Download Investigating the Zeeman Effect: A Deeper Look

Investigating the Zeeman Effect: A Deeper Look
Lauren Woolsey
Ay201b – 6 May 2013
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Introduction
The Zeeman effect was first discovered in the laboratory by Pieter Zeeman, a Dutch physicist (see Figure
1), who effectively proved the theory put forth by Hendrik Lorentz that light emitted in the presence of a
magnetic field, if polarized, would be affected. The pair were awarded the Nobel Prize in Physics in 1902.
This discussion will focus specifically on specific examples of Zeeman splitting in astronomical contexts.
Figure 1: At left, Pieter Zeeman, the scientist who discovered this effect in the laboratory. At right, a
photograph he took showing an absorption line without and with the presence of a magnetic field.
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Spectral lines and Zeeman splitting: an overview
When astronomers and physicists talk about spectral lines, they are most commonly talking about
absorption or emission lines created when an electron changes energy levels. These types of lines are often
in ultraviolet, optical, and infrared wavelengths, as they are very energetic. However, lines in a spectrum
can also be created through more subtle processes. A well-known example in astronomy is the 21-cm line of
neutral Hydrogen (HI), which is emitted when the spin of the electron flips from being parallel to the spin of
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the proton in the nucleus to being anti-parallel to the proton’s spin. These two states have slightly different
energies, and lines emitted through this electron spin-flip transition for different atoms and molecules are
often in radio wavelengths. We will discuss in §4 why long wavelengths are best for observations of the
Zeeman effect in the interstellar medium (ISM).
Figure 2: An example of Zeeman splitting where one electron transition that would normally produce one
emission line has been split into nine lines due to a background magnetic field
Above, Figure 2 demonstrates how this effect works for an example electron energy level transition from
S1 to 3 P2 . There are two different types of polarization, σ and π, based on the difference between initial
and final angular momentum quantum number mj . The σ components are elliptically polarized, and the
circular component of the polarization is what is observed in the Stokes I and Stokes V, discussed in §4. The
amount of splitting between two mj values for a single line is given by:
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∆νZ = g
µB B
h
(1)
where µB is the Bohr magneton, h is the Planck constant, and g is the Lande g-factor (Heiles et al., 1993).
Specific examples are presented in the following two sections.
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The Sun as a nearby example of Zeeman splitting
The Sun is an example of the strong Zeeman effect, where splitting of the lines is strong enough to be seen
in the overall intensity and the amoung of splitting is accurately found using equation 1. This doesn’t seem
like an impressive feat, as the whole point of Zeeman splitting is that the spectral lines are split. However,
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Figure 3: An example of the strong Zeeman effect on the Sun, where the strength of magnetic fields along
the line of sight reach 100 Gauss. On the left, we see the extreme ultraviolet Sun, and the regions of highest
intensity, active regions, show up in the magnetogram on the length as locations of strong field strength with
neighboring regions of opposite polarity, and these also trace sunspots in optical images
as we will explore in §4 and §5, the effect is usually much more subtle, requiring specific regions of the
interstellar medium and observations of a select few spectral lines. In Figure 3 we compare the Sun observed
in the ultraviolet with a magnetogram of the Sun, which measures the amount of splitting of the Ni I 676.8
nm line and the polarity of this splitting. For a closer look, we turn to Figure 4, which shows an observation
of a sunspot. The vertical bar on the left side is the slit for the spectrograph, which shows splitting in the
region of the slit that falls within the sunspot, where magnetic fields are very strong. Such observations of
the magnetic field strength at the Sun’s photosphere are commonly used as boundary conditions for codes
that model the 3D magnetic field of the Sun out to 1 AU. This is an essential aspect to get correct, because
the Sun’s magnetic field plays an important role in the heating of the corona and acceleration of the Solar
wind (see research projects at http://www.cfa.harvard.edu/~lwoolsey/research.html).
