# Detecting reconnection sites using the Lorentz Transformations for
electromagnetic fields

Giovanni Lapenta  Mathematics Department, KULeuven University, Belgium.  Space
Science Institute, Boulder, Colorado, USA.

(Received 2020 December 13; Revised 2021 February 21; Accepted 2021 February
25)

###### Abstract

We take a pragmatic definition of reconnection to find locations where a
reconnection electric field causes a $\mathbf{E}\times\mathbf{B}$ drift that
carries two components of the magnetic field towards their elimination. With
this in mind as our target, we observe that such locations can be found using
a new indicator: the velocity of the Lorentz transformation that eliminates
two components of the local magnetic field. Serendipitously, the indicator
naturally becomes subluminal in the close proximity of a point where two
components of the magnetic field vanish and it is hard zero at the vanishing
location. Everywhere else the velocity of this Lorentz frame change far
exceeds the speed of light. This property can be quickly applied in practice
because computing the frame change is a local operation that requires only the
knowledge of the local magnetic and electric field: it can be applied in a
simulation or in observational data from a field instrument. We further show
that the points identified can be classified in 6 categories that extend the
usual types of magnetic nulls to the case of 3D reconnection in presence of a
guide field. The approach is used to identify secondary electron scale
reconnection sites in a turbulent outflow from a primary reconnection site in
a highly resolved massively parallel fully kinetic particle in cell
simulation. Numerous points are found and their detailed analysis is reported.

reconnection — Lorentz transformation — Jacobian matrix — kinetic PIC
simulation

##  1 Introduction

Reconnection is easiest to visualize in 2D. The magnetic field in the plane
reverses sign at a reconnection point and the $\mathbf{E}\times\mathbf{B}$
drift carries the frozen-in plasma toward the reconnection site where the
magnetic field becomes zero right at the location of reconnection (Biskamp,
2000). For its appearance, the reconnection point is called an _x-point_. In
this process, energy is converted from magnetic to kinetic, not just at the
x-point itself but as part of the whole process of inflow and outflow.

The x-points are easily identified in 2D as the locations where the in-plane
magnetic field vanishes, $\\{B_{j}\\}_{j=1,2}=0$, its null points. There are
two types of null points, the aforementioned x-points and the o-points. The
_o-points_ are formed in the outflow of reconnection when two connection
outflow merge to form a bundle of magnetic field lines and plasma called
plasmoid (Finn & Kaw, 1977) or within turbulent outflows (Fu et al., 2017).
The two types of null points are easily distinguished using the two
eigenvalues of the Jacobian matrix $J_{\mathbf{B}}=\partial B_{i}/\partial
x_{j}$ in the 2D plane ($\\{x_{j}\\}_{j=1,2}=(x,y)$) (Lau & Finn, 1990). We
recall that the Jacobian matrix for the magnetic field must have the sum of
the eigenvalues zero, since the trace of the matrix is the divergence of the
magnetic field, obviously zero. We then have two possibilities: First, we have
two real eigenvalues of opposite sign (with their sum zero): this is the case
of the x-points. Second, the eigenvalues are complex conjugate, in this case
we have an o-point. A similar classification can be obtained using the out of
plane component of the vector potential, $A_{z}$: in this case the x and o
points are obtained as null points of the gradient $\nabla A_{z}=$ and the
distinction between x and o points from the eigenvalues of the Hessian matrix
(Servidio et al., 2009).

In 2D, the presence of an out of plane magnetic field does not complicate the
story (Fu et al., 2016) and we can distinguish _antiparallel reconnection_
when the asymptotic reconnecting field is entirely on the plane from
_component reconnection_ when there is a out of plane field, called _guide
field_. In both cases the analysis is the same but in component reconnection
the x and o points no longer are nulls in the true sense: they are only nulls
for the in-plane field.

The extension to 3D is far from easy (Priest & Forbes, 1999). A theoretical
basis is provided by the condition required to break field line connectivity
(Hesse & Schindler, 1988) that can be used as a practical indicator of
reconnection in 2D (Goldman et al., 2016) and in 3D (Lapenta et al., 2015)
kinetic simulations. The topological equivalent to o and x points are the 3D
null points, $\\{B_{j}\\}_{j=1,2,3}=0$ (Lau & Finn, 1990) but 3D reconnection
does not happen only near null points (Priest & Démoulin, 1995) (a condition
reminiscent of 2D anti-parallel reconnection), and more complex topological
configurations are possible, for example with reconnection developing along
separator lines that are lines connecting two null points (Parnell et al.,
2010) (a conditions reminiscent of 2D guide field reconnection) or within flux
ropes with significant twist as suggested based on magnetospheric observations
(Ergun et al., 2016) and solar coronal modesl (Török & Kliem, 2005) or typical
of fusion devices (Freidberg, 2008) and expected in many astrophysical systems
like jets (Lapenta et al., 2006; Li et al., 2006) reproduced also in
laboratory experiments (Intrator et al., 2009; Bellan, 2018).

We take here a pragmatic approach: we define reconnection, even in 3D, as a
location where the $\mathbf{E}\times\mathbf{B}$ drift carries the frozen-in
plasma towards a reconnection site where two magnetic field components become
zero right at the location of reconnection. This definition, as it turns out,
can much more easily be generalized to 3D using the Lorentz transformation of
the Maxwell equations.

The Lorentz transformations of the electromagnetic field (Misner et al., 1973)
can always be used to find a frame where the electric and magnetic fields are
parallel (Landau & Lifshitz, 1975; Beklemishev & Tessarotto, 1999). However,
this frame change does not keep the direction of the electric field invariant.
We show here that it is in fact often not possible to eliminate the components
of the magnetic field in the plane perpendicular to the direction of the
electric field in the laboratory frame. A real solution cannot be found and
the solution becomes complex with a speed exceeding the speed of light.

The key finding of the present work is that the speed of the frame change that
eliminates two components of the magnetic field drops to subluminal speed in
close proximity of a reconnection site, becoming exactly zero right at it. It
is perhaps not surprising that the Lorentz frame change that removes two
components of the magnetic field can be used to identify the process of
reconnection defined also as the mechanism that eliminates two components of
the magnetic field via an $\mathbf{E}\times\mathbf{B}$ drift. The logic is not
circular because only at a site with ongoing reconnection defined as above
such a Lorentz transformation exists for subluminal speeds.

The key suggestion of the present manuscript is that this property of the
speed of the Lorentz transformation that eliminates two components of the
magnetic field can be used as a convenient way to detect reconnection sites.
Reconnection sites defined as locations where the $\mathbf{E}\times\mathbf{B}$
drift brings two components of the magnetic field to their annihilation.

We introduce then an indicator (in short called _Lorentz frame indicator_)
defined as the local speed of the Lorentz transformation that eliminates the
two components of the magnetic field normal to the electric field and observe
that this results in a very selective indicator that is lower than the speed
of light only in very few locations. The computation of this indicator is of
similar complexity to the identification of null points but detects also other
types of more complex 3D reconnnection topologies. The new indicator is
simpler compared to other methods to find these complex 3D magnetic
reconnection topologies such as the squashing factor (Titov et al., 2002; Finn
et al., 2014) and the field integrated parallel potential (Lau & Finn, 1990;
Richardson & Finn, 2012) that requires complex integration along the field
lines and even of the agyrotropy (Scudder & Daughton, 2008) that requires one
to compute the eigenvalues of the pressure tensor. The proposed indicator is a
local operation that requires only the knowledge of the local magnetic and
electric field: it can be applied in a simulation or in observational data
from a field instrument.

When a suspected reconnection point is identified, we show that it can be
classified in 6 categories that extend the usual 2D classes of o-points and
x-points. The x-points correspond to reconnection sites in their proper
definition and some correspond to the o-points. Some are yet more complex 3D
topologies. We show a simple analysis of the magnetic field properties that
can provide the type.

To demonstrate the approach proposed, we identify secondary electron scale
reconnection sites in a turbulent outflow from a primary reconnection site in
a highly resolved massively parallel fully kinetic particle in cell (PIC)
simulation. We find numerous points and analyze the processes happening in
their vicinity. We compute the agyrotropy, the parallel electric field and
study the electron flow relative to the ion flow in the vicinity.

The manuscript is organized in Section 2 that introduces the proposed
indicator of reconnection based on on Lorentz transformations. Section 3
details the classification procedure that should be used to identify the types
of points found by the Lorentz frame indicator using the Jacobian of the
magnetic field on the reconnection plane. Section 4 provides an overview of
how the Lorentz indicator behaves in the well known and understood 2D case:
this is of course a case where the new method is not really needed but it is
interesting to see how it behaves where the solution is well known. Section 5
shows the Lorentz frame indicator in the simpler but illustrative case of 2D
reconnection where a local analytical formula can be obtained for the
neighborhood of a point of interest. Section 6 demonstrates the application to
a massively parallel particle in cell simulation showing what reconnection
points are found and analyzing their properties. Section 7 summarizes our
findings and draws our conclusions.

##  2 Detection based on the Lorentz transformation of electromagnetic fields

Let us consider one specific point, $\mathbf{x}_{0}$, and a specific time,
$t_{0}$, and there define the Lorentz transformations for a frame moving with
a velocity
$\mathbf{v}_{L}=\aleph\mathbf{E}(\mathbf{x}_{0},t_{0})\times\mathbf{B}(\mathbf{x}_{0},t_{0})$,
where $\aleph$ is a free scalar factor to be determined. We choose a system of
coordinates in the laboratory frame that more easily highlights the direction
of $\mathbf{v}_{L}$. We choose our unit normals as:
$\hat{\mathbf{n}}_{3}=\mathbf{E}(\mathbf{x}_{0},t_{0})/E(\mathbf{x}_{0},t_{0})$,
$\hat{\mathbf{n}}_{2}=\mathbf{v}_{L}/v_{L}$ and
$\hat{\mathbf{n}}_{1}=\hat{\mathbf{n}}_{2}\times\hat{\mathbf{n}}_{3}$. The
corresponding coordinate system is $(\xi_{1},\xi_{2},\xi_{3})$. Note that this
choice differs for convenience from the classical choice in plasma physics of
aligning the third axis with $\mathbf{B}$: in this coordinate system by
construction, $\mathbf{B}$ has no component in the direction
$\hat{\mathbf{n}}_{2}$ but it has a component in the other two:
$B_{\xi_{3}}=B_{\parallel}$ _where parallel is meant with respect to the
electric field $\mathbf{E}$_ and $B_{\xi_{1}}=B_{\bot}$ _where $\bot$ is meant
again with respect to the electric field $\mathbf{E}$_. Figure 1 shows the
axis choice for a case of an x-point and o-point.

