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Universitat Hamburg

Fachbereich Physik

Structure and X-ray emission

of the accretion shock

in classical T Tauri stars

Diplomarbeit

Hans Moritz Gunther

completed in Hamburg on October 31th, 2005

angefertigt unter der Betreuung von Prof. J. H. M. M. Schmitt

[email protected] Sternwarte, Gojenbergsweg 112, 21029 Hamburg

Abstract

This work deals with young stellar objects, which are still surrounded by anaccretion disk, the classical T Tauri stars (CTTS). One of the main predictionsof the magnetically funnelled infall model for CTTS is the existence of a hotaccretion spot. I present a computer simulation of the shock structure, whichexplicitly considers non-equilibrium effects. Observable quantities like the lineflux are calculated for many X-ray lines. I find, that basically only two param-eters of the infalling gas govern the structure of the shock, the infall velocityand the pre-shock density. Comparing to a stellar model atmosphere I estimatethe depth at which the shock occurs. For shocks with velocities above 400 km/sand particle densities below 1011 cm−3 the stellar pressure causes deviationsfrom the model used in deeper layers. Line fluxes taken from the literature forTW Hya agree with to my predictions and allow to determine the infall velocityas 525 km/s and the infall density as 1012 cm−3. I confirm that the emittingplasma is metal depleted, but neon enhanced. The spot fills 0.1− 0.2 % of thestellar surface and the mass accretion rate is 1 · 10−10M/yr. This is signifi-cantly less than results obtained previously at other wavelenghts and indicatesthat the shock has an inhomogeneous cross section because X-ray observationprobe only the hottest part of the post-shock region whereas measurements inthe UV, optical and infrared are able to detect cooler components, too. FurtherI calculate a lower bound on the absorbing column density NH = 1020 cm−2

from the model, which is compatible to observations. The model does not fitBP Tau well, although I tried to correct the observations for different extinctionvalues. This is attibuted to BP Tau’s known strong coronal activity. Possiblevalues for the shock parameters are discussed, the most likely ones point to475 km/s and 1012 cm−3.

Zusammenfassung

Eine der wesentlichen Voraussagen der Modellvorstellung vom magnetischkontrollierten Einfall auf klassische T Tauri Sterne (CTTS) ist eine heißeEinschlagsregion. Ich stelle hier eine Computersimulation vor, die explizitauch Nicht-Gleichgewichtsphanomene berucksichtigt. Als beobachtbare Großewird fur viele Spektrallinien im Rontgenbereich die Intensitat berechnet.Es zeigt sich, dass im Wesentlichen nur zwei Parameter die Struktur desSchocks bestimmen: die Einfallgeschwindigkeit und die Dichte das einfallen-den Gases vor der Schockfront. Im Vergleich mit einer Modellatmosphareermittle ich die Tiefe, in der sich die Schockfront bildet. Fur Schocks mitEinfallgeschwindigkeiten uber 400 km/s und Einfalldichten unter 1011 cm−3 be-wirkt der Druck der Sternatmosphare Abweichungen in den unteren Schichtenvom hier benutzten Modell. Linienflusse fur TW Hya aus der Literatur stimmenmit meinen Vorhersagen uberein und erlauben es, die Einfallgeschwindigkeit zu525 km/s und die Dichte zu 1012 cm−3 zu bestimmen. Ich kann bestatigen,dass im beobachteten Plasma Metalle abgereichert sind, aber Neon verstarktauftritt. Die Einschlagsregion fullt 0.1 − 0.2 % der Sternoberflache und derMassenfluss betragt 1 · 10−10M/Jahr. Diese Zahlen sind deutlich geringerals vorhergehende Resultate aus anderen Wellenlangenbereichen und deutenan, dass ein Querschnitt durch den Schock moglicherweise nicht homogenist, weil im Rontgenbereich nur die heißesten Bereiche der Region nach demSchock beobachtet werden, im UV, optischen oder infraroten aber auch kaltereRegionen erfasst werden. Aus dem Modell berechne ich eine untere Grenze furdie absorbierende Saulendichte im Schock von NH = 1020 cm−2, dieser Wertist mit Beobachtungen vertraglich. BP Tau wird durch das Modell nicht gutbeschrieben, obwohl die Beobachtungen fur Absorption korrigiert wurden. DerUnterschied erklart sich aus der schon bekannten starken koronalen Aktivitatvon BP Tau. Mogliche Werte fur die Schock Parameter werden diskutiert, diebesten sind 475 km/s und 1012 cm−3.

4 CONTENTS

Contents

1 Introduction 61.1 Star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 The accretion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Plan of report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Basic physics 112.1 Gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Continuum treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Basic equations of hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Simplifying assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.4 Simplified fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.5 The shock front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.6 Post-shock region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Microscopic physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Ionisation state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.3 Excitation state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.4 Free-Free-Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Program design and verification 243.1 Program design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Design concept and programming language . . . . . . . . . . . . . . . . . . . 243.1.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.3 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Ionisation/Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Accuracy and program parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Results 334.1 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Temperature and density profiles . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Ionisation states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.3 Influence of the initial temperature . . . . . . . . . . . . . . . . . . . . . . . 364.1.4 Influence of density and infall velocity . . . . . . . . . . . . . . . . . . . . . 364.1.5 Depth of shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.6 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.7 Deviation from equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

CONTENTS 5

4.2 Emission lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Origin of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.2 Line ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Application to TW Hya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.1 Abundance independent line ratios

Infall velocity and density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.2 Element abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3.3 Filling factor and mass accretion rate . . . . . . . . . . . . . . . . . . . . . . 46

4.4 Application to BP Tauri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Discussion 515.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 TW Hydrae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 BP Tauri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 CHAPTER 1. INTRODUCTION

Chapter 1

Introduction

Figure 1.1: Sky disk of Nebra (Credit:Landesamt fur Archaologie Sachsen-Anhalt).The group of 7 golden circles between ’sun’ and’moon’ is commonly referred to as the Pleiades(Meller 2002).

1.1 Star formation

Far back in time the first humans already saw asky very similar to our own when they looked upat night. Artifacts brought to us by archaeolo-gists from the dawn of time like the 3600 yearold sky-disk of Nebra (see figure 1.1) prove thatthe heavens above us were always inspected bymankind. At first the sky seems to be immovableand unchangeable, a place of gods, but using so-phisticated technics we know nowadays that starsare born and die, although on eternal time scales

Figure 1.2: Aloha Orion, imaged in the in-frared with the WFCAM of the United KingdomInfrared Telescope (UKIRT) on Mauna Kea, is alarge molecular cloud. Credit: Joint AstronomyCentre; image processing by C. Davis, W.Varricatt

compared to human life. They emerge from giantmolecular clouds like the one in Figure 1.2. Theseclouds begin to collapse under their own gravi-tation and, because they are not homogeneous,the mass concentrates in several distinct regions,which later will become star systems, possibly bi-naries or even hosts for planets (a comprehensivereview of the whole development can be found inFeigelson and Montmerle 1999). In this phase,which takes a few thousand years, the objects

1.1. STAR FORMATION 7

are called ’infalling protostars’. They are stillhidden in a large surrounding matter envelope,which makes it difficult to observe them, onlythe infrared (class 0 sources) and radio emissionpenetrates this barrier. Conservation of the ini-tial angular momentum forces the structure torotate around an axis, but nothing supports theenvelope along that axis. So material falls to theplane of the evolving protostar, keeping its an-gular momentum, and develops a thick disk, thestar becomes an ’evolved protostar’. Its age isnow of the order 105 years, its infrarad signatureclass I. The material continues to contentrate inthe disk plane. At the same time the star growsby mass flux from the disk. Only in the lastdecade evidence emerged that the disk has nodirect contact to the star at a boundary layer,but an inner radius of ∼ 0.5 AU (e.g. Muzerolleet al. 2003). In the 1990’s theoretical consider-ations were published proposing a magneticallyfunnelled accretion stream stretching above thisgap (Uchida 1983; Koenigl 1991), which is ex-plained in more detail in section 1.2. When theenvelope does not hide the system any longer,the infrared signature of the source changes toclass II and it is called a classical T Tauri star(CTTS) following the prototype T Tauri (see fig-ure 1.3) in the Taurus-Auriga star forming re-gion. This phase lasts a few million years, un-til the disk is not replenished any longer fromthe envelope and begins to vanish. In this statethe stars are weak-line T Tauri stars (WTTS).When the disk has completely dissolved eitherby accretion or because it was blown away by awind (and maybe it formed even planets) the starreaches the main sequence. The energetic radia-tion from these stars is reflected in the remaininggas from the original cloud, giving us the oppor-tunity to watch wonderful pictures on the nightsky. An example is a young group of stars called’Pleiades’ (figure 1.4), by a strange coincidencethese are also the stars depicted on the ancientsky-disk of Nebra (figure 1.1) (Meller 2002).More massive stars go through a similar sequenceas the one just described, but this work dealswith relatively low mass (M∗ < 3M) TTS. Theyshow late-type spectra (see e.g. Muzerolle et al.

Figure 1.3: False Colour image of the collapsinggas cloud in the T Tau system. Credit: C. &F. Roddier (IfA, Hawaii), Canada-France-HawaiiTelescope.

1998b, 2003). Traditionally the distinction be-tween CTTS and WTTS is drawn by their Hαequivalent width. WTTS have weak (less than10 A) Hα lines, CTTS stronger ones. Nowadaysmore reliable tracers of accretion are known, themost prominent being the so called ’veiling’. Thismeans that lines which are normally in absorp-tion in comparable stars get veiled by hot con-tinuum emission generated in the spot where theaccreting matter hits the star. This has been ob-served in the infrared (Muzerolle et al. 2003) andin the ultraviolet (UV) band (Costa et al. 2000;Gullbring et al. 2000). According to the currentmodel (section 1.2) the gas moves from the diskto the star at near free-fall velocity. Again thisis a tracer for accretion and it has been founddirectly in lines as broad red wings (Edwardset al. 1994; Lamzin et al. 2004). It was explainedthat the gas and dust forms a disk around thestar because of angular momentum conservation,now observational evidence proves that materialis transferred to the star, so either the star spinsup considerably or angular momentum leaves the


Figure 1.4: The Pleiades are young stellar ob-jects. The blue nebula are rests of the gascloud which reflect the star light. Credit:Aron Charad, National Optical AstronomyObservatory

system. The second possibility is realised: Awind can be detected by analysing line profilesfor ions in a low ionisation state (e.g Lamzin et al.2004; Dupree et al. 2005). All this is thought tobe driven by a magnetic field, so it is plausible tocheck for emission at higher energies and in factX-ray emission was detected in many CTTS us-ing the Einstein (Feigelson and Decampli 1981;Feigelson and Kriss 1989) and ROSAT satelites(e.g. Feigelson et al. 1993; Neuhaeuser et al. 1995;Gregorio-Hetem et al. 1998). At first sight the X-ray emission seems to originate from coronal ac-tivity in a scaled-up version of our own Sun, butobservations with the new generation of satelites,which provide a much better spectral resolutionpoint to the existence of a dense plasma mostlikely in the accretion spot. Only two CTTShave been studied in detail so far using Chandra(Kastner et al. 2002) and XMM-Newton (Stelzerand Schmitt 2004; Schmitt et al. 2005).

1.2 The accretion model

Observations indicate, that the circumstellar diskaround CTTS has an inner radius of several stel-lar radii (Muzerolle et al. 1998a, 2003). The cur-

rent model states, that here, where the Keplerianrotation rate of the disk equals the stellar rota-tion period, magnetic field lines from a stellardipole filed can penetrate the disk. The geom-etry is shown in figure 1.5. At the inner rim ofthe disk gas is ionised by radiation from the star(Glassgold et al. 2000), so it starts to interactwith the magnetic field. The gravitation drags allmaterial towards the star, but the ions are forcedto follow the magnetic field lines and rise abovethe disk plane in an ’magnetically funnelled’ ac-cretion flow. Along the magnetic field they areaccelerated gravitationally until the impact onthe star with nearly free-fall velocity. Extensivetheoretical models have been calculated and showthat this mechanism can transport angular mo-mentum from the star to the disk, so the staris not accelerated to unusual fast rotation (Shuet al. 1994). Furthermore, this magnetic fieldconfiguration drives simultaneously a wind fromthe disk (Shu et al. 2000; Konigl and Pudritz2000).The accretion spot is presumably located at highlatitudes, but the exact position and the form ofthe regions is still a matter of debate. It couldbe a ring at constant latitude or several distinctspots. Romanova et al. (2004) have done a fully3 dimensional magneto-hydrodynamic (MHD)simulation of the funnel flow and find two or moreessentially banana shaped spots near the pole de-pending on the inclination of magnetic and rota-tional axis of the star. An example from theirsimulation is shown in figure 1.6. Two accretionfunnels form (left panel) and the position of thespots for an misalignment between rotational andmagnetical axis Θ = 15 settells close the mag-netic poles. The spot is inhomogeneous, density(right panel) and velocity increase towards thecentre.This general accretion model describes whichdensity and infall velocities are to be expectedin an accretion spot. The inflowing gas pene-trates the stellar atmosphere until a shock frontdevelops which heats up the gas. It emits theline and continuum radiation, that we observe asveiling. It cools down and mixes with the sur-rounding stellar atmosphere in the end. Only

1.2. THE ACCRETION MODEL 9

Figure 1.5: Model of disk accretion, from Hartmann (1998)

Figure 1.6: Left: Accretion funnel streams: Red marks sample magnetic field lines. The arrowfollows a particle stream. Right: The shape of the accretion spot resembles a banana. µ marksthe axis of the magnetic moment of the star, Ω its rotation axis and Θ the angle between them.The density of the accretion stream is colour coded from blue (low) to red (high). from: Romanovaet al. (2004)


a detailed modelling of shock front and radi-ating region can explain the observed features.Concentrating on the optical and infrared emis-sion Calvet and Gullbring (1998) published asuccessful model and similar work was done forX-ray by Lamzin (1998). Since that time theavailable atomic data has improved and compu-tational contrains are lifted due to much fasterhardware. In this thesis I present a new simu-lation of the accretion shock and the followingregion, which considers non-equilibrium effects,uses up-to-date databases and resolves individ-ual lines in the X-ray region. The results pre-sented are applied to high-resolution X-ray spec-troscopy of the CTTS TW Hya (Kastner et al.2002; Stelzer and Schmitt 2004) and BP Tau(Schmitt et al. 2005).

1.3 Plan of report

In the following section the underlying physics forthe model is summarised and extensive deriva-tions of the implemented formulas are given. Atthe same time necessary assumptions and simpli-fications are pointed out. In section 3 the struc-ture of the implementation and its computationalparameters are explained. The results are pre-sented in section 4, first the models are analysedin general and the condition on the emitting re-gions are discussed, then they are applied to datafrom TW Hya and BP Tau. The pre-shock con-ditions for these stars are identified. In section 5my findings are compared to literature values andpossible physical implications are discussed. Ashort summary of the main points is given in sec-tion 5.4.

11

Chapter 2

Basic physics

In this section I derive the physical formulas, which are implemented in the simulation. The firstpart, section 2.1, tells about the macroscopic physics, the hydrodynamics, the second part, section2.2, look at the physics of the single atom or ion and describes ionisation and recombination and allprocesses which produce radiation. In both sections the relevant simplifications are explained anddiscussed.

