Numerical Modelling of Astrospheres

Numerical modelling of Astrospheres

J Light orcid.org 0000-0002-9788-5540

Dissertation accepted in partial fulfilment of the requirements for the degree Master of Science in Astrophysical Sciences at the North-West University

Supervisor: Prof SES Ferreira Co-supervisor: Dr K Scherer

Graduation May 2020 24330310

Abstract

A stellar wind creates a cavity in the interstellar medium, which is known as an astrophere. In this work, radiative cooling is included in a standard 2D magneto-hydrodynamic numerical model to simulate the evolution of an astrosphere. First, a parameter study is done to determine the effects that an interstellar medium magnetic field, the mass loss rate, the outflow velocity and the interstellar medium density have on astrospheric evolution. It is found, depending of course on the model parameters, that the cooling creates a thin dense shell surrounding the astrosphere during the early stages of the astrospheric evolution. The cooling also increases the compression ratio of the bow shock, while the added interstellar medium magnetic field reduces the compression. The astrospheres surrounding luminous blue variable stars HD 99953 and AG Carinae are also computed. These stars show variability in their outflow parameters at the inner boundary. It is found that the variations in the outflow parameters have no effect on the astropause or the bow shock of the astrosphere. For AG Carinae, the variations in the mass-loss rate and outflow velocity at the inner boundary are larger compared to HD 99953. These larger variations of AG Carinae also have no influence beyond the termination shock as with HD 99953. Luminous blue variable stars also undergo eruptive events with large increases in mass loss over a short period of time. These eruptions only slightly influence the bow shock. Eruptions that increase the outflow velocity and mass loss create an astrosphere that has a larger size and during the early stages of evolution the eruptions cause the termination shock to oscillate between the astropause and the inner boundary. These eruptions also increase the distance to the astropause slightly; however, at later stages, there are no visible signs of the influence of these eruptions. Keywords: Astrosphere, magneto-hydrodynamic, luminous blue variables, radiative cooling, interstellar medium, stellar wind, stellar cycle Acknowledgements

I would like to acknowledge and thank the following people for supporting me throughout my Master’s degree:

Firstly, I would like to thank the National Research Foundation and the National As- trophysics and Space Science Programme for providing ﬁnancial support.

Next, I want to thank my supervisor, Prof. S.E.S. Ferreira, and my co-supervisor, Dr. K. Scherer.

I also want to thank the staﬀ at the Centre for Space Research at the North-West Uni- versity. In particular, Mrs. Petro Sieberhagen and Lendl Fransman for administrative support and Matthew Hollerman for IT support. As well as Cecile van Zyl for the language editing.

Finally, I would like to thank my family and friends and a special thank you to my wife, Stacy, for always supporting me throughout this journey. Contents

1 Introduction1

2 Astrospheres3

1 Astrospheric structure...... 3

2 A numerical model to simulate astrospheric expansion...... 4

2.1 Radiative cooling...... 7

2.2 Interstellar magnetic ﬁeld...... 11

3 Modelling of astrospheres 13

1 Eﬀect of ISM magnetic ﬁeld on the astrosphere structure...... 14

2 Eﬀect of mass loss rate on astrospheric evolution...... 17

3 Eﬀect of outﬂow velocity of a stellar wind on astrosphere evolution.... 22

4 Eﬀect of ISM density on astrosphere...... 26

5 Summary and conclusion...... 27

4 Luminous Blue Variable: HD 99953 30

1 Introduction...... 30 CONTENTS

2 Modelling of HD 99953...... 31

3 HD 99953 evolution without periodic outﬂow...... 33

3.1 HD 99953 simulations with no radiative cooling or ISM magnetic ﬁeld...... 33

3.2 Eﬀect of radiative cooling on the evolution of HD 99953...... 36

3.3 HD 99953 evolution with the eﬀect of radiative cooling and ISM magnetic ﬁeld on astrosphere evolution...... 38

4 HD 99953 evolution with periodic outﬂow...... 42

4.1 HD 99953 simulated with no radiative cooling or ISM magnetic ﬁeld but periodic variation in other parameters...... 42

4.2 HD 99953 simulations including the eﬀect of radiative cooling and ISM magnetic ﬁeld on astrospheric evolution...... 45

5 Summary and conclusions...... 48

5 Luminous Blue Variable: AG Carinae 50

1 Introduction...... 50

2 Modelling of AG Carinae...... 51

3 AG Carinae evolution without periodic outﬂow...... 52

4 AG Carinae evolution with periodic outﬂow...... 55

5 Summary and conclusions...... 59

6 Eruptions 62

1 Evolution of an astrosphere with an eruption that increases the mass loss rate...... 63

2 Astrospheric evolution with an eruption that increases mass loss rate and velocity...... 67

3 Summary and conclusion...... 70

4 Future work and improvements...... 71 CHAPTER 1

Introduction

The cavities created around stars due their stellar winds interacting with the surrounding interstellar medium can be described as in Parker(1961). A review of this interaction between the wind and the surrounding medium can be found in Holzer & Axford(1970). These cavities created around stars by their stellar winds are called astrospheres. For the Sun, this is called the heliosphere and the evolution of the heliosphere has been modelled by e.g. Scherer & Ferreira(2005) and Ferreira et al.(2012) using a similar model as will be utilised here. See also work by Pogorelov et al.(2009), Opher et al. (2011) and Decin et al.(2012).

Different compared to other numerical simulations of astrospheres, is that in this work, variability will be introduced in the mass loss rate and the outflow velocity to study the astrospheres around luminous blue variable type stars. Also different from traditional heliospheric models is that for astrospheric evolution, radiative cooling must also be introduced. The radiative cooling used in this work is described in Dalgarno & McCray (1972) and Dyson & Williams(1997). The cooling is added in the numerical model used by Fahr et al.(2000) and Scherer & Ferreira(2005) through the cooling function from Mellema & Lundqvist(2002). An interstellar medium (ISM) magnetic field is also included in the model resulting in similar effects as discussed by van Marle et al.(2015b).

Chapter2 will serve as an introduction, and Chapter3 of this work will show the eﬀects that some of the parameters may have on the overall structure of an astrosphere. The parameters that will be considered in this work are the strength of the ISM magnetic

1 2

field, the mass loss rate, outflow velocity of the stellar wind and the density of the ISM. The effects that these have on the evolution of the astrospheres will be shown and in particular show that radiative cooling might influence astrospheric evolution. The effect that the ISM magnetic field may have on astrospheric evolution will also be discussed.

In Chapters4 and5, simulations will be done to calculate astrospheres around luminous blue variable (LBV) type stars. These types of stars show variable behaviour in their luminosities, temperatures and mass loss rates (e.g. Puls et al., 2008; Vink, 2012). It will be shown whether the variation of the outflow parameters has any effect on the evolution of the astrospheres. In particular, this work will consider two stars, namely HD 99953 (in Chapter4) and AG Carinae (in Chapter5). These stars show variability in their mass loss rates and outflow velocities (Haucke et al., 2018; Stahl et al., 2001). For HD −8 −1 −7 −1 99953 the mass-loss varies between 8 × 10 M .year and 2.2 × 10 M .year with −5 −1 a period of several days, and for AG Carinae it varies between 3.16 × 10 M .year −4 −1 and 1.58 × 10 M .year for a 10-year period.

In Chapter6, the effect that the eruption phase of a luminous blue variable may have on the astrosphere surrounding a star will be modelled. These stars undergo variations in luminosities in periods of years to decades, but can also undergo a sudden ejection of a large amount of matter over very short periods of time, namely giant eruptions (Sterken, 2003; Smith, 2007). This work will consider a luminous blue variable type star in a quiescent phase with a constant mass loss rate and outflow velocity. After 100 000 years, a sudden increase in mass loss rate and outflow velocity will be assumed, followed by the star returning to the quiescent phase. The effect of this eruption will be shown on the evolution of the astrosphere. CHAPTER 2

Astrospheres

1 Astrospheric structure

The large-scale structures of astrospheres have three distinct features, these are the termination shock (TS), the astropause (AP) and the bow shock (BS). See Scherer et al. (2015) for a complete discussion. The region between the TS and the AP is called the inner astrosheath (IAS), and the area between the AP and the BS is called the outer astrosheath (OAS). The TS is created where the hyper- or supersonic stellar wind is decelerated to subsonic velocities. The TS converts the kinetic energy of the matter into thermal energy, increasing the temperature of the IAS after the termination shock, e.g. Richardson et al.(2008). The IAS is an area where the ﬂow is mostly incompressible and subsonic. For the Sun, the eﬀective temperatures are ≈106 K(Zank et al., 2009). The AP is a tangential discontinuity between the stellar wind and the ISM, there is no mass transport across the AP and the velocity normal to the AP vanishes (Scherer et al., 2015). On either side of the AP, the thermal pressure from the stellar wind and the thermal pressure from the ISM are equal. The BS exists due to the expansion of the astrosphere into the ISM. The bow shock can also be created due to the relative motion between the ISM and the star, assuming the relative motion is supersonic. It is irrelevant whether the star or the ISM is moving supersonically; as long as the relative motion between the ISM and the star is supersonic, a bow shock will be created for a pure hydrodynamic case. (see Scherer et al., 2016).

3 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 4

2 A numerical model to simulate astrospheric expansion

In this work, the evolution of an astrosphere is simulated using the model of Fahr et al. (2000), Scherer & Ferreira(2005) and Ferreira & de Jager(2008). I added a periodic ﬂuctuation in stellar wind density and outﬂow velocity at the inner boundary to the existing model. This is discussed further in chapters4 and5. In these chapters the results of simulations of LBV-type stars will be shown.

Simulations and discussions on the evolution of astrospheres have been performed, e.g. by Arthur(2012); van Marle et al.(2015a); Mackey et al.(2016); Meyer et al.(2017). These groups all use different numerical schemes to solve the magneto-hydrodynamic equations. Arthur(2012) used a different numerical code to simulate the expasion of an astrosphere in an HII which is in turn expanding into the ambient medium. This is done to determine the X-ray emmision from the shocked stellar wind. van Marle et al.(2015a) show numerical simulations of bow shocks for giant stars moving supersonically through the ISM and the effects that an ISM magnetic field and the presence of interstellar dust grains may have on these bow shocks. Mackey et al.(2016) simulated the red supergiant W26 to observe whether the stellar wind is confined by photoionization. Meyer et al. (2017) discusses the effect that an interstellar magnetic field of 3.5 − 7 µG has on the optical emmision of bow shocks. This work differs in the sense that the parameters used for the simulations are very similar but this work focuses on the effect that the periodic variation of the mass loss rate and stellar wind density as found with LBV-type stars, at the inner boundary, would have on the shape and size of the astrosphere. Finally the eruptive events of the LBV-type stars with their significant increase in mass loss over a short period of time will be simulated to show how these events change the shape and size of the astrospheres created by these stars.

This models solve the well-known magneto-hydrodynamic (MHD) equations to simulate astrospheric evolution:

∂ρ + ∇ · (ρv) = 0 ∂t ∂(ρv) 1 + ∇ · (ρv⊗v + P ∗I − B ⊗ B) = 0 ∂t 4π ∂e 1 + ∇ · [(e + P ∗)v − B(B · v)] = −n n Λ(T ) (2.1) ∂t 4π e H ∂B + ∇×(B×v) = 0 ∂t ∇·B = 0 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 5 with ρ the density, v the velocity, B the magnetic ﬁeld, P ∗ = P + B2/8π describing the pressure, I is the unit matrix and ⊗ is the dyadic product. The total energy is given by

ρ|v2| P B2 e = + + , (2.2) 2 γ − 1 8π

where γ is the adiabatic index and in this work only a one-ﬂuid case is considered with γ = 5/3. These equations describe the conservation of mass, momentum and energy, respectively, along with the induction equation and the requirement that the magnetic ﬁeld must be divergence-less. The pressure term P ∗ describes the total pressure of the ISM, which is 2 made up of P the gas pressure and B /8π the magnetic pressure. The term −nenH Λ(T )

accounts for the radiative cooling, with ne the electron density and nH the proton density. The cooling eﬃciency is given by Λ(T ) and is discussed in the following section.

Figure 2.1 shows the computed density profile of an astrosphere after 200 000 years simulation time. The mass loss rate, which is the amount of mass that is ejected from −6 −1 the star, is assumed to be 3.27 × 10 M .year at the inner boundary of the astrosphere, which is set as 0.03 pc. The inner boundary of the astrosphere is taken to be larger than the radius of the star. In this scenario, the outflow velocity, with which the matter flows from the inner boundary of the astrosphere into the ISM, is taken to be 1000 km.s−1. These parameters correspond to the parameters used by, Arthur (e.g. 2012); van Marle et al. (e.g. 2015a); Mackey et al. (e.g. 2016); Meyer et al. (e.g. 2017). The ISM density used here of 1 particles.cm−3 is the density between the assumed ISM density of 0.57 particles.cm−3 used in Meyer et al.(2017) and the assumed ISM density of 1.89 particles.cm−3 used in van Marle et al.(2015a). This astrosphere is a pure hydrodynamic case where there is no magnetic field included by setting B = 0 in equations 2.1. There is also no radiative cooling included. The ram pressure from the stellar wind 1 2 given by Pram = 2 ρv balances with the ISM pressure as the stellar wind expands into the ISM leading to the formation of the TS and AP. The density increases at the TS and the outflow velocity decreases from a supersonic to a subsonic flow. After the TS, there is an increase in temperature due to the kinetic energy of the stellar wind being converted into thermal energy. A large increase in density is found at the AP, which is the contact discontinuity between the stellar wind and the ISM. The BS forms due to the relative motion between the stellar wind and the ISM and shocks the ISM resulting in heating of the OAS.