Figure 1: Orientation of the axis of the reference frame moving at speed
$\mathbf{v}_{L}$ chosen to compute the Lorentz transformation that eliminates
the magnetic field at the central location of the axis system. We choose our
unit normals as: $\hat{\mathbf{n}}_{3}=\mathbf{E}/E$,
$\hat{\mathbf{n}}_{2}=\mathbf{v}_{L}/v_{L}$ and
$\hat{\mathbf{n}}_{1}=\hat{\mathbf{n}}_{2}\times\hat{\mathbf{n}}_{3}$. The
corresponding coordinate system is $(\xi_{1},\xi_{2},\xi_{3})$. The example
shows a o-point (panel a) and a x-point (panel b) formed by field lines laying
on the $(\hat{\mathbf{n}}_{1},\hat{\mathbf{n}}_{2})$ plane.

The Lorentz transformations for the electromagnetic fields in these
coordinates from the laboratory frame to the frame moving with velocity
$\mathbf{v}_{L}$ give (Lorentz, 1904; Misner et al., 1973):

$\begin{split}B_{\xi_{1}}^{\prime}&=\gamma\left(B_{\bot}-\frac{\aleph
E^{2}}{c^{2}}B_{\bot}\right)\\\ B_{\xi_{2}}^{\prime}&=\gamma B_{\xi_{2}}=0\\\
B_{\xi_{3}}^{\prime}&=\gamma B_{\xi_{3}}\end{split}$ (1)

where $c$ is the speed of light and $\gamma$ is the Lorentz factor
$\gamma=1/\sqrt{1-v_{L}^{2}/c^{2}}$. The vector
$\mathbf{v}_{L}\times\mathbf{E}$ is entirely directed along
$\hat{\mathbf{n}}_{1}$ by choice of the coordinate system (and this explains
why we chose them as we did).

We can eliminate the perpendicular magnetic field component
$B_{\xi_{1}}^{\prime}$ in the moving frame if we choose $\aleph=c^{2}/E^{2}$:

$\mathbf{v}_{L}=c^{2}\frac{\mathbf{E}\times\mathbf{B}}{E^{2}},$ (2)

as can be shown with simple algebraic manipulations substituting eq. (2) into
the Lorentz transformations, eq. (1). In this case, by construction, the
parallel (to the electric field) magnetic field remains but the perpendicular
magnetic field vanishes. This outcome is precisely what happens at a
reconnection site where the magnetic field normal to the reconnection electric
field is carried towards its elimination.

Note that this choice of transformations differs from the textbook frame
change that brings the electric and magnetic field to be parallel (Landau &
Lifshitz, 1975). In that case a frame change is always possible and it leads
to the relativistic version of the $\mathbf{E}\times\mathbf{B}$ drift:

$\mathbf{v}_{rel}=\frac{\bm{\gimel}}{2\gimel^{2}}\left(1-\sqrt{1-4\gimel^{2}}\right),$
(3)

with $\bm{\gimel}=\mathbf{E}\times\mathbf{B}/(E^{2}+B^{2})$ (Beklemishev &
Tessarotto, 1999).

In our choice we restrict the transformation to eliminate the magnetic field
in the plane normal to the laboratory electric field: the Lorentz
transformation can become impossible because the fields become complex and the
speed $v_{L}>c$. In fact in most practical situations, $v_{L}$ exceeds the
typical $\mathbf{E}\times\mathbf{B}$ drift by $B^{2}/E^{2}\approx c^{2}$, far
exceeding the speed of light by orders of magnitude and making this type of
coordinate transformation impossible. However, this typical condition is not
true in the vicinity of magnetic reconnection: there the coordinate
transformation that eliminates the local $B_{\bot}$ becomes subluminal,
$v_{L}<c$ and in fact at reconnection sites proper $v_{L}=0$.

This property of the Lorentz transformations can be understood on an intuitive
basis: Reconnection is a process where a reconnection electric field causes a
motion that brings the magnetic field lines and the plasma towards a region
where the magnetic field energy is converted into kinetic energy and the
magnetic field lines break (Zweibel & Yamada, 2016). By breaking we mean the
magnetic field in the line, as it passes through a reconnection site, in that
particular instant and location, disappears so that the line stops there and
reforms outside the reconnection site. With this interpretation of
reconnection, it is then clear that as one approaches the reconnection site
the speed of the Lorentz transformation that eliminates the magnetic field
(while keeping the electric field direction unchanged) locally drops and
reaches zero when the field is actually annihilated in the laboratory frame.

This is best visualized first in 2D (Biskamp, 2000; Priest & Forbes, 1999),
see Fig. 1. Magnetic field lines in this case have 2 singular points, x-points
and o-points. Both can be created by reconnection: the x-point is the location
of magnetic field breaking and reforming and the o-points form in the outflow
of reconnection when two reconnection jets form a plasmoid between two
reconnection sites. The x-point and the o-point become null points of the in-
plane magnetic field. They are, however, not necessarily null points as an out
of plane magnetic field can be present. In this situation, Sect. 4 shows how
the indicator $v_{L}$ behaves in the vicinity of the two types of points,
dropping always below the speed of light in the vicinity of the singular
points, being exactly zero at the x and o points. This is true regardless of
the presence of an out of plane magnetic field (called guide field in the
literature). In the notation used here the guide field is field component
$B_{\xi_{3}}$

This peculiar property of the Lorentz transformation becomes especially
convenient as a reliable indicator of reconnection in the generalisation to 3D
dimensions where component reconnection in absence of true null points cannot
be easily identified.

The Lorentz transformation-based indicator makes the task easy. Finding these
regions becomes simply the task of identifying where the $v_{L}$ indicator
drops to subluminal speeds, and to zero at the reconnection site proper. When
we consider the $v_{L}$ indicator in 3D, the velocity of the Lorentz
transformation that brings the magnetic field to zero is again dropping to
subluminal speed in the vicinity of true 3D nulls and hard zero right at null
points proper (for the obvious reason that the laboratory frame already has
zero magnetic field). But the speed $v_{L}$ drops to subluminal and zero also
at separators or at any location of our pragmatic definition of reconnection,
i.e. at locations where the $\mathbf{E}\times\mathbf{B}$ brings two magnetic
field components to be annihilated, regardless of the size of the field in the
third direction (guide field).

Short of studying the whole topological connectivity of the magnetic field
(Bungey et al., 1996; Priest et al., 1997), there is no simple way to
determine the presence of reconnection happening without nulls. Nulls can be
detected in 3D but other types of reconnection require to track the full
complexity of the magnetic topology in 3D. This is obviously not possible in
local measurements taken from a spacecraft and it is highly inconvenient even
in simulations. The Lorentz frame indicator $v_{L}$, instead, is a local
measure that can be computed point by point and drops to subluminal speeds in
the vicinity of any 3D feature extending to 3D component reconnection. The
Lorentz frame indicator can then easily identify the 3D equivalent of an
x-point (the x-line) or an o-point (a flux rope). At the actual topological
singularity of the magnetic field, the indicator becomes hard zero. The latter
is, of course, unlikely to happen exactly at any point on a computational grid
even in a highly resolved simulation. But locating the exact point where
$v_{L}$ is hard zero is not needed in practice because a whole finite size
region around an x or o point (and their 3D extension) become subluminal,
greatly facilitating the detection in simulations and observations.

##  3 Classification of 3D Lorentz subluminal regions

Once a region of subluminal $v_{L}$ Lorentz frame indicator is found, the
determination still needs to be made as to whether it is a x-point or o-point
like feature. This task is easy in 2D. As noted above, in this case the
Jacobian has zero trace, i.e. the magnetic field has zero divergence. X-points
correspond to a Jacobian matrix with two real eigenvalues (they forcibly have
opposite sign because the trace of the matrix, corresponding to the sum of the
eigenvalues, is zero). O-points correspond to a Jacobian matrix with complex
conjugate eigenvalues.

The situation in 3D is more complex. The Lorentz frame indicator finds a point
where a subluminal transformation eliminates two components of the magnetic
field and a direction $\hat{\mathbf{n}}_{3}$ for the reconnection electric
field. In the perpendicular plane
$(\hat{\mathbf{n}}_{1},\hat{\mathbf{n}}_{2})$, the electric field causes the
$\mathbf{E}\times\mathbf{B}$ drift that conveys the magnetic field lines
projected in the pane $(\hat{\mathbf{n}}_{1},\hat{\mathbf{n}}_{2})$ to their
break and reconnection. However there is an out of plane magnetic field,
$B_{\xi_{3}}$, called guide field and aligned with the reconnection electric
field. The in-plane $(\xi_{1},\xi_{2})$ field than does not have zero
divergence: the full 3D field has zero divergence, including the contribution
of $\partial B_{\xi_{3}}/\partial\xi_{3}$.

The vector field in the $(\hat{\mathbf{n}}_{1},\hat{\mathbf{n}}_{2})$ plane
has then a more complex structure. Around a in plane null a generic 2D vector
field (not endowed with a zero divergence) has 6 different possibilities
(Helman & Hesselink, 1989) shown in Fig. 2.

Figure 2: Six possible configurations around a null of a generic vector field
on a 2D plane. The six configurations depend on the eigenvalues of the
Jacobian matrix. For each type of 2D null, the real and imaginary part of the
two eigenvalues of the Jacobian matrix are indicated and the standard name is
given as commonly used in the relevant literature (Helman & Hesselink, 1989).

From the eigenvalues of the Jacobian matrix on this 2D plane
$J_{B_{\xi_{1},\xi_{2}}}=\\{\partial
B_{\xi_{i}}/\partial\xi_{j}\\}_{(i,j=1,2)}$, we can find for each location
flagged by the Lorentz frame indicator what is the type of behavior of the
vector field.

The possibility for non-zero divergence of the vector field on the 2D plane
$(\hat{\mathbf{n}}_{1},\hat{\mathbf{n}}_{2})$ has increased the diversity of
behavior. We recognize immediately the closest relatives to the x-point as the
saddle and of the o-point as the center. These are the only two points still
possible in a divergence free field. The other 4 classes of points are allowed
by the non zero 2D divergence (compensated for by the derivatives in third
dimension). These points have a configuration that is reminiscent of that of
spine and spiral nulls but they are not nulls in 3D in the sense discussed in
Greene (1988); Lau & Finn (1990). The types of reconnection uncovered by the
Lorentz transformation include all the nulls but extend the set to many more
cases with more complex 3D topology.

##  4 Speed of the Lorentz frame eliminating the magnetic field in the
vicinity of a 2D magnetic topological singularity

Before considering the 3D full complexity, it is educational to consider first
the well known 2D case where we know by immediate visual inspection where an x
point or an o point is, just by looking at the out of plane component of the
vector potential (aka the flux function). It is useful to explore in 2D the
vicinity of a x-point and o-point the behavior of the velocity required by the
Lorentz transformations to eliminate the local value of the magnetic field.