2.1 Gas dynamics

Different physical processes are important in different zones. The gravitation of the star acceleratesthin gas onto the stellar surface (’pre-shock zone’). On the surface it hits the atmosphere and astrong shock develops (’shock zone’). This is called an accretion shock. In a first layer the bulkkinetic energy is converted into undirected thermal energy (’shock front’). This process is veryefficient for ions but not for electrons. Two interacting fluids develop: The ions and the electrons.Energy exchange between the species is slow and therefore they are described as two different, butinteracting components. This model is commonly referred to as ’Two fluid approximation’. Thetemperatures of ions and electrons then converge (’post-shock zone’) as the gas cools down becauseis it emits radiation (therefore also called ’cooling zone)’.

2.1.1 Continuum treatment

If the scale height L of all relevant physical processes is much larger than the mean free path lof a single particle a continuum treatment is possible. Otherwise the gas-dynamic equation, theBoltzmann equation, needs to be solved. Only for the first case e.g. the pressure p or the particlenumber density n are well defined in a small volume element V as the average of some microscopicquantity of each particle.The mean free path can be estimated by multiplying the formula for the mean free collision time tcfor charged particles from Spitzer (1965,equation 5-26) with some average velocity v:

l ≈ vtc ≈ v0.29

Λ

m1/2(kT )3/2

ne4≈ 1 m (2.1)

m is the mass (e.g. of an electron), e its charge. k is Boltzmann’s constant and Λ the Coulomblogarithm which is defined later (equation 2.32). (I use the thermal electron velocity for T = 106 Kand n = 1014 cm−3). At this point is only important that the numerical evaluation of this formulafor typical conditions for electrons gives l ≈ 1 m. The same is true for protons because the velocity

12 CHAPTER 2. BASIC PHYSICS

is proportional to the inverse square root of the mass so the mean free path is independent of it.’Typical’ condition are taken from Calvet and Gullbring (1998) and Lamzin (1995) who obtaindensities in the shock region of about n ≈ 10−11 g

cm3 which translates roughly into a number densityof ions nion ≈ 1014cm−3 in their simulation of accretion shocks with temperatures ≈ 106 K.

This value is much smaller than the dimension of the radiation emitting region. The viscous shockfront is only of the order of a few mean free paths, so its internal structure cannot be resolved in acontinuum approximation and the resulting hydrodynamical quantities will be calculated in section2.1.5.

2.1.2 Basic equations of hydrodynamics

If the mean free path is small, all properties of the gas are described by a few macroscopic quantities.In general they depend on position ~r and time t. These quantities can be scalars like temperatureT = T (~r, t) and mass density ρ = ρ(~r, t), vectors like the bulk velocity ~v = ~v(~r, t) or tensors like thestress tensor Pik = Pik(~r, t). To simplify the notation the dependence on space and time coordinatesis no longer explicitly written.The mass density ρ is then directly proportional to the ion number density

ρ = µmHnion (2.2)

mH is the mass of a hydrogen atom. The mean molecular weight µ is a dimensionless numbercalculated from

µ =∑

i

ξimi

mH(2.3)

where mi is the mass of one atom, ξi the relative abundance of species i and i runs over all atomicspecies in the gas. The stress tensor Pik is given by component i of the force acting on the unitsurface element normal to direction k. It can be decomposed in

Pik = −δikP + πik (2.4)

P is the scalar thermodynamic pressure, πik the viscous stress tensor and δik the Kronecker-δ, whichequals 1 for i = k and 0 otherwise. The sign difference on the right hand side arises, because Pikrepresents forces acting on a volume element V , but the pressure P is a force acting out of thevolume on its surroundings.The equation of state for a gas, if it is known, provides a relation between preassure and temperature.For the perfect gas it reads:

P = nkT (2.5)

where k is Boltzmann’s constant.The kinetic energy of a single particle is Ekin = m

2~u2 with ~u being its speed and m its mass. For

larger ensembles it is useful to decompose the velocity in a bulk part ~v which represents the velocityof the centre of mass and a random part which is called thermal motion ~w. The energy per atom/ionE consists of the kinetic part

Ekin =1

2µmH~v

2 (2.6)

and a thermal part

Etherm =nf2kT (2.7)

2.1. GAS DYNAMICS 13

Electrons and ions have three degrees of freedom associated with the three spatial dimensions sonf = 3. Especially for processes with constant external pressure it is useful to define the enthalpyH = Etherm + P

nbecause a gas needs to exert the force P against the external pressure to expand.

For a volume V this results in the work PV . So the work per atom/ion required to expand the gasfrom zero to the specific volume V = 1

nis P

n. For an perfect monoatomic gas:

H = Etherm +P

n=

3

2kT + kT =

5

2kT =

5P

2n(2.8)

In general

H =γ

γ − 1

P

n(2.9)

γ is the adiabatic index, which is γ = 5/3 for monoatomic gases. Now the fluxes of conserved

quantities are calculated following the derivation of Shu (1991,chapter 4). ~A is a surface normal ofthe volume V pointing outward.

mass flux There is no way to produce or destroy mass, so the change of the total mass in a volumeV equals the flux through its surface A.

d

dt

∫

V

ρdV = −∮

∂V

ρ~vd ~A = −∫

V

∇(ρ~v)dV

The second step is done with the help of Gauß’s theorem. On the left hand side integrationand differentiation are exchanged to get ∂ρ

∂t+∇~jm = 0 using the mass flux

~jm = ρ~v (2.10)

This is valid for every differential volume element so it is valid everywhere. Here no fusionor fission takes place. So the number of atoms stays constant, too and it is useful to defineatom/ion flux by number:

~j = n~v (2.11)

momentum flux The momentum ~p =∫Vρ~vdV of a fluid changes by momentum flux or by forces

acting on the fluid. The pressure acts on the surface of the volume V , the gravitation is avolumetric force (~g is the gravitational acceleration and points towards the gravitating mass).For momentum component i:

d

dt

∫

V

ρvidV = −∮

A

(ρvi)~vd ~A+

∮

∂V

PikdAk +

∫

V

ρgidV

This can be transformed similarly to the mass flux to reveal the momentum flux tensor as

jpik = (µmHnvivk + Pik) (2.12)

energy flux In a volume V there are nV particles. The time rate of change of the total energy(kinetic and thermal) is given by the flux of energy through the surface and by the work done

on this volume. Work is done by stress and gravitation. Additionally there is a heat flux ~Fthermand a radiation flux. Energy is lost by emission (with a rate Γrad) and gained by absorption(Λrad) of radiation.

d

dt

∫

V

n(Ekin + Etherm)dV = −∮

A

n(Ekin + Etherm)~vd ~A+

∮AviPikdAk +

∫Vρ~v~gdV −

∮A~Fthermd ~A+

∫V

(Γrad − Λrad)dV


The equation for the energy flux per atom/ion

~jE = (Ekin + Etherm)~v − ~vPikn

+~Fthermn

(2.13)

now looks liked

dt[n(Ekin + Etherm)] +

∂

∂xk(njEk) = ρ~v~g + Γrad − Λrad

From standard thermodynamics the adiabatic speed of sound c is known as

c =

√γPV

µmH=

√γkT

µmH(2.14)

In media streaming with some bulk velocity v it is useful to define the Mach number M as the ratiobetween bulk velocity and speed of sound

M =v

c(2.15)

because there are fundamental differences between subsonic (M < 1) and supersonic (M > 1)motions.

2.1.3 Simplifying assumptions

In order to do the simulation some approximations and assumptions are necessary. They are de-scribed here and their applicability is discussed if possible a priory.

Direction of flux and choice of coordinate system

Figure 2.1: The coordinate system: The z-axis points inward and material flows along this direction.

The spatial extent of the accretion region is very small compared to the total stellar surface. Thishas been shown by previous simulations (Lamzin 1995; Calvet and Gullbring 1998; Romanova et al.


2004) and confirmed by observations (Calvet and Gullbring 1998; Costa et al. 2000; Muzerolle et al.2000; Stelzer and Schmitt 2004; Schmitt et al. 2005). The surface can be taken as flat and thegas moves normal to it. The origin of the coordinate system is placed at the accretion shock frontwith the positive direction of the z-axis towards the centre of the star (see figure 2.1). Assuming nogradients and no flux parallel to the surface, all partial derivatives in the x- and y-direction are zero.Obviously this cannot be true everywhere because the accretion column is finite and has a surfacewith gradients towards the normal stellar atmosphere, but it greatly simplifies the calculations.Surface effects include thermal and radiative conduction or anisotropy pressure. If the diameterof the accretion region is large compared to its depth in z direction, then surface effects should besmall. Accretion spot sizes from the references cited above and a shock depth from this simulation(setion 4.1.5) indicate that condition is easily satisfied.

Time dependence

In the model there is no explicit time dependence. Everything is in a quasi static state. This meansthe structure of the shock does not change with time, e.g. the energy contained in a specified depthis constant and because of that not only the energy itself is conserved, but also the energy flux. Itis questionable if such a state really exists or if turbulent fluxes create an ever changing structure.On the one hand the high infall velocities contain enough energy to fuel turbulent phenomena, buton the other hand they should quickly wash out small fluctuations. Under assumption of a quasistatic state there are no temporal gradients and all partial derivatives to time can be set to zero.This also simplifies total time derivatives (written in one dimension).

d

dt=

∂

∂t+∂z

∂t

∂

∂z= v

∂

∂z(2.16)

Magnetic fields and heat conduction

As described in section 1.2 the current model states that the infalling material is channelled bymagnetic field lines and moves along them, so the direction of the magnetic flux is parallel to thebulk motion and there is no magnetic force. So it is sufficient to use normal hydrodynamics andnot magneto-hydrodynamics. Nevertheless, there will be a small scale chaotic magnetic field. Thissuppresses free electron movement and consequently there is no or only little heat conduction, sincein a plasma the electrons transport most of the heat due to their longer mean free path (Zel’Dovichand Raizer 1967). I use Ftherm = 0 in the simulation. Lamzin (1998) includes a small magnetic fieldin his energy and momentum equations and finds its strength is not coupled to macroscopic shockparameters. In section 4.3 I will discuss the importance of heat conduction if it is not suppressed.

Gravitation

Gravitational forces are negligible. The difference of the potential energies for an ion between thestellar surface and the bottom of the accretion region in a few hundred kilometers depth is only0.2 eV, which is very small compared to the thermal energy of a few eV and the kinetic energyof hundreds of eV. The electron mass is much smaller and so the potential and kinetic energy arethree orders of magnitude smaller, hence gravitational energy is very small compared to thermaland kinetic, too.


Viscous forces

Internal stress is small in thin gases and is neglected. On the right hand side of equation (2.4) Ikeep only the scalar component.

2.1.4 Simplified fluxes

These approximations simplify the equations of the fluxes. Scalar expressions representing the z-component of a quantity are used instead of vector expressions, because there is no flux and nogradient in x and y direction:

~jm =

00jm

So equation (2.10) reads nowjm = ρv (2.17)

or the particle flux densityj = nv (2.18)

The momentum flux (2.12) simplifies to

jp = µmHnv2 + P = µmHnv

2 + nkT (2.19)

In the two fluid approximation the total momentum flux is the sum of the ion and the electroncomponent. The term melectronv

2 can be neglected because melec mion.

jp = µmHnv2 + nkTion + neleckTelec (2.20)

The energy flux per atom/ion (2.13) gives

jE = (Ekin + Etherm)v +vP

n= v(Ekin +H) (2.21)

2.1.5 The shock front

The shock front is very thin. It takes only a few mean free paths for the ions to decelerate andtransfer their kinetic energy to thermal motion. Other degrees of freedom than thermal ones arenot excited so fast, so the ionisation state does not change. Zel’Dovich and Raizer (1967) define thethickness δ of the shock front as

δ =v0 − v1

(∂v/∂x)max

Here v0 is the bulk velocity before the gas reaches the shock front and v1 after passing through thefront. They also show that the order of magnitude for the front can be approximated in terms ofthe Mach number M (equation 2.15) and the mean free path l (chapter 7 Zel’Dovich and Raizer1967):

δ ≈ lM

M2 − 1

For strong shocks the thickness seems to vanish, but of course there is a lower bound. The thicknesscannot be shorter than a few mean free paths because there obviously need to be interactionsbetween the particles to decelerate them. Therfore, the volume of the viscous shock front is small


and the internal structure unimportant, which allows a treatment as mathematical discontinuity.The fluxes of the conserved quantities mass (2.17), momentum (2.20) and energy (2.21) are sufficientto calculate the state behind the shock front. This derivation follows the book of Zel’Dovich andRaizer (1967) (chapter 7)

ρ0v0 = ρ1v1 (2.22)

P0 + ρ0v20 = P1 + ρ1v

21 (2.23)

h0 + µmHv20 = h1 + µmHv

21 (2.24)

The state before the shock is marked by index 0, behind the shock front by index 1. Equation (2.24)is rewritten in terms of ρ, v and P with (2.8).

5P0

2ρ0

+v2

0

2=

5P1

2ρ1

+v2

1

2(2.25)

In equation (2.22) ρ1 can be eliminated and the resulting equations are written in terms of dimen-sionless variables:

v =v1

v0P =

P1

P0x =

P0

ρ0v20

x contains only the variables before the shock and is known. Now (2.23) is

x+ 1 = P x+ v

Substituting this in (2.25) leads to

5x + 1 = 5v(x + 1)− 4v2

and0 = 4v2 − 5 · (x + 1)v + (5x+ 1)

with two solutions

v =5x+ 1

4and v = 1

The second solution describes a situation with steady flow and without shock, the first one yieldsthe desired result. All parameters behind the shock can now be calculated:

P1 =3ρ0v

20 − P0

4(2.26)

v1 =P0 − P1

ρ0v0+ v0 (2.27)

ρ1 =ρ0v0

v1(2.28)

These equations are called ’Rankine-Hugoniot-jump conditions’.There are much less viscous forces between electrons. Therefore the electronic component is notprocessed via the Rankine-Hugoniot conditions, but to first approximation just gets compressedadiabatically. This means the temperature raises according to

Te1 = Te0(ρ1

ρ0)(γ−1) (2.29)

with γ being the adiabatic index like in equation (2.9). Evaluation these equations shows thatthe temperature of the electrons is orders of magnitude lower then the ion temperature after theaccretion shock front.


