Λ Bootis Stars ISM Gas Accretion As an Explanation for Chemical Peculiarity

λ Bootis stars ISM gas accretion as an explanation for chemical peculiarity

Supervisors: Author: prof. dr. I.E.E. Inga Kamp Michael J. van Schaik prof. dr. Dieter Breitschwerdt Abstract λ Bootis stars are a type of chemically peculiar star. Around 2% of A-type stars have a λ Bootis spectrum. There are many different theories about the origin of the peculiarity. Few studies have been made into quantifying the fraction of λ Bootis stars that can be explained with each method. Our aims are to investigate the validity of the ISM accretion theory and explore the effects of bow shocks. This is done by modelling the accretion onto the star with Bondi-Hoyle accretion. An ISM gas model is used to provide densities and velocities of the gas. To this model we have added molecular clouds. The effect of bow shocks are investigated through a scaling factor for the density and relative velocity. We find that bow shocks can increase the probability of finding a λ Bootis up to a factor 103. The inclusion of molecular clouds had no significant effects on our conclusions. The ISM accretion theory matches the observed ratio of λ Bootis stars for an accretion threshold of −11 10 M /yr when we include the effect of bow shocks.

1 Contents

1 Introduction 3 1.1 What are λ Bootis stars?...... 3 1.2 Theories on the origin of λ Bootis stars...... 5 1.2.1 Diﬀusion/mass loss...... 5 1.2.2 Binaries...... 6 1.2.3 Accretion/Diﬀusion...... 6 1.3 Bow shocks...... 7 1.4 This thesis...... 8

2 Methodology 9 2.1 Bow shock model...... 9 2.2 Gas data model...... 11 2.2.1 Comparing gas data model at two diﬀerent evolutionary stages...... 11 2.2.2 Molecular Clouds...... 11 2.3 Bondi-Hoyle accretion...... 14 2.4 Large scale calculation...... 16

3 Results 18 3.1 1D analysis...... 18 3.2 Accretion areas and their sphere of influence...... 18 3.3 Accretion Rates...... 18 3.3.1 Influence of molecular clouds on accretion rates and area...... 20 3.3.2 Influence of bow shocks on accretion rates and area...... 23 3.3.3 Influence of accretion threshold...... 23

4 Discussion 27 4.1 Comparison of model with observations...... 27

5 Conclusion 29

2 1 Introduction 1.1 What are λ Bootis stars? A λ Bootis star is a type of chemically peculiar main-sequence star. They range from late B, through A to early F in spectral type (Kamp and Paunzen, 2002). Chemical peculiar stars have spectra with absorption lines that do not correspond to what one would expect given the star’s temperature, mass and metallicity. There are many phenomena that can cause chemical peculiarities: accretion, diffusion, magnetism, convection and pulsations, all these effects present themselves in the stellar mass range that A type stars occupy (Griffin, 2008). In the spectra of around 10% to 20% of main-sequence A and B stars chemical peculiarities have been observed (Folsom et al., 2012). These peculiarities are not found in the entire star, they are limited to the stellar atmosphere (Mathys, 2004; Smith, 1996). These peculiarities make these chemically peculiar stars interesting to study. What exactly makes them different? Why these stars peculiar and others not? Around 2% of the stars in the Galactic field and in open clusters within the spectral range B8-F4 are showing λ Bootis type spectra (Paunzen, 2001). Contrary to other chemical peculiar stars like Am, Ap and Fm stars, that have over-abundances of heavy elements, λ Bootis stars have under-abundances for some of those elements (Michaud and Charland, 1986). In particular the MgII line at 4481 Ångström is weaker compared with the same line in chemically normal stars. For a long time, the defining criterium of λ Bootis stars was that they were A type stars with a weak MgII spectral line at 4481 Ångström (Gray, 1988). Many more features and characteristics have been discovered since. In Fig.1 the relative weakness of various Fe and Ca (middle three spectra) lines compared with the normal spectral lines of equivalent normal stars (top and bottom) is shown (Paunzen and Heiter, 2014).

Figure 1: A comparison of spectra from three accepted λ Bootis stars (middle three) together with two normal stars (top and bottom). From Paunzen and Heiter(2014).

3 Fe-peak elements can be a factor 100 under-abundant, while lighter elements like C, N, O and S have roughly solar abundances (Venn and Lambert, 1990). Other elements like Mg, Ca, Ti and Sr are similarly under-abundant (Venn and Lambert, 1990). There is possibly a positive correlation between the Na abundance in the stars and the local interstellar column density of Na I (Paunzen et al., 2002). Ages of λ Bootis stars vary between 107 and 109 yr (Iliev et al., 2002). Young pre-main-sequence Herbig Ae/Be stars have also been observed with typical λ Bootis spectra (Folsom et al., 2012). In their sample about half of the stars showed chemical peculiarities that would be typical for λ Bootis stars, which is much more than 2% of the A type stars overall (Folsom et al., 2012). The rotational velocity distribution is the same as for normal A-type stars (Paunzen, 2004).

Figure 2: HR diagram of the 52 members and 17 candidates that are most likely members of the λ Bootis class. Different symbols show the different pulsation types. In green are the evolutionary tracks for stars of different mass. The red dotted line marks the γ Dor instability strip. The δ Sct strip is delimited by the solid red lines. From Murphy and Paunzen(2017).

An overview of current λ Bootis stars has been compiled by Murphy and Paunzen(2017) and Fig.2 shows how the stars are distributed in the HR diagram. The stars are distributed over the main-sequence between 1.5 and 2.6 M . There are different pulsation types present, but also constant stars. δ Sct stars are pulsating stars, whose pulsations are due to the κ mechanism. The outer envelopes of δ Sct stars are not completely ionized yet. When the star heats up, this gas ionizes and becomes more opaque. When the opacity increases, the star contracts and becomes hotter until the pressure is large enough that the star starts to expand again, releasing all the trapped energy. In this pulsation cycle, the star will be less luminous as it is shrinking while it is more luminous when it expands. γ Dor stars pulsate through gravity modes. These pulsations start at the base of the envelope convection zone. It is assumed that the convective luminosity is constant over the pulsation cycle. Any extra luminosity is then blocked at the base of the convective zone. This drives pulsations for γ Dor stars (Guzik et al., 2000). The described pulsation mechanisms depend on the properties and composition of the star. The various pulsation types originate from different parts in the star. When looking at the different pulsation modes, one can infer interior properties of the star and gain information that would otherwise not be available from the spectrum. In this way it was found that the peculiarities of chemically peculiar stars are bound to their surface (Turcotte, 2005). Stars in the mass range 1.4-2.6 M have shallow convective

4 zones. Due to this, these stars can have a chemically distinct surface compared to the rest of the stars interior. This can be seen in the large number of diﬀerent chemically peculiar A stars (Folsom et al., 2012). When looking at the distribution of λ Bootis on the HR diagram (Fig.2), it quickly becomes clear that most of them are within the δ Sct instability strip. This looks to be coincidental as not every λ Bootis star shows the typical pulsations of a δ Sct star or of a γ Dor star (Murphy and Paunzen, 2017). This means that the λ Bootis stars do not have the same origin as δ Sct stars and γ Dor stars. On the HR diagram it can also clearly be seen that there are λ Bootis stars much hotter than one would expect from a star on the instability strip. Based on all these characteristics, a working deﬁnition was made of properties that a star has to meet to belong to the λ Bootis class.

• The λ Bootis stars are early-A to early-F type stars with an approximate spectral type range (based on the hydrogen lines) of B9.5 to F0 with possible members as late as F3.

• The λ Bootis stars are characterized by weak Mgii λ4481 lines, such that the ratio Mgii λ4481/Fei λ4383 is signiﬁcantly smaller than in normal stars. In addition the spectra exhibit a general metal- weakness. The typical shell lines, such as Feii λ4233, tend to be weak as well. But the λ Bootis stars do not show the typical shell spectral characteristics. • The following classes of stars should be excluded from the λ Bootis group even if they show weak λ4481 lines: shell stars, protoshell stars, He-weak stars (easily distinguished on the basis of their hydrogen line temperature types), and other CP stars. FHB and intermediate Population II stars may be distinguished from the λ Bootis stars on the basis of their hydrogen line proﬁles. High- v sin i stars should be considered as λ Bootis candidates only if the weakening of λ4481 is obvious with respect to the standards with high values of v sin i.