When we require the 3D electric and magnetic fields to depend only on the 2D
variables $x,y$ (sometimes this condition is referred to as 2.5D), the
topological conditions of reconnection become rigorous and can be easily
characterized using the out of plane vector potential component $A_{z}(x,y)$.
In the vicinity of an o-point of x-point, a similarity solution can be
constructed to represent the local magnetic topology (Imshennik & Syrovatskii,
1967):

$A_{z}(x,y)=\frac{B_{0}d_{i}}{2}\left(\frac{x^{2}}{d_{x}^{2}}\pm\frac{y^{2}}{d_{y}^{2}}\right)$
(4)

where we assumed the location of the magnetic topological feature in the
origin of the coordinates. The sign between the two terms determines whether
the magnetic point is of type o (+) or x (-). The values of $d_{x}$ and
$d_{y}$ determine the axis of the hyperbolic (x-point) or elliptic (o-point)
magnetic field lines. In the case of o-points, their ratio determines the
aspect ratio of the flux rope surrounding the point. In the case of x-point,
the ratio determines the reconnection geometry and is typically of order 10, a
consequence of the well known ”universal” fast kinetic reconnection rate (Shay
et al., 1999; Birn & Priest, 2007).

The magnetic field can then be computed simply as
$\mathbf{B}=-\hat{\mathbf{z}}\times\nabla A_{z}+\hat{\mathbf{z}}B_{g}$. In
presence of a local reconnection electric field
$\mathbf{E}=\hat{\mathbf{z}}E_{r}$ (assumed uniform as required for steady
reconnection) one can compute easily the Lorentz frame indicator as:

$v_{L}(x,y)=cB_{0}d_{i}\frac{\sqrt{x^{2}/d_{x}^{4}+y^{2}/d_{y}^{4}}}{E_{r}}$
(5)

Figure 3 shows the results for a typical choice of $d_{x}=10d_{i}$,
$d_{y}=d_{i}$ and $E_{r}/v_{A}B_{0}=0.1$. As can be seen, in the central o or
x point the speed of the Lorentz transformation is zero but it increases away
from it. The speed becomes luminal at an ellipse around the point: all points
within the ellipse $v_{L}=c$ have subluminal Lorentz transformation speed and
can be used to detect the vicinity to the o or x point. This is an important
property because it is hard to find the hard zero for $v_{L}$ but it is much
simpler to determine areas where the speed becomes subluminal. Figure 3 shows
these regions to be tightly confined around x and o points.

a)

b)

Figure 3: Speed of the Lorentz transformation that eliminates the local field.
The speed is normalized to the speed of light $c$. The dotted line correspond
to the frame speed equal to the speed of light. The solid black lines are the
magnetic field lines. The case of the o-point (panel a) and x-point (panel b)
are shown. The Lorentz frame indicator is identical in both cases.

From the perspective of the Lorentz frame indicator there is no difference
between o and x points.

Note that this derivation assumed the presence of only a out of plane
reconnection electric field. This is true only in a small region of electron
skin depth size in the vicinity of the reconnection point. At larger
distances, there is an in plane electric field (Chen et al., 2011).The
extension to any orientation of the electric field is straight-froward. In 2D,
the process of reconnection is constrained by the geometry to remain within
the 2D plane. We need now to define the Lorentz transformation that eliminates
the in-plane field only: $\mathbf{B}_{2D}=-\hat{\mathbf{z}}\times\nabla
A_{z}$:

$\mathbf{v}_{L}=c^{2}\frac{\mathbf{E}\times\mathbf{B}_{2D}}{E^{2}},$ (6)

and the modulus of the frame speed is computed as:

$v_{L}=c^{2}\frac{\sqrt{E^{2}B_{2D}^{2}-(\mathbf{E}\cdot\mathbf{B}_{2D})^{2}}}{E^{2}}$
(7)

that is obviously zero when a point is a null for the in plane magnetic field.
The 2D case for the Lorentz frame indicator is therefore directly related to
the standard method based on finding the nulls of the in plane field (Servidio
et al., 2009).

As an illustration of how the Lorentz indicator looks in a well known 2D
reconnection case, we consider here a case often used in the literature: a
current sheet where the tearing instability (Coppi, 1965) develops in many
sites that coalesce into progressively larger islands (Finn & Kaw, 1977).
Specifically we consider a case of the type reported in (Eriksson et al.,
2014, 2015): the initial state is a force free equilibrium:

$\begin{split}B_{x}(y)&=B_{0}\tanh(y/\delta)\\\
B_{z}(y)&=\Bigl{(}B_{0}^{2}+B_{g}^{2}-B_{x}^{2}\Bigr{)}^{1/2}\end{split}$ (8)

with a very narrow current sheet $\delta/d_{i}=0.1$ (to promote quick short
scale tearing) and uniform plasma pressure and density. The plasma properties
represent the the solar wind Custer observation reported in where more details
can be found (Eriksson et al., 2015): guide field $B_{g}/B_{0}=3.9$ and
temperature $T_{e}=T_{i}=10$ eV. We use $m_{i}/m_{e}=256$ and a domain
$200\times 30d_{i}$. The problem is simulated using the code iPic3D (Markidis
et al., 2010)111Available at www.ipic3d.org using $2560\times 384$ cells and
770 billion particles.

As it turns out, the speed of the Lorentz null-B frame is vastly larger than
the speed of light in most of the domain: several orders of magnitude.
Visualization of a field having values of many orders of magnitude except for
a few near singular locations where it drops near zero is visually arduous. We
prefer then to renormalize this indicator as:

$\mathscr{L}=\log_{10}\left(\frac{v_{L}}{c}\right)$ (9)

The advantage of using the indicator $\mathscr{L}$ is that it varies more
smoothly, being zero at the speed of light, positive for superluminal speeds
and negative for subluminal speeds.

Figure 4 shows the Lorentz indicator in a sub-domain (blown up to see more
clearly the features). We show two full islands and half of two neighboring
ones and three x points in between. As it can be seen the Lorentz indicator
identifies all of these feature as well as providing considerable information
about the local structure. Strictly speaking the reconnection site proper is
defined where $v_{L}=0$. This statistically happens in between grid points but
the grid point closest to it will have the lowest $v_{L}$. Note that the
information of the Lorentz indicator is not just to quantitatively count the
reconnection sites but it also provides a qualitative information on how
active the reconnection is and how widely it affects its vicinity: more active
reconnection regions appear as brighter, wider patches in the indicator.

Figure 4: False color representation of the Lorentz indicator $\mathscr{L}$ in
a sub-domain $x=[40,110]$ of a simulation of an initial force free current
sheet in the solar wind (Eriksson et al., 2015). Superimposed we show the
isocontours of the vector potential $A_{z}$. Note the asymmetric colorscale
chose to make the area around small vlaues of $v_{L}$ more visible. The pixels
with $v_{L}<1$ (amaranth to garnet) is hardly visible if the surrounding
region is not highlighted too.

Figure 4 also show the usual vector potential. In 2D the established procedure
is to use the vector potential to find x and o points. It is informative to
check what the Lorentz indicator does in the proximity of every topological
feature identified by the vector potential. As can be seen, the o-point at the
island centers and the reconnection x-points between magnetic islands are all
captured.

As noted above, the proposed Lorentz indicator is no replacement for null
point finders. Its aim is different: it wants to find locations where an
electric field carries magnetic field lines, breaks them and pulls them apart.
This in 2D can only happen in a x-point (or in a o-point where magnetic field
lines can be piled up). In 3D other possibilities arise and in the next
section we will see what happens in a 3D simulation.

##  5 Detection of reconnection in simulated 3D turbulent reconnection
outflows

We test the reconnection indicator for the Lorentz null magnetic field frame
transformation in a turbulent outflow of a 3D reconnection region. Previous
work has identified that even in initially laminar conditions, the process of
reconnection in 3D leads to outflows rich in free energy due to shears,
superalfvenic electron beams, thin currents. These sources of energy cause
secondary instabilities: drift (Divin et al., 2015), Kelvin-Helmholtz(Fermo et
al., 2012), interchange (Nakamura et al., 2002), Buneman (Goldman et al.,
2008), (drift)-tearing (Daughton et al., 2011). The combined effect of these
instabilities is to make the flow turbulent. The turbulent properties of the
outflow have been investigated in previous 3D simulations (Pucci et al., 2017)
and shown to agree remarkably with in situ measurements of the turbulent
spectrum of electric and magnetic field observed by the Cluster mission
(Eastwood et al., 2009).

We use here a simulation very similar to that reported in the aforementioned
reference by Pucci et al. (2017). Previous work has pointed out that the
turbulent outflow is the possible host for the formation of secondary
reconnection sites (Lapenta et al., 2015). Often reconnection affects only
electrons but leaving ions unaffected (Phan et al., 2018; Stawarz et al.,
2019). Analyses based on kinetic indicators of reconnection, such as slippage,
non-ideal terms of the generalized Ohm’s law and agyrotropy support that
conclusion (Lapenta et al., 2015). So far direct inspection of the suspect
sites was the main way to investigate whether the points flagged by these
indicators were likely to be real reconnection sites or not. This worked in
presence of null points, but with difficulty and relying on the painstaking
selection of multiple field lines to follow. However, secondary reconnection
sites in reconnection outflow are likely to be often unrelated to nulls and
resemble more the twisting of field lines wrapped in flux ropes, as described
in Ergun et al. (2016). The visual inspection of field lines is of limited use
because at any given time the field lines all form bundles whose internal
rotation is linked to reconnection but makes it impossible to observe a field
line before and after breaking up and reconnecting.

The Lorentz subluminal frame change indicator changes this situation. We can
compute the frame velocity leading to the elimination of the magnetic field in
the plane $(\xi_{1},\xi_{2})$ and flag the regions where the speed drops below
the speed of light.

To study the application of the Lorentz frame indicator we consider a 3D fully
kinetic simulation of reconnection outflows conducted again with the iPic3D
PIC code (Markidis et al., 2010). We consider the same type of simulation
recently reported in (Lapenta et al., 2015): it uses a standard Harris
equilibrium (Harris, 1962):

$\displaystyle\mathbf{B}(y)=B_{0}\tanh(y/L)\hat{\mathbf{x}}+B_{g}\hat{\mathbf{z}}$
(10) $\displaystyle n(y)=n_{0}\operatorname{sech}^{2}(y/L)+n_{b}$ (11)

defined by the thickness $L/d_{i}=0.5$ and with the parameters
$m_{i}/m_{e}=256$, $v_{the}/c=0.045$, $T_{i}/T_{e}=5$. A guide field
$B_{g}/B_{0}=0.1$ and a background plasma of $n_{b}/n_{0}=0.1$ is imposed,
where $B_{0}$ is the asymptotic in plane field and $n_{0}$ is the peak Harris
density. To convert code units to physical units we chose as reference:
$B_{0}=20$\mathrm{n}\mathrm{T}$$ and
$n_{0}=$1\text{\times}{10}^{6}$$\mathrm{m}^{-3}$$.

