11institutetext: Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210023, PR China 11email<EMAIL_ADDRESS>22institutetext: School of
Astronomy and Space Science, University of Science and Technology of China,
Hefei 230026, PR China 33institutetext: National Astronomical Observatories,
Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing
100101, PR China 44institutetext: School of Astronomy and Space Science,
University of Chinese Academy of Sciences, Beijing 101408, PR China
55institutetext: College of Science, China Three Gorges University, Yichang
443002, PR China

# Evolution of the local spiral structure of the Milky Way revealed by open
clusters

C. J. Hao 1122 Y. Xu 1122 L. G. Hou 33 S. B. Bian 1122 J. J. Li 11 Z. Y. Wu
3344 Z. H. He 1122 Y. J. Li 11 D. J. Liu 1155

(Received 19 February 2021 / Accepted 27 June 2021)

The structure and evolution of the spiral arms of our Milky Way are basic but
long-standing questions in astronomy. In particular, the lifetime of spiral
arms is still a puzzle and has not been well constrained from observations. In
this work, we aim to inspect these issues using a large catalogue of open
clusters. We compiled a catalogue of 3794 open clusters based on Gaia EDR3. A
majority of these clusters have accurately determined parallaxes, proper
motions, and radial velocities. The age parameters for these open clusters are
collected from references or calculated in this work. In order to understand
the nearby spiral structure and its evolution, we analysed the distributions,
kinematic properties, vertical distributions, and regressed properties of
subsamples of open clusters. We find evidence that the nearby spiral arms are
compatible with a long-lived spiral pattern and might have remained
approximately stable for the past 80 million years. In particular, the Local
Arm, where our Sun is currently located, is also suggested to be long-lived in
nature and probably a major arm segment of the Milky Way. The evolutionary
characteristics of nearby spiral arms show that the dynamic spiral mechanism
might be not prevalent for our Galaxy. Instead, density wave theory is more
consistent with the observational properties of open clusters.

###### Key Words.:

Galaxies: evolution – Galaxy: structure – Galaxy: kinematics and dynamics –
open clusters and associations: general

##  1 Introduction

Accurately revealing the spiral structure of the Milky Way and its evolution
has attracted the attention of many astronomers. After the M 33 galaxy was
explored by Hubble (1926), it was speculated that our Milky Way is also
possibly a spiral galaxy. Since then, much effort has been dedicated to
uncover the spiral structure of the Galaxy with different kinds of tracers and
methods (e.g., Lindblad, 1927; Oort et al., 1958; Becker, 1963, 1964; Becker &
Fenkart, 1970; Georgelin & Georgelin, 1976; Moffat et al., 1979; Russeil,
2003; Paladini et al., 2004; Dias & Lépine, 2005; Xu et al., 2006; Moitinho et
al., 2006; Vazquez et al., 2008; Reid et al., 2009; Moitinho, 2010; Hou & Han,
2014, 2015; Reid et al., 2019; Xu et al., 2018, 2021; Poggio et al., 2021).
However, because the Sun is deeply embedded in the Galactic disc, multiple
structures at different distances along the line of sight are superimposed;
this makes very difficult to accurately decompose these structures and depict
the actual spiral structure (Xu et al., 2018b). Moreover, another big
challenge is reliably tracing the Galaxy’s morphology to the past to
ultimately understand its evolutionary history.

Open clusters (OCs) are one of the good spiral tracers and have some specific
advantages for understanding the properties of spiral structure of the Galaxy.
An open cluster is a group of stars that formed from the same giant molecular
cloud, which are roughly the same age. Unlike the other good tracers for
investigating the spiral arms of the Galaxy (e.g., high-mass star formation
region (HMSFR) masers, massive O–B type (OB) stars, and HII regions), the ages
of OCs cover a wide range, from a few million years (Myr) to tens of billions
of years, making them potentially good tracers for investigating the spiral
structure outlined by young to old objects, and in particular the evolution of
spiral arms. In addition, the large number of cluster members makes it
possible to derive more accurate values of cluster parameters (e.g.,
distances, proper motions, and radial velocities) than those of individual
star. The spiral structure of the Galaxy has been explored by many research
works using OCs as tracers.

The relationship between OCs and spiral arms was first discussed by Becker
(1963, 1964), who studied a sample of 156 OCs with photometric distances at
that time. Becker suggested that the distribution of those OCs with earliest
spectral type of O–B2 followed three spiral arm segments in the solar
neighbourhood. In comparison, the distribution of older OCs with earliest
spectral type from B3 to F did not present arm-like structures and seemed to
be random. These results were confirmed by Becker & Fenkart (1970) and Fenkart
& Binggeli (1979) with larger samples of OCs. While a diferent explanation was
proposed by Janes & Adler (1982) and Lynga (1982), these authors suggested
that the distributions of OCs seemed more like some clumpy-like concentrations
or complexes rather than a spiral pattern. Dias & Lépine (2005) used a sample
of 212 OCs to showed that the young OCs (ages $<1.2\times 10^{7}$ yr) still
remain in the three major arm segments in the solar neighbourhood, that is,
the Perseus, Local, and Sagittarius-Carina Arms. The OCs $\sim$ 20 Myr in age
are leaving the spiral arms and moving to the inter-arm regions, while for the
older OCs, their distributions do not present clear clump-like or spiral-like
structures. Dias & Lépine (2005) also estimated the spiral pattern rotation
speed of the Milky Way with OCs and confirmed that a dominant fraction of the
OCs are formed in spiral arms. These results were recently updated by Dias et
al. (2019) with the second data release of Gaia (Gaia Collaboration et al.,
2018). Moitinho et al. (2006) analysed the stellar photometry data for many
OCs; a sample of 61 OCs was used to trace the disc structure in the third
Galactic quadrant. The detailed spiral structure in the third Galactic
quadrant was also explored by Vazquez et al. (2008) by taking advantage of the
data of young OCs, blue plumes, and molecular clouds. For a review about the
previous efforts to use OCs to study the Galactic structure, we recommend
Moitinho (2010). Later, Camargo et al. (2013) studied a sample of young OCs,
which are related with the Perseus Arm and even the Outer Arm (also see
Bobylev & Bajkova, 2014). By taking advantage of the second data release of
Gaia, Cantat-Gaudin et al. (2018, 2019, 2020) have derived the members and
mean parameters for more than 1800 Galactic OCs. The projected distribution of
these OCs onto the Galactic disc was compared with a spiral arm model of our
Galaxy (Reid et al., 2014).

In brief, it is commonly accepted that young OCs can be used to trace the
nearby spiral arms, while older OCs have more scattered distribution, which
may be used to represent the distribution of older stellar components in the
Galactic disc. However, the evolutionary properties and the lifetime of the
spiral arms of the Galaxy are still puzzles and have not been well constrained
from observations. With its specialty, OC serves as good tracer for better
understanding these issues. Fortunately, the number of OCs and the accuracy of
OC parameters (parallaxes, proper motions, and radial velocities) have been
improved significantly in the past few years, largely as a result of the
efforts of Gaia (see Table 1). In this work, we aim to compile a large
catalogue of OCs from references and accurately derive their parallax
distances, proper motions, and radial velocities from the latest Gaia Early
Data Release 3 (hereafter Gaia EDR3) (Gaia Collaboration et al., 2016, 2020).
This largest catalogue of OCs available to date provides a good chance to
study the above issues about spiral structure of the Galaxy, not only to map
the nearby spiral arms, but also to understand their evolution, and further,
to place observational constraints on the different evolutionary mechanisms.

This article is organised as follows. In Sect. 2, we describe the sample of
OCs compiled in this work. The results and discussions are presented in Sect.
3, including the spiral structure in the solar neighbourhood traced by OCs,
the kinematic properties of OCs, the evolutionary properties of their scale
heights, and a regression analysis. Conclusions are given in Sect. 4.

Table 1: A list of references for the previously known OCs.

| OCs |
---|---|---
Reference | Number | Data used
Dias et al. (2002) | 2167 | WEBDA(1) et al.
Dias et al. (2014) | 1805 | UCAC4(2)
Kharchenko et al. (2013) | 3006 | 2MASS(3) & PPMXL(4)
Schmeja et al. (2014) | 139 | 2MASS
Scholz et al. (2015) | 63 | PPMXL & UCAC4
Cantat-Gaudin et al. (2018) | 1229 |  Gaia DR2
Cantat-Gaudin et al. (2019) | 41 |  Gaia DR2
Castro-Ginard et al. (2018) | 23 |  Gaia DR2 & TGAS(5)
Castro-Ginard et al. (2019) | 53 |  Gaia DR2
Castro-Ginard et al. (2020) | 582 |  Gaia DR2
Liu & Pang (2019) | 76 |  Gaia DR2
He et al. (2021) | 74 |  Gaia DR2
Ferreira et al. (2020) | 25 |  Gaia DR2
111(1) WEBDA: http://obswww.unige.ch/webda/; (2) UCAC = United States Naval
Observatory CCD Astrograph Catalog; (3) 2MASS = Two Micron All Sky Survey; (4)
PPMXL: https://dc.zah.uni-heidelberg.de/ppmxl/q/cone/info; (5) TGAS = Tycho-
Gaia Astrometric Solution.