• λ Bootis stars are also characterized by broad hydrogen lines, and in many cases, these hydrogen lines are exceptionally broad. In the late-A and early-F λ Bootis stars, the hydrogen line proﬁles are often peculiar, and are characterized by broad wings, but shallow cores. • The distribution of rotational velocities of the λ Bootis stars cannot be distinguished from that of the normal Population I A-type stars.

This definition is taken as is from Paunzen(2004) and is based on the definition found in Gray(1988). A detailed description for the classification of λ Bootis stars can be found in (Paunzen, 2004).

1.2 Theories on the origin of λ Bootis stars The λ Bootis phenomenon has been named after the ﬁrst star that was found to have such a spectrum (Morgan et al., 1943): λ Bootis. Ever since that ﬁrst observation, there have been speculations over the origin of the star’s peculiar spectrum. There are multiple theories about the origin of the λ Bootis stars and these will be discussed below.

1.2.1 Diffusion/mass loss Michaud and Charland(1986) formulated the diffusion/mass loss theory to explain the λ Bootis phenomenon. They compare the gravitational acceleration to the radiative acceleration of the different chemical elements within the star. The radiative acceleration is not the same on all elements (e.g. Mn has a relatively high acceleration and C a normal one), resulting in a chemically peculiar surface, see Fig.3. This theory has been used to explain the chemical over-abundances of Ap and Am stars. Stellar winds are a mechanism for stars to remove mass from their surface layers. Over time this mass loss will slowly erase the over-abundances in the surface. The elements that were once over-abundant will in time −13 become under-abundant in this way. With a mass loss rate of 10 M /yr, Michaud and Charland (1986) calculated that it will take 109 yr to reach the observed abundances of iron peak elements in λ Bootis stars. This method thus fails to give a satisfying explanation for the young λ Bootis stars. Furthermore, there is a large sensitivity to rotational or turbulent mixing happening in the star, as the mixing in the star will remove the surface peculiarities by mixing them with the chemically normal interior of the star (Charbonneau, 1993). λ Bootis stars have the same rotational velocities as A-type

5 stars (Paunzen, 2004). Stars in the mass range 1.4-2.6 M have a rotational velocity distribution with a peak between 140-170 km/s (Zorec and Royer, 2012). Higher rotational velocities have more mixing in the stars.

Figure 3: A diagram of the diffusion/mass loss theory showing how different radiative accelerations can create chemical separation within the star. The red ball represents an element with a high radiative acceleration (e.g. Mn) and the black ball one with a lower radiative acceleration (e.g. C). The gray area represents the layer of the star where stellar winds draw material from. a) is a young star where there are no peculiarities. In b) the difference in radiative pressures has made the star peculiar with over-abundances in the surface. c) Stellar winds have created an under-abundance in the surface.

1.2.2 Binaries Another theory to explain λ Bootis stars is that it are actually a binaries being misidentified as single objects. In Faraggiana and Bonifacio(1999) an investigation is made into the combined spectrum of a binary. They show how the continuum emission from one star veils the metal lines of the other, making them appear weaker in the composite spectrum. The combined spectrum has the averaged colour and effective temperature from the two stars (Faraggiana and Bonifacio, 2005, 1999). This theory is further tested by Stütz and Paunzen(2006), who conclude that the spectroscopic binary model cannot be used to explain the majority of the observations. This was done by creating synthetic spectra and comparing those with observations. In their research, 90% of observed, well-investigated λ Bootis spectra could not be explained with the binary theory. There is a second theory involving stars in a binary system proposed by Andrievsky(1997). These stars would have to be in very close orbits with orbital periods less then one day. As the stars lose angular momentum, they merge together. During the merging process, some of the gas can be swept up into a circumstellar shell which can get accreted onto the star. This can have a strong influence on the surface abundances, as is explained in the next section.

1.2.3 Accretion/Diffusion Another explanation for the abundance patterns is accretion of dust-poor gas onto the star as proposed by Venn and Lambert(1990). Dust consists of heavier elements, so by removing the dust from the accreted gas, you get gas that is metal-poor. When this metal-poor gas gets accreted onto the surface of a star, it will change the abundance pattern of the stellar surface if the star has a shallow convection zone. Because the gas has low metal-abundances, the surface will see a lowered abundance of those same elements. The shallow convective mixing zones of A type stars (10−6 in fractional mass assumed) keep the peculiar material on the surface long enough for the spectra to be influenced by it (Turcotte and Charbonneau, 1993). The specific abundance pattern of the stellar surface will depend on the star’s metallicity and the abundance pattern of the accreted gas. Dust particles are made out of refractory elements, exactly the same elements that observations found to be under-abundant in λ Bootis. This

6 dust can be pushed away from the star through radiation pressure, leaving only the metal-depleted gas to be accreted. The origin of this gas can vary: circumstellar, interstellar and even hot Jupiters (Venn and −11 Lambert, 1990; Kamp and Paunzen, 2002; Jura, 2015). Accretion rates of 10 M /yr are necessary to overcome internal mixing for rotational velocities of up to 100 km/s and a 10−6 fractional mass of the star in the mixing zone (Turcotte, 2002). This happens on a timescale of MCZ /M˙ which means roughly 0.1 Myr in this case. When accretion stops, the surface contamination is mixed with the deeper interior of the star in roughly 1 Myr (Turcotte and Charbonneau, 1993). The exact time depends on the mass of the mixing region and the flux in and out of the mixing zone due to the rotation speed of the star and diffusion. A natural explanation for the distribution of λ Bootis stars on the HR diagram in Fig.2 can be found with this theory. The hottest stars in Fig.2 will have radiation and stellar winds that are too strong for the gas inflow to overcome and therefore the gas cannot be accreted onto the star. The incoming gas encounters outgoing stellar winds. These winds have more momentum than the typical accretion flow and will thus stop it from reaching the star and being accreted onto the surface (Kamp and Paunzen, 2002). At the cold end, the gas does get accreted, but the mixing zones in the star become too massive for any contamination to stay bound to the surface. The contamination will be mixed with the rest of the star, thus removing the surface peculiarity. This effect is further enhanced by the weaker radiation field that removes less of the dust from the accreted gas, resulting in a less peculiar gas that gets accreted (Murphy and Paunzen, 2017). The high frequency of λ Bootis stars in Herbig Ae/Be can be explained this way through the accretion of the gaseous elements from the circumstellar disk (Folsom et al., 2012). The inflow of gas still includes helium. Murphy and Paunzen(2017) argue that this fresh flow of helium could cause the δ Sct pulsations seen in Fig.2.

Figure 4: (a) A star moving through the ISM, (b) moving into and clearing a cavity in a cloud. From Kamp and Paunzen(2002).

Kamp and Paunzen(2002) showed that, using typical values for the density and relative velocity between a star and an interstellar gas cloud (Fig.4), it is possible that Bondi-Hoyle accretion provides enough accretion to overcome mixing. For this to happen the star needs to spend enough time in a cloud for the surface contaminations to become large enough to be observable. The star clears a shock-like cavity inside of the cloud. When the star leaves the cloud, the contaminations are slowly mixed through the star again and it returns to being a normal star. The idea of ISM accretion as explanation for λ Bootis spectra needs to be explored further.