2.1.6 Post-shock region

While passing the shock front the gas decelerates to subsonic velocities and the ions are much hotterthen the electrons. Therefore, heat is transfered from the ionic to the electronic component in thepost-shock zone. Simultaneously the gas radiates and cools down, so the energy of the gas is nolonger conserved and equation (2.21) is not valid anymore as it does not include the radiative energyflux. Instead the energy per particle must be described by an ordinary differential equation (ODE)for the ions and another for the electrons.Starting from the thermodynamic relation

TdΣ− PdV = dU (2.30)

where Σ denotes the entropy and U the internal energy I will derive the ODE for the ionic component.For simplification everything will be expressed per heavy particle (atom or ion).TdΣ = dQ is the flux of heat through the boundaries of the system. The system ”ions” looses heatby collisions with the colder electrons. Microscopically heat is just the energy of unordered motion.The fraction of the kinetic energy of a single particle transferred per collision cannot exceed themass ratio melec

mion. It is most efficient for the light ions. Hydrogen is by far the most abundant element

and its atomic mass number is one. The hydrogen ions have a much larger cross sections than theatoms and hydrogen is nearly completely ionised in the whole shock region anyway, so contributionsof heavier elements are neglected. Zel’Dovich and Raizer (1967,Chapter VII, §10) give the heatoutflow ωei per unit volume per unit time as (in cgs-units)

ωei =3

2knTion − Telec

T3/2elec

Λ

252

The number density of hydrogen ions is the number density of all heavy particles n multiplied withthe abundance ξH of hydrogen and the ionisation fraction x1

H (these quantities are defined in section(2.2) more strictly), so

ωei =3

2kξHx

1H

Tion − TelecT

3/2elec

Λ

252(2.31)

Λ is the Coulomb-logarithmΛ ≈ 9.4 + 1.5 lnTelec − 0.5 lnnelec (2.32)

In the last two equations temperatures are meant to be in Kelvin. Substituting (2.31) in (2.30)written per heavy particle and taking the time derivative results in

dU

dt+ P

dV

dt= T

dΣ

dt=dQ

dt=

1

nωei

Using the stationary condition (2.16) transforms this in an ordinary differential equation only de-pendent on x. Differentiation with respect to x will be indicated by ′.

vU ′ + PvV ′ =1

nωei

The internal energy U is in this case the thermal energy from equation (2.7), the pressure P can berewritten with (2.5). The specific volume V is the inverse of the number density V = 1

n.

v

(3

2kTion

)′+ vnkTion

(1

n

)′= −ωeixen (2.33)


It is convenient to write the electron number density as nelec = xen. xe is the number of electronsper heavy particle.The derivation for the electron component is similar. As the heat loss of the ion gas is a gain forthe electrons this term enters with opposite sign and an additional loss term Qcol due to radiationappears.

v

(3

2xekTelec

)′+ vxenkTelec

(1

n

)′= (ωei −Qcol)xen (2.34)

In principle Qcol depends on the local temperature, density and ionisation state of the gas. Theseeffects are small and they are neglected in solving the ODEs for small time steps. There are fourindependent variables left (n,v,Tion and Telec). n can be eliminated easily because of particle numberconservation (equation 2.18). The momentum flux conservation (2.20) is a second algebraic equationand can be solved for v. Unfortunately it is quadratic in v with two solutions. I will use the followingabbreviations

M = µmH

τi = kTion

τe = xekTelec

τ = τi + τe

(2.35)

These solutions are

v± =jp ±

√j2p − 4Mj2τ

2Mj(2.36)

v+ and v− are equal for v0 =√

τM

which is the isothermal speed of sound. It is lower than theadiabatic one in (2.14). For subsonic velocities v− is the physical decelerating solution. v+ is theaccelerating branch. I will need the derivative of v with respect to x.

dv

dx=

jτ ′√j2p − 4Mτj2

=j (τ ′iτ

′e)√

j2p − 4Mτj2

(2.37)

Now (2.33) and (2.34) can be rewritten:

3

2

τ ′iτi

+v′

v= −ωeixej

τiv2(2.38)

3

2

τ ′eτe

+v′

v= − xej

τev2(ωei −Qcol) (2.39)

(2.40)

Solving for τ ′i and τ ′e leaves a set of two coupled ODEs which needs to be solved numerically.

τ ′i =

xejv2

[−ωei

τi− 2j

3v√j2p−4Mτj2

(ωei

(1 + τe

τi

)−Qcol

)]

32τi

+ j

v√j2p−4Mτj2

(1 + τe

τi

) (2.41)

I state τ ′e here without backsubstituting τi since the formula is complex.

τ ′e = τ ′iτeτi

+2xej

3v2

[ωei

(1 +

τeτi

)− qcol

](2.42)


To extract the physical parameter Telec x′e must be known from the ionisation and recombination

processes (given in section 2.2).

T ′e =τ ′exek− Telec

x′exe

2.2 Microscopic physics

Observations in astrophysics measure radiation, which is emitted by several different processes.They can be separated into free-free, free-bound and bound-bound processes. They all depend onthe ionisation and excitation state of the atom or ion in question and possibly on some macroscopicparameters namely the electron density.The abundance ξ of all elements stays constant, because in the accretion shock no fission or fusiontakes place. The abundance of element A is defined as

ξA =nA

n(2.43)

where n is the number density of all heavy particles and nA the number density of atoms and ionswith atomic number A.An atom with element number A can be in A + 1 different ionisation states. By convention theneutral state is labeled as ’I’, e.g. ’C I’. The fraction of all atoms of element A with ion charge Zis written xAZ . All ions are in exactly one state at any time so

A∑

Z=0

xAZ = 1 (2.44)

for all A. Using equation (2.43) the number density nAZ of ions with charge Z and element numberA can be calculated as

nAZ = nξAxAZ (2.45)

It changes when ionisation or recombination processes occur.This section lists the simplifications in the treatment of the microscopic physics and explains thedifferent processes which are considered in the simulation.

2.2.1 Simplifications

Many different processes take place in a plasma. To model the overall behaviour the physically mostimportant ones need to be selected. In this simulation the plasma is treated as purely collision-dominated. The collision rate is small enough so that excitation and deexcitation processes aremuch faster than ionisation and recombination, therefore they can be treated separately. At eachtime the excitation is taken to be in its equilibrium state.

Velocity distribution

According to Spitzer (1965,chapter 5.3) particle velocities are Maxwellian distributed after a fewmean free path lengths. His formula for the collision time scale tc of charged particles like electronsor protons with themselves is

tc =0.290

Λ

m1/2(kT )3/2

ne4(in cgs-units) (2.46)

2.2. MICROSCOPIC PHYSICS 21

Here e is the elementary charge and all other symbols have their usual meaning. For hydrogen ionsthe time scale is (neglecting the dependence of Λ on T and n, see equation (2.32)):

tc(hydrogen ions) ∼ 8 · 10−6

(T

106 K

)3/2 (1014/cm3

n

)[in s] (2.47)

Right behind the shock front the temperature may be higher and the density two orders of magnitudelower, but even in this case the ions will be Maxwell distributed after a few hundred meters, electronsarrive at a Maxwellian distribution about 50 times faster than ions, because their mass is much lower.In the shock front itself they stay cool and keep their old Maxwell distribution.A Maxwellian velocity distribution Φ(v) is characterised by its temperature T :

Φ(v)dv =4√π

( m

2kT

) 32v2 exp

(v2m

2kT

)dv (2.48)

where k is the Boltzmann constanta and m the mass of the particle. The most probable velocity is

vprob =√

2kTm

. Hydrogen, the lightest atom, is already ∼ 1800 times more massive than an electron,

therefore its velocity can be neglected if the temperatures of electrons and ions are similar. In thisapproximation the relative velocity ~v2

rel between electron and proton is just the electron velocity. In(2.48) the velocity contributes only quadratically and the Maxwellian averaged of a quantity a iswritten < a >. The averaged quadratic relative velocity is

~v2rel = < (~vion − ~velec)2 >

= < v2ion + v2

elec − 2~vion~velec >

= < v2ion > + < v2

elec > −2 < ~vion~velec >

(2.49)

< ~vion~velec > equals 0 for uncorrelated velocities. This can be used to fold the two Maxwell distri-butions for electrons and ions. The result is again a Maxwell distribution and can be treated as ifonly the electrons move with the temperature Teeff = Telec + Tion

melecmion

.

Electromagnetic fields

Electric and magnetic fields are neglected. This is possible because the electrons have a high mobilityand any effective charge density is balanced quickly by in- or outflow of electrons. Zel’Dovichand Raizer (1967,in chapter VII,§13) argue that there is actually a small charge separation in theshock wave only on length scales of the Debye-radius which is smaller than ≈ 10−2 cm for typicalconditions in the accretion plasma. It was already mentioned that magnetic fields are interestingglobally, because they channel the material onto the accretion spot, but the field strength is toosmall to produce effects, which are directly observable in X-rays. In this model electrons and ionsmove with identical bulk velocity so there is no current which could produce a strong local magneticfield.

Optical depth

Photons emitted in an optically thick plasma are scattered after emission or absorbed and laterreemitted. This can change the spectral distribution and is an energy source in the absorbingregions. Much simpler are optically thin plasmas and in this model the assumption is made that


every photon escapes from the plasma. Hot plasma usually has a very small continuum opacity andthe X-ray emitting region is not very large. However, radiation may be reprocessed in the accretioncolumn and high energy photons will be absorbed and create a region of ionised plasma in front ofthe shock (Lamzin 1995; Calvet and Gullbring 1998).

2.2.2 Ionisation state

Ions are ionised by electron collisions (bound-free) and recombine by electron capture (free-bound).So the number density of ionisations ni→i+1 per unit time from state i to i + 1 is proportional tothe number density of ions ni in ionisation state i and the number density of electrons ne. Forconvenience I do not write a superscript identifying the element in question.

ni→i+1 = Ri→i+1neni (2.50)

Recombination is the reverse process

ni→i−1 = Ri→i−1neni (2.51)

Ri→j is called a rate coefficient. It describes ionisation for j = i+1 and recombination for j = i−1.For element z R1→0 = Rz+1→z+2 = 0 because 1 represents the neutral atom which cannot recombineany further and z + 1 the completely stripped ion which cannot lose more electrons. The crosssection σ for each process depends not only on the ion, but also on the relative velocity of ion andelectron as well. The rate coefficient is proportional to the Maxwellian averaged cross section:

R ∼< σ(v) > (2.52)

On the one hand the number of ions in state i decreases by ionisation to i+ 1 or recombination toi− 1, on the other hand it increases by ionisations from i− 1 to i and by recombination from i+ 1.So for each element z there is a set of equations

dnidt

= ne(Ri−1→i ni−1 − (Ri→i+1 +Ri→i−1)ni +Ri+1→i ni+1) (2.53)

Through the factor ne the equations for all elements are coupled and together with the condition ofnumber conservation they provide a complete system of coupled differential equations. This can besimplified considerably with the assumption that ne is constant (which is always true if hydrogenis completly ionised) for small time intervals ∆t because this results in one independent set ofequations per element. Dividing by the number density of the element in question and using 2.45leads to

dxidt

= ne(Ri−1→i xi−1 − (Ri→i+1 +Ri→i−1) xi +Ri+1→i xi+1) (2.54)

This approximation is good if most hydrogen is ionised because its abundance ξH is much largerthan for any other element. This is the case in the post-shock zone and I assume that this is truealready in front of the shock (as shown in Lamzin 1995; Calvet and Gullbring 1998).In a quasi static state all time derivatives vanish. The z + 1 equations for element z can be writtenin matrix form. Only z of them are linearly independent, so the first row can be replaced with athe normalisation condition (2.44). Here the arrows are left out for short notation, R12 should be

2.2. MICROSCOPIC PHYSICS 23

read as R1→2.

100...

=

1 1 1 1 · · ·neR12 −ne(R21 +R23) neR32 0 · · ·

0 neR23 −ne(R32 +R34) neR43 · · ·...

. . .. . .

. . .. . .

x1

x2

x3...

(2.55)

There are two important ways to ionise an atom or ion (Raymond 1988). On the collision theelectron can directly ionise (direct ionisation) or excite the target above the ionisation thresholdand instead of falling back to its groundlevel the excited electron can then autoionise (excitation- autoionisation). Mazzotta et al. (1998) use a single fitting formula from Arnaud and Rothenflug(1985) and Arnaud and Raymond (1992) to calculate the rate coefficients. Recombination is possiblein three ways (Raymond 1988): Direct recombination means that the electron is captured to a lowstate. Its energy is emitted by a photon. Instead it could loose at least some energy by excitingan electron of the target. If that electron is emitted again the net result is just elastic scattering,but if it is still bound there are now two excited electrons which will emit photons. This is called adielectronic recombination. The third possibility is a charge transfer reaction, where one electronpasses from one atom, usually hydrogen or helium, to a higher ionised ion. This effectively raisesthe ionisation of the first and lowers it for the second. Raymond (1988) states that this may bean important channel in many astrophysical applications, but Arnaud and Rothenflug (1985) arguethat in hot plasmas there are not enough neutral atoms. All these are free-bound processes.

2.2.3 Excitation state

Spectral lines are excited mostly by electron collisions. They decay in one or more steps to theground state and emit photons in spectral lines. The line strength of several lines depends oncollision strength and the statistical weight of ground state (Raymond 1988). The energy of thetransition determines the wavelength of the line. These constants can be calculated from theoreticalconsiderations and from measurements. They are compiled in a number of databases like CHIANTI(Dere et al. 1998; Young et al. 2003) or APED (Smith et al. 2001). Brickhouse (1999) and Raymond(1988) point out how important the accuracy of theses rates is to obtain reliable simulations andstrongly advice to use up to date databases.Not all deexcitations produce lines. Some emit two photons and the energy per photon can vary.This is the so called ’Two photon continuum’.These processes are summarised as bound-bound reactions.

2.2.4 Free-Free-Reactions

In a plasma electrons that pass ions are accelerated in their electric field. Maxwell’s equationsshow that this causes the emission of Bremsstrahlung. It is possible to calculate classical and semi-classical approximations. The loss is proportional to the square of the ion charge, but for ions thatare not completely stripped, electrons with higher energy may impact more closely to the nucleusand feel a less screened field. I use the implementation of CHIANTI (Young et al. 1998). For veryhot plasmas (temperatures above 107 K) this is the dominant source of emission (Raymond 1988).

24 CHAPTER 3. PROGRAM DESIGN AND VERIFICATION

Chapter 3

Program design and verification

This section covers the design of the program, itsinternal structure and the data sources for physi-cal constants in section 3.1. It then describes thetests which check the correctness of the imple-mentation in 3.2 and finishes with a discussionabout the influence of some programme parame-ters on the execution time an the accuracy of thesimulated results (section 3.3).

3.1 Program design

3.1.1 Design concept and pro-

gramming language

The task set for this thesis consists of two parts.Firstly I needed to write the simulation pro-gram and secondly there is the analysis of theresults. The simulation part requires a languagewith full programming capability in a stable andfast environment. The existence of many cannedsolvers for differential equations and efficient ma-trix handling is an advantage. There is a softwarepackage which does the full excitation/deexcita-tion calculations and delivers the line intensities:CHIANTI (Young et al. 2003; Dere et al. 1997).This is published in IDL (Interactive data lan-guage). It would have been time consuming anderror prone to redo this in a different languageand it is always difficult to construct complexinteracting programs from different underlyingsystems. Additionally IDL is well suited to dodata analysis and plotting. Therefore, I decidedto implement as much as possible in IDL. Forthe progam design this means that no change ofany data format is necessary between simulation

and analysis. Data can be saved in ASCII ornative IDL format and some information can bekept in variables over the whole time. A huge li-brary of mathematical routines for equation solv-ing is supplied and arrays are a native data for-mat which can be handled fast and conveniently.On the other hand, modern object oriented de-sign technics are only partially supported. IDLsupplies basic objects in the version 6.1, but somemethods do not work in an object oriented en-vironment. Instead data is passed via commonblocks without explicit control over their lifetime.I could implement only one building block as anactual object: The output. All data to be kept ispassed into this object which organises the struc-ture and the sorting in different files. There aresome arguments to use dimensionless parametersof order unity, but this approach conflicts withCHIANTI which requires density and tempera-ture input in physical units. For consistency Ichose to use cgs-units throughout the program.