The density compression ratio of a shock is calculated by taking the ratio of the densities either side of a shock. This ratio is calculated in terms of the Mach number. If the 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 6

Astrophere after 200 000 years 101 Number density 1000 BS )

3 800 100 OAS m c . s ) e 1 l c i s

t 600 . r AP a m p k

( 1 (

y y t t i i s c n 400 o l e e d

V r e b 2

m 10

u IAS 200 N TS

3 0 10 Velocity

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 2.1: Computed astrosphere corresponding to a mass loss rate of 3.27 × −6 −1 −1 10 M .year and an outﬂow velocity of 1000 km.s . The positions of the TS, AP and BS are shown. compression ratio has a value of 1, then the shock would disappear. For a hydrodynamic case without cooling, the maximum compression ratio for a monatomic gas, with an adiabatic index of 5/3, is 4 (Choudhuri, 1998). According to Shu(1992), the addition of cooling allows for arbitrarily high compression ratio values and very strong density compressions is a distinguishing feature of radiative shocks compared to non-radiative shocks. The compression ratio of the BS in a pure hydrodynamical case with no cooling is found to be close to 4 and is shown in Figure 2.2. The solid red line shows the compression ratio of the BS for the computed astrosphere shown in Figure 2.1 where the ISM density is assumed to be 1 particles.cm−3 in the model. For a denser ISM, with a density of 10 particles.cm−3, the compression ratio is shown by the solid blue line. The higher ISM density decreases the compression ratio. Decreasing the ISM density increases the compression ratio, which is shown by the solid black line, where an ISM 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 7

Compression ratio vs time 5

2 Compression Ratio

ISM density of 0.5 particles. cm 3 ISM density of 1 particles. cm 3 ISM density of 10 particles. cm 3 0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 2.2: The compression ratio of the BS for the astrosphere shown in Figure 2.1 shown by the red line. The compression ratios for an astrosphere with the same stellar parameters but with an ISM density of 0.5 particles.cm−3 and 10 particles.cm−3 are shown by the black and blue line, respectively. density of 0.5 particles.cm−3 is assumed in the model. In the next section, it will be shown that radiative cooling inﬂuences the compression.

2.1 Radiative cooling

Radiative cooling is important for astrospheric simulations, as discussed in van Marle &

Keppens(2011). The cooling is added in the model by the additional term, −nenH Λ(T ), in the conservation of energy equation. The cooling function used in this work is taken from Mellema & Lundqvist(2002). The radiative cooling process is explained in Dal- garno & McCray(1972), and a short summary of the process follows hereafter. A 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 8 heated, partially ionised gas is able to cool through the emission of radiation. The thermal kinetic energy of the gas is converted into radiation by collisional processes. The collisional processes excite an atom, ion or molecule. This excited atom, ion or molecule will radiate away the gained energy through a photon. If the surrounding environment is optically thin, the photon is able to escape the environment removing the gained energy from the environment and reducing the total thermal kinetic energy of the environment. The process is described by Dyson & Williams(1997) as

A + B→A + B∗ where the collision of A with B results in B being in an excited state. After some time, this results in B∗→B + hv where hv is the photon being radiated away from the collision and possibly out of the environment.

The inclusion of radiative cooling in a model can have two additional effects on the simulation of astrospheres. As stated in van Marle & Keppens(2011), the first effect is a reduction in the energy, due to the loss of energy through radiation. The reduction in energy can lead to a change in the morphology of the gas, as high thermal pressure areas disappear. The second effect is the formation of radiative cooling instabilities due to either the temperature or density dependence of the cooling. If the instability was created due to density dependant cooling, the high density areas lose more energy than their surroundings. The loss of energy leaves the high density area with a lower temperature leading to a lower local pressure. A lower local pressure leads to these areas being compressed and increasing the density. This further increases the cooling rate and the process gets repeated.

Figure 2.3 shows an astrosphere at six different times during its evolution in a pure hydrodynamical case, now including the effects of radiative cooling. In the first panel, the number density is shown as a function of distance. The second panel shows the temperature as a function of distance from the inner boundary. When compared to Figure 2.1 this figure shows how radiative cooling affects the evolution of an astrosphere. After 15 000 years, the solid blue line, the astrosphere is in the process of starting to be cooled; as such, the OAS is being compressed as the energy of the shocked ISM is being radiated away. The orange dashed line shows the astrosphere after 25 000 years. The astrosphere at this stage is also shown in figure 2.4 compared to the same astrosphere without cooling added in the model. The cooling creates a sharp peak at approximately 3.6 pc, where the number density drastically increases from 10 to 4 × 101. The large increase in density of particles (per cm3) is said to form a shell surrounding the astrosphere. The second panel 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 9 )

3 Astrosphere with cooling at different times m

c 1 . 10 s e l c i t 100 r a p ( 1 y 10 t i 15 000 years s

n 25 000 years e 10 2 50 000 years d 100 000 years r

e 200 000 years b 10 3 300 000 years m u

N 107 )

K 6

( 10

e r 5 u 10 t a r

e 104 p m

e 103 T

102

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 2.3: Computed density profiles of an astrosphere at different times showing how radiative cooling affects the outer structure of the astrosphere when this is included in the model. shows the cooling of the OAS as the temperature of the thin thick shell is now greatly reduced in this region. At 50 000 years, green dotted line, there is an increase in the thickness of the OAS compared to 25 000 years and the compression ratio also decreases. The temperature after 50 000 years shows the increase in the thickness of the OAS. As time moves on, the astrosphere has already been cooled and the outer structure begins to decompress. This becomes more apparent after 100 000 years, red dashed-dotted line, where a fine structure forms in the OAS. Shown by the green line and to a lesser effect the red line, there seems to be now a hot outer astrosheath (HOAS) and a cold outer astrosheath (COAS), with the HOAS being near the BS and the COAS, the cooled remnant of the thin shell. The COAS can be seen in the temperature profile as the sudden decrease in temperature. The compression ratio has also decreased dramatically from 50 000 to 100 000 years. The compression ratio due to radiative cooling becomes 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 10 less apparent at later stages in the evolution, as seen in Figure 2.3, after 200 000 years, shown by brown dashed-dotted line, where the profile now looks very similar to the pure hydrodynamic case. At this stage and for these parameters, the OAS resembles that of an astrosphere without cooling. After 200 000 years, purple dashed-dotted line, there are no visible COAS or HOAS, only an OAS that is quite similar to the one in Figure 2.1. A direct comparison between a cooling and a no-cooling scenario can be seen in Figure 2.4, which shows the evolution after 25 000 and 300 000 years, respectively.

Cooling vs no cooling

101 ) 3 m c

. 0 s 10 e l c i t r a p (

y 1 t

i 10 s n e d

r e b 2 m 10 u N

25 000 years with no cooling 25 000 years with cooling 10 3 300 000 years with no cooling 300 000 years with cooling

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 2.4: The computed density profiles which shows the difference between an astrosphere with radiative cooling and without at two different stages of its evolution.

Figure 2.4 shows two different astrospheres corresponding to solutions with and without radiative cooling, respectively, and in a pure hydrodynamical case at an early stage of evolution and a later stage. Times of 25 000 and 300 000 years were chosen. The solid lines show the astrosphere with cooling and the dotted lines without cooling. At 25 000 years, the astrosphere with radiative cooling assumed in the model has a thin and dense outer shell showing the full effect of cooling, while at 300 000 years, the astrosphere 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 11 has already cooled and enough time has passed for the outer structure to be determined only by the pressures of the stellar wind and the ISM. The astrosphere without cooling at 25 000 years and 300 000 years is similar to Figure 2.1. Also from Figure 2.4 follows that the cooling has affected the TS distance, but the TS compression ratio remains unchanged. The change in the TS distance is due to the lower thermal pressure of the IAS and OAS, allowing the stellar wind to be able to push further into the ISM. At this stage, the OAS is reduced to a thin structure due to the cooling. The decrease in the thermal pressure leads to compression and the density has to increase. Once the astrosphere has cooled, it is unable to continue cooling and the increase in temperature requires a decrease in density. This creates an OAS that is similar, but not necessarily the same as for the case without cooling. The cooling still results in an astrosphere that has a smaller size than one without cooling, even after the OAS has decompressed over time. As shown in Figures 2.3 and 2.4, cooling has a significant effect on the outer shell’s density and compression ratio, n.

Compression ratio vs time 45 ISM density of 0.5 particles. cm 3 ISM density of 1 particles. cm 3 3 40 ISM density of 10 particles. cm

Compression Ratio 15

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 2.5: The compression ratio as a function of time for the pure hydrodynamic astrosphere including cooling shown in Figure 2.3.

Figure 2.5 shows the compression ratio, n, as a function of time for a pure hydrodynamic case including cooling. This ﬁgure shows the compression ratio, n, of the outer shell for three diﬀerent ISM densities assumed in the model. The black line shows the n with an ISM density of 0.5 particles.cm−3, the red line shows n for an ISM density 2. A NUMERICAL MODEL TO SIMULATE ASTROSPHERIC EXPANSION 12 of 1 particles.cm−3 and the blue line shows n for an ISM density of 10 particles.cm−3 assumed in the model. As already mentioned, the addition of cooling leads to the creation of the thin outer shell, which increases the compression ratio drastically and exists until cooling is no longer important. Shown in Figure 2.5 is that the compression ratio is greatly dependent on the ISM density. Comparing the scenario where the ISM has a density of 1 particles.cm−3, the red line, to the scenario where the ISM has a density of 10 particles.cm−3, the blue line, the ISM density is increased by a factor of 10 and the BS compression ratio decreases by a factor of ∼2.5. When the ISM density is decreased to 0.5 particles.cm−3, shown by the black line, the compression ratio of the BS is increased from ∼35 to ∼44, indicating the sensitivity to this parameter.

2.2 Interstellar magnetic ﬁeld

When the stellar wind of a star expands into the ISM, its evolution is influenced by the pressures from the ISM, namely the thermal pressure, if the ISM is hot (8000 - 10 000 K), and the ram pressure created by the expansion of the wind into the ISM. If the ISM has a magnetic field, a third pressure is added, namely the magnetic field pressure of the ISM (van Marle et al., 2015b; Scherer et al., 2016). The effect that an ISM magnetic field has on the evolution of an astrosphere is shown by van Marle et al.(2015b). As the wind expands, it initially pushes the magnetic field lines, which are embedded in the ISM, outwards. These magnetic field lines are pressed together in the shocked outer shell and this increases the magnetic pressure. The increased magnetic pressure is not enough to stop the expansion of the stellar wind, but significantly reduces the compression ratio. The effect of the ISM magnetic field pressure is now shown in the next chapter. CHAPTER 3

Modelling of astrospheres

This chapter builds on the previous chapter and considers some of the parameters that may have an effect on the overall structure of an astrosphere. These include ISM magnetic field pressure, mass loss rate, outflow velocity and ISM density. The first section in this chapter introduces various ISM magnetic field strengths in the model to determine their effect on the evolution of the astrosphere. It was shown in the previous chapter that for the pure hydrodynamic case, radiative cooling increases the compression ratio of the BS at the early stages of evolution, but at later stages there is very little evidence apart from a decrease in size and a slight increase in density at the AP. Radiative cooling is especially important at the BS, where it increases the compression ratio of the BS to values larger than the expected 4. The ISM magnetic field is added to show the combined effect of an ISM magnetic field pressure and cooling on the astrospheric evolution.

The following sections thereafter show how other parameters, namely the mass loss rate, outflow velocity and ISM density influence the evolution of an astrosphere. For interest, results with no-cooling and cooling but no ISM magnetic field as well as the combined effect, cooling and the ISM magnetic field on the evolution of the astrosphere are discussed for completeness.

13 1. EFFECT OF ISM MAGNETIC FIELD ON THE ASTROSPHERE STRUCTURE14

1 Eﬀect of ISM magnetic ﬁeld on the astrosphere structure

As discussed in Chapter2 and by van Marle et al.(2015b), the ISM magnetic field may influence the outer structure of the astrosphere as it increases the total pressure of the ISM. This can be seen from the third equation in equation 2.1, which describes the conservation of energy. The pressure term is the combination of the gas pressure of the ISM and the magnetic pressure from the ISM magnetic field. The inclusion of an ISM magnetic field will therefore increase the total ISM pressure that the stellar wind has to push back. This section compares astrospheres in a pure hydrodynamical case, as in Chapter2, with astrospheres in a magnetohydrodynamic case with three different ISM magnetic field strengths. This will be done with and without radiative cooling to show how the addition of the ISM magnetic field may change the effect of the cooling and how the astrosphere evolves when both effects from the magnetic field and cooling are present. Very important to note is that due to constraints of the model, the ISM magnetic field in this model will be azimuthal everywhere, therefore reducing the model to 1D, but showing the maximum effect of that an ISM magnetic field may have on astrospheric evolution.

The effect of the added pressure from the ISM magnetic field can be seen in Fig- ure 3.1 showing different scenarios corresponding to different field strengths. These four scenarios have an ISM density of 1 particles.cm−3 and a stellar wind density of ∼10 particles.cm−3 at an inner boundary of 0.03 pc corresponding to a mass loss rate −6 −1 of 3.27 × 10 M .year . Results are shown after 200 000 years. The blue dotted line shows a pure hydrodynamical astrosphere where the magnetic field strength is set to zero, B = 0, in equations 2.1. Note that radiative cooling is not included in the model here. In the pure hydrodynamical case, the TS is located at ∼ 3.75 pc, with the AP at ∼ 12 pc and the BS at ∼ 13.5 pc with a compression ratio of ∼ 4. The solid orange line has an ISM magnetic field with a strength of 3µG. As the stellar wind expands outwards into the ISM, the ISM is compressed and now the ISM magnetic field counteracts this compression, increasing the size of the OAS. (see also van Marle et al., 2015b). Increas- ing the size of the OAS decreases the compression ratio of the BS. For such a magnetic field strength value, there is no effect on the inner shock structure, e.g. the TS distance. The green dashed line corresponds to a magnetic field strength of 10µG. The larger magnetic field strength increases the size of the OAS drastically compared to the weak magnetic field and the hydrodynamical case, and there is also a further decrease in the compression ratio of the BS. As the pressure increases from the added magnetic pressure due to the higher ISM magnetic field, the stellar wind ram pressure is still able to push the ISM as far back as in the weaker magnetic field case; however, the higher pressure makes a thicker shell at the AP and the AP is slightly closer to the inner boundary. An 1. EFFECT OF ISM MAGNETIC FIELD ON THE ASTROSPHERE STRUCTURE15

Different ISM magnetic field strengths after 200 000 years 101 ) 3 100 m c . s e l c i t r a p

( 1

10 y t i s n e d

r e b 10 2 m u N

0 G 3 G 10 G 3 10 20 G

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.1: Four different computed astrospheres showing the effect that an ISM magnetic field has on the evolution of the astrosphere. The blue dotted line is a pure hydrodynamical case. The orange solid line has an ISM magnetic field strength of 3 µG, the green dashed line has 10 µG and the red dashed-dotted line has a theoretical value of 20 µG. Results are shown at 200 000 years’ simulation time and no radiative cooling is assumed in the model. even stronger magnetic field strength of 20µG is shown with the red dashed-dotted line; this theoretical case shows an astrosphere under the influence of an extremely strong magnetic field. A magnetic field of that strength increases the total pressure of the ISM drastically, resulting in a TS being closer to the inner boundary than in the other cases and the size of the IAS has also decreased dramatically. Even though the size of the IAS has decreased, the ISM magnetic field is so strong that it decompresses the OAS to such an extent that the size of the OAS is about three times the size of the one in the hydrodynamical case.

