Quiet Sun Magnetic Fields
J. Sánchez Almeida
Instituto de Astrofísica de Canarias, E-38200 La Laguna, Tenerife, Spain
Abstract. The seemingly un-magnetized part of the solar surface is not really un-magnetized. It is occupied by magnetic
structures producing low polarization which, therefore, escape detection in traditional measurements. Since most of the solar
surface belongs to this category, the quiet Sun magnetic fields can easily carry most of the magnetic flux and energy existing
in the photosphere at any given time. Consequently, they are a potentially important ingredient of the solar magnetism.
Most of the physical properties of the quiet Sun are still uncertain (distribution of field strengths, area coverage, influence
on higher atmospheric layers, etc.).It is clear, however, that the topology of the field is complex, with field lines of very
different properties coexisting in each resolution element. This fact hampers the detection of the quiet Sun magnetic fields.
I argue that the best present measurements detect, at most, 30 % of the existing magnetic flux. Then the quiet Sun contains at
least as much magnetic flux as all active regions and the network during the solar maximum.
INTEREST TO STUDY THE
MAGNETISM OF THE QUIET SUN
Even during the maximum of the solar cycle, most of
the solar surface appears as non-magnetic in traditional
magnetic field determinations (e.g., the gray background
in the magnetogram shown in Figure 1). This so-called
quiet Sun does not produce enough polarization to show
up in such measurements, however, one cannot infer
from this fact that the magnetism of the quiet Sun regions is non-existing or un-important. Rather, the limited
sensitivity of the standard measurements, together with
the large surface coverage, indicate that the quiet regions
may be very important in terms of the global magnetic
properties of the Sun. A simple order-of-magnitude estimate illustrates the point. Magnetographs show a total
unsigned magnetic flux across the solar surface of some
7 1023 Mx at solar maximum (e.g., Schrijver and Harvey [1]). If one divides this flux by the area of the solar
surface, the flux density turns out to be of the order of 12
G. This figure is close to the noise level of the standard
measurements (some 7 G for the magnetogram in Figure
1; see Jones et al. [2]). Consequently, signals below the
usual sensitivity and covering most of the solar surface
may contain as much magnetic flux as sunspots, active
regions and the network all together.
If the quiet Sun carries a substantial fraction of the
solar magnetic flux, weak polarization signals should
appear upon improvement of sensitivity of the magnetometers. Such weak signals are actually observed. When
the noise is in the few G level and the angular resolution about 1" , then most of the solar surface becomes
magnetic. Such sensitivity and angular resolution is fre-
.
FIGURE 1. Typical magnetogram. Black and white represent magnetic signals with two different polarities. The gray
background shows no signal above the sensitivity and corresponds to the part of the solar surface denoted along the text as
quiet Sun. For details of this magnetogram, see Jones et al. [2]
quently achieved by the new generation of solar spectropolarimeters which consistently show polarization signals almost everywhere (e.g., Grossmann-Doerth et al.
[3]; Lin and Rimmele [4]; Lites [5]).
Apart from the mere existence of this type of magnetism, little is known about the properties of the quiet
Sun magnetic fields. They constitute a potentially impor-
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
293
tant ingredient of the solar magnetism that deserves careful study. One would like to know what is the amount of
magnetic flux and energy that the quiet Sun contains. In
particular, it would be important to find out whether and
to what extent the quiet Sun magnetic fields are related
to the other manifestations of the solar magnetism (active
regions or network). Does it result from the decay of active regions? Does it emerge as is from sub-photospheric
layers? Is it created in-situ by a dynamo driven by the
granulation? Does it follow the solar cycle? Is it connected to the magnetic fields in the corona, from where
the solar wind emanates? The importance of these and
similar questions provides a clear rationale to study the
magnetism of the quiet Sun.
A conspicuous observational feature characterizes the
line polarization produced by the quiet Sun magnetic
fields. The weak line polarization signals emerging from
quiet Sun regions turn out to be highly asymmetric
(Sánchez Almeida et al. [6]; Grossmann-Doerth et al.
[3]; Sigwarth et al. [7]). They do not show the characteristic line shape to be expected if the magnetic and
velocity fields were constant in the resolution element.