The implementation is simplified by the fact thatthe physics falls conveniently into two parts. Idescribed the physical basis of the hydrodynamicpart in 2.1 and the microscopic part in 2.2. Inprincipal both systems of equations are coupledvia the electron density, but in equation 2.54 theadvantage of keeping it constant for small timesteps is obvious. If this approach is taken al-ready for the microscopic physics alone there isno reason to keep the coupling to the hydrody-namics. In this way, the largest system of coupledODEs, which needs to be solved, is the rate equa-tion for the heaviest element consisting of ∼ 30equations, which is easily managable by typicalcanned ODE solvers.

3.1. PROGRAM DESIGN 25

Figure 3.1: Diagram of programme control, data flow (arrows) and output data (dashed arrows).See text for details

3.1.2 Numerical simulation

The simulation needs physical parameters of theinflowing gas as input (gas density and speed,temperatures of ion and electron component,abundances and ionisation states). In the shockfront it is compressed according to the analyt-ical Rankine-Hugoniot jump conditions. Afterpassing through this phase the numerical simula-tion starts. At the beginning of every time stepthe ionisation state changes via ionisation andrecombination. According to this the CHIANTIcode calculates the radiative losses. This in turnis an input to the procedure ’hydrodynamics’,which solves the gas dynamic equations. In thismanner the program calculates the shock struc-ture from the surface deeper into the stellar at-mosphere and after every step the data is sentto the output object. The simulation stops whenthe gas cools down to 12000 K (This is calledthe ’maximum depth’ of the shock.) In section4.1.5 it will be shown that the gas cools very fastin the end and because of that the temperaturewhich stops the simulation is unimportant. Theresults are written to disk and program controlreturns to the calling level. In the flow chart 3.1the program and data flow is shown.

In decoupling the hydrodynamic and the ionisa-tion equations an inaccuracy is introduced. Thisis smaller for smaller step sizes, because thenthe results of the hydrodynamics calculation af-fect the microscopic part earlier and the otherway round. On the other hand, smaller stepsare computationally more time-consuming. Anadaptive step size control adjusts the spatial sizeof steps in such a way that the maximum changeof all hydrodynamic variables or ionisation statesper step is smaller than a given maximum value.Figure 3.2 illustrates how it is implemented. Thebasic concept from 3.1 is extended. After pro-cessing the Rankine-Hugoniot conditions the ini-tial step size is set. Then the program simu-lates a complete step (microscopic and macro-scopic physics). If any variable changes by morethan the specified maximum (a program param-eter) this step is discarded, the step size reducedand the step recalculated. Otherwise it is ac-cepted and the data given to the output object.Now there is a check if one of the aborting con-ditions is matched. The normal stop occurs ifthe temperature drops below 12000 K. In this re-gion certainly no X-ray emission is formed anylonger and the structure is dominated by photo-


Figure 3.2: Flow chart for the adaptive step size control. The acceptable value for the maximumchange is a program parameter

spheric processes which are not included in thesimulation. The program aborts with an errormessage if it detects a NaN or reaches a maxi-mum depth or a maximum number of steps (toavoid endless loops while simulating larger grids).The change for the gas dynamic variables (den-sity, bulk velocity, temperature of ion and elec-tron component, electron to ion ratio and energyloss rate) is calculated relative to the last timestep. For the ion fraction it is defined as the ab-solute difference to the last time step. This isnecessary to avoid problems with very small ionabundances at the edge of numerical precision.No distinction is made between abundances ofdifferent elements. If e.g. the maximum changeallowed is 0.1 than ∆nHI = 0.11 will force theprogram to redo the step with a smaller step sizethe same way as e.g. ∆nCoXXV I = 0.11. Thisis necessary to obtain a good accuracy for theline data of every element. After each step, ac-cepted or discarded, the algorithm interpolateslinearly and sets the new step size in such a waythat the predicted change will be half the max-imum. For only slowly varying quantities thisassures that after the first the next step is mostlikely accepted as well and computing time is not

wasted. There is another programme parameterwhich controls the minimum step size. Under nocircumstances a step can be smaller than this inorder to prevent situations where the continuumtreatment is no longer possible.

3.1.3 Data sources

The module for microscopic physics needs atomicinput data. All parts which are handled bythe CHIANTI software, the excitation/deexci-tation calculations, of course use the CHIANTIdatabase (Young et al. 2003; Dere et al. 1997).The dielectronic recombination coefficients aretaken from Mazzotta et al. (1998). They takedata from various sources (see their paper andreferences therein) and fit them with a singleformula with tabulated coefficients. For the ra-diative recombination and the ionisation (colli-sional and auto-ionisation) rate I use a code fromD.A. Verner, which is available in electronic formon the web1. The data sources for the radia-tive recombination coefficients are given in table3.1. The ionisation data is taken from Arnaudand Raymond (1992) for Fe, from Arnaud and

1http://www.pa.uky.edu/∼verner/fortran.html

3.2. VERIFICATION 27

H-like, He-like, Li-like, Na-like Verner and Ferland (1996)Other ions of C, N, O, Ne Pequignot et al. (1991) refitted by Verner and

Ferland (1996) formula to ensure correct asymp-totes

Fe XVII-XXIII Arnaud and Raymond (1992)Fe I-XV refitted by Verner and Ferland (1996) formula to

ensure correct asymptotesOther ions of Mg, Si, S, Ar, Shull and van Steenberg (1982)

Ca, Fe, NiOther ions of Na, Al Landini and Monsignori Fossi (1990)

Other ions of F, P, Cl, K, Ti, Landini and Monsignori Fossi (1991)Cr, Mn, Co (excluding Ti I-II,

Cr I-IV, Mn I-V, Co I)All other species interpolations of the power-law fits by Verner

Table 3.1: Articles with data used by the Verner recombination code

Rothenflug (1985) for H, He, C, N, O, Ne, Na,Mg, Al, Si, S, Ar, Ca, Ni (with some correctionsdescribed in Verner and Iakovlev (1990)) andfrom interpolation/extrapolation by Verner forother elements. The code is supplied in Fortranlanguage. I translated it to C with the f2c con-verter, because C matches the IDL portable call-ing convention for calling shared objects with-out large modifications. Voronov (1997) pub-lished a newer data compilation, but accordingto Mazzotta et al. (1998) it suffers from a mis-print in one of its sources which is only partiallycorrected. This would give rates which are nearlya factor of 2 wrong.The Verner codes work from H (Z = 1, Z is theelectric charge of the nucleus) up to Zn (Z = 30),but for the dielectronic recombination Mazzottaet al. (1998) the coefficients are only publishedup to Ni (Z = 28)

3.2 Verification

Errors in programming inevitably occur and ex-tensive testing is necessary to verify that the sim-ulation reproduces the physics correctly.

3.2.1 Gas dynamics

If no energy is lost due to radiation and the flowis subsonic the equations from section 2.1 pre-dict a steady flow. I ran the hydrodynamic mod-ule with numerous input parameters over a widerange and it always responded as expected. Inthe post shock zone there are never supersonicvelocities because the material is decelerated inthe initial shock. The module issues an error ifstarted with supersonic gas velocities.Furthermore, the heat transfer between the twocomponents of the gas, ions and electrons istested. Under certain simplifying assumptions(although maybe unphysical) I can derive an an-alytic solution how the temperature depends ondepth or time. The following example is donefor a pure, completely ionised hydrogen gas. Theionisation state is set constant because otherwisea calculation of the microscopic physics would benecessary as well. In equation (2.31) the abun-dance of hydrogen ξH = 1 and the ionisationx1H = 1 are replaced by their numerical values

and the equation now reads

ωei =3

2k

Λ

252

Tion − TelecT

3/2elec

(3.1)

Density n and bulk velocity u are constant andbecause of that the Coulomb-logarithm Λ fromequation (2.32) is constant, too. Placing all this


in equation (2.33) now results in an easily solv-able differential equation:

v3

2k∂Tion∂z

= −n3

2k

Λ

252

Tion

T3/2elec

+ n3

2k

Λ

252

1√Telec

(3.2)The homogeneous part is solved with an expo-nential ansatz and the complete solution is de-rived by varying the constant. This solution is

Tion(z) = A0 exp(−x nΛ

v252T3/2elec

) + Telec (3.3)

A0 is set by the initial conditions. In figure3.3 a test run is plotted. Instead of the spacex the x-axis shows the time from t = v

xfor

a plasma with v = 50 km/s and mass den-sity ρ = 10−12 g/cm3. The temperature slowlyreaches the constant electron temperature frombelow. For a large temperature difference moreheat is transferred and the ion temperature risesfaster. As the ion temperature rises, the ex-change gets slower. Simulation and analytic so-lution from equation (3.3) coincide. This showsthat the code models the heat transfer correctly.Calculations for constant ion temperature givea picture which is qualitatively similar, but be-cause the dependence of ωei and Λ is more com-plicated I do not present an analytical solutionhere. If both temperatures are free, they con-verge against their mean value because electronsand ions have the same number of degrees of free-dom so they get the same energy per Kelvin ac-cording to equation 2.7. These tests give me con-fidence that the hydrodynamics module works asexpected.

3.2.2 Ionisation/Recombination

My choice of data is very similar to that ofMazzotta et al. (1998). To check my implemen-tation I took ionisation equilibria from three dif-ferent sources. First the published data fromMazzotta et al. (1998), second the ionisationequilibrium calculated from the matrix equation2.55 and third the microscopic physics module ofthe simulation program started with any ionisa-tion state and simulating a long time until ev-erything relaxes to the equilibrium state. The

Figure 3.3: Simulation and analytic solution fora situation of gas with different electron and iontemperature, velocity v = 50 km/s and mass den-sity ρ = 10−12 g/cm3

second and the third check internal consistencyand stability of the numerical algorithm for ion-isation and recombination. The resulting ioni-sation equilibria are equal to a precision far be-low 1%. Mostly they also match the Mazzottaet al. (1998) values. In figure 3.4 and figure 3.5the ionisation fractions for the elements with thelargest differences are shown. Mazzotta et al.(1998) uses data without the corrections fromVerner and Iakovlev (1990) so I expect devia-tions on at least some elements. These checks

Figure 3.4: Ionisation fractions for selected ionsof C; solid lines: Mazzotta et al. (1998), dashedlines: simulation

3.2. VERIFICATION 29

Figure 3.5: Same as in 3.4 for Cl

show that the rates for ionisation and deexci-tation produce the correct equilibria. What isstill missing is a proof of the correct behaviourin time. Unfortunately, there are no laboratorymeasurements under comparable circumstancespublished. It is difficult to set up a collisionallydominated plasma and measurements are oftendone by lasers which introduce important pro-cesses of stimulated emission (see for exampleBurgess et al. 1980). I did not want to write an-other simulation just to find that for some reasonthe laboratory values depend on other uncontrol-lable parameters like the purification of the gasused, so an analytic solution for the time depen-dence of the ionisation fraction of pure hydrogenwas used. This has only one electron and there-fore only one ionised state which makes the solu-tion easy for a gas of constant temperature andproton density n. Writing down the rate equa-tion (2.54) and realising that the electron densityne equals the density of ionised hydrogen n2 leadsto (a dot denotes a time derivative)

n2 = n1neR1→2 − n2neR2→1

= n1n2R1→2 − n22R2→1

(3.4)

With the normalisation condition n1 + n2 = nthis further simplifies to a differential equationof just one parameter (n2):

n2 = (n− n2)n2R1→2 − n22R2→1 (3.5)

The stationary equilibrium is

n2(t =∞) = n · R1→2

R1→2 +R2→1(3.6)

DGL (3.5) is of Bernoulli’s type. So the substi-tution with z = n−1

2 linearises it.

−z = nR1→2z − (R1→2 +R2→1)

z(t) = A exp(−nR1→2t) solves the homogeneouspart and variation of the constant gives a specialsolution of the inhomogeneous equation: z(t) =R1→2+R2→1

nR1→2. The general solution is then the sum

of both.

z(t) =R1→2 +R2→1

nR1→2

+ A exp(−nR1→2t) (3.7)

Here A is a constant set by the initial condi-tion. Now I return to the original problem in3.5 and define the typical relaxation time scaleτ−1 = nR1→2. Furthermore, the whole equationis divided by the density n to obtain the ion frac-tions x.

x2(t) =x2(∞)

1 +(x2(∞)x2(0)

− 1)

exp(−t/τ)

=x2(0)x2(∞)

x2(0)− (x2(0)− x2(∞)) exp(−t/τ)(3.8)

Several different situations can be modeled andcompared to this formula. A first test startsat any temperature with neutral hydrogen.Equation (3.4) shows that nothing should hap-pen because there are no initial electrons and sono reactions take place. The simulation gives astatic solution as well, but this is an unphysi-cal question, because in reality at least a fewatoms are always in the state H ii and anywaythere is a small chance to ionise one of the part-ners in a neutral-neutral collision. In real stellarplasma the contribution of this process is negligi-ble and is not implemented in the code. More in-teresting is the situation for starting with a com-pletely ionised plasma. At relatively low tem-peratures parts of it recombine and the fractionof x1 = 1 − x2 = nHI/nH converges to its equi-librium value. Figure 3.6 shows a nearly perfect


Figure 3.6: Time evolution of the HI fraction ina pure hydrogen gas. The analytic solution iscalculated with equation 3.8

match between theory and simulation. Furthertests for other elements show a behaviour that isqualitatively similar, but analytic solutions forheavy elements are hard to find. This is thelast step for the verification of the microscopicphysics module: The ion fractions converge to-wards the correct values in the correct time.

