Bifurcation between stable and unstable configurations

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The magnetic field of normal stars

As I mentionned in the introduction (see Sect. 1.4), until recently the only type of magnetic fields known in hot stars were the strong magnetic fields, with a longitudinal field strength above 100 G (Auri`ere et al. 2007; Wade et al. 2014b) associated with a quite simple and stable topology (e.g, L¨uftinger et al. 2010; Silvester et al. 2014).
In 2009, a longitudinal magnetic field much weaker than any previous detection in hot stars has been measured in the normal A star Vega (Ligni`eres et al. 2009; Petit et al. 2010). The measurement leads to a longitudinal magnetic field value less than 1 Gauss. This field is much weaker than the lower limit of ∼ 100 G of the strong magnetic fields. Some scenarios was developed to explain the dichotomy between strong and weak magnetic fields (Auri`ere et al. 2007; Braithwaite & Cantiello 2013). They predict that the new kind of weak magnetic field exists in all hot stars that do not host a strong magnetic field (see Sect. 1.4). Vega may thus well be the first confirmed member of a new family of magnetic stars: the weakly magnetic hot stars.
The motivation to progress on this topic is strong because the discovery of a new, po-tentially widespread class of weakly magnetic A stars offers important new information about the dichotomy between strong and weak magnetic fields in tepid stars. In an at-tempt to interpret this division, Auri`ere et al. (2007) proposed a scenario based on the stability of a large scale magnetic configuration in a differentially rotating star, leading to estimating a critical field strength above which magnetic fields can remain stable on long time scales, while magnetic fields below this limit would likely be destroyed by the internal shear. More detailed models including 2D and 3D numerical simulations (Gaurat et al. 2015; Jouve et al. 2015) tend to confirm the existence of a critical field in such configurations, where the pre-main sequence contraction is a possible way to force differential rotation. On the other hand, the magnetic dichotomy might simply be the result of two different magnetic field generation processes. Braithwaite & Cantiello (2013) proposed that Vega-like magnetic stars are the result of the slow evolution of magnetic configurations characterized by weak initial magnetic helicity and argue that it should be widespread among most intermediate-mass and massive stars. Meanwhile, Ferrario et al. (2009) proposed that the small fraction of strong magnetic fields could be produced in early stellar merging events.
Studying stars with the same precision as Vega can confirm or disprove the prediction of the scenarios to explain the dichotomy and bring constraints for these theories. In particular, studying the weakly magnetic hot stars can help us to understand the origin and the properties of this kind of magnetism.

Vega

Introduction

Vega is a normal bright A0V star (V=0.03). Since it is often used as a standard star, it is well studied and its spectral and stellar parameters are thus well known. It is a fast rotator, however due to its inclination (i ∼ 7o) its v sin i is low: v sin i =22 km s−1 (Takeda et al. 2008). Its temperature is Teff =9519 K and its log g is 3.88. Due to the flattening at the pole, Teff increases from the pole to the equator. In this study, we have taken the temperature of the pole as a reference for Teff .
The longitudinal magnetic field of Vega is very weak (less than 1 Gauss) and the mag-netic map of the surface of Vega obtained thanks to Zeeman-Doppler Imaging technique (ZDI, Petit et al. 2010) revealed a magnetic spot concentrated around the rotational pole of Vega with a strength of about 7 G (Petit et al. 2014a).
B¨ohm et al. (2015) found temporal variabilities with a period similar to the rotational period of Vega. This rotational modulation reveals the existence of spots on the surface of Vega close to the equator. These spots can have different explanations: they can be chemical spots or magnetic spots (like the ones of the Sun). However, due to the high rotational velocity, the atmosphere of Vega is not stable enough to generate chemical spots. Therefore, the weak magnetic field of Vega is a likely explanation for this spot. This confirms the finding of the ZDI map, which shows weak magnetic spots close to the equator. The magnetic field of Vega is more complex than the one of the strongly magnetic stars.
In addition,Butkovskaya et al. (2011) found a long periodic variation of Vega, with a period of 21 years, by measuring the equivalent width and the flux density of various lines. This variability may also be due to the weak magnetic field of Vega, for example if it undergoes a magnetic cycle. Studying the evolution of the magnetic field of Vega over several years can help us to understand the origin of this variability.