The coordinates are chosen with the initial Harris magnetic field along $x$
with size $L_{x}=40d_{i}$, the initial gradients along $y$ with
$L_{y}=15d_{i}$. The third dimension, where the initial current and guide
field are directed, is initially uniform with $L_{z}=10d_{i}$. Open boundaries
are imposed in $x$ and $y$ and periodicity is assumed along $z$.

The conditions are identical to what is reported in (Pucci et al., 2017) but
the grid resolution is higher with 1200x450x300 cells, resolving better the
electron skin depth, in the reconnecting background plasma $\Delta x\approx
0.2d_{e,b}$ (the initial Harris plasma is quickly swept away by reconnection).
The time step resolves also well the electron cyclotron frequency, even in the
strongest field, $\omega_{ce,Bmax}\Delta t\approx 0.3$ (and even better in
more average fields). Each plasma species is described by two populations of
$5^{3}$ computational particles per cell, one represents the Harris density
($n_{0}\operatorname{sech}^{2}(y/L))$ and the other the background $n_{b}$. A
total of 81 billion particles and 162 million cells is used in a supercomputer
parallel topology of 50x30x20 (i.e. 30,000) processors.

Reconnection is initialized with a uniform perturbation along $z$, very
localized in the center of the domain (Lapenta et al., 2015). The evolution of
the system is described in details in our previous works (Lapenta et al.,
2015, 2016; Pucci et al., 2017; Lapenta et al., 2020): the initial
perturbation quickly produces a primary reconnection region that initiates the
evolution leading to a strong outflow that interacts with the surrounding
plasma forming pileup regions. The outflow develops turbulence (Vapirev et
al., 2013; Divin et al., 2015; Pucci et al., 2017; Lapenta et al., 2020) and
secondary reconnection sites (Lapenta et al., 2015; Liu et al., 2018). We
report here the conditions at cycle 18000, when we stop the simulation as the
pile up region is reaching the boundary and about to exit.

The resulting data-set gives us information about all fields and moments on
the points of the simulation grid. In each, we can compute the Lorentz frame
indicator $\mathscr{L}$. A volume rendering is shown in Fig. 5.

Figure 5: Volume rendering of the Lorentz frame indicator $\mathscr{L}$. A
negative value implies subluminal speed for the Lorentz transformation that
eliminates the magnetic field in the reconnection plane. A myriad locations in
the outflow as well as the main central reconnection regions are flagged as
locations where the Lorentz transformation can eliminate the magnetic field at
subluminal speed.

A very large number of tiny locations are identified. Visual inspection is
challenging and a hawkish eye is needed to spot the smallest. The color scale
used is the best among the many possible to highlight the regions but the task
is still not for every eye. Some of the Lorentz subluminal regions shown in
Fig. 5 are indeed just one pixel.

In fact, in principle, the location of a singular point is a point only,
infinitesimally small. Obviously the likelihood of one such point sitting
exactly on a grid location is vanishingly small, but luckily, as shown in Sect
4, the whole neighborhood of a magnetic point with singular topology has
subluminal $v_{L}$ and we can still identify them by considering all grid
points where the speed is subluminal: $\mathscr{L}<0$.

For this reason, we resort to automatic sorting. We task a computer algorithm
with identifying every point on the grid where the Lorentz transformation
speed that eliminates the magnetic field becomes subluminal. We consider the
regions distinct only if the points are at least one electron skin depth apart
and in each region we take the point with the minimal speed. Note that this
method takes only into account the grid points. A possible improvement for
future work is to interpolate the fields within each cell and look for the
minimum Lorentz speed anywhere within the cell. This is similar to the
approach needed to find all null points in a grid-based field: the nulls are
within the cells not at the grid nodes (Haynes & Parnell, 2007; Fu et al.,
2015; Olshevsky et al., 2016). For the Lorentz frame indicator the need is
less acute because the indicator remains subluminal in an electron skin depth
scale region making the identification easier. We limit ourselves here only to
the grid points because the results do not seem to indicate the need for a
more refined sub-grid interpolation. However, this conclusion is valid only in
the present example where the electron scale is well resolved. In simulations
that do not resolve the electron skin depth well, the grid point-based
algorithm can miss regions of subluminal indicator that never comprise a grid
point and are fully enclosed within a single cell. In that case it becomes
necessary to rely on subgrid interpolations to locate the true minimum of
$v_{L}$ within each cell.

In the search, we exclude the central region $x/d_{i}\in[18,22]$ shown in Fig.
5 as a large area of subluminal speed. This is the central reconnection region
caused by our initial perturbation: we know it is there because we put it
there, we look only for the secondary reconnection regions.

The analysis finds in this case 54 secondary points, listed in Table 1. Of
these, the vast majority are x-like saddle points (39), 6 are attracting foci,
5 repelling foci, 3 are repelling and 1 is an attracting point. There is no
perfect center: a center is the limit of attracting and repelling foci when
the real part is exactly zero, or in other words when the divergence of the in
plane magnetic field component is exactly zero. In 3D, the field will always
have a minimum of variation in the third dimension and perfect centers can’t
be expected. However, site number 5 is a repelling focus with relatively small
real part, resulting in a nearly visual center (see Fig. 8 below).

Figure 6: False color rendering of the electron speed $V_{ex}(x,y,z=L_{z}/2)$
in the mid-plane along $z$. The location and type of the points where the
Lorentz frame indicator becomes subluminal is shown, corresponding to the
points of singular topology listed in Table 1. Panel a gives a complete
overview and panel b and c blow up the two busiest separatrix regions:top-left
and bottom-right. The point detected are numbers as in Table 1 and marked with
their type: center (C), saddle (S), repelling (R), attracting (A), repelling
focus (P) and attracting focus (N).Note that the points are not on the plane
show and each have a different $z$ but are projected on the plane shown.

Site | Type | $x/d_{i}$ | $y/d_{i}$ | $z/d_{i}$ | $B_{\xi_{3}}/B_{0}$ | $\displaystyle\frac{B_{\xi_{3}}}{||J_{B}||_{2}d_{i}}$
---|---|---|---|---|---|---
1 | S | 35.80 | 13.00 | 6.87 | 0.80 | 7.19
2 | S | 33.03 | 4.53 | 7.67 | 0.66 | 3.22
3 | S | 34.60 | 4.40 | 3.40 | 0.69 | 4.40
4 | S | 6.00 | 2.67 | 5.43 | 0.81 | 2.23
5 | P | 8.27 | 10.73 | 5.97 | 0.64 | 1.91
6 | S | 27.60 | 6.80 | 5.23 | 0.52 | 2.32
7 | S | 7.97 | 11.13 | 5.87 | 0.67 | 2.20
8 | S | 7.90 | 11.20 | 6.00 | 0.67 | 2.25
9 | S | 5.27 | 12.03 | 2.33 | 0.76 | 2.29
10 | N | 36.53 | 13.20 | 3.00 | 0.84 | 3.82
11 | S | 32.90 | 4.47 | 7.60 | 0.67 | 3.14
12 | S | 33.20 | 4.43 | 7.73 | 0.66 | 2.93
13 | S | 3.97 | 2.67 | 6.40 | 0.80 | 6.05
14 | P | 33.30 | 4.47 | 8.00 | 0.67 | 3.17
15 | S | 6.13 | 11.57 | 2.63 | 0.72 | 1.99
16 | R | 35.53 | 7.30 | 4.90 | 0.48 | 4.29
17 | P | 5.33 | 11.90 | 2.20 | 0.75 | 2.29
18 | S | 35.33 | 3.60 | 1.87 | 0.77 | 3.25
19 | N | 4.57 | 6.97 | 5.50 | 0.48 | 4.00
20 | S | 6.87 | 9.93 | 8.27 | 0.50 | 2.73
21 | S | 34.70 | 12.43 | 4.57 | 0.76 | 3.63
22 | S | 35.50 | 7.07 | 1.47 | 0.48 | 3.79
23 | S | 36.60 | 13.17 | 3.10 | 0.84 | 3.41
24 | S | 5.77 | 11.93 | 2.40 | 0.74 | 2.13
25 | A | 4.77 | 3.23 | 1.03 | 0.72 | 4.80
26 | S | 6.37 | 3.27 | 2.00 | 0.75 | 3.10
27 | S | 31.53 | 4.67 | 0.70 | 0.66 | 2.10

Site | Type | $x/d_{i}$ | $y/d_{i}$ | $z/d_{i}$ | $B_{\xi_{3}}/B_{0}$ | $\displaystyle\frac{B_{\xi_{3}}}{||J_{B}||_{2}d_{i}}$
---|---|---|---|---|---|---
28 | S | 35.50 | 3.47 | 7.70 | 0.78 | 2.79
29 | S | 6.03 | 11.30 | 5.50 | 0.67 | 2.57
30 | R | 4.07 | 9.40 | 2.93 | 0.59 | 3.24
31 | N | 5.00 | 7.20 | 8.13 | 0.41 | 1.81
32 | S | 4.63 | 2.30 | 2.73 | 0.84 | 2.07
33 | S | 8.20 | 3.70 | 5.80 | 0.74 | 1.50
34 | S | 33.83 | 4.90 | 5.77 | 0.62 | 2.20
35 | S | 36.60 | 13.20 | 3.03 | 0.85 | 3.37
36 | S | 6.27 | 3.10 | 1.50 | 0.76 | 2.26
37 | R | 33.00 | 4.50 | 8.43 | 0.65 | 1.87
38 | S | 8.17 | 10.97 | 6.93 | 0.66 | 1.61
39 | S | 34.10 | 8.10 | 7.63 | 0.34 | 1.64
40 | N | 3.87 | 2.43 | 5.23 | 0.81 | 7.38
41 | S | 36.40 | 2.93 | 1.87 | 0.84 | 2.39
42 | S | 33.43 | 4.77 | 5.60 | 0.63 | 1.59
43 | P | 5.00 | 12.40 | 7.10 | 0.77 | 1.80
44 | S | 36.23 | 3.27 | 5.33 | 0.80 | 2.43
45 | N | 36.47 | 13.27 | 2.87 | 0.84 | 1.44
46 | S | 32.30 | 4.97 | 8.03 | 0.61 | 1.42
47 | S | 32.20 | 4.83 | 8.00 | 0.62 | 1.37
48 | S | 35.77 | 2.70 | 7.93 | 0.82 | 1.44
49 | P | 5.27 | 11.93 | 2.20 | 0.76 | 1.14
50 | S | 6.70 | 11.50 | 5.07 | 0.71 | 0.94
51 | S | 13.47 | 6.93 | 6.97 | 0.28 | 0.82
52 | S | 5.93 | 11.60 | 8.93 | 0.71 | 1.10
53 | S | 8.40 | 10.87 | 7.27 | 0.65 | 0.54
54 | N | 36.33 | 12.60 | 4.17 | 0.77 | 1.08

Table 1: Singular topological points identified by the Lorentz transformations
as locations were a subluminal transformation eliminates the field
$B_{\perp}$. The points are numbered as in the Fig. 6 their type and location
is provided. The type is indicated as center (C), saddle (S), repelling (R),
attracting (A), repelling focus (P) and attracting focus (N), for the types
illustrated in Fig. 6

The positions of the topological points identified by the Lorentz
transformations are projected in the $(x,y)$ plane of the simulation in Fig.
6. As noted above, we have excluded the points in the central region
$x/d_{i}\in[18,22]$. As can be seen, the points are concentrated in the
separatrix region where the inflow and outflow interact. The points are
located both in the flow region going towards the primary reconnection site
(yellow-red on the left side and green-blue on the right side) and the outflow
from the reconnection site (opposite colors). This is a region of strong
turbulence (Lapenta et al., 2020) caused by drift modes and Kelvin-Helmholtz
instability due to the presence of density and velocity gradients (Divin et
al., 2012) and by drift tearing modes due to the strong thin current layers in
the region (Daughton et al., 2011). Note that the outflow from reconnection
remains collimated only in the central region (Goldman et al., 2011) but turns
to a turbulent flow that fills the domain around the pile up region in the
outflow. The presence of a mild but significant guide field in this simulation
explains the bend in the outflowing jet (Goldman et al., 2011) and the
asymmetry between the upper and lower separatrices (Kleva et al., 1995; Ricci
et al., 2004; Birn & Priest, 2007).