3.3 Accuracy and program

parameters

Numerical errors are introduced in different ways.I will discuss sources of errors from the atomicdata to the final analysis of simulation results.The hydrodynamic calculations only require afew very well known fundamental constants likeBoltzmann’s constant k and the masses of atomsor electrons. The uncertainty in these data isnegligible. The situation is more complicatedfor the microscopic physics. Here large sets ofatomic data are needed: Coefficients for colli-sional ionisation and recombination and all thedata in the CHIANTI database for excitation/de-excitation calculations which form the emissionlines. Several authors stress the importanceto use accurate atomic data (Brickhouse 1999;Kaastra 1998), but laboratory measurements ex-ist only rarely. del Zanna (2002) compares dif-

ferent plasma codes in the X-ray region and con-cludes that CHIANTI in general works as wellas others. The accuracy of single line data de-pends on the particular transition in questionand ranges from better then 1% up to a factor ofmaybe two (Landi et al. 2002). New and more ac-curate data is published frequently. I did the sim-ulation with CHIANTI v.4.2 (Young et al. 2003)and the analysis of line ratios with the new ver-sion CHIANTI v.5.0 (Landi in press).The overall energy loss due to radiation is thesum of bound-bound, free-bound, and free-freeprocesses. The contribution of bound-boundemission is typically larger than the others byone or two orders of magnitude. So the error isbasically given by the line emission. Each linehas a different accuracy and its importance de-pends on the temperature, which makes it is dif-ficult to judge the total error. CHIANTI pro-vides a routine (bb rad loss.pro) to calculate theenergy loss via line emission. It uses the approx-imate method of Mewe et al. (1986) with an ac-curacy ∼ 10 %, but this is numerically unstableif used out of equilibrium which is necessary inthis simulation. The simulation uses more de-tailed CHIANTI methods, which should resultin more exact results, that is the error should beless than 10 %.In section 3.1.2 the errors introduced by sepa-rating the equations for gas dynamics and mi-croscopic processes are already discussed. Twoprogram parameters adjust this. Any parame-ter is allowed to change only by some specifiedmaximum fraction. The smaller this value themore steps are done and the smaller the error,but also the more time is needed. In the be-ginning right behind the initial shock and at theend of the cooling zone lots of ionisations andrecombinations occur. A small maximum changeleads to very small step sizes in these regions, butthey emit only a few photons and their accuracyis not as important as that of the large regionsin the middle of the shock, because all observa-tional quantities like line ratios or intensities areintegral quantities over the entire shock region.Figure 3.7 showns how the simulated shock depthdepends on the minimum step size (left) for typ-

3.3. ACCURACY AND PROGRAM PARAMETERS 31

Figure 3.7: The dependence of the depth of a shock (left) and the number of steps (right) on theminimum steps size

Figure 3.8: The dependence of the intensity of a Fe xvii line (left) and the number of steps (right)on maximum change per step

ical shock parameters and a maximum change of10%. For small values of the minimum step sizethe resulting depth is the same, for very largesteps (about 105 cm) the results are different.With a minimum step size of 107 cm the wholecooling region is simulated in only two steps.Obviously these results cannot be very physical.Other quantities like line intensities behave simi-larly. In the right hand plot of figure 3.7 the num-ber of steps, which is a good measure for the com-puting time, is shown. Increasing the minimumstep size makes the execution faster. Choosing aminimum step size conservatively I take 102 cm.This so small that it should give good results, but

saves already about one third of the time com-pared to setting no minimum size. The accuracyis then basically set by the maximum relativechange per step that controls how fast changesfrom gas dynamics effect the microscopic physicsand the other way round. In figure 3.8 the inten-sity of one line of Fe xvii (at 15.02 A) is shown.Figure 3.9 explains where the difference of theintensity in figure 3.8 arises. In calculations withlarger changes per step the step size is larger. Inthe dash-dotted and dashed lines the sharp edgesbetween the steps are clearly visible. The largesteps mean that the ion does not recombine fastenough. The intensity of a line is proportional to


Figure 3.9: Ionisation fraction of Fe XVII calcu-lated with different accuracy

the ion fraction multiplied by the density whichis nearly constant in the regions which are of in-terest here. So the calculated line intensity istoo large. For small changes per step the curvelooks smooth as expected for a freely flowing gas.This indicates that the model is close to a possi-ble physical situation. Maximum changes above40% are for illustrative purposes only: The lin-ear interpolation that predicts the next step sizeis no longer a good approximation and the num-ber of steps actually rises again (right part of3.8) because the next step size begins to fluctu-ate widely.For smaller maximum changes the number ofsteps rises fast (3.8 right panel), but the phys-ical parameters do not reach a constant value.

Smaller changes per step deliver different (andpresumably more accurate) results. A compro-mise is needed between the required accuracy andthe computation time available. I chose a max-imum relative change per step of 5%, in mostcases it is smaller because the program sets thestep size in a way that a change of 2.5% is ex-pected in order to make the next step pass theaccuracy test with a high chance.The integration of the single ODEs within themodules is done by the built-in IDL function’lsode’ to local relative tolerance of 10−7. It isdifficult to combine all error estimates to a globalerror because they contribute differently at dif-ferent times and positions. Some errors are self-regulating, if e.g. in one time step the ionisa-tion state of any atom is calculated too high forany reason in the next time step there will be astronger recombination and it drops again. Thisis the best case, usually errors propagate and ef-fect other variables as well. It is important tokeep this point in mind and analyse which vari-ables are more stable than others. Lines formedin the central cooling region are presumably moreaccurate than lines from ions which occur only asintermediate step in the ionisation process rightbehind the shock front because here changes oc-cur on the same scale as the step size. Comparingline ratios instead of absolute fluxes has the ad-vantage that errors of the total intensity divideout. If, for example, lines are formed in the sametemperature region then they will originate in thesame volume of gas and their ratio is independentof the volume emission measure.

4.1. MODEL PARAMETERS 33

Chapter 4

Results

In this section I describe and discuss the resultsof the shock simulation. In section 4.1 the struc-ture of the shocks is discussed. This includes thetemperature, density and ionisation profiles andallows to check some assumptions made earlierabout the geometry and the thermal heat flux. Itis also shown that the shock structure basicallydepends only on the density and velocity of theinfalling gas. The second section 4.2 deals withthe emission from the shock and describes how todistinguish different scenarios by analysing lineratios. For the example of Ne ix it is shown ingreat detail in which region of the shock severallines are formed. These results are applied toobservations of the CTTS TW Hydrae in section4.3, the infall velocity and density are identifiedand further shock characteristics (filling factorand mass accretion rate) are calculated. In sec-tion 4.4 I try to fit my simulation to data fromBP Tau, but no satisfying match is achieved.

4.1 Model parameters

The shock structure cannot be observed directlybecause it is not spatially resolved. Accordinglythere is only the simulation to guide our under-standing about e.g. temperature, density andionisation profiles.It has to be remembered that some simplifica-tions are made (see section 2.1.3), especially thatthe simulation has no explicit time dependenceand no turbulent flux.The shock itself occupies a very thin layer (seesection 2.1.5). The temperature of the ion com-ponent rises at the shock front and energy istransferred to the electrons so they heat up as

well. In the following figures I will use a shockwith typical behaviour as an example. It is calcu-lated with values for v0 similar to typical free-fallvelocities for CTTS and densities n0 motivatedby X-ray measurements (Stelzer and Schmitt2004). In the pre-shock region ions and elec-trons have time to equalise their temperatures,the value is taken from photoionisation calcula-tions by Calvet and Gullbring (1998). The inputparameters are summarised in table 4.1. Figure4.1 shows a sketch illustrating the accretion ge-ometry and the nomenclature I use.

4.1.1 Temperature and densityprofiles

Figure 4.2 shows simulated temperature and thedensity profiles. The temperature shown in theplot is the temperature of the ion component.Directly behind the shock front, defined as depth0 km, it cools down fast because it transfers en-ergy to the electrons by collisions. After a fewkilometers both components have nearly identi-cal temperatures and henceforth there is basi-cally only radiative heat loss. This is the post-shock zone where the radiation originates, whichis analysed in this thesis. During the process ofcooling the density rises and because of momen-tum conservation the gas slows down simultane-ously (equation 2.20). The emissivity of a lineis proportional to the number of electrons andto the number of ions in question, for constantionisation fractions this means that it is aboutproportional to the square of the density. Asmore energy is lost from the system the densityincreases and it follows that the energy loss rate

34 CHAPTER 4. RESULTS

parameter symbol valueinfall velocity v0 400 km/sinfall density n0 1012 cm−3

ion temperature Tion0 20000 Kelectron temperature Telec0 20000 K

initial ionisation states Arnaud and Rothenflug (1985)

Table 4.1: Initial pre-shock conditions for a ’typical’ shock discussed in the following sections

shock front

pre−shockzone

zone

cooling orpost−shock

maximumdepth

stellar interior

surfacestellar

(z=0)

Figure 4.1: Illustration showing the different zones in the shock. Arrows show the particle flux.

Figure 4.2: Temperature and density profiles fora shock calculated with the initial conditionsfrom table 4.1

does as well. So the gas cools down very fast inthe end. The simulation stops when the temper-ature drops below 12000 K (’maximum depth’),where the calculation is not valid any longer be-cause the energy that is transported from thecentral region of the star to its surface becomesimportant and also the optical depth increases,consequently radiation transfer has to be mod-elled. Both problems are not included in thissimulation. In this region the accreted materialmixes with the stellar atmosphere.The typical shock reaches about 130 km depth,which is much smaller than a stellar radius. Inthis way the simplifying assumption of a planarstar surface in section 2.1.3 is justified a poste-riori. These calculations agree qualitatively withLamzin (1998) and Calvet and Gullbring (1998)and lie quantitatively between their results.The region where the electron temperature differssignificantly from the ion temperature is much


smaller then the maximum depth. A two fluidtreatment is not necessary for most parts of theshock, which was already assumed in previouscalculations (Lamzin 1998; Calvet and Gullbring1998). In figure 4.3 the electron and the ion tem-perature are plotted in comparison. At the shockfront the electrons stay relatively cool becausethey are only compressed adiabatically. The en-ergy flows from the ions to the electrons and al-ready in a depth of 2 km their temperatures arenearly equal at the order of two million Kelvin.Electrons and ions have the same number of ki-netic degrees of freedom and therefore get thesame thermal energy per particle. Their num-ber density is similar and so the temperatureequalises at about half the initial four million de-grees.

Figure 4.3: Ion and electron temperatures in theupper post-shock region for a shock calculatedwith the initial conditions from table 4.1

4.1.2 Ionisation states

The simulation starts with the ionisation equi-librium for Tion from Arnaud and Rothenflug(1985). In fact, this may underestimate the de-gree of ionisation, because the shock is very hotand emits hard radiation which photoionises theinfalling gas. Calvet and Gullbring (1998) cal-culate this pre-shock stage by a photoionisationcode using their shock emission as an input andfind that hydrogen is ionised completely at least

100 km above the stellar surface.Moreover we know that the large scale struc-ture of the infalling matter column cannot be de-scribed in a one dimensional model. In strict 1Dapproximation all emission gets absorbed by thematter in the (infinite) infalling column. Thegeometrical configuration of the funnel flow isstill unknown and so is the proportion of emit-ted photons ionising atoms in the pre-shock ac-cretion flow. To solve this problem by a simu-lation including non-equilibrium effects requiresvast computing resources because in principle ithas to be done iteratively. Fortunately, the de-pendence of the calculated shock properties onthe initial ionisation state is only weak and anystate which provides a reasonable number of elec-trons to start further ionisation processes will do.The temperature behind the shock is several or-ders of magnitude larger than that of the infallinggas, therefore the remaining atoms are ionised tohigh stages very quickly. This stops when the col-lisional ionisation equilibrium corresponding tothe local temperature is reached. The ionisationcross section depends on the element, but gener-ally the situation is close to equilibrium after afew km which is short compared to the maximumdepth of the shock. The required ionisation en-ergy cools the gas only slightly. Stripping O i toO vii needs about 400 eV. This is comparable tothe kinetic energy of a single particle before theshock, but elements which are easier to ionise likehydrogen and helium are by far more abundant,so the energy loss is much smaller on average. Inthe main cooling zone the ionisation state staysclose to the local collisional equilibrium state.When the temperature drops, recombinations oc-cur until the simulation stops. In real stars therecombinations go on until stable photosphericconditions are reached.As an example figure 4.4 shows how the ionisa-

tion depends on depth. The charge of the mostabundant iron ion is plotted. Initially this is Fe iii(charge 2) and in the hot environment behind theshock it quickly ionises to Fe xv. Then the gascools down and the ions recombine slowly main-taining the ionisation equilibrium. At ∼120 kmthe temperatrue drops so fast (see figure 4.2) that


Figure 4.4: Charge of the most abundant iron ionin different depth in a shock calculated with theinitial conditions from table 4.1. Iron ionises fastand then slowly recombines while the gas cools.

the recombinations cannot keep step and the ionsdrop out of equilibrium. A more detailed discus-sion for the example of neon can be found in sec-tion 4.2.1Several simulations were performed to investi-gate the influence of the initial ionisation stages.Virtually no emission is produced by the ini-tial ions, because the ionisation processes happenvery fast. The main influence of the initial ioni-sation state is that slightly higher temperaturesare reached if the ionisation degree is higher andso the cooling time is longer. Therefore, the inte-grated intensity of all lines which originate in themain cooling zone between 10 and 120 km depthis slightly higher, but the first few ionisationsdo not need much energy so the effect is onlymarginal. Simulations starting with the samegas temperature but different ionisations showthat line ratios are nearly equal and strong linesare only affected within a few percent. Still it ispossible that the slightly higher temperature justbuilds up, e.g. Ne x for a short time where usu-ally Ne ix is the highest ionisation state. This in-duces the presence of additional weak Ne x lines.If the ionisation is lower than assumed in table4.1, lines formed in the hot parts of the plasmaare weaker.

4.1.3 Influence of the initial tem-

perature

The initial temperature is only importantthrough its influence on the initial ionisationstate. Using the equations (2.6) and (2.7) thethermal energy per particle and its bulk kineticenergy can be compared. The thermal energy isabout a few eV, the kinetic energy a few hun-dred of eV mostly converted into heat by theshock. Equation (2.31) shows that the energytransfer from the hotter to the cooler compo-nent is fastest where the temperature differenceis largest. Compared to the ≈ 106 K of the iongas just behind the shock front a difference inelectron temperature of 103 K is insignificant.

4.1.4 Influence of density and in-

fall velocity

Using the ionisation state and the temperaturesfrom table 4.1 I calculated a grid of simula-tions with the parameters velocity ranging from200 km/s to 600 km/s in steps of 25 km/s anddensities between 1010 cm−3 and 1014 cm−3 with13 points equally spaced on a logarithmic scale.The infall velocity and the density are the physi-cally most interesting parameters. In sections 4.3and 4.4 only these two are varied to fit observa-tions to the simulation. They have a strong im-pact on the characteristic properties of a shock.The maximum post-shock temperature of the ioncomponent T1, for example, depends quadrati-cally on the infall velocity v0, but not on the den-sity. This can be seen directly by transformingequation (2.5):

T1 ∼P1

n1∼ n0v

20

n1∼ v1u0 ∼ v2

0 (4.1)

In the second step P1 is expressed with equa-tion (2.26) neglecting the initial pressure P0, inthe third step I use the particle conservationlaw in the form (2.22). In the last the com-bined equations (2.26) and (2.27) again takingP0 = 0. Figure 4.5 shows contour lines of con-stant maximum ion temperature. They are par-allel to lines of constant infall velocities, only for


very small densities the approximation above isno longer strictly valid and the lines curve. Theyare spaced with constant temperature differencesand their difference in velocity space decreasesaccording to (4.1). A second example for a shock

Figure 4.5: The lines mark parameters with con-stant ion temperature behind the shock in unitsof 106 K.

property that varies depending on the infall den-sity and velocity is the length of the cooling zone,which will be discussed in the next setion.