Figures 2b-f include a set of observed Stokes V profiles (degree of circular polarization versus wavelength
within the range of wavelengths of a spectral line). These
profiles have to be compared with the perfectly antisymmetric signal that arise when magnetic and velocity
fields are spatially resolved (Figure 2a). The existence of
these so-called asymmetries of the Stokes profiles proves
that the magnetic field in the quiet Sun varies within the
typical 1" ( 725 km) resolution elements of the present
observational setups. Moreover, the fact that the individual spectral lines generate net circular polarization indicates that part of this variation has to occur along the lineof-sight (LOS), within a fraction of the vertical extent of
the photosphere (see the discussion in Sánchez Almeida
[8], Sect. 2.2). The extreme character of some of the observed asymmetries suggests that the variation within the
resolution element is not mild. For example, very often
two different polarities seem to coexist in each resolution
element (Sánchez Almeida et al. [6]; Sánchez Almeida
and Lites [9]; Socas-Navarro and Sánchez Almeida [10];
Lites [5]). All magnetic field measurements are based
on the correct interpretation of the observed polarization. The complexity the magnetic field in the quiet Sun
warns against simplistic interpretations of this polarization. Oversimplifications often lead to the omission of
magnetic structures and the underestimation of the magnetic flux existing in the quiet Sun. This work analyzes
several observational biases that arise (and may be potentially important) when the interpretation assumes that
a uniform magnetic field occupies each resolution element. I set lower limits to the missing flux due to (a) the
coupling between magnetic field strength and density, (b)
the presence of a wide range of field strengths, and (c) the
existence of both polarities in each resolution element.
FIGURE 2. Ideal versus observed Stokes V profiles. If the
magnetic field and the velocity were constant within the resolution element, then the observed Stokes V profiles b-f (the
solid lines) should be like the profile in a, i.e., perfectly antisymmetric. The dots in figures b-f correspond to synthetic
spectra able to reproduce the observed line profiles (Sánchez
Almeida and Lites [9]).
BASIC ASSUMPTIONS OF THE
STANDARD MEASUREMENTS
As it was pointed out above, the measurements of magnetic field direction, strength, etc., rely on the correct interpretation of the observed line polarization. In practice,
this interpretation is based on a number of assumptions
on the structure of the magnetic field that are at variance with the complications of the line polarization observed in the quiet Sun. A prototypical example of magnetic measurement is the magnetogram. Magnetograms
employ the so-called magnetograph equation,
Bm
Ccal V
(1)
which relates the observable V (Stokes V signal at a fixed
wavelength within a spectral line) and an estimate of
294
1. the magnetic field vector B is constant within the
resolution element (more precisely, it is either constant or zero), and
2. the temperature and density that characterize the
thermodynamic of the atmosphere are not modified
by the presence of magnetic fields,
Thin
Thick
τ=1
the magnetograph signal B m is equal to the magnetic flux
density B ,
Bm
B Σ
B n ds
Spatial direction -->
ds
(2)
FIGURE 3. Cartoon of two irregular magnetic concentrations embedded in a non-magnetic background. The spatial
scale of the irregularities is either smaller (left) or larger (right)
than the photon-mean-free-path. The solid line points out the
geometrical depth from where the observed photons escape.
The dashed lines correspond to LOS.
Σ
The integrals extend to the surface of the resolution element Σ, which is perpendicular to the unit vector along
the LOS n. Note that B represents the magnetic flux
in the plane perpendicular the line-of-sight divided by the
area of the resolution element.
Since the hypotheses (1) and (2) above are generally
not satisfied in the quiet Sun,
B Bm 1
Height in the atmosphere -->
the magnetic flux density B m . The symbol Ccal stands
for a calibration constant. Under several hypotheses, in
particular,
COUPLING BETWEEN DENSITY AND
MAGNETIC FIELD STRENGTH
(3)
Any magnetic structure that lasts long enough tends to
evolve to a mechanical equilibrium configuration. The
characteristic time of the process is of the order of size
of the structure divided by the propagation speed of those
perturbations that relieve the lack of equilibrium. This
time scale is extremely short in the photosphere: using the sound speed for the propagation speed (some
10 km s 1 ), the time scale is of the order of 100 s for
a structure with the size of a granule, and 0.1 s for a 1 km
wide magnetic concentration. Consequently, one should
expect that most magnetic concentrations satisfy physical constrains characteristic of structures in equilibrium.