1.3 Bow shocks When an object moves at supersonic velocities through a compressible medium, a shock will occur. A shock is an abrupt change in the density, pressure and temperature of the medium. In the case of a bow shock, this shock is detached from the object much like the water in front of a ship. The ﬂow behind the shock is divided into two parts: the subsonic region in the centre and the supersonic region in the wings. These two regions are divided by the sonic line. In the universe these bow shocks occur at many length scales: Earth has one with the solar wind, stars as they move through the ISM and galaxies and clusters cause a shock because of their movement through the IGM. The presence of a bow shock signiﬁcantly alters both the densities and velocities of the gas around the star. Gáspár et al.(2008) investigated whether the IR excess of the star δ Velorum could be due to a bow shock instead of a debris disk, as was assumed before. Considering the fact that most stars move at supersonic velocities, we would expect most stars to have bow shocks. The increase in post shock

7 Figure 5: 24 µm image of δ Velorum in log scaling showing the bow shock. Top left: original observation of the star. Top right: the bow shock becomes visible when the the star is removed from the image. Bottom left: the shock close to the star with the direction and outline indicated by the arrow and contour. Bottom right: contours of the observed intensity from the bottom left. Figure and caption from Gáspár et al.(2008). temperature heats up the dust which emits in the IR. It is thus possible for stars to show an IR excess in their spectrum because of the presence of a bow shock. This makes IR excess stars good candidates to do bow shock research on. Unfortunately, most IR excess stars are distant and cannot be resolved, so it is not possible to observe the shock directly. A resolved star can be seen in Fig.5. The IR emission of the dust is clearly visible in the two distinct wings of the bow shock. So, while diﬃcult to observe directly, it is possible for a star to have a bow shock.

1.4 This thesis In this thesis we aim to understand what fraction of λ Bootis stars can be explained with the ISM accretion method proposed by Kamp and Paunzen(2002). The full parameter space of relative velocities (between ISM and star) and densities is investigated for the solar neighbourhood. The occurrence rate of stars with a sufficient accretion rate for a λ Bootis spectrum is compared to their observationally determined occurrence rate. We also take a look at the effects that bow shocks have on the accretion of ISM onto stars. In section2 we take a detailed look at our methodology and the different physical models we used are described in more detail. In section3 we present our results. These results are discussed and analysed in section4. Our conclusions and suggestions for future works are in section5.

8 2 Methodology

To get a better understanding of the origin of the λ Bootis phenomenon, it is important to take apart the diﬀerent origin theories and look at the contribution from each. In this project, the fraction originating from ISM gas accretion is investigated further. This is the method proposed by Kamp and Paunzen (2002). In this scenario, when a star travels through the ISM, the radiation pressure from the star stops the dust particles from accreting onto the star. This strongly depletes the iron-peak elements while leaving elements like C, N, O and S, resulting in a λ Bootis spectrum. The rate of accretion is calculated through Bondi-Hoyle accretion (BH-acc). BH-acc is the accretion of gas onto a point source with mass moving through a gaseous medium. In BH-acc, the radius up to which particles are accreted onto the star is calculated and used to give an accretion rate (Bondi and Hoyle, 1944). Kamp and Paunzen(2002) applied this to only a single test case and in this thesis, a more complete picture will be established by using a gas simulation model of the local environment (Schulreich et al., 2017) to provide gas densities and velocities for the BH-acc. A bow shock model is used to relate the ISM gas conditions to the conditions of the accreted gas.

2.1 Bow shock model Schulreich and Breitschwerdt(2011) have made a bow shock model that uses the shape of the bow shock and the upstream conditions to calculate the properties of the flow region and the shape of the blunt body. A blunt body has separated flow over a large fraction of its surface. This body can be anything from galaxies to marbles. For this thesis, I want to investigate how the gas density and the velocity of the gas with respect to the star changes with the introduction of a bow shock. This solution for the hypersonic blunt body problem is mathematically difficult since there is both a sub- and supersonic portion of flow behind the shock. The highly nonlinear partial differential equations of hydrodynamics that need to be solved are elliptic in the subsonic region while they are hyperbolic in the supersonic region. Schulre- ich and Breitschwerdt(2011) follows the analysis of Schneider(1968) to solve this problem which has the advantage of being valid in the entire flow field behind the shock. Fig.7 shows the relevant coordinates involved. An in depth analysis can be found in Schulreich and Breitschwerdt(2011). The difficulty here is that it is an inverse problem, meaning that we need to know either the shape of the bow shock to see what the shape of the body will be or change the bow shock morphology until a desired body shape is reached. The latter is the approach followed here. The de- Figure 6: Shock-oriented coordinate system of sired shape of the body was one where the body’s boundary-layer type, where z and r are Cartesian shape is similar to the heliosphere. We tried to coordinates for plane flow, x and y are the distances match the shape of the shock to observed bow along the shock surface and normal to it, with u and shocks like the one in Fig.5. Fig.7 shows the v as the corresponding velocity components, U∞ is shape of the body and shock. the free-stream velocity, and φ is the stream func- A second problem is that the model does not tion. Flow quantities immediately behind the shock give values for the density and velocity of the gas in the point N are denoted by a hat (∧), ad in the within the object itself, as it is modeled as a blunt point S by an asterix (*)(adapted from Schneider body. However, the star and the accretion are (1968)).Figure and caption from Schulreich and Bre- both within the blunt body object of the simula- itschwerdt(2011). tion. One way to solve this is to use mass conservation and see how much matter flows through the bow shock and use the missing matter as an average density within the entire object. The velocity at the edge of the object will then be used as the velocity of the gas within. This likely underestimates the density while overestimating the velocity of the

9 gas. In the wings of the bow shocks, the velocity is higher while the density is lower compared to the stagnation point, where the ﬂow is subsonic and the density the highest. This method was investigated but not used due to the fact that the location where these values were obtained is far removed from the accretion radius. Gáspár et al.(2008) ﬁnd the bow shock in Fig.5 extends to around 3000 AU, while typical accretion radii are 1-100 AU. Hence this approach will not give representative values inside of that accretion radius. One of the plots from the blunt body bow shock model is shown in Fig.7.

Figure 7: A map of the shock quantities, where the object is the white ﬁeld at the bottom of each panel and the ISM the white ﬁeld on top. Top left: pressure (dyn cm−2). Top right: log(density) (g/cm−3). Bottom left: log(temperature) (K). Bottom right: Mach number, the black line inside of the shock separates the subsonic and the supersonic regions.

Another way is to look at the values for the density and velocity at the stagnation point and use those. The stagnation point is always within the accretion radius. Total mass ﬂow must be conserved according to the Rankine-Hugoniot mass conservation condition:

ρ1 u1 = ρ2 u2 ≡ m (1) Subscripts in these equations refer to the location, subscript 1 means the upstream value and 2 refers to the downstream value. ρ is the density, u is the velocity of the gas and m is the mass. Using the bow shock model, it is possible to obtain the post shock density. Using the bowshock model and Eq.1, it is possible to derive a scaling factor for the gas velocity and density post shock. The density is increased −3 by a factor 4.5 in the bow shock model under typical conditions (Vrel 7-28 km/s, density 10-50 cm ). With this factor and Eq.1 we ﬁnd that the velocity needs to be scaled by a factor 0.22. Thi, however, does not take into account that the gas velocity increases in the wings of the shock. Due to this, there is a slight increase in the velocity at the outer edges of the accretion radius. To account for this, the velocity is scaled to a factor 0.25. Fig.7 shows the density in the top right panel.

10 2.2 Gas data model Taking typical values for the relevant parameters (density and relative velocity) in BH-acc is possible, but much more can be learned from examining the full range and different combinations of densities and velocities. A model of the local environment is needed for this. The modelled values of the ISM from Schulreich et al.(2017) can be used here. Schulreich et al.(2017) have created a model of the ISM to provide insight into the formation of the Local Bubble. They investigated whether the 60Fe enhancement found on earth was the result of a set of supernovae that also resulted in the formation of the Local Bubble. Their model is based on the RAMSES code from Teyssier(2002), which uses the adaptive mesh refinement technique to give good a resolution where it is needed while conserving computing power. The model covers a space representing 3 kpc divided into 1024*1024*1024 data points. This pc scale resolution is well suited for the timescales and distances involved in BH-acc. The simulation has a maximum run time of 194 Myr. It rotates along with the local standard of rest around the centre of the Milky Way and neglects the shearing effects due to the galactic differential rotation. The initial gas distribution is being matched to the the observations by Ferrière(1998) and Schulreich and Breitschwerdt (2016). To conserve on computational time, only the slice through the galactic plane is used.