##  2 Sample

To better understand the spiral structure of the Galaxy with OCs, an updated
OC catalogue with accurately measured parameters from Gaia EDR3 is necessary.
The recently published Gaia EDR3 contains the positions, parallaxes, and
proper motions for more than 1.5 billion stars of different types, ages, and
evolutionary stages in the whole sky (Gaia Collaboration et al., 2020). For
the sources with magnitude $G<$ 15, the parallax uncertainties could be
0.02–0.03 mas, and the proper motion uncertainties could be 0.02–0.03 mas
yr-1. Gaia EDR3 provides excellent database to improve the parameter
accuracies for a large number of OCs.

Figure 1: Distributions of Galactic OCs. (a) YOCs (cyan dots; ages $<$ 20 Myr)
and HMSFR masers (white triangles) projected onto the Galactic disc. The dot
size is proportional to the number of cluster members. The solid curved lines
trace the centres (and dotted lines the widths enclosing 90% of the masers) of
the best-fit spiral arms given by Reid et al. (2019). The distance
uncertainties of the masers are indicated by the inverse size of the symbols.
The yellow dashed lines denote the best-fit spiral arm centres from the
distribution of YOCs. The Galactic centre (red star) is at (0, 0) kpc and the
Sun (red symbol) is at (0, 8.15) kpc. The background is a new concept map of
the Milky Way credited by Xing-Wu Zheng & Mark Reid BeSSeL/NJU/CFA. The
distributions of the older clusters are shown in panel (b), (c), and (d) for
the OCs with ages of 20–200 Myr (purple dots), 200–1000 Myr (brown dots), and
$>$ 1000 Myr (red dots), respectively. The dot size is also proportional to
the number of cluster members. The parallax uncertainties of the OCs shown are
all smaller than 10%.  Table 2: Best-fit parameters of nearby spiral arms
using HMSFR masers and OCs of different age groups.

| Arm | $\varphi$ | $R_{\rm ref}$ | $\beta_{\rm ref}$ | Arm width |  |
---|---|---|---|---|---|---|---
|  | (deg) | (kpc) | (deg) | (kpc) |  |
| Sagittarius-Carina | 17.1 $\pm$ 1.6 | 6.04 $\pm$ 0.09 | 24 | 0.27 $\pm$ 0.04 |  |
HMSFR masers | Local | 11.4 $\pm$ 1.9 | 8.26 $\pm$ 0.05 | 9 | 0.31 $\pm$ 0.05 |  |
| Perseus | 10.3 $\pm$ 1.4 | 8.87 $\pm$ 0.13 | 40 | 0.35 $\pm$ 0.06 |  |
| Sagittarius-Carina | 16.2 $\pm$ 0.4 | 6.72 $\pm$ 0.07 | 2 | 0.27 $\pm$ 0.02 |  |
YOCs ($<$ 20 Myr) | Local | 10.8 $\pm$ 0.6 | 8.18 $\pm$ 0.06 | 6 | 0.24 $\pm$ 0.06 |  |
| Perseus | 9.6 $\pm$ 0.8 | 9.42 $\pm$ 0.04 | 20 | 0.33 $\pm$ 0.03 |  |
| Sagittarius-Carina | 15.7 $\pm$ 0.5 | 6.80 $\pm$ 0.07 | 2 | 0.30 $\pm$ 0.03 |  |
OCs (20–60 Myr) | Local | 10.7 $\pm$ 0.8 | 8.24 $\pm$ 0.07 | 6 | 0.31 $\pm$ 0.03 |  |
| Perseus | 9.0 $\pm$ 1.0 | 9.45 $\pm$ 0.06 | 20 | 0.36 $\pm$ 0.04 |  |
| Sagittarius-Carina | 15.2 $\pm$ 0.5 | 6.79 $\pm$ 0.07 | 2 | 0.33 $\pm$ 0.04 |  |
OCs (60–100 Myr) | Local | 8.9 $\pm$ 0.9 | 8.09 $\pm$ 0.06 | 5 | 0.40 $\pm$ 0.03 |  |
| Perseus | 9.0 $\pm$ 0.9 | 9.43 $\pm$ 0.07 | 20 | 0.39 $\pm$ 0.05 |  |

222 For each spiral arm, the best-fit pitch angle ($\varphi$), initial radius
($R_{\rm ref}$), reference Galactocentric azimuth angle ($\beta_{\rm ref}$),
and arm width are listed. The OCs are divided into three subsamples: (1) YOCs
($<$ 20 Myr, 633 clusters); (2) OCs with ages from 20 to 60 Myr (334
clusters); and (3) OCs with ages from 60 to 100 Myr (262 clusters).

Up to now, thousands of OCs have been identified, but scattered in references.
Dias et al. (2002) assembled a large catalogue of OCs based on many previous
studies, which contains 2167 objects. Dias et al. (2014) gave the proper
motions and membership probabilities for 1805 optically visible OCs. By taking
advantage of near-infrared photometric data of approximately 470 million
objects in Two Micron All Sky Survey (2MASS, Skrutskie et al., 2006) and
proper motions from the PPMXL catalogue (Roeser et al., 2010), Kharchenko et
al. (2013) completed a survey of all previously known OCs called the Milky Way
Star Clusters Catalog (MWSC), which lists 3006 objects, including known OCs
and candidates, globular clusters, associations, and asterisms. Subsequently,
the MWSC catalogue was complemented by 139 new OCs at high Galactic latitudes
(Schmeja et al., 2014) and 63 new OCs identified by Scholz et al. (2015).
After the publication of Gaia DR2 (Gaia Collaboration et al., 2018), the
membership probabilities of $\sim$ 1200 OCs presented in previous catalogues
were recalculated by Cantat-Gaudin et al. (2018). These authors also
discovered 60 new OCs. Besides, the radial velocities of 861 OCs in Cantat-
Gaudin et al. (2018) were obtained by Soubiran et al. (2018). Castro-Ginard et
al. (2018) developed a machine-learning approach and found 23 new OCs from
Gaia DR2. Soon after, Castro-Ginard et al. (2019) detected 53 new OCs in the
direction of the Galactic anti-centre. Cantat-Gaudin et al. (2019) identified
41 new OCs in the direction of the Perseus Arm. Meanwhile, 76 new OCs were
reported by Liu & Pang (2019). Recently, 582 new OCs were discovered by
Castro-Ginard et al. (2020), 25 new OCs were detected by Ferreira et al.
(2020), and 74 new OCs were found by He et al. (2021), respectively.

As the first step of this work, we compiled a large
catalogue333https://cdsarc.u-strasbg.fr/ftp/vizier.submit//hcjgaia2021_v2/,
which includes 3794 Galactic OCs collected from references (see Table 1). The
procedure is described as follows. Firstly, 2469 OCs were selected from
Kharchenko et al. (2013), which provides details of member stars for each of
the OCs. Then, the member stars with probabilities greater than 60% were
picked out to conduct positional cross-matching with Gaia EDR3 using a
matching radius of 1 arcsec. Making use of the sample-based clustering search
method (Hao et al., 2020), which was an adaptation of the OC search method
presented in Castro-Ginard et al. (2018), we yielded 2457 OCs. For the OC
catalogue of Dias et al. (2002), we cross-matched it with the catalogue of
Kharchenko et al. (2013), and 315 OCs not listed in Kharchenko et al. (2013)
were obtained. The sample-based clustering search method was applied to these
targets, and we got 309 OCs. Subsequently, we cross-matched the catalogue of
Cantat-Gaudin et al. (2018) with these 2766 OCs and then obtained 153 OCs. For
these 153 OCs, only the member stars with probabilities greater than 60% were
picked out. The same procedure was conducted on the catalogue of Cantat-Gaudin
et al. (2019). Then, we obtained detailed parameters of 658 OCs from the
catalogues of Castro-Ginard et al. (2018, 2019, 2020), including the member
stars. In addition, we collected 76 new OCs from the catalogue of Liu & Pang
(2019), 25 new OCs from Ferreira et al. (2020), and 74 new OCs from He et al.
(2021), respectively. Finally, we obtained a total of 3794 OCs. The parameters
of these OCs were determined using Gaia EDR3.

