# THERMAL AND MAGNETIC FIELD STRUCTURE OF NEAR EQUATORIAL CORONAL HOLES

Hiremath, K. M. and Hegde, M  Indian Institute of Astrophysics, Bangalore,
India<EMAIL_ADDRESS>
###### Abstract

We use full-disk, SOHO/EIT 195 $\AA$ calibrated images to measure latitudinal
and day to day variations of area and average photon fluxes of the near
equatorial coronal holes. In addition, energy emitted by the coronal holes
with their temperature and strength of magnetic field structures are
estimated. By analyzing data of 2001-2008, we find that variations of average
area (A), photon flux (F), radiative energy (E) and temperature (T) of coronal
holes are independent of latitude. Whereas inferred strength of magnetic field
structure of the coronal holes is dependent on the latitudes and varies from
low near the equator to high near both the poles. Typical average values of
estimated physical parameters are: $A\sim 3.8(\pm 0.5)\times
10^{20}~{}cm^{2},F\sim 2.3(\pm 0.2)\times
10^{13}~{}photons\;cm^{-2}\;sec^{-1},E\sim 2.32(\pm 0.5)\times
10^{3}~{}ergscm^{-2}sec^{-1}\ and\ T\sim 0.94(\pm 0.1)\times 10^{6}~{}$ K.
Average strength of magnetic field structure of coronal hole at the corona is
estimated to be $\sim$ $0.08\pm 0.02$ Gauss. If coronal holes are anchored in
the convection zone, one would expect they should rotate differentially.
Hence, thermal wind balance and isorotation of coronal holes with the solar
plasma implies the temperature difference between the equator and both the
poles. Contrary to this fact, variation of thermal structure of near
equatorial coronal holes is independent of latitude leading to a conclusion
that coronal holes must rotate rigidly that are likely to be anchored
initially below the tachocline confirming our previous study (ApJ, 763, 137,
2013).

sun:equatorial coronal holes, sun:thermal structure, Coronal Hole:thermal
structure, Coronal Hole:magnetic field structure

##  1 INTRODUCTION

Coronal holes (CH), with unipolar magnetic field structures (Harvey & Sheeley
1979; Harvey et al. 1982), are the lowest density plasma structures mainly
detected in either UV or X-ray radiations of the sun’s atmosphere and are
associated with rapidly expanding magnetic fields and the acceleration of the
high-speed solar wind (Krieger et al. 1973; Neupert & Pizzo, 1974; Nolte et
al. 1976; Zirker 1977; Cranmer 2009 and references therein; Wang 2009;
Wiegelmann, Thalmann, Solanki 2014). Possible link of sunspot (Hiremath and
Mandi 2004 and references there in; Hiremath 2009) activity on the Earth’s
atmosphere and climate are well recorded in the literature. Recently,
evidences are building up that coronal holes also trigger responses in the
Earth’s upper atmosphere and magnetosphere (Soon et al. 2000; Lei et al. 2008;
Shugai et al. 2009; Sojka et al. 2009; Choi et al. 2009; Ram, Liu & Su 2010;
Krista 2011; Verbanac et al. 2011; Fathy et.al. 2014; Machiya and Akasofu
2014). Very recently, Hiremath, Hegde and Soon (2015) came to a conclusion
that, in addition to influence of sunspots, emission of coronal holes also
trigger and maintain Indian Monsoon rainfall.

From the ISRO (Indian Space Research Organization) funded project, authors′
interest in the study of coronal holes is to examine radiative responses of
these structures on the Indian summer Monsoon rainfall. For this purpose,
estimation of radiative flux and energy of the CH at 1 AU during their
evolution passage on the solar disk is necessary. Keeping these main
objectives in mind, by using SOHO/EIT 195 $\AA$ calibrated images, thermal
structure of CH such as photon flux, energy and hence temperature at the sun
and at 1 AU are estimated.

Dynamics such as rotation rates (Hiremath and Hegde 2013 and references there
in) and in situ plasma conditions of the CH are also of considerable interest
for the solar community as the fast solar wind most likely originates in these
regions (Stucki et al. 2002; Hegde et.al. 2015). Ion temperatures in a polar
coronal hole from the line width measurements are estimated by Tu et al.
(1998). Centre-to-limb variation of radiance of the transition region and
coronal lines have been obtained by Wilhelm et al. (1998). Doppler shifts
measurements of CII, O VI and Ne VIII lines are obtained by Warren, Mariska
and Wilhelm(1997). Peter & Judge (1999) also studied these and few other lines
and found that the redshift to blue shift transition occurs at electron
temperatures of about 5 $\times 10^{5}$ K. Analysis of SXT/Yohkoh images (Hara
et. al. 1996) have shown that the estimated temperature structure (1.8 -2.4
$\times 10^{6}$ K) in CH is of same order as that in the ambient medium.
Mogilevsky, Obridko and Shilova (1997) also arrive at a similar conclusion.
From the analysis of Big Bear solar observatory magnetograms and SOHO EIT
images, Zhang et.al. (2007) came to a conclusion that temperature of coronal
hole and ambient medium is not entirely different.

In recent years, space based observations such as Yohkoh and SOHO missions are
extensively used for estimation of temperature structure of CH (Hara et al.
1994; Moses et al. 1997). One physical parameter that senses thermal structure
of CH is electron temperature that is estimated by different spectroscopic
methods. A detailed assessment of observations in coronal holes and the
deduced temperatures is published by Habbal, Esser and Arndt (1993). See also
a detailed review of estimation of coronal hole temperature by Habbal (1996)
and Wilhelm (2012) respectively. Electron temperatures in CH can be measured
with the help of a Magnesium line ratio of a temperature-sensitive pair (see
Wilhelm 2006). With the assumption that density and temperature of the gas
from which spectral lines are formed are constant along the line of sight,
Habbal et al. (1993) estimated the temperature structure of CH. Using two SOHO
spectrometers, CDS and SUMER, electron temperature of CH is measured as a
function of height above the limb in a polar coronal hole (David et al. 1998;
Wilhelm et al. 1998). Doschek & Laming (2000) found that increase in the
emission-line ratio of the polar coronal hole is primarily due to increase of
the electron temperature with height. Marsch, Tu and Wilhelm (2000) found that
the hydrogen temperature increased only slightly from 1 $\times 10^{5}$ to 2
$\times 10^{5}$ K in the height range from 12,000 to 18,000 km. Stucki et al.
(2000) presented that with increasing formation temperature, spectral lines
show on average, an increasingly stronger blue shift in coronal holes relative
to the quiet sun at equal heliospheric angle. Furthermore, Xia, Marsch and
Wilhelm (2004) reported that the bases of coronal holes seen in chromospheric
spectral lines with relatively low formation temperatures displayed similar
properties as normal quiet sun regions. More recently, studies of Wilhelm
(2006; 2012) suggest that, in a polar coronal hole region, the electron
temperature in plumes is estimated to be $\sim 8\times 10^{5}$ K and, $\sim
1.13\times 10^{6}$ K in the inter plume regions, at the height of 45 Mm above
the limb.

From the silicon and iron coronal lines, Doscheck and Feldman (1977) conclude
that CH temperature must be less than 1 MK. From the jets of coronal holes,
Nistico et.al. (2011) compute electron temperature of CH by the filter ratio
method at 171 $\AA$ and 195 $\AA$ and estimated the temperature structure with
a magnitude in the range of $\sim$ 0.18-1.3 MK. By using two lines Mg IX (368
Å) and Mg X (625 Å) and, with a similar line ratio technique, Doyle et.al.
(2010) derived the different coronal hole temperature structures during solar
maximum ($\sim$ 1.04 MK) and during minimum ($\sim$ 0.82 MK) respectively. The
EUV (Fe XV at 284 Å) and radio (at 169 and 408 MHz) observations (Chiuderi,
Avignon and Thomas 1977) suggest that CH consists of a hot (10 % of the CH
surface with $\sim$ $2\times 10^{6}$ K) and a cold ($\sim$ $0.8\times 10^{6}$
K) regions to explain observations of both the EUV and radio coronal hole
temperature structures. From the analysis of EUV (SOHO/CDS) and radio emission
(164-410 MHz, by the Nancay Radioheliograph, France), and with a model,
Chiuderi et. al. (1999) estimated the CH temperature structure to be $\sim$
$9\times 10^{5}$ K. From the spectroscopic diagnostics of Mg VIII (430.47
$\AA$ and 436.62 Å) ion, observed by Sky lab space probe, Dwivedi and Mohan
(1995), estimated the coronal hole electron temperature to be $\sim$ $8\times
10^{5}$ K. Dwivedi, Mohan and Wilhelm (2000), from the SUMER observations,
estimate the CH temperature to be $\sim$ $(6.5-7.5)\times 10^{5}$ K.
Observations (Esser et.al. 1999) of Ly$\alpha$, $\lambda$1216, Mg X (625 Å),
and O VI (1038 Å) spectral lines with the UVCS instrument on board SOHO, from
$1.35-2.1R_{\odot}$, yield that proton temperature of CH slowly increases
between 1.35 and $2.7R_{\odot}$ and does not exceed $3\times 10^{6}$ K in that
region.

