Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results
Indirect estimations of diameters of the smallest flux tubes outside sunspots are made using SOT/Hinode observations of Fei 6301.5 and 6302.5 lines. These estimations are based on the comparison of measured effective magnetic field strength Bₑff in named lines. It is shown that Bₑff (6301.5)/Bₑff (6...
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irk-123456789-1199452017-06-11T03:03:25Z Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results Botygina, O.O. Gordovskyy, M.Yu. Lozitsky, V.G. Indirect estimations of diameters of the smallest flux tubes outside sunspots are made using SOT/Hinode observations of Fei 6301.5 and 6302.5 lines. These estimations are based on the comparison of measured effective magnetic field strength Bₑff in named lines. It is shown that Bₑff (6301.5)/Bₑff (6302.5) ≈ 1.3 in the range Bₑff = 40-300 G, and Bₑff (6301.5)/Bₑff (6302.5) ≈ 1.0 for Bₑff ≤ 10-20 G. The first case corresponds to the two-component magnetic field with kG flux tubes and weak background field, whereas the second one corresponds to background field without flux tubes. Assuming that the field range Bₑff = 10-40 G corresponds to the case with only one flux tube in each pixel, the flux tube diameters should be 15-30 km. Possible influence of the brightness contrast and the Zeeman saturation could change this estimation by approximately 20%. 2016 Article Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results / O.O. Botygina, M.Yu. Gordovskyy, V.G. Lozitsky // Advances in Astronomy and Space Physics. — 2016. — Т. 6., вип. 1. — С. 20-23. — Бібліогр.: 17 назв. — англ. 2227-1481 DOI:10.17721/2227-1481.6.20-23 http://dspace.nbuv.gov.ua/handle/123456789/119945 en Advances in Astronomy and Space Physics Головна астрономічна обсерваторія НАН України |
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Indirect estimations of diameters of the smallest flux tubes outside sunspots are made using SOT/Hinode observations of Fei 6301.5 and 6302.5 lines. These estimations are based on the comparison of measured effective magnetic field strength Bₑff in named lines. It is shown that Bₑff (6301.5)/Bₑff (6302.5) ≈ 1.3 in the range Bₑff = 40-300 G, and Bₑff (6301.5)/Bₑff (6302.5) ≈ 1.0 for Bₑff ≤ 10-20 G. The first case corresponds to the two-component magnetic field with kG flux tubes and weak background field, whereas the second one corresponds to background field without flux tubes. Assuming that the field range Bₑff = 10-40 G corresponds to the case with only one flux tube in each pixel, the flux tube diameters should be 15-30 km. Possible influence of the brightness contrast and the Zeeman saturation could change this estimation by approximately 20%. |
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Article |
| author |
Botygina, O.O. Gordovskyy, M.Yu. Lozitsky, V.G. |
| spellingShingle |
Botygina, O.O. Gordovskyy, M.Yu. Lozitsky, V.G. Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results Advances in Astronomy and Space Physics |
| author_facet |
Botygina, O.O. Gordovskyy, M.Yu. Lozitsky, V.G. |
| author_sort |
Botygina, O.O. |
| title |
Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results |
| title_short |
Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results |
| title_full |
Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results |
| title_fullStr |
Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results |
| title_full_unstemmed |
Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results |
| title_sort |
estimation of the flux tube diameters outside sunspots using hinode observations. preliminary results |
| publisher |
Головна астрономічна обсерваторія НАН України |
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2016 |
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http://dspace.nbuv.gov.ua/handle/123456789/119945 |
| citation_txt |