2.2.1 Comparing gas data model at two different evolutionary stages The gas data model can run for different amounts of time. It is of interest to see if the ISM changes in ways that will have effect on the amount of accretion calculated. To do this, two simulation run times are chosen: 181 Myr, corresponding to the formation of the Local Bubble, and 194 Myr, which is the longest the simulation has run and it should correspond to the present situation. As mentioned earlier, it is expected that 1 Myr after accretion has stopped, the star will have returned to a normal abundance pattern. An A type star lives for much longer than that, so it will be able to experience an evolution in its surroundings. In Fig.8, the density is plotted as is from the gas data model.

Figure 8: Gas densities of the central slice obtained from the gas model by Schulreich et al.(2017). A density of 10 [cm−3] is 2.32 · 10−23 [g cm−3]

2.2.2 Molecular Clouds The gas data model only considers atomic gas as it does not model chemistry. The highest densities in the ISM are found in molecular clouds, however. To see if there is a large difference due to this, a simple recipe is constructed that adds molecular clouds to the model. The values for the filling factor, the mass and the density, are taken from Sparke and Gallagher(2007). The filling factor for molecular H 2 should be <0.1% with particle density of 200 cm−3 or higher, see Table1. We are interested in the ratio of mass between molecular clouds versus atomic gas in the galactic plane. Table1 gives us this information on all the components that make up a significant mass fraction. In the galactic plane, around 39% of the mass is in the form of molecular H2 (using Eq.2 to determine the

11 Table 1: The observed components of the Milky Way with a substantial contribution to the mass density (Sparke and Gallagher, 2007). They note that the value for molecular H2 is very uncertain but do not provide errors.

Molecular H2 Cold Atomic Hi Warm Atomic Hi Ionized Hii Filling Factor <0.1% 2%-3% 35% 20% 9 9 9 9 Mass (M ) 2 · 10 3 · 10 2 · 10 1 · 10 Vertical Extent (pc) 80 100 250 1000 7 7 6 6 Mass Density (M /pc) 2.5 · 10 3 · 10 8 · 10 1 · 10 Density (cm−3) >200 25 0.3 0.15

the fraction of mass for molecular H2). This means that the mass that is in the gas model represents only 61% of the total mass. This mass is placed on the densest and coldest regions of the model so that at most 0.1% of the model is enhanced, matching the filling factor of the molecular H2. The 0.1% represents a maximum so we stay below it for safety. The same limits were used for both ages of the model. With the filling factor accounted for, the second step is to match the mass ratio. To do this, all densities within the marked region are multiplied by a factor 51.3 to give the ratio observed in the Milky Way. This factor was the one closest to get the 39% mass fraction in molecular clouds (Eq.2). A compromise needed to be made between both slices to get close to observations, this meant that the added mass in the 181 Myr slice overshoots our target while the filling factor in the 194 Myr is lower than observed. The effect of our addition on the mass and peak density can be seen in Table2. The first two columns give the total mass in the data slice, both of the base model (Mbase) and that with added molecular clouds (Maddedclouds). The maximum densities are in the third (base) and fourth (with molecular clouds) columns. The last column gives the filling factor of the enhanced regions.

Mass density(molecular clouds) fraction = (2) P Mass density(all components)

Table 2: The eﬀect on mass and maximum density from the addition of molecular clouds to the gas data model.

−3 −3 Mbase (M ) Maddedclouds (M ) Max ρ (cm ) Max ρcloud (cm ) Fill. fact. clouds (%) 181 9.8·109 18.6·109 39 1997 0.08 194 9.4·109 14.6·109 30 1539 0.05

12 Figure 9: Density plot of the central slice of the data cubes. Top: the density as provided by the model. Bottom: the density with added molecular clouds. Both plots on the left are of the 194 Myr slice, while the plots on the right are of the 181 Myr slice. The black boxes in the 195 Myr slice are zoomed in upon in Fig.10.

As we can see in Fig.9, the extra density is concentrated in small clumps, exactly as we would expect for molecular clouds. Zooming in on one region in Fig. 10 shows in more detail how the extra density is placed in a small clump. There are slight diﬀerences in the modelled ﬁlling factor between the two slices and the total mass added, but these are small compared to the uncertainty of the mass in molecular clouds (Sparke and Gallagher, 2007). To stay consistent, both slices had the molecular clouds added in the same manner. In Table.2, we see that that the 181 Myr slice has about 5% more mass from the start but the highest density is 30% (9 cm−3) higher. This might explain the larger increase in mass seen in the 181 Myr slice with the addition of molecular clouds, as the molecular clouds get placed in dense and cold areas.

13 Figure 10: A zoom in on the regions marked in the 194 Myr slice in Fig.9. Left: basic model. Right: model enhanced with molecular clouds. Same colour scale as Fig.9.

2.3 Bondi-Hoyle accretion With the gas model fully set up, we can now relate the gas parameters to gas accretion onto the star. Ideally, here one would use a hydrodynamical simulation for all the different combinations of parameters. This, however, would be too time consuming and beyond the scope of this research project. To examine the wide range of scenarios, BH-acc is a good choice as it allows for a quick calculation instead of a complete simulation (Edgar, 2004). Next, we will explain the physical picture behind BH-acc. Consider an object with mass moving into a cloud. As it gets closer, the gas will get pulled towards, and possibly into, the object. In Fig. 11, we see two of the flow lines and three regions of interest in the case where there are no collisions between the gas streams. The flow lines go from A → B and C → D and represent the gas particles just missing the object. Together they divide the chart into the three regions. In region I all the flow lines will collide with the object. Region II is a single-stream area, as there is only one flow line at any given point. The last region III is then a multi-stream region with flows from above and below the axis crossing each other without interacting.

Figure 11: A side view of the diﬀerent regions in Bondi-Hoyle accretion without collisions in the gas. Image from Bondi and Hoyle(1944). .

14 When the gas does collide, the picture changes to Fig. 12. Here we see major changes in particular to region III, which is now split into IIIa and IIIb. Region I stays the same and region II is expanded. This expansion is due to the collisions in the multi-stream region III. These collisions stop the ﬂow from continuing, thus, the single-stream region II is expanded. The actual collisions happen in region IIIa, which should have a thickness of around the mean free path of the particle, which is smaller with higher densities than it is around the axis. The size of region IIIb is dependent on the gas pressure inside, pushing out, and the momentum of the gas coming in from region II through IIIa that is pushing inwards. Part of the gas in region IIIb will be gravitationally bound to the star. This is the matter that will be accreted onto the star. The remainder that is not bound will ﬂy away and be lost.

Figure 12: A side view of the diﬀerent regions in Bondi-Hoyle accretion with collisions in the gas. Image from Bondi and Hoyle(1944).

The mass accretion rate (M)˙ as derived by Bondi and Hoyle(1944) is ˙ 2 2 −3 M = 2.5 · π · G · M · ρ∞ · Vrel (3)

It describes the mass accreted by a point object of mass M moving with supersonic velocities (Vrel) through a gaseous medium with density ρ∞ (the density upstream). G is the gravitational constant. This is the steady state solution for an object that moves from empty space into a cloud. In this thesis, this is a star moving through the ISM and encountering clouds. Here, 1.5 M is chosen as stellar mass, as it is a safe lower bound for observed λ Bootis stars. The star and the gas both have their own velocities, but it is the relative diﬀerence between them that is needed. Equation3 can be re-written in terms of the accretion radius:

˙ 2 M = π · ρ∞ · Vrel · racc (4) with √ −2 racc = 2.5 · G · M · Vrel (5) Together Eq. (4) and (5) illustrate that Bondi Hoyle accretion can be understood as an increased surface area that collects the gas. There is an extra consideration about the relative velocity and the density, namely the presence of bow shocks. As mentioned earlier these have a significant effect on these two parameters. It is important to actually look at the gas that gets accreted because the values for ρ and V can be quite different compared to the upstream values. A scaling factor is used to incorporate the effect of the bow shock on these two values. The density is increased by a factor 4.5 in the bow shock model under typical conditions −3 (Vrel 7-28 km/s, density 10-50 cm ). With this factor and Eq.1, we find that the velocity needs to be scaled by a factor 0.22. However, this does not take into account that the gas velocity increases in the wings of the shock. Due to this there is a slight increase in the velocity at the outer edges of the accretion radius. To account for this, the velocity is scaled to a factor 0.25. Fig.7 shows the density in the top right panel. This accretion mechanism depends on gas being able to lose angular momentum in region IIIa. It does this through collisions. Kamp and Paunzen(2002) use the following equation to quantify this: √ −16 −2 Ncoll = 6.3 · 10 · ρ · 2.5 · G · M · Vrel (6)

15 Eq. (6) represents the amount of collisions (Ncoll) over a semi-circle where is the radius is equal to the accretion radius from Eq. (5). When Ncoll is suﬃciently large, accretion is allowed to happen. At least three collisions are needed to reduce the angular momentum (Kamp and Paunzen, 2002). The two streams cross each other behind the star as shown in Fig. 12. The result of this is that the density in Eq. (6) needs an extra factor 2 to represent the two streams crossing each other behind the star. Each stream has a density ρ∞, so in the region where the streams cross the density is doubled.

