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The possibility of recovery of the vertical distribution of the magnetic
field in umbrae by line width differences

S.G. Mozharovsky

Ussuriysk Astrophysical Observatory of FEB RAS, Ussuriysk, Russia

sw@newmail.ru

Abstract. This article studies the possibility of recovery of the vertical
distribution of the magnetic field in the photospheric layers of sunspot
umbra by the Difference of the Line profile Absolute Widths of neutral
zirconium ? 6127, 6143, and 6134 å (i.e. by the DLAW method). The model
calculations show that the response function (RF) of spectral line width at
some line profile depth level to variations in the magnetic field and LOS
velocities attributed to the same height. This raises the prospect of a
parallel analysis of the vertical distribution of magnetic fields and LOS
velocities. On the other hand, our research has discovered a difficulty in
interpreting the data, since the RF and effective heights of response
(HORs) strongly depend on the values of the gradient of the physical
parameter under investigation themselves. This means that the magnetic
field strengths measured by using Stokes profiles (by any method, not just
by the DLAW) cannot be attributed to one and the same optical height, if
the magnetic field gradient changes. A final conclusion about the prospects
of the DLAW method requires the organization and execution of a series of
special observations.

Keywords: "Sunspots, Umbra"; "Spectral Line Profiles"; "Spectral Line
Formation"; "Magnetic Field Measurements"; "Magnetic Field Gradient"

1. Introduction

The study of sunspot magnetic fields is a topical issue in solar physics
(Borrero and Ichimoto, 2011). There is a simple idea for measuring absolute
values of a magnetic field. If we take any two spectral lines with
different Lande factors (factors of the magnetic splitting), the difference
in the absolute widths of their intensity profiles will be proportional to
the field strength. This idea should work, for example, for the well-known
pair of lines of the multiplet number 1 Fe I ? 5250 and 5247 å. These lines
have similar atomic parameters, except for the Lande factor. Therefore,
changes in temperature theoretically should not affect the magnetic field
measurements made by the method of subtracting the absolute width of the
profiles of these lines.
It is considered that different points of the line profile have different
effective heights of formation (HOFs). In this case, measurement of the
widths at different levels of line depth will allow measurement of the
distribution of the magnetic field strength with height in the photospheric
layers, just as the bisector of the spectral line profile allows us to
trace the distribution with height of the bulk line of sight (LOS)
velocity.
Umbrae observations are always accompanied by a large amount of scattered
and blurred light from the penumbra. Therefore, in practice by applying the
method of the difference in the absolute widths of line profiles Fe I ?
5250 and 5247 å to measure magnetic fields, we are likely to get an
"average temperature in the hospital" - values, considerably contaminated
by values of the magnetic fields of the surrounding penumbra. The situation
is different if we do the observations in the lines of the multiplet number
2 Zr I ? 6127, 6143 and 6134 å. These are triplets with different Lande
factors. They have similar atomic parameters, as do iron lines. However,
these lines practically vanish in the penumbra and quiet photosphere. The
depth of their profiles dmax is more than 40% in the umbra spectrum and
does not exceed 5% in the spectrum of sunspot penumbra. So, stray light
from the penumbra and from the quiet photosphere will not affect the
measurement of the magnetic field. The influence of umbral bright points
will also weaken if they exist in the umbra.
One might ask why we need to calculate the field strength using the
difference between the widths of the lines, when we can just get the Stokes
profiles and make a SIR (Stokes Inversion by Response functions) (Ruiz Cobo
and del Toro Iniesta, 1992) or make the measurements by the Magnetograph
method?
The distribution of the physical parameters according to height in sunspot
umbra is rather complex. Figuratively speaking, we have a system of
equations with a large number of unknowns. To solve such a system we must
take a sufficient number of equations. Different techniques are used for
this purpose: observations in different spectral lines are combined,
different Stokes parameters are involved, and constraints are applied with
the help of models of physical processes. Thus, measurement of the
difference in profile of absolute widths of spectral lines with similar
atomic parameters is an additional way of measurement, another equation
that can get an unambiguous description of the medium under study. And this
additional way of measurement may be combined with some of the inversion
methods, in particular with the SIR method.
The model calculations performed in this work showed that the changes in
magnetic field with height are really reflected in the differences of the
absolute widths of line profiles of zirconium at different levels of
profile depth. On the other hand, only an analysis of high quality
observations will permit an assessment of how informative and complementary
to traditional methods of measurement this method is.
Unfortunately, the author does not currently have the possibility of making
such observation. So one of the purposes of this article is to draw
observers' attention to this potentially promising method. Sufficiently
detailed and varied model calculations have been carried out so that
researchers can evaluate the possible prospects and disadvantages of the
proposed method in comparison with existing methods.