Estimation of the flux tube diameters outside sunspots using Hinode observations. Preliminary results / O.O. Botygina, M.Yu. Gordovskyy, V.G. Lozitsky // Advances in Astronomy and Space Physics. — 2016. — Т. 6., вип. 1. — С. 20-23. — Бібліогр.: 17 назв. — англ. |
| series |
Advances in Astronomy and Space Physics |
| work_keys_str_mv |
AT botyginaoo estimationofthefluxtubediametersoutsidesunspotsusinghinodeobservationspreliminaryresults AT gordovskyymyu estimationofthefluxtubediametersoutsidesunspotsusinghinodeobservationspreliminaryresults AT lozitskyvg estimationofthefluxtubediametersoutsidesunspotsusinghinodeobservationspreliminaryresults |
| first_indexed |
2025-07-08T16:58:18Z |
| last_indexed |
2025-07-08T16:58:18Z |
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1837098755180986368 |
| fulltext |
Estimation of the �ux tube diameters outside sunspots
using Hinode observations. Preliminary results
O.O.Botygina1∗, M.Yu.Gordovskyy2†, V.G. Lozitsky1‡
Advances in Astronomy and Space Physics, 6, 20-23 (2016) doi: 10.17721/2227-1481.6.20-23
© O.O.Botygina, M.Yu.Gordovskyy, V.G. Lozitsky, 2016
1Astronomical Observatory, Taras Shevchenko National University of Kyiv, Observatorna str., 3, Kyiv, 04053, Ukraine
2Jodrell Bank Centre for Astrophysics, University of Manchester, Alan Turing building, Manchester M13 9PL, UK
Indirect estimations of diameters of the smallest �ux tubes outside sunspots are made using SOT/Hinode obser-
vations of Fei 6301.5 and 6302.5 lines. These estimations are based on the comparison of measured e�ective magnetic
�eld strength Beff in named lines. It is shown that Beff(6301.5)/Beff(6302.5) ≈ 1.3 in the range Beff = 40�300G,
and Beff(6301.5)/Beff(6302.5) ≈ 1.0 for Beff ≤ 10�20G. The �rst case corresponds to the two-component magnetic
�eld with kG �ux tubes and weak background �eld, whereas the second one corresponds to background �eld without
�ux tubes. Assuming that the �eld range Beff = 10�40G corresponds to the case with only one �ux tube in each
pixel, the �ux tube diameters should be 15�30 km. Possible in�uence of the brightness contrast and the Zeeman
saturation could change this estimation by approximately 20%.
Key words: Sun: magnetic �elds, photosphere
introduction
The problem of true sizes of the smallest discrete
�ux tubes outside sunspots is far from being solved.
It is well known that sunspots are the largest struc-
tures on the Sun with the strongest magnetic �elds.
Their diameters, typically, are 5�100Mm, and mag-
netic �eld strengths are in the range 2�3 kG, some-
times reaching 4�5 kG [1, 5, 14]. Solar pores (i. e.
small sunspots without penumbra) have diameters,
typically, 1�3Mm and magnetic �eld of 1.5-2 kG.
Similar values of �eld strength are found in spa-
tially unresolved magnetic �ux tubes using line-ratio
method; true �ux tube diameters have been esti-
mated in the range 100�300 km (e. g., [15]). Such
estimations have been obtained using the magneto-
graphic observations with direct resolution 1700 km,
assuming negligible contribution of background �eld.
Similar method and assumptions have been used by
Wiehr [17]. Assuming that �eld Beff = 2G corre-
sponds to a single �ux tube within the aperture of
2400 km, he estimated their diameters to be around
dmin = 62 km.
It is necessary to note, that both assumptions
(about negligible contribution of background �eld
and Beff = 2G) are too rough. Lozitsky [7, 8],
Gordovskyy & Lozitsky [4] have shown that the
best interpretation of measurements made in about
ten spectral lines with very di�erent Lande factors
from [3, 6] can be given using the two-component
model with non-zero background �eld. Similar con-
clusions have been made in [9, 12] based on mag-
netographic observations of quiet regions in 5250.2
and 5247.1 lines. They showed that the magnetic
�eld strength in �ux tubes, Bft, is 1.5�2.2 kG, while
the strength of the background �eld, Bbackgr, and the
�lling factor of �ux tubes, f , are related via simple
formula: Bbackgr/f ≈ 1 kG. Therefore, dmin = 40�
50 km, assuming Beff = 4G and size of 1Mm of enter
aperture for direct observations.