Gaia EDR3 provides parallax uncertainties for almost all of its stars. To
obtain precise parallax for each of these OCs, we only extracted the member
stars whose parallax uncertainties are smaller than 10%. Not all of the 3794
OCs have member stars with parallax uncertainties better than 10%. Hence, a
high-accuracy subsample of 3611 OCs was extracted, in which, 1742 OCs have
radial velocities. For these 1742 OCs, the means, standard errors, and
uncertainties of their radial velocities were determined using the method of
Soubiran et al. (2018). After that, the member stars brighter than G = 17 mag
were used to determine the position and proper motions for each of these OCs.
For the bright member stars, the uncertainties are better than 0.05 mas for
the position and 0.07 mas yr-1 for the proper motions, respectively (Gaia
Collaboration et al., 2020). In terms of photometry, for the stars of G = 17
mag, the associated photometric error is 0.001 mag and 0.012 mag for GBP, and
0.006 mag for GRP (Gaia Collaboration et al., 2020). The determined parameters
of the OCs, together with the reference(s) where the OC was selected, all are
given in the on-line catalogue of this work.

Some recent works (e.g., Cantat-Gaudin et al., 2018; Cantat-Gaudin & Anders,
2020; Monteiro et al., 2020) suggest that there are some false positive or
non-existing clusters in the previous catalogues. Most of these controversial
or erroneous objects are the alleged old and high-altitude clusters (Cantat-
Gaudin & Anders, 2020). We checked the 146 false positive or nonexistent
clusters given by Dias et al. (2002), and we found that 81 of these are not
included in our catalogue. In this work, our aim is to study the spiral arms
in the Galactic disc, and we primarily adopt relatively young OCs (e.g., ages
$<$ 100 Myr). For the 65 possibly false positive clusters suggested by Dias et
al. (2002) and kept in our compiled catalogue, only 16 of these are $<$ 100
Myr in age. In comparison, the total number of OCs $<$ 100 Myr in age in our
catalogue is 1229. Hence, we believe that this problem does not influence the
following analysis results about the spiral structure of the Galaxy. Besides,
the ages for the 2837 OCs in our compiled catalogue were already given in
previous works; these are adopted directly. For the remaining 957 ($\sim$ 25%)
OCs, most of these ($\sim$ 70%) are recently identified OCs based on the Gaia
data. We determined their ages and the errors following the methods described
in Liu & Pang (2019), Hao et al. (2020), and He et al. (2021). The age
parameter for each of the 3794 OCs and the corresponding reference(s) are also
given in the on-line catalogue.

##  3 Results and discussions

###  3.1 Spiral structure in the solar neighbourhood

Figure 1 presents the projected distributions of OCs onto the Galactic disc.
Figure 1(a) shows that the 633 young OCs (YOCs; i.e., ages $<$ 20 Myr) in our
catalogue are concentrated in distinct structures, which are almost concordant
with the known spiral arms outlined by HMSFR masers (Reid et al., 2019) and/or
bright OB stars (Xu et al., 2018, 2021). A large number of YOCs are located in
the Perseus, Local, Sagittarius–Carina, and Scutum–Centaurus Arms, which seem
to be the nurseries for many OCs. These arm features traced by YOCs extend as
far as $\sim$3 to 8 kpc along different spiral arms. These properties are in
general consistent with previous results (e.g., see Becker & Fenkart, 1970;
Dias & Lépine, 2005; Moitinho et al., 2006; Vazquez et al., 2008; Dias et al.,
2019; Cantat-Gaudin et al., 2018, 2020). In Figure 1(a), it is also shown that
the more populous OCs (i.e., those with more member stars) seem to be more
concentrated on the major spiral arms than the less populous ones. In
addition, spur-like features indicated by the concentration of some YOCs are
visible in the inter-arm regions, which is consistent with the results shown
using HMSFR masers (Reid et al., 2019) and bright OB stars (Xu et al., 2018,
2021). This all suggests that our Milky Way is not a pure grand-design spiral,
but perhaps a multi-armed spiral galaxy showing complex substructures (Xu et
al., 2016).

The distributions of older OCs are shown in Figure 1(b), (c) ,and (d).
Consistent with previous results (e.g., Moitinho et al., 2006; Vazquez et al.,
2008; Bobylev & Bajkova, 2014; Reddy et al., 2016; Dias & Lépine, 2005;
Cantat-Gaudin et al., 2018, 2020), the OCs are gradually dispersed in the
Galactic disc as they age. Structural features are still discernible for the
populous OCs with ages of tens of million years, but the older OCs (e.g., ages
$>$ 200 Myr) present more diffuse distributions. Indeed, when using the OCs
older than 1000 Myr as tracers, spiral arm features are almost unrecognisable.
These properties indicate that as the YOCs age they gradually migrate away
from their birthplaces, probably accompanied by the evolution of spiral arms.

Figure 2: Proper motion vectors of the YOCs. A motion scale of 40 km s-1 is
indicated in the bottom left corner of the plot. The ages of OCs are colour
coded. The background is the same as that of Reid et al. (2019). Figure 3:
Average peculiar motions of the OCs in the direction of Galactic rotation as a
function of cluster age. The age bin is 5 Myr. The magenta symbols and error
bars indicate the average PMs and their associated errors.

Inspired by the above phenomena, we considered that as OCs gradually age the
spiral arms traced by OCs of different ages are probably not the same;
instead, they can be used to reflect the evolution of the spiral arms of the
Galaxy. A subsample of OCs below 100 Myr in age is extracted from our
catalogue and used to explore this issue. To ensure that there are plenty of
OCs in each age interval, we divide those OCs that are below 100 Myr into
three different age groups of $<$ 20 Myr (633 clusters), 20–60 Myr (334
clusters), and 60–100 Myr (262 clusters). From their distributions, the three
nearby spiral arms (i.e., the Sagittarius-Carina, Local, and Perseus Arms) are
discernible. For the more distant Scutum–Centaurus Arm, as fewer OCs are
located in it, this arm cannot be well traced and hence is omitted in the
following analysis. Then, in order to make a comparison with the maser
results, we determine the parameters of spiral arms with the same method of
Reid et al. (2019), which is used in analysing their maser parallax data. To
fit the pitch angle and arm width for a spiral arm traced by YOCs, a
logarithmic spiral-arm model is adopted (Reid et al., 2014),

$\\\ \\\ \ln(R/R_{\mathrm{ref}})=-(\beta-\beta_{\mathrm{ref}})\tan\varphi,$
(1)

where Rref is the reference Galactocentric radius, $\beta_{\rm ref}$ is the
reference azimuthal angle, and $\varphi$ is the pitch angle. The zero point of
the Galactocentric azimuthal angle, $\beta$, is defined as a line towards the
Sun from the Galactic centre, and the azimuthal angle increases towards the
direction of Galactic rotation. The quantity R is the Galactocentric radius at
a Galactocentric azimuth $\beta$. The fitting results are listed in Table 2.

In comparison to the values derived by using HMSFR masers, the pitch angles of
the three nearby spiral arms traced by OCs tend to be smaller; the differences
become larger when older OCs were used as tracers, thus implying a tighter
winding of the spiral arms traced by older OCs. This is possibly a true
feature for the Milky Way; however, the differences of pitch angles are not
significant when considering the fitting uncertainties and higher quality data
of OCs would be necessary to confirm this. Besides, arm width is also an
important parameter for a spiral arm. We note that the fitted arm widths for
the OCs with ages of 60–100 Myr are inclined to be larger than those of the
HMSFR masers. In comparison, the arm widths derived from YOCs are in general
consistent with HMSFR masers.

Figure 4: Evolution of the scale height of OCs with OC age. The 3376 OCs
within 4.0 kpc of the Sun are divided into 32 bins according to their ages.
The blue line indicates the evolution of the scale height with cluster age.
The blue dots correspond to the typical values of log10(age) in each bin. The
solid magenta dots show the distances of the OCs to the Galactic plane as a
function of cluster age.