4.1.5 Depth of shock

In figure 4.2 the temperature and density profilesfor a typical shock are plotted. All calculatedshocks exhibit the same qualitative behaviour,but large quantitative differences, especially thelength of the cooling zone varies over 3 ordersof magnitude. A higher infall velocity leads to adeeper shock because the material reaches highertemperatures and consequently needs longer tocool down, so it flows for longer times and reachesdeeper regions. A second correlation is more sur-prising: Lower infall densities result in shockswith a larger spacial extent. Gas emits energyroughly proportional to the square of the den-sity, therefore lowering the density will increasethe time the gas needs to cool down to stellartemperatures. Figure 4.6 shows contour lines ofconstant maximum depth, which are plotted aconstant factor apart. They appear to be evenly

spaced on linear scales, which proves exponentialdependence of the depth on the infall density andvelocity. High velocities and small densities im-

Figure 4.6: The length of the cooling zone (’max-imum depth’) dependent on the logarithm of theinfall density log(n0) and the infall velocity v0:The length of the cooling region is labeled. Thedotted area marks regions where the deeper partsof the simulated shock are not included in lineemissivity calculations.

ply a long time needed to cool the gas down.In the hydrodynamics approach used in this sim-ulation there is no direct influence of the star onthe gas, therefore caution has to be applied wheninterpreting these results, especially for thin in-falling gases. After the shock has occurred thereis essentially free-flow during cooling. If the gashas a high ram pressure this picture works well.The surrounding atmosphere is pushed away andit mixes with the accreted material only aftercooling. For thin gases the ram pressure is muchlower and additionally they need more time tocool down. They may be strong enough to pushthe thin upper atmosphere, but the density ofa star increases with depth. So in deeper lay-ers the approximation of a free flowing gas isno longer valid and the whole calculation breaksdown. The depth where this happens depends, ofcourse, also on the parameters of the star. Rightbehind the shock the conditions will be as pre-dicted and data for lines originating in this regioncan be analysed as usual, but emission from lay-


ers below a few hundred km will be affected.The shock front forms when its ram pressureequals roughly the gas pressure of the stellar at-mosphere. The ram pressure is

pram = ρv20 (4.2)

After passing through the shock front the shockpressure actually decreases because the gasdecelerates to about v0

4(equation 2.26 and 2.27,

neglecting the initial pressure P0). This is notbalanced by the increase of thermodynamic pres-sure according to equation (2.5) although thetemperature rises significantly. This means thatthe shock gas will have a pressure comparableto or even lower that the surrounding medium.The deeper it penetrates into the star the largerthe pressure of the surrounding atmospherebecomes. To estimate the orders of magnitudeof pressure I use an atmospheric model fromthe PHOENIX programme. PHOENIX is ageneral purpose stellar atmospheric modellingcode which uses more than 500 million lines forits radiation transport calculations. I use here adensity profile from AMES-cond-v2.6 with effec-tive temperature Teff = 4000 K, surface gravitylog g = 4.0 and solar metalicity (Brott andHauschildt (2005) based on Allard et al. (2001))to compare realistic atmospheric pressures tothe shock ram pressure. The stellar parameterschosen resemble those of typical CTTS (seesection 1). Inserting them into equation (4.2)

I obtain pram ≈ 3500 n0

1012cm−3

(v0

400 km/s

)dyn.

In figure 4.7 the pressure in the atmosphere isshown. The origin of the stellar depth has beenset for this plot where the stellar pressure equals100 dyn. For the standard shock from table4.1 the ram pressure equals the atmosphericpressure at about 650 km, figure 4.6 shows thatthe cooling length is about 100 km. The upperblack bar in figure 4.7 illustrates this situation.The bar starts where the ram pressure equalsthe thermodynamic pressure of the surroundingatmosphere and it extends of 100 km. Thestellar pressure at the end of the cooling zone(at 650 km+100 km=750 km) differs from thatat the shock front roughly by a factor of two.For lower densities and higher infall velocities

Figure 4.7: Pressure in a stellar atmosphere ofa star with Teff = 4000 K, logg=4.0 and solarmetalicity from PHOENIX (thin solid line). Thebars and labels illustrate two examples for theposition of a shock in the atmosphere (see text).

like e.g. n0 = 1011 cm−3 and v0 = 500 km/s thisratio reaches quickly two orders of magnitude(lower black bar in figure 4.7). In this case itis not justified to use this shock model. Thepressure rises exponentially, so independent ofthe starting point only shocks with small coolinglength can be described by the hydrodynamicmodelling used here. For larger stars, i.e. smallerlog(g), the pressure does not rise as fast as in fig-ure 4.7, but still the pressure difference betweenthe shock front and the maximum depth is atleast an order of magnitude if the shock extendsmore than 1000 km. Because of that all linefluxes are integrated between the shock front andthe maximum depth or 1000 km whichever is less.

4.1.6 Thermal conductivity

Thermal conduction tends to smooth out tem-perature gradients. It is not included in the sim-ulation (see section 2.1.3), still the temperaturegradients in the models now allow to estimate itsimportance. It is possible to derive the thermalconduction heat flux from kinetic gas theory butI use the formula from Spitzer (1965,chapter 5.5)


here.

Fcond = κ0T5/2dT

dz(4.3)

Here κ0 = 2 · 10−4 δTΛ

erg K7/2

s cmis the coefficient of

thermal conductivity and δT is a correction factorto relate real gases to ideal behaviour. For hy-drogen δT is of the order 0.1. I now compare thethermal heat flux according to equation (4.3) tothe energy flux by the bulk motion which trans-ports heat and bulk kinetic energy. For a typi-cal shock their ratio is shown in figure 4.8. It is

Figure 4.8: Ratio between thermal heat conduc-tion Fcond and energy flux by bulk motion Ffluxfor a typical shock (table 4.1)

large right behind the front, reaches a minimumand increases slowly again towards the maximumdepth. The thermal flux is important close tothe shock front because it transports the ther-mal energy produced there upstream in the pre-shock zone and heats the inflowing gas to the20.000 K assumed in table 4.1. It will distributesome energy downstream of the shock front, too,but the main part of the cooling zone is affectedonly marginally. In the end the temperature gra-dient again becomes much steeper, but in thisregion the gas is quite cool already and thereforethe absolute magnitude of Fcond small accordingto equation 4.3.The energy flux due to bulk motion (the two leftterms in equation 2.13) is proportional to thenumber density. Accordingly the thermal heat

conduction from equation (4.3) is more impor-tant for low density scenarios, but apart fromlowest density cases it never reaches more thana few percent of the energy transport by bulkmotion in the main part of the cooling zone.If there are small chaotic magnetic fields theycould be frozen in the bulk motion of the mate-rial. They will further suppress thermal conduc-tion which requires motion of particels relativeto the bulk motion and accordingly perpendic-ular to this small-scale magnetic field. Due tothe high temperatures all atoms are ionised andtherefore all particles are charged and affectedby magnetic fields. So the assumption of section2.1.3 to neglect heat conduction is justified at thelevel of a few percent.

4.1.7 Deviation from equilibrium

In contrast to earlier works (Calvet andGullbring 1998; Lamzin 1998) my simulationmodels the ionisation and recombination processand does not assume that the ions are alwaysin collisional equilibrium. The code of CHIANTI4.2 (Young et al. 2003) was modified to work withthe calculated non-equilibrium ionisation insteadof the tabulated equilibrium states, but strictlythe atomic physics used is only valid for the as-sumption of local ionisation equilibrium (Porquetet al. 2001; Mewe and Schrijver 1978a,b). Toquantify the deviation ∆I from equilibrium I de-fine for element A:

∆IA =1

2

∑

ionisation levels Z

|xAZequilib(T )− xAZcalc|

(4.4)The factor 1

2normalises this quantity to unity.

∆IA is shown for the shock parameters of table4.1 for selected elements in figure 4.9. The ioni-sation cross sections and therefore the time scaleto reach collisional ionisation equilibrium differsbetween elements. The largest discrepancies canbe found directly behind the shock and close tothe maximum depth. In the main cooling zonethe difference is relatively small, in fact hydrogenand neon are so close to their collisional ionisa-tion equilibrium that their lines are below x-axisin figure 4.9. Oxygen reaches the state of O vii


Figure 4.9: Difference of the ionisation state forselected elements to their equilibrium value fora typical shock (table 4.1). 0 means all ions arein their equilibrium state, 1 that all ions are inionisation states which cannot be found in thecorresponding collisional equilibrium.

immediately, it takes a bit longer to build upO viii and even longer to get completely ionised.At a depth of 30 km the equilibrium state be-tween these three ionisation stages is reached,which can be seen as very small ∆IA in figure4.9. The gas already cools down again and theoxygen recombines from here on. The cross sec-tion is so low that it is always more highly ionisedthan in equilibrium. As the density rises and thetemperature drops the recombination acceleratesand at 125 km nearly all oxygen is again in thestate O vii which is close to its equilibrium valueat this point. In the last few kilometers the hy-drodynamic parameters change so quickly thatall elements drop out of equilibrium again. Foriron the structure is similar in the beginning, butbetween the depth of 95 km and 125 km thereare some wiggles in the curve. They correspondto the region in figure 4.4 where iron recombinesand quickly passes from Fe xiii to Fe ix. Theyshould be regarded as numerical because the in-terpolation of the tabulated equilibrium values toa continuous temperature may already introduceerrors on this scale.

4.2 Emission lines

Only the emission from the shock is directly ob-served. Therefore line ratios and line fluxes arevital to the analyses of observational data. Lineratios can deliver information on the density andtemperature of the emitting region, because theabundance of the ions is temperature dependent.In hotter regions they will be higher ionised andthe ratios of lines of different ionisation stagesis a helpful tool to measure this. Unfortunatelythe elemental abundance of the infalling gas isnot known a priory. Some authors suggest thatit could be depleted of grain forming elements(Stelzer and Schmitt 2004; Argiroffi et al. 2005;Drake et al. 2005), so it is best to use differ-ent ions of the same element. Examples of thisinclude the ratios of the Lyman α to the reso-nance line of the so called ’helium-like triplet’for Ne, O or N. The helium-like triplets offer aa temperature and density diagnostic which iseven independent of the ionisation calculation.The basic atomic physics theory has been devel-oped since the article Gabriel and Jordan (1969)by various authors. The most recent compi-lation is due to Porquet et al. (2001,see refer-ences therein for a more comprehensive litera-ture listing). The helium-like triplet consists ofthe recombination line (label r, atomic transi-tion 1s2 1S0 − 1s2p 1P1), the intercombinationline (label i), which in fact again is a doublet oftwo close unresolvable lines (atomic transitions1s2 1S0 − 1s2p 3P2,1), and the forbidden line (la-bel f, atomic transition 1s2 1S0− 1s2p 1S1). Twoline ratios are widely used: One is density sensi-tive

R(nelec) =f

i(4.5)

and one is temperature sensitive

G(Telec) =f + i

r(4.6)

Unfortunately the line strengths and because ofthat the ratios change as a function of the back-ground radiation field, too. For observationaldata this leads to some ambiguity between highdensities and high background radiation fields.Nevertheless the helium-like triplets have proved

4.2. EMISSION LINES 41

to be a useful tool in the analyses of T Tauri stars(Kastner et al. 2002; Stelzer and Schmitt 2004;Schmitt et al. 2005; Argiroffi et al. 2005).Ratios between lines of different elements allowmeasurements of the relative elemental abun-dance and the emitted energy per area for anysimulated line in comparison to measured valuesgives the size of the emitting region.

4.2.1 Origin of lines

In comparing different line ratios one has to becareful not to rely on too simple ideas. As I re-marked earlier photons carry information aboutthe physical conditions of the emitting region(e.g. about the electron density using the R-ratio from equation (4.5)), but not about its ori-gin. If we compare ratios from different models,the regions, where the photons in question aremainly emitted, may be different. Figure 4.10shows data for the Ne ix triplet from three simu-lations with different infall velocities. The x-axisis always labeled with normalised depth. Thefirst row shows the relative abundance of the im-portant neon ions. In the case v0 = 225 km/smost Ne is not ionised up to Ne ix, which ishelium-like. Only in the beginning the tempera-ture is high enough to form it and it takes sometime to reach the abundance maximum at depth∼ 0.3. With the higher velocity (v0 = 350 km/s)a higher temperature is reached (see lower row offigure 4.10, where temperature and density pro-files are shown) and nearly all Ne is in the form ofNe ix over the entire length of the cooling zone.With even more input energy (v0 = 500 km/s)the ionisation to Ne ix occurs nearly instantlyafter passing through the shock front, but someneon is even ionised to Ne x and recombines later,so that near the end of the shock a second max-imum in the Ne ix abundance is reached. Themiddle row shows the volume emissivity for theforbidden and the intercombination line of Ne ix.For small velocities the ratio probes the den-sity at a small distance to the shock front, atv0 = 350 km/s the emission peaks just behindthe shock front, but has a tail extending up tothe region of rapid recombination and in the high

velocity case the emission is strong over the en-tire region because when the temperature dropsand the line should get weaker the abundance ofNe ix increases due to recombining Ne x (see up-per right panel of figure 4.10). Thus the line ratioobserved includes density information integratedover the entire shock region. (The absolute scaleof the emissivity is not the same for the threecolumns. It differs by several orders of magnitudedue to a huge difference of the impacting kineticenergy.) The general procedure for comparisonbetween simulation and observation must be tocompare all observed lines to the emission inte-grated over the entire simulated shock to selectthe shock parameters realised.

4.2.2 Line ratios

Among the most useful line ratios are the R-ratio(equation 4.5) and the G-ratio (equation 4.6) forthe helium-like triplets of oxygen and neon. Foreach model in the grid I calculate these ratiosfrom the total line flux. Figure 4.11 shows con-tour plots for the R and G ratios for oxygen. Inprinciple observations of just these two line ra-tios allow to find the infall velocity v0 ( usingthe G-ratio) and density n0 (using the R-ratio)for any star, but if more line ratios are used theconclusions will be statistically more significant.The ratio f/i depends strongly on the backgroundradiation field. The figure shows the ratio com-puted for three different background black-bodyradiation fields. The top three lines belong to theratio 0.05. The difference between Trad = 6000 Kand Trad = 8000 K is small, but the differenceto the line for Trad = 10000 K already notice-able. A v0 and n0 which results in a ratio of0.3 with Trad =6000 K show a ratio of only 0.1for Trad = 10000 K. f/i of 0.5 or higher can-not be produced if a background radiation withTrad = 10000 K is present. This illustrates theproblem that a small f/i ratio does not necessar-ily originate in a high density region, but couldbe a sign of strong UV-radiation. For neon theplots look similar, however the f/i ratio is onlyaffected by stronger radiation fields with Trad be-tween 10000 K and 15000 K.


Figure 4.10: Each column shows results from a simulation with different infall velocity v0. Thex-axis is labelled by normalised depth, starting with 0 at the shock front. The first row shows therelative abundance of important Ne ions, the second the volume emissivity of the forbidden (dashedline) and intercombination (solid line) lines of the helium-like triplet Ne ix (in different arbitraryunits for each column, because they differ by several orders of magnitude) and the third row theelectron temperature [Te in K] (solid line, left scale) and the ion density [n in cm−3] (dashed line,right scale).

4.3. APPLICATION TO TW HYA 43

Figure 4.11: Simulated R (left) and G (right) ratios for oxygen. Both show contours for threedifferent background radiation temperatures. For clarity only the contours for Trad = 6000 K arelabelled.