In particular, consider an atmosphere where the magnetic
field strength is not uniform. In order to keep horizontal mechanical balance, the variations of magnetic pressure associated with the variations of magnetic field have
to be compensated by gas pressure variations. In other
words, the total pressure Pt depends only on the height in
the atmosphere z,
How much larger? We do not know it yet, since it depends on (still unknown) details of the structure of the
quiet Sun magnetic fields. However, one can estimate the
deficit of Bm with respect to B in specific cases. Bias
arising form the breakdown of conditions (1) and (2) are
analyzed in the forthcoming sections.
Consider a magnetic atmosphere whose physical properties are not uniform in the resolution element. The
polarized spectrum emerging from such irregular atmosphere reflects some sort of ill-defined volume average of
the local properties of the atmosphere. The nature of this
average is easy to work out in two extreme cases corresponding to irregularities whose spatial scales are either
much smaller or much larger than the typical photonmean-free-path (see Figure 3). When the irregularities
are optically thin then the average proceeds by first averaging the local absorption and emission, and then producing the polarized spectra corresponding to the mean
atmosphere (this is the MISMA approximation put forward by Sánchez Almeida et al. [6]). When the irregularities are optically thick (but still spatially unresolved
with the present instrumentation), the spectrum of each
irregularity is first produced independently, and then the
mean among these spectra renders the observable spectrum. (For irregularities of intermediate scales the synthesis is far more complicated since the two sorts of average are no longer uncoupled.) These ideas on the nature
of the volume average corresponding to micro and macro
irregularities will be used below.
Pt z
B2 z
Pgz
8π
(4)
so that the magnetic field strength B B and the gas
pressure Pg have to be anti-correlated. Assume, for the
sake of simplicity, that the temperature does not vary
with B. Then Equation (4) renders a simple relationship
between the local density ρ and the local magnetic field
strength,
ρ ρ0 1 BBmax2
(5)
295
where Bmax 8π Pt is the maximum possible magnetic field strength, which corresponds to fully evacuated
plasma (ρ 0). The symbol ρ 0 in Equation (5) stands for
the density of unmagnetized plasma (B 0).
Now consider an irregular atmosphere where the magnetic field strength is not constant in planes perpendicular to the LOS. The emission and absorption of photons depend on the local density of the plasma that emits
and absorbs. According to Equation (5), the more magnetic the plasma is the less dense and, consequently, the
less polarization is expected to emit. This suggest that
very strong magnetic fields (B B max ) produce negligible polarized light and therefore may be missed in the
magnetic field determinations. Such simplistic argument
holds only if the light emitted by plasmas with different
magnetic fields comes from the same atmospheric layers.
This is the case when dealing with optically thin magnetic irregularities (Figure 3, left), since the mean opacity determines a single optically depth independent of the
field strength (see last paragraph of the previous section).
If the magnetic field fluctuations are optically thick, then
the reduction of opacity associated with the decrease of
density allows to see deeper in the more magnetic irregularities, and the global increase of density with depth may
easily compensate the deficit of polarized emission 1 .
Sánchez Almeida [11] studies this coupling between
field strength and mass density. It is found to be responsible for the observed decrease of magnetic signals having B 09 Bmax . By correcting for the low sensitivity to
large field strengths, Sánchez Almeida [11] sets a lower
limit for the missing flux density,
B Bm 2
two lines as a function of the magnetic field strength (top
row). The two synthetic lines, whose atomic parameters
have been chosen to represent typical lines used in magnetic studies2 , differ only because of their Zeeman splittings. The visible line fully splits for B 15 kG whereas
0.5 kG suffices in the IR (see the gray gap between the
white and black bands in Figures 4a and 4b). In order to
compute the spectra produced by the full distribution of
field strengths, one would need to average the images 4a
and 4b along the vertical direction 3 . In general, for an arbitrary distribution of field strengths, the average has to
be weighted with the probability of finding a given field
strength. For example, Figures 4c and 4d show Stokes
V profiles resulting from two different probabilities having only sub-kG fields and mostly kG fields. They correspond to averages of the lower and upper parts of the
images, respectively (B 05 kG and B 05 kG). The
Stokes V profiles in Figures 4c and 4d are normalized
to the flux density in the resolution element. First, note
how the IR signal in Figure 4d decreases with increasing
mean magnetic field strength. This is due to the spread in
wavelength of the IR signals for kG fields. Such spread
ceases for sub-kG fields where the signals accumulate at
a constant wavelength (see Figure 4b). Should the quiet
Sun contains both kG regions and sub-kG regions, one
would preferentially detect those producing the largest
signals, which in the IR correspond to the sub-kG fields.
