
in print elsewhere.
Mathematical notation is fairly standard throughout, except for a few exceptions. Both Ω0
and Ω are used for the Keplerian frequency at the centre of the shearing box. The subscript
for solids is either d or p, referring to dust and to particles. I decided not to standardise
mathematical notation because of the risk of introducing subtle errors in the text.
Acknowledgements
A number of people helped me during my thesis work. My supervisors, Hubert and Thomas,
were always there for me with ideas, suggestions and support. They also encouraged me to
develop my own science, which I am deeply grateful for. I was paid one month by NORDITA
in Copenhagen in the time between finishing the master’s thesis and beginning the PhD in
Heidelberg. I am grateful to Axel Brandenburg and Anja Andersen for arranging this. Anja
and Axel are also thanked for their continuous support and many recommendation letters.
In this connection I am grateful to “Christian & Ottilia Brorsons Rejselegat” for paying part
of the moving expenses when coming to Heidelberg. I am also thankful for several rounds of
support from “Det Johansenske Familielegat”.
I have throughout my thesis work had the opportunity to interact with a number of brilliant
astrophysicists both at the MPIA and at other places in Germany and around the globe. At
the MPIA the collaboration with Kees Dullemond and Frithjof Brauer was especially fruitful
because our topics are so close and yet so different. I would generally like to thank Christian
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