##  6 Analysis of the detected sites.

The method described above identified 54 potential points of interest where
reconnection might be involved. A detailed analysis of all 54 of them is
provided in the supplemental material as a movie where each frame is one
detected point. Let us focus on a few of the most characteristic cases.
Figures 7-8 show all six cases illustrated in Fig. 6 and listed in Table 1.

Some features are common to all sites.

First, there is always right on spot or in very close proximity a significant
agyrotropy. Agyrotropy signifies a complex structure of the pressure tensor
with significant off-diagonal terms and has come to be a reliable indicator of
reconnection (Scudder & Daughton, 2008; Aunai et al., 2013; Swisdak, 2016).
The off-diagonal terms of the pressure tensor contribute non ideal terms to
the generalized Ohm’s law and allow the violation of the frozen in condition
(Biskamp, 2000; Birn & Priest, 2007; Hesse, 2006; Ricci et al., 2004). A key
discovery of the Magnetospheric MultiScale (MMS) mission was the observational
detection a non gyrotropic electron pressure tensor, confirming the theoretic
prediction (Burch et al., 2016). This is a critical need to allow electrons to
slip with respect to the field lines, which is central to the definition of
reconnection. Note that agyrotropy is typically highest just outside a
reconnection site and not quite at it (Scudder & Daughton, 2008; Burch et al.,
2016).

Second, the Lorentz frame indicator $\mathscr{L}$ is minimum at the selected
locations, by construction of the selection method, but the speed of the
Lorentz transformation that eliminates the magnetic field remains subluminal
(i.e. the indicator $\mathscr{L}<0$) in a non singular and smooth area around
the point measures, allowing for an easier identification.

Third, in the vicinity of the identified sites there is a significant
reconnection electric field. By definition we choose our coordinate system to
align the $\xi_{3}$-axis with the reconnection electric field and with the
guide magnetic field. The reconnection electric field reported in SI units
measures the strength of the reconnection rate. It is in fact perfectly
possible to have a quiescent magnetic singularity, one not accompanied by a
significant electron flow around it, this is a potential reconnection site
that is at the moment not reconnecting (Olshevsky et al., 2015). As can be
seen, however, the reported cases all show significant reconnection electric
fields, in fact a field of $100$$\mathrm{m}\mathrm{V}\mathrm{/}\mathrm{m}$ is
exceptionally high when compared with in situ data reported by MMS (Ergun et
al., 2016).

Fourth, as shown in the table 1, all points have a very substantial guide
magnetic field. This is supported by the recent observation done by MMS where
electron-scale secondary reconnection sites are observed to have
preferentially a sizable guide field (Stawarz et al., 2019). This might be
here more a consequence of the fact that the domain has no large scale region
with small field and reconnection happens where it can. Were there regions of
low magnetic field perhaps reconnection would happen also at low guide field.
The typical outflow from a primary reconnection site is magnetized for the
electrons except for the primary reconnection site itself, so one must expect
relatively large guide field in secondary reconnection sites embedded in the
magnetic field of the outflow. We mean by large, here, larger than 1/10 the
reconnecting field.

Finally but most importantly, the electron flow around these sites is clearly
strong and singular. The reconnection site forms a pattern of flow that
disrupts the nearby pattern and determines around it a flow that is clearly
aware of the presence of the reconnection site and is affected by it.
Detecting the electron flow around a reconnection site is the most common way
to identify reconnection. By this standard then the identified sites break the
frozen-in condition. Not all selected sties show a trivially interpretable
flow pattern. This is due to the 3D nature of the flow that includes a out of
plane component not shown in the figure. In all sites (see also movie in the
supplemental section), the flow is primarily carried by the electrons. We show
in the panels on the third row the electron flow speed in the reconnection
plane ($\mathbf{v}_{e(\xi_{1},\xi_{2})}$) and in the fourth row the relative
electron-ion speed
($\mathbf{v}_{e(\xi_{1},\xi_{2})}$-$\mathbf{v}_{i(\xi_{1},\xi_{2})}$).
Obviously in all cases the ions are not moving in a significant manner
compared with the electrons, compatible with the secondary reconnection sites
being site of electron-scale reconnection (Phan et al., 2018).

If these are general properties, each site has its own peculiarities best
appreciated by looking at all 54 of them in the movie provided in the
supplemental material. Some sites are more related to 2D x-points, some more
to 2D o-point. In the first case, one can consider the location a reconnection
site proper while the latter can be considered the center of a flux rope. One
case sticks out among the 54: P site 25, shown in Fig. 8 where the real part
of the eigenvalues while not hard zero is relatively small allowing us to
consider this focus point almost as a center.

Figure 7: Conditions around three types of detected points: a saddle (S, site
1), and repelling (P, site 14) and an attractive (N, site 19 focus). For each
column corresponding to a different point, four panels are shown from top to
bottom. Top row: in plane magnetic field lines superimposed over the electron
agyrotropy of the electron pressure tensor, $\mathcal{A}$. Second row: False
color representation of the Lorentz frame indicator $\mathscr{L}$ with
superimposed arrow plot of the in plane magnetic field,
$\mathbf{B}_{(\xi_{1},\xi_{2})}$. Third row, false color and arrow plot of the
electron flow velocity $\mathbf{v}_{e(\xi_{1},\xi_{2})}$ (the color indicates
the modulus of the speed normalized by the speed of light). Fourth row, false
color representation of the reconnection electric field $E_{\xi_{3}}$, with
superimposed arrow plot of the in plane relative electron-ion flow velocity,
$\mathbf{v}_{e(\xi_{1},\xi_{2})}-\mathbf{v}_{i(\xi_{1},\xi_{2})}$. The data
reported here is provided for all 54 detected sites in the form of a movie
where each frame contains the same four panels shown here: Video Figure 8:
Conditions around three types of detected points: a repelling (R, site 16), an
attractive focus (A, site 25) point and a repelling focus (P, site 5) that is
in fact nearly a center as the spiraling dominates over the outward going of
the field lines. The four rows have the same type of information as in Fig. 7
Figure 9: Cumulative probability distribution of the velocity $v_{L}$ of the
local Lorentz frame transformation that eliminates the magnetic field,
indicating the probability of finding a pixel with speed $v_{L}$ below a
certain value $V$ (blue). Conditional probability of finding a pixel with
agyrotropy exceeding a threshold value of 0.2 among the points with $v_{L}<V$
(red).

The evidence provided partially in Figs. 7-8 and completely in the movie in
the supplemental material is that there is always a substantial agyrotropy in
the vicinity of a detected point, as observed also in in-situ studies (Chen et
al., 2019). Figure 9 makes this assessment quantitative. We report the
cumulative probability of finding a point with $v_{L}$ less than a certain
value $V$. As can be seen the probability of finding subluminal points is
exceedingly small: topological singularities are a rare find. To be precise,
in the simulation reported, the probability of finding a pixel with subluminal
Lorentz frame speed $v_{L}$ is $2.1296\text{\times}{10}^{-06}$ (approximately
the same of a royal flush in the game of poker). This is then also the
probability of finding a reconnection site. Other methods that do not use the
Lorentz frame indicator have to face these adverse odds. A spacecraft flying
in situ will also have a similar hard time finding precisely such points. The
proposed Lorentz frame indicator, instead, finds them naturally by
construction and it can detect also the neighborhood where the speed while not
subliminal drops by several orders of magnitude compared with the rest of the
domain, signaling a nearby reconnection site.

Figure 9 additionally shows the conditional probability that the agyrotropy in
the points with $v_{L}<V$ is larger than 0.2 (a value to be considered quite
substantial as the typical maximum value observed near a reconnection site is
0.4 (Scudder & Daughton, 2008), we are using the half maximum value). As can
be seen this conditional probability reaches nearly one for small values of V
but then drops to only 50% when the selectivity of $v_{L}$ becomes small. The
meaning is simple: there is in the vicinity of singular point a significantly
high agyrotropy but only in 50% of the cases the high agyrotropy is right on
the point. This is not unexpected because the agyrotropy is large in the
vicinity of a reconnection site but not exactly at it (Scudder & Daughton,
2008).

##  7 Conclusions

Finding reconnection sites in 3D is a notoriously difficult problem. Part of
the difficulty is that it is difficult to even come to a universal definition
of what 3D reconnection is.

We take here a pragmatic definition that is suitable for the type of
reconnection one is interested in finding in a modern large scale fully
kinetic simulation. We define reconnection as the process where a reconnection
electric field produces a drift that carries two components of the magnetic
field towards a point where they are eliminated and new field lines are
formed.

Once we take this pragmatic definition, we observe that in the well understood
2D case of x and o point, this condition is accompanied by a peculiar property
of the Lorentz transformations of the electromagnetic fields under the
condition of eliminating the magnetic field in the plane normal to the
electric field in the laboratory frame: at the o and x points the speed of the
Lorentz transformation that eliminates the magnetic field is zero and in the
proximity of it it drops to subluminal speeds. Everywhere else in the domain
the speed of this transformation exceeds the speed of light, signifying that
there is no transformation that eliminates the normal component of the
magnetic field. This observation makes the detection of x and o point very
simple: in a simulation they can be found by looking for the points where the
speed of this specific type of Lorentz transformation becomes subluminal (i.e.
the transformation is possible).

In 2D this condition can be studied analytically (see Sect. 4) in the vicinity
of o and x points and the extension to 3D is straightforward.