Figure 3.2 compares the pure hydrodynamical astrosphere with three astrospheres with diﬀerent ISM magnetic ﬁeld strengths, but now with the addition of radiative cooling. 1. EFFECT OF ISM MAGNETIC FIELD ON THE ASTROSPHERE STRUCTURE16

Different ISM magnetic field strengths after 200 000 years 101 ) 3 100 m c . s e l c i t r a p

( 1 10 y t i s n e d

r e b 2

m 10 u N

0 G 3 G 10 G 10 3 20 G

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.2: Similar to Figure 3.1; however, radiative cooling is included.

The pure hydrodynamical case, shown again with the blue dotted line, is the same as the one in Figure 3.1 with the addition of cooling. As shown already, the astrosphere is completely cooled in ≈ 50 000 years and the structure of the OAS is very similar to a no-cool scenario. Results in Figure 3.2 are very similar to Figure 3.1. Once an ISM magnetic ﬁeld is present, the ISM magnetic pressure starts dominating over the thermal pressure of the ISM and the eﬀect of cooling is barely visible. This is better shown in Figure 3.3.

Figure 3.3 shows the compression ratios for the four astrospheres in Figure 3.2 as a function of time. As also shown in Figure 3.3, for the pure pure hydrodynamical case, cooling has a large effect on the compression ratio as shown in the solid black line. An ISM magnetic field of 3µG, solid blue line, reduces the density of the shell created by the cooling and reduces the compression ratio drastically. At later stages, the inclusion of an ISM magnetic field yields a compression ratio that is smaller compared to a pure 2. EFFECT OF MASS LOSS RATE ON ASTROSPHERIC EVOLUTION 17

Compression ratio vs time 35 0 G 3 G 10 G 20 G 30

15 Compression Ratio

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 3.3: The compression ratios of the four astrospheres shown in Figure 3.2 as a function of time. hydrodynamical case including cooling. A strong magnetic ﬁeld of 10µG, solid red line, and 20µG, solid green line, decompresses the OAS leading to a compression ratio close to 1. After 200 000 years, these magnetic ﬁelds causes the compression ratio to decrease to such a degree that the shock is almost eliminated.

2 Eﬀect of mass loss rate on astrospheric evolution

This section will show the eﬀect that various mass loss rates have on the shape and structure of an astrosphere. Changes in the mass loss rate result in changes in the ram pressure of the stellar wind. This can be seen from the density of the stellar wind, which can be calculated using the continuity equation, 2. EFFECT OF MASS LOSS RATE ON ASTROSPHERIC EVOLUTION 18

M˙ ρ = , (3.1) 4πr2v

with the mass loss rate of the star being given by M˙ , r is the distance, or in this case the radius of the star, and v is the bulk speed. In this work, the radius of the star is taken to be the inner boundary of the astrosphere, which for this chapter it is set at 0.03 pc. A large inner boundary is chosen to allow for only a fully developed stellar wind to expand into the ISM. The ram pressure can be calculated from this, and is given by

1 P = ρv2, (3.2) ram 2

from which it is clear that changing the mass loss rate, changes the ram pressure. Since the ram pressure balances with the ISM pressure to create the TS, a higher ram pressure from a larger mass loss rate will increase the distance between the inner boundary and the TS. Higher ram pressure also results in a higher kinetic energy that is able to be converted into thermal energy at the TS, yielding a hotter IAS.

Figure 3.4 shows four astrospheres corresponding to four diﬀerent mass loss rates as- −7 −1 sumed in the model. The mass loss rate for the blue dotted line is 3.27×10 M .year , this mass loss rate corresponds to a stellar wind density of ∼1 particles.cm−3 at the inner boundary, assuming an outﬂow speed of 1000 km.s−1 and the radius of the inner boundary at 0.03 pc. The ISM density in this case is taken to be 10 particles.cm−3, which is higher compared to Figure 3.2. With such ISM density, the resulting astrosphere is relatively small in size compared to Figure 3.2. The solid orange line corresponds to a mass −6 −1 −3 loss rate of 3.27×10 M .year leading to a stellar wind density of ∼10 particles.cm at the inner boundary, which is set at 0.03 pc in this model. A mass loss rate of −6 −1 −3 6.54×10 M .year leads to a stellar wind density of ∼20 particles.cm , shown with −5 −1 the green dashed line. Finally, a mass loss rate of 3.27×10 M .year is shown by the red dashed-dotted line, where the stellar wind density is ∼100 particles.cm−3 at 0.03 pc. A comparison between the four mass loss rates shows that an increase in the mass loss rate leads to an increase in the ram pressure and therefore an increase in the overall size of the astrosphere, as is expected from equation 3.2. The distance between the inner boundary and the TS increases with increasing ram pressure, as the higher ram pressure can push the ISM further back before balancing with the ISM pressure. The shocked stellar wind in the IAS has a higher thermal pressure for the larger mass loss rate due 2. EFFECT OF MASS LOSS RATE ON ASTROSPHERIC EVOLUTION 19

Different mass loss rates after 200 000 years 102

) 1 3 10 m c . s e l c i

t 0

r 10 a p (

y t i s n e 10 1 d

r e b m u N 10 2 1 particles. cm 3 at 0.03 pc 10 particles. cm 3 at 0.03 pc 20 particles. cm 3 at 0.03 pc 100 particles. cm 3 at 0.03 pc

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.4: Computed astrospheres with four diﬀerent mass loss rates, 3.27 × −7 −1 −6 −1 10 M .year shown by the blue dotted line, 3.27×10 M .year , the solid orange −6 −1 −5 −1 line, 6.54 × 10 M .year the green dashed line and 3.27 × 10 M .year the red dashed-dotted line, with no radiative cooling or ISM magnetic ﬁeld. to some of the ram pressure being converted into thermal pressure and the larger mass loss rates lead to higher ram pressures.

Figure 3.5 shows the distance from the inner boundary at 0.03 pc to the BS for the −7 −1 four astrospheres shown in Figure 3.4. The mass loss rate of 3.27 × 10 M .year that has a stellar wind density of 1 particles.cm−3 at 0.03 pc with an outﬂow velocity of −1 −6 −1 1000 km.s is shown by the solid blue line. A mass loss rate of 3.27 × 10 M .year −6 −1 is shown by the red dotted line, 6.54 × 10 M .year is shown by the green dashed- −5 −1 dotted line and 3.27 × 10 M .year is shown by the black dashed line. This shows that larger mass loss rates creates a BS further from the inner boundary resulting in a larger cavity. For example a factor of 10 increase in mass loss rate results in a ∼50% larger astrosphere after 200 000 years. 2. EFFECT OF MASS LOSS RATE ON ASTROSPHERIC EVOLUTION 20

Bow shock distance vs. time 14 1 particles. cm 3 at 0.03 pc 10 particles. cm 3 at 0.03 pc 20 particles. cm 3 at 0.03 pc 100 particles. cm 3 at 0.03 pc 12

6 Distance (pc)

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 3.5: The distance from the inner boundary to the BS as a function of time for the four diﬀerent computed astrospheres shown in Figure 3.4.

With the addition of radiative cooling and an ISM magnetic field, the outer structure of the astrosphere changes as described earlier. The outer structure is the OAS and the BS. Figure 3.6 shows four astrospheres with different mass loss rates as in Figure 3.4, but now with the addition of radiative cooling and an ISM magnetic field, but for 50 000 years’ simulation time. Figure 3.7 shows the same four astrospheres as in Figure 3.6, but now after 200 000 years. The cooling function being used is the one of Mellema & Lundqvist(2002) and the ISM magnetic field has a strength of 3 µG. For the case of a low mass loss rate, which is shown by the blue dotted line at 200 000 years shown in Figure 3.7, the ISM magnetic field has very little effect on the shape of the OAS when compared to Figure 3.4. This is due to the ISM pressure dominating the ram pressure. The added magnetic pressure decreases the size of the OAS slightly. Note the difference in the scaling of the x-axis between Figure 3.4 and Figures 3.6 and 3.7. It can be seen in Figures 3.6 and 3.7 that the lower mass loss stops cooling earlier compared to the other 2. EFFECT OF MASS LOSS RATE ON ASTROSPHERIC EVOLUTION 21

Different mass loss rates after 50 000 years

102 ) 3 m c

. 101 s e l c i t r a p ( 100 y t i s n e d

e 10 1 b m u N

1 particles. cm 3 at 0.03 pc 10 2 10 particles. cm 3 at 0.03 pc 20 particles. cm 3 at 0.03 pc 100 particles. cm 3 at 0.03 pc

0 2 4 6 8 10 Distance (pc)

Figure 3.6: The same four astrosphered shown in Figure 3.4 with the inclusion of radiative cooling and an ISM magnetic ﬁeld. Result is shown after 50 000 years. cases. A thin, very dense shell is visible for the ∼10 particles.cm−3, ∼20 particles.cm−3 and very pronounced shell for the ∼100 particles.cm−3 scenario in Figure 3.6. A higher mass loss rate creates a higher stellar wind density and the cooling time is proportional to the density leading to a longer cooling time. The mass loss rate scenario corresponding to a scenario of ∼20 particles.cm−3 at a radius of 0.03 pc and 200 000 years, shown as the green dashed line, has already cooled as in Figure 3.6, but a thin structure is still visible in Figure 3.7 as the astrosphere expands. The ∼100 particles.cm−3 scenario has the higher mass loss rate and the OAS is still cooling at 200 000 years and the OAS is very thinly compressed and has an extremely high density. Even though an ISM magnetic ﬁeld is present, it is unable to counteract the compression from the cooling due to the relatively high ISM density. 3. EFFECT OF OUTFLOW VELOCITY OF A STELLAR WIND ON ASTROSPHERE EVOLUTION 22

Different mass loss rates after 200 000 years

102 ) 3 m

c 1

. 10 s e l c i t r a

p 0 (

10 y t i s n e d

r 10 1 e b m u N

2 10 1 particles. cm 3 at 0.03 pc 10 particles. cm 3 at 0.03 pc 20 particles. cm 3 at 0.03 pc 100 particles. cm 3 at 0.03 pc

0 2 4 6 8 10 Distance (pc)

Figure 3.7: Similar to Figure 3.6 with the result shown after 200 000 years.

3 Eﬀect of outﬂow velocity of a stellar wind on astrosphere evolution

This section will show the effect that different outflow velocities have on the evolution of astrospheres. Radiative cooling and an ISM magnetic field will be added, similar to the previous section, to determine whether changing the outflow velocity changes how cooling and the magnetic field affect the astrosphere. The effect that the outflow velocity has on an astrosphere is shown in Figures 3.8 and 3.9, where four different outflow velocities are shown. The stellar wind density for all four cases is ∼10 particles.cm−3 at the inner −6 −1 boundary of 0.03 pc. This corresponds to a mass loss rate of 1.65 × 10 M .year where the inner boundary for the model has been set at 0.03 pc. An ISM density of 10 particles.cm−3 is assumed. 3. EFFECT OF OUTFLOW VELOCITY OF A STELLAR WIND ON ASTROSPHERE EVOLUTION 23

Different outflow velocities after 200 000 years

101 ) 3 m c . s e l 100 c i t r a p (

y t i s

n 10 1 e d

r e b m u N 10 2

500 km. s 1 1000 km. s 1 1500 km. s 1 2000 km. s 1

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.8: Comparison of astrospheres created with varying outflow velocities. The blue dotted line has an outflow velocity of 500 km.s−1, the orange solid line, 1000 km.s−1, the green dashed line, 1500 km.s−1 and the red dashed-dotted line, 2000 km.s−1 at the inner boundary. There is no radiative cooling or ISM magnetic field included.

Figure 3.8 shows the four astrospheres created with varying outflow velocities with no radiative cooling or ISM magnetic field included. The blue dotted line has an outflow velocity of 500 km.s−1. The low outflow velocity leads to a low ram pressure as can be seen from equation 3.2 in the previous section. The low ram pressure balances with the pressure from the ISM at a distance of approximately ∼1.25 pc to create the TS. The AP is located at ∼4.8 pc. The larger outflow velocity of 1000 km.s−1, shown by the orange solid line, extends the distance from the inner boundary to the TS to a distance of ∼1.9 pc, while the AP is situated at a distance of ∼7 pc. The compression ratio of the bow shock has also increased somewhat. The green dashed line has an outflow velocity of 1500 km.s−1. The TS is moved further from the inner boundary to a distance of ∼2.3 pc with an AP at a distance of ∼9 pc. The compression ratio for an outflow velocity 3. EFFECT OF OUTFLOW VELOCITY OF A STELLAR WIND ON ASTROSPHERE EVOLUTION 24 of 1500 km.s−1 is larger than all the lower outflow velocities. This pattern continues when looking at the red dashed-dotted line, which shows an astrosphere with an outflow velocity of 2000 km.s−1. The TS is shifted further into the ISM and further away from the inner boundary of the astrosphere. The AP distance from the inner boundary seems almost linear with increasing outflow velocity as the 2000 km.s−1 case has an AP even further from the inner boundary and about 65% further than, for example, the 1000 km.s−1 scenario. The compression ratio of the bow shock also increases slightly. Increasing the outflow velocity increases the ram pressure allowing the astrosphere to push the ISM further back before balancing out with the ISM pressure since the density decreases as the stellar wind is blown outwards.

Different outflow velocities after 200 000 years 102

1 ) 10 3 m c . s e l c

i 0

t 10 r a p (

y t i s

n 1

e 10 d

r e b m u

N 10 2

500 km. s 1 1000 km. s 1 1500 km. s 1 2000 km. s 1

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.9: Comparison of astrospheres created with varying outflow velocities. The blue dotted line has an outflow velocity of 500 km.s−1, the orange solid line, 1000 km.s−1, the green dashed line, 1500 km.s−1 and the red dashed-dotted line, 2000 km.s−1 with the inclusion of radiative cooling and an ISM magnetic field.