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The Zeeman effect in the ISM
A discussion of the Zeeman effect must start with the species most often observed. Heiles et al.
(1993) compile an extensive list of candidates for Zeeman observations, which I reproduce for only the
B
most commonly used candidates in Table 1. Here, b = 2gµ
Hz µG−1 such that Equation (1) becomes
h
∆νZ = 2b B.
In Table 1, the density nH represents the environmental density where these species are observable. In
all of the lines for which we have a hope of detecting Zeeman splitting, a pattern is clear: we need very long
wavelengths so that telescopes have enough resolution to make a detection. For HI, only the outer parts of
molecular clouds are traced by excess 21-cm emission. OH is found in molecular clouds and is observable for
densities up to 2500 cm−3 . Zeeman splitting in OH emission has also been observed in masers with strong
magnetic fields, as have Hyperfine transition lines of H2 O.
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Figure 4: A closer look at the absorption lines of a sunspot, where splitting is seen across the sunspot due
to strong line-of-sight magnetic field strength
Species
HI
OH
OH
H2 O
Table 1: Common Species used for ISM Zeeman Observations
Transition
Line ν (GHz) b (Hz µG−1 ) nH (cm−3 )
2 1
1.42
2.8
S 2 , F=1-0
2 3
Π 2 , F=1-1
1.665
3.27 clouds: 103 ; masers:
2 3
Π 2 , F=2-2
1.667
1.96 clouds: 103 ; masers:
Hyperfines of (616 - 523 )
22.235
0.0029
masers: up to
low
107
107
109
In masers, the magnetic field can be strong enough to classically split the spectral line and the total
magnetic field strength can be observed. However, the magnetic fields commonly observed in the interstellar
medium in molecular clouds are low enough that the splitting of the spectral line is not great enough to
produce a measureable effect. In this case, we must turn to Stokes Parameters. As mentioned earlier, the
σ transtions produce elliptical polarity. The right circular polarization (RCP) and left circular polarization
(LCP) can be measured, where the sum of these is the Stokes I parameter and the difference is the Stokes V
parameter. With no magnetic field present, the Stokes V spectrum should be a flat line at zero. However, if
a magnetic field is present, the “V-spectrum” can be fit by the derivative of the “I-spectrum” scaled up by
the strength of the magnetic field along the line-of-sight as follows (Sarma et al., 2013):
V = ∆νZ
b dI
dI
= B
dν
2 dν
(2)
Note, Sarma et al. (2013) use different notation, this follows Heiles et al. (1993). Example observations
are presented in Figure 5 and Figure 6, and the pairing of Stokes I and Stokes V reflect the left column of
plots in the interactive module.
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Figure 5: One of the first observations of the Zeeman effect in molecular clouds, using the OH 1665 MHz
and OH 1667 MHz lines (Goodman et al., 1989)
Figure 6: A very recent observation of Zeeman splitting in the star-forming region S88B (Sarma et al., 2013)
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The interactive module
The module created for investigating the Zeeman effect has several options that the user can change.
They are seen as a pull-down menu and five slider bars in Figure 7, and can be described as follows:
• Lines: the spectral line from the list presented in Table 1 to be observed.
• Temperature: a hotter environment produces thermally broadened spectral lines.
• Turbulent Velocity: turbulence adds non-thermal Doppler broadening to spectral lines.
• Line-of-Sight Magnetic Field: the key parameter for changing the Zeeman effect observed.
• Y-axis Bounds for Stokes V: the magnitude of the Zeeman effect ranges depending on inputs, so this
allows the user to “zoom in” to a range where the effect is observable.
• Signal-to-Noise Ratio: Longer observations on a telescope produce a higher signal-to-noise, so the
exposure time we need to see a clear signal for given inputs may differ.