Observations

The data were obtained with the spectropolarimeters Narval and ESPaDOnS and were collected in the polarimetric mode measuring Stokes V (circular polarization). Vega was observed over several years: 2008, 2009, 2010, 2011, 2014, and 2015 (see Table 5.1), to determine if its magnetic field changes over the years. Due to a mechanical problem on one rhomboedra, the data taken on 21 and 22 August 2011 are not exploitable in polarization. This unfortunately corresponds to 42% of the total data gathered that year.

Data Analysis

In the absence of any detectable polarized signatures in individual spectral lines of Vega, I applied the LSD procedure (see Sect. 2.5.2) to each spectrum. The line list is extracted from the VALD atomic data base (Kupka & Ryabchikova 1999; Piskunov et al. 1995) using the respective effective temperature and log g of Vega. This line list is extracted using Teff =10000K and log g=4.0. I rejected the lines whose depth is less than 1% of the continuum and the lines blended with the hydrogen lines. The final mask contains 1041 lines.
Due to the weakness of the signatures of the magnetic field on Vega, we co-added the observations taken each year to reach a high S/N ratio. To coadd the LSD profiles, I weighted each individual spectra proportionally to its squared S/N:
where wi and S Ni are the weight and S/N of the ith spectrum.
I renormalized the yearly averaged spectra with the continuum task of IRAF, because the automatic spectral normalization of Libre ESprit is not sufficiently precise. If the continuum is not very flat, it can influence the LSD profiles.
I obtained a definite detection of a magnetic field for the yearly averaged profiles of 2008, 2009, and 2010 (see Table 5.2). For these years, the FAP is less than 3.603×10−6. For 2011 and 2014, I obtained marginal detections (see Table 5.2) with a FAP of ∼ 3×10−5. Due to the low S/N in the Stokes V profiles (see Table 5.2), I obtained a non-detection for 2015 with a detection probability of ∼50% and a FAP of 4.791×10−1. Nevertheless, a Stokes V signature is visible every year (see Fig. 5.1) When comparing the results obtained for each year, I find no significant variability in the Zeeman signatures in the Stokes V profiles (see Fig. 5.1). The magnetic field of Vega seems stable over the years of observations. Thanks to the center-of-gravity method (Semel et al. 1993), I calculated the longitudinal magnetic field for each yearly averaged profile using a mean Land´e Factor of 1.27 and a mean wavelength of 500 nm corresponding to the normalization of the LSD profiles. The 6 values of the longitudinal magnetic field are coherent, with a precision of 0.2-0.5 G (see Table 5.2).
FIGURE 5.1: Yearly average Stokes V LSD profiles: in black for 2008, in red for 2009, in green for 2010, in blue for 2011, in yellow for 2014, and in brown for 2015.
This stability of the magnetic field of Vega is in agreement with the scenario that explains the dichotomy between strong and weak fields, which argues that the origin of the weak field is the same as the strong fields, i.e. it is a fossil origin and not a dynamo origin.
Butkovskaya et al. (2011) find variation in the intensity of the lines of Vega with a period of 21 years. I compared the depth of the intensity LSD profiles for each year. No significant variability is detected in the intensity LSD profiles (see Fig. 5.2) between 2008-2015, contrary to Butkovskaya et al. (2011)’s claim. Even if the observations did not cover the full period of 21 years, if I follow the modulation of the lines found by Butkovskaya et al. (2011), the depth of the intensity profiles should decrease between 2008 and 2011 and increase between 2011 and 2015. I do not observe this behaviour in the intensity LSD profiles (see Fig. 5.2).
However, the LSD technique can influence the shape of the intensity profiles. To further check if there is a variation in the depth of the lines of Vega, I compared the depth of the individual lines used by Butkovskaya et al. (2011), i.e. Fe at 516.7 nm and Mg at 518.3 nm for each year of observations(see Fig. 5.3).
For the Fe line at 516.9 nm Butkovskaya et al. (2011) found between their observations in 2008 and 2010 a depth variation of ∼ 6% and for the Mg line at 518.3 nm a depth variation of 3%. I found no significant variability in these two lines (see Fig. 5.3). For the Fe line at 516.9 nm I obtained a depth variations of 4% however this variations did not follow the behavior predicted by Butkovskaya et al. (2011). The deepest Fe line is the one of 2015 instead of the one of 2011. The Fe line for the other years of observations seems to follow the variability predicted by Butkovskaya et al. (2011). For the Mg line at 518.3 nm, the variation is ∼ 1.5%.