###  3.2 Kinematic properties of OCs

With the distances, proper motions, and radial velocities for the 1742 OCs in
the high-accuracy sample, we can derive their three-dimensional (3-D) velocity
information following the methods of Xu et al. (2013) and Reid et al. (2019).
The 3-D velocities were straightforwardly calculated using the linear speeds
projected onto the celestial sphere and radial velocities, which were obtained
from the proper motions and distances. Then, the peculiar motions (PMs; i.e.,
non-circular motions) of the OCs were estimated by subtracting the effect of
Galactic rotation and the solar PMs using the updated Galactic parameters. In
this approach, we adopted a Galactic rotation speed of 236 $\pm$ 7 km s-1,
where the Sun is at a distance of 8.15 $\pm$ 0.15 kpc to the Galactic centre
(Reid et al., 2019). Solar PMs of U⊙ = 10.6 $\pm$ 1.2 km s-1, V⊙ = 10.7 $\pm$
6.0 km s-1, and W⊙ = 7.6 $\pm$ 0.7 km s-1 were assumed (Reid et al., 2019),
which are the velocity components towards the Galactic centre in the direction
of Galactic rotation and towards the North Galactic Pole, respectively. The
sources with an uncertainties larger than 10 km s-1 were eliminated in the
following analysis.

In Figure 2, we present the PMs of YOCs in the solar neighbourhood. The PMs of
YOCs seem to be random. As the gravitational potential of the Galactic bar is
weak in these regions, the PMs of YOCs should be intrinsic. We find that the
average peculiar motion is 1.3 $\pm$ 0.2 km s-1 towards the Galactic centre
and $-$6.8 $\pm$ 0.2 km s-1 in the direction of the Galactic rotation. Hence,
there is an average lagging motion for YOCs behind the Galactic rotation.
Similar properties were recently noticed for HMSFR masers (Reid et al., 2019)
and OB stars (Xu et al., 2018b). We also find that the average lagging motion
tends to increase as the OCs age (Figure 3).

As discussed in Sect. 3.1, the YOCs gradually migrate away from their birth
arms. With the kinematic data, the mean time for the YOCs to traverse their
natal spiral arms could be estimated. The average arm width w is taken as 0.31
kpc, which is the mean width of the segments of the Perseus, Local, and
Sagittarius-Carina Arms (Reid et al., 2019). We select the YOCs that are
located in spiral arms according to the best-fit model of Reid et al. (2019),
which is consistent with our fitting results with YOCs as shown in Figure 1
and Table 2. As shown in Figure 2, the YOCs approximately migrate either
towards the Galactic centre direction or Galactic anti-centre direction.
Hence, the travel speed of YOCs is defined as the average of the absolute
values of the derived mean velocities. The travel velocity for the YOCs
located in spiral arms, $v$, is estimated to be 14.6 $\pm$ 0.2 km s-1. The
mean time taken by a YOC to leave its natal spiral arms is simply estimated as
$t=w\div v=$ 20.8 $\pm$ 2.2 Myr.

###  3.3 Scale height of OCs

In this section, we focus on the vertical distribution of OCs. The high-
accuracy subsample with 3611 OCs (Sect. 2) is adopted. Statistically, about
43.4% of the 3611 OCs in our sample are located above the International
Astronomical Union (IAU) defined Galactic plane ($b=0^{\circ}$). In previous
works, asymmetric vertical distributions of gas or stars above and below the
IAU-defined Galactic plane were commonly found (e.g., Anderson et al., 2019;
Reid et al., 2019), which are generally interpreted as the Sun slightly
deviating from the Galactic midplane towards the north Galactic pole, that is,
the height of the Sun $z_{\odot}$ ¿ 0 pc. Many efforts have been made to
determine the value of $z_{\odot}$ (e.g., Cantat-Gaudin et al., 2020, and
references therein). By using the OCs with accurate Gaia parallaxes, we can
refine the Galactic plane. We divide a subsample of 3376 OCs within 4.0 kpc of
the Sun into 32 groups according to the OC ages, which are used to determine
z⊙. Then, the z⊙ values of the 32 groups are averaged to obtain the mean
vertical displacement of the Sun with respect to the Galactic plane. Here, the
values of z⊙ and the scale height are determined using the following function
(Maíz-Appelániz, 2001):

$\\\ \\\
N(z)=N\times\exp{\left[-\frac{1}{2}\left(\frac{z-z_{\odot}}{h}\right)^{2}\right]},$
(2)

where $N$ is the central space density of OCs at z = z⊙ with respect to the
Galactic plane ($b=0^{\circ}$) and $h$ is the scale height. By analysing the
vertical distribution of OCs with the accurate Gaia parallaxes, the value of
z⊙ is refined to be 17.5 $\pm$ 3.8 pc above the true Galactic plane. This
value is consistent with the results given by Buckner & Froebrich (2014) and
Karim & Mamajek (2017), but smaller than that of Cantat-Gaudin et al. (2020).
The value of z⊙ = 17.5 $\pm$ 3.8 pc is adopted in this work to adjust the
$z$-offsets of the OCs and calculate the scale heights.

Figure 4 shows the evolution of the scale height with cluster age for the 3376
OCs within 4.0 kpc of the Sun, which is consistent with the dispersion of the
projected OCs as time goes by. The scale height is about 42 pc for the OCs
with a mean age of $t_{\rm OC}\sim$ 1.5 Myr and increases to about 130 pc for
the OCs with $t_{\rm OC}\sim$ 850 Myr. For the OCs with ages greater than
$t_{\rm OC}\sim$ 1 Gyr, the scale heights increase obviously. For the YOCs
(ages $<$ 20 Myr), the scale height is estimated to be 70.5 $\pm$ 2.3 pc.
While for the OCs with ages of 20–100 Myr, the value is 87.4 $\pm$ 3.6 pc. The
deviations of the scale heights of the very young OCs from those of HMSFR
masers (19 $\pm$ 2 pc; Reid et al., 2019) and O–B5 stars (34 $\pm$ 3 pc; Maíz-
Appelániz, 2001) are small, but increases gradually as OC age. These results
are consistent with the properties discussed in Sect. 3.1. The OCs within an
age of a few million years are still located in or very close to their
birthplaces, hence have a small scale height. The spiral structure traced by
these OCs resembles that of HMSFR masers or OB stars. But as the OCs age, the
OCs gradually migrated further from their birthplaces and have larger scale
height.

Figure 5: Distributions of the regressed OCs. The OC ages have been grouped
into: (a) 20–40 Myr, (b) 40–60 Myr, (c) 60–80 Myr, (d) 80–100 Myr, and traced
back to 20, 40, 60, and 80 million years ago, respectively. The solid curved
lines trace the centres (and dotted lines the widths enclosing 90% of the
masers) of the best-fit spiral arms given by Reid et al. (2019), and have been
rigidly rotated back to the past for comparison. The background is a new
concept map of the Milky Way (credited by: Xing-Wu Zheng & Mark Reid
BeSSeL/NJU/CFA), which is also rigidly rotated back to the past. The adopted
pattern speed is 27 km s-1 kpc-1. Table 3: Best-fit results of the spiral arm
parameters for the present-day OCs and the regressed OCs.

Age groups | 0–20 Myr | 20–40 Myr | 40–60 Myr | 60–80 Myr | 80–100 Myr |
---|---|---|---|---|---|---
Regress to: |  | 20 MYA | 40 MYA | 60 MYA | 80 MYA |
|  |  | Pitch angle: $\varphi$ (degree) |  |  |
Sgr–Car Arm | 16.2 $\pm$ 0.4 | 16.3 $\pm$ 1.0 | 16.2 $\pm$ 1.1 | 16.2 $\pm$ 1.0 | 16.0 $\pm$ 1.0 |
Local Arm | 10.8 $\pm$ 0.6 | 10.8 $\pm$ 0.9 | 10.5 $\pm$ 1.0 | 10.5 $\pm$ 0.9 | 10.6 $\pm$ 1.0 |
Perseus Arm | 9.6 $\pm$ 0.8 | 9.3 $\pm$ 0.9 | 9.5 $\pm$ 0.9 | 9.6 $\pm$ 0.9 | 9.5 $\pm$ 1.0 |
|  |  | Arm width: $w$ (kpc) |  |  |
Sgr–Car Arm | 0.27 $\pm$ 0.02 | 0.24 $\pm$ 0.09 | 0.24 $\pm$ 0.09 | 0.25 $\pm$ 0.09 | 0.25 $\pm$ 0.10 |
Local Arm | 0.24 $\pm$ 0.06 | 0.28 $\pm$ 0.03 | 0.20 $\pm$ 0.05 | 0.23 $\pm$ 0.10 | 0.22 $\pm$ 0.09 |
Perseus Arm | 0.33 $\pm$ 0.03 | 0.29 $\pm$ 0.09 | 0.29 $\pm$ 0.09 | 0.30 $\pm$ 0.10 | 0.31 $\pm$ 0.10 |

444For each of the spiral arms near the Sun, we list its best-fit pitch angle
($\varphi$) and arm width (w), including their 1$\sigma$ errors. Sgr–Car:
Sagittarius-Carina Arm; MYA: million years ago.