In reality, the assumption that Beff = 4G cor-
responds to one �ux tube per pixel (or aperture) is
rather questionable. In the studies discussed above,
Beff = 4G is supposed to exceed the noise level of
the magnetograph. However, the noise level depends
on various instrumental parameters, for instance,
the integration time. Therefore, the use of speci�c
spectral manifestations would provide more reliable
tool for spatially unresolved �eld diagnostics. In
the line-ratio method, Beff(5247.1)/Beff(5250.2) >
1 (or Beff(6301.5)/Beff(6302.5) > 1) corresponds
to the two-component magnetic �eld � with kG
�ux tubes and weak background �eld. On the
contrary, in case of Beff(5247.1)/Beff(5250.2) =
1 and Beff(6301.5)/Beff(6302.5) = 1, the �eld
should be nearly uniform. Obviously, a sin-
gle �ux tube per pixel should be a transitional
case between Beff(6301.5)/Beff(6302.5) > 1 and
Beff(6301.5)/Beff(6302.5) = 1; additional �ux tubes
would further change this ratio. This idea is used in
the present work to estimate sizes of �ux tubes using
Hinode spectropolarimetric observations.
∗olga.botygina@gmail.com
†mykola.gordovskyy@manchester.ac.uk
‡lozitsky@observ.univ.kiev.ua
20
Advances in Astronomy and Space Physics O.O.Botygina, M.Yu.Gordovskyy, V.G. Lozitsky
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1.2
0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2
B(6301), kG
B
(6
3
0
2
),
k
G
Fig. 1: Beff(6302.5) vs. Beff(6301.5) outside sunspots
measured using the splitting of centres-of-mass of I+V
and I−V Stokes pro�les. Dashed black line corresponds
to Beff(6301.5) = Beff(6302.5). Dashed-dotted line cor-
responds to the linear �t between −400 and 400G.
B(6302), G
B
(6
3
0
1
),
G
0 100 200 300
300
400
200
100
0
Fig. 2: Beff(6301.5) vs. Beff(6302.5) outside sunspots
measured using the bisector splitting of I ± V pro-
�les. Dashed black line corresponds to Beff(6301.5) =
Beff(6302.5). Dashed-dotted line corresponds to the lin-
ear �t between −400 and 400G.
observations
Space observatory Hinode (Dawn in Japanese)
was launched on September 23, 2006. The main
objective of Hinode is high-precision measurements
of small changes in the intensity of the solar mag-
netic �elds. The observatory has three scienti�c
instruments: the Solar Optical Telescope (SOT),
the EUV Imaging Spectrometer (EIS), and the X-
Ray Telescope (XRT) (see [16]). The diameter of
the primary mirror of the telescope is 50 cm, which
provides direct di�raction limit of the spatial res-
olution at the level of 0.32 arcsec = 230 km on
the Sun in the red spectral region (λ=6302Å). De-
composition of sunlight into a spectrum was made
by Fabry-Perot interferometer. The equipment of
Hinode allow observations of the Sun in 10 spec-
tral lines, including Mgi 5172.7Å, Fei 5247.1, 5250.2,
5250.6, 5576.1Å, Nai 5896Å, Fei 6301.5, 6302.5Å,
Tii 6303.8Å, Hα 6563Å. In the spectropolarimetric
mode (receiving distributions of Stokes parameters
I,Q, U, V over the line pro�le) spectral resolution is
about 90mÅ at 6300Å. In this work we analyse non-
sunspot magnetic �eld measurements in area of three
active regions: 29 January 2015, 8 April 2015 and 9
May 2015, NOAA 2268, 2320 and 2339, respectively.
As it was explained above, for determination
of the �ux tube size the dependence Beff(6301.5)
vs. Beff(6302.5) should be considered. Two ex-
amples of such dependence for non-sunspot �elds
based on Hinode data are given in Figs. 1,2. In
these Figures, dashed-dotted line presents ratio
Beff(6301.5)/Beff(6302.5) = 1.3, whereas cross-
hatching line � Beff(6301.5)/Beff(6302.5) = 1.0.
Two methods were used for Beff determination. First
is traditional �centre of gravity� method that ap-
plies to I + V and I − V pro�les. This method in-
cludes the central parts of pro�les which have the
maximum intensity. Second method bases on deter-
mination of splitting of bisectors of the same pro-
�les, excluding line core and far wings where gra-
dient of intensity, dI/dλ, is low. From Fig. 1 it
follows that ratio Beff(6301.5)/Beff(6302.5) = 1.3
begins from about 10G. So, we can suppose that
minimal e�ective magnetic �eld, which correspond
to the presence of only one �ux tube inside aper-
ture, Beff,min, is about 10G. From Fig.2 we can
see that ratio Beff(6301.5)/Beff(6302.5) = 1.3 is
actual in �eld range Beff = 40�300G, whereas
Beff(6301.5)/Beff(6302.5) = 1 � for Beff ≈ 20G.