Figure 3.9 shows the same four astrospheres as is shown in Figure 3.8; however, the eﬀects of radiative cooling and an ISM magnetic ﬁeld of 3 µG are included. The magnetic 3. EFFECT OF OUTFLOW VELOCITY OF A STELLAR WIND ON ASTROSPHERE EVOLUTION 25 )

3 Different outflow velocities after 200 000 years 101 m c . s e l 100 c i t r a

p 1 ( 10

y t i s

n 10 2 e 1

d 500 km. s

1 r 1000 km. s

e 1500 km. s 1

b 3 10 2000 km. s 1 m

u 108 N

) 10 K (

6 e 10 r u t

a 5

r 10 e p 4

m 10 e T 103

102 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.10: Similar to Figure 3.9 showing the 4 astrospheres with different outflow velocities; however, with an ISM density of 1 particles.cm−3. pressure from the ISM magnetic field does not have as great an effect as it did in section 1, since the ISM is denser with a density of 10 particles.cm−3. To fully appreciate the effect that an ISM magnetic field may have, Figure 3.10 shows the same as in Figure 3.9, but with a lower ISM density of 1 particles.cm−3 also now showing the temperature profile of the astrosphere. The lower ISM density shows the effect of the ISM magnetic field creating a larger OAS for all the scenarios. The different scenarios, like in previous sections, have already cooled and after 200 000 years the OAS has a thin-shell remnant as it expands. The OAS still shows signs of the COAS and HOAS that are created by the interplay of the cooling and the ISM magnetic field. The COAS and HOAS can be observed from the temperature of the astrosphere shown in Figure 3.10. Just after the AP, the temperature is rapidly decreasing for the thin shell-like remnant, but then increases, which shows the HOAS. The COAS is a cold thin dense area, while the HOAS is larger as the OAS expands. The COAS can also be seen from the density profile as 4. EFFECT OF ISM DENSITY ON ASTROSPHERE 26 a thin dense area at the AP, which is a result of the thin-shell, due to radiative cooling much earlier.

4 Eﬀect of ISM density on astrosphere

As can already be seen from comparing Figure 3.9 to Figure 3.10, the ISM density around a star affects the size of the astrosphere as a higher ISM density increases the ISM pressure that the ram pressure from the stellar wind has to balance out in order to create the TS and the astrosphere. This section shows the effect that different ISM densities have on the evolution of an astrosphere. The different ISM densities will also influence the effect of radiative cooling on the outer structure of the astrosphere as the cooling is dependent on density. The mass loss rate for all scenarios in this −6 −1 section is taken as 1.65 × 10 M .year , which corresponds to a stellar wind density of ∼10 particles.cm−3 at the inner boundary of 0.03 pc. The outflow velocity of the astrospheres in this section is taken as 1000 km.s−1 at the inner boundary.

The four scenarios corresponding to the four different ISM densities at 200 000 years are shown in Figure 3.11, where there is no radiative cooling and no ISM magnetic field assumed in the model. The size of the different astrospheres is only dependent on the ISM density. The higher ISM density increases the ISM pressure creating a smaller astrosphere. The compression ratio of the TS is unaffected by the increasing ISM density; however, the TS distance from the inner boundary of the astrosphere is decreased since the ram pressure from the stellar wind has to deal with a larger ISM pressure for higher ISM densities. The larger ISM density requires a larger ram pressure to balance. The stellar wind’s density decreases as it is blown further from the inner boundary of the astrosphere, lowering the ram pressure of the stellar wind seen from equation 3.2. The balance between the ram pressure and the ISM pressure is therefore achieved later for the lower ISM densities forming the TS further from the inner boundary of the astrosphere.

Figure 3.12 shows four different astrospheres with varying ISM densities similar to Figure 3.11, assuming an ISM magnetic field of 3 µG and radiative cooling in the model. Shown in the different panels are different stages of evolution, namely 50 000, 100 000, 150 000 and 200 000 years, respectively. The higher ISM density of 50 particles.cm−3 shows no sign of cooling at all up to 200 000 years, since the density is too high for any energy to be radiated away. The effect of the ISM magnetic field is also not visible as the gas pressure from the higher density dominates over the ISM magnetic pressure. The same is true for a lower density of 10 particles.cm−3. The lower ISM densities of 1 particles.cm−3 and 0.5 particles.cm−3 shown by the solid orange and blue dotted lines, respectively, show a greater effect from the ISM magnetic field and cooling. The cooling creates a clear 5. SUMMARY AND CONCLUSION 27

Different ISM densities after 200 000 years

102

) 1 3 10 m c . s e l

c 0 i 10 t r a p (

y t i s 10 1 n e d

r e b

m 2

u 10 N

0.5 particles. cm 3 1 particles. cm 3 10 3 10 particles. cm 3 50 particles. cm 3

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.11: The eﬀect that ISM density has on the shape and distance the astrosphere expands into the astrosphere is shown with ISM densities 1 particles.cm−3 for the blue dashed line, 10 particles.cm−3 for the solid orange line, 50 particles.cm−3 for the green dashed line and 100 particles.cm−3 for the red dashed-dotted line. These results do not include radiative cooling nor an ISM magnetic ﬁeld.

COAS and HOAS structure (as discussed earlier) with an increase in density at the AP while the ISM magnetic ﬁeld decompresses the OAS after it has cooled leading to an OAS that is larger in size when compared to the hydrodynamical cases in Figure 3.11.

5 Summary and conclusion

This chapter illustrated the effects that some parameters have on the evolution of an astrosphere in the pure hydrodynamical case and including radiative cooling and ISM magnetic field. The parameters, namely the ISM magnetic field strength, mass loss rate, 5. SUMMARY AND CONCLUSION 28

Different ISM densities

102

101

100

10 1

10 2

) 50 000 years 3 3 10 102 m c . s

e 1

l 10 c i t r

a 100 p (

y t i 10 1 s n e d

r 10 e b 100 000 years m 10 3 u 2 N 10

101

100

10 1

10 2

10 3 150 000 years 102

101

100

10 1

0.5 particles. cm 3 10 2 1 particles. cm 3 10 particles. cm 3 10 3 200 000 years 50 particles. cm 3

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Distance (pc)

Figure 3.12: Similar to Figure 3.11, showing the effect of ISM density for four different densities with the inclusion of radiative cooling and an ISM magnetic field. Results are shown at four different times, namely 50 000 years, 100 000 years, 150 000 years and 200 000 years. 5. SUMMARY AND CONCLUSION 29 outflow velocity and ISM density were varied and the effects these variations have on the evolution of an astrosphere were shown.

As was shown, the inclusion of an ISM magnetic field in the model increased the size of the OAS and decreased the BS compression ratio, as was shown in Figure 3.1. Figure 3.2 showed that the inclusion of a strong ISM magnetic field negates the effects of cooling. A stronger ISM magnetic field reduced the compression ratio, as was shown in Figure 3.3, to less than the pure hydrodynamic case, shown in Figure 2.2.

An increase in the mass loss rate increased the ram pressure of the stellar wind allowing it to create a larger astrosphere, as was shown in Figures 3.4 and 3.5. It is apparent that increasing the mass loss rate increases the size of the astrosphere. It was found that the lower mass loss rate allowed an astrosphere to cool more rapidly and efficiently, creating a very dense thin shell surrounding the astrosphere during earlier stages in the evolution. Higher mass loss rates also created the thin dense shell due to the cooling; however, this shell appeared much later in the evolution of the astrosphere. Once the shell was formed, the magnetic pressure from the ISM magnetic field decompresses the OAS, increasing the thickness in the shell, but decreasing the compression. The thickness in the shell also increases with time. This can be seen when comparing Figure 3.6, which shows four astrospheres computed with four different mass loss rates after 50 000 years, with Figure 3.7, which shows the same four astrospheres after 200 000 years. At this stage, the thin shell decompressed and increased in thickness and depending on parameters, influenced the cooling time.

The effect of four different ISM densities on model solutions was shown in Figure 3.11. Figure 3.12 showed the same at four different times, 50 000, 100 000, 150 000 and 200 000 years. ISM densities of 50 particles.cm−3 and 10 particles.cm−3 showed no effect from radiative cooling or an ISM magnetic field due to the high ISM pressure. The lower ISM densities of 0.5 particles.cm−3 and 1 particles.cm−3 showed a thin dense shell that was formed early in the astrosphere’s evolution. The thickness of the OAS increased and a COAS and HOAS were formed. The magnetic pressure also increased the thickness of the OAS. CHAPTER 4

Luminous Blue Variable: HD 99953

1 Introduction

A luminous blue variable (LBV) is an evolved, very luminous, unstable hot supergiant star that suffers irregular eruptions or more rarely giant eruptions. See e.g. Humphreys & Davidson(1994) for a complete review of LBVs. These stars are at one of their final evolutionary phases and have high mass loss rates due to stellar winds (e.g. Solovyeva et al., 2019). LBVs undergo four major types of intrinsic photometric variability: microvariations, S Dor phases, stochastic variability and eruptions (Sterken, 2003). These types of variability are not unique to LBVs. Microvariations are a common feature among supergiants and not just LBVs. See Vink(2012) and references therein. The S Dor phases are divided into short S Dor phases, which are on the time-scale of years and long S Dor phases, which have time-scales of decades, during which there is significant spectroscopic and photometric variability. During the short and long S Dor phases, the LBV has approximately constant luminosity and the variations are due to temperature changes. (see Puls et al., 2008). The effective temperatures of LBVs vary from over ≈30 000 K to only ≈15 000 K for the luminous LBVs and less luminous LBVs respectively (e.g. Vink, 2012). During the S Dor phase, the mass loss rate of these stars is of −5 −1 −3 −1 the order 10 M .year to 10 M .year with terminal velocities in the range of 100 to 500 km.s−1 (Puls et al., 2008). The eruptions are accompanied by both luminosity and mass increases, with ejection of several solar masses of material (Guzik & Lovekin, 2014). These eruptions are discussed further in Chapter6.

30 2. MODELLING OF HD 99953 31

This chapter looks at the astrosphere that forms around the luminous blue variable HD 99953. Chapter3 illustrated the effects that mass loss rate, outflow velocity, ISM density, and ISM magnetic field have on the evolution and shape of an astrosphere. This chapter looks at the effect that radiative cooling and ISM magnetic field have on the astrosphere created around HD 99953 and the importance of these effects in simulations. HD 99953 is an early B supergiant, which shows variable stellar parameters (Haucke et al., 2018), which is why it was chosen for this work.

In this chapter, the astrosphere corresponding to parameters associated with the quiescent and the maximum phases of HD 99953 is first shown as extreme scenarios sepa- rately and then these are compared to the case where the star transitions between the two phases periodically. The comparison between the astrosphere created with the varying stellar parameters and the astrospheres created during the quiescent and maximum phases only, will show the effect that the periodic variation of the stellar parameters have on the astrosphere. For all cases, the pure hydrodynamical case without radiative cooling will be compared to the case where there is an ISM magnetic field present as well as radiative cooling. HD 99953 has an outflow velocity ranging from 250 km.s−1 to 700 km.s−1 with an outflow number density ranging from 1.115 particles.cm−3 to 1.095 particles.cm−3 at the inner boundary (Haucke et al., 2018).

2 Modelling of HD 99953

This section gives the stellar parameters for the astrospheric cavity that is formed around HD 99953, a variable and pulsating blue supergiant. At present, this is a poorly studied star. See Haucke et al.(2018) and Fraser et al.(2010) for the derivation of the stellar parameters. This type of star shows similar behaviour to an LBV-type star, which exhibits periodic variation in the mass loss rate and outflow velocity. The simulations are first done without an interstellar magnetic field or any radiative cooling effects to visualise the pure hydrodynamical case. Thereafter, radiative cooling and an ISM magnetic field will be included to see the effects that these have on this particular astrosphere. The astrospheric evolution will be modelled using different parameters to show the cavity during a quiescent and a maximum phase. The eruption phase of this star will be done in Chapter6, since this can be analogous with a supernova explosion. The periodic behaviour of the star will also be modelled by varying the outflow parameters between the parameters for a quiescent phase to a maximum phase. Different periods will also be modelled to show the effect that the period length has on the astrosphere’s evolution.

Figure 4.1 shows the warm dust structure around HD 99953, which was heated by radiation from the star in infra-red, (Haucke et al., 2018). These authors also established 2. MODELLING OF HD 99953 32

Figure 4.1: Image taken from Haucke et al.(2018) showing the wind-blown and warm dust structures around HD 99953 using WISE.

−8 −1 the outflow parameters for HD 99953. The minimum mass loss rate is 8×10 M .year with a minimum outflow velocity of 250 km.s−1 and increases to a maximum mass loss −7 −1 −1 rate of 2.2 × 10 M .year with a maximum outflow velocity of 700 km.s . For this work, the period is chosen to be 17 years for illustrative purposes. The mass loss rate is converted to a density, the stellar wind density, by using the outflow velocity corresponding to that specific mass loss rate as well as the inner boundary distance from which the stellar wind is simulated to begin. An inner boundary distance of 0.03 pc is assumed in the model. The minimum mass loss rate yields a stellar wind density of 1.115 particles.cm−3 at the inner boundary. Using the maximum outflow parameters, the stellar wind density obtained for the maximum phase is 1.095 particles.cm−3 at the inner boundary. The large variation in the mass loss rate and the outflow velocity yields almost the same stellar wind density due to the variation in the outflow velocity. The outflow temperature for both the minimum and maximum outflows is set at 1 × 105 K. 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 33

The ISM density around HD 99953 is assumed to have a density of 1 particles.cm−3. With such a low density, the eﬀects from the ISM magnetic ﬁeld and cooling will be more visible.

3 HD 99953 evolution without periodic outﬂow

This section will look at a range of possible computed astrospheres assuming known parameters of HD 99953. First, parameters assumed corresponding to the quiescent and maximum phases, respectively, and without any periodic behaviour are used to establish upper and lower limits of the astrosphere before adding periodic behaviour in section4. The effects of radiative cooling and an ISM magnetic field of 3µG are also added later to show the effects these have on the astrosphere around HD 99953. This magnetic field strength is found taken from

3.1 HD 99953 simulations with no radiative cooling or ISM magnetic ﬁeld

A computed astrosphere created without radiative cooling and an ISM magnetic ﬁeld is shown in Figure 4.2 for an outﬂow velocity of 250 km.s−1 and a mass loss rate of −8 −1 8 × 10 M .year . These parameters correspond to HD 99953 being in a quiescent state, where the parameters are at a minimum. The astrospheres shown here have similar main features as is described in Chapter2, namely a TS, an AP and a BS.

