The upper limit of the magnetic desert 

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The mergers hypothesis

Another explanation to generate a magnetic field in hot stars is the merging of two pre-main sequence stars. During the merging of young proto-stars, a magnetic field can be generated through a dynamo process due to the strong shears (Ferrario et al. 2009). Mergers of stars are rare and that could explain the observed small fraction of magnetic hot stars. This scenario predicts that no magnetic field exists in close binaries (Schneider et al. 2016) and that the produced merger star shows significant nitrogen enrichment at its surface (Glebbeek et al. 2013). However, a double magnetic close binary system, ǫ Lupi, was discovered by Shultz et al. (2015). Moreover, no observational proofs could confirm this scenario.

The fossil field origin

Fossil magnetic fields are products of a seed field (Mestel 1999). In the fossil field theory, the magnetic field observed at the surface of hot stars is a remnant of the magnetic field of the molecular cloud from which the star was formed. During the early stages of stellar formation, the proto-star is fully convective, a dynamo can enhance and sustain the seed field. As the radiative core appears and the convective zone disappears in the center of the star, the magnetic field relaxes onto a large-scale dipole, like the ones observed at the surface of hot stars. In addition, it is possible that the appearance of the convective core, just at the end of the stellar formation, produces a tilt of the dipole and explains why we observed oblique dipoles in basically all hot stars (Featherstone et al. 2009). Figure 1.1 shows a scheme of the fossil field scenario.
One of the issue of this theory was the survival of the magnetic field: it is difficult to find a magnetic field that is stable enough to survive during the lifetime of the stars. Indeed, a magnetic field is only stable on the Alfv´en timescale, that corresponds to the time needed for a magnetic Alfv´en wave to cross the star. This time depends on the magnetic strength, but for a typical magnetic hot star it is around a few years. However, Braithwaite & Spruit (2004) and Duez & Mathis (2010) demonstrated thanks to simulations and analytical work that magnetic fields can be stable over the stellar lifetime if it is a mix between toroidal and poloidal components. They modeled a star without rotation with a random magnetic field. At the beginning, the field decreases rapidly. However at some point, the decreasing stops and the field reaches a stable configuration that is the same whatever the initial conditions. The configuration is approximatively axisymmetric and made of toroidal and poloidal components. At the stellar surface, it appears as a dipole, just like the ones observed on hot stars.
In addition, Alecian (2012) showed that Herbig Ae/Be stars, which are the precursors of the magnetic Ap/Bp stars, host magnetic fields with a similar occurrence rate and configuration as main sequence hot stars. This indicates that the fields observed in hot stars are already present at the pre-main sequence phase. It thus provides support for the fossil field theory.
Therefore, it is now well established that the magnetism in hot stars is of fossil origin. However, the creation and evolution of these fields are not known in detail, during the stellar formation, and require further investigations. One remaining problem of the fossil magnetic field theory is that it does not explain why only 7% of hot stars host a magnetic field (Grunhut & Neiner 2015).

The dichotomy between strong and weak fields

All magnetic massive and intermediate-mass stars discovered until recently have a strong dipolar magnetic field, with a typical strength 3of kG (Power et al. 2007). These fields are stable over time. The fossil field origin is well established for these fields.
Recently, Ligni`eres et al. (2009) discovered an ultra-weak magnetic field in the normal A star Vega. The spectropolarimetric time series was interpreted in terms of a surface magnetic field distribution using the Zeeman-Doppler Imaging technique (ZDI, Petit et al. 2010), unveiling a peak local field strength of about 7 G (Petit et al. 2014a). The results of that study support the view that Vega is a rapidly rotating star seen nearly pole-on, and the reconstruction of the magnetic topology at two epochs revealed a magnetic region of radial field orientation, closely concentrated around the rotation pole. Another ultra-weak magnetic field was discovered in the chemically peculiar Am star Sirius (Petit et al. 2011). However, the shape of the signature in Sirius is peculiar with a prominent positive lobe without negative lobe. This signature is not expected in the normal Zeeman theory. These two stars may well be the first confirmed members of a much larger, as yet unexplored, class of weakly magnetic hot stars.
What is the origin of these weak magnetic fields? Two different theories can explain the dichotomy between strong fields and ultra-weak fields: the first one assumes that the two types of magnetic fields are produced by two different processes and have dif-ferent properties. The second assumes that the strong and weak fields have a common origin but, during the stellar evolution, there is a bifurcation between weak and strong magnetic fields. The latter explanation was used in various scenarios to explain the dichotomy between strong and weak magnetic fields.