Based on this, we propose an indicator of magnetic singular points where
reconnection defined as above takes place. The indicator is given in eq. (2)
and can be computed locally, without needing any field aligned integration or
any local differentiation. This indicator can be computed cell by cell and in
situ observations point by point. When a point of indicator $\mathscr{L}<0$ is
identified, the local type of magnetic singularity can be identified studying
the Jacobian of the magnetic field. This operation requires non local
information to compute the derivatives: This poses no trouble in a simulation
but for in situ observations it requires the use of multi-spacecraft missions
where the use of the curlometer technique allow the computation of gradients
(Dunlop & Eastwood, 2008). The approach is currently used to find magnetic
nulls (Fu et al., 2015) and it could be extended for the purpose proposed
here. However, unlike finding nulls (Haynes & Parnell, 2007; Fu et al., 2015)
or computing the Poincare-index (Xiao et al., 2006) that require only the
magnetic field, the Lorentz indicator is based also on the electric field. The
electric field tends to be more noisy. This did not prove to be a issue in the
simulation cases studied here but it might be a consideration for application
to in situ measurements.

We tested the approach for highly resolved 2D and 3D fully kinetic simulations
of reconnection. We find numerous detected sites and we studied the properties
of each potential reconnection site. We find that in virtually 100% of them
there are other concurring indicators of reconnection, in particular a large
agyrotropy.

The proposed approach is not meant to replace the well established method of
location of magnetic nulls (Haynes & Parnell, 2007; Fu et al., 2015). Nulls
are just one way in which reconnection can happen in 3D. Our approach is
different and aims at finding locations where an electric field is carrying 2
components of B to an annihilation point. Active reconnecting nulls have this
property too and indeed the proposed indicator finds the nulls as well because
the nulls are a limit case of the approach proposed for the case where not
just 2 but all 3 components of the magnetic field vanish at one point.
However, the proposed indicator finds also other types of reconnection,
unrelated to nulls.

The pragmatic definition of reconnection used here might not satisfy all
proposed definitions of reconnection: It is meant instead to find regions
where the processes we have become accustomed to associate with fast kinetic
reconnection take place. With this goal in mind we show for one 3D simulation
the ability of the method to identify numerous locations where such processes
happen: strong electron drift in near complete absence of ion motion (electron
scale reconnection), electron temperature agyrotropy and the presence of a
reconnection electric field in the direction normal to the plane of the two
reconnecting magnetic field components.

While our aim is operational, the physics interpretation of the Lorentz frame
indicator is tantalizing. Magnetic field lines do not exist and are a figment
of our imagination but if we want to give them an existence, they would be
moving with the frozen in plasma at the local $\mathbf{E}\times\mathbf{B}$. We
find that in proximity of reconnection, a frame exists where an observer
moving in the same direction of the magnetic field lines could eliminate the
magnetic field without exceeding the speed of light while remaining in the
same plane normal to the laboratory electric field. The existence of such a
frame appears to be intrinsically linked with the presence of reconnection.

## Acknowledgments

The author is grateful to John Finn for stimulating discussions on magnetic
topology and 3D reconnection and with Fabio Bacchini for the illuminating
discussion on Lorentz frame transformations. This project has received funding
from the KULeuven Bijzonder Onderzoeksfonds (BOF) under the C1 project
_TRACESpace_ , from the European Union’s Horizon 2020 research and innovation
programme under grant agreement No. 776262 (AIDA) and from the NASA grant
80NSSC19K0841. Computing has been provided by NASA at the NAS and NCCS high
performance computing facilities, by the Flemish Supercomputing Center (VSC)
and by the PRACE Tier-0 program. This research used resources of the National
Energy Research Scientific Computing Center, which is supported by the Office
of Science of the US Department of Energy under Contract No. DE-
AC02-05CH11231.

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Publications of the Astronomical Society of Australia Forbes, D. A. et al.

xxxx 2024 10.1017/pasa.xxxx.xx

Research Paper

D. A. Forbes, Email<EMAIL_ADDRESS>
D. A. Forbes et al. Globular Clusters in Local Volume Dwarf Galaxies
Publications of the Astronomical Society of Australia 00, 1–12.
https://doi.org/10.1017/pasa.xxxx.xx

(Received xx xx xxxx; revised xx xx xxxx; accepted xx xx xxxx)

# Keck/KCWI Spectroscopy of Globular Clusters in Local Volume Dwarf Galaxies

Duncan A. Forbes1 Daniel Lyon2 Jonah Gannon1 Aaron J. Romanowsky3,4 and Jean
P. Brodie1,2 1Centre for Astrophysics & Supercomputing, Swinburne University,
Hawthorn VIC 3122, Australia  2School of Chemistry and Physics, Science and
Engineering Faculty, Queensland University of Technology, Gardens Point
Campus, Brisbane, QLD 4001, Australia  3Department of Physics and Astronomy,
San José State University, One Washington Square, San Jose, CA 95192, USA
4Department of Astronomy & Astrophysics, university of California, Santa Cruz,
CA 95064, USA

###### Abstract

A number of nearby dwarf galaxies have globular cluster (GC) candidates that
require spectroscopic confirmation. Here we present Keck telescope spectra for
15 known GCs and GC candidates that may be associated with a host dwarf
galaxy, and an additional 3 GCs in the halo of M31 that are candidates for
accretion from a now disrupted dwarf galaxy. We confirm 6 star clusters (of
intermediate-to-old age) to be associated with NGC 247. The vast bulk of its
GC system remains to be studied spectroscopically. We also confirm the GC
candidates in F8D1 and DDO190, finding both to be young star clusters. The 3
M31 halo GCs all have radial velocities consistent with M31, are old and very
metal-poor. Their ages and metallicities are consistent with accretion from a
low mass satellite galaxy. Finally, three objects are found to be background
galaxies – two are projected near NGC 247 and one (candidate GCC7) is near the
IKN dwarf. The IKN dwarf thus has only 5 confirmed GCs but still a remarkable
specific frequency of 124.

###### keywords:

galaxies: dwarf, galaxies: star clusters: general, galaxies: halos, galaxies:
formation

##  1 Introduction

Globular clusters (GCs) are among the oldest structures in the Universe with a
range of metallicities, compact sizes (half-light radii $\sim$ 3 pc) and a
log-normal luminosity function (Brodie & Strader 2006). There are many
outstanding issues concerning their formation and destruction, enrichment
history and relationship with host galaxy properties (Forbes et al. 2018).
They reveal a remarkable linear scaling relation between their number and the
total halo mass of their host galaxy (e.g. Burkert & Forbes 2020). For low GC
numbers (in low mass galaxies) the relation has considerable scatter as
stochastic effects dominate.

Less massive galaxies tend to host fewer GCs than more massive ones. However,
one could argue that the importance of GCs increases for dwarf galaxies since
the number of GCs per unit galaxy starlight (specific frequency, SN), and the
GC system mass to stellar mass ratio, can be much higher than for more
luminous/more massive galaxies (e.g. Georgiev et al. 2010; Forbes et al.
2020). Once formed in a vigorous star formation event, GCs are imprinted with
a formation age and metallicity that is unchanged over cosmic time. The GCs of
dwarf galaxies provide important assembly history clues of not only their host
galaxy (Georgiev et al. 2010), but also giant galaxies since their metal-poor
GC subpopulation is thought to be largely accreted from disrupted dwarf
galaxies (Forbes & Remus 2018).

The low number of GCs in dwarfs means that it is very important to confirm the
status of each given the addition/removal of a single GC candidate can make a
large difference to its GC system (in some cases the system is either one GC
or none). Although imaging searches have been carried out for many local
dwarfs (e.g. Georgiev et al. 2010), spectroscopy is needed to confirm a common
radial velocity with the host galaxy and an old age (i.e. $\geq$ 8 Gyr) to
confirm its status as a GC rather than young massive cluster.

Follow-up spectroscopy is time consuming and requires accurate coordinates
from the imaging study. We note here the incorrect coordinates for a single GC
candidate in the nearby dwarf galaxy KK65, imaged by HST, as reported by
Sharina et al. (2005). Their coordinates for KK65-3-1095 are 07:42:29.4,
+16:34:29 (J2000). Examining WFPC2 and ACS imaging in the Hubble Legacy
Archive reveals that no bright object lies at these coordinates. Sharina et
al. (2005) indicate that the GC is located within the main body of the galaxy,
projected 0.33 kpc from its centre. There is indeed a single resolved GC-like
object within the body of the galaxy. It is however located at 07:42:31,
+16:33:30 (J2000), and deserves spectroscopic follow-up.

Here we present spectroscopy of some known GCs and additional GC candidates
using the Keck Cosmic Web Imager (KCWI) integral field unit on the Keck II
telescope. The targets were selected to be visible during twilight and/or
during intervals in our main observing programmes. We focused on blue GCs
which might be metal-poor. The current record holder for a metal-poor GC is
EXT8 in the halo of M31 with [Fe/H] = –2.90 (Larsen et al. 2020). Prior to
this measurement it was thought that a floor exists in metallicity of [Fe/H]
$\sim$ –2.5 for GCs. Our targets include GC candidates in NGC 247 (D = 3.3
Mpc), DDO190 (D = 2.9 Mpc), IKN (D = 3.8 Mpc), along with known GCs in F8D1 (D
= 3.8 Mpc) and Sextans A (D = 1.4 Mpc). We also include 3 known GCs in the
halo of M31 (which may have been accreted from a dwarf satellite galaxy). As
well as deriving radial velocities (to confirm association with the possible
host galaxy) we derive stellar population properties and compare the GCs in an
age-metallicity diagram.

##  2 Data

###  2.1 The Targets

In Figure 1 we show a montage of the 15 target objects. The imaging comes from
HST/ACS, or, if not available we show colour images from Legacy
(https://www.legacysurvey.org/dr9/description/) or SDSS
(https://classic.sdss.org/home.php) surveys. We have 8 targets for NGC 247: 6
with HST imaging and 2 with only ground-based Legacy imaging. HST imaging
exists for the sources in DDO190 and F8D1, and for two in M31 (PA-41 and
B336). In the latter two cases, the objects are located in the halo of M31 and
clearly resolved with individual bright stars seen in their outer regions. For
H12 in the halo of M31 and the known GC around Sextans A we only have ground-
based imaging. Further details of the individual targets, and their possible
host galaxy, are discussed below.

###  2.2 Data Acquisition

We observed 15 known GCs and GC candidates using the Keck Cosmic Web Imager
(KCWI) on the Keck II telescope over several years. Table 1 summarises the
observations. It lists the target object, its possible host galaxy and its
B-band absolute magnitude and stellar mass (with dwarf galaxy values from
Weisz et al. 2011), coordinates of the target, date of observation, KCWI
slicer used (M = Medium, L = Large), Grating (BL, BH3 or BM), central
wavelength setting and exposure time.