The necessity for the Stokes I and Stokes V plots was made clear in §4 (see Figures 5 and 6), and we will
now discuss the other two plots briefly. The line profile without Zeeman effect is meant to provide a baseline
for comparison with the Stokes I parameter. If there is no magnetic field, the top two plots (Stokes I and
line profile without Zeeman effect) will be identical. The flux is normalized to peak at a unitless value of 1
because we have not made any assumptions about the source brightness and distance or the telescope used.
These details are beyond the scope of this module and are not necessary to understand the Zeeman effect
in a general sense. It is worth noting, however, that observations using radio telescopes commonly report
“flux” or “intensity” in terms of antenna temperature. See http://ay201b.wordpress.com/2011/04/12/
course-notes/#definitions_of_temperature for details.
The final plot, in the bottom right of Figure 7, represents a rough idea of how the magnetic field strength is
related to the density of OH and H2 O masers. Observations indicate a weak proportionality of the magnitude
1/2
of BLOS with nH (Crutcher et al., 2010). This relation may extend to 103 cm−3 , or dense dark clouds in
the ISM. Crutcher et al. (2010) explain that the relation arises from flux freezing, where magnetic field lines
are dragged along by the dense medium above a threshold density.
This should serve as a good starting point to start playing with the module and discovering the difficulty
of making different observations of the magnetic field in the interstellar medium. To get you started, Table
2 lists typical values of structures in the ISM. The turbulent velocity is approximated using one of Larson’s
Laws (Larson, 1981):
σv ≈ 1.1 (L [pc])0.38 km s−1
(3)
Region
Giant Molecular Cloud
Dark Cloud
Core
Table 2: Typical Properties of ISM Structures
Size, L (pc) Density, n (cm−3 ) Temperature (K)
100
100
50
10
1000
20
0.3
10000
10
σv (km/s)
0.5
0.3
0.2
Experiment with similar values in the interactive module to get a sense of how difficult observations of
the Zeeman Effect in the interstellar medium are. Note the scaled intensity of the Stokes V to the Stokes I
and the lowest signal-to-noise ratio you can “get away with” and still clearly see a signal in the Stokes V.
How difficult is it to see true splitting of the lines? Remember, when you have the environment you need to
observe the Zeeman effect in the ISM, it’s gold, but the regions of the ISM where this is possible make up
only a small fraction of the total volume. Good luck!
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Figure 7: A screenshot of the interactive module
References
R. M. Crutcher, “Magnetic fields in molecular clouds: Observations confront theory,” ApJ, 520, 1991 pp.
706-713.
R. M. Crutcher, “Magnetic Fields in Interstellar Clouds from Zeeman Observations: Inference of Total Field
Strengths by Bayesian Analysis,” ApJ, 725(1), 2010 pp. 466-479.
A. A. Goodman, R. M. Crutcher, C. Heiles, P. C. Myers, T. H. Troland, “Measurement of magnetic field
strength in the dark cloud Barnard 1,” ApJ, 338, 1989, pp. L61-L64.
J.-L. Han, R. Wielebinski, “Milestones in the Observations of Cosmic Magnetic Fields,” Chinese J. of A &
A, 2(4), 2002, pp. 293-324.
C. Heiles, A. A. Goodman, C. F. McKee, E. G. Zweibel, “Magnetic fields in star-forming regions:
Observations,” in Protostars and Planets III, Tuscon: University of Arizona Press, 1993.
R. B. Larson, “Turbulence and star formation in molecular clouds,” MNRAS, 194, 1981, p.p. 809-826.
C. F. McKee, E. G. Zweibel, A. A. Goodman, C. Heiles, ”Magnetic fields in star-forming regions: Theory,”
in Protostars and Planets III, Tuscon: University of Arizona Press, 1993.
G. B. Rybicki, A. P. Lightman, Radiative Processes in Astrophysics, Germany: Wiley-VCH, 2004.
A. P. Sarma, C. L. Brogan, T. L. Bourke, M. Eftimova, T. H. Troland, “VLA OH Zeeman observations of
the star forming region S88B,” ApJ, 767(1), 2013.
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