###  3.4 Regression analysis

Based on multiwavelength observations, we have some knowledge about the spiral
structure of our Milky Way (e.g., Xu et al., 2018b). However, the lifetime of
the spiral arms of the Galaxy of the galaxy is still a puzzle; the question is
whether they are long-lived or short-lived. Many of the older OCs (ages $>$ 20
Myr) in our sample have accurate parallaxes, proper motions, and radial
velocities. With an appropriate model, we may trace their trajectories back
tens of millions of years ago, until they were very young (i.e., ages $<20$
Myr), and resided in their natal spiral arms in the past. With this method,
the spiral structure in the past could be explored and compared with its
current view. Such an approach provides a good and special way to inspect the
evolutionary history of nearby spiral pattern and helps to address questions
such as whether the spiral arms remain stable or have they evolved with time.
Actually, tracing OCs back to their birthplaces has been proposed by some
previous works (e.g., Dias & Lépine, 2005; Dias et al., 2019) and used to
determine the pattern speed of the Galaxy by assuming a stable spiral pattern.
In the following, we adopt a regression analysis method to inspect the
stability of nearby spiral structure.

Since the YOCs ($<$ 20 Myr) can well trace the spiral arms, we perform the
regression analysis using 20-Myr intervals. The OCs in the age ranges of 20–40
Myr, 40–60 Myr, 60–80 Myr, and 80–100 Myr are regressed to 20, 40, 60, and 80
million years ago, respectively, by adopting a commonly used model of OC
motions in the Milky Way (Wu et al., 2009). This model has been widely used to
calculate the orbits of different objects (e.g., Odenkirchen et al., 1997;
Allen et al., 2006), which employs an axisymmetric Galactic gravitational
potential (Allen & Santillan, 1991), consists of a spherical central bulge and
a disc, plus a massive, spherical halo extending to a distance of 100 kpc from
the Galactic centre (Paczynski, 1990; Flynn et al., 1996). The total mass of
the model is 9.0 $\times$ 1011 M⊙, and the local total mass density in the
solar neighbourhood is 0.15 M⊙ pc-3. The results of regression are shown in
Figure 5. We also fit spiral arms to the distributions of regressed OCs with
the same method described in Sect. 3.1. The fitting results are listed in
Table 3.

At present, about 72% of the YOCs are located in the spiral arms defined by
HMSFR masers. In comparison, the percentages are about 39% $-$ 59% for the
older clusters in the four different age groups (20$-$40 Myr, 40$-$60 Myr,
60$-$80 Myr and 80$-$100 Myr). After regression, many OCs are traced back to
spiral arms in the past as shown in Figure 5. We found that the percentages of
regressed OCs in spiral arms increase to 71%$-$85%, which are comparable to
the present-day value. The vertical scale height of the present-day OCs in the
age groups of 20–100 Myr is about 87 pc. After regression, the scale height
decreases to about 69 pc, close to the value for the present-day YOCs ($\sim$
71 pc). In addition, we also notice that the best-fit pitch angles and arm
widths of the Sagittarius-Carina, Local, and Perseus Arms at different
regression times are consistent with each other within the uncertainties,
which are all are very close to the present-day values traced by YOCs (Table
3). Hence, the spiral-arm features depicted by regressed OCs are roughly
concordant with those traced by present-day YOCs. These results jointly
suggest that the spiral arms near the Sun probably are compatible with a long-
lived pattern, which has remained stable in the past 80 Myr. The Local Arm,
which was thought to be a spur or branch but now is suggested as a major
spiral arm segment, also has probably existed for a long time. However,
although the spiral pattern can be roughly recognised over almost 80 Myr, it
is not enough to support a long-lived stable pattern. It is also possible that
the pattern could be in the last stages of disintegration. In the next works,
we expect to further inspect this issue with more OCs and an improved model of
OC motions in the Milky Way.

The evolutionary characteristics of nearby spiral arms revealed by OCs could
provide some observational constraints on the formation mechanisms of the
spiral arms of the Galaxy, which are hotly debated all along. There are two
major competing scenarios: the dynamic spiral mechanism and the density wave
theory. Proponents of the former argue that the spiral arms are short-lived,
transient, and recurrent (Toomre & Toomre, 1972; Sellwood & Carlberg, 1984).
Besides, the dynamic spiral mechanism may be able to explain the offsets in
some local regions, but it does not predict the systematic positional offsets
between spiral arms delineated by different kinds of tracers. Our above
results are compatible with the long-lived and rigid, rotating nature of
nearby spiral arms, and hence seem to conflict with the predictions of the
dynamic spirals, indicating that this mechanism might be not prevalent for our
Milky Way. In comparison, the (quasi-stationary) density wave theory suggests
that the spiral arms are long-lived and rigidly rotated; this theory also
predicts systematic spatial offsets between the spiral arms traced by HMSFRs
and old stars (Lin & Shu, 1964, 1966; Shu, 2016). In this work we show that
the evolutionary properties of spiral arms indicated by OCs are consistent
with the expectations of the density wave theory. Besides, the results from
the study addressing the metallicity gradient step near the Galactic
corotation radius with OCs and cepheids also suggested long-lived spiral arms
for the Milky Way (Lépine et al., 2003). However, we noticed that the density
wave theory expects that the Milky Way has four major spiral arms and the
Local Arm should not exist (Yuan, 1969). The presence of the long-lived Local
Arm ($\gtrsim$ 80 Myr) implied by our analysis enhances the challenge
previously posed on this issue (Xu et al., 2013, 2016). Lépine et al. (2017)
has interpreted the Local Arm as an outcome of the spiral corotation
resonance, which traps arm tracers and the Sun inside it (also see Barros et
al., 2021).

Density wave theory predicts that the Galactic spiral pattern should remain
unchanged over a long time and rotate at a constant speed (Shu, 2016). With
the regression analysis method, we can constrain the spiral pattern speed by
comparing the OC locations after regression with the positions of spiral arms
rigidly rotated back to that time. Based on this, the present-day spiral arms
are rotated back to 20, 40, 60, and 80 million years ago, respectively, and
then compared with the distributions of the OCs after regression (Figure 5).
Under the assumption of a constant pattern speed, which is independent of the
Galactocentric distance, the pattern speed is limited to be 26–28 km s-1
kpc-1, which is consistent with the values given by for example Dias & Lépine
(2005); Dias et al. (2019), and Gerhard (2011). The results also suggest that
the regression analysis method used in this work is feasible.

##  4 Conclusions

In summary, we have studied the evolution of the spiral arms of the Galaxy
using a large catalogue of OCs based on Gaia EDR3. A regression analysis
method applied to OCs indicates that the spiral arms near the Sun are
compatible with a long-lived spiral pattern, which might have been essentially
stable for at least the past 80 million years. In particular, the Local Arm is
also probably long-lived in nature, and well traced by many YOCs, HMSFR
masers, and OB stars, all of which suggest that the Local Arm is probable a
major arm segment of the Milky Way. These evolutionary characteristics of
spiral arms are more consistent with the expectations of the density wave
theory. The dynamic spiral mechanism might be not prevalent for the Milky Way.
These results are expected to be confirmed by conducting further studies with
more OCs in the Milky Way.

###### Acknowledgements.

We appreciate the anonymous referee for the instructive comments which help us
a lot to improve the paper. This work was funded by the NSFC, grant numbers
11933011, 11873019, 11673066, 11988101 and by the Key Laboratory for Radio
Astronomy. L.G.Hou thanks the support from the Youth Innovation Promotion
Association CAS. We acknowledge Dr. Mark J. Reid for allowing us to use his
program, utilised to fit the pitch angles of the Galactic spiral arms. We used
data from the European Space Agency mission Gaia
(http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC; see
http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for DPAC has been
provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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# Wormholes in the Non-minimally Coupled Gravity with Electromagnetism

Özcan SERT<EMAIL_ADDRESS>Department of Physics, Faculty of Arts and
Sciences, Pamukkale University, 20070 Denizli, Türkiye

###### Abstract

Wormholes are geometric passages in spacetime which can connect two faraway
regions of the same universe or two different universes with a shortcut.
Although they are currently considered as hypothetical structures, they might
be important observable structures that allow even time travel. But their
general relativistic construction requires an exotic matter with negative
energy density. The construction of wormholes containing realistic matter is a
very important challenge. Therefore, in this work we investigate wormholes in
the model which is non-minimally coupled electromagnetic fields to gravity.
Then we give some traversable wormhole geometries with realistic dust matter.