2. Materials and Methods


1. Spectral lines used

Table 1 summarizes the data on the spectral lines used in this paper. For
comparison, Table 2 shows similar data for the well-known pair of lines Fe
I ? 5250 and 5247 å. The numbers of multiplets are given, according to
Moore (1945).

Table 1. Data on the lines of the Zr I multiplet number 2. Meanings of the
columns are explained in the text.
|W6127-W613|60.1 |3042 |
|4 | | |
|W6143-W613|37.4 |2640 |
|4 | | |
|W6127-W614|22.7 |4058 |
|3 | | |


In Table 3, we see a very big difference in the results. However, if the
ZR6143 line width is made wider by 5.7 må than actually observed, then all
three measured field strengths would be equal about to 3042 gauss.
The width deficit of ZR6143 for 6127-6143 and 6143-6134 pairs with
different signs is included in the correction term Eij of relation (4). The
deficit can be explained if we reduce the value of the ZR6143 oscillator
strengths relative to other lines. According to the calculations, the
absolute width of the Zr I line profiles in the sunspot umbra varies by
about 6 må with changes in lg(gf) of 0.10 dex. Such a change in lg(gf) does
not contradict the known data for these lines (see Table 1). For
measurement of field strength up to 10 gauss, i.e. for a difference in
widths of 0.2 må, we need to know the relative values of lg(gf) of these
three lines with an accuracy of 0.1[dex]•0.2[må]/6[må]=0.003[dex]. To
determine the ratio of the lg(gf) with such precision we can use umbral
observation of the zirconium lines with known field strength (measured by
line Ti I ? 6064 å for example).
Correction of the lg(gf) value for one of the lines violates the similarity
in behavior of the lines relative to temperature. Thus, the DLAW method
for the zirconium lines is not ideal. It requires accounting for the
difference that occurs from the influence of different temperatures on the
three lines. This difference is slight. We can hope that a smooth change in
temperature will affect the difference between the widths of these lines in
a fairly simple way in comparison with the much stronger and more complex
influence of temperature on the difference between ordinary lines, i.e.
lines that differ considerably in their parameters of atomic electron
transitions. So, we can try to use the difference in the line widths of the
multiplet number 2 of Zr I (or multiplet number 1 of Fe I) as a means of
analyzing changes in photospheric temperature, but such a study is beyond
the scope of this article. Analysis of the temperature correction term Eij
from relation (4) will be discussed in Section 5.

3. Model calculations related to the measurement of the magnetic field and
its gradient


1. Relationship between a section of line wing and height in the
photosphere

The idea proposed in the article gives hope of tracing the distribution of
magnetic field strength along the height of the photosphere by measuring
the line profile difference widths at different levels of profile depth.
Also, it can be expected that the response of the line wings to changes of
LOS velocity (i.e., the displacement of bisector) are attributed to the
same height as that of the magnetic field changes. Model calculations allow
this to be checked. As can be seen from the Figure 2, the heights of
response of the width difference of the zirconium lines to magnetic field
strength changes and heights of response of bisector on the LOS velocity
changes practically coincide.

[pic][pic]
Figure 2. Comparison of RFs for profile points that match the given levels
of profile depth d. (a) The responses for the profiles width difference
between lines ZR6127 and ZR6134. RFs obtained by changing of the magnetic
field strength B at 10 gauss in a probe layer. (b) The responses of the
ZR6127 line bisector, caused by the introduction to a probe layer LOS
velocity of 50 m•s-1. The width of the probe layer is 0.05•log(?5).