Figure 4.2 shows this astrosphere at four different times during its evolution. The blue dotted line is the structure of the astrosphere at 50 000 years, the solid orange line is at 100 000 years, the green dashed line at 150 000 years and the red dashed-dotted line is at 200 000 years. The low mass loss rate combined with the low outflow velocity, creates an astrosphere that is slowly evolving into the ISM. Even though there is no relative motion assumed between the star and the ISM, the shock is still referred to as the bow shock. The astrosphere at different stages shows the expansion of the astrosphere into the ISM with the TS moving further into the ISM. The expansion of the astrosphere into the ISM increases the size of the IAS as well as the OAS, while the compression ratio of the BS decreases. As the astrosphere evolves and expands into the ISM; the compression ratio of the BS decreases as energy is lost from the stellar wind.

Figure 4.3 shows the computed astrosphere that is expected during the maximum phase −7 −1 of the star. It has a mass loss rate of 2.2 × 10 M .year and an outﬂow velocity of 700 km.s−1. The higher outﬂow velocity allows the astrosphere to expand much further 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 34

HD99953 for 250 km. s 1 outflow velocity

100 ) 3 m c . s e l c i t r 10 1 a p (

y t i s n e d

r e b 2 m 10 u N

50 000 Years 100 000 Years 150 000 Years 200 000 Years

0 2 4 6 8 10 Distance (pc)

Figure 4.2: An astrosphere that is created assuming the quiescent phase of HD 99953 −8 −1 and with the minimum mass loss rate of 8 × 10 M .year and the corresponding minimum outflow velocity of 250 km.s−1. There is no radiative cooling or ISM magnetic field included. Figure shows resulting astrosphere at 50 000, 100 000, 150 000 and 200 000 years. into the ISM than in the quiescent case due to the larger ram pressure. Comparing Figure 4.3 with Figure 4.2, the astrosphere created during the maximum phase of HD 99953 is considerably larger than the one created during the quiescent phase. The distance from the inner boundary to the TS of the astrosphere has more than doubled due to the ram pressure being so much larger during the maximum phase. The higher outflow velocity of the maximum phase results in a much larger astrosphere as well as a BS with a larger compression ratio when compared to the quiescent phase. The compression ratio of the BS also decreases as the astrosphere expands further into the ISM, similar to Figure 4.2. The size of the OAS also increases.

Figure 4.4 shows the velocity of the wind for HD 99953 at the BS of the astrosphere for the quiescent and maximum phases. The outﬂow velocity of HD 99953 is 250 km.s−1 at 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 35

HD99953 for 700 km. s 1 outflow velocity

100 ) 3 m c . s e l c

i 1

t 10 r a p (

y t i s n e d

r 10 e b m u N

50 000 Years 3 10 100 000 Years 150 000 Years 200 000 Years

0 2 4 6 8 10 Distance (pc)

Figure 4.3: An astrosphere created by a star with the maximum mass loss rate of −7 −1 −1 2.2 × 10 M .year at the corresponding maximum outﬂow velocity of 700 km.s . There is no radiative cooling and no ISM magnetic ﬁeld. The results are shown for 50 000, 100 000, 150 000 and 200 000 years. the inner boundary of 0.03 pc for the quiescent phase and 700 km.s−1 for the maximum phase. As the astrosphere expands into the ISM, the kinetic energy of the stellar wind decreases due to the ISM pressure. The stellar wind velocity at the BS decreases expo- nentially as a function of time. The velocity at the BS for the maximum phase remains higher than the velocity for the quiescent phase.

As the computed astrosphere expands into the ISM, the size of the cavity increases. The distance from the inner boundary set at 0.03 pc to the BS increases as the cavity gets larger. Figure 4.5 shows the distance from the inner boundary to the BS for the quiescent and maximum phase of HD 99953, respectively, showing the evolution into the ISM. 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 36

Velocity at Bow Shock

250 km. s 1 700 km. s 1

100

80 ) 1 s . m k

( 60

y t i c o l e

V 40

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 4.4: The velocity at the BS for the quiescent and maximum phase as a function of time. The blue line is the computed astrosphere for the quiescent phase and the red line is the computed astrosphere for the maximum phase.

3.2 Eﬀect of radiative cooling on the evolution of HD 99953

This section includes the eﬀect of radiative cooling in the model to show the eﬀect on the evolution of the astrospheres that were computed in the previous section using the radiative cooling function taken from Mellema & Lundqvist(2002). The cooling creates a thin dense shell around the astrosphere and increases the compression ratio of the BS drastically.

Figure 4.6 shows the evolution of the astrosphere with the inclusion of radiative cooling. This astrosphere is the same astrosphere that is shown in Figure 4.2 which has an outflow velocity of 250 km.s−1. This set of stellar parameters allows the astrosphere to cool very efficiently. Comparing the different profiles shows that, as time increases, the size of the 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 37

Bow shock distance vs Time

250 km. s 1 7 700 km. s 1

3 Distance (pc)

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 4.5: The distance from the inner boundary to the BS for the quiescent phase, shown by the blue line, and for the maximum phase, shown by the red line, as a function of time as the astrosphere evolves.

OAS does as well and the compression ratio decreases. The size of the OAS is the same as for the case shown in Figure 4.2, where there is no cooling.

For the maximum phase with an outﬂow velocity of 700 km.s−1 at the inner boundary, the results are shown in Figure 4.7. The radiative cooling in the OAS creates the thin dense shell as seen after 50 000 years, blue dotted line. The thickness of the OAS increases as the astrosphere evolves creating a COAS and HOAS structure. The green dashed line shows the astrosphere after 150 000 years and the thickness of the OAS increases while the compression ratio of the BS decreases. After 200 000 years, the red dashed-dotted line, the shell is not as dense and the thickness of the OAS is now of the same size as shown in Figure 4.3 after 200 000 years. The density of the COAS (shell-like remnant) near the AP decreases as the astrosphere evolves. The eﬀect of cooling is more apparent 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 38

HD99953 for 250 km. s 1 outflow velocity

100 ) 3 m c . s e l c i t

r 1

a 10 p (

y t i s n e d

r e b

m 10 2 u N

50 000 Years 100 000 Years 150 000 Years 200 000 Years

0 2 4 6 8 10 Distance (pc)

Figure 4.6: The evolution of an astrosphere that is similar to Figure 4.2 with the inclusion of radiative cooling. in Figure 4.7 for the higher outﬂow velocity of 700 km.s−1 as compared to Figure 4.6 for the lower outﬂow velocity of 250 km.s−1 at the inner boundary of 0.03 pc.

3.3 HD 99953 evolution with the eﬀect of radiative cooling and ISM magnetic ﬁeld on astrosphere evolution

This section includes the effect of radiative cooling and an ISM magnetic field pressure in the model when astrospheric evolution is calculated. The ISM magnetic field is assumed to be 3 µG. Due to the spherical geometry and limitations of the model, the inclusion of a magnetic field reduces the model to 1D with the orientation of the field such that it results in maximum magnetic pressure. It will be shown that the outer structure and shape of the astrosphere change with the inclusion of radiative cooling and an ISM magnetic field. 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 39

HD99953 for 700 km. s 1 outflow velocity

101 ) 3 m c

. 0 s 10 e l c i t r a p (

y 1 t 10 i s n e d

r e b 2

m 10 u N

50 000 Years 3 100 000 Years 10 150 000 Years 200 000 Years

0 2 4 6 8 10 Distance (pc)

Figure 4.7: An astrosphere at four diﬀerent times with an outﬂow velocity at the inner boundary of 0.03 pc of 700 km.s−1. This is similar to Figure 4.3 with the inclusion of radiative cooling.

Figure 4.8 shows the same astrosphere as in Figure 4.2, but with the effects of an ISM magnetic field and radiative cooling now included in the model. Comparing Figure 4.8, which has cooling and an ISM magnetic field to Figure 4.6, which has no magnetic field and only cooling, the effect shows that the magnetic field creates a more distinct COAS and HOAS. The blue dotted line depicts the astrosphere at 50 000 years. There is a clear increase in density at the AP. Due to radiative cooling during the early stages of the evolution, the OAS is compressed, while the ISM magnetic field opposes this compression increasing the size of the OAS. This effect is evident when comparing the 50 000 years case to the 200 000 years case. Comparing Figure 4.8 to Figure 4.6, the effect of added magnetic field pressure is very clear as the thickness of the OAS is increased drastically and the compression ratio of the BS is reduced. After 100 000 years, the outer structure of the astrosphere has not changed as it is already cooled; however, the ISM magnetic 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 40

HD99953 for 250 km. s 1 outflow velocity

100 ) 3 m c . s e l c i t

r 10 1 a p (

y t i s n e d

r e b

m 10 2 u N

50 000 Years 100 000 Years 150 000 Years 200 000 Years

0 2 4 6 8 10 Distance (pc)

Figure 4.8: This is the astrophere for the minimum mass loss rate of 8 × −8 −1 −1 10 M .year with a corresponding outflow velocity of 250 km.s . The inner boundary is set at 0.03 pc. This includes the effect of radiative cooling as well as the interstellar magnetic field of 3 µG. Results are shown for 50 000, 100 000, 150 000 and 200 000 years.

field has decompressed the OAS, increasing the size of the OAS. The green dashed line showing the astrosphere after 150 000 years and the red dashed-dotted line depicting the astrosphere after 200 000 years show that the OAS is almost double the size of the similar case without radiative cooling and an ISM magnetic field depicted in Figure 4.2. The inclusion of the ISM magnetic field and cooling reduces the compression ratio of the BS.

In Figure 4.9, the outflow velocity was increased from 250 km.s−1 in Figure 4.8 to 700 km.s−1. This shows an astrosphere including radiative cooling and an ISM magnetic field at the expected maximum phase of HD 99953. Also shown in the bottom panel is the temperature profile. Similar to the velocity increase from Figure 4.2 to 4.3, this increase in the velocity pushes the astrosphere further into the ISM due to a higher 3. HD 99953 EVOLUTION WITHOUT PERIODIC OUTFLOW 41

) 1 3 HD99953 for 700 km. s outflow velocity m c

. 100 s e l c i t r 1 a 10 p (

y t i s 10 2 n

e 50 000 years d 100 000 years r

e 150 000 years 3 b 10 200 000 years m u 107 N )

K 6

( 10

e r u

t 5

a 10 r e p

m 104 e T

103

0 2 4 6 8 10 Distance (pc)

Figure 4.9: An astrosphere created by a star with the maximum mass loss rate of −7 −1 −1 2.2 × 10 M .year at the corresponding maximum outflow velocity of 700 km.s . First panel shows the density profile and second panel shows the temperature profile. The results are shown for 50 000, 100 000, 150 000 and 200 000 years. ram pressure. There is a sharp increase in the density at the astropause due to the ISM magnetic field and cooling as the astrosphere is cooled during the very early stages because of the low ISM density and the low outflow velocities. After 50 000 years, the astrosphere has already cooled and the effects from the ISM magnetic field start to dominate expanding the size of the OAS. The maximum phase shows the same trends as in the quiescent phase shown in Figure 4.8, where the astrosphere is cooled during its early evolution and thereafter the ISM magnetic field dominates the shaping of the outer structure. The second panel of Figure 4.9 shows the temperature profile of the astrosphere. This shows the constant temperature in the IAS and the decrease of the temperature from the astrosphere to the ISM. Beyond the AP, the temperature is lower compared to the rest of the OAS and the remnant of the shell-like structure is still visible. 4. HD 99953 EVOLUTION WITH PERIODIC OUTFLOW 42

4 HD 99953 evolution with periodic outﬂow

This section shows computed examples of astrospheres assuming parameters of HD 99953 that periodically vary between its quiescent and maximum phase. This is achieved by varying the outflow parameters between the minimum and maximum at the inner boundary. The mass loss rate and the outflow velocity are the two parameters that are varied in this section. The inner boundary distance does not change during the shift from quiescent to maximum phase, remaining at 0.03 pc. The stellar wind density −8 −1 calculated for the quiescent phase using a mass loss rate of 8 × 10 M .year with an outflow velocity of 250 km is 1.115 particles.cm−3, while a maximum phase with a mass −7 −1 −1 loss rate of 2.2 × 10 M .year and a corresponding outflow velocity of 700 km.s yields an stellar wind density of 1.095 particles.cm−3 resulting in a number outflow density that is approximately equal for both phases at the inner boundary, 0.03 pc. The stellar wind density is therefore almost constant throughout these simulations at 1.115 particles.cm−3 and the velocity varies. The period of the variation is changed to see the effect that the period duration might have on the astrosphere where the period specifies the number of years that the velocity remains in the minimum and the maximum phases, respectively. After the specified period, the velocity changes to the maximum or minimum, respectively. This creates a step-like variation in the outflow velocity. The periods chosen for this work are 17, 30 and 100 years for illustrative purposes. While the outflow velocity is varied during the periods, the stellar wind density, outflow temperature is kept constant. Any variations that arise in the stellar wind density are only due to the step-like variation in the outflow velocity. Astrospheres without radiative cooling and an ISM magnetic field will be shown first and compared to astrospheres that have an ISM magnetic field and cooling, as was done in section3.

4.1 HD 99953 simulated with no radiative cooling or ISM magnetic ﬁeld but periodic variation in other parameters

This section now contains a periodic variation in the outflow velocity in the model. The initial period that is used is 17 years. Different periods will be included in the following section to see the effect that different periods have on the astrosphere. Results in this section do not include radiative cooling or an ISM magnetic field and are for illustrative purposes.

Figure 4.10 shows an astrosphere where the outﬂow velocity varies between the quiescent and maximum phase of HD 99953 with a period of 17 years where the blue dotted line shows the astrosphere after 50 000 years, the solid orange line is after 100 000 years, the green dashed line is after 150 000 years and ﬁnally the red dashed-dotted line is 4. HD 99953 EVOLUTION WITH PERIODIC OUTFLOW 43

HD99953 for periodic velocity with 17 year period

100 ) 3 m c . s e l c

i 1

t 10 r a p (

y t i s n e d 2 r 10 e b m u N

50 000 Years 10 3 100 000 Years 150 000 Years 200 000 Years

10 1 100 101 Distance (pc)

Figure 4.10: Astrosphere with no radiative cooling or ISM magnetic field with the inclusion of step-wise periodicity, with a period of 17 years. Astrosphere shown at 50 000, 100 000, 150 000 and 200 000 years. Number density and distance are on a log scale. after 200 000 years. To better visualise the periodicity, the distance and the number density are shown on a logarithmic scale. After 50 000 years, it is clear that the overall structure of the astrosphere with periodic outflow is similar to one without periodic outflow shown in Figure 4.2 and 4.3. The periodicity in the outflow velocity has caused a variation in the stellar wind density as the faster wind during the maximum period is able to catch up with the slower wind of the quiescent phase. The slower and faster winds cause clumping in the density, where there are areas of larger density and areas of lower density. The periodicity has no effect on the outer structure of the astrosphere and a negligible effect past the TS as the IAS is an incompressible region. The distance it takes for the periodic effects to be reduced to negligible from the inner boundary such that the variation in the stellar wind density becomes negligible is the dissipation distance. For a period of 17 years and variation between the quiescent and maximum phase of HD 4. HD 99953 EVOLUTION WITH PERIODIC OUTFLOW 44

99953, the dissipation distance is approximately 0.1 pc. Past the dissipation distance, the periodicity has little eﬀect on the astrosphere except for the size of the astrosphere, which is seen in Figure 4.11.