In most of the afore mentioned studies, estimated electron temperature, along
the slit of the observations over the region of CH, does not represents
temperature structure of whole CH. Using filter ratio technique, although one
can use DEM method for estimation of temperature of CH, this method yields
ambiguity (Zhang, White and Kundu, 1999; Landi, Reale & Testa 2012; Guennou et
al. 2012). Moreover, line ratio method might not be appropriate ( see section
4.5, Slemzin, Goryaev and Kuzin 2014) for estimation of coronal hole
temperature as there is a appreciable contribution of scattered light due to
brighter ambient corona, especially for the coronal holes. However, in this
study we present a simple method for estimation of radiative flux, energy and
temperature structure of CH at the sun and near Earth.

As solar wind due to coronal hole is directly proportional to area (Hegde em
et.al. 2015; Rotter et.al 2012; Karachik and Pevtsov 2011; Abramenko et. al.
2009; Shugai, Veselovsky and Trichtchenko 2009; Vršnak, Temmer and Veronig
2007; Nolte et.al. 1976), it is interesting to examine how these parameters
vary during the evolution of coronal hole. To be specific, for example, as the
Earth’s ionosphere responds with the solar wind due to coronal hole it is
important to estimate physical parameters such as area, temperature structure,
etc.,

As for dynamics, except some of the studies (Shelke & Pande 1985; Obridko &
Shelting 1989; Navarro-Peralta & Sanchez-Ibarra 1994; Insley et al. 1995),
other studies (Wagner 1975; Wagner 1976; Timothy & Krieger 1975; Bohlin 1977;
Hiremath and Hegde 2013; Japaridze et.al. 2015) indicate rigid body rotation
rates of the coronal holes. With large number of data and accurately estimated
average longitude from the central meridian of the coronal holes, especially
Hiremath and Hegde (2013) came to a conclusion that irrespective of area and
latitudes coronal holes rotate rigidly. We do not mean that all the coronal
holes rotate rigidly. This is because all means, even the coronal holes near
the poles. However, we have restricted the data (also with additional three
criteria as described in section 2) of coronal holes that occur between 40 deg
North to 40 deg South, that is near equatorial coronal holes.

As the coronal holes are unipolar magnetic flux tubes (Harvey & Sheeley 1979;
Harvey et al. 1982), condition of infinite conductivity of the corona leads to
isorotation of the coronal hole flux tubes with the ambient plasma rotation.
If the coronal holes rotate differentially, then thermal wind balance equation
(Brun, Antia and Chitre 2010) yields temperature difference between the
equator and the poles. On the other hand, if coronal holes rotate rigidly,
there is no temperature difference between the equator and poles. Hence, in
order to confirm whether coronal holes rotate rigidly or differentially,
information regarding latitudinal variation of thermal structure of the
coronal holes is necessary.

Physics of MHD waves (that emanate from the coronal holes) is important not
only for understanding the heating of corona but also useful for understanding
the fast solar wind. Many MHD models (Davila 1985; Cally 1986, 1987; Ofman
2005 and references there in) were developed to probe these phenomena where in
strength and geometry of magnetic field structure of coronal hole are
necessary. Although a general consensus is emerged that geometry of coronal
hole magnetic field structure is unipolar, to the knowledge of authors and
till date, no study is available that estimates magnetic field strength of
coronal hole in 195 $\AA$ that probably originates (Fig 2, Yang et. al. 2009)
around 1.1 solar radius.

In addition to importance of physical parameters of CH for study of solar-
terrestrial relationship, one has to also address the following questions in
order to resolve the fundamental problems presented in the previous
paragraphs. To summarize the same: (i) How the physical properties, such as
area, radiative flux, energy and temperature structure vary during the
evolution passage of CH over the observed solar disk? (ii) Do these physical
parameters of CH are dependent or independent of heliographic latitude? (iii)
From the information of latitudinal variation of temperature structure of CH,
is it possible to get any information whether CH rotate rigidly or
differentially? (iv) What is strength of magnetic field structure of CH at the
height of coronal region around 1.1$R_{\odot}$ where the line 195 $\AA$
originates.

In order to seek the answers to afore mentioned problems, we use near
equatorial coronal holes for the the present study.

In section 2, we present the data of near equatorial coronal holes and method
of analysis, and the results of this analysis are presented in section 3. In
section 4, with a brief discussion, conclusions of this study are presented.

##  2 DATA AND ESTIMATION OF DIFFERENT PHYSICAL PARAMETERS OF CH

We use 195 $\AA$ full-disk images obtained by EIT on board SOHO, although on
board instrument also observes full-disk EUV images in other wave length (171
$\AA$, 284 $\AA$ and 304 $\AA$) pass bands. A detailed description of the
instrument is provided by Delaboudiniére et al. (1995). The obtained images
are in FITS format and individual pixels are in units of data number (DN/sec).
DN is defined to be output of the instrument electronics which corresponds to
the incident photon signal converted into charge within each CCD pixel
(Madjarska & Wiegelmann 2009). Further details of SOHO/EIT 195 $\AA$ images,
their calibration, method of detection of CH with estimation of heliographic
coordinates (such as latitude $\theta$ and longitude $l$ ) and, computation of
total DN counts (TDN) of coronal holes are described in our previous study
(Hiremath & Hegde 2013). In the present study, we mainly concentrate on the
data of near equatorial coronal holes that are distributed with in 40 deg
north to 40 deg south. Additional three more criteria used in selection of the
data are: (i) in order to minimize the projection effects (especially coronal
holes near both the eastern and the western limbs), we considered only the
coronal holes that emerge within 65∘ central meridian distance, (ii) the
coronal holes must be compact, independent, and not elongated in latitude and,
(iii) during coronal holes passage across the solar disk, it should not merge
with other coronal hole.

###  2.1 Computation of Area of CH

As described in the previous study (Hiremath and Hegde 2013), once boundary of
a CH is detected, total number of pixels (TNP) with in the detected boundary
is estimated and area $A$ of coronal hole and its measured uncertainty $\delta
A$ are computed as follows.

$\displaystyle A=c_{1}\frac{TNP}{cos\;l}\hskip 19.91684ptcm^{2}\,,$ (1)
$\displaystyle\delta A=c_{1}(TNP)(tan\;lsec\;l)\delta l\hskip
19.91684ptcm^{2}\,,$ (2)

where the multiplicative constant term $c_{1}(=3.573\times 10^{16}$) is
estimated from the resolution of pixel size and the factor $1/(cos\;l)$ ($l$
is viewing angle or longitude from the central meridian) is a correction
factor for the projectional effect for the CH that are close to the limb.

We have also corrected the projectional effects from the formula
$A_{c}=c_{1}A_{obs}/cos(\delta)$ (where $A_{c}$ is corrected area, $A_{obs}$
is observed area and
$cos(\delta)=sin(B_{0})sin(\theta)+cos(B_{0})cos(\theta)cos(l)$, with $\theta$
and $l$ are heliographic latitude and heliographic longitude from the central
meridian of the CH respectively, whereas $B_{0}$ is the heliographic latitude
of the center of the solar disk at the time of observation) that takes into
account both the latitude and longitudinal projections. However, we got the
same results. This is obvious as the data set is not in the higher latitudes.