2. Influence of the magnitude and sign of the vertical gradient of the
magnetic field on spectral line profiles

Magnetic field strength in sunspot umbra is not constant with height.
According to a survey of Solanki (2003) at the level of the photosphere
field strength decreases with height at a rate of 1-3 G•km-1. If the
measurements are taken at two levels of the line profile depth, we will
verify the possibility of evaluating the gradient of the magnetic field
with height using a numerical simulation.
A series of calculations of the zirconium line profiles for a continuous
series of gradients linear with X=lg(?5) was carried out. Field strength at
a depth of X=-1.5 always remained at 2700 G. At a depth of X=1, it ranged
from 4200 to 1200 G in steps of 100 gauss, and at a depth of X=-4, on the
contrary, from 1200 to 4200 G.
[pic]
[pic]

Figure 3. The difference between the field strengths measured: (a) by the
DLAW methods for line pair ZR6127 - ZR6134, at levels of the profile depth
d = 20 and 10%; (b) from the positions of the centers of gravity of the
Stokes V profiles for Fe I lines ? 6173 and 6302 å. Both graphs are
constructed for the three models built from the SW75 model by addition of
model parameter ??.

Geometrical heights for the SW75 model are equal to -90, and 360 km at
levels X=1 and X=-4, respectively. Thus, the gradient ranged from +6.7 to
-6.7 G•km-1. The calculation was carried out for three models with SW75
temperature stratification with different effective temperatures Teff =
3627, 3909 and 4425 K, which are obtained through the use of model
parameters ?? equal to +0.10, 0 and -0.15, respectively. The graphs in
Figure 3 (a) relating to models with different effective temperature do not
coincide. This indicates dependence on the temperature of the measured
gradient. However, if we compare the DLAW method with conventional methods
of measurement of magnetic field gradient (see for example Figure 3(b)), we
cannot say that the proposed method is worse. Thus, at zero gradients in
the model taken for the calculation, the DLAW method and traditional method
both predicts the presence of a gradient of a certain value depending on
the temperature of model (See Figure 3 (a) and (b)). That is, in any method
of determining a magnetic field gradient with height, the temperature
distribution must be considered.
Another unpleasant thing in Figure 3 (a) is that in the case of a positive
gradient, i.e. when the field strength grows in the upward direction, the
method predicts a negative gradient. At first glance, this indicates that
the method is incorrect. However, if we look at the graph of the absolute
width of a single line at any given profile depth - see Figure 4, we find a
similar curve having extremum shifted towards positive gradients.
[pic]

Figure 4. The dependence of the spectral line absolute width on the
vertical gradient of the magnetic field in a sunspot umbra. The dependence
is defined for a given level of profile depth in the case of the
calculations for the SW75 model.
This means that other methods, for example the method SIR, which operates
with the individual points on the profiles Stokes V and I, must encounter
treatment of the same situation.

[pic]

Figure 5. The Stokes profiles (a) of the intensity RI=I/IC and (b) of the
circular polarization RV=V/IC for several values of the vertical gradient
of the magnetic field of opposite sign. With the growth of the positive
values of the gradient, the value of the slope of the wings RI and RV first
increases and then starts to decrease.

How the intensity and circular polarization profiles of the lines change at
different positive and negative magnetic field gradients can be seen in
Figure 5. At some positive gradient the slope of zirconium line wing for
the SW75 model reaches a maximum and then begins to decrease again with a
further increase in gradient. This leads to an effect that is contrary to
habitual concepts - the response of the profile located closer to the wings
forms higher in the photosphere than the response of the profile section
located closer to the line core.
We tested this conclusion by calculation of RFs for different points of the
profiles at four positive gradient values - see Figure 6.
[pic]

Figure 6. Comparison of RFs of different points on the line wings for 4
values of a positive gradient of the magnetic field. Graphs are given to
assess the relative effective HORs of different points on the profile depth
scale. Calculations were carried out with the following parameters: the
SW75 umbra model, Vmi=0, Blog(?)=1.5=2700 G, ?=15œ. Parameters of the probe
layer, used for RF calculations: ?B=+10G, ?X=0.05•log(?).