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Conclusion

I did not find long-term variability in the magnetic field of Vega, nor in its inten-sity LSD profiles. However, the observations did not cover the whole period found by (Butkovskaya et al. 2011), we should to observe again Vega during the next years.
The stability in the magnetic field of Vega is in favor of a fossil origin of the field, as predicted by the scenarios to explain the dichotomy. These scenarios are based on a common origin of strong and weak magnetic fields, but during the evolution the magnetic field distribution splits into two distinct families.

UZ Lyn

Introduction

It is probable that Vega is the first detection of a new type of magnetic stars among intermediate-mass stars. The scenarios to explain the dichotomy (Auri`ere et al. 2007; Braithwaite & Cantiello 2013) predict that this kind of ultra-weak fields exists in all stars that do not host a strong magnetic field. Confirming that this new type of mag-netism is common among intermediate-mass stars will provide important constraints on the magnetic field of a typical intermediate-mass star. Compared to strong fossil magnetism, which only concerns a small fraction of stars, such constraints would have a strong impact for stellar evolution models. It is therefore important to perform deep spectropolarimetric studies of chemically normal A stars to determine if ultra-weak magnetic fields are typical of A stars. It is however challenging, due to the high S/N required to detect these weak fields and thus the large amount of observing time spent per target. As a consequence, we need to carefully choose the targets for such studies.
One good target for this study is UZ Lyn. UZ Lyn is a bright star and it is a long-period spectroscopic binary (Lehmann et al. 2003) with an orbital period of 3.6 years. The primary is a normal A2V star with a temperature Teff =9310±100 K and log g=4.1 ± 0.1 (Lehmann et al. 2003). Therefore, its stellar parameters are close to the ones of Vega. UZ Lyn is only slightly cooler than Vega. The spectral type of the secondary component is not known, but Lehmann et al. (2003) estimated its mass to 0.46 M⊙. The chemical abundances of the primary are close to the solar ones (Caliskan & Adelman 1997; Lehmann et al. 2003), like for Vega (Takeda et al. 2008).
UZ Lyn is thus a good target to test if ultra-weak magnetic fields like the one of Vega are common in normal A-star type or if Vega is a peculiar case.

Observations

UZ Lyn was observed 39 times with the Narval spectropolarimeter in circular polariza-tion mode during semester 2013B, 6 times during semester 2014A, and 20 times during semester 2014B, in the frame of a program to search for ultra-weak magnetic fields in A stars (PI: F. Ligni`eres), for a total of 65 observations. UZ Lyn was also observed twice in the frame of the Binarity and Magnetic Interactions in various classes of Stars (BinaMIcS) project (a project to study the magnetic field of close binaries, PI: E. Ale-cian, see Neiner et al. 2015a) with Narval (PI: C. Neiner). I thus have a total of 67 observations.

Data Analysis

I ran the LSD technique (see Sect. 2.5.2) on the individual spectra using a line mask corresponding to the temperature and log g of the primary. The mask contains 879 lines after removing the hydrogen lines, the lines blended with the hydrogen lines and the ones that I did not see in the spectra. I used a velocity bin of 5.4 km s−1 to improve the S/N and to have around 20 velocity bins in the line profile.
The S/N of the individual Stokes V profiles is around 30000-40000 except for some nights during which the weather was not good enough. This S/N is not high enough to detect a Zeeman signature in the individual LSD profiles, therefore I co-added the spectra taken on the same night, when possible, to improve the S/N. I obtained non-detections for the nightly-averaged LSD profiles, expect for the observation taken on 12 October 2013. For this night I obtained a marginal detection with a detection probability of 99.03% inside the stellar lines and a detection probability of 25% outside the stellar lines. The result of the nightly-averaged LSD profiles on 12 October 2013 is shown in Fig. 5.4.
The signature is more prominent on the blue side of the line profile (see Fig. 5.4).
Using the centre-of-gravity method (Rees & Semel 1979) with a mean wavelength of 500 nm and a mean Land´e factor of ∼ 1.234 corresponding to the normalization parameters used in the LSD, I calculated the longitudinal field value (Bl) corresponding to these Zeeman signatures over the velocity range [-50:50] km s−1. The measured longitudinal magnetic field is -4.3 ± 4.2 G. The Null polarization is 5.2 ± 4.1 G for the night of 12 October 2013.
For the other nightly-averaged LSD profiles, the detection probability is between 5% and 88% inside the stellar lines, which corresponds to a non-detection. Therefore, on 36 nights of observations, I obtained one marginal detection and 35 non-detections.
To improve the S/N further, I co-added all LSD profiles, even the ones with a low S/N because they have a low influence in the grand-average profile due to their low weight. Due to the binary, the radial velocity changes over the observations. To co-add the LSD profiles, I thus realigned the LSD profiles on the same radial velocity. To know the radial velocity of the observation, I fitted the core of the intensity profile by a Gaussian.