###  2.2 Computation of Average Radiative Flux of CH at $L_{1}$

For estimation of radiative flux $F$ of CH at the Lagrangian point $L_{1}$ in
the space, we use information from the SOHO/EIT instrumental response curve
(see the Figure 3, a postscript file $calib.ps$ is obtained from the website
http://umbra.nascom.nasa.gov/eit/eit_guide/). However, according to SOHO/EIT
website information, CH data in 195 $\AA$ is also sensitive to other three
wavelength (171 $\AA$, 284 $\AA$ and 304 $\AA$) bands whose contributions to
the instrumental responses are to be computed judiciously in the following
way. For this purpose, by integration of area under curve (Fig 3), one has to
estimate response values of $R_{1}$, $R_{2}$, $R_{3}$ and $R_{4}$ for all the
four (171 $\AA$, 195 $\AA$, 284 $\AA$ and 304 $\AA$) wavelength channels.
First we manually digitize all the four response curves (see Table 1) and by
using Trapezoidal rule method, integration of area under curve is computed.
Finally a grand average response R (=($R_{1}$+$R_{2}$+$R_{3}$+$R_{4}$)/4) is
computed. Results of average responses for different channels are presented in
Table 1.

Table 1: Digitized values of the instrumental response curves

$\lambda$ | R1 | R2 | R3 | R4
---|---|---|---|---
$\AA$ |  |  |  |
170 | 2e-15 | 1.0e-16 | 1.0e-16 | 1.0e-16
180 | 1.5e-12 | 3e-13 | 7e-16 | 5e-15
190 | 5e-14 | 6e-12 | 8e-16 | 4e-15
200 | 1.5e-14 | 8e-13 | 9e-16 | 4e-15
210 | 6e-15 | 1.5e-13 | 1e-15 | 5e-15
220 | 8e-15 | 7e-14 | 1.5e-15 | 6e-15
230 | 6e-15 | 5e-14 | 1.5e-15 | 7e-15
240 | 3e-15 | 3e-14 | 2.5e-15 | 9e-15
250 | 2.5e-15 | 2e-14 | 4e-15 | 1.1e-14
260 | 1.5e-15 | 1.5e-14 | 2e-14 | 1.5e-14
270 | 1.5e-15 | 9e-15 | 1.1e-13 | 3e-14
280 | 9e-16 | 7e-15 | 3e-13 | 7e-14
290 | 6e-16 | 4e-15 | 2e-14 | 1.5e-13
300 | 4e-16 | 2e-15 | 3e-16 | 4e-13
310 | 1.5e-16 | 9e-16 |  | 3.8e-13
320 |  | 4e-16 |  | 9e-14
330 |  | 2e-16 |  | 9e-15
340 |  |  |  | 1.2e-15
avg= | 4.43465e-15 | 1.75493e-14 | 1.40760e-15 | 2.60490e-15

††footnotetext: Note: Unit of instrumental responses ($R_{1}\,to\,R_{4}$) is
(DN $sec^{-1}$)/(photons $cm^{-2}$ $sec^{-1}$ $steradian^{-1}$ $\AA^{-1}$)

As the EIT instrumental response function $R$ is in the unit of DN
$sec^{-1}$/(photons $cm^{-2}$ $sec^{-1}$ $steradian^{-1}$ $\AA^{-1}$), one can
divide measured TDN (total number of DN counts of CH) by the instrumental
response function $R$ in order to get the radiative flux emitted by CH. Hence,
total radiative flux $F$ emitted by whole region of CH is

$\displaystyle
F=c_{2}\frac{TDN}{R}sin\theta\,photons\,cm^{-2}sec^{-1}Sr^{-1}\,,$ (3)

and its error $\delta$ F is

$\displaystyle\delta
F=c_{2}{\Big{[}}\left(\frac{TDN}{R}\right)cos\theta\delta\theta+sin\theta*\delta\left(\frac{TDN}{R}\right)\Big{]}\,,$

where the multiplicative constant factor $c_{2}=2.38\times 10^{-6}$ is
computed from average of all the four wavelengths (171$\AA$, 195$\AA$,
284$\AA$ and 304 $\AA$) in order to eliminate the term $\AA^{-1}$ in the
instrumental response $R$. Similarly equation of radiative flux F1 at the
Lagrangian point L1 (near earth) is

$\displaystyle F1=c_{2}\frac{TDN}{R}sin\theta Sr\,photons\,cm^{-2}sec^{-1},$
(4)

where

$Sr={A\over D^{2}}$ ($A$ is area of CH, Sr is steradian angle and $D$ is
distance between the sun and the orbit of SOHO satellite). Uncertainty in the
radiative flux $\delta F1$ of CH is computed as follows

$\displaystyle\delta F1=c_{2}{\Big{[}}\left(\frac{TDN}{R}\right)cos\theta
Sr\delta\theta+\left(\frac{TDN}{R}\right)sin\theta Sr\\_err$

$+\delta(\frac{TDN}{R})sin\theta
Sr{\Big{]}}\,photons\,cm^{-2}sec^{-1}Sr^{-1}\,,$ (5)

where Sr_err is error in steradian.

###  2.3 Computation of Average Radiative Energy emitted by CH at $L_{1}$

Total radiative energy $E$ emitted by CH is

$\displaystyle E=h\nu F_{1}\,ergscm^{-2}sec^{-1}\,,$ (6)

where $h\nu$ ($h$ is Planck’s constant and $\nu$ is frequency of radiation) is
a quanta of photon energy of the EUV radiation. Uncertainty $\delta E$ in the
energy is

$\displaystyle\delta E=h\nu\delta F1\,ergscm^{-2}sec^{-1}.$ (7)

###  2.4 Computation of Average Temperature structure of CH in the Corona

Spectroscopic methods yield temperature along the observed slit. Whereas here
we measure the average temperature of the observed whole coronal hole which we
call as “temperature structure”.

From the information of radiative energy ($I_{CHL}$) of CH at the Lagrangian
point $L_{1}$, following ratio yields the radiative energy ($I_{CHS}$) of CH
in the corona

$\frac{I_{CHS}}{I_{CHL}}=\frac{\int(Ecos\theta
d\theta)A/{R_{CH}}^{2}}{\int(Ecos\theta
d\theta)A/{R_{CHL}}^{2}}=\frac{{R_{CHL}}^{2}}{{R_{CH}}^{2}}\,,$ (8)

where $R_{CHL}$ is the distance between sun’s center and the Lagrangian point
$L_{1}$ and, $R_{CH}$ is distance between the centre of the sun and the height
at which CH is formed in the corona. By knowing the values of $R_{CH}$, as
observed CH in 195 $\AA$ is formed at the height of $\sim$ 1.1 $R_{\odot}$
(where $R_{\odot}$ is radius of the sun; see Fig 2 of Yang et.al. 2009), and
$R_{CHL}$, the ratio of RHS of equation (8) is estimated to be $\sim$
$3.14\times 10^{4}$. Hence, we get $I_{CHS}=3.14\times
10^{4}I_{CHL}=3.14\times 10^{4}(\frac{TDN}{R}sin\theta)$. Assuming that plasma
of CH is in thermodynamic equilibrium, total energy radiated by CH is equated
with the Planck’s law and average temperature structure $T$ and its
uncertainty $\delta T$ of CH are computed as follows

$\displaystyle T=\frac{hc}{\lambda
kln(\frac{2hc^{2}}{\lambda^{5}I_{CHS}}+1)}\,K,$ (9)

$\displaystyle\delta T=\left(\frac{\lambda
k}{hc\;ln(\frac{2hc^{2}}{\lambda^{5}I_{CHS}}+1)}\right)\frac{2hc^{2}(\delta
I_{CHS})T^{2}}{\lambda^{5}I_{CHS}^{2}}\,K$ (10)

where $c$ is velocity of light, $\lambda$ is wavelength and $k$ is Boltzmann
constant respectively and, $I_{CHS}$ is radiative energy of CH at the corona.