In the first row the RFs to change in field strength for line ZR6127 are
shown. Points with d=32%, i.e. close to the line core show that the
response comes with almost the same height (log(?5) — -2.3) at any value of
the gradient. The wings are formed deeper when a gradient is less than +3
G•km-1, and conversely higher when a gradient is more than this value. The
ZR6134 line has a lower sensitivity to the magnetic field, so all the
regularities for it are repeated at higher values of field and gradient
(see second row in Figure 6). As can be seen from the third row in Figure 6
the altitude dependence on the width difference of the point position on
the profile changes its sign at a gradient of about 2 G•km-1. This sign
change can be seen in Figure 3 (a) in the form of an extremum of the curve
at the same (2 G•km-1) value of the gradient.
Let us draw conclusions from the above. Growth of the magnetic field
upwards increases the absorption coefficient for the line wings in the
upper photosphere. This growth at some gradient can compensate for the
decrease in the absorption in direction line core - wings. Thus, the
optical thickness above a level at a certain geometrical height depends on
the magnetic field gradient. Changes in the optical thickness will change
the position of the RFs on the scale of geometric heights and will change
the HORs of line profile points. The measured absolute width of the profile
and the difference between the widths of the line pairs with similar atomic
parameters will strongly depend on the magnitude and sign of the magnetic
field gradient. Expressed in technical terms, "feedback" is created. Thus,
to properly convert the difference widths on two levels of profile depth d
to the magnetic field gradient, this feedback must be taken into account.
This fact is not a shortcoming of the method. Magnetic field gradients and
LOS velocity gradients significantly change the response function RF. This
should be adequately taken into account in each of the inversion methods
used: in the methods that involves the measurement of the differences in
the absolute width of the line profile (DLAW), and in the methods that deal
only with the Stokes profiles of individual spectral lines, as in the case
of the conventional method SIR.


4. Analysis of the temperature correction term Eij


1. Response of the width difference of the zirconium line profiles to
changes in temperature

Let us consider RF_T, the response function of the studied parameter to
temperature changes - see Figure 7 (a).
[pic]
[pic]

Figure 7. RFs of difference between the width of two lines (a) and RFs of
width of one line (b) of the zirconium to temperature changes at 10K in a
probe layer with thickness 0.05•log(?5). RF_Ts are calculated for a number
of levels of line profile depth d. In calculating RF_T a simplified scheme
was used. When the temperature was varied in the local layer, the values of
the gas and the electron pressure was not recalculated.
Local variations in temperature of 10K produce almost the same magnitude of
change in the width differences, as changes in the magnetic field of 10
gauss. This follows from comparison of the absolute values of RF_T in
Figure 7 (a) and values of RF_B in Figure 2 (a). Thus the difference
between the widths contains information about the changes both in the field
strength and in the temperature. If we want to reconstruct the distribution
of the field strength along the height, we need to split the information
and restore the temperature distribution first. Figure 7 (b) convinces us
that this is possible, since the dependence of the width of the profile on
temperature is about 10 times stronger than the dependence of the width
difference.
It should be noted that the character of the functions RF_T and RF_B
differs considerably. Unlike RF_B, RF_T changes its sign along the optical
depths scale. Furthermore RF_T has other effective HORs, than RF_B has.
Therefore the effect of local perturbation of temperature at a given height
will affect the profiles quite differently from that of a local
perturbation of the field strength at the same height.


2. Intensity profiles of the zirconium lines for photosphere models with
different effective temperature

For further analysis of the temperature correction term Ed,ij it is
important to visually imagine how the profiles of zirconium and the
relationship between them change with changes in temperature of the
radiating medium.
Figure 8 shows graphs of such profiles. We draw attention to two features
in the graphs.
1) The ratio of the maximum amplitudes of lines ZR6127, ZR6143, and
ZR6134 changes with the effective temperature of a model. The
excitation potential of the lower level ?, and the oscillator
strengths gf are slightly different for these three lines. These
values are included in the selective absorption coefficient as a
factor gf•10-??, that results in change in the relative intensity of
profiles when the value ?=5040/T is changed.


