In the remaining chapters, we tackle a few of the open inference problems in the context of Galactic astrophysics. We focus on studies of the Galactic magnetic field and the related components of the interstellar medium.
In Chapter 2, we study the applicability of a technique developed recently by Junklewitz & Enßlin (2011) to detect magnetic helicity from observations of Faraday rotation and synchrotron emission. Using several sets of simulated observations in the Galactic setting, we test under which conditions the technique can be expected to yield reliable results. Of special importance are the assumptions that one makes about the distribution of free thermal electrons and cosmic ray electrons throughout the Milky Way. As we pointed out in Sec. 1.1.3, little is known about these distributions and studying the impact of false assumptions is therefore important. Thus, only conclusions regarding a combination of the electron densities and the magnetic field can be drawn.
To apply the technique to observational data, a continuous map of the Galactic Faraday depth is needed. As a first attempt, we use observational data from extragalactic radio sources along with thecritical filter formalism of Enßlin & Weig (2010) to construct such a map. However, due to the use of extragalactic sources, the map created in this way contains not only the Galactic contribution to Faraday rotation but also unknown extragalactic contaminations. These should ideally have been assigned to the error budget of the data. To overcome the problem of an unknown error budget, we develop theextended critical filter formalism in Chapter 3. The idea is to make use of the correlation structure of the signal, itself reconstructed from the entirety of the available data under the assumption of statistical isotropy, to separate fluctuations in the data that are consistent with this correlation structure from other ones that can then be assigned to the error budget. Thus, we are able to make use of data with an uncertain amount of uncertainty. We study the performance of this methodology in a variety of test cases with simulated signals and controlled noise characteristics.
In Chapter 4, we employ this newly developed algorithm to create an improved map of the Galactic Faraday depth. For this, we want to assign any extragalactic contributions to the error budget of the data points. However, the size of the extragalactic contributions is not known. At best, an expectation for the average extragalactic contribution can be deduced, but it is expected to vary vastly from one data point to the other. Additionally, there are several observational reasons why a few of the data points are likely to be faulty. For these as well, we should drastically increase the error budget, but we have no way of telling in advance which of the data points they are. Therefore, we are in a situation that is ideally suited for theextended critical filter formalism. Before using it to reconstruct a map, we assemble an extensive data catalog of Faraday rotation observations of extragalactic sources, consisting of newly provided data as well as data from the literature. We finally present our statistical results and discuss their possible astrophysical implications.
After this, we train our view on the reconstruction of Galactic emission maps. The emission intensity due to most physical processes discussed in Sec. 1.1.2 can be expected to be orders of magnitude higher within the Galactic disk than within the halo. Also, it
1.3 Outline 29
is always positive. Thus, we find ourselves in exactly the situation described in Sec. 1.2.5, in which a log-normal description is appropriate. Therefore we develop in Chapter 5 a technique to reconstruct log-normal signal fields from observational data in cases in which the signal covariance is unknown.
Another aspect introduced in that chapter is the notion of spectral smoothness. For statistically isotropic signals, the covariance is determined by the power spectrum. In many cases, this power spectrum is expected to be a smooth function of the scale-length. We study in Chapter 5 how to incorporate this expectation in a fully Bayesian way with the usage of an appropriate prior, thus alleviating the need for an ad-hoc treatment of spectral smoothness for thecritical and extended critical filter. We again study the performance of the resulting algorithm in a series of simulated scenarios.
Finally, we summarize our conclusions in Chapter 6 and give a brief outlook on future work.
Chapter 2
Probing magnetic helicity with
synchrotron radiation and Faraday
rotation
Note: This chapter has been published in Astronomy & Astrophysics (Oppermann et al. 2011a).
2.1
Introduction
Helicity is of utmost interest in the study of astrophysical magnetism. Mean field theories for turbulent dynamos operating in the Galactic interstellar medium have been successful in explaining how the observed magnetic field strengths are maintained (e.g. Subramanian 2002). These theories predict that helicity is present on small scales in interstellar magnetic fields. Observationally detecting or excluding helicity in these fields would therefore either strongly suggest that these theories are valid or indicate that there are some flaws in them. However, since helicity is a quantity that describes the three-dimensional structure of a magnetic field and most observation techniques produce at best two-dimensional images leading to an informational deficit, it has thus far largely eluded observers. Previous work on the detection of magnetic helicity in astrophysical contexts has focused mainly on either magnetic fields of specific objects, such as the Sun (see e.g. Zhang 2010, and references therein) or astrophysical jets (cf. e.g. Enßlin 2003; Gabuzda et al. 2004), or cosmological primordial magnetic fields (e.g. Kahniashvili & Ratra 2005; Kahniashvili et al. 2005). Two exceptions are the work by Volegova & Stepanov (2010), in which the use of Faraday rotation and synchrotron radiation for detecting magnetic helicity was suggested for the first time, and the work of Kahniashvili & Vachaspati (2006), in which the use of charged ultra high energy cosmic rays of known sources is suggested for probing the three- dimensional structure of magnetic fields through which they pass. However, the sources of ultra high energy cosmic rays are not known yet and the applicability of this test is therefore limited.
32 2. Probing magnetic helicity
TheLITMUS (LocalInferenceTest forMagnetic fields whichUncovers heliceS) proce- dure for the detection of magnetic helicity suggested by Junklewitz & Enßlin (2011) probes the local current helicity density B~ ·~j, which for an ideally conducting plasma becomes
~
B·~j ∝B~ ·∇ ×~ B~. (2.1)
Here, the magnetic field is denoted by B~ and the electric current density by~j. The test uses measurements of the Faraday depth and of the polarization direction of synchrotron radiation to probe the magnetic field components along the line of sight and perpendicular to it, respectively. Its simple geometrical motivation should make it applicable in a gen- eral setting, provided these quantities can be measured. The results depend only on the properties of the magnetic field along a line of sight and are therefore purely local in the two-dimensional sky projection. Our aim is to test this idea on observational as well as on simulated data, thereby determining the conditions under which the test will yield useful results.
This paper is organized as follows. In Sect. 2.2, the basic equations used in theLITMUS
test are reviewed. They are applied to observational data describing the Galactic magnetic field in Sect. 2.3, with special emphasis on a sophisticated reconstruction of the Faraday depth, described in Sect. 2.3.2. Section 2.4 is devoted to a thorough general assessment of the test’s reliability. To this end it is applied to simulated observations of increasing complexity. Section 2.4.1 describes the application in a flat sky approximation, whereas Sect. 2.4.2 examines all-sky simulations, finally arriving at complete simulations of the Galactic setting in Sect. 2.4.2, where realistic electron distributions are added. We discuss our results and conclude in Sect. 2.5.