Bifurcation between stable and unstable configurations

To explain the dichotomy between strong and weak magnetic fields, Auri`ere et al. (2007) proposed a scenario based on the stability of large scale magnetic configurations in dif-ferentially rotating stars. If the magnetic field is weak enough, it cannot prevent the winding-up of the poloidal field into a toroidal field. The increasingly toroidal config-uration is expected to become unstable to a pinch-type instability, called the Tayler instability, that replaces the large scale field configuration by a new configuration struc-tured at the length scale of the instability. On the contrary, if the initial magnetic field is strong enough, Maxwell stresses will tend to impose uniform rotation and eventually lead to a stable configuration. Starting with an initial distribution of magnetic fields ranging from low to high dipolar strengths, this mechanism predicts a sharp decrease of the longitudinal field – i.e. the surface averaged of the line-of-sight field component – between stable and unstable configurations. An order of magnitude estimate of the critical field separating the stable and unstable configuration yields: Bc = (4πρ)1/2rΩ. This estimate turns out to be quite close to the observed minimum field of Ap stars and the predicted increase of the critical field with the rotation is also compatible with the few existing data. See Figure 1.2 for an illustration of this scenario.
Regarding the 7% occurrence issue, Auri`ere et al. (2007)’s scenario does not assume that at the end of their formation process some stars are magnetic and others not. It proposes instead that all stars are initially magnetic and that the differentiation leading to the 7% incidence rate is due to a subsequent bifurcation between stable and unstable configurations.
Models of a magnetic field in a differentially rotating radiative zone (Gaurat et al. 2015; Jouve et al. 2015) show that instabilities destroy the large scale magnetic field when the poloidal field is weaker than a critical value. These models are compatible with the scenario developed by Auri`ere et al. (2007).

Failed fossil field

Braithwaite & Cantiello (2013) also proposed a scenario to explain the dichotomy. In this scenario, called failed fossil field, the magnetic fields evolve dynamically toward an equilibrium in absence of any driving from differential rotation, convection or merid-ional circulation. At first, the magnetic field is not in equilibrium, and evolves on its own dynamic timescale. As it does so, magnetic energy is lost and the field strength drops. While in the strongly magnetic stars an equilibrium is quickly reached and the field essentially stops evolving (a fossil field), in weakly magnetic stars the field is still evolving. If the time to reach equilibrium is shorter than the age of the star, it is called a failed fossil field. The time to reach the stable configuration depends on the rotation of the star. For slow rotators, this time is longer than for fast rotators. This theory predicts that this kind of unstable magnetic field exist in the fraction of massive stars that do not have strong magnetic fields, younger stars should tend to have stronger fields and faster rotators have stronger fields.
For both theories presented above, there is a lack of observational constraints. New ob-servations are needed to provide constraints to differentiate between theories to explain the bifurcation between strong and weak magnetic fields.