2.4 Large scale calculation Using the previous theories, equations and data, the fraction of stars that are accreting sufficiently to exhibit a λ Bootis spectrum can be calculated. First, the accretion rate is calculated in every point of the gas slice for one particular velocity (V∗) of a star. Only the points where there are sufficient collisions and the accretion rate is at least higher than the accretion threshold are accepted as having sufficient accretion for a λ Bootis spectrum to be established, others are set to 0, meaning no λ Bootis spectrum can be generated. The accretion threshold is a parameter that can be changed for the calculation. The −11 −14 accretion threshold will be somewhere between 10 and 10 M /yr (Turcotte and Charbonneau, −14 1993). An accretion threshold of 10 M /yr is assumed unless otherwise stated as it is the minimum accretion rate found in Turcotte and Charbonneau(1993). Without this minimum we would see λ Bootis stars in our calculations for which there is no theoretical basis. The stars are given a V∗ in the X and Y direction (Fig. 13). The distributions for V∗ are obtained from Binney(2010). These distributions are in the U and V planes, however. In this thesis we will take the U distribution for the X plane of the gas model and the V distribution for the Y plane. Together with the gas velocities (Vgas), the relative velocity (Vrel) between the ISM and the star can be calculated. During the calculations for the accretion rates at different velocities (section 3.3.3) it was found that if V∗ is larger than 25 km/s, then there would be no accretion at all. This is understandable considering the accretion −3 rate scales like Vrel and Vrel is the difference between V∗ and Vgas. So to save on computation time, an upper bound of 25 km/s was placed on V∗ (Fig. 14). 65% of the stars are above this limit. Stars moving faster than this would return no accreting stars without having to fully go through the Bondi-Hoyle calculations. A lower bound of 1 km/s was also assumed for V∗. This is to ensure that the stars are actually moving at supersonic speeds in molecular clouds (sound speed 0.3 km/s) and cold neutral matter (sound speed 0.96 km/s), which is necessary for both BH-acc and for the presence of bow shocks. Less then 0.001% of stars have a V∗ below 1 km/s. In this way it is possible to create a map of where and how much accretion takes place at every point in the gas data model. The filling factor of accreting stars will tell us about the frequency of λ Bootis stars in the Figure 13: The velocity distributions for V∗. Top computational volume. Higher accretion thresh- panel: velocity distribution in the X direction corresponding with the U plane with σ =35 km/s. Bot- old and higher V∗ will reduce the frequency, while U a higher density will increase the area where ac- tom panel: velocity distribution in the Y direction cretion takes place. corresponding with the V plane with σV =20 km/s. The effects of accretion stay for about 1 Myr Distributions obtained from Binney(2010) after the accretion stopped (Turcotte and Char-

16 Figure 14: Absolute V∗ distribution in the X,Y plane. Two black lines are placed at 1 and 25 km/s. bonneau, 1993). This means that there could be stars outside of actively accreting areas that are still λ Bootis stars. This is investigated by drawing a circle around each accreting area with an radius equal to the distance travelled by the star in 1 Myr. For slow stars, this radius will be small, while for faster stars it will be larger. This shows us the regions where we could expect to see λ Bootis stars. The λ Bootis abundances form in 0.1 Myr. In that time, a star with V∗=25 km/s (our maximum) will travel 2.5 pc. This is less then the resolution (3 pc) of our gas model. This means that a star will show a λ Bootis spectrum in every place where it can acrete in our model.

17 3 Results 3.1 1D analysis First we investigate how the relevant quantities evolve when the star travels along a straight line through the ISM (Fig. 15). This was done using only the gas model and BH-acc, without any modifications like enhanced densities (section 2.2.2) or taking bow shocks (section 2.1) into account. All figures and calculations use the gas data model from Schulreich et al.(2017) to provide densities and the ISM velocities. The accretion rate, the accretion radius and the number of collisions in the gas behind the star are all small when the relative velocity is high, as expected from Eq. (3), Eq. (4) and Eq. (6). The density has only a minor effect on the accretion rate, the accretion radius and the number of collisions. Interesting to note is that the amount of collisions (from Eq. (6)) is strongly peaked compared to the accretion rate and the accretion radius. The amount of collisions has significant effects on where accretion is allowed as we see multiple areas (16% total) where the accretion rate is above the minimum accretion rate (10−14 M ) for a λ Bootis spectrum to be established, yet only 6% has enough collisions (Eq. (6)) to allow BH-acc. In Fig. 15, the regions where this condition is satisfied are outlined in red. The accretion radius over the whole line averaged is 57 AU with a deviation of 17.5 AU. In the red outlined areas this becomes an average of 169 AU with standard deviation of 150 AU. This large std is a result of the peak at around 2500 pc which is much larger then the others. We observe something similar for the density where overall the average is 1.2 cm−3 with a standard deviation of 0.92 cm−3. In the red areas this becomes an average of 1.7 cm−3 with a standard deviation of 4.5 cm−3. The density in the accretion allowed area around 2500 pc is significantly lower compared to the other areas. This result would indicate that there would be negative densities, which is not physical.

3.2 Accretion areas and their sphere of influence In Fig. 16, we see the difference between the accreting areas and the possible location where a λ Bootis could be detected in a 2D plane of the 194 Myr data cube. This result is without any extra effects like bow shocks or added molecular clouds. All accretion rates below the minimum are filtered out, as are those where there are not enough collisions happening. In this case, V∗ is 4 km/s in both the X and Y direction as this value was also used for the 1D analysis. The detection area is around 4 times as large as the accretion area. However, half of the stars are moving into this area towards the cloud and do not have λ Bootis spectrum yet, while half is coming out of the cloud and does have a λ Bootis spectrum. The shapes of the matching areas are still very similar. This calculation took around 6 hours (on a dual core, 1 GHz processor), due to the size and resolution of the data cube. As a result, this evaluation was not possible to do for statistically significant number of V∗.

3.3 Accretion Rates There are two methods to present the number of stars in our simulation volume that would have a λ Bootis spectrum. One way is to make a map of the accretion rate (e.g. the left side of Fig. 16). This shows the morphology of accreting zones and spatial distribution of the accretion rates. This plot was made by taking a star with a predetermined V∗ and running it through the gas model to generate a map of where the accretion takes place for that V∗. This can be used to calculate and show how much accretion is taking place and where. By varying V∗, the effects on the accretion area and accretion amount can be studied. However, it is difficult to see directly from the map what the probability is of finding a λ Bootis star, which is a goal of this thesis. To solve this problem, we use a second approach. We run our calculation many times, each time a new random V∗ is used. These V∗ are taken from Binney(2010). The fraction of area where there is enough accretion for a λ Bootis spectrum to arise is calculated for each individual star. The probability that a star has a λ Bootis spectrum is the average Vpeculiar of all these random stars. The accretion threshold (if accretion is below this, accretion is not high enough to show a λ Bootis spectrum in the star) can be changed to investigate the effect on the probability of finding a λ Bootis star. In this way, we can investigate the probability to find a λ Bootis star at different accretion thresholds.