###### pacs:

Valid PACS appear here

##  I Introduction

Wormholes are one of the most fascinating prediction of General Relativity
(GR). Inspired by the recent observations of the other predictions of GR such
as black holes and gravitational waves, the possibility of their existence
encourages scientists to study them increasingly. They are tunnels which can
connect two faraway regions of the same universe or two different universes
with a shortcut. They might even turn into time machines that could allow time
travel for highly advanced civilizations MorrisThorne1988 ;
MorrisThorneYurtsever1988 .

Wormhole structures were firstly noticed by Einstein and Rosen ER1935  looking
for a solution to the two-body problem in General Relativity and discussing
whether material particles are singularities in the field. In order to find a
unified description of gravity and electromagnetism, Einstein and Rosen tried
to obtain a non-singular description of material particles. They obtained
regular spherically symmetric and static solutions by modifying General
Relativistic field equations (by multiplying them the square of the
determinant of the metric) and changing variable in the solutions. Thus, they
had a solution to this singularity problem by replacing an elementary particle
with a finite bridge. They thought that the elementary particle corresponded
to the bridge connecting the geometries (Schwarzschild or Reissner-Nordström)
at two opposite ends or sheets, known as Einstein-Rosen (ER) bridge. Recently,
after many years, the ER bridge was associated with Einstein-Podolsky-Rosen
paradox as ”ER=EPR” conjecture by Maldacena and Susskind Maldacena2013  in
order to explain the connection between any two entangled quantum systems.
According to this conjecture the ER bridge (or wormhole geometry connected by
two quantum systems) ER1935  is equivalent to the EPR entanglement of two
quantum systems EPR1935  and this led to the idea that quantum particles could
communicate via wormholes recently.

This Einstein-Rosen bridge was named as wormhole by Wheeler for the first time
in the 1950s Wheeler19572 ; Wheeler19571 . However this is not a traversable
wormhole as it does not allow two-way travel due to the presence of the event
horizon Fuller1962 . The traversable wormholes allowing travel among distant
universes first constructed by Morris and Thorne MorrisThorne1988  which are
free from singularities and event horizons. This work further developed and
revealed that the traversable wormhole could allow time travel by Morris,
Thorne, and Yurtsever MorrisThorneYurtsever1988 . Later, many general
relativistic wormhole studies Hawking1988 ; Visser1989 ; Lemos2003 ;
Damour2007 ; Lobo2005 ; Sushkov2005 ; Bueno2018 ; JavedOVgun2022 ; Ovgun2016 ;
RahamanPant2019 ; Garcia2012 ; Lobo2008  inspired by these seminal papers.
However, these traversable wormhole solutions can be constructed by exotic
matter with negative energy density or pressure which violate the null energy
condition in General Relativity. It is a challenging problem to find
traversable solutions without negative energy density and satisfying the null
energy condition. In order to construct traversable wormhole solutions with a
realistic matter, alternative modified theories of gravitation can be
considered. Some wormhole solutions of these modified theories satisfying the
null energy condition can be found in Lobo2009 ; Garcia2010 ; Montelongo2011 ;
Harko2013 ; Bhawal1992 ; Mehdizadeh2015 ; Anchordoqui1997 ; Rosa2022 .

A natural generalization of the Morris-Thorne wormhole is to study charged
cases that arise in the presence of electric and magnetic fields. In
particular, the presence of intense electrical charge in the throat of
wormhole creates a repulsive force, preventing wormhole from collapsing and
helping its throat to open. Moreover, charges support stability of wormhole
playing the role of exotic matter Kim2001 ; Kuhfittig2021 ; Ernesto2004 .
Furthermore, charged modified gravity models also play the role of exotic
matter by increasing energy of wormholes, therefore they support the stability
of wormholes Ovgun2023 ; Sharif2023 ; Jusufi2017 ; Moraes2019 ; Sharif2021 .
On the other hand, the non-minimally coupled gravity models with
electromagnetism and gravity are very intriguing, since they have solutions to
the cosmological and astrophysical problems of gravity. Some properties and
spherically symmetric and static solutions of the models are found in
Dereli20111 ; Dereli20113 ; Sert12Plus ; Sert13MPLA ; Dereli20112 ;
Sert16regular ; Dereli19901 , some compact star solutions in Sert2017 ;
Sert2018 ; Sert20182  and cosmological solutions in AADS ; SertAdak2019 .

In this study we construct traversable wormhole solutions satisfying the null
energy condition in the non-minimally coupled models. We give the solutions
for both cases with electric field and magnetic field separately. We show that
magnetic solutions can be obtained from electrical solutions by a duality
transformation. Therefore these solutions are dual to each other. After all
the calculations, we obtain the non-minimal models in $R^{\pm\alpha}F^{2}$
form satisfying these solutions.

##  II Field Equations of the Non-minimally Coupled Gravity

In the presence of matter, Lagrangian of the model consisting of the non-
minimal coupling between gravity and electromagnetism can be written as
follows Dereli20111 ; Dereli20113 ; Sert12Plus ; Sert13MPLA ; Dereli20112 ;
Sert16regular ; Sert2017 ; Sert2018

$L=\frac{1}{2\kappa^{2}}R*1-Y(R)F\wedge*F+L_{mat}+\lambda_{a}\wedge T^{a}\;.$
(1)

Here $R$ is the curvature scalar, $*1$ is the volume 4-form, $Y(R)$ is an
arbitrary real function of $R$, $F=dA$ is the Maxwell tensor with potential
1-form $A$ , $J$ is the source 3-form of the electromagnetic field, $L_{mat}$
is the matter Lagrangian of the isotropic fluid. By taking variation of this
Lagrangian, the gravitational field equation is found as

$\displaystyle-\frac{1}{2\kappa^{2}}R^{bc}\wedge*e_{abc}=Y(F_{a}\wedge*F-F\wedge\iota_{a}*F)-\frac{k}{\kappa^{2}}*R_{a}+(\rho+p)u_{a}*u+p*e_{a}\
$ (2)

together with the constraint

$\displaystyle Y_{R}\mathcal{F}=-\frac{k}{\kappa^{2}}\ .$ (3)

Here $p$ is the pressure and $\rho$ is the energy density of the matter,
$u=u_{a}e^{a}$ and $u_{a}$ is the time-like vector field for an observer and
$k$ is a real constant which determines the strength of the coupling between
gravity with electromagnetism. We can define
$\mathcal{F}=*(F\wedge*F)=F_{mn}F^{mn}$ or $F\wedge*F=-\mathcal{F}*1$ by
taking Hodge star of the Maxwell invariant.

It is important to note that the gravitational field equation (2) of the model
does not involve derivatives higher than second-order by the constraint (3)
and therefore this field equation does not have any ghost degrees of freedom.
A second way to obtain the second-order field equation for this model is to
add this constraint (3) to the action of the model (1) by a Lagrange
multiplier and the new action gives the field equation (2) after the
variation. In order to check the energy conditions we write the gravitational
field equation (2) as

$\displaystyle G_{a}=\kappa^{2}\tau_{a}^{eff}$ (4)

where $G_{a}=-\frac{1}{2\kappa^{2}}R^{bc}\wedge*e_{abc}$ is the Einstein
tensor and we can define the effective energy-momentum tensor $\tau_{a}^{eff}$

$\displaystyle\tau_{a}^{eff}$ $\displaystyle=$ $\displaystyle
Y(F_{a}\wedge*F-F\wedge\iota_{a}*F)-\frac{k}{\kappa^{2}}*R_{a}+(\rho+p)u_{a}*u+p*e_{a}$
(5) $\displaystyle=$
$\displaystyle\kappa^{2}\tau_{a}^{NM}+\kappa^{2}\tau_{a}=\tau_{ab}^{eff}*e^{b}$
(6)

as the sum of the non-minimal energy-momentum tensor $\tau_{a}^{NM}$ and
matter energy-momentum tensor $\tau_{a}$. Here we define that
$\tau_{00}^{eff}=\rho^{eff}$ is the effective energy density,
$\tau_{11}^{eff}=P_{r}^{eff}$ is the effective radial pressure,
$\tau_{22}=\tau_{33}^{eff}=P_{t}^{eff}$ is the effective tangential pressure,
while $\tau_{00}=\rho$ and $\tau_{11}=\tau_{22}=\tau_{33}=p$ are the energy
density and pressure for matter. There are also the following Maxwell field
equations

$\displaystyle d(*YF)=0,\hskip 42.67912ptdF=0\;.$ (7)

Additionally, we can obtain

$\displaystyle\frac{1-k}{\kappa^{2}}R=(\rho-3p)\;$ (8)

by taking the trace of the gravitational field equation (2).