[pic]

Figure 8. Intensity profiles of the Zr I lines, depending on the model
parameter ??, where ?=5040/T. Effective temperature of models are 5677,
5099, 4628, 4238, 3909, 3627, 3383, and 3170 K. For each model the value
of the umbra contrast I is given, i.e. the ratio of the continuum
intensities for the umbra and quiet photosphere. The temperature of the
model for extreme left profiles (??=-0.4) corresponds to the temperature of
the penumbra. Conditions in the sunspot umbrae correspond to profiles with
?? =0.0 and 0.1. The values ?? = +0.2 and +0.3 for the physical conditions
on the Sun are unreal and profiles were calculated to visualize trends.

2) Line profiles for cold models have extended wings. This reflects an
increase in the damping parameter a with decrease in effective
temperature. The parameter a also depends on the excitation potential
of the lower level ?. The form of this dependence is more complicated.
It can be found, for example, in the book of D. Gray (1976), equation
(11.36). This means that the ratio of the widths in the wings, i.e.
for values of d from 0% to 4-8% varies with temperature differently
from the width ratio along the remainder of the line profile.
Thus, when using the DLAW method it is best not to use fragments of
profiles adjoining the wings and the core of intensity profiles.


3. The dependence of the temperature correction term Ed,ij on the
effective temperature of a model

Even with zero magnetic fields, the width of the ZR6127, ZR6143, and ZR6134
line profiles differs slightly. This difference varies with profile depth
d. When adding a magnetic field into the consideration, this initial
difference remains as a constant component. We describe it as the term
Ed,ij in equation (4), intended for calculation of the magnetic field.
[pic]
[pic]
[pic]

Figure 9. Graphs of temperature correction term Eij from relation (4) is
shown. Graphs are shown for different levels of profile depth d, for which
the difference between the widths of the lines is measured. The graphs are
calculated for a number of models with temperature distribution SW75, which
is modified by use of the model parameter ??. The value of the correction
term in må can be converted to gauss with the help of coefficients 49, 68
and 177 G/må for drawings (a), (b) and (c), respectively.

The term depends primarily on the temperature, so we call it the
temperature correction term. Figure 9 shows graphs of the temperature
correction term for a set of photosphere models with different effective
temperature. The graphs are calculated from formula (4) by substituting the
specified value of the magnetic field and the widths of the computed
profiles.
It is important that in the most suitable profile fragments for measurement
of magnetic fields (8% linear. For different pairs of lines, these curves are similar; they differ
in scale and the initial shift.
[pic]
[pic]
[pic]

Figure 10. The dependence of the temperature correction term Eij (a) of the
magnetic field B, (b) of the angle of the magnetic field vector to the line
of sight ?, (c) of microturbulent velocity Vmi.

A slope of Eij(d) will lead to a false gradient of the magnetic field with
height if the field is determined by the DLAW method without taking into
account the term Eij. The graphs in Figure 3 (a) have been obtained without
the correction term Eij(d) and this alleged gradient (the difference
between the field strengths measured at levels d = 10% and 20%) is marked
by thin lines in the Figure 3 (a).
It should be noted that the above calculations are made for certain values
of oscillator strengths, namely, -1.06, -1.18 and -1.30 for the lines
ZR6127, ZR6143, and ZR6134 respectively. These oscillator strengths are
chosen so that the lines have the same equivalent width in calculations for
the SW75 model and the absence of a magnetic field. The real lg(gf) values
can differ from those used in the calculations by up to 0.1 dex. Changes in
0.1 dex will shift the curves for Eij(d) along the abscissa scale by ~ 6 må
and deforms them in a certain way.

4. The dependence of the correction term Ed,ij on the other parameters
of the model calculations

Correction term Eij varies not only with temperature. According to Figure
10, Eij changes of ~ 1.5 må can be caused by a change in field strength
from 3000 to 1000 G or by a change in the angle of the magnetic field
vector to the line of sight from 0œ to 90œ. Changes on 1.5 må corresponds
to the seeming change in the magnetic field at ~ 75 G. Thus, Eij(T=Const,
B,?) << Eij(T, (B,?)=Const).