Goal of the thesis

Understanding the magnetism of massive and intermediate-mass stars is critical to make progress in stellar evolution theory. Magnetic fields are key actors in the evolution of all stellar objects. Although the influence of magnetic fields on stellar evolution has been recognized for a long time, progress have been challenged by a lack of observational constraints combined with the difficulty in modeling magnetohydrodynamics processes. This is particularly true in the range of intermediate-mass and massive stars. Until recently, only a small (∼7%) fraction of hot stars were known to be magnetic (Grunhut & Neiner 2015) usually with a simple topology (i.e. dipolar) and their magnetic fields are stable in time. To explain these properties, the current theory, the fossil field theory, describes this magnetism as remnant of an early phase of the star’s life, but leaves many basic questions unanswered, such as the small fraction of magnetic stars, and in practice provides no constraint to stellar evolution theory.
In last decades, progress was achieved to understand this magnetism thanks to a new generation of spectropolarimeters. First, the similar detection rate of strong magnetic fields among intermediate-mass and massive stars suggest a common origin to the mag-netic fields of all hot stars. Then, the observations of Ap/Bp stars revealed the lower limit to the magnetic fields of intermediate-mass stars and the existence of a magnetic desert between the strong magnetic fields and the weak magnetic fields like the one detected on Vega. The scenarios to explain this dichotomy are based on the stability of magnetic fields. The strong and weak fossil magnetisms originate from the bifurcation between stable and unstable magnetic configurations (Auri`ere et al. 2007; Braithwaite & Cantiello 2013). However, more observational and numerical works are needed in order to discriminate between the different scenarios.
Understanding the origin of the weak magnetism is an exciting new challenge of stellar magnetism and can provide new constraints for theory. Do all supposed non-magnetic stars actually host a weak magnetic field? For example, the detection of low frequency modulation of the light-curve compatible with a rotational modulation found in ∼70% of the A-type Kepler stars could be explained by the presence of star spots or other magnetic co-rotating features (Balona 2011; B¨ohm et al. 2015). A large occurrence of weak magnetic fields in hot stars would also have a direct impact on stellar evolution models by providing the first direct constraints on the value of the magnetic field of a typical intermediate-mass or massive star.
My PhD was undertaken in this context as part of the ANR Imagine project (Investi-gating MAGnetism of IntErmediate-mass and massive stars, PI:F. Ligni`eres). The aims of this project is to investigate the magnetism of hot stars thanks to observations and modeling, to reach a better understanding of the properties of the magnetic hot stars and the physical processes that occur in these stars.
My PhD thesis consisted in analyzing observational data taken with high-resolution spectropolarimeters, principally with Narval installed at the Pic du Midi Observatory, to detect magnetic fields. This instrument and the techniques I have used are presented in Chapter 2. One part of my thesis was dedicated to the study of strong magnetic fields and my work is described in chapters 3 and 4. I analyzed the observations of a massive O star: ζ Ori A (see chapter 3) because only a few O stars are known to be magnetic and each new discovery of a magnetic O star helps to understand the magnetic properties of massive stars. While we know that a magnetic desert exists among intermediate-mass stars, we did not know if it extends to massive stars. Bouret et al. (2008) claim that ζ Ori A hosts a weak dipolar magnetic field (∼ 100 G). Confirming or refuting this result would bring constraints on the existence of the magnetic desert in massive stars. I was also involved in a project to determine the upper limit of the magnetic desert thanks to observations of Ap/Bp stars. The goal of this project is to test the dependence of the upper limit with rotation (see chapter 4). The other part of my thesis is dedicated to the search for ultra-weak fields like the one of Vega or Sirius in hot stars, to bring constraint to the various scenarios that explain the dichotomy and to have a better understanding of the magnetic properties of this kind of fields. The results are presented in chapters 5 and 6. I present the results of the spectropolarimetric study of normal stars (see chapter 5): Vega, UZ Lyn and some B stars. Then, I present the result for chemically peculiar stars (see chapter 6). I studied three Am stars: β UMa, θ Leo and Alhena, and one HgMn star: α And. In chapter 7 I then summarize the conclusions of my work and present the perspectives for future studies.

Detecting magnetic fields

We use an indirect method to detect magnetic fields thanks to the emitted light of the stars. The detection of a magnetic field is based on the Zeeman effect, that influences the lines in the spectrum of the star. Indeed, a spectral line that is sensible to the magnetic field is divided in several components when it is formed in a magnetic environment. However, this splitting is difficult to detect in stars due to their weak magnetic field and to line broadening by stellar rotation. Nevertheless, the components of a Zeeman triplet are polarized, and we can use this polarization to diagnose the field.

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
Contents vii
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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