Thus, in �eld range between 40 and 10G we can ex-
pect transition from two- to mono-component mag-
netic �eld structure. In the next section the use of
this diagnostic peculiarity for estimation of �ux tube
diameter is described.
interpretation of the data
We assume that the measured e�ective magnetic
�eld Beff represents the magnetic �ux going through
each pixel. It is valid for lines with low magnetic
sensitivity, which does not experience the e�ect of
magnetic saturation. (The e�ect of magnetic satura-
tion is the loss of the measured magnetographic sig-
nal in very strong �elds (kG range). It appears when
spectral line components corresponding to the strong
21
Advances in Astronomy and Space Physics O.O.Botygina, M.Yu.Gordovskyy, V.G. Lozitsky
�eld move outside the analysed spectral range.) In
addition, we believe that the magnetic �eld is longi-
tudinal (has only the line-of-sight component) and
two-component everywhere, i. e. consists of unre-
solved �ux tubes with strength B0 on their axes and
background �eld with strength Bbackgr. Therefore,
Beff = (1− f)Bbackgr + fkB0, (1)
where f is a �lling factor for �ux tubes and k is
the factor that takes into account the radial distri-
bution of magnetic �eld B(x) in �uxtube (k = 1
corresponds to a rectangular pro�le and k < 1 cor-
responds to non-rectangular pro�les, i. e. similar to
those observed in sunspots). According to [9], the
pro�le B(x) for �ux tubes in quiet regions on the
Sun is approximately the same as in solar pores, and
can be approximated by the expression
B(x) = B0
(
1− x4
)
, |x| ≤ 1, (2)
where x is the relative distance from the �ux tubes
axis, x = r/r0, r is linear distance from the tube
axis, r0 is the tube radius. In case of pro�le (2),
k = 2/3. Let us transform the expression (1), taking
into account that Bbackgr ≪ kB0:
f ≈ Beff
Bbackgr/f + kB0
,
where Bbackgr/f ≈ 1 kG [9]. Then, assuming that
one �ux tube with diameter d is inside instrument's
aperture, after simple transformations we obtain:
d = 2
√
fminS0
π
≈ 2
√
Beff,minS0
π (Bbackgr/f + kB0)
, (3)
where fmin and Beff,min are minimal �lling factor and
the e�ective magnetic �eld, respectively. These val-
ues correspond to the presence of only one �ux tube
inside aperture. S0 is equivalent area of the input
aperture for direct observations. Taking into account
that direct resolution for SOT/Hinode is 230 km, we
have S0 = 4.15 · 104 km2, assuming a circular en-
trance aperture of the instrument. As it was ex-
plained above, Beff,min = 10�40G. Also, taking into
account the estimation of B0 in [9, 10], we will sup-
pose B0 = 2.2�2.3 kG. Substitution of these param-
eters into (3) gives the following estimations:
d ≈ 14.5 km for Beff,min = 10G,
d ≈ 20.5 km for Beff,min = 20G,
d ≈ 29 km for Beff,min = 40G.
Thus, despite the magnitude of the uncertainty
Beff,min, values of d hit in the relatively narrow range,
approximately 15�30 km. These values are smaller
than similar estimations obtained earlier in [9, 17],
so, it is necessary to consider other e�ects that may
change the value of d.