###  2.3 Data Reduction

All data were reduced using the standard KCWI data reduction pipeline
(Morrissey et al. 2018), following the method described in Gannon et al.
(2020). After reduction, data cubes were cleaned and trimmed to their good
spatial and spectral wavelength ranges and then processed as below. The output
was standard star calibrated, non-sky subtracted data cubes. The reduced KCWI
datacubes were displayed using Qfitsview. A aperture was defined to include
the total light from each GC. A background annulus (which includes both sky
and galaxy backgrounds) was also defined around each GC and subtracted. A 1D
spectrum was then extracted. Barycentric corrections were then applied and
each spectrum was median combined to produce the final science spectrum. The
final spectra have S/N ratios $\sim$20 per Å, sufficient for deriving both
radial velocities and stellar populations. The exceptions are the spectra of
GC candidates in DDO190 and F8D1 which have a lower S/N of $\sim$10 per Å.

###  2.4 Data Analysis

We used the code pPXF (Cappellari et al. 2017) to measure recessional
velocities and stellar populations from each spectrum. We follow the method
described in Gannon et al. (2020) which involves fitting the spectrum using
the Coelho et al. (2014) synthetic stellar library in 256 different input
parameter combinations to pPXF. A modest range of additive and multiplicative
polynomials were used. When present, emission lines are simultaneously
modelled with the absorption features.

Table 1: Observations. Host properties and GC candidates coordinates and KCWI
setup.

Target | Host Galaxy | Host MB | Host M∗ | R.A. | Dec. | Date | Slicer | Grating | R | Central Wavelength | Exp.Time
---|---|---|---|---|---|---|---|---|---|---|---
|  | (mag) | (108 M⊙) | (J2000) | (J2000) |  |  |  |  | (Å) | (s)
GC1 | Sextans A | -13.7 | 1.4 | 10:10:43.8 | -04:43:28.9 | 2020/11/10 | M | BM | 4000 | 4925 | 2420
GC1 | NGC 247 | -19.2 | 30 | 00:47:07.6 | -20:45:29.1 | 2023/10/23 | M | BL | 1800 | 4550 | 6240
GC2 | NGC 247 | -19.2 | 30 | 00:47:07.6 | -20:45:20.3 | 2023/10/23 | M | BL | 1800 | 4550 | 6240
GC3 | NGC 247 | -19.2 | 30 | 00:47:07.5 | -20:45:19.8 | 2023/10/23 | M | BL | 1800 | 4550 | 6240
GC4 | NGC 247 | -19.2 | 30 | 00:47:13.4 | -20:47:22.8 | 2023/10/23 | M | BL | 1800 | 4550 | 5040
GC5 | NGC 247 | -19.2 | 30 | 00:47:14.2 | -20:47:22.1 | 2023/10/23 | M | BL | 1800 | 4550 | 5040
GC6 | NGC 247 | -19.2 | 30 | 00:47:14.2 | -20:47:17.7 | 2023/10/23 | M | BL | 1800 | 4550 | 5040
5P | NGC 247 | -19.2 | 30 | 00:46:54.6 | -20:41:29.5 | 2023/10/23/ | M | BL | 1800 | 4550 | 2520
10P | NGC 247 | -19.2 | 30 | 00:47:09.9 | -20:30:19.0 | 2023/10/23 | M | BL | 1800 | 4550 | 2520
GCC1 | DDO190 | -14.1 | 0.6 | 14:24:45.0 | 44:31:36.1 | 2020/02/16 | M | BH3 | 9000 | 5080 | 1200
GC1 | F8D1 | -12.2 | 1.4 | 09:44:39.4 | 67:26:05.9 | 2021/01/13 | M | BL | 1800 | 4550 | 1800
GCC7 | IKN | -10.8 | 2.0 | 10:08:08.9 | 68:28:36.8 | 2021/02/17 | L | BL | 900 | 4550 | 3600
B336 | M31 | -21.2 | 1000 | 00:40:47.6 | 42:08:41.8 | 2020/11/10 | M | BH3 | 9000 | 5070 | 3600
H12 | M31 | -21.2 | 1000 | 00:38:03.6 | 37:43:59.0 | 2020/10/20 | M | BH3 | 9000 | 5070 | 5400
PA-41 | M31 | -21.2 | 1000 | 00:53:39.6 | 42:35:15.0 | 2022/01/29 | L | BL | 900 | 4550 | 900
|  |  |  |  |  |  |  |  |  |  |

Figure 1: Montage of sample GCs and GC candidates. From left to right and down
are images of NGC 247, DDO190, F8D1, IKN, M31 and Sextans A sources. Single
filter HST/ACS imaging is shown if available, otherwise colour images are from
Legacy or SDSS ground-based imaging. The orientation of the images is North
up, East left unless otherwise shown. A 10” bar is shown in each panel.

##  3 Results and Discussion

In Figures 2 and 3 we present spectra of the star clusters (i.e. old GCs and
younger star clusters) associated with their host galaxy. As well as the
original spectrum (black) we show the ppxf best fit model (magenta). Figure 4
shows the 3 GC candidates that we find to be background galaxies. These
candidates are projected on the sky near NGC 247 (5P and 10P) and the IKN
galaxy. The wavelength range and resolution of the spectra varies (see Table 1
for setup used) but each spectrum always includes H$\beta$.

Figure 2: KCWI spectra of 6 GC candidates around NGC 247. They range from
young massive star clusters to old globular clusters, all associated with NGC
247. Emission lines in the spectra of GC1 and GC3 may be the result of
incomplete galaxy background subtraction. Key absorption lines are
highlighted.

Figure 3: KCWI spectra of GCs associated with other galaxies. They include GCs
associated with DDO190, F8D1, M31 and Sextans A. Key absorption lines are
highlighted.

Figure 4: KCWI spectra of three GC candidates that are background galaxies.
Candidates are 5P and 10P projected near NGC 247 and GCC7 near the IKN galaxy.
Key emission lines are highlighted.

In Table 2 we summarise our results. This includes the radial velocity (RV)
and stellar population parameters, along with uncertainties. The radial
velocity errors we report only include the formal measurement uncertainty from
ppxf and do not include systematics which would be on the order of 10–30 km/s.
Stellar population results come from the median of the parameter distributions
with uncertainties taken as the 16th and 84th percentiles of these
distributions. Below we discuss our results for individual galaxies. We note
the caveat that any integrated light spectra of metal-poor GCs can be affected
by the presence of horizontal branch stars, which may bias results towards to
younger ages.

Table 2: Results. Recession Velocities and Stellar Populations (and errors).

Target | Host | RV | Age | [Z/H] | Comment
---|---|---|---|---|---
|  | (km/s) | (Gyr) | (dex) |
GC1 | Sextans A | 296 (2.0) | 8.3 (+1.2, -0.9) | -2.11 (+0.06, -0.03) |
GC1 | NGC 247 | 83 (4.6) | 5.0 (+0.4,-0.7) | -1.27 (+0.25, -0.15) |
GC2 | NGC 247 | 54 (2.8) | 7.7 (+0.6, -0.7) | -1.03 (+0.09, -0.05) |
GC3 | NGC 247 | 41 (6.0) | 4.3 (+0.4, -0.5) | -2.04 (+0.22, -0.07) |
GC4 | NGC 247 | 77 (5.1) | 9.2 (+0.5, -0.7) | -0.79 (+0.06, -0.05) |
GC5 | NGC 247 | 86 (5.0) | 10.6 (+0.7, -2.0) | -0.49 (+0.07, -0.19) |
GC6 | NGC 247 | 91 (4.0) | 6.7 (+0.4, -1.0) | -1.59 (+0.08, -0.04) |
5P | NGC 247 | z = 1.58 | – | – | AGN
10P | NGC 247 | z = 0.91 | – | – | AGN
GCC1 | DDO190 | 160 (7.0) | 2.6 (+1.8, -1.3) | -2.17 (0.12, -0.05) |
GC1 | F8D1 | -108 (23) | 0.48 (+0.5, -0.2) | -1.06 (+0.56, -0.55) |
GCC7 | IKN | z = 0.23 | – | – | galaxy
B336 | M31 | -595 (2.3) | 7.0 (+0.6, -0.7) | -2.01 (+0.06, -0.05) |
H12 | M31 | -379 (1.9) | 8.2 (+0.8, -1.0) | -1.88 (+0.05, -0.04) |
PA-41 | M31 | -104 (4.4) | 7.8 (+1.5, -1.3) | -1.93 (+0.08, -0.06) |
|  |  |  |  |

###  3.1 Individual Galaxies

$\bullet$ Sextans A
Sextans A is classified as IBm with a radial velocity of 324 $\pm$ 1 km/s from
NED. Its single GC has been studied by Beasley et al. (2019). They found RV =
305 $\pm$ 15 km/s, age = 8.6 $\pm$ 2.7 Gyr and [Fe/H] = –2.38 $\pm$ 0.29. The
more recent work by (Gvozdenko et al. 2024) found the same age with a
metallicity [Fe/H] = –2.14 $\pm$ 0.04. Thus it is an old, very metal-poor GC
associated with this dwarf galaxy. From our KCWI spectrum we measure RV = 296
$\pm$ 2.0 km/s and stellar populations that are similar to those reported,
i.e. age = 8.3${}^{+1.2}_{-0.9}$ and [Z/H] = –2.11${}^{+0.06}_{-0.03}$. This
gives us further confidence in our stellar population fitting. We note that
the total metallicity is expected to be similar, or higher by $\sim$0.3 dex,
than the iron metallicity depending on the (unknown) alpha-element abundance.

$\bullet$ NGC 247
NGC 247 is an edge-on SBd spiral in the nearby Sculptor Group, with ongoing
star formation activity and a stellar mass of $\sim 3\times 10^{9}M_{\odot}$.
From a spectrum of the nucleus and using ppxf, Kacharov et al. (2018)
determined its star formation history and found it to be very broad. At old
ages (7–10 Gyr) it displays a wide range of metallicities (–2.3 $<$ [Fe/H] $<$
–0.5). This enriches to –1 $<$ [Fe/H] $<$ –0.5 around 2–3 Gyr ago. However,
there are also some indications of star formation $\sim$ 10 Gyr ago that was
metal-rich ([Fe/H] $\sim$ 0), and $\leq$ 1 Gyr ago that was metal-poor ([Fe/H]
$\sim$ –1.5). Thus it appears to have quite a complex star formation/chemical
enrichment history.

The galaxy has a radial velocity of 156 $\pm$ 2 km/s. To our knowledge there
are only 3 spectroscopically confirmed GCs from Olsen et al. (2004). Their
GCs, named NGC247-1, -2 and -3 (in their table 4) have metallicities of [Fe/H]
= –1.13, –0.41 and –0.89 respectively. No ages were reported. They estimated a
total of 60 $\pm$ 30 GCs, so that the majority remain to be confirmed. A young
‘fuzzy’ star cluster with an age of $\sim$ 300 Myr and [Z/H] $\sim$ –0.6 was
reported recently by Romanowsky et al. (2023).