Table 2: Daily variation of different physical parameters of Coronal Holes

01-05 Jan 2001
---
Area |  $\delta$Area | FS1 |  $\delta$FS2 | FAU3 |  $\delta$FAU4 | ES5 |  $\delta$ES6 | EAU7 |  $\delta$EAU8 | T9 |  $\delta$T10
E+20 | E+20 | E+13 | E+13 | E+8 | E+8 | E+3 | E+3 | E-2 | E-2 | E+6 | E+6
5.124 | 0.680 | 5.637 | 0.445 | 1.319 | 0.115 | 5.694 | 0.470 | 1.332 | 0.118 | 1.075 | 0.697
5.286 | 0.391 | 5.868 | 0.515 | 1.416 | 0.084 | 5.927 | 0.312 | 1.431 | 0.112 | 1.057 | 0.605
5.058 | 0.199 | 5.354 | 0.514 | 1.237 | 0.152 | 5.407 | 0.341 | 1.249 | 157 | 1.045 | 0.614
4.788 | 0.122 | 4.571 | 0.215 | 0.999 | 0.047 | 4.617 | 0.312 | 1.009 | 0.107 | 1.034 | 0.601
4.732 | 0.101 | 5.658 | 0.238 | 1.223 | 0.561 | 5.715 | 0.390 | 1.235 | 0.523 | 1.048 | 0.615
01-06 Jan 2001
Area |  $\delta$Area | FS |  $\delta$FS | FAU |  $\delta$FAU | ES |  $\delta$ES | EAU |  $\delta$EAU | T |  $\delta$T
E+20 | E+20 | E+12 | E+12 | E+7 | E+7 | E+2 | E+2 | E-3 | E-3 | E+6 | E+6
2.790 | 0.106 | 1.537 | 0.403 | 1.958 | 0.521 | 1.651 | 0.155 | 1.978 | 0.526 | 1.191 | 0.167
3.423 | 0.060 | 2.273 | 0.596 | 3.553 | 0.936 | 2.967 | 0.230 | 3.588 | 0.945 | 1.162 | 0.222
3.145 | 0.025 | 2.242 | 0.532 | 3.220 | 0.766 | 2.430 | 0.227 | 3.252 | 0.774 | 1.164 | 0.325
2.754 | 0.079 | 2.016 | 0.411 | 2.535 | 0.524 | 1.661 | 0.204 | 2.561 | 0.053 | 1.153 | 0.342
1.642 | 0.092 | 2.018 | 0.515 | 0.861 | 0.013 | 4.058 | 0.116 | 0.870 | 0.019 | 0.960 | 0.412
1.552 | 0.177 | 2.283 | 0.514 | 0.808 | 0.012 | 2.406 | 0.115 | 0.817 | 0.077 | 0.955 | 0.356

11footnotetext: Average radiative flux (photons cm${}^{-2}sec^{-1}Sr^{-1}$) of
CH measured on the Sun

22footnotetext: Uncertainty in radiative flux of CH measured on the Sun

33footnotetext: Average radiative flux (photons cm${}^{-2}sec^{-1}$) of CH
estimated at 1AU

44footnotetext: Uncertainty in radiative flux of CH measured at 1AU

55footnotetext: Average radiative energy (ergs $cm^{-2}sec^{-1}Sr^{-1}$) of CH
measured on the Sun

66footnotetext: Uncertainty in radiative energy of CH measured on the Sun

77footnotetext: Average radiative energy (ergs $cm^{-2}sec^{-1}$) of CH
estimated at 1AU

88footnotetext: Uncertainty in radiative energy of CH measured at 1AU

99footnotetext: Average temperature (K) of CH measured on Sun

1010footnotetext: Uncertainty in temperature of CH measured on Sun

Fig 1(a) Fig 1(b)

Figure 1: Left side Fig 1 (a) shows full-disk SOHO/EIT 195 $\AA$ image of
01-01-2001, 00:24:11 UT with CH (in the north eastern hemisphere and close to
center). This full disk image is kindly provided by Dr. Gurman. Whereas, Fig
1(b) illustrates with a given threshold, contour map of the CH.

Fig 2(a) Fig 2(b)

Figure 2: Figure 2(a) illustrates, number of coronal holes for different life
spans on the solar disk. Whereas, for different latitude bins in both the
hemispheres, Fig 2(b) illustrates, number of coronal holes.
---
Figure 3: For different observed channels (171$\AA$, 195$\AA$, 284$\AA$ and
304$\AA$), this figure illustrates SOHO/EIT instrumental response curves.

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Figure 4: Irrespective of their latitudes, between +65∘ to -65∘ longitudes
from the central meridian, variation of of different physical parameters of
coronal holes such as Area, Flux (on the sun and near Earth at 1 AU) and
Temperature during their evolution on the solar visible disk. In all the
figures, blue bar plot represents the observed values of Area, Flux,
Temperature respectively and the red continuous line represents a least-square
fit $Y(t)=C_{0}+C_{1}t+C_{2}t^{2}$ to the observed values. $Y(t)$ is the
estimated different physical parameters, $t$ is day of observation, and
$C_{0}$, $C_{1}$ and $C_{2}$ are the constant coefficients determined from the
least square fit. Whereas the red dashed lines represent the one standard
deviation (which is computed from all the data points) error bands. $\chi^{2}$
is a measure of goodness of fit.

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Figure 5: Irrespective of their latitudes, between +45∘ to -45∘ longitudes from the central meridian, daily variation of different physical parameters of coronal holes such as Area, Flux (on the sun and near Earth at 1 AU) and Temperature during their evolution on the solar visible disk. In all the figures, blue bar plot represents the observed values of Area, Flux, Temperature respectively and the red continuous line represents a least-square fit of the forms $Y(t)=C_{0}+C_{1}t+C_{2}t^{2}$ to the observed values. $Y(t)$ is the estimated different physical parameters, $t$ is day of observation, and $C_{0}$, $C_{1}$ and $C_{2}$ are the constant coefficients determined from the least square fit. Whereas the red dashed lines represent the one standard deviation (which is computed from all the data points) error bands. $\chi^{2}$ is a measure of goodness of fit.  Table 3: Estimated CH parameters for +65∘ to -65∘ Parameters | C0 |  $\delta$C0 | C1 |  $\delta$C1 | C2 |  $\delta$ C2 | $\chi^{2}$
---|---|---|---|---|---|---|---
A$\times 10^{20}$ | 3.57 | 0.83 | 0.15 | 0.30 | -0.02 | 0.02 | 0.040
F$\times 10^{11}$ | 6.06 | 0.30 | 2.97 | 0.70 | -0.20 | 0.06 | 0.054
F1$\times 10^{7}$ | 3.70 | 0.90 | 0.23 | 0.10 | -0.03 | 0.01 | 1.721
T$\times 10^{6}$ | 0.91 | 0.30 | 0.02 | 0.04 | -0.001 | 0.004 | 0.017
Table 4: Estimated CH parameters for +45∘ to -45∘ Parameters | C0 |  $\delta$C0 | C1 |  $\delta$C1 | C2 |  $\delta$ C2 | $\chi^{2}$
---|---|---|---|---|---|---|---
A$\times 10^{20}$ | 3.70 | 0.98 | 0.80 | 0.70 | -0.02 | 0.01 | 0.152
F$\times 10^{11}$ | 5.40 | 1.20 | 4.30 | 0.90 | -0.50 | 0.09 | 0.103
F1$\times 10^{7}$ | 1.80 | 1.80 | 1.30 | 0.90 | -0.14 | 0.09 | 0.052
T$\times 10^{6}$ | 0.90 | 0.30 | 0.50 | 0.10 | -0.05 | 0.01 | 0.003
Table 5: Estimated CH parameters for +65∘ to -65∘ Parameters | C0 |  $\delta$C0 | C1 |  $\delta$C1 | $\chi^{2}$
---|---|---|---|---|---
A$\times 10^{20}$ | 3.80 | 0.50 | -0.50 | 0.10 | 1.409
F$\times 10^{12}$ | 1.30 | 0.08 | 0.50 | 0.20 | 6.986
F1$\times 10^{7}$ | 4.40 | 0.30 | 0.70 | 0.50 | 0.405
CT$\times 10^{6}$ | 0.93 | 0.40 | 0.17 | 0.14 | 0.041
CP$\times 10^{-2}$ | 0.97 | 0.20 | 0.17 | 0.15 | 0.041
Photospheric $|B|$ | 4.27 | 0.22 | 3.67 | 0.90 | 4.474
Corona $|B|$ | 0.075 | 0.020 | 0.09 | 0.04 | 0.610
MP$\times 10^{-3}$ | 0.18 | 0.04 | 0.77 | 0.15 | 2.478
TP$\times 10^{-2}$ | 0.97 | 0.10 | 0.03 | 0.10 | 0.041
True temperature$\times 10^{6}$ | 0.94 | 0.10 | 0.03 | 0.01 | 0.041
Table 6: Estimated CH parameters for +45∘ to -45∘ Parameters | C0 |  $\delta$C0 | C1 |  $\delta$C1 | $\chi^{2}$
---|---|---|---|---|---
A$\times 10^{20}$ | 3.80 | 0.50 | +0.50 | 0.10 | 1.409
F$\times 10^{12}$ | 3.00 | 0.20 | 2.60 | 0.20 | 3.744
F1$\times 10^{7}$ | 5.20 | 0.10 | 5.10 | 1.00 | 1.559
CT$\times 10^{6}$ | 0.90 | 0.10 | 0.40 | 0.01 | 0.857
CP$\times 10^{-2}$ | 0.93 | 0.10 | 0.50 | 0.10 | 0.857
Photospheric $|B|$ | 4.21 | 0.22 | 1.32 | 0.90 | 4.162
Corona $|B|$ | 0.08 | 0.01 | 0.03 | 0.01 | 4.162
MP$\times 10^{-3}$ | 2.40 | 0.10 | 3.40 | 0.20 | 0.032
TP$\times 10^{-2}$ | 0.01 | 0.001 | 0.0002 | 0.0001 | 2.481
True temperature$\times 10^{6}$ | 1.01 | 0.01 | 0.02 | 0.03 | 2.481