Table of contents :

1 Introduction 
1.1 Hot stars
1.2 Magnetic fields in hot stars
1.3 Origin of the magnetic field in hot stars
1.3.1 The dynamo hypothesis
1.3.2 The mergers hypothesis
1.3.3 The fossil field origin
1.4 The dichotomy between strong and weak fields
1.4.1 Bifurcation between stable and unstable configurations
1.4.2 Failed fossil field
1.5 Goal of the thesis
2 Detecting magnetic fields 
2.1 Polarization
2.2 Zeeman effect
2.3 Measuring polarization
2.4 The spectropolarimeter Narval
2.5 Data analysis
2.5.1 Data reduction with Libre-Esprit
2.5.2 Least Square Deconvolution technique
2.5.3 Detection probability
2.5.4 Longitudinal magnetic field
2.5.5 Oblique rotator model
I Strong magnetic fields 
3 The supergiant ζ Ori A 
3.1 Introduction
3.2 Observations
3.2.1 Narval spectropolarimetric observations
3.2.2 Archival spectroscopic observations
3.3 Checking for the presence of a magnetic field
3.4 Separating the two components
3.4.1 Identifying the lines of each component
3.4.2 Spectral disentangling of Narval data
3.4.3 Disentangling using the archival spectroscopic data
3.5 Measuring the longitudinal magnetic field of ζ Ori Aa
3.5.1 Using the Narval data and correcting for the companion
3.5.2 Using synthetic intensity profiles
3.5.3 Using disentangled spectroscopic data
3.6 No magnetic field in ζ Ori Ab
3.6.1 Longitudinal magnetic field values for ζ Ori Ab
3.6.2 Upper limit on the non-detected field in ζ Ori Ab
3.7 Magnetic field configuration
3.7.1 Rotational modulation
3.7.2 Field strength and geometrical configuration
3.7.3 Stokes V modeling
3.8 Magnetospheres
3.8.1 Magnetospheric parameters
3.8.2 Hα variations
3.9 Discussion and conclusions
4 The upper limit of the magnetic desert 
4.1 Introduction
4.1.1 Choice of targets
4.2 Data Analysis and longitudinal field measurements
4.2.1 HD12447
4.2.2 HD19832
4.2.3 HD22470
4.2.4 HD28843
4.2.5 HD32650
4.2.6 HD96707
4.3 Dipolar magnetic field
4.3.1 HD19832
4.3.2 HD32650
4.3.3 HD96707
4.3.4 HD22470
4.3.5 HD28843
4.4 Conclusion
II Ultra-weak magnetic fields
5 The magnetic field of normal stars 
5.1 Vega
5.1.1 Introduction
5.1.2 Observations
5.1.3 Data Analysis
5.1.4 Conclusion
5.2 UZ Lyn
5.2.1 Introduction
5.2.2 Observations
5.2.3 Data Analysis
5.2.4 Conclusions
5.3 B stars
5.3.1 Introduction
5.3.2 Choice of targets
5.3.3 Observations
5.3.4 ι Her
5.3.5 γ Peg
5.3.6 Conclusion for B stars
5.4 Conclusion
6 Weak magnetic fields in chemically peculiar stars 
6.1 The Am stars: β UMa and θ Leo
6.1.1 Introduction
6.1.2 Selected targets
6.1.3 Data analysis
6.1.4 Results
6.1.4.1 LSD profiles with complete line mask
6.1.4.2 Possible instrumental artifacts at high SNR
6.1.4.3 Establishing the Zeeman origin of Stokes V signatures
6.1.5 Discussion
6.1.5.1 Peculiar Stokes V signatures in Am stars
6.1.5.2 Origin of the magnetism of Am stars
6.1.5.3 Towards a systematic exploration of weak magnetic fields in Am stars
6.2 The Am star of Alhena
6.2.1 Introduction
6.2.2 Observations
6.2.3 Magnetic analysis
6.2.4 Discussion and conclusion
6.3 The HgMn star: α And
6.3.1 Introduction
6.3.1.1 The target: α And
6.3.2 Observations and data analysis
6.3.3 Data Analysis
6.3.4 The secondary: α And B
6.3.5 Conclusion
6.4 Conclusion
7 Conclusions and perspectives

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