On the other hand, the sensitivity of the visible line is
more balanced for weak and strong fields (Figure 4c).
Note, however, that the visible signals are weaker, skewing the determinations to larger flux densities.
The measurement of the physical properties of the
quiet Sun fields is still in a primitive phase. Nevertheless we already know that IR lines and visible lines render very different magnetic field strengths (see SocasNavarro and Sánchez Almeida [10][12], where you can
also find how to determine B independently of B m ). The
difference goes in the sense of the bias described above,
being the IR measurements those yielding the lower field
strengths. Since the ranges of observed field strengths in
the visible and in the IR have almost no overlap, one
can assume that the two observations detect different
magnetic structures. The observed IR and visible flux
densities are similar (see the data collected by Sánchez
Almeida et al. [13]). This fact suggests that approximately half of the flux is missing in estimates based on
only IR lines or only visible lines. Using the notation of
the previous section, this detection of only half of the ex-
(6)
BIAS DEPENDING ON THE MAGNETIC
FIELD STRENGTH
The magnetic field strength is not constant within the
typical resolution elements, which biases the magnetic
flux density determinations in yet another way. If a wide
range of field strengths is present then, depending on the
magnetic sensitivity of the spectral line used for measuring, one tends to select a particular part of the distribution of field strengths. In particular, if the whole range
of magnetic field strengths from zero to the maximum
possible value is present (0 B B max 2 kG), highly
split Infra-Red (IR) lines tend to choose the sub-kG part
of the distribution whereas visible lines are more sensitive to the kG fields. The mechanism is illustrated in
Figure 4, which shows the synthetic Stokes V profiles of
2
Fe I 6302.5 Å for the visible line and Fe I 15648 Å for the IR line.
This approximation considers spatially unresolved structures that are
optically thick (Figure 3, right), but the bias described in this section
affects optically thin structures as well (Sánchez Almeida and Lites
[9]).
3
1
Sunspots correspond to this second case. They are strongly evacuated
but produce polarized light.
296
isting flux can be summarized as,
B Bm 2
(say 50%), then
(7)
B Bm 035
05 1 12
(8)
where B represents the unsigned flux density to be
observed if one could resolve the opposite polarities 4 .
Again, the equation represents a lower limit to the ratio
B Bm since only those magnetic structures where
the annihilation between opposite polarities is not perfect
leave a residual Stokes V able to reveal the presence of
two polarities. (The imperfect cancellation is due to the
existence of a Doppler shift associated with the change
of polarity.)
FIGURE 4. Top: Stokes V profiles (Stokes V versus wavelength) for a range of magnetic field strengths (B, in kG units).
Image a shows the behavior of a magnetically sensitive visible
line (e.g., Fe I 6302.5 Å). Image b represents the same line if it
were in the near IR (say, at 1.5 µ m). Bottom: average Stokes V
profiles if the magnetic field strengths existing in the resolution
element were mostly weak (the solid lines) or strong (the dotted
lines). They have been computed by averaging the top images
for B 05 kG (the solid lines) and B 05 kG (the dotted
lines). The Stokes V profiles in c and d have been normalized
to the same flux density in the resolution element: 1 G.
FIGURE 5. Synthetic magnetograms of magnetic fields produced by the turbulent dynamo numerical simulations of Cattaneo [14] and Emonet and Cattaneo [15]. Left, snapshot of
the original simulation. Right, same magnetogram but observed
with 1"angular resolution. 90% of the original signals go away
after the spatial smearing. The gauge, shown for reference, corresponds to 1".