HD99953 after 200 000 years

100 ) 3 m c . s e l 1 c 10 i t r a p (

y t i s n e

d 2 10 r e b m u N

10 3 250 km. s 1 outflow velocity 700 km. s 1 outflow velocity 17 year period

10 1 100 101 Distance (pc)

Figure 4.11: The eﬀect periodicity has on an astrosphere. The solid line shows an astrosphere that has periodicity between the quiescent and maximum phase of HD 99953, with a period of 17 years. The dotted line is the quiescent phase and the dashed line is the maximum phase of HD 99953. The number density and distance are on log scale. There is no ISM magnetic ﬁeld or radiative cooling included.

Figure 4.11 shows how the periodicity between the quiescent and the maximum phase changes the size of the astrosphere when compared to the size of an astrosphere created during a pure quiescent state as in Figure 4.2 and one created during a pure maximum state as in Figure 4.3. The solid green line shows the astrosphere that was created with periodic outflow velocity between 250 and 700 km.s−1 and should be between these two solutions. The blue dotted line is an astrosphere created during the quiescent phase with the outflow velocity of 250 km.s−1 and the orange dashed line is an astrosphere of HD 99953 during maximum phase, where the stellar wind has an outflow 4. HD 99953 EVOLUTION WITH PERIODIC OUTFLOW 45 velocity of 700 km.s−1. For all three cases, the stellar wind density is kept constant at 1.115 particles.cm−3. This shows that periodic scenario creates an astrosphere that has a size that is between one that is created by the quiescent only phase scenario and one that is created by the maximum only phase scenario. The astrosphere created by periodic outflow resembles more an astrosphere created during the maximum phase than one created during the quiescent phase. The period for the periodicity in this case is 17 years. The termination shock distance for the quiescent phase after 200 000 years is approximately 0.8 pc from the outer boundary of the star or the inner boundary of the astrosphere. The termination shock distance for the maximum phase at the same stage in evolution is found at 1.5 pc. The periodic astrosphere has a termination shock at around 1.3 pc. During the maximum phase of the periodic outflow, the astrosphere pushes further into the ISM than during the quiescent phase; however, the maximum phase does not last very long and the ram pressure drops due to the drop in the outflow velocity. The drop in ram pressure does not allow the termination shock to push further into the ISM. During the quiescent phase, the distance between the termination shock and the inner boundary decreases slightly due to the period being of such a length that the faster wind of the following maximum phase pushes the termination shock further away from the position that the shock has during a pure quiescent state. This leads to a stable termination shock that is situated closer to the shock that is formed during a pure maximum phase rather than being centred exactly in-between a pure quiescent and maximum phase. The distance from the inner boundary at 0.03 pc to the TS for the assumed quiescent phase is 0.792 pc, while for the maximum phase it is 1.441 pc. At this stage, the distance to the TS, for the astrosphere with periodic outflow included in the model with a period of 17 years, is 1.250 pc.

4.2 HD 99953 simulations including the eﬀect of radiative cooling and ISM magnetic ﬁeld on astrospheric evolution

This section contains similar results as in section 4.1; however, it includes an ISM magnetic field of 3µG and radiative cooling as described in Chapter2. This section again investigates how periodic outflow may effect astrospheric evolution including the effect that the ISM magnetic field and cooling have on the structure.

Figure 4.12 shows an astrosphere where the outflow velocity varies between the quiescent phase with a velocity of 250 km.s−1 and the maximum phase with a velocity of 700 km.s−1. This is similar to Figure 4.10; however, there is now an ISM magnetic field present and there is also radiative cooling assumed in the model. The cooling and added magnetic pressure from the ISM magnetic field affect the structure of the OAS as discussed earlier, showing a remnant of the shell-like structure due to the cooling much 4. HD 99953 EVOLUTION WITH PERIODIC OUTFLOW 46

HD99953 for periodic velocity with 17 year period

100 ) 3 m c . s e l 1 c 10 i t r a p (

y t i s n e

d 2 10 r e b m u N

3 50 000 Years 10 100 000 Years 150 000 Years 200 000 Years

10 1 100 101 Distance (pc)

Figure 4.12: Astrosphere that has an outﬂow density of 1.115 particles.cm−3 and the velocity varies between 250 and 700 km.s−1 with a period of 17 years. The distance and number density are on log scale. The density varies due to the change in the velocity. As the astrosphere keeps expanding, it becomes similar to an astrosphere without any periodicity. earlier. The cooling and magnetic pressure have no eﬀect on the variation in the stellar wind density before the TS.

Figure 4.13 shows three astrospheres with periodic outflow corresponding to three different periods. This figure will highlight the effect that a change in the period of the periodic outflow has on astrospheric evolution. The three different periods represent a short S Dor phase, which has a timescale of a few years and a long S Dor phase, which has a period of variability of a few decades. The solid blue line has a period of 17 years similar to Figure 4.12. The orange dashed line has a period of 30 years and the green dotted line has a period of 100 years. All the scenarios have a similar astrosphere in size due to the variation in the outflow velocity being dissipated before the TS. The difference in period has no effect on the astrosphere past the TS. The difference between 4. HD 99953 EVOLUTION WITH PERIODIC OUTFLOW 47

HD99953 after 200 000 years

100 ) 3 m c .

s 1

e 10 l c i t r a p (

y t i s n 2 e 10 d

r e b m u N

10 3 17 year period 30 year period 100 year period

10 1 100 101 Distance (pc)

Figure 4.13: A comparison between astrospheres at 200 000 years with three different periods. Solid blue line is 17-year period, orange dashed line is 30-year period and green dotted line is 100-year period. The number density and distance are on a log scale. the three periods is the dissipation distance of the variations in the number density. A dissipation distance for the 17-year period is found to be approximately 0.1 pc. The dissipation distance for the 30-year period is 0.15 pc, which is only slightly larger than the shorter period. For a period of 100 years, the dissipation distance greatly increases to 0.6 pc. After 0.6 pc, the variation in the stellar wind density due to the variation in the outflow velocity still continues; however, it is already significantly dissipated. A tiny variation in the density continues from the dissipation distance up until the TS where it dissipated completely. The tiny variation inside the IAS is negligible as the flow is incompressible there. This shows that any period less than a 100 years in the variation of the outflow velocity has no effect on the structure of the astrosphere, since the variation is unable to penetrate beyond the TS.

The TS distance and AP distance created by the astrosphere with periodic velocity 5. SUMMARY AND CONCLUSIONS 48

Termination Shock and Astropause Distance

Periodic Termination Shock Periodic Astropause

2 Distance (pc)

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 4.14: The distance from the inner boundary to the TS (solid) and AP (dashed- dotted) for the periodic phase scenario, where it switched between the quiescent and maximum phase, with a period of 17 years. with a period of 17 years as a function of time are shown in Figure 4.14. This shows that the periodic variation in outflow velocity at the inner boundary does not affect the distance from the inner boundary to the AP. During the early stages of evolution, the variation in the stellar parameters do affect the distance from the inner boundary to the TS. After 25 000 years, the distance that the TS is formed from the inner boundary increases almost linearly with time.

5 Summary and conclusions

In this chapter, the effect that a periodic outflow (where the stellar wind velocity and the mass loss rate at the inner boundary of the computed astrosphere vary periodically 5. SUMMARY AND CONCLUSIONS 49 between the assumed minimum and maximum parameters) has on a computed astrosphere surrounding the star was investigated. Results were shown in Figure 4.12, which included cooling and an ISM magnetic field of 3 µG in the model. The stellar parameters are those assumed for HD 99953. For the star HD 99953, the outflow velocity was the only parameter that was varied at the inner boundary between the minimum outflow velocity of 250 km.s−1 and the maximum velocity of 700 km.s−1. The periodic outflow created an astrosphere that in size, was between the size of an astrosphere created by the assumed minimum and maximum parameters; however, it more closely resembled the size of the astrosphere seen during the maximum phase as shown in Figure 4.11. Before the TS, the small oscillations that appeared in the number density and velocity due to the periodic outflow vanished when a period of 17 years was assumed in the case of HD 99953. The effect that radiative cooling and the added magnetic pressure from the ISM magnetic field had on the astrosphere compared to a pure hydrodynamic case was that a thin dense shell was created surrounding the astrosphere during the early years of its evolution as shown in Figure 4.9. Thereafter, the magnetic pressure aided the decompression of the OAS, increasing the thickness of the OAS. Three different periods of 17, 30 and 100 years were also assumed for HD 99953 for illustrative purposes. The effect that these different periods had on the structure of the astrosphere was shown in Figure 4.13. These figures showed that a larger period has an effect on the size of the oscillations in the number density before the TS. After the TS, the astrospheres had similar structure and size. The larger periods also increased the dissipation distance with a period of 17 years having had a dissipation distance of 0.1 pc while a period of 100 years had a dissipation distance of 0.6 pc as shown in Figure 4.13. The small oscillations in the number density did initially affect the distance from the inner boundary to the TS of the computed astrosphere. The periodic variation of the outflow parameters at the inner boundary had no effect on the AP or BS. CHAPTER 5

Luminous Blue Variable: AG Carinae

1 Introduction

A very well-studied, bright and active LBV type star is AG Carinae (Stahl et al., 2001). For this reason, it is chosen for this chapter and used to illustrate the eﬀect that the periodic outﬂow has on the evolution of the astrosphere around AG Carinae. This star shows behaviour of a typical LBV type star. The star has photometric, spectroscopic, and polarimetric variability on a range of timescales spanning days to decades, e.g. Groh et al.(2009).

In this chapter, the astrosphere of AG Carinae will be computed, first with the assumed stellar parameters for the minimum and maximum phases, only to show the extreme scenarios as in the previous chapter. Later, this will be compared to the scenario where the stellar parameters are varied periodically between the minimum and the maximum phase at the outflow boundary. The comparison will show the effect that the variation has on the evolution of the computed astrosphere. For all the cases, the pure hydrodynamical case without radiative cooling will be compared to a hydrodynamical case that includes cooling and a case where there is cooling as well as an ISM magnetic field, to show the effects that cooling and an ISM magnetic field may have on the evolution of the astrosphere. For AG Carinae, the outflow velocity varies from 150 km.s−1 during the minimum to 300 km.s−1 during the maximum at the inner boundary of the astrosphere, which is taken as 0.03 pc. This is significantly lower than the velocities for HD

50 2. MODELLING OF AG CARINAE 51

99953. The mass loss rate of AG Carinae during the minimum phase is assumed to be −5 −1 3.16 × 10 M .year , while for the maximum phase the mass loss rate is assumed to −4 −1 be 1.58 × 10 M .year at the inner boundary. See Stahl et al.(2001) for the stellar parameters used in this work. AG Carinae shows diﬀerent periodic behaviour compared to HD 99953, as the outﬂow velocity is lower, while the mass loss rate is much higher at the inner boundary.

2 Modelling of AG Carinae

In this section, the stellar parameters for the LBV AG Carinae are discussed. As mentioned earlier, AG Carinae shows periodic variation on multiple time scales (Groh et al., 2009). The time scale considered for the periodic variation of the stellar parameters in this work is a period of 15 years, as it represents a time-scale between a short S Dor phase and a long S Dor phase. The actual period for AG Carinae to go from visual minimum to maximum and back again is 10 years (Stahl et al., 2001).

Figure 5.1: The mass loss rate of AG Carinae as a function of temperature. Analysed by Stahl et al.(2001). Taken from Vink & de Koter(2002). 3. AG CARINAE EVOLUTION WITHOUT PERIODIC OUTFLOW 52

AG Carinae shows variability in its mass loss as seen from Figure 5.1 from Vink & de Koter(2002). AG Carinae is one of the most studied LBVs as quantitative spectroscopic analysis has been done at various epochs, and mass loss rates have been reported for different S Dor phases, e.g. Vink(2012). The solid line shows the mass loss of AG Carinae as a function of the effective temperature while the star is going from visual minimum to maximum. The dashed line is the behaviour of the mass loss for the transition from visual maximum to minimum. The mass loss rate varies from 3.16 × −5 −1 −4 −1 10 M .year to 1.58 × 10 M .year at the inner boundary. During the period −5 −1 where the mass loss rate is 3.16 × 10 M .year , the outflow velocity at the inner boundary is 150 km.s−1. The mass loss rate, outflow velocity and the inner boundary with a distance of 0.03 pc yield an outflow number density of 645.0 particles.cm−3 at the inner boundary during the minimum phase. For the maximum phase, which has a −4 −1 −1 mass loss rate of 1.58 × 10 M .year and an outflow velocity of 300 km.s at the inner boundary, which is kept at 0.03 pc, the outflow number density is computed as 1612.0 particles.cm−3 at the inner boundary. The ISM density around AG Carinae is assumed to be 10.0 particles.cm−3.

3 AG Carinae evolution without periodic outﬂow

The computed astrosphere around AG Carinae for the assumed minimum and maximum phases respectively is presented in this section. The inner boundary of the astrosphere is set at 0.03 pc. For the minimum phase, the astrosphere is simulated with an outflow velocity of 150 km.s−1 and an outflow number density of 645.0 particles.cm−3 at the inner boundary. The ISM density is taken to be 10.0 particles.cm−3. When an ISM magnetic field is present, it has a strength of 3µG.