##  3 RESULTS

As in the previous study (Hiremath & Hegde 2013), we follow the similar
criteria in selection of coronal holes and adopt similar method in binning the
CH data for different latitudes. A typical coronal hole in the north-eastern
quadrant of the SOHO/EIT image observed on 1st Jan 2001 is presented in Fig
1(a). Whereas separated coronal hole with its boundary detected from the
threshold criterion (see the details in section 2 of Hiremath & Hegde 2013) is
presented in Fig 1(b). We define apparent life span $\tau$ (as actual life
span is larger) as number of observed days that have first and last appearance
of CH on the same side of solar disk. In this way, for the observed coronal
holes with different life spans (minimum of 4 days to maximum of 10 days) that
appear at different latitudes and between +65∘ to -65∘ longitude from the
central meridian are illustrated in Fig 2(a). With the constraint that coronal
holes that occur between +40∘ (northern hemisphere) to -40∘ (southern
hemisphere) latitude zones, 113 coronal holes with different life spans
ultimately yield totally 796 data points for the present analysis. As the
minimum life span of coronal hole is 4 days, transient coronal holes (that
have life span $\leq$ 2 days; Kahler and Hudson 2001) that might have
different physical properties are not included in this analysis. Irrespective
of their life spans and after binning the coronal holes between 0-5∘, 5-10∘,
etc., average latitudes of CH are computed. Fig 2(b) illustrates the idea of
how totally 796 observed data points are distributed in different latitude
bins.

For the typical two coronal holes (first CH is observed in the southern
(latitude $\sim$ $30^{o}$) hemisphere and second CH is observed in the
northern (latitude $\sim$ $13^{o}$) hemisphere respectively), following the
methods presented in section 2, daily estimated different physical parameters
are presented in Table 2. For the sake of comparison of temperature structure
of CH computed from our method, by employing filter ratio technique, we also
computed temperature structure. For example, Hinode/XRT data of 12/02/2007 and
15/08/2007 taken in Al-mesh/Al-poly filters are considered for temperature
measurement. We followed the filter ratio method of Narukage et al. (2011) and
temperature structure of these two CH is estimated. We find that temperature
structure of CH estimated by our method and temperature structure of CH
estimated by method of Narukage et al. (2011) is of the same order. With this
confidence in mind that both the methods yield the same temperature
structures, for all the 113 coronal holes considered for this study, we
compute all the physical parameters (as defined in section 2) and are
illustrated in Figs. 4-11.

Our first objective is to understand daily evolution of coronal hole during
its passage over the observed solar disk. For example, irrespective of their
life spans and latitudes, for longitudes between +65∘ to -65∘, that are
combined together in both the latitudes, Fig 4. illustrates the daily
evolution of area (A), radiative flux (F) on the sun, radiative flux near the
Earth (to be specific at Lagrangian point $L_{1}$ where SOHO space craft is
positioned) ($F_{1}$) and temperature structure (T) of CH on the sun.
Polynomial of degree 2 (quadratic) yields the best fit for the daily variation
of CH. Different estimated parameters ($C_{0}$, $C_{1}$ and $C_{2}$) of the
polynomial fit with their respective uncertainties ($\delta C_{0}$, $\delta
C_{1}$ and $\delta C_{2}$) and, $\chi^{2}$ (a measure of goodness of fit)
values are presented in Table 3 (+65 to -65 deg longitudes) and in Table 4
(+45 to -45 deg longitudes) respectively.

It is to be noted that , daily variations of radiative flux of CH may be
useful for the Earth’s ionospheric studies (Bauer 1973; Hinteregger 1976;
Roble and Schmidtke 1979; Richards, Fennelly, Torr 1994; Lilensten et.al.
2007; Dudok de Wita and Watermanna 2010; Kretzschmar et.al. 2009). According
to Bauer (1973), one of the principal ionizing radiation (others are solar
x-ray photons, galactic cosmic rays, solar cosmic ray protons, etc.)
responsible for formation of planetary ionospheres is solar EUV. Although all
the EUV photons from sun’s whole disk can ionize the planetary atmospheres,
EUV photons from coronal holes have more momentum that probably disturb the
ambient planetary ionospheres. That means, probably transient behavior of EUV
photons of the coronal holes leads to transient or sudden disturbances in the
planetary ionospheres.

From observations of EUV images one can argue that much brighter surrounding
coronal hole must be responsible for ionizing the Earth’s upper atmosphere
rather than a dark coronal hole. However, there is a difference in these two
images. Compared to surrounding brighter region of coronal holes, radiation
emitted from the coronal hole is mainly associated with fast solar wind,
accompanied by increase in temperature, is more effective in ionizing the
Earth’s atmosphere and also leading to geomagnetic disturbances.

###  3.1 Estimation Life Span of CH

As for solar physics point of view, it is interesting to estimate the actual
life span of coronal holes. If we consider daily variation of area of CH on
the solar disk, best fit yields the quadratic equation with respect to time.
As a first approximation, we can crudely estimate average life span of CH
although individual coronal holes may have different life spans. After
neglecting the non-linear term (as it is very small, see Figures 4 and 5 and,
Table 3 and 4) and assuming that daily variation of area curve is symmetric
(with respect to maximum area), for area of CH to be zero when it decays
completely, average life span is estimated to be $\sim$ 46 days.

###  3.2 Estimation of Magnetic Diffusivity of Corona

From this daily evolution area curve, we find the average area $\sim$
$10^{20}$ $cm^{2}$ from which average diameter (assuming that area $A$ is
circular) $L$ of CH is estimated to be $\sim$ $6\times 10^{9}$ cm. Further
making assumption that CH is a magnetic flux tube whose area evolution is only
dictated by magnetic diffusion, magnetic diffusivity (=$L^{2}\over{\tau}$,
where $\tau$ is life span) of the corona is estimated to be $\sim\,10^{13}$
$cm^{2}$/sec which is very close to estimates of previous studies (Krista
2011; Krista et.al. 2011; Hiremath and Hegde 2013). Similarly neglecting the
second non-linear coefficient (which is not significant as the error is of
same order) in the empirical law of area that is derived from the least square
fit (Fig. 4), one can estimate the area derivative $dA\over{dt}$ and by
employing the derived area derivative law
($dA\over{dt}$=$\eta{d^{2}A\over{dr^{2}}}$) from the previous study (Hiremath
and Hegde 2013; see the section 4), magnetic diffusivity $\eta$ of the corona
is estimated to be of similar order ($10^{13}$ $cm^{2}$/sec), as estimated
above. Interestingly and it is to be noted that magnetic diffusivity estimated
by our both the methods and magnetic diffusivity estimated by the previous
studies (Krista 2011; Krista et.al. 2011) are of the same order.

###  3.3 Thermal and Magnetic Field Structures of the Coronal Holes

Knowing thermal structure is very important in understanding the fast solar
wind emanating from the coronal hole (Hegde et al. 2015 and references there
in). It is also important to examine whether thermal structure of coronal is
independent or dependent on the latitude which in turn may give clue as to why
high latitude coronal holes have high solar wind velocity compared to the low
latitude or equatorial coronal holes (McComas, Elliott and Steiger 2002)
during the minimum period. It is also interesting to know whether from thermal
pressure of coronal hole (if CH is a magnetic flux tube, the CH pressure is
combination of plasma and magnetic pressure) one can separate the magnitude of
average magnetic field structure of coronal holes at the corona. These
important observed physical parameters such as thermal and magnetic structures
are essential to probe the depth (Hiremath and Hegde 2013) and structure of
the coronal holes. In the following study, first by estimating the radiative
flux, with the assumption of thermodynamic equilibrium, we estimate the
temperature structure. This temperature structure we call as total
temperature. This assumption of thermodynamic equilibrium is almost akin to
the same assumption employed in differential emission measure (for example
Hahn, Landi and Savin 2011 ) in estimating temperature structure of coronal
holes. Further by using the observed density of CH, we compute the total
pressure. The obvious reason for calling total pressure is that CH is a
magnetic flux tube whose total pressure is the combination of thermal and
magnetic pressures. This concept basically is invoked from the Parker’s (1955)
idea wherein if magnetic flux tube is in hydrostatic equilibrium, a
combination of gas and magnetic pressure in the flux tube is balanced by the
external gas thermal pressure. In the following section, by using information
of estimated total temperature and pressure, we not only derive the actual
temperature structure but also estimate the average magnetic field structure
of the coronal hole.