The degree of cancellation due to mixed polarities may
be far more severe than the conservative limit set by the
inequality (8). Figure 5 illustrates the huge decrease of
polarization signals that can be induced by the presence
of mixed polarities. It shows a synthetic magnetogram
emerging from the numerical simulations of turbulent
dynamo by Cattaneo [14] and Emonet and Cattaneo [15].
In this simulation the magnetic field grows out of the
kinetic energy of the granular motions. It disappears by
Ohmic diffusion when field lines of two polarities intertwine in the whirls of the granulation downdrafts. This
turbulent dynamo mechanism may explain the origin of
the quiet Sun magnetism and, in addition, it does not
seem to contradict any observational constraint (Sánchez
MIXED POLARITIES IN THE
RESOLUTION ELEMENT
Stokes V profiles like those in Figures 2f-g point out
the existence of two different polarities in the resolution
element. The sign of the circular polarization reverses
upon change of polarity, therefore, the existence of unresolved mixed polarities reduces the Stokes V signal
and, via Equation (1), hides part of the existing magnetic
flux. Since the presence of mixed polarities seems to be
very common, this effect may severely bias the magnetic
flux density determinations. Socas-Navarro and Sánchez
Almeida [10] find that some 35% of the weak Stokes V
profiles produced by the quiet Sun require mixed polarities to be reproduced. Assuming that each one of these
mixed polarities cancels a fraction of the observed flux
If B and B are the flux densities of the two opposite
polarities, then B B B whereas Bm B B . We consider that at least 35% of the resolution elements
have B Bm 05 Bm , which leads to B Bm 035 05 Bm
and, consequently, to Equation (8).
4
297
Almeida et al. [13]). Figure 5 shows, side-by-side, both
the original magnetogram at full resolution (some 15 km
or 0".02) plus the magnetogram smeared to 1" resolution,
typical of real observations. Most of the signals are gone.
(The two images are shown in a common scale for direct
comparison.) Only 10% of the original signals survive
the spatial smearing so that, for this particular numerical
simulation, Bm 01 B or
B Bm 10
the solar wind. However, there is a moral to be extracted
from the difficulties to detect photospheric quiet Sun
magnetic fields. It is a caveat of application to all atmospheric layers, including the corona. We know that
an important structuring of the fields possibly exist
at scales that cannot be observed due to technical
limitations.
• Because of the insufficient spatial resolution, many
magnetic structures (not necessarily irrelevant or
secondary) elude detection.
• A rather complete knowledge of the topology and
structure of the magnetic field is needed for a proper
interpretation of the observations. Simplistic interpretations are bound to severe bias.
•
(9)
CONCLUSIONS
The quiet Sun is magnetic. Contrarily to the implicit assumptions of routine magnetic field determinations, the
magnetic field of the quiet Sun is not uniform within the
typical 1" resolution elements. This fact biases the measurements so that a fraction of the existing magnetic flux
eludes detection. I have considered three among the possible biases associated with the lack of enough resolution, namely, the drop of polarized emission associated
with local magnetic field strength enhancements, the sensitivity of spectral lines to specific ranges of fields, and
the existence of mixed polarities in the resolution element. Lower limits to the missing flux due to these effects are given in Equations (6), (7) and (8). Since the
three of them are independent, their contributions have
to be added up to estimate the total effect, i.e.,
B Bm
3
∑
B Bm i
ACKNOWLEDGMENTS
ISO/Kitt Peak data used here (Figure 1) are produced
cooperatively by NSF/NOAA, NASA/GSFC, and
NOAA/SEL. Thanks are due to F. Cattaneo, T. Emonet,
R. Grappin, S. Habbal, H. Socas-Navarro, and R. Woo
for clarifying discussions. This work has been partly
funded by the Spanish MCT, project AYA2001-1649.
REFERENCES
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(10)
2.
where B is the true flux density and B B m i
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3.
3
B Bm
∑
i 1
i 1
B Bm i 2 32
4.
5.
6.
(11)
7.
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