Using imaging from the Subaru telescope, Santhanakrishnan (2016) identified 14
additional GC candidates (named 1P – 14P). An additional 6 candidates were
initially identified in the Subaru imaging and these were found to be included
in a single HST/ACS image of NGC 247 (PI: Dalcanton). These objects appear in
the HST imaging as star clusters with resolved stars in their outer regions.
See Romanowsky et al. (2023) for further details of the Subaru imaging. Using
KCWI we obtained data for all 6 HST star clusters (in two pointings) and for
the candidates 5P and 10P (in two separate pointings), giving a total of 8 GC
candidates. Of the HST sources, we find all 6 to be bona fide star clusters
associated with NGC 247. They have a range of ages and metallicities 4.3 $<$
age $<$ 10.6 Gyr –0.49 $<$ [Z/H] $<$ –2.0. Several of them might be better
described as intermediate-aged star clusters. We note that GC1 and GC3 have
emission lines from imperfect galaxy background subtraction, however these
were excluded in the ppxf fit. Two candidates (those from the ground-based
imaging), 5P and 10P, are found be to background AGN and not GCs.

A relevant comparison galaxy is NGC 2403 which lies at a distance of 3.2 Mpc
and has a stellar mass of 7 $\times$ 109 M⊙. Using KCWI spectroscopy and a
similar analysis process we measured a range of ages (0.4 $<$ age $<$ 12.5
Gyr) and metallicities (–0.61 $>$ [Z/H] $>$ –2.07) for half a dozen star
clusters (Forbes et al. 2022). Thus similarly to NGC 247, the star clusters of
NGC 2403 appear to have formed in-situ over an extended period of time.

$\bullet$ DDO190
DDO190 (UGC9240) is an Im dwarf with a RV = 150 $\pm$ 4 km/s. Sharina et al.
(2005) identified 1 very blue candidate in HST/WFPC2 imaging which they named
U9240-3-4557 (we use GCC1 here). They resolved the candidate and measured a
half-light radius of 3.6 pc and MV = –7.2. From our KCWI spectrum of this
candidate we measure RV = 160 $\pm$ 7 km/s. This RV is well within the
velocity range of the rotating HI gas (W50 = 45 km/s) in DDO190 (Ott et al.
2012) and thus we confirm a GC associated with the galaxy. Our spectrum
(Figure 3) is limited in wavelength and of low S/N but an young age (2.6 Gyr)
and low metallicity ([Z/H] = –2.17) is indicated (as might be expected from
its blue colour).

$\bullet$ F8D1
F8D1 is a low surface brightness dwarf galaxy in the M81 group (D = 3.8 Mpc).
The galaxy has been studied using HST imaging by Zemaitis et al. (2023). It
reveals a tidal tail indicating an ongoing interaction and a TRGB distance of
3.67 Mpc. It has a metallicity [M/H] = –1.14 $\pm$ 0.09. Its large half-light
radius and low surface brightness make it an ultra-diffuse galaxy (UDG). F8D1
has a single GC has a spectroscopic RV measured by Chiboucas et al. (2009) of
–125 $\pm$ 130 km/s. The RV for the galaxy is reported in NED as –125 with a
reference to Kourkchi & Tully (2017). We suspect the latter is actually the RV
of the GC as reported by Chiboucas et al. (2009) and not the host galaxy. The
M81 group has a range of RVs with M81 itself at RV = –36 $\pm$ 3 km/s. We
measure –108 $\pm$ 23 km/s for the GC, which we associate with F8D1. Our low
S/N spectrum of F8D1 GC reveals several strong Balmer lines and we find a very
young age of 480 Myr. Its recent formation suggests that it has formed as
result of the ongoing interaction.

$\bullet$ IKN
The IKN dwarf, in the M81 group (D = 3.8 Mpc), is a very low surface
brightness dwarf and a candidate for a UDG. It has 5 confirmed old GCs
(Tudorica et al. 2015) giving it a remarkable specific frequency of SN = 124
(Georgiev et al. 2010). Based on SDSS DR10 imaging, Tudorica et al. (2015)
identified 2 GC candidates requiring spectroscopic follow-up. Candidate GCC6
lies very close to a bright star. Here we present a spectrum of GCC7 and find
it to be a background galaxy at z $\sim$ 0.2.

$\bullet$ M31
Three known GCs in M31 were observed. They are B336, H12 and the outer halo GC
PA-41 from the PAndAS survey (Huxor et al. 2014). The Revised Bologna Catalog
(RBC) version 5 http://www.bo.astro.it/M31/) lists B336 with a RV = –609 $\pm$
27 km/s. Its metallicity is very low but with a large error, i.e. [Fe/H] =
–2.5 $\pm$ 0.6. For H12, Zhou et al. (2011) measured RV = –412 $\pm$ 33 km/s
and old ages (13.6${}^{+1.4}_{-1.7}$ Gyr), metal-poor ([Fe/H] =
–1.80$\pm$0.18) and alpha enhanced ([$\alpha$/Fe] $\sim$ 0.5) in their Cas
system. PA–41, also known as SDSS9, has a very blue colour of (g–i)0 = 0.69
from de Tullio Zinn & Zinn (2013), suggesting it might also be quite metal-
poor. We are not aware of any recession velocity reported for PA-41. According
to Mackey et al. (2019), PA-41 is likely associated with a stellar stream
(C/D) that came from a disrupted satellite galaxy. We confirm association with
M31 for all 3 GCs and derive old ages of $\sim$8 Gyr and very low
metallicities [Z/H] $\sim$ –2 (although not as low as EXT8).

###  3.2 Age-metallicity Relation

The ages and metallicities of GCs formed within a galaxy broadly follow an
age-metallicity relation (AMR) that describes the chemical enrichment in that
galaxy over cosmic time (e.g. Horta et al. 2021), although significant
variations due to local conditions can be present (e.g. infalling halo gas may
lower the mean metallicity of in-situ gas). An AMR can be approximated by a
closed box chemical enrichment model. For example, Forbes (2020) assigned
Milky Way GCs to several disrupted dwarf satellite galaxies on the basis of
their ages and metallicities. The corresponding AMRs could then be used to
estimate the original mass of the disrupted dwarf.

Figure 5: Age-metallicity diagram of sample GCs. red asterisks show GCs
associated with NGC 247, and red squares with other host galaxies. The 3 red
lines on the right hand side indicate the metallicities of 3 GCs in NGC 247
from Olsen et al. (2004). The solid line shows the age-metallicity relation
inferred for Gaia-Enceladus dwarf galaxy from Forbes (2020) and triangles for
the model AMR from Horta et al. (2021) for a stellar mass $\sim 10^{9}$ M⊙
galaxy. The NGC 247 star clusters show a broad range of age and metallicity
similar to that reported for the nuclear region by Kacharov et al. (2018). The
GCs associated with other galaxies tend to be less enriched than the AMRs
shown, consistent with formation in a low mass ($<$ 109 M⊙) galaxy.

In Figure 5 we show our age and metallicity measurements for the old GCs and
younger star clusters. We also show the AMR of the Gaia-Enceladus dwarf galaxy
(we have assumed that [Z/H] $\sim$ [Fe/H]). This dwarf is thought to have had
a stellar mass of $\sim 10^{9}$ M⊙, hosted $\sim 28$ GCs and was accreted by
the Milky Way some 10 Gyr ago (Forbes 2020). We also include the AMR model by
Horta et al. (2021) for a galaxy with a similar stellar mass. NGC 247 has a
stellar mass of 3 $\times$ 109 M⊙. The plot shows that the NGC 247 star
clusters reveal a broad range of ages and metallicities similar to that
reported by Kacharov et al. (2018). We also include markers for the 3 NGC 247
GCs with [Fe/H] metallicities (ages were not published) from Olsen et al.
(2004). Given that the total GC system may contain up to 60 $\pm$ 30 GCs
(Olsen et al. 2004), it would be worthwhile to obtain more spectroscopic
samples to investigate whether distinct age-metallicity relations exist for
old GCs vs the more recently formed star clusters.

The GCs associated with other galaxies have a wide range in ages and
metallicities. They all occupy a location in the diagram below the AMR of a
109 M⊙ galaxy. For the 3 M31 halo GCs, this suggests that if they formed in a
now accreted dwarf satellite galaxy that this galaxy had an original mass $<$
109 M⊙.

##  4 Summary

Here we present KCWI spectra of 15 known GCs and GC candidates in several
nearby dwarf galaxies and in the halo of M31. We derive radial velocities and
ages and metallicities for each object.

For the known GC associated with Sextans A we find a similar radial velocity
and stellar populations to Beasley et al. (2019), of an old age and very low
metallicity. Of the 8 GC candidates that might be associated with NGC 247, we
find 2 (5P and 10P) to be background AGN. The other 6 candidates around NGC
247 are confirmed as star clusters with a range from ages from intermediate to
old (i.e. bona fide GCs). They also reveal a range of metallicities and
scatter widely about an age-metallicity relation. For the IKN galaxy the 1 GC
candidate is found to be a z $\sim$ 0.2 background galaxy. Thus IKN’s total GC
system remains at 5, which still has a remarkable specific frequency of SN =
124. For the GC candidate in DDO190 confirm its association. Our low S/N
spectrum indicates a young age ($\sim$ 3 Gyr) and low metallicity ([Z/H]
$\sim$ –1). This suggests that the candidate is a young star cluster
(consistent with its very blue colour). For the known GC in F8D1, we measure a
more accurate velocity confirming association. Its stellar population
properties, again from a low S/N spectrum, suggests very recent formation as a
result of the ongoing tidal interaction. The 3 M31 halo GCs are confirmed by
their radial velocities and are found be to very metal-poor and old. They may
have been accreted from a disrupted low mass ($<$ 109 M⊙) dwarf galaxy.

###  4.1 Bibliography

## Data Availability

Raw data is available in the Keck Observatory Archive (KAO).

##  5 Acknowedgements

We thank the referee for several useful comments. We thank L. Buzzo, L.
Haacke, S. Janssens, A. Ferre-Mateu for help and useful discussions. We thank
J. Carlin for initial ground-based images of NGC 247 that aided in the
selection of spectroscopic targets. We thank V. Santhanakrishnan for
identifying GC candidates around NGC 247, and S. Larsen and A. Gvozdenko for
M31 targets. This work was supported by a NASA Keck PI Data Award,
administered by the NASA Exoplanet Science Institute. We thank the ARC for
financial support via DP220101863. Here we make use of NED
(https://ned.ipac.caltech.edu/) and Qfitsview
(https://www.mpe.mpg.de/ott/QFitsView/). The data presented herein were
obtained at Keck Observatory, which is a private 501(c)3 non-profit
organization operated as a scientific partnership among the California
Institute of Technology, the University of California, and the National
Aeronautics and Space Administration. The Observatory was made possible by the
generous financial support of the W. M. Keck Foundation. The authors wish to
recognize and acknowledge the very significant cultural role and reverence
that the summit of Maunakea has always had within the Native Hawaiian
community. We are most fortunate to have the opportunity to conduct
observations from this mountain.

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