Figure 5.2 shows the computed astrosphere around AG Carinae for the minimum phase of the star. The ﬁgure shows three cases for the minimum phase, namely a pure hydrodynamical case without radiative cooling assumed in the model shown by the blue dotted line, a hydrodynamical case that has radiative cooling added in the model shown by the orange dashed line, and a case where cooling is added in the model along with an ISM magnetic ﬁeld that has a strength of 3µG, which is shown by the solid green line. The pure hydrodynamical case shows an astrosphere with the same structure as the one shown in Figure 2.1. The addition of cooling reduces the size of the IAS and the TS forms further from the inner boundary than in the hydrodynamical case without cooling added. The cooling created the thin dense shell during the earlier stages of the astrospheric evolution and at this later stage the OAS has started expanding. The compression ratio of the BS is much larger for the case including cooling in the model 3. AG CARINAE EVOLUTION WITHOUT PERIODIC OUTFLOW 53

Ag Car after 200 000 years for 150 km. s 1 outflow velocity

2 ) 10 3 m c . s e l c i t r 101 a p (

y t i s n e d

r 0

e 10 b m u N

10 1 Hydrodynamic case Hydrodynamic case including cooling Magnetohydrodynamic case including cooling

0 2 4 6 8 10 12 14 Distance (pc)

Figure 5.2: The astrospheres around AG Carinae for a pure hydrodynamic case (blue dotted), a pure hydrodynamic case with cooling added in the model (orange dashed) and a magnetohydrodynamic case with cooling and an ISM magnetic field of 3µG assumed in the model (solid green). Results are shown for the assumed quiescent phase of AG Carinae with an outflow velocity of 150 km.s−1 and an outflow number density of 645.0 particles.cm−3 at the inner boundary. compared to the one without cooling. When there is an ISM magnetic field present with a strength of 3µG, there is little difference to the structure of the astrosphere.

For the assumed maximum phase of AG Carinae, the computed astrosphere is shown in Figure 5.3. The results show the computed astrosphere with an outﬂow velocity of 300 km.s−1 and outﬂow stellar wind density of 1612.0 particles.cm−3 at the inner boundary, which is at 0.03 pc. Similar to Figure 5.2, the pure hydrodynamic scenario is shown by the blue dotted line. The astrosphere is larger than the computed astrosphere for the minimum phase of AG Carinae, but has a similar structure. The orange dashed line shows a hydrodynamical scenario with the addition of radiative cooling in the model. There is a thin dense shell around the astrosphere due to cooling. This can be seen from 3. AG CARINAE EVOLUTION WITHOUT PERIODIC OUTFLOW 54

Ag Car after 200 000 years for 300 km. s 1 outflow velocity

103 ) 3 102 m c . s e l c i t r a 1 p 10 (

y t i s n e d

e 100 b m u N

10 1 Hydrodynamic case Hydrodynamic case including cooling Magnetohydrodynamic case including cooling

0 2 4 6 8 10 12 14 Distance (pc)

Figure 5.3: The same three astrosphere scenarios shown in Figure 5.2 but now during the assumed maximum phase of AG Carinae with an outflow velocity of 300 km.s−1 and outflow stellar wind density of 1612.0 particles.cm−3 at the outflow boundary of 0.03 pc. the large density for the case including cooling compared to the pure hydrodynamic case. It also shows that the OAS forms a thin shell compared to the case without cooling. The size of the IAS is smaller once the cooling is added into the model. The solid green line shows the astrosphere with the addition of an ISM magnetic field in the model. There is no visible effect by introducing the ISM magnetic field in the model. Comparing the minimum phase, Figure 5.2, to the maximum phase, Figure 5.3, the size of the astrosphere of the maximum phase is larger and the compression ratio is larger. The density in the COAS is also higher for the maximum phase. 4. AG CARINAE EVOLUTION WITH PERIODIC OUTFLOW 55

4 AG Carinae evolution with periodic outﬂow

This section will show the effect that the periodic variability between the minimum and maximum phases shown in the previous section will have on the evolution of the astrosphere. The periodic variability applied to AG Carinae is the same periodicity used in this work for HD 99953, where it is a step-wise periodicity between the computed minimum and maximum parameters. The period chosen for this work is 15 years for illustrative purposes. Since the number density at the inner boundary differs for the minimum and maximum phase, the periodic variation will vary the outflow number density and velocity while the temperature is kept constant. The ISM density for these results is also taken to be 10.0 particles.cm−3.

Ag Car for 15 year period

103

2 ) 10 3 m c .

s 1

e 10 l c i t r a p

( 0 10 y t i s n e d

r 10 e b m u

N 10 2

50 000 Years 100 000 Years 10 3 150 000 Years 200 000 Years

10 1 100 101 Distance (pc)

Figure 5.4: Evolution of the computed astrosphere around AG Carinae with a periodic variation between the minimum and maximum phase with a period of 15 years. There is no radiative cooling or ISM magnetic ﬁeld present. The number density and distance are on log scale. 4. AG CARINAE EVOLUTION WITH PERIODIC OUTFLOW 56

The periodic outflow of AG Carinae is shown in Figure 5.4, first for a scenario without radiative cooling and an ISM magnetic field present in the model. At the inner boundary −5 −1 of 0.03 pc, the minimum mass loss rate used in the model is 3.16×10 M .year with an outflow velocity of 150 km.s−1, while for the maximum phase, the mass loss rate is −4 −1 −1 assumed to be 1.58 × 10 M .year with an outflow velocity of 300 km.s also at the inner boundary, which is kept at 0.03 pc. The high mass loss rate and low outflow velocities at the inner boundary that is varied create a large variation in the number density throughout the astrosphere. The blue dotted line shows the astrosphere after 50 000 years where there is no visible stable TS or IAS. The oscillations in the number density do not persist past the AP. As the astrosphere evolves, after 100 000 years, solid orange line, the overall size of the astrosphere has increased by ∼50 percent. There is still no definitive and stable TS and the oscillations in the number density persists through the IAS. The oscillations start to decrease in the IAS after 150 000 years, green dashed line, but there is still no stationary and stable TS. The same behaviour is found after 200 000 years. For the evolution of the computed astrosphere, the oscillations in the number density from the inner boundary to the AP is very large.

A scenario including radiative cooling is shown in Figure 5.5. This astrosphere has the same parameters with the inclusion of radiative cooling and is shown at the same stages during its evolution as the astrosphere shown in Figure 5.4. The cooling aﬀects the outer structure of the astrosphere creating a thin dense shell around the astrosphere as the OAS is compressed. This is evident from the early stages of the evolution as can be seen from the blue dotted line showing the computed astrosphere after 50 000 years. There is a clear TS and a very narrow IAS leading to the AP where there is a very dense and very thin shell on the outside of the astrosphere. The oscillations seen in Figure 5.4 before the AP are greatly reduced. The astrosphere after 100 000 years, solid orange line, shows a larger IAS. The thin shell around the astrosphere is still present. Unlike the minimum and maximum phase, the thin shell does not increase in thickness as the astrosphere evolves for the astrosphere where the outﬂow parameters are varied periodically at the inner boundary of 0.03 pc.

The inclusion of an ISM magnetic field has decompressed the OAS in the previous sections. Figure 5.6 is the same astrosphere shown in 5.4 with the inclusion of radiative cooling and an ISM magnetic field of 3 µG. Comparing Figure 5.6 with Figure 5.5 shows that the addition of the ISM magnetic field is not important. Due to the high mass loss rate and high density of the ISM, the addition of the ISM magnetic field shows no effect on the OAS as the thin dense shell is still present after 200 000 years. Again, the oscillations in the number density are dissipated before the TS with very small oscillations still visible in the IAS. After 50 000 years, blue dotted line, the IAS starts to 4. AG CARINAE EVOLUTION WITH PERIODIC OUTFLOW 57

Ag Car for 15 year period

103 ) 3 102 m c . s e l c i t r

a 1 p 10 (

y t i s n e d

r 0

e 10 b m u N

1 10 50 000 Years 100 000 Years 150 000 Years 200 000 Years

10 1 100 101 Distance (pc)

Figure 5.5: The evolution of the same astrosphere as shown in Figure 5.4 with the inclusion of radiative cooling. No ISM magnetic ﬁeld. form and is very small. The solid orange line showing the astrosphere after 100 000 years shows a TS with a larger IAS and a very thin dense shell around the astrosphere.

The oscillations in the number density visible on Figure 5.6 are due to the variation in the number density and outflow velocity at the inner boundary of AG Carinae. Figure 5.7 shows the computed astrosphere’s density profile, velocity profile, Mach number and temperature as a function of distance after 200 000 years. The first panel shows the density profile, the small oscillations in the number density that persist up to the TS. While the number density varies at the inner boundary between the minimum and the maximum phase, the velocity also varies. The second panel on Figure 5.7, however, shows very little variation in the outflow velocity with only small oscillations visible very close to the inner boundary. The large number densities involved from the faster outflow velocity catching up with the slower outflow velocity averages the outflow velocity out throughout the astrosphere. There are larger clumps where the number density does 4. AG CARINAE EVOLUTION WITH PERIODIC OUTFLOW 58

Ag Car for 15 year period

103 ) 3

m 10 c . s e l c i t r a

p 1

( 10

y t i s n e d

e 100 b m u N

10 1 50 000 Years 100 000 Years 150 000 Years 200 000 Years

10 1 100 101 Distance (pc)

Figure 5.6: The same astrosphere shown in Figure 5.4 with the addition of radiative cooling and an ISM magnetic field. Distance and number density are on a log scale. vary slightly until the TS. The Mach number as a function of distance from the inner boundary is shown in the third panel. At the inner boundary, the outflow is supersonic up to the TS where the Mach number drops below 1. The temperature profile of the wind is shown in the fourth panel. The temperature of the wind oscillates with the oscillation in the outflow number density and outflow velocity near the boundary. From this profile, the effect of radiative cooling is clearly visible as the temperature decreases drastically at a distance of ∼7 pc. The temperature is lower than the temperature of the ISM for the size of the OAS.

The size of the astrosphere created during the periodic phase of AG Carinae is shown in Figure 5.8. The distance from the inner boundary to the TS (solid) and BS (dotted) is shown. There are small oscillations in the early stages of the astrosphere simulated with the periodic outﬂow due to the variation in stellar parameters at the inner boundary. 5. SUMMARY AND CONCLUSIONS 59 3 Ag Carinae after 200 000 Years for 15 year period m c . s e l 2

c 10 i t r

a 1 p

10 y t i

s 0

n 10 e d

r 10 1 e b m u N 108 1 s . 7 m 10 c

y t i

c 6 o

l 10 e V

105

103

102

101

100 Mach Number 10 1

106

105

104

103

2 Temperature (K) 10

0 2 4 6 8 10 pc

Figure 5.7: The computed astrosphere around AG Carinae after 200 000 years. The first panel is the density profile, second panel is the velocity profile, third panel is the Mach number and the fourth panel is the temperature.

These oscillations also oscillate the distance that the TS is formed from the inner boundary. At later stages in the evolution, these oscillations are decreased and dissipated. The high densities involved create a very thin dense shell with the TS and BS close to each other in the early stages of the evolution of the astrosphere. After 50 000 years, the distance between the TS and BS increases.

5 Summary and conclusions

As shown in this chapter, the periodic outflow of AG Carinae was different from the periodic outflow of HD 99953, which was discussed in the previous chapter. First the outflow velocity of AG Carinae varied between the minimum velocity of 150 km.s−1 and a maximum velocity of 300 km.s−1 at the inner boundary of 0.03 pc. Secondly, the number 5. SUMMARY AND CONCLUSIONS 60

Termination Shock and Bow Shock Distance

Periodic Termination Shock 7 Periodic Bow Shock

3 Distance (pc)

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 5.8: The distance from the inner boundary to the TS and BS for the computed astrosphere of AG Carinae with periodic variation between the assumed quiescent and maximum phase with a period of 15 years. There is an ISM magnetic ﬁeld strength of 3µG and radiative cooling added in the model.

density for AG Carinae was much larger than for the computed astrosphere of HD 99953. At the inner boundary for the minimum phase of AG Carinae, the stellar wind density −3 −5 −1 was 645.0 particles.cm as the mass loss rate was assumed to be 3.16×10 M .year . For the maximum phase, the stellar wind density was 1612.0 particles.cm−3 as the mass −4 −1 loss rate was 1.58 × 10 M .year . The periodic outflow for AG Carinae had an assumed period of 15 years for illustrative purposes. This period falls between a short S Dor phase and a long S Dor phase. The addition of cooling in the model reduced the size of the astrosphere greatly when comparing Figure 5.4, which had no cooling or magnetic field, to Figure 5.5, which included cooling. The thin dense shell that formed around the astrosphere of AG Carinae was very dense and had a very high compression ratio of ∼30. Figure 5.7 showed that there were only small oscillations in the outflow velocity close to the inner boundary, thereafter the velocity remained constant. There were small 5. SUMMARY AND CONCLUSIONS 61 oscillations in number density up to the TS. AG Carinae showed the same behaviour as HD 99953 for the periodic outflow. The periodic outflow created an astrosphere that was situated between the minimum and maximum astrospheres in size. However, for AG Carinae, the periodic astrosphere did not show an increase in the thickness of the OAS as the astrosphere evolves because of the large compression. As shown in Figure 5.6, not even the introduction of an interstellar magnetic field of 3µG could reduce the compression. CHAPTER 6

Eruptions

As discussed in previous chapters, LBV-type stars undergo microvariations, a short S Dor phase, a long S Dor phase and some have giant eruptions. During these eruptions, the mass that is lost from the star is much greater than during the quiescent phase or S Dor phases and the luminosity increases by many magnitudes (Humphreys & Davidson, 1994). The best documented cases of this eruptive behaviour are the LBV-type stars, η Carinae and P Cygni, (see Sterken, 2003). The mass lost during the ﬁrst eruption of η −1 Carinae in the 1840s was 10 M corresponding to a mass loss rate of 0.1−0.5 M .year . See Puls et al.(2008) and references therein. These mass loss rates are much greater than −3 −1 the current assumed mass loss rate of η Carinae, which is 1.6 ± 0.3×10 M .year (van Boekel et al., 2003). The mass loss rate during eruptions can be determined by the total mass and kinetic energy of the nebulae surrounding these stars. See Smith & Hartigan (2006) and references therein for a discussion on how this was done. P Cygni underwent an eruption phase and the mass lost during this phase is assumed to be ∼0.1M , showing that the mass loss rate during the eruption phase for P Cygni was much less than those for η Carinae (Smith, 2007). The increase in mass loss rate during eruptions for η Carinae and P Cygni is given in Clark et al.(2009) and references therein, as well as the parameters for other LBV-type stars observed to undergo eruptions. According to Smith (2007), the mass lost during the eruptive phases of η Carinae and P Cygni illustrates the likely range of mass ejection in LBV outbursts. Based on the current assumed mass loss rate of η Carinae, the mass loss rate increased by an order of 103 during the giant eruption.