####  3.3.1 Latitudinal Variation of Thermal Structure of Coronal Hole

Many previous studies (Zhang et al. 2007; Landi 2008 etc.) concentrated on
single coronal hole and estimated average temperature structure. As for our
knowledge, this is a first study which uses many coronal holes for estimating
the thermal structure. Following the previous study (Hiremath and Hegde 2013)
in selecting the data, first we estimate the different physical parameters of
CH that occur between +65∘ to -65∘ longitudes from the central meridian.
Following the method described in section 2 and irrespective of their areas
and number of observed days on the solar disk, for different latitudes,
variation of different physical parameters such as area, radiative flux (on
the sun and near Earth) and total temperature structure are presented in Fig
6. For example, one can notice from the least-square fit (of the form
$Y(\theta)=C_{0}+C_{1}sin^{2}\theta$, where $\theta$ is latitude, $C_{0}$ and
$C_{1}$ constants and, Y represents different physical parameters) of area-
latitude curve that, on average, equatorial coronal holes have large area
compared to high latitude coronal holes. Whereas other illustrations of
radiative flux and temperature structures follow inverse latitudinal
variations, rather than following similar area-latitude relationship.
Inconsistency of both these results probably could be due to contribution of
coronal holes that are near +65∘ to -65∘ longitudes, close to the limb. Hence,
in order to completely minimize such projectional effects, further we
restricted the data of CH that occur between +45∘ to -45∘ longitudes from the
central meridian and the same results that are presented in Fig 6 are also
presented in Fig 9. It is to be noted that although there is more variation,
especially in case of Figures 9(c) and 9(d), over all trend of least square
fits for all the Figures is same (of the form $c_{1}+c_{2}sin^{2}\theta$).
Whereas, in case of illustrations 6, except Figures 6(b)-6(d), law of least
square fit for Figure 6(a) is different (of the form
$c_{1}-c_{2}sin^{2}\theta$). However, one can notice from Fig 9 that
inconsistency in latitudinal variation of area and flux curves is removed and
all the illustrations have same latitudinal variations.

Latitudinal variation of different physical parameters of CH are subjected to
least square fit (of the the form $Y(\theta)=C_{0}+C_{1}sin^{2}\theta$) and
estimated coefficients $C_{0}$ and $C_{1}$ with their respective uncertainties
and $\chi^{2}$ values are presented in Tables 5 and 6 respectively. In both
the tables, with the units in cgs, first column consists of area (A),
radiative flux (F) on the sun, flux (F1) near earth, total temperature (CT),
total pressure (CP), strength of coronal hole magnetic field structure at the
photosphere ($|B|$) and at the corona ($|B_{c}|$) and, magnetic pressure (MP)
of the coronal hole at corona, actual pressure (TP) and temperature of the
coronal hole at the corona. Whereas second to fifth columns represent the
coefficients and their uncertainties as estimated by the least square fit. The
last column in both the tables represents the value of $\chi^{2}$ (a measure
of goodness of fit).

Although variation of thermal structure (for example observed inferences:
David et.al. 1998; Landi 2008; Landi and Cranmer 2009; Hahn, Landi and Savin
2011 and, theoretical inferences: Osherovish et.al. 1985) of the coronal hole
at different heights in the corona is available, studies on latitudinal
variation of thermal structure of coronal holes are not available. In this
respect, we believe that, our results on latitudinal variation of thermal
structure of coronal holes will be very useful to the solar community.

####  3.3.2 Estimation of Strength of Magnetic Field Structure of the Coronal
Holes

From the latitudinal variation of total temperature and the observed density
structure (Doschek et al. 1997 etc.) of coronal holes, with respect to
latitude, total pressure $P$ ($=2n_{e}kT$, where $n_{e}$ is number of electron
density, $k$ is Boltzmann constant and $T$ is estimated total temperature; it
is assumed that coronal hole plasma has same number of electrons and protons
density hence the number $2$ is multiplied) is estimated and is presented in
Fig 7(a) (for the +65∘ to -65∘ longitudes). In order to minimize the
projectional effects, same parameter (for the +45∘ to -45∘ longitudes from the
central meridian) is presented in Fig 10(a). One can notice from both the
figures that latitudinal variation of total pressure of the coronal hole
depends upon the latitude such that equatorial coronal holes have low pressure
compared to high latitude coronal holes. In case one accepts that coronal hole
is a magnetic flux tube, then total pressure in the coronal hole is sum of
plasma and magnetic pressures. Offcourse, best analogy (except strong magnetic
fields in sunspots) between magnetic flux tubes (sunspots) and coronal hole
can be obtained from the previous (Fla et.al. 1984; Davila 1985; Cally 1986,
1987; Osherovich et.al., 1985; Ofman 2005 and references there in; Obridko and
Solovev, 2011) MHD models.

Hence, if we accept that plasma pressure of coronal hole is independent of
latitudes, then one possible interpretation for latitudinal variation of total
pressure of the coronal holes could be due to contribution from the magnetic
pressure. That means if one knows the latitudinal variation of magnetic field
structure of coronal holes, one can compute the magnetic pressure and can be
subtracted from the estimated total pressure. Unfortunately, to the knowledge
of the authors, there are no such studies that give the information of
latitudinal variation of magnetic field structure of coronal holes at the
corona. For this purpose, we adopt the following method in computing the
magnetic field structure and hence magnetic pressure of the coronal hole at
the corona.

First we make the reasonable assumption that coronal hole is a magnetic flux
tube that probably anchored below the photosphere (Gilman 1977; Golub et.al
1981; Jones 2005; Hiremath and Hegde 2013). As the magnetic flux tube is
embedded in the solar atmosphere, increase with height from photosphere to
corona results in decrease of surrounding ambient plasma pressure and hence
tube must expand. In the following, this statement can further be corroborated
from the previous study (Hegde, Hiremath and Doddamani 2014). With the SDO
data, for the coronal hole observed in three wavelengths 174 $\AA$, 193 $\AA$
and 211 $\AA$, average area of coronal hole are : $0.5\times 10^{20}$ $cm^{2}$
for 174 $\AA$, $0.98\times 10^{20}$ $cm^{2}$ for 193 $\AA$ and, $1.06\times
10^{20}$ $cm^{2}$ for 211 $\AA$ respectively. Simultaneously DN counts
(radiative intensity) also reduce from 174 $\AA$, 193 $\AA$ and 211 $\AA$
respectively. Successive increase of line formation (Yang et.al. 2009) for 174
$\AA$, 193 $\AA$ and 211 $\AA$ at different heights are: 1.01$R_{\odot}$,
1.05$R_{\odot}$ and 1.3$R_{\odot}$. One can notice that within the 30% of
solar radius from the photosphere, coronal hole’s area increases twice the
area of coronal hole at the photosphere. Considering these facts it is
reasonable to consider the coronal hole is expanding from photosphere to
corona where 195 $\AA$ line originates.

Further if one accepts that coronal hole is a Parker’s flux tube (Parker
1955), then magnitude of magnetic field structure $B$ inside the coronal hole
is directly proportional to ${P_{e}}^{1/2}$ (where ${P_{e}}$ is external
ambient pressure of the plasma). Hence, with this simple relationship, one can
estimate strength of magnetic field structure $B_{c}$ of the coronal hole at
the corona if one knows the strength of magnetic field structure and ambient
plasma pressures at different heights. To be specific, the resulting
derivation will be (that is obtained from the above simple relationship)
Bc=Bpho ($P_{ce}\over P_{pho}$)1/2 (where $B_{c}$ and $P_{ce}$ are strength of
magnetic field structure of the coronal hole and ambient plasma pressure at
the corona and, Ppho and Bpho are strength of magnetic field structure of the
coronal hole and ambient plasma pressure at the photosphere). As for strength
of magnetic field structure $B_{pho}$ of coronal hole at the photosphere, for
the same time of observation and latitude, we consider the inferred values at
the photosphere (using Solar Monitor website). This inferred field from the
photospheric magnetograms is the line of sight component. Then with the above
formula and, by using the ambient external pressure at the photosphere and the
corona (Aschwanden 2004), magnetic field structure $B_{c}$ (and hence magnetic
pressure $B_{c}^{2}\over{4\pi}$) of the coronal hole at the corona (around 1.1
$R_{\odot}$, where the 195 $\AA$ line is originated) is estimated.