##  III WORMHOLE SOLUTIONS

We consider the following spherically symmetric line element to describe the
traversable wormholes in 3+1 dimensions:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f^{2}(r)dt^{2}+\frac{dr^{2}}{1-b(r)/r}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}$ (9)

Here the function $f(r)$ is known as the redshift function and it must be
finite or not have an event horizon in the range from the throat $r_{0}$ to
infinity in order to obtain traversable wormholes. The other metric function
$b(r)$ is known as the shape function and it must satisfy the condition
$(b-b^{\prime}r)/b^{2}>0$ MorrisThorne1988 ; Hochberg1997  for traversability
of wormholes. This condition corresponds to a flaring out surface, that is
getting wider smoothly in diameter from throat of wormhole. The condition
becomes $b^{\prime}(r_{0})<1$ for $r=r_{0}$ and $b(r)<r$ for $r>r_{0}$. Also a
wormhole metric must be asymptotically flat. In accordance with the spherical
symmetry of the geometry, we will consider the following Maxwell tensor 2-form
which has electric and magnetic field component in radial direction

$\displaystyle F$ $\displaystyle=$ $\displaystyle E(r)e^{1}\wedge
e^{0}+B(r)e^{2}\wedge e^{3}.$ (10)

The electric field $E(r)$ or magnetic field $B(r)$ can be turned off depending
on the source by setting $E=0$ or $B=0$ in the resulting differential
equations, respectively. For this ansatz the Maxwell equations (7) lead to the
following equations.

$\displaystyle E(r)=\frac{q_{e}}{r^{2}Y}\;,\hskip
42.67912ptB(r)=\frac{q_{m}}{r^{2}}$ (11)

Here $q_{e}$ and $q_{m}$ are integration constants corresponding to the
electric and magnetic monopole charge of the source. These ansatz give us the
following differential equations for the gravitational field equation (2)

$\displaystyle\frac{b^{\prime}}{r^{2}}$ $\displaystyle=$
$\displaystyle\rho^{eff}\;,$ (12)
$\displaystyle(1-\frac{b}{r})\frac{2f^{\prime}}{rf}-\frac{b}{r^{3}}$
$\displaystyle=$ $\displaystyle P_{r}^{eff}\;,$ (13)
$\displaystyle\frac{f^{\prime\prime}}{f}(1-\frac{b}{r})+\frac{f^{\prime}}{rf}(1-\frac{b^{\prime}}{2}-\frac{b}{2r})-\frac{b^{\prime}}{2r^{2}}+\frac{b}{2r^{3}}$
$\displaystyle=$ $\displaystyle P_{t}^{eff}\;,$ (14)

with the constraint

$\displaystyle\frac{dY}{dR}(E^{2}-B^{2})=\frac{k}{2\kappa^{2}}\;.$ (15)

Here

$\displaystyle\rho^{eff}$ $\displaystyle=$
$\displaystyle-k\Big{[}(\frac{f^{\prime\prime}}{f}+2\frac{f^{\prime}}{rf})(1-\frac{b}{r})+\frac{f^{\prime}}{2f}(-\frac{b^{\prime}}{r}+\frac{b}{r^{2}})\Big{]}+\kappa^{2}Y(E^{2}+B^{2})+\kappa^{2}\rho\;,$
(16) $\displaystyle P_{r}^{eff}$ $\displaystyle=$ $\displaystyle
k\Big{[}(\frac{f^{\prime\prime}}{f}(1-\frac{b}{r})+(\frac{f^{\prime}}{2f}+\frac{1}{r})(-\frac{b^{\prime}}{r}+\frac{b}{r^{2}})\Big{]}-\kappa^{2}Y(E^{2}+B^{2})+\kappa^{2}p\;,$
(17) $\displaystyle P_{t}^{eff}$ $\displaystyle=$
$\displaystyle-k\Big{[}-\frac{f^{\prime}}{rf}(1-\frac{b}{r})+\frac{b^{\prime}}{2r^{2}}+\frac{b}{2r^{3}}\Big{]}+\kappa^{2}Y(E^{2}+B^{2})+\kappa^{2}p\;,$
(18)

and the Ricci curvature scalar becomes

$\displaystyle
R=\frac{2f^{\prime\prime}b}{rf}-\frac{2f^{\prime\prime}}{f}+\frac{f^{\prime}b^{\prime}}{rf}++\frac{3f^{\prime}b}{r^{2}f}-\frac{4f^{\prime}}{rf}+\frac{2b^{\prime}}{r^{2}}\;.$
(19)

For a geometry describing interior of a wormhole, it is important to obtain
tideless and regular solutions. Therefore we consider the simple redshift
function $f(r)=1$, which give regular and zero tidal force for stationary
observers. Then the above equations turn out to be

$\displaystyle\frac{b^{\prime}}{r^{2}}$ $\displaystyle=$
$\displaystyle\kappa^{2}Y(E^{2}+B^{2})+\kappa^{2}\rho=\rho^{eff}\;,$ (20)
$\displaystyle-\frac{b}{r^{3}}$ $\displaystyle=$ $\displaystyle
k(-\frac{b^{\prime}}{r^{2}}+\frac{b}{r^{3}})-\kappa^{2}Y(E^{2}+B^{2})+\kappa^{2}p=P_{r}^{eff}\;,$
(21) $\displaystyle-\frac{b^{\prime}}{2r^{2}}+\frac{b}{2r^{3}}$
$\displaystyle=$
$\displaystyle-k(\frac{b^{\prime}}{2r^{2}}+\frac{b}{2r^{3}})+\kappa^{2}Y(E^{2}+B^{2})+\kappa^{2}p=P_{t}^{eff}\;,$
(22)

with the constraint equation
$\frac{dY}{dR}(E^{2}-B^{2})=\frac{k}{2\kappa^{2}}$ and the Ricci scalar (19)
simplifies to $R=\frac{2b^{\prime}}{r^{2}}$ .

###  III.1 Solutions with Electric Field

We firstly looking for solutions without magnetic field by setting $B(r)=0$ or
$q_{m}=0$ in the equations (11,20,21,22,15). Then we find the following set of
solutions for the unknown functions:

$\displaystyle b(r)$ $\displaystyle=$
$\displaystyle\frac{(1-3k)}{6(k+1)}r^{4}R^{\prime}+\frac{1}{6}(k+1)r^{3}R\;,$
(23) $\displaystyle Y(r)$ $\displaystyle=$
$\displaystyle\frac{8\kappa^{2}q_{e}^{2}}{R^{\prime}(1-3k)r^{5}}\;,$ (24)
$\displaystyle E^{2}(r)$ $\displaystyle=$ $\displaystyle\frac{r^{6}R^{\prime
2}}{64q_{e}^{2}\kappa^{4}}(9k^{2}-6k+1)\;,$ (25) $\displaystyle p(r)$
$\displaystyle=$
$\displaystyle\frac{1}{24\kappa^{2}}(rR^{\prime}(3k-1)+R(8k-4))\;,$ (26)
$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{1}{8\kappa^{2}}(rR^{\prime}(3k-1)+4R)$ (27)

in terms of the Ricci scalar

$\displaystyle R=C_{1}+C_{2}r^{-\frac{4(2k-1)}{3k-1}}\;$ (28)

where $C_{1}$ and $C_{2}$ are integration constants. When we write the shape
function $b(r)$ (23) in terms of the radial coordinate $r$, we see that

$\displaystyle
b(r)=\frac{C_{1}}{6}r^{3}+\frac{C_{2}(3k-1)}{2(k+1)}r^{\frac{k+1}{3k-1}}$ (29)

and the integration constant $C_{1}$ should be zero; $C_{1}=0$, to obtain an
asymptotically flat geometry. Also, in order to satisfy the first flare-out
condition $b(r_{0})=r_{0}$, the integration constant $C_{2}$ should be

$\displaystyle C_{2}=\frac{2(k+1)r_{0}^{\frac{2(k-1)}{3k-1}}}{3k-1}$ (30)

Also the second flare out condition gives us the inequality
$b^{\prime}(r_{0})={\frac{k+1}{3k-1}}<1$ at the throat, which has the solution
$-\infty<k<\frac{1}{3}$ or $1<k<\infty$. By substituting these integration
constants above, we obtain

