A REMARK ON FINITE METRIC SPACES
Vašek<EMAIL_ADDRESS>
Concordia University, Montreal and Charles University, Prague

and

Ida<EMAIL_ADDRESS>
Charles University, Prague

> Abstract: Richmond and Richmond (Amer. Math. Monthly 104 (1997), 713–719)
> proved the following theorem: If, in a metric space with at least five
> points, all triangles are degenerate, then the space is isometric to a
> subset of the real line. We prove that the hypothesis is unnecessarily
> strong: In fact, $\Theta(n^{2})$ suitably placed degenerate triangles
> suffice.
>

##  1 Results

Given a metric space $(V,\mbox{\em dist\/})$, we follow [1] in writing $[rst]$
to signify that $r,s,t$ are pairwise distinct points of $V$ and $\mbox{\em
dist\/}(r,s)+\mbox{\em dist\/}(s,t)=\mbox{\em dist\/}(r,t)$. Following [6], we
refer to three-point subsets of $V$ as triangles; if $[rst]$, then the
triangle $\\{r,s,t\\}$ is called degenerate.

Now let $M$ be a metric space $(V,\mbox{\em dist\/})$. Trivially, if there is
a linear order $\preceq$ on $V$ such that $r\prec s\prec t\;\Rightarrow[rst]$,
then all triangles in $M$ are degenerate. Richmond and Richmond [6] proved the
converse under a mild lower bound on $\lvert V\rvert$:

######  Theorem 1 ([6]).

Let $M$ be a metric space $(V,\mbox{\em dist\/})$ such that $\lvert
V\rvert\geq 5$. If all triangles in $M$ are degenerate, then there is a linear
order $\preceq$ on $V$ such that $r\prec s\prec t\;\Rightarrow[rst]$.

Here, the lower bound on $\lvert V\rvert$ cannot be reduced: consider
$V=\\{a,b,c,d\\}$ and $\mbox{\em dist\/}(a,b)\\!=\\!dist(b,c)\\!=\\!\mbox{\em
dist\/}(c,d)\\!=\\!\mbox{\em dist\/}(d,a)\\!=\\!1$, $\mbox{\em
dist\/}(a,c)\\!=\\!\mbox{\em dist\/}(b,d)\\!=\\!2$.

We are going to prove that the hypothesis of Theorem 1 can be relaxed as soon
as the lower bound on $\lvert V\rvert$ is increased by $1$ and that it can be
relaxed further and further as $\lvert V\rvert$ gets larger and larger. To
formulate this result, consider an arbitrary $3$-uniform hypergraph $(V,{\cal
E})$. Clearly, there is a unique minimal superset ${\cal E}^{\star}$ of ${\cal
E}$ such that no six-point subset of $V$ contains precisely $19$ members of
${\cal E}^{\star}$. We let ${\rm cl}\,{\cal E}$ denote this ${\cal
E}^{\star}$.

######  Theorem 2.

Let $M$ be a metric space $(V,\mbox{\em dist\/})$ such that $\lvert
V\rvert\geq 6$ and let $(V,{\cal E})$ be a $3$-uniform hypergraph such that
${\rm cl}\,{\cal E}$ consists of all triangles in $M$. If all triangles that
belong to ${\cal E}$ are degenerate, then there is a linear order $\preceq$ on
$V$ such that $r\prec s\prec t\;\Rightarrow[rst]$.

The hypothesis of Theorem 2 is far weaker than the hypothesis of Theorem 1. To
see this, take a triangle $S$ in $V$ and let ${\cal E}$ consist of all
triangles in $V$ that have a nonempty intersection with $S$. Here ${\rm
cl}\,{\cal E}$ consists of all all triangles in $V$ and, with $n$ standing for
$\lvert V\rvert$, we have $\lvert{\cal E}\rvert=\binom{n}{3}-\binom{n-3}{3}$,
which is only $\Theta(n^{2})$. Thus far fewer than $\binom{n}{3}$ degenerate
triangles in an $n$-point metric space suffice to force the conclusion of
Theorem 1 as long as these triangles are suitably placed. Even if the
degenerate triangles are placed inconveniently, already $\binom{n}{3}-n+7$ of
them always force the conclusion of Theorem 1. This follows directly from the
conjunction of Theorem 2 and the next theorem:

######  Theorem 3.

Let $(V,{\cal E})$ be a $3$-uniform hypergraph with $n$ vertices. If
$\lvert{\cal E}\rvert\geq\binom{n}{3}-n+5$, then ${\rm cl}\,{\cal E}$ consists
of all three-point subsets of $V$.

To see that the lower bound on ${\cal E}$ in Theorem 3 cannot be reduced,
consider ${\cal E}$ consisting of all three-point subsets of
$V=\\{1,2,\ldots,n\\}$ except for all $\\{1,2,j\\}$ with $3\leq j\leq n-2$.
Here, $\lvert{\cal E}\rvert=\binom{n}{3}-n+4$ and ${\rm cl}\,{\cal E}={\cal
E}$.

Theorem 2 is an immediate corollary of Theorem 1 and the following theorem:

######  Theorem 4.

Let $M$ be a metric space $(V,\mbox{\em dist\/})$ such that $\lvert
V\rvert\geq 6$ and let $(V,{\cal E})$ be a $3$-uniform hypergraph.If all
triangles that belong to ${\cal E}$ are degenerate, then all triangles that
belong to ${\rm cl}\,{\cal E}$ are degenerate.

##  2 Proofs

Most of the work involved in proving Theorem 4 is subsumed in its following
special case:

######  Lemma 1.

Let $M$ be a metric space $(V,\mbox{\em dist\/})$ such that $\lvert
V\rvert=6$. If $19$ triangles in $M$ are degenerate, then all $20$ triangles
in $M$ are degenerate.

###### Proof.

By assumption, there is a triangle $T$ in $V$ such that the $19$ triangles
distinct from $T$ are degenerate. In particular, for each of the three
elements $\lambda$ of $T$, all triangles in $V-\\{\lambda\\}$ are degenerate
and so, by Theorem 1, there is a linear order $\leq_{\lambda}$ on
$V-\\{\lambda\\}$ such that

$r<_