After binning in different latitude zones, latitudinal variation of average
strength of magnetic field structure of coronal hole at the photosphere are
presented in Fig 7b (for +65∘ to -65∘ longitudes) and in Fig 10b (for +45∘ to
-45∘ longitudes). It is interesting to note that, as we reasoned above, indeed
magnitude of magnetic field structure (and magnetic pressure) of the coronal
hole at the photosphere increases from equator to higher latitudes although
curve of Fig 10(b) appears to be flatter. Estimated strength of magnetic field
structure of the coronal hole and its magnetic pressure at the corona are
presented in Figures 7(c) and 7(d) (for +65∘ to -65∘ longitudes) and in
Figures 10(c) and 10 (d) (for +45∘ to -45∘ longitudes). One can notice that,
on average, magnitude of magnetic field structure of coronal hole at the
corona is estimated to be $\sim$ 0.08($\pm$0.02) Gauss.

Offcourse, one can argue from the spatially resolved individual coronal holes
that this estimated field strength depends on whether the coronal holes are in
new active regions where the fields are strong or in old expanded unipolar
regions where the fields are weak. However, our estimated strength of magnetic
field structure is derived from the least square fit (first coefficient of the
linear fit) with many coronal holes rather than obtaining the spatial
information.

There are also other possibilities that this estimated strength of magnetic
field is possibly different than the actual strength of magnetic field
structure for the following two reasons. Firstly observed magnetic field
structure of the coronal hole estimated from the photospheric magnetogram is
longitudinal and hence inferred magnetic field of the coronal hole at corona
is also a longitudinal component. Whereas for the radial component of magnetic
field (such a field structure is invoked in modeling of coronal hole),
strength of longitudinal component of magnetic field of coronal hole appears
to be under estimated as radial field is $B_{l}/cos(\lambda-\phi)$ (where
$B_{l}$ is longitudinal component of magnetic field, $\lambda$ is latitude
(that varies 0 to 90 deg from equator to pole) and $\phi$ is inclination angle
of rotational axis of the sun). Secondly and according to Parker’s (1955) Flux
tube model, estimated strength of magnetic field depends upon square root of
ambient pressure which ultimately is model dependent. Hence, it can not be
ruled out that some amount of uncertainty (0.01, $\sim$ $13\%$) persists that
is reflected in the estimated strength of longitudinal component of magnetic
field from the least square fit. However, it is important to be noted that we
haven’t come across any study that estimates strength of magnetic field of the
coronal hole observed in 195 $\AA$.

By knowing latitudinal variation of strength of magnetic field structure $B$
of the coronal hole, the estimated magnetic pressure is subtracted from the
total pressure of the coronal hole and actual thermal pressure of the coronal
hole is computed. By knowing electron density and thermal pressure (Figures
8(a) and 11(a)), actual temperature structure of corona hole is computed and
latitudinal variation of the same is presented in Fig 8(b) (for +65∘ to -65∘
longitudes) and in Fig 11(b) (for +45∘ to -45∘ longitudes) respectively. From
these figures, we find that variation of temperature structure of coronal
holes is independent of solar latitudes.

6(a) 6(b)

|
---|---

6(c) 6(d)

Figure 6: For different latitudes between +65∘ to -65∘ longitudes from the
central meridian, variation of different physical parameters (blue bar plot)
such as area, radiative flux emitted by CH on the sun and at Earth and,
apparent temperature structure of CH respectively. Red continuous line
represents a least-square fit of the form $Y(\theta)=C_{0}+C_{1}sin^{2}\theta$
to different observed parameters (where $\theta$ is the latitude, $C_{0}$ and
$C_{1}$ are constant coefficients determined from the least square fit).
Whereas the red dashed lines represent the one standard deviation (which is
computed from all the data points) error bands. $\chi^{2}$ is a measure of
goodness of fit.

7(a) 7(b)

|
---|---

7(c) 7(d)

Figure 7: For different latitudes, between +65∘ to -65∘ longitudes from the
central meridian, variation of different physical parameters (blue bar plot)
such as total pressure, magnitude of magnetic field structure at the
photosphere and at the corona and, magnetic pressure of CH respectively. Red
continuous line represents a least-square fit of the form
$Y(\theta)=C_{0}+C_{1}sin^{2}\theta$ to different observed parameters (where
$\theta$ is the latitude, $C_{0}$ and $C_{1}$ are constant coefficients
determined from the least square fit). Whereas the red dashed lines represent
the one standard deviation (which is computed from all the data points) error
bands. $\chi^{2}$ is a measure of goodness of fit.

8(a) 8(b)

Figure 8: For different latitudes, between +65∘ to -65∘ longitudes from the
central meridian, variation of thermal pressure and temperature structure of
CH. Red continuous line represents a least-square fit of the form
$Y(\theta)=C_{0}+C_{1}sin^{2}\theta$ to different observed parameters (where
$\theta$ is the latitude, $C_{0}$ and $C_{1}$ are constant coefficients
determined from the least square fit). Whereas the red dashed lines represent
the one standard deviation (which is computed from all the data points) error
bands. $\chi^{2}$ is a measure of goodness of fit.

9(a) 9(b)

|
---|---

9(c) 9(d)

Figure 9:  For different latitudes, between +45∘ to -45∘ longitudes from the
central meridian, variation of different physical parameters (blue bar plot)
such as area, radiative flux emitted by CH on the sun and near the Earth of CH
respectively. Red continuous line represents a least-square fit of the form
$Y(\theta)=C_{0}+C_{1}sin^{2}\theta$ to different observed parameters (where
$\theta$ is the latitude, $C_{0}$ and $C_{1}$ are constant coefficients
determined from the least square fit). Whereas the red dashed lines represent
the one standard deviation (which is computed from all the data points) error
bands. $\chi^{2}$ is a measure of goodness of fit.

10(a) 10(b)

|
---|---

10(c) 10(d)

Figure 10:  For different latitudes, between +45∘ to -45∘ longitudes from the
central meridian, variation of different physical parameters (blue bar plot)
such as total pressure, magnitude of magnetic field structure at the
photosphere and at the corona and, magnetic pressure of CH respectively. Red
continuous line represents a least-square fit of the form
$Y(\theta)=C_{0}+C_{1}sin^{2}\theta$ to different observed parameters (where
$\theta$ is the latitude, $C_{0}$ and $C_{1}$ are constant coefficients
determined from the least square fit). Whereas the red dashed lines represent
the one standard deviation (which is computed from all the data points) error
bands. $\chi^{2}$ is a measure of goodness of fit.

11(a) 11(b)

Figure 11:  For different latitudes, between +45∘ to -45∘ longitudes from the
central meridian, variation of thermal pressure and temperature structure of
CH. Red continuous line represents a least-square fit of the form
$Y(\theta)=C_{0}+C_{1}sin^{2}\theta$ to different observed parameters (where
$\theta$ is the latitude, $C_{0}$ and $C_{1}$ are constant coefficients
determined from the least square fit). Whereas the red dashed lines represent
the one standard deviation (which is computed from all the data points) error
bands. $\chi^{2}$ is a measure of goodness of fit.

##  4 Discussion and conclusions

There are two very interesting results that need worthy to be discussed here.
First result is the magnitude of magnetic field structure of coronal holes
that increases from equator to both poles of the sun. One can argue from the
previous studies (Harvey et al 1982; Webb and Davis 1985; Abramenko et.al
2009) that this result could be due to occurrence of coronal holes during the
evolution of solar cycle as the polar coronal holes have stronger magnetic
fields than the coronal holes at low latitudes. Although inferred result of
latitudinal variation of strength of magnetic field structure of coronal holes
from the present study is consistent with the results of previous studies,
question remains why high latitude coronal holes occur with high average
magnetic field strengths than the low latitude coronal holes.