$\displaystyle b(r)$ $\displaystyle=$ $\displaystyle
r_{0}(\frac{r}{r_{0}})^{\frac{k+1}{3k-1}}\;,$ (31) $\displaystyle R$
$\displaystyle=$
$\displaystyle\frac{2(k+1)r_{0}^{\frac{2(k-1)}{3k-1}}}{3k-1}r^{-\frac{4(2k-1)}{3k-1}}\;,$
(32) $\displaystyle Y(r)$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}q_{e}^{2}(3k-1)r_{0}^{-\frac{2(k-1)}{3k-1}}}{(k+1)(2k-1)}r^{-\frac{4k}{3k-1}}\;,$
(33) $\displaystyle E^{2}(r)$ $\displaystyle=$
$\displaystyle\frac{(k+1)^{2}(2k-1)^{2}r_{0}^{\frac{4(k-1)}{3k-1}}}{q_{e}^{2}\kappa^{4}(3k-1)^{2}}r^{-\frac{4(k-1)}{3k-1}}\;,$
(34) $\displaystyle p(r)$ $\displaystyle=$ $\displaystyle 0$ (35)
$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\frac{2(1-k^{2})r_{0}^{\frac{2(k-1)}{3k-1}}}{(3k-1)\kappa^{2}}r^{-\frac{4(2k-1)}{3k-1}}\;.$
(36)

In normal matter case, the null energy condition states that
$\tau_{ab}k^{a}k^{b}\geq 0$ or $\rho+p\geq 0$ for any null vector $k^{a}$ and
the weak energy condition states that $\tau_{ab}u^{a}u^{b}\geq 0$ or
($\rho\geq 0$ and $\rho+p\geq 0$) for any time-like vector $u^{a}$. Because of
$p=0$ in the solution, the energy density $\rho(r)$ should be positive in
order to satisfy these energy conditions. We obtain the inequality
$\frac{1-k^{2}}{3k-1}\geq 0$ from the condition on the energy density (36) and
it has the solution interval $-\infty<k\leq-1$ or $\frac{1}{3}<k\leq 1$ for
the $k$ parameter. Also the function $Y(r)$ is indefinite for $k=-1$ value.
Then the intersection of these two intervals obtaining from the flare out
conditions and the energy conditions is $k<-1$. Thus we see that the solutions
in the range $k<-1$ allow wormholes involving normal matter, while the
solutions in the range $1<k<\infty$ lead to wormholes involving exotic matter.

On the other hand, in modified gravity models the generalized null energy
condition $\tau_{ab}^{eff}k^{a}k^{b}\geq 0$ is violated in order for these
models to have wormhole geometries Hochberg1997 ; Lobo2009 ; Garcia2010 ;
Capozziello2012 ; Francisco2013 ; Harko2013 ; Capozziello2014 ;
Capozziello2015 . This model also violates the condition for the effective
energy-momentum tensor (5), since $\rho^{eff}+P_{r}^{eff}\leq 0$ and
$\rho^{eff}+P_{t}^{eff}\leq 0$ in this interval. Thus the modified terms by
the non-minimal couplings play the role of the exotic matter that violates the
energy condition to have the worm hole geometry. In order to find the non-
minimal function $Y(R)$, we calculate the inverse function $r$ as

$\displaystyle
r=\left(\frac{3k-1}{2k+2}\right)^{\frac{1-3k}{8k-4}}r_{0}^{\frac{2k-2}{8k-4}}R^{\frac{1-3k}{8k-4}}$
(37)

and substitute it in (33), we find

$\displaystyle Y(R)=R_{0}R^{\alpha}$ (38)

where $\alpha=\frac{k}{2k-1}$ and $R_{0}$ is

$\displaystyle
R_{0}=\frac{q_{e}^{2}\kappa^{2}}{2k-1}\left(\frac{3k-1}{k+1}\right)^{\frac{3k-1}{2k-1}}r_{0}^{\frac{-2(k-1)}{2k-1}}2^{\frac{k}{1-2k}}\;.$
(39)

Then the corresponding gravitational model to the electrically charged
solutions become

$L=\frac{1}{2\kappa^{2}}R*1-R_{0}R^{\alpha}F\wedge*F+L_{mat}+\lambda_{a}\wedge
T^{a}\;.$ (40)

  * •

Solutions for $k=-2$ :

As an example to the solutions in the range that satisfy the energy
conditions, we set $k=-2$. Then the integration constant (30) becomes
$C_{2}=\frac{2}{7}r_{0}^{6/7}$ and we can calculate the functions
(31,32,33,34,35,36) in the solution above as

$\displaystyle R=\frac{2r_{0}^{6/7}}{7r^{20/7}}\;,\hskip
28.45274ptb(r)=r_{0}(\frac{r}{r_{0}})^{1/7}\;,\hskip
28.45274ptY(r)=-\frac{7q_{e}^{2}\kappa^{2}}{5r_{0}^{6/7}r^{8/7}}\;,\hskip
28.45274pt\ \ \ $ (41) $\displaystyle
E^{2}(r)=\frac{25r_{0}^{12/7}}{49q_{e}^{2}\kappa^{4}r^{12/7}}\;,\hskip
28.45274pt\rho(r)=\frac{6r_{0}^{6/7}}{7\kappa^{2}r^{20/7}}\;,\hskip
28.45274ptp(r)=0\;.\ \ \ $ (42)

We see that the energy density is positive and satisfies the null energy
condition. In order to find the non-minimal function $Y(R)$, we write $r$ in
terms of $R$ as $r=2r_{0}^{6/20}R^{-7/20}$ and the non-minimal function
becomes

$\displaystyle
Y(R)=-(\frac{7}{2})^{2/5}\frac{q_{e}^{2}\kappa^{2}7^{7/5}}{5r_{0}^{6/7}}R^{8/20}\;.$
(43)

###  III.2 Solutions with Magnetic Field

Secondly we consider solutions with only magnetic field by setting $E=0$.
Similar to the above, we look for tideless solutions with the simple redshift
function $f(r)=1$. Then we obtain the same functions with (31,32,35,36) after
the similar manipulations above, but the following non-minimal function and
the magnetic field

$\displaystyle Y(r)$ $\displaystyle=$
$\displaystyle\frac{(2k-1)(k+1){r_{0}}^{\frac{2(k-1)}{3k-1}}}{q_{m}^{2}\kappa^{2}(3k-1)}r^{\frac{4k}{3k-1}}\;,$
(44) $\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{q_{m}}{r^{2}}\;.$
(45)

Then the non-minimal function of the model for this case becomes

$\displaystyle Y(R)=\frac{1}{R_{0}R^{\alpha}}$ (46)

in terms of Ricci scalar $R$, where $\alpha=\frac{k}{2k-1}$ and $R_{0}$ has
just been given in (39) with $q_{e}=q_{m}$. It is remarkable that the non-
minimal function (46) is inverse of the previous non-minimal function (38).
Thus we find the dual magnetic wormhole solutions of the previous electric one
for the following Lagrangian 4-form

$L=\frac{1}{2\kappa^{2}}R*1-\frac{1}{R_{0}R^{\alpha}}F\wedge*F+L_{mat}+\lambda_{a}\wedge
T^{a}\;.$ (47)

Therefore, in addition to the electrical and magnetic wormhole solutions, we
also show that the validity of the duality relation of the non-minimal model
Sert13MPLA . The duality transformation is $(F,*F)\rightarrow(*YF,-YF)$ and
$(Y\rightarrow\frac{1}{Y})$ and it is the modified version of the duality
transformation of Maxwell equations that is $(F,*F)\rightarrow(*F,-F)$.

##  IV Conclusion

In general relativity, wormhole solutions of Einstein field equation entail
exotic matter which violates the null energy condition $\rho+p\geq 0$. The
main problem is to construct a realistic wormhole geometry as motivation for
many studies. However, the field equations of modified theories of gravity can
accept normal matter, provided that the modified null energy condition
$\rho^{eff}+P_{i}^{eff}\geq 0$ is still violated Hochberg1997 ; Lobo2009 ;
Garcia2010 ; Capozziello2012 ; Francisco2013 ; Harko2013 ; Capozziello2014 ;
Capozziello2015 . Thus, in order to have the wormhole geometries, the role of
the exotic matter plays the modified terms, which is the non-minimal couplings
in this non-minimal model. Thus, we have investigated wormhole solutions in
the non-minimally coupled gravity model with electromagnetism. We have
constructed traversable wormhole geometries with realistic pressure-less dust
matter for the different values of the parameter $k$ in the range $k<-1$. We
find that the matter satisfy all the energy conditions in the range of the
parameter. But the solutions in the range $1<k<\infty$ lead to wormhole
geometries involving exotic matter. The solutions may have electric field or
magnetic field optionally and they may be transformed into each other by the
duality transformation.

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