4.3 Application to TW Hya

TW Hya was identified as a CTTS early, butit is not submerged in a dark, molecular cloudlike many other CTTS. TW Hya is the clos-est known CTTS, its distance is only 57 pc(Wichmann et al. 1998). Photospheric observa-tions show variability between magnitude 10.9and 11.3 in the V-band and broad Hα profiles(FWHM ∼ 200km/s) were observed by Rucinskiand Krautter (1983) and Muzerolle et al. (2000),and it apparently belongs to a group of objectswith similar age, the TW Hydrae association(TWA) (Webb et al. 1999). TW Hydrae’s massand radius are usually quoted as M∗=0.7 M,R∗=1.0 R and the age as 10 Myr from Webbet al. (1999). These values carry considerable un-certainties because they are based on pre-main-sequence evolutionary tracks from D’Antona andMazzitelli (1997), which carry uncertainties upto a factor of two by themselves. Using the’typical error’ from Webb et al. (1999) the age,for example, could be wrong by an order ofmagnitude. An alternative is given by Batalhaet al. (2002), who place TW Hya on the HR-diagram from Baraffe et al. (1998) and deter-mine stellar parameters from the optical spec-trum, which fits an older (30 Myr) and smallerstar (R∗=0.8 R). The spectral type of TW

Hya is K7 V-M1 V (Webb et al. 1999; Batalhaet al. 2002). The system is seen nearly face-on(Kastner et al. 1997; Wilner et al. 2000; Alencarand Batalha 2002). Moreover, TW Hya ex-hibits a hot wind (Dupree et al. 2005), whichis likely driven by the same magnetic processes,which control the accretion (see e.g. Shu et al.1994) and displays variations in the line profilesand the veiling, which have been interpreted assigns of rotation of an accretion spot (Alencarand Batalha 2002; Batalha et al. 2002). TWHya has been observed in the UV with IUEand in the X-ray with ROSAT by Costa et al.(2000). After the advent of high-resolution X-raysatellites there have been two observations withChandra/HETGS (Kastner et al. 2002) and withXMM-Newton/RGS (Stelzer and Schmitt 2004).I will use these observations as a basis for myanalysis (see the references for details on the ob-servations) and process both in parallel to checkconsistancy of my results.

4.3.1 Abundance independent

line ratiosInfall velocity and density

As explained above ratios between lines of thesame element allow to find the best model param-eters independent of the elemental abundance.


Line ratio S&S Kastner v0 = 525 v0 = 575

N R-ratio 0.30± 0.40 0.00 0.00N G-ratio 0.58± 0.37 0.75 0.73

N Lymanα/N vir 4.73± 2.30 2.70 3.26O R-ratio 0.04± 0.06 0.00± 0.27 0.02 0.02O G-ratio 0.56± 0.11 1.14± 0.70 0.72 0.69

O Lymanα/O viir 1.99± 0.30 2.44± 1.03 1.64 2.19Ne R-ratio 0.47± 0.09 0.44± 0.12 0.33 0.32Ne G-ratio 1.10± 0.13 0.93± 0.14 0.80 0.75

Ne Lymanα/Ne ixr 0.30± 0.08 0.67± 0.10 0.27 0.51

Fe xvii(17.09A+17.05A)/15.26A 2.68± 1.26 4.92 4.62Fe xvii17.09A/17.05A 0.71± 0.44 0.81 0.79

Table 4.2: The line ratios used in the fitting process for both observations are shown with theirerrors, S&S data is from Stelzer and Schmitt (2004), Kastner from Kastner et al. (2002). The tworightmost columns show results for the best-fit scenario from the simulation, the velocities of themodels are given in km/s (n0 = 1012 cm−3 for both models).

The XMM-Newton data (Stelzer and Schmitt2004) contains the helium-like triplets of N,O andNe. For these three elements the Lyman α linesare included, too; I do not use the oxygen Lymanβ line, because it is blended with an Fe xviiiline and relatively weak anyway. No strict fit-ting is carried out, I select the model from thegrid which produces the smallest χ2 in a stan-dard weighted χ2 maximum likelihood estima-tion (Press 2002). Since the measurement er-rors are not gaussian and the model, which iscompared to the data, non-linear, the standardway to assign probabilities to χ2 values is notvalid. Poisson distributions produce more out-liers then Gauß distributions, so the χ2 valuesare expected to be higher than usual. Only aMonte-Carlo-simulation could provide the likeli-hood of the model.Figure 4.12 shows the χ2 values for a comparison

between simulation and observation using nineline ratios (R-,G-ratio, Lymanα/r for N,O,Ne,see table 4.2). The best model has the parame-ters v0 = 525 km/s and n0 = 1012 cm−3 with anunreduced χ2 = 12.6 (7 degrees of freedom), butfigure 4.12 shows that a model with slightly lowerv0 and n0 is also acceptable. Neon lines confinethe fit strongest because they are observed withrelatively high count rates and their values carry

Figure 4.12: Contours are labelled with χ2 val-ues for TW Hya, data from Stelzer and Schmitt(2004)

therefore only small statistical uncertainties. Thedensity is restricted mainly by f/i ratios, the ve-locity by the Lyman α/r-ratio. One ratio doesnot quite fit the general picture. Figure 4.13shows the simulated neon G-ratio. The observedvalue is 1.1 ± 0.13, which points to much lowerinfall velocities (the 1 σ range is dotted in thefigure). If I fit the remaining eight ratios only Iget very similar contour lines, but χ2 is now only6.4 (6 degrees of freedom). On the one hand this


Figure 4.13: Simulated Neon G-ratios, the dottedarea marks the observation ±1σ.

would be acceptable even for gaussian errors, onthe other hand there is no obvious reason whythe neon lines should have a behaviour which isdifferent from nitrogen or oxygen. The fit doesnot depend on the background radiation temper-ature in the range 6000 K to 10000 K althoughthe predicted f/i-ratio for oxygen changes.The data from Chandra (Kastner et al. 2002)includes a different wavelength interval. Onlythe triplets from O and Ne are contained, butChandra has a better resolution and this allowsthe measure several iron lines. I use the threelines of Fe xvii at 15.26 A and the doublet at17.05 A and 17.09 A, and calculate the ratio ofthe two doublet members and between their andthe third line (see table 4.2). The contour linesfor χ2 are plotted in figure 4.14. The best fitmodel has v0 = 575 km/s, which is slightly fasterthan before, and n0 = 1012 cm−3, the same asthe other data. The unreduced χ2 is only 8.3(6 degrees of freedom). Again there is no differ-ence between 6000 K and 10000 K taken as tem-perature for a black-body radiation background.Like the other data the neon G-ratio points tolower infall velocities, although it is compatiblewith v0 = 400 km/s, which is faster that for thefirst data set. Both fits have a tight lower boundon the density of the infalling gas. It cannot bemuch below n0 = 1012 cm−3, but a slightly higherdensity or lower velocity is acceptable. I estimate

Figure 4.14: Contours are χ2 values (10,30,50and 100) for TW Hya, data from Kastner et al.(2002)

the error as one grid point, that is ±25 km/s forv and ±0.33 for log(n0) with n0 in cm−3.The general model states that the gas impacts onthe star with free fall velocity:

vfree =

√2GM∗R∗

≈ 600

√M∗M

√RR∗

km

s(4.7)

I evaluate this for the parameters from Webbet al. (1999) to 500 km/s and I get 550 km/swith the stellar radius from Baraffe et al. (1998).The equation 4.7 does not take into account thatthe gas is not coming from infinity, but froman surrounding disk, hence actually the velocitymight be lower. This sets a tight upper boundand I take v0 = 525 km/s from here on which isjust large enough be be compatible with the fitspersented earlier.

4.3.2 Element abundances

With fixed model parameters I can now com-pare the emissivities for two lines of differentelements. I take the ratio between observationand simulation for each line and compare thoseratios. Thus I can determine how the ratio ofthe two elements in question differs from theratio the simulation is based on (Allen 1973).Unfortunately, this method does not deliver ab-


Element Stelzer and Schmitt (2004) Kastner et al. (2002)C 0.25± 0.03N 0.7± 0.1 0.9± 0.2O 0.33 0.33Ne 4± 0.3 4± 1.0Fe 0.27± 0.03 0.3± 0.06

Table 4.3: This table shows the abundance of elements relative to (Allen 1973). The errors arestatistical only, I estimate the systematic error to 15%, oxygen carries no statistical error, becausemy method provides only ratios, which are all relative to oxygen.

solute abundances. Only carbon, nitrogen, oxy-gen, neon and iron have significant lines in thisobserved wavelength range. I do the calculationrelative to oxygen, which offers four unblendedlines (He-like-triplet and Lyman α). The datafrom Stelzer and Schmitt (2004) and Kastneret al. (2002) is consistent within the statisticalerrors (see table 4.3). Additionally, there aresystematic errors, which include the uncertaintyin the model parameters chosen; they probablydominate the statistical error from the line fit-ting. Neon is enhanced compared to oxygen bya factor of 12, nitrogen by a factor of 2-3, onthe other side iron and carbon are slightly (byabout a quarter) less abundant than in our localneighbourhood. I need to keep in mind that allthese number are relative to oxygen. They areroughly in accordance with the numbers given inKastner et al. (2002) and Stelzer and Schmitt(2004). These authors use the continuum foran absolute calibration. They conclude indepen-dently that oxygen is reduced compared to solarabundance by a factor of 3. I scale my analysiswith this value and summarise my results in ta-ble 4.3.The new abundances change the total emissivityof the plasma, because some lines get weaker andsome stronger. So I did new simulations with theelemental abundance from Allen (1973) correctedby the factors in table 4.3 and went through thefitting procedure from section 4.3.1 again. Theresulting values for n0 and v0 differ only by onegrid point, which is the estimated model uncer-tainty. The shock structure is similar qualita-tively, but the cooling length rises from∼ 600 km

to ∼ 950 km. All line ratios between ions of thesame element agree within their errors with theoriginal predictions, the total intensity, of course,now fits the new elemental abundances.

4.3.3 Filling factor and mass ac-

cretion rate

The simulation gives the integrated energy fluxin a line per unit area. Taking the ratio betweenthe observed energy flux fobs (at the distance tothe earth) and the simulated one fsim per unitarea (at the shock front) allows to calculate theaccretion spot size Aspot. Ignoring interstellar ex-tinction for the time being the total energy flux isconserved, but the flux per area decreases as theradiation dilutes for spherically symmetric emis-sion:

4πd2fobs(d) = Aspotfsim(R∗)

Here d represents the distance between the starand the observer and R∗ the stellar radius. Thisequation can be solved for the spot size

Aspot =fobs(d)

fsim(R∗)4πd2 (4.8)

The filling factor f expresses the proportion ofthe stellar surface, that is covered by the spot:

f =Aspot4πR2

∗=

fobs(d)

fsim(R∗)

d2

R2∗

(4.9)

To set bounds on the time scale of disk dispersaland to estimate the total energy, which can bereleased by accretion, the mass accretion rate iscalculated. It is the product of the spot size and


the mass flux per unit area, which in turn is theproduct of the gas density ρ0 = µmHn0 and theinfall velocity v0, conventionally the unit chosenfor the mass flux is solar masses per year.

dM

dt= Aspotρv0 = AspotµmHn0v0 (4.10)

The formulae are applied to all unblended linesin the observations and the weighted average de-termined. The statistical errors are ∼ 10%, I es-timate the systematical error to 15%. The spotsize is then 6 − 7 · 1019 cm2, which yields fill-ing factors of 0.1% and 0.2% for R∗ = 1 R andR∗ = 0.8 R respectively. The mass flux is about1 · 10−10 M/yr.Now it is possible to assess the hydrogen columndensity LH , which the radiation passes betweenemission and observation. Depending on the po-sition of the emitting atom three independentsources contribute: The interstellar hydrogendensity, which is beyond the scope of this simu-lation, the pre-shock gas or the surrounding stel-lar atmosphere and the post-shock gas betweenthe point of emission and the shock front (seeillustrative figure 4.15). The total column den-sity of the cooling zone is LHpost = 5 · 1020cm−2,it increases nearly linearly from the shock frontto the maximum depth because the density isnearly constant in the main cooling zone (see fig-ure 4.16). Emission produced shortly behind theshock therefore passes less material. Then the ra-diation passes further material between the shockfront and the interstellar space. Either it fol-lows the path of the infalling matter or crossesthe boundary between pre-shock gas and undis-turbed stellar photosphere (see left panel of fig-ure 4.16). Which possibility is realised to whatproportion depends on the size of the spot. Inthe case of large spots most photons emitted out-wards will be affected by extinction from the pre-shock material with

LH ≈ 1012cm−3 · 1000 km ≈ 1020cm−2

The 1000 km are an estimate based on figure 4.7.The distance from the shock front to the point,where the stellar atmosphere has only a pressureof 100 dyn, can be read off there as about 500 km.

Figure 4.16: Column density LH in cm−2 be-tween shock front and point of emission, depth0 km is the shock front

The accretion funnels curve (see section 1.2) andthe radiation, propagating linearly, leaves thematter stream somewhere above the stellar sur-face, otherwise highly energetic radiation wouldbe absorbed by ionising the infalling material be-fore the shock completely (Lamzin 1995). If theradiation crosses the boundary to the stellar at-mosphere it gets absorbed in a column density ofmore than 1021 cm−2. In figure 4.17 (right panel)is shown the column density of the undisturbed

Figure 4.17: Column density LH in cm−2 be-tween interstellar space and the shock front, theshock front is at 500 km depth according to thePHOENIX model, see section 4.1.5


shock front

max. depth

Figure 4.15: Illustration of different possible photon paths (dashed arrows) and examples wherephotons are absorbed

stellar atmosphere depends on the depth. Theshock front is buried at about 500 kmA lower bound on the total column density isLH ≈ 1020 cm−2 for a photon, which is emittedjust behind the shock front and then penetratesthe pre-shock gas.

4.4 Application to BP Tauri

BP Tau is a CTTS as evidenced by its excesscontinuum emission and by its Balmer emis-sion (Bertout et al. 1988). It is a memberof the Taurus-Auriga star forming region (dis-tance ∼ 140 pc), where several TTS are lo-cated, Wichmann et al. (1998) argue this pointbecause of common proper motion, although theHIPPARCOS paralax suggests BP Tau could bea single foreground star in a distance of 57 pc(further discussion in Drake et al. 2005). BPTau’s spectrum fits the type K7 and was studiedextensively via optical monitoring by Gullbringet al. (1996), who found small variations on thetime scale of hours with light curves differentfrom solar flares. They interpret this as varyingaccretion rate. A consecutive simultaneous mon-itoring in the optical and the X-ray was carriedout by Gullbring et al. (1997) over five nights andshowed that variations in these bands are uncor-related and led them to the conclusion that the

X-ray emission is the result of normal activityand not a hot accretion spot.Stellar parameters have been estimated for BPTau in different articles. Simon et al. (2000)compare two different possibilities for mass de-termination. Placing the star on the pre-main-sequence evolutionary tracks from D’Antona andMazzitelli (1997) they obtain M∗ = 0.7M, us-ing dynamical observations of CO in the disk theresulting mass is higher: M∗ = 1.2M. The ra-dius is estimated to 2-3 R (Bertout et al. 1988;Muzerolle et al. 2003). According to equation(4.7) the free-fall velocity for BP Tau is then450 km/s or 550 km/s depending on the valuechosen for the mass.BP Tau was after TW Hya the second CTTS tobe observed with a high-resolution X-ray satelite.Schmitt et al. (2005) published the data I usehere.In the same manner as for TW Hya (see section4.3.1 for details) abundance independent ratiosare compared. Only neon and oxygen lines arestrong enough here, for both elements Schmittet al. (2005) give the line fluxes in the helium-like triplet and the Lymanα lines, for oxygen ad-ditionally in Lymanβ. Applying all possible ra-tios directly does not give a satisfactory fit. Theχ2 = 99 (5 degrees of freedom) is unacceptablehigh and its contour lines point to shock densitiesand velocities far off scale as best model. Fitting

4.4. APPLICATION TO BP TAURI 49

Figure 4.18: χ2 values for BP Tau, data from Schmitt et al. (2005), using the helium-like tripletonly (left) and additional Lymanα lines corrected for an absorbing column density.