This important observed and inferred information of latitudinal variation of
strength of magnetic field probably suggests coronal holes’ origin that is not
consistently understood. As the coronal holes are unipolar magnetic field
structures, their origin can be understood from the global nature of magnetic
field structure of the sun. Probably, we believe, following conjecture on
genesis of coronal holes is consistent with the result of latitudinal
dependency of magnitude of magnetic field structure of the coronal holes.

During the period of minimum solar activity, white light pictures taken during
total solar eclipse, one can notice the dipole like magnetic field structure
delineated along the intensity patterns (rays) that are originated from the
two poles. Infact observational (Stenflo 1993) and theoretical studies
(Hiremath and Gokhale 1995 and, references there in) can not rule out such a
large-scale global dipole like magnetic field structure, may be of primordial
origin. Offcourse we have to make a clear distinction between the “steady” and
“time dependent” parts of solar magnetic field structure. “Steady” part of
solar magnetic field structure has a time scale of billion of years (Hiremath
and Gokhale 1995 and reference therein). Whereas time dependent part of solar
magnetic field structure has a 22 yrs time scale. For understanding genesis of
coronal hole, in the following conjecture, we invoke the large-scale “steady”
part of the magnetic field structure in order to be compatible with the
inferred latitudinal variation of magnetic field structure of the coronal
hole.

Observational (Stenflo 1993) estimates yield the intensity to be $\sim$ 1 G.
Whereas, for matching of 22 year of magnetic periodicity, previous study
(Hiremath and Gokhale 1995 and, references there in) estimates sun’s global
dipole like magnetic field structure to be $\sim$ 0.01 G. This result is also
consistent with the recently estimated average magnetic field structure of the
$\sim$ 0.028 G (Kotov 2015). Although genesis of coronal holes is debatable,
for the consistency of inferred coronal hole magnetic field structure whose
intensity increases from equator to pole, we present probable mechanism of CH
formation as follows.

If the large-scale coronal poloidal magnetic field structure is perturbed,
Alfven waves are produced whose interference pattern leads to formation of
coronal hole structure. If $B_{p}$ is intensity of large-scale steady part of
magnetic field structure, then resulting amplitude $\delta B_{p}$ of Alfven
waves is same order as that of intensity of original magnetic field structure.
That is $\delta B_{p}$ is $\sim$ $B_{p}$. Although steady part of poloidal
part of magnetic field structure probably consists of combined (uniform,
dipole and quadrupole) filed structure (Hiremath and Gokhale 1995), for the
sake of simplicity, let us consider dipole like field structure only. For a
particular radius, intensity or magnitude of magnetic field structure varies
as $sin^{2}\lambda$ (where $\lambda$ is observed latitude; here $\lambda$=0∘
is equator and $\lambda$=90∘ is pole) whose magnitude increases from equator
to pole. That means, as presented in Figures (7(c) and 10(c)), coronal holes
originated at the equator must have less magnitude of magnetic field structure
compared to the coronal holes that originate near the poles. If one accepts a
naive concept that coronal holes are formed due to Alfven wave perturbations
of poloidal component of magnetic field structure, as the amplitude of
magnetic field structure of Alfven waves is at least same order as that of
steady part of magnetic field structure, hence it is not surprising that
average (see the fits) strength of magnetic field structure is same as
strength (in the range of 0.01-1 G) of steady magnetic field structure as
estimated by observational (Stenflo 1993; Kotov 2015) and theoretical
(Hiremath and Gokhale 1995) studies. It is to be noted that this speculative
explanation for the origin of coronal holes has to be treated cautiously
unless some other studies also agree with our conjecture that coronal holes
originate from the Alfven perturbations of global large-scale magnetic field
structure.

Another interesting result is that variation of thermal structure (especially
actual temperature) of coronal holes is independent of latitudes. As coronal
hole is a magnetic flux tube, for the steady state of magnetic field structure
and no gain of magnetic flux during coronal hole evolution, condition of
infinite conductivity leads to isorotation of coronal holes with the
surrounding ambient plasma rotation. That means coronal hole magnetic flux
bundle follows the path of isorotational contours. To be precise, coronal
holes that might have formed due to Alfven wave perturbations travel along the
large scale magnetic field structure parallel to isorotational contours.
Infact, in the previous study (Hiremath and Gokhale 1995), we have shown that
large-scale magnetic field structure, may be of primordial origin, consists of
combined magnetic field structure in the radiative core and current free
(combination of dipole and quadrupole like field structures that are embedded
in the uniform) field structure in the convective envelope and both the
structures in turn isorotate with the internal solar plasma rotation as
inferred by the helioseismology.

From the above discussion, it is clear that rotation rate of coronal holes and
rotation rate of ambient solar plasma depend upon each other. Helioseismic
inferences (Hiremath 2013, 2016 and references there in) yield rigid body
rotation in the radiative core and differential rotation in the convective
envelope. That means if coronal holes originate only in the convective
envelope, they must rotate differentially. Otherwise coronal holes likely to
rotate rigidly if coronal holes originate in the radiative core. On the
theoretical (Gilman 1977; Golub et. al. 1981; Jones 2005) and observational
(Hiremath and Hegde 2013) inferences, it is argued that coronal holes probably
originate below the convective envelope which in turn implies that coronal
holes rotate rigidly (as the radiative core rotates rigidly). On other hand,
by evolving magnetic diffusion equation, Wang and Sheeley (1993) simulate the
rotation rate of the coronal holes from the current free nature of the coronal
field that is rotating with the distorted active region fields.

If one considers curl of momentum equation in the cylindrical coordinates and
for steady angular velocity gradient, angular velocity of solar plasma is
balanced (Chandrasekhar 1956; Brun, Antia and Chitre 2010 and references there
in) by the combined forces due to stretching/tilting of vorticity due to
velocity gradients, advection of vorticity by the flows, turbulent and Reynold
stresses, Maxwell stresses, baroclinic forces, etc. If one assumes other
forces are negligible, then gradient of angular velocity is balanced by the
baroclinic forces only. In cylindrical coordinates, the relationship can be
expressed in the form of equation ${{\partial\Omega}\over{\partial z}}={g\over
rc_{p}}{{\partial<S^{{}^{\prime}}>}\over{\partial z}}$, where $\Omega$ is
angular velocity of the solar plasma, $g$ is acceleration due to gravity,
$c_{p}$ is specific heat at the constant pressure, ${g\over c_{p}}$ is the
adiabatic temperature gradient, $r$ is the radial variation and
$<S^{{}^{\prime}}>$ is entropy of the ambient medium. This equation is called
thermal wind balance equation (Brun, Antia and Chitre 2010). That means,
unless there is a temperature difference between the pole and equator, angular
velocity of the solar plasma can not be differential with respect to latitude
and also can not be maintained. As the coronal holes isorotate with the solar
plasma, this equation also implies that unless there is a temperature
structure that varies from equator to pole, coronal holes can not rotate
rigidly. However, present study yields the temperature structure (and hence
entropy) of coronal holes independent of solar latitude. Hence, above thermal
wind balance equation implies that ${\partial\Omega}\over{\partial z}$=0. That
means coronal holes must rotate rigidly or rotation rate of coronal holes is
independent of solar latitude. Infact this reasoning also matches with the
recent results (Hiremath and Hegde 2013) of rigid body rotation rates of
coronal holes derived from the SOHO 195 $\AA$ data.

To conclude this study, for the years 2001-2008, near equatorial coronal holes
detected from the SOHO/EIT images are used to understand the area evolution
and, latitudinal variation of thermal and magnetic field structure. Different
estimated physical parameters of the coronal holes are: area $\sim$ $3.8(\pm
0.5)\times 10^{20}$ $cm^{2}$, radiative flux at the sun $\sim$ $2.3(\pm
0.2)\times 10^{13}$ photons $cm^{-2}$ $sec^{-1}$, radiative energy $\sim$
$2.32(\pm 0.5)\times 10^{3}$ ergs $cm^{-2}sec^{-1}$, temperature structure
$\sim$ $0.94(\pm 0.1)\times 10^{6}$ K and magnitude of magnetic field
structure estimated to be $\sim$ $0.08\pm 0.02$ G.

Acknowledgments

This work has been carried out under “CAWSES India Phase-II program of Theme
1” sponsored by Indian Space Research Organization(ISRO), Government of India.
SOHO is a mission of international cooperation between ESA and NASA.

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