62 1. EVOLUTION OF AN ASTROSPHERE WITH AN ERUPTION THAT INCREASES THE MASS LOSS RATE 63

In this chapter, the effect that these eruptive events might have on the astrospheres surrounding these stars will be studied. The astrospheres are modelled during a quiescent phase without any variation in the mass loss rate, outflow velocity or outflow temperature at the inner boundary. A large increase in the mass loss rate will be introduced into the model after 100 000 years for a short period of time, which for this work is assumed to be 100 years. During this period, the mass loss rate will be drastically increased in the model by a factor of 103 similar to the increase in the mass loss rate for η Carinae as well as a factor of 102 for illustrative purposes. The astrosphere will be compared to the case without any eruptive event taking place to show the difference in size and structure of the astrosphere. For an increase in the mass loss rate of a factor of 103 while keeping the outflow velocity constant, the stellar wind density at the inner boundary of 0.03 pc will also increase by a factor of 103. The stellar wind density and the stellar wind velocity will be increased at the inner boundary during the eruptive phase to show the effect that a faster wind will have on the astrosphere. The increase in velocity at the inner boundary will decrease the stellar wind density at the inner boundary as can be seen from equation 3.1. This chapter includes the effect of radiative cooling in the model and an ISM magnetic field strength of 3µG is assumed in the model.

1 Evolution of an astrosphere with an eruption that increases the mass loss rate

In this section, only the mass loss rate will be increased at the inner boundary of 0.03 pc. −6 −1 The computed astrospheres in this section have a mass loss rate of 1 × 10 M .year and an outflow of 500km.s−1 at the inner boundary of 0.03 pc. This mass loss rate and outflow velocity lead to a stellar wind density of 6.123 particles.cm−3 at the inner boundary of 0.03 pc. These parameters are assumed in the model as they fall in the range of mass loss rates of an LBV-type star as given in Vink & de Koter(2002). The astrosphere is computed with these parameters up to the age of 100 000 years. At this stage, an eruptive event is included in the model where the mass loss rate is increased by a factor of 102 and 103 at the inner boundary for a duration of 100 years. During this eruptive period, the outflow velocity at the inner boundary is kept constant at 500km.s−1 for this section. This leads to an increase in the stellar wind density at the inner boundary of 102 and 103, respectively. The ISM density surrounding these astrospheres is assumed to be 10.0 particles.cm−3.

The computed astrosphere having no eruptive event, the computed astrosphere that has an eruptive event increasing the mass loss rate by a factor of 102, and the computed astrosphere that has an eruptive event that increases the mass loss rate by a factor of 103 1. EVOLUTION OF AN ASTROSPHERE WITH AN ERUPTION THAT INCREASES THE MASS LOSS RATE 64

Evolution of astrosphere with eruptive event

101

100

10 1

) 100 000 years

3 10 2 m

c 1

. 10 s e l c i t r 0 a 10 p (

y t i s

n 10 1 e d

r e b 2 150 000 years

m 10 u N 101

100

10 1

2 10 200 000 years

101

100

10 1

Mass loss increase by 102 Mass loss increase by 103 10 2 250 000 years No eruption

0 2 4 6 8 10 Distance (pc)

Figure 6.1: The evolution of an astrosphere with an eruptive event at 100 000 years. −6 −1 The eruptive event increases the mass loss rate from 1 × 10 M .year , shown by −4 −1 the green dashed line, to 1 × 10 M .year , shown by the blue dotted line, and to −3 −1 1 × 10 M .year , shown by the solid orange line, respectively. The event lasts for 100 years. 1. EVOLUTION OF AN ASTROSPHERE WITH AN ERUPTION THAT INCREASES THE MASS LOSS RATE 65 can be seen in Figure 6.1. This figure shows the evolution of these three computed astrospheres. The astrosphere without any eruptive event is shown by the green dashed line, the astrosphere that has an eruptive event increasing the mass loss rate by 102 is shown by the blue dotted line and the astrosphere that has an eruptive event that increases the mass loss rate by 103 is shown by the solid orange line. For the duration of the eruptive events, the event that increases the mass loss rate by 102, the stellar wind density at the inner boundary increases from 6.123 particles.cm−3 to 6.123 × 102 particles.cm−3, while for the event that increases the mass loss rate by 103 the stellar wind density at the inner boundary is 6.123 × 103 particles.cm−3. The first panel in Figure 6.1 shows these three computed astrospheres after 100 000 years as before this stage the three astrospheres are exactly the same. The eruptive event can be seen from the large increase in density near the inner boundary of the blue dotted and solid orange lines. The second panel shows the three simulated astrospheres 50 000 years after the eruption has occurred. Eruptions of this magnitude seem to have very little effect outside of the AP. Only the eruptive event that increased the mass loss rate by a factor of 103 has an effect beyond the AP. There are variations in the density in the IAS as there are waves travelling back and forth as the much denser wind collides with the AP and some of which is reflected backwards. The third panel shows the astrospheres 100 000 years after the eruption. At this stage, the oscillations in the density in the IAS have decreased and the computed astrosphere, which underwent an eruption that increased the mass loss rate by a factor of 102 is not distinguishable from the astrosphere that had no eruption assumed in the model. The fourth panel shows the evolution of the three astrospheres 150 000 years after the eruption. The only noticeable difference in the three astrospheres is the slight increase in the thickness of the OAS. 1. EVOLUTION OF AN ASTROSPHERE WITH AN ERUPTION THAT INCREASES THE MASS LOSS RATE 66

Termination Shock, Astropause and Bow Shock Distance

No Eruption Termination Shock Eruption (102) Termination Shock Eruption (103) Termination Shock 5 No Eruption Astropause Eruption (102) Astropause Eruption (103) Astropause No Eruption Bow Shock Eruption (102) Bow Shock 4 Eruption (103) Bow Shock

3 Distance (pc) 2

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 6.2: The distance from the inner boundary of 0.03 pc to the TS, AP and BS for the computed astrospheres where there was no eruptive event (blue), an eruptive event that increased the mass loss rate by a factor of 102 (red) and an eruptive event that increased the mass loss rate by a factor of 103 (green). The eruptive event lasted for 100 years.

The effect of the eruptions is more visible when one looks at the evolution of the astrospheres in terms of the distance from the inner boundary to the TS, AP and BS. This is shown in Figure 6.2. In Figure 6.2, the astrosphere that has no eruptive event is shown in blue, the event that increases the mass loss rate by 102 for the eruption is shown in red, and the event that increases the mass loss rate by 103 is shown in green. The solid lines show the three computed astrospheres’ TS, the dashed-dotted lines show the astrospheres’ AP and the dotted line show the three astrospheres’ BS. This figure shows that the eruptions have a very small effect on the size of the astrosphere as the distance from the inner boundary to the BS for all three astrospheres is very similar. The AP distance is also largely unaffected by the eruption. However, the eruptions do have a large effect on the TS distance. The eruptions cause the distance from the inner 2. ASTROSPHERIC EVOLUTION WITH AN ERUPTION THAT INCREASES MASS LOSS RATE AND VELOCITY 67 boundary to the TS to oscillate around until it settles after about ∼100 000 years after the eruption.

2 Astrospheric evolution with an eruption that increases mass loss rate and velocity

In this section, the computed astrospheres undergo an eruptive event that increases the 2 3 −6 −1 mass loss rate at the inner boundary by a factor of 10 and 10 , from 1×10 M .year −4 −1 −3 −1 to 1 × 10 M .year and 1 × 10 M .year , respectively. During these eruptive events, the outflow velocity at the inner boundary of 0.03 pc will also be increased from 500km.s−1 to 750km.s−1. In the previous section, when the mass loss rate was increased by a factor of 102 or 103, the stellar wind density also increased by a factor of 102 or 103, respectively, due to the outflow velocity being kept constant. In this section, the increase in the mass loss rate and outflow velocity leads to a stellar wind density at the inner boundary that is less than the equivalent eruptive event in the previous section as can be seen from equation 3.1. The astrospheres have an assumed stellar wind density at the inner boundary of 6.123 particles.cm−3 when there is no eruptive event. −6 −1 During the eruptive event that increases the mass loss rate from 1 × 10 M .year −4 −1 −1 −1 to 1 × 10 M .year and the velocity from 500km.s to 750km.s , the stellar wind density at the inner boundary is 4.082 × 102 particles.cm−3. The larger eruptive event that increases the mass loss rate by 103 and the outflow velocity by 250km.s−1 has a stellar wind density at the inner boundary of 4.082×103 particles.cm−3 for the duration of the event. The ISM density for this section is also assumed to be 10.0 particles.cm−3.

The computed astrospheres, one where there is no eruption, one with an eruption that increases the mass loss rate by 102 and the outflow velocity from 500km.s−1 to 750km.s−1 and one with an eruption that increases the mass loss rate by 103 and also the outflow velocity from 500km.s−1 to 750km.s−1 are shown in Figure 6.3. This figure shows the same three astrospheres shown in Figure 6.1; however, during the eruptions in this figure the outflow velocity was also increased by a factor of 1.5 at the inner boundary. In this figure, the first panel shows the three computed astrospheres at the onset of the eruption. There is a large increase in the density, but different from eruptions shown in Figure 6.1, where there was no increase in the outflow velocity. The increased density is followed by a lower dense area as the mass during the eruption is ejected away from the star at a higher velocity than the continuous stellar wind. The second panel shows the three scenarios 50 000 years after the eruption and for the smaller eruption, which increased the mass loss rate by 102, the computed astrosphere has a structure very similar to the corresponding scenario in Figure 6.1. For the larger eruption, however, 2. ASTROSPHERIC EVOLUTION WITH AN ERUPTION THAT INCREASES MASS LOSS RATE AND VELOCITY 68

Evolution of astrosphere with eruptive event

101

100

10 1

10 2

) 100 000 years 3 10 3 m

c 1

. 10 s e l c i t

r 0

a 10 p (

y t i

s 10 1 n e d

e 2

b 10 150 000 years m u N 101

100

10 1

2 10 200 000 years

101

100

10 1

Mass loss increase by 102 Mass loss increase by 103 10 2 250 000 years No eruption

0 2 4 6 8 10 Distance (pc)

Figure 6.3: The evolution of an astrosphere with an eruptive event at 100 000 years. −6 −1 The eruptive event increases the mass loss rate from 1 × 10 M .year , shown by −4 −1 the green dashed line, to 1 × 10 M .year , shown by the blue dotted line, and to −3 −1 1 × 10 M .year , shown by the solid orange line, respectively. The eruption also increases the outﬂow velocity from 500km.s−1 to 750km.s−1 at the inner boundary. The event lasts for 100 years. 2. ASTROSPHERIC EVOLUTION WITH AN ERUPTION THAT INCREASES MASS LOSS RATE AND VELOCITY 69 there are larger oscillations in the density in the IAS, and the TS is pushed further from the inner boundary compared to Figure 6.1. The AP distance has also increased. In the third panel, where the astrospheres have continued to evolve for 100 000 years after the eruption, the oscillations in the density for both the small and large eruption have decreased drastically. The size of the astrosphere that had the large eruption is larger than the one that experienced the small eruption and the one without an eruption. The fourth panel shows the astrospheres 150 000 years after the eruptions had taken place and there are no signs of any eruption having taken place when one compares the astrosphere that had the small eruption, which increased the mass loss rate by 102 and the velocity by a factor of 1.5, to the astrosphere that had no eruption. The sizes of these astrospheres are very similar and there is no diﬀerence in their structure.

Termination Shock, Astropause and Bow Shock Distance

6 No Eruption Termination Shock Eruption (102) Termination Shock Eruption (103) Termination Shock No Eruption Astropause Eruption (102) Astropause 5 Eruption (103) Astropause No Eruption Bow Shock Eruption (102) Bow Shock Eruption (103) Bow Shock 4

3 Distance (pc) 2

0 0 25 50 75 100 125 150 175 200 Years (103)

Figure 6.4: The distance from the inner boundary of 0.03 pc to the TS, AP and BS for the computed astrospheres where there was no eruptive event (blue), an eruptive event that increased the mass loss rate by a factor of 102 (red) and an eruptive event that increased the mass loss rate by a factor of 103 (green). The eruptive event lasted for 100 years 3. SUMMARY AND CONCLUSION 70

The radii of the TS, AP and BS for this scenario are shown in Figure 6.4. From this figure, it is clear that the astrosphere created with the inclusion of a small eruption has a very similar size to an astrosphere where there is no eruption. An increase of 102 in the mass loss rate and an increase in the outflow velocity by a factor of 1.5 at the inner boundary for 100 years is unable to affect the outer structure of the astrosphere. The distance from the inner boundary to the TS does oscillate around the distance that the TS is for the case without an eruptive event. For the case where the eruption increases the mass loss rate by 103 and the velocity by a factor 1.5, the oscillations of the TS are much greater. This larger eruption also increases the distance between the inner boundary and the AP as well as the distance to the BS.

3 Summary and conclusion

The eruption phases of LBV-type stars eject large amounts of stellar matter in a short period of time. This was modelled in this section by computing astrospheres that had −6 −1 −1 a mass loss rate of 1 × 10 M .year and an outflow velocity of 500km.s at the inner boundary of 0.03 pc. At 100 000 years, an eruptive event was introduced, which increased the mass loss rate by factors of 102 and 103, respectively, while keeping the outflow velocity constant. This event lasted for 100 years and the results were shown in Figures 6.1 and 6.2. These figures showed that these eruptions have no effect on the outer structure of the astrospheres, except for slightly increasing the distance from the inner boundary to the BS when the eruption increased the mass loss rate by 103. The eruptive events caused oscillations in the inner structure of the astrosphere. The high mass expanded outwards pushing the TS closer to the AP, after which it was bounced back and a wave was formed being reflected back and forth from the inner boundary to the TS. This wave caused the oscillations in the distance the TS was from the inner boundary.

Figures 6.3 and 6.4 showed the same two eruptive events that increased the mass loss rate by factors 102 and 103, while also increasing the outﬂow velocity at the inner boundary from 500km.s−1 to 750km.s−1. The increased outﬂow velocity created a lower dense area once the eruptive event has occurred. The increased velocity along with the increased mass loss rate during the eruptive events created larger astrospheres compared to the other scenarios. The oscillations in the TS are larger than the other scenarios, but were still dissipated with time as the continued evolution of the astrosphere after the eruptive event averaged out the density in the inner structure. 4. FUTURE WORK AND IMPROVEMENTS 71

4 Future work and improvements

Due to the limitations of the model only being able to calculate one- and two-dimensional results, the results in this work are all one-dimensional. In future this work can be improved by extending the model to calculate in three-dimensions giving a clearer view of the influences of the ISM magnetic field and cooling. This will also provide insights into the effect that varying ISM magnetic field inclinations will have on the evolution of the modelled astrospheres. Bibliography

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