LH = 2 · 1021 cm−2 LH = 7 · 1021 cm−2

Line ratio Observation Simulation Observation Simulation

O R-ratio 0.40± 0.18 0.02 0.48± 0.22 0.04O G-ratio 1.16± 0.30 0.76 1.36± 0.36 0.87

O Lymanα/O viir 1.06± 0.28 1.05 0.27± 0.07 0.30O Lymanα/Lymanβ 2.68± 0.76 8.42 6.59± 1.87 9.89

Ne R-ratio 0.39± 0.26 0.32 0.41± 0.27 0.58Ne G-ratio 1.18± 0.46 0.86 1.24± 0.48 0.99

Ne Lymanα/Ne ixr 2.38± 0.71 0.10 1.58± 0.47 0.01

Table 4.4: This table shows the line ratios used in the fitting for BP Tau. The data is fromSchmitt et al. (2005) and corrected for the extinction given in brackets. The corresponding best-fitscenarios from the simulation are printed for comparison. Their parameters are v0 = 475 km/s andn0 = 1012 cm−3 and v0 = 400 km/s and n0 = 1011.7 cm−3 respectively. For the fit of the left scenariothe oxygen Lymanα to Lymanβ ratio was ignored.

just the R- and G-ratios of the triplets resem-bles a shock with the parameters v0 = 275 km/sand n0 = 1011.7 cm−3 (see left panel figure 4.18),here χ2 equals 4.0. One possible explanationis that emission from BP Tau passes an hydro-gen column density of LH = 1 − 2 · 1021 cm−2

(Hartigan et al. 1995; Gullbring et al. 1998;Robrade and Schmitt submitted), which affectsdifferent wavelengths in different ways. In thispicture it is justified to use the triplet linesonly because they have very similar wavelengthsand suffer therefore from comparable extinction.To test on this I correct the observed fluxes

with the method from Morrison and McCammon(1983) (with corrections in Balucinska-Churchand McCammon (1992)) as it is implementedin the ’Package for Interactive Analyses of LineEmission (PINTofALE)’ (Kashyap and Drake2000). This correction lowers χ2 to 54 (5 degreesof freedom). These high values indicate that theproposed model may not explain the emissionfrom BP Tau.Checking each line ratio on its own (see 4.4) re-veals that especially the ratio oxygen Lymanβto Lymanα presents problems. Ignoring this linedelivers acceptable shock parameters:


v0 = 475 km/s and n0 = 1012 cm−3 with χ2 = 17(see right panel of figure 4.18 an table 4.4). Withthe method described in section 4.3.2 no reliableelemental abundance can be measured, the modeldoes not fit the observations well enough, al-though it shows that neon is enhanced comparedto oxygen. I take the abundances from Robradeand Schmitt (submitted) and the shock param-eters presented above to estimate some shockproperties. They carry considarable uncertain-ties and are only valid under the assumption thatmagnetically funnelled infall takes place on BPTau at all. The best guess for these parametersis f = 0.05% and 2− 3 · 10−10 M/yr.

A shock with v0 = 275 km/s as suggested by thehelium-like triplet lines can be excluded becauseit requires a filling factor larger than unity to ex-plain the total flux.A second approach is to attribute the oxygen Lyβto Lyα ratio to larger values for LH . Using allavailable lines, but varying LH shows that col-umn densities above LH = 7 · 1021 cm−2 fit thedata with χ2 ∼ 20 (ratios in table 4.4). Theshock would have the infall parameters v0 =400 km/s and n0 = 1011.7 cm−3 in this case, thisleads to a larger surface filling factor f = 0.5%and mass accretion rate of 2 · 10−9 M/yr be-cause the energy flux observed represents only asmall fraction of energy emitted.

5.2. TW HYDRAE 51

Chapter 5

Discussion

In this chapter I compare my results obtained inthe previous sections for TW Hya and BP Tau topublished literature values and discuss the devi-ations. For TW Hya the two basic issues are themetal depletion but neon enhancement observedand the mass accretion rate, which is determineddifferently using different wavelengths. This willbe looked at in detail in section 5.2. In the caseof BP Tau it is assumed that the X-ray emissionis dominated by coronal activity which presentsproblems for the comparison to my simulation(section 5.3).

5.1 Simulation

As shown in section 4.1, my simulation pro-duces a consistent picture of a shock. It agreeswith previous calculations done independentlyby other authors (Lamzin 1998; Calvet andGullbring 1998). Assumptions made in the be-ginning about the plane geometry and the ther-mal heat flux can be justified a posteriori. Othersimplifications remain to be solved. To omiteither the quasi-stationary condition or to in-clude magnetic fields or turbulent flow requiresa full more dimensional magneto-hydrodynamic(MHD) simulation. To extend the simulation tolonger wavelengths is only possible if radiationtransfer is calculated because in these regimesthe optical depth will become important. Thecomparison to observational data in section 4.3proves that all these complications are not nec-essary to reproduce the observed X-ray spectra.

5.2 TW Hydrae

The best fit between my simulation and the ob-servations from TW Hya is obtained using ashock with the parameters v0 = (525± 25) km/sand log(n0) = 12 ± 0.3 (n0 in cm−3). Previousauthors have assumed velocities closer to ∼300 km/s (Lamzin 1998; Calvet and Gullbring1998). Observational evidence points to lowervalues, too. Stelzer and Schmitt (2004) converttheir observed temperature into a velocity in avery simple way and they get ∼ 350 km/s. Inthe UV there are several lines with red wings,which extend up to ∼ 400 km/s (Lamzin et al.2004), but then the gas accelerates strongly closeto the stellar surface and because of particle num-ber conservation the density will be lower in thehigh velocity region, so, depending on the ge-ometry of the accretion funnel, it may well bethat the emission measure in this region is verysmall. In this case the observed lines will haveweak extensions to larger velocities, which arenot identified observationally yet. So I regardthese observation as a lower bound; an upperbound is provided by the free-fall velocity (equa-tion 4.7) of ∼ 500− 550 km/s. The results frommy simulation are compatible with both boundsand can be taken as input for the elemental abun-dance fitting procedure in section 4.3.2 reliably.The abundances measured agree well with pre-vious analyses of the data used (Kastner et al.2002; Stelzer and Schmitt 2004). A similar pat-tern of metal depletion has been observed in thewind by Lamzin et al. (2004) of TW Hya andit was also noted by Argiroffi et al. (2005) us-ing X-ray observations of the quadruple system

52 CHAPTER 5. DISCUSSION

TWA 5 in the vicinity of TW Hya. Stelzer andSchmitt (2004) interprete the abundances as asign of grain depletion, where the grain formingelements condensate and mainly those elementswhich stay in the gas phase like the noble gasneon are accreted. This is discussed in detailby Drake et al. (2005) who collect evidence thatmetal depletion can be seen in the the infraredand UV, too, where the spectral distribution in-dicates well advanced coagulation into larger or-biting bodies which may resist the inward motionof the accreted gas (for details see references inthat article).Drake (2005) points to the absorption problem,but he believes that shocks are buried underNH > 1023 cm−2 because he disregards the pos-sibility that photons pass through the thinnerpre-shock gas and not the surrounding stellar at-mosphere. In section 4.3.3 I discussed the hy-drogen column density NH which the photonsemitted in the cooling zone pass on their wayto the observer and estimated a lower limit ofLH = 1020 cm−2. The actually measured columndensity is 3.5 · 1020 cm−2 (Robrade and Schmittsubmitted). This even allows for situations wherethe photons have to penetrate more more pre-shock gas than I assumed or are emitted deeperinside the cooling zone. However the picture de-scribed (illustrated in figure 4.15) implies thatthe spots are located in regions of the stellar sur-face pointing towards the earth.

Measurements of TW Hya in different wave-length regions lead to conspicuously distinctmass accretion rates. Generally the estimatesoutdo my simulation, which gives the accretionrate (1±0.25)·10−10 M/yr and the filling factor0.1%-0.2%. In two articles Alencar and Batalha(2002) and Batalha et al. (2002) deploy opticalspectroscopy and photometry, they state a massaccretion rate between 10−9 and 10−8 M/yr anda filling factor of a few percent. In the UV thepicture is inconsistent. On the one hand two em-pirical relations for line intensities as accretiontracers (Johns-Krull et al. 2000) indicate massaccretion rate above 3 · 10−8 M/yr (data fromValenti et al. (2000) evaluated by Kastner et al.(2002)), on the other hand fitting blackbodies on

the UV-veiling by Muzerolle et al. (2000) sug-gests a significantly lower value: Only 0.3% of thesurface area need to be covered by a spot to pro-duce the observed accretion luminosity, this leadsto the accretion rate 4 · 10−10 M/yr. A similarprocedure has been applied by Costa et al. (2000)earlier, but the estimated filling factor is 5%there. The previous X-ray analyses by Kastneret al. (2002) (dM

dt= 10−8 M/yr) and Stelzer and

Schmitt (2004) (dMdt

= 10−11 M/yr) suffer fromthe problem that they use filling factors extractedfrom UV-measurements (from Costa et al. (2000)and Muzerolle et al. (2000) respectively) and amass density calculated from X-ray observationswhich does not necessarily represent the same re-gion. My simulation now is the first possibilityrely on the X-ray only. The different values formass accretion rates are summarised in figure 5.1.

The quoted estimates do not seem consis-

Figure 5.1: Estimated mass accretion rates. +:optical and UV measurements; *: X-ray, for thedata from Stelzer and Schmitt (2004) the dottedline connects to the UV measurement, which de-livers the filling factor, 3: this simulation; datasources: see text

tent with the assumption that the differences arepurely statistical, but ask for a physical explana-tion. In longer wavelengths one observes plasmaat cooler temperatures and in general the fill-ing factor and mass accretion rates are higher.This can be solved naturally by an inhomoge-neous spot with parts that impact with free-fall

5.3. BP TAURI 53

Figure 5.2: Accretion spots are not homogenous according to Romanova et al. (2004), µ marks theaxis of the magnetic moment of the star, Ω its rotation axis, the angle between them is 30. Thedensity (left) and velocity (right) before the shock front are colour coded, their reference values aregiven in the text.

velocity and parts that are decelerated by mag-netic or other processes. The fast particles wouldbe responsible for a shock region with high tem-peratures which is observed in X-rays, whereas inother spectral bands cooler areas can be detected,too, and therefore the total area and the ob-served total mass flux is larger. Accretion spotswith this kind of properties are predicted by themagneto-hydrodynamic simulations recently per-formed by Romanova et al. (2004). They con-centrate on the geometry of the accretion funneland the disk and do not perform simulation ofthe structure of the spot. The parameters theychoose produce higher density spots (referencevalue n0 = 3 · 1012) and slower infall velocities(reference value v0 = 300 km/s). A cross sectionthrough the accretion funnel is shown in figure5.2, which agrees with the considerations above.

5.3 BP Tauri

The results of the simulation for BP Tau are pre-sented in section 4.4. They show that the under-lying infall model does not describe the situationin BP Tau well. I tried already there to cor-rect for an absorption column and this increasesthe quality of the fit. On the other hand si-multaneous optical and X-ray observations showuncorrelated variations (Gullbring et al. 1997).

Robrade and Schmitt (submitted) conclude thatthe X-ray activity from BP Tau is dominated bycoronal activity, but has a contribution from ac-cretion that explains the unusual f/i-ratios. Alllines are then expected to be contaminated by acoronal contribution, although the lines originat-ing in hotter regions, that is lines like Lymanαor Lymanβ, should be affected more than thetriplet lines. In this picture variations are at-tributed to normal coronal activity like flaressimilar the one reported in Schmitt et al. (2005).Shock and coronal contributions cannot be dis-entangled from the observation, consequently thesimulation has to be compared to the total emis-sion, and of course a good fit cannot be expected.The best possibility is then to use v0 = 475 km/sand n0 = 1012 cm−3, which gives the lowest χ2

and has the infall velocity close to free-fall. Theestimated filling factor (f = 0.05%) and massaccretion rate (2 − 3 · 10−10 M/yr) are thenupper limits because they assume that all emis-sion in the lines considered is produced in theshock front. Like for TW Hya the filling factorobserved rises with larger wavelengths. Ardilaand Basri (2000) measure fUV = 0.44% usingUV data from the IUE satelite and Calvet andGullbring (1998) get fopt/IR = 0.7% with opticaland near-infrared data. They both determine amass accretion rate ∼ 10−8M/yr. While thisprobably lends some support for the second sce-


nario in section 4.4, a large (f = 0.5%) shockwith a high absorption column, LH is confinedby other methods to 1− 2 · 1021 cm−2 (Robradeand Schmitt submitted). This simulation cannotdistinguish between these alternatives, but dueto the observation that the X-ray variations arenot correlated to optical ones the first alternativeseems more plausible, although it is an unsatis-fying situation not to be able to determine exactshock properties because of the coronal compo-nent.

5.4 Summary

I presented a computer program, which simulatesthe accretion spot in CTTS. I derived the equa-tions used and explained how the program is de-signed. The analysis of the simulation resultsclearly shows that the model of magnetically fun-nelled accretion explains the basic features in theX-ray emission of CTTS and a simulation with-out radiative transfer is adequate.

• The emission of TW Hya is dominated byaccretion. The parameters of the pre-shockgas are determined from modelling the shockand comparing it to data to v0 = (525 ±25) km/s and log(n0) = 12 ± 0.3 (n0 incm−3).

• This allows to calculate the global parame-ters filling factor and mass accretion rate as(1±0.25)·10−10 M/yr and f = 0.1−0.2 %.

• Checking this with literature values showsthat the size of the shock region depends onthe wavelength where it is observed.

• In BP Tau coronal activity produces mostof the X-ray emission, which clearly showsup in the high χ2 values for the fits.Nevertheless BP Tau is a actively accretingCTTS and some emission is produced in aspot.

• From selected lines bounds on the propertiesof this shock are derived.

• This supports the thesis that accretion shockare a common feature on CTTS.

• New X-ray observations can now be con-verted easily into shock properties using thetools developed in the course of this diplomathesis, so to obtain a statistically significantsample of stars further high resolution ob-servations of accreting stars are required.

5.4. SUMMARY 55

Acknowledgements

The implementation of my program uses many routines of CHIANTI. CHIANTI is a collaborativeproject involving the NRL (USA), RAL (UK), MSSL (UK), the Universities of Florence (Italy) andCambridge (UK), and George Mason University (USA).This research has made use of NASA’s Astrophysics Data System.

Some English words would be blank here without LEO, an online-dictionary operated by theUniversitat Munchen.

I want to express my gratitude to Prof. J. H. M. M. Schmitt, who proposed the topic for thisdiploma thesis and coached me throughout its course and to his college Prof. P. Hauschildt, whoadvised me on some hydrodynamic problems and agreed to act as second corrector.

I thank my colleagues at the Hamburger Sternwarte. I found an excellent infrastructure there, espe-cially in terms of office space and computing equipment. I had a very pleasent year and the peopleI met answered my questions patiently. Still some need to be emphasised, U. Wolter, because heshared his office with me and was the first person to be bothered with questions, B. Fuhrmeister,who advised me on some IDL problem and on my English expression and J. Robrade, who sacrificedhis parts of his labour time nearly daily to explain X-rays and stars to me.

My family and my girlfirend are the last, not the least, on the list. They bear a physicist and alwaysencouraged me to go on and do what I like.


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