discussion
Similar to the equation (1), the intensity of cir-
cular polarization, Stokes V , can be written as:
Vobs = (1− f)Vbackgr + fVft. (4)
In the weak-�eld approximation:
V =
dI
dλ
∆λH ∝ Ic
drλ
dλ
B, (5)
where I is the Stokes I parameter, ∆λH is the Zee-
man splitting, rλ = Iλ/Ic is relative intensity, Ic is
intensity in the nearest spectral continuum, B is the
magnetic �eld. In principle, for non-sunspot mag-
netic �elds, formula (5) is suitable for evaluating Vobs
and Vbackgr, but not suitable for Vft. The main rea-
son is that, due to Zeeman saturation, Stokes V in-
creases slower than drλ/dλ. Let us introduce the
factor accounting for Zeeman saturation, Zs. Thus,
Vft ∝ Ic,ft
drλ
dλ
ZsBft, (6)
where Zs ≤ 1. From (6) it also follows, that Vft
depends on Ic,ft, i. e. intensity of the spectral con-
tinuum in �ux tubes. According to the observations
in [2, 11] Ic,ft/Ic,backgr > 1, potentially, if we assume
that �ux tubes coincide with solar �ligree or facu-
lae knots, this ratio can be as high as 2. Now, let
us introduce the parameter describing the �ux tube
continuum contrast, Kft, as Kft = Ic,ft/Ic,backgr. Ob-
viously, as long as �lling factor f is small for non-
sunspot areas, Ic,obs ≈ Ic,backgr. Our �nal assump-
tion is:(
drλ
dλ
)
obs
≈
(
drλ
dλ
)
backgr
≈
(
drλ
dλ
)
ft
.
After substituting these parameters into formula (4)
and making elementary transformations, we get
Bobs = (1− f)Bbackgr + fKftZsBft. (7)
Assuming Bobs ≡ Beff , formulas (4) and (7) yield the
following expression:
d ≈ 2
√
Beff,minS0
π(Bbackgr/f + kKftZsB0)
. (8)
Let us compare the values of d for two cases: Kft = 2
and Kft = 1 assuming Zs = 1; the corresponding di-
ameters are d2 and d1, respectively. From (8) it is
obvious that
d2
d1
=
√
Bbackgr/f + kB0
Bbackgr/f + 2kB0
≈
√
1000 + 1500
1000 + 3000
≈ 0.8.
22
Advances in Astronomy and Space Physics O.O.Botygina, M.Yu.Gordovskyy, V.G. Lozitsky
Thus, two-fold enhancement of continuum contrast
in �ux tubes changes our estimations by only about
20%. True diameters of �ux tubes in this case are
d2 = 0.8× (14.5− 29) ≈ 12− 23 km.
Similar calculations show that signi�cant Zeeman
saturation of Zs = 0.5 would result, on the contrary,
in increasing of d by about 20%. Therefore, the in-
crease of the continuum contrast and Zeeman satu-
ration, have opposite e�ect on the value of d and,
in the general, should not greatly a�ect our estima-
tions. Hence, we can conclude that actual diameters
of �ux tubes should be, approximately, in range 15�
30 km, i. e. much smaller that di�raction limit of the
most modern solar instruments.
summary
For three active regions on the Sun we have made
indirect estimations of diameters of the smallest �ux
tubes outside sunspots using SOT/Hinode polari-
metric observations. These estimations are based
on comparison of measured magnetic �eld strength
Beff in Fei 6301.5 and 6302.5 lines. It is shown that
Beff(6301.5)/Beff(6302.5) ≈ 1.3 in the �eld range
Beff =40�300G, and Beff(6301.5)/Beff(6302.5) ≈ 1.0
for Beff ≤ 10�20G. First case corresponds to two-
component magnetic �eld with kG �ux tubes and
weak background �eld, whereas the second one corre-
sponds to one-component �eld without �ux tubes. If
we assume that only one �ux tube exists in each pixel
in the �eld range Beff = 10�40G, then the diameters
of �ux tubes should be 20�30 km. Possible in�uence
of the brightness contrast and the Zeeman saturation
could change this estimation by about 20%. From
our results it follows, that even in era of GREGOR
telescope [13]), with 1.5m aperture and about 80 km
resolution, the smallest magnetic �ux tubes might
not be resolved spatially. Naturally, these smallest
�ux tubes cannot be resolved by other modern solar
telescopes and, therefore, appropriate interpretation
of such observations will require multi-component
�eld models.
acknowledgement
The authors are extremely thankful to Dr. My-
roslav Stodilka for helpful discussions and anony-
mous referee for useful critical notes and comments.
Hinode telescope is Japanese project which was con-
structed and launched by ISAS/JAXA, in coopera-
tion with NAOJ (Japan), NASA (USA) and STFC
(UK) Telescope is operated by named organizations
as well as ESA and NTS (Norway). This study was
funded by the Taras Shevchenko National University
of Kyiv, No.16BF023-01, No.16BF023-02 projects.
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