5. Conclusion and discussion

The initial idea of the possibility of creating a device that measures
magnetic field strength through the use of the Difference in the zirconium
Line Absolute Widths (i.e., the DLAW method) has undergone significant
changes.
On the one hand, a large number of factors affecting measurement results
were revealed. Thus, the DLAW method does not have the simplicity,
linearity, and independence of other physical parameters that were expected
initially. On the other hand the method has shown the potential to restore
the distribution of magnetic field strength with height in a new way. The
DLAW method allows one to measure the distribution of the magnetic field
along the depth of the spectral line intensity profile as a first step. As
a second step, it is necessary to establish a link between points on a
given level of profile depth and points at geometric height.
Until now, the main way of restoring the distribution of magnetic field
strength with height has been by solution of an inverse problem for the
Stokes profiles of spectral lines. Comparison of the results of different
methods will test both methods, and possibly get more accurate results.
Until now, the main way of restoring the distribution of magnetic field
strength with height was SIR method based on solution of an inverse problem
for the Stokes profiles of separated spectral lines. We can test SIR using
the difference between the absolute widths of the spectral lines with
similar atomic parameters and method of inversion that unlike SIR. And the
application of the two methods can yield more accurate results of the
magnetic field restoring.
Results of the study of the DLAW method predict an opportunity to detect
bulk elements of radiative medium with simultaneous local variations in the
magnetic field and the LOS velocity which are sufficient enough for the
perturbation of line profiles.
The task of establishing a link between points at a given level of profile
depth and geometric heights is not easy. The presence of the magnetic field
gradient itself stretches or compresses or even changes the sign of
relations between the line profile depths and geometric heights. It is not
known to what degree exactly this relation can be set for real, not
simulated, data. However, this problem is not only a problem of the DLAW
method. This is a general property of line profiles that is met with by any
method for the recovery of the physical parameters with height by Stokes
profiles.
Research into the DLAW method has been undertaken for models that are
homogeneous in the plane of the Sun surface. A real sunspot umbra is
inhomogeneous. How inhomogeneities distort the intensity profiles of the
zirconium lines must be explored using experimental data, which we do not
have. To interest researchers who have the opportunity to make observations
of sunspots umbrae in the lines of the multiplet number 2 Zr I we list once
again the merits of the proposed idea. ZR6127, ZR6143, and ZR6134 lines are
well suited for the study of sunspot umbra, they are not distorted by
scattered light of the penumbra and quiet photosphere since the profiles of
these lines is negligibly weak in a penumbra. The ZR6127, ZR6143 profiles
show distinct magnetic splitting, despite the small Lande factor. This is
due to the fact that zirconium is a heavy element and its lines have a
narrow Doppler profile. The parameters of the atomic transitions of these
three lines are close. In some cases, for example, in the analysis of the
LOS velocities the profiles can be superimposed on each other by its wings
(not forgetting the difference between the widths associated with the
magnetic field) and thus increase the signal to noise ratio. The small
difference in the temperature sensitivity of the profiles can be used in
order to analyze the distribution of temperature in the umbra. Finally,
differences between the widths of the profiles contain information about
the distribution of the magnetic field with height and extracting this
information is a routine task.
Among the disadvantages of the DLAW method for determining magnetic field
strength, the requirement for a large signal to noise value, the need to
accurately measure the position of the wings of the lines (0.2 må and
better), and the need for accurate alignment of the spatial scans of the
spectrum for the three lines which are sufficiently far from each other
should be noted. At the same time, the scans should be recorded
simultaneously. The width difference of lines does not allow determination
of the angle of the magnetic field vector to the line of sight. Therefore
it is desirable to combine the recording of intensity profiles with the
recording of other Stokes parameters.
The exact relation of the oscillator strength of the ZR6127, ZR6143, and
ZR6134 lines is not currently known, but this drawback will be eliminated
in the investigation process.

Acknowledgements The author expresses his thanks to Dr. V.N. Obridko for
valuable discussion and helpful advice. Also author thanks the referee of
Solar Physics journal for critical and helpful comments.

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