18 Figure 15: 1D evolution of the various parameters and quantities along the centre line of the central slice. At each point, the calculations were performed with a V∗ of 4 km/s the X direction. The ISM data is taken from the 194 Myr model. The red lines outline where there are 3 or more collisions, which allows BH-acc to take place. The blue line in the ﬁrst panel is set at the minimum accretion threshold −14 of 10 M found in Turcotte and Charbonneau(1993).

19 Figure 16: Left: Accreting zones for a single star with a velocity of 4 km/s in both the X and Y direction, for the 194 Myr slice. Right: The red regions represent the spatial areas where one could observe a star with a λ Bootis spectrum using the accreting zones from the left image and a timescale of 1 Myr for the pattern to disappear. In the blue regions, no λ Bootis spectra will be observed.

To get the complete picture, it is useful to take a look at both ways of analysing the results. The maps show how the accretion areas and rates are distributed spatially and a graph shows how many λ Bootis stars we can expect to ﬁnd. In the next sections, we present our results in both manners. We have chosen to use similar axes to illustrate the diﬀerences between the results.

3.3.1 Influence of molecular clouds on accretion rates and area This section compares the results of the base model to the results of the base model with added molecular clouds for two time steps of the gas model. The base calculation is the gas model unchanged together with BH-acc without bow shocks. A comparison is made between both time steps, 181 Myr and 194 Myr, of the gas data model to see if the age has an impact. In Figs. 17 and 18, we see the impact of V∗ on the accretion rate and area. The exact accretion probability and accretion rate can be found in Tables 3 and4, respectively. The highest V ∗ of 15 km/s was chosen such that the the accretion areas became visible in Figs. 17 and 18, any higher and the areas would be too small to see. Two extra velocities were chosen at 2/3 (intermediate) and 1/3 (lowest) of that value. The highest velocity therefore has almost no (less than 0.1%) accretion area. Interesting to note is that the presence of the molecular clouds did not increase the accretion area for the chosen velocities. The accretion probability did not change, but the maximum accretion rate is around a factor 51 higher in the calculations with molecular clouds (Table 4). This was expected because to add the molecular clouds, we increased the density by a factor 51.3 in certain areas and the accretion rate scales linearly with density (Eq. (3)). It can happen that the place with the highest accretion rate in the base does not get enhanced with molecular clouds because the temperature is too high. When comparing Fig.9 with Figs. 17 and 18, we observe that that there is an increased accretion rate in the locations where molecular clouds have been added. As the velocity decreases, the accretion area increases as expected from Eq. (3). In Table3, we see the filling factors of the accretion area. Stars moving at the velocity tabulated have the indicated probability of actively accreting enough material for a λ Bootis spectrum to form. Table4 presents the maximum accretion rates found. For the fastest star in the base slices, the maximum accretion rate is barely (factor −14 5) above the accretion threshold (10 M /yr). For the lowest V∗, the highest accretion rate increases −12 to around 10 M /yr. When comparing the base model with the model that includes molecular clouds, the maximum accretion rate is increased by roughly a factor 50 in every case. In general, lower stellar velocities give more accreting area and a higher accretion rate in those areas. The 181 Myr slice has the same order of magnitude values as the 194 Myr slice, but has up to 40% smaller values for the accretion area and rate. The difference between the two slices becomes larger as V∗ increases.

20 21

Figure 17: Accretion rates in the 194 Myr slice with a star moving at diﬀerent velocities. Left: 5 km/s in X and Y. Middle: 10 km/s in X and Y. Right: 15 km/s in X and Y. Top row is without molecular clouds. Bottom row is with molecular clouds. 22

Figure 18: Accretion rates in the 181 Myr slice with a star moving at diﬀerent velocities. Left: 5 km/s in X and Y. Middle: 10 km/s in X and Y. Right: 15 km/s in X and Y. Top row is without molecular clouds. Bottom row is with molecular clouds. Table 3: Accretion probability without bow shocks for diﬀerent combinations of V∗, gas model run time and inclusion of molecular clouds.

5 km/s in X and Y 10 km/s in X and Y 15 km/s in X and Y 181 Myr base 16.39 % 1.45 % 0.08 % 181 Myr mol. 16.39 % 1.45 % 0.08 % 194 Myr base 15.34 % 1.47 % 0.05 % 194 Myr mol. 15.34 % 1.47 % 0.05 %

Table 4: Accretion rate (M /yr) without bow shocks for diﬀerent combinations of V∗, gas model run time and inclusion of molecular clouds.

5 km/s in X and Y 10 km/s in X and Y 15 km/s in X and Y 181 Myr bow shock 1.27 ·10−12 1.59 ·10−13 4.70 ·10−14 181 Myr bow shock + mol. 6.50 ·10−11 8.13 ·10−12 2.41 ·10−12 194 Myr bow shock 9.83 ·10−13 1.22 ·10−13 3.64 ·10−14 194 Myr bow shock + mol 5.01 ·10−11 6.27 ·10−12 1.85 ·10−12

3.3.2 Influence of bow shocks on accretion rates and area The results of the increase in density and the lowering of the velocity due to bow shocks can be seen in Figs. 19 and 20 and Tables5 and6, and by comparing them to the results in the previous section. Even at the highest V∗ (same velocities used as before), the accretion area is more than 50% of the area in the simulation. As the velocity decreases, the accretion area increases in size. The highest accretion filling factor is 87.83% with bow shocks, compared to 16.39% without bow shocks for the set of V∗ that was used. Molecular clouds do not increase the accretion area, only the maximum accretion rate by a factor 50 again (Table6). The largest accretion rates are orders of magnitude larger, going as high as −8 −14 10 M /yr. The accretion rates in the model with the effects of bow shocks range from 10 M /yr to −8 10 M /yr. Figs. 19 and 20 show a smooth gradient in the accretion rate. The accretion rate gradually tapers off from the local maximum to the edge of the accretion area. There are no discontinuous jumps in the accretion rate. There is still a difference between both slices: the accretion area is about 5% smaller for the 194 Myr slice, while the highest accretion rate is 25% smaller. However, these differences are small compared to those found in the previous section.

Table 5: Accretion probability for the model with bow shocks for diﬀerent combinations of V∗, gas model run time and inclusion of molecular clouds.

5 km/s in X and Y 10 km/s in X and Y 15 km/s in X and Y 181 Myr bow shock 87.83 % 71.51 % 54.00 % 181 Myr bow shock + mol. 87.83 % 71.51 % 54.00 % 194 Myr bow shock 83.96 % 67.95 % 52.21 % 194 Myr bow shock + mol. 83.96 % 67.95 % 52.21 %

3.3.3 Inﬂuence of accretion threshold

−14 Up to now, the accretion rate threshold was set to 10 M /yr. If the accretion rate is lower, there will not be enough accretion to create a λ Bootis spectrum in the star. However, this threshold is the lowest −11 value found in literature. Turcotte(2002) calculated that rates up to 10 M /yr or higher would −14 −10 be needed. Based on this, we chose our accretion limits to range from 10 M /yr to 10 M /yr. It is worthwhile to investigate the eﬀects on the accretion probability when the accretion threshold is changed and to see how the probability compares with the observed probability of λ Bootis stars in the solar neighborhood of 2%. In Fig. 21, the eﬀect of the accretion threshold on the accretion probability is shown. Each data point was made by calculating the accretion area for 2000 randomly distributed X and Y velocities drawn from

23 24

Figure 19: Accretion rates in the 194 Myr slice with a star moving at diﬀerent velocities. Left: 5 km/s in X and Y. Middle: 10 km/s in X and Y Right: 15 km/s in X and Y. Top row is without molecular clouds. Bottom row is with molecular clouds. 25

Figure 20: Accretion rates in the 181 Myr slice with a star moving at diﬀerent velocities. Left: 5 km/s in X and Y. Middle: 10 km/s in X and Y Right: 15 km/s in X and Y. Top row is without molecular clouds. Bottom row is with molecular clouds. Table 6: Accretion rate (M /yr) with bow shocks for diﬀerent combinations of V∗, gas model run time and inclusion of molecular clouds.

5 km/s in X and Y 10 km/s in X and Y 15 km/s in X and Y 181 Myr bow shock 3.67 ·10−10 4.59 ·10−11 1.36 ·10−11 181 Myr bow shock + mol. 1.86 ·10−8 2.34 ·10−9 6.94 ·10−10 194 Myr bow shock 2.83 ·10−10 3.54 ·10−11 1.04 ·10−11 194 Myr bow shock + mol. 1.44 ·10−8 1.80 ·10−9 5.35 ·10−10

the distributions in Fig. 13 and averaging these accretion areas. This was done for both simulation run times (181 Myr and 194 Myr) and for the presence of molecular clouds and/or bow shocks. As expected, the accretion probability decreases with increasing accretion threshold. As we saw before when taking the effect of bow shocks into account, the accretion probability is 1.5 to 5 orders of magnitude larger compared to the base model or the base model with molecular clouds. Furthermore, the slope is less steep at the higher accretion thresholds. Bow shocks are essential in reproducing the 2% −11 of λ Bootis stars found in observations. My simulations require an accretion threshold of 10 M /yr to reproduce the 2%, which was the threshold calculated by Turcotte(2002). At higher thresholds, not all λ Bootis stars could be explained even with bow shocks. At lower thresholds, the probability is vastly higher than that observed. Calculations for the base model and the molecular clouds model show the same accretion probability −11 up to an accretion threshold of 10 M /yr or higher. At this threshold, the calculation with molecular clouds is about 0.01% higher, roughly the same as the filling factor used for molecular clouds. At an −10 accretion threshold of 10 M /yr, the difference between the models is 2 orders of magnitude, but this only corresponds to an accretion probability increase of 0.001% for the molecular clouds.

Figure 21: The probability of ﬁnding a star actively accreting gas as a function of the accretion threshold. A black dashed line is placed at 2% to represent the observed ration of λ Bootis stars.

26 4 Discussion

We went for a combination of the blunt body bow shock model and the theoretical values from the Rankine-Hugoniot equations. Using the blunt body bow shock model requires too many parameters that were not readily available, to give the most accurate results. What is the shape of the bow shock? What is the shape of the blunt body? Many different combinations of upstream velocities and Mach numbers. The lack of resolved λ Bootis stars and A type stars with bow shocks in general made this fitting process imprecise and increased the uncertainty of the results from the bow shock model. The shape of the object was matched to the heliosphere. This is an approximation, as the stars that we are considering are more massive and travel at different velocities through different ISM conditions. Furthermore, the problem considers a blunt body, while we are interested in what happens inside of the body close to the star. However, from the plots that were made for some different scenarios with typical parameters (velocities, densities, Mach numbers, etc.), we could use the values for the density and velocity of the gas downstream. The density was found to increase by a factor 4.5 in the shock stagnation point compared to the upstream density. With the mass conservation Rankine-Hugoniot condition (Eq. (1)), we were able to scale the velocity of the gas in the shock. λ Bootis stars can exist for 1 Myr outside of a place where they are actively accreting. In our results we mostly only took a look at the filling factor of accreting areas. In Fig. 16, we calculated a 50% difference between the area where λ Bootis stars could be observed and the accretion area. How far the star travels outside of an accreting area scales with V∗. This will change the ratio between the area were we can observe λ Bootis stars and where stars can accrete enough to form a λ Bootis spectrum. Due to this, we did not include this result in the rest of our calculations, as we only had a single data point for it and felt that we could not attribute any statistical significance to that one result. It is important to note that not all stars inside this area are λ Bootis stars, there are also the incoming stars that move towards the accretion area and as such still have a non peculiar surface. It was unfortunate that this calculation could not be investigated in depth due to the restraints on computational time, as it would be interesting to see how these results change with different velocities. Faster stars are able to travel farther from the accretion area than slower stars, this might result in a less negative correlation between the velocity of the star and the probability that it is a λ Bootis star. The effects of molecular clouds were smaller than expected. The filling factor for molecular clouds is only 0.1%. This means that their effect is noticed only in very particular places. Only in the densest and coldest places of the gas model would they exist. These places were already favourable to accretion due to their high density. At the lower accretion thresholds in Fig. 21, the effects of molecular clouds are not visible, while at the high thresholds they are. This is due to the extra density that makes the accretion rate high enough to overcome the threshold. What this means, is that the majority of λ Bootis stars originate in regions where atomic gas is dominant instead of being driven through accretion inside of molecular clouds. There is a small difference between the gas models: the accretion probability is up to 4% lower in the 194 Myr slice. The largest accretion rates are up to 25% smaller. This is most likely due to lower mass and maximum density found in that slice (Table.2). The effects are more pronounced at higher velocities or accretion thresholds where stars could only accrete enough to show a λ Bootis spectrum under the most favourable conditions. The higher density and maximum density in the 181 Myr slice (Table2) could be the deciding factor for why that slice has more λ Bootis when the accretion threshold is high. Overall, the same conclusions can be drawn from both slices, meaning that our conclusions are not due to one specific realization of the gas data used.

4.1 Comparison of model with observations According to observations, roughly 2% of the stars in the relevant spectral domain are λ Bootis stars in the galactic ﬁeld, as well as in open clusters (Paunzen, 2001). Our results give insight into the fraction of λ Bootis stars that could be due to ISM accretion. The percentage of observed λ Bootis stars that can be explained with our calculations is shown in Fig. 21. Without bow shocks, this fraction is sizeable (around 50%) at the lowest two accretion thresholds. At higher accretion thresholds, the amount of observed stars that can be explained rapidly drops from 10% to 0.01%. We can also see that molecular clouds have little impact on the viability of the ISM accretion theory. When the molecular clouds are taken into account only 1-2% of observed λ Bootis stars can be explained in addition to the base calculation.

27 −11 Turcotte(2002) found that an accretion rate of 10 M /yr would be needed to create a λ Bootis spectrum. In Fig. 21, we see that at that accretion rate the model with bow shocks could explain all of −10 the observed stars. At the highest threshold (10 M /yr) around 25% of stars can still be explained, proving it a robust channel that could work even for stars that have slightly larger mixing zones then −14 −12 used by Turcotte(2002). At accretion rates between 10 M /yr and 10 M /yr, there are too many λ Bootis stars in our calculations. One explanation for this is that these accretion threshold are too low to form λ Bootis stars in reality. Another reason could be that the effect of our bow shocks is too strong and they play a less important role in reality. By using the properties of observed stars, we can test our calculations. The star that Gáspár et al.(2008) investigated does not belong to the λ Bootis class, but it has a known relative velocity −3 (Vrel=35 km/s) and ISM density (3.5 cm ). What is even more important, is that it has a resolved bow shock. This makes it a good test case for the ISM accretion model. If we apply our scaling factor to this star’s density (4.5) and Vrel (0.25), it should not have enough accretion to show a λ Bootis spectrum. We have calculated the accretion for this star, taking the effects of the bow shock into account, and find −12 −11 an accretion rate of 1.7 · 10 M /yr (Eq. (3)). This is still below the accretion threshold of 10 M /yr. Our calculations for this case agree with the observed results.

28 5 Conclusion

In this thesis we set out to investigate how many of the observed λ Bootis stars could be explained with the ISM accretion/diffusion model. The presence and effect of a bow shock was introduced as a new possible element. This was done using Bondi-Hoyle accretion (Bondi and Hoyle, 1944) and a model of the local interstellar environment (Schulreich et al., 2017) to which we added molecular clouds. Without the presence of bow shocks, only a small fraction of λ Bootis stars can be explained with ISM accretion. When taking bow shocks into account, we find that our results match observational data well as both give a frequency of around 2% of the stars. The added molecular clouds did not change the amount of λ Bootis stars that can be explained by more than 0.1%, while the adopted accretion threshold did have an effect of orders of magnitude. For many stars, though, the angular momentum could not be reduced behind the star, thus preventing the star from accreting through Bondi-Hoyle accretion. In conclusion, we find that ISM accretion is one viable avenue for the emergence of a λ Bootis spectrum in a star and can even explain the observed occurrence rate among A-type stars. For future works, it would be interesting to see results a complete magneto-hydrodynamic simulation of a bow shock together with Bondi-Hoyle accretion or with accretion modelled directly into that simulation. In this work, we only worked with the central slice of the gas data model in a 2D environment. Exploring the full 3D space of the model could provide additional insight. On the observational side, a lot of progress could be made: if more λ Bootis stars can be observed with resolved bow shocks then that information can be used to enhance the accuracy of our results, in particular for the bow shock model. In our analysis we did not look at the effect of variations in the dust fraction or metallicity in the gas, which varies from location to location in reality. That information together with a model of the stellar interior could be used to calculate the abundances of specific elements within the star, thus providing insight into the different abundance patters that have been observed in λ Bootis stars. After many years since the discovery of the first λ Bootis star we come ever closer to solving their mystery. We hope that this thesis sheds some light on how they get their spectrum.

29 Acknowledgements

I would like to thank my ﬁrst supervisor, Inga Kamp, for coming up with an interesting idea for a thesis that could be done in collaboration with the TU Berlin and for her feedback which helped me present a quality thesis. My thanks also go out to my second supervisor, Dieter Breitschwerdt, for giving me a place at a new university and. Both of them have my thanks for all the eﬀort that went into scheduling Skype meetings. I am thankful for Michael Schulreich and his help and assistance in setting up the gas data model. And lastly, special thanks go out to my wife Sietske Bouma for all the tea, support and proofreading she did to help me write this thesis.

30 References

Andrievsky, S. M. (1997). On the possible origin of λ Boo stars. A&A, 321:838–840.

Binney, J. (2010). Distribution functions for the Milky Way. MNRAS, 401:2318–2330. Bondi, H. and Hoyle, F. (1944). On the mechanism of accretion by stars. MNRAS, 104:273. Charbonneau, P. (1993). Particle transport and the Lambda Bootis phenomenon. I - The diﬀusion/mass- loss model revisited. ApJ, 405:720–726.

Edgar, R. (2004). A review of Bondi-Hoyle-Lyttleton accretion. NewAR, 48:843–859. Faraggiana, R. and Bonifacio, P. (1999). How many lambda Bootis stars are binaries? A&A, 349:521–531. Faraggiana, R. and Bonifacio, P. (2005). C, N, O in λ Boo stars and in composite spectra. A&A, 436:697–700.

Ferrière, K. (1998). Global Model of the Interstellar Medium in Our Galaxy with New Constraints on the Hot Gas Component. ApJ, 497:759–776. Folsom, C. P., Bagnulo, S., Wade, G. A., Alecian, E., Landstreet, J. D., Marsden, S. C., and Waite, I. A. (2012). Chemical abundances of magnetic and non-magnetic Herbig Ae/Be stars. MNRAS, 422:2072–2101. Gáspár, A., Su, K. Y. L., Rieke, G. H., Balog, Z., Kamp, I., Martínez-Galarza, J. R., and Stapelfeldt, K. (2008). Modeling the Infrared Bow Shock at δ Velorum: Implications for Studies of Debris Disks and λ Boötis Stars. ApJ, 672:974–983. Gray, R. O. (1988). The spectral classiﬁcation of the Lambda Bootis stars. AJ, 95:220–228.

Griﬃn, R. E. M. (2008). The A-type stars: Normal or ... what? Contributions of the Astronomical Observatory Skalnate Pleso, 38:93–102. Guzik, J. A., Kaye, A. B., Bradley, P. A., Cox, A. N., and Neuforge, C. (2000). Driving the Gravity-Mode Pulsations in γ Doradus Variables. ApJL, 542:L57–L60.

Iliev, I. K., Paunzen, E., Barzova, I. S., Griﬃn, R. F., Kamp, I., Claret, A., and Koen, C. (2002). First orbital elements for the lambda Bootis spectroscopic binary systems HD 84948 and HD 171948. Implications for the origin of the lambda Bootis stars. A&A, 381:914–922. Jura, M. (2015). Lambda Boo Abundance Patterns: Accretion from Orbiting Sources. AJ, 150:166.

Kamp, I. and Paunzen, E. (2002). The λ Bootis phenomenon: interaction between a star and a diﬀuse interstellar cloud. MNRAS, 335:L45–L49. Mathys, G. (2004). Rotation and Properties of A- and B-Type Chemically Peculiar Stars (Invited Review). In Maeder, A. and Eenens, P., editors, Stellar Rotation, volume 215 of IAU Symposium, page 270.

Michaud, G. and Charland, Y. (1986). Mass loss in A and F stars - The Lambda Bootis stars. ApJ, 311:326–334. Morgan, W. W., Keenan, P. C., and Kellman, E. (1943). An atlas of stellar spectra, with an outline of spectral classiﬁcation.

Murphy, S. J. and Paunzen, E. (2017). Gaia’s view of the λ Boo star puzzle. MNRAS, 466:546–555. Paunzen, E. (2001). A spectroscopic survey for lambda Bootis stars. III. Final results. A&A, 373:633–640. Paunzen, E. (2004). The λ Bootis stars. In Zverko, J., Ziznovsky, J., Adelman, S. J., and Weiss, W. W., editors, The A-Star Puzzle, volume 224 of IAU Symposium, pages 443–450.

31 Paunzen, E. and Heiter, U. (2014). A Spectral Atlas of lambda Bootis Stars. Serbian Astronomical Journal, 188:75–84. Paunzen, E., Iliev, I. K., Kamp, I., and Barzova, I. S. (2002). The status of Galactic ﬁeld λ Bootis stars in the post-Hipparcos era. MNRAS, 336:1030–1042.

Schneider, W. (1968). A uniformly valid solution for the hypersonic ﬂow past blunted bodies. Journal of Fluid Mechanics, 31(2):397–415. Schulreich, M. M. and Breitschwerdt, D. (2011). Astrophysical bow shocks: an analytical solution for the hypersonic blunt body problem in the intergalactic medium. A&A, 531:A13.

Schulreich, M. M. and Breitschwerdt, D. (2016). Assessing the link between recent supernovae near Earth and the iron-60 anomaly in a deep-sea crust. In Supernova Remnants: An Odyssey in Space after Stellar Death, page 113. Schulreich, M. M., Breitschwerdt, D., Feige, J., and Dettbarn, C. (2017). Numerical studies on the link between radioisotopic signatures on Earth and the formation of the Local Bubble. I. 60Fe transport to the solar system by turbulent mixing of ejecta from nearby supernovae into a locally homogeneous interstellar medium. A&A, 604:A81. Smith, K. C. (1996). Chemically Peculiar Hot Stars. Ap&SS, 237:77–105. Sparke, L. and Gallagher, J. (2007). Galaxies in the Universe: An Introduction. Cambridge University Press. Stütz, C. and Paunzen, E. (2006). On the λ Bootis spectroscopic binary hypothesis. A&A, 458:L17–L20. Teyssier, R. (2002). Cosmological hydrodynamics with adaptive mesh reﬁnement. A new high resolution code called RAMSES. A&A, 385:337–364.

Turcotte, S. (2002). Mixing and Accretion in λ Bootis Stars. ApJL, 573:L129–L132. Turcotte, S. (2005). Using the Seismology of Non-magnetic Chemically Peculiar Stars as a Probe of Dynamical Processes in Stellar Interiors. Journal of Astrophysics and Astronomy, 26:231. Turcotte, S. and Charbonneau, P. (1993). Particle transport and the Lambda Bootis phenomenon. II - an accretion/diﬀusion model. ApJ, 413:376–389.

Venn, K. A. and Lambert, D. L. (1990). The chemical composition of three Lambda Bootis stars. ApJ, 363:234–244. Zorec, J. and Royer, F. (2012). Rotational velocities of A-type stars. IV. Evolution of rotational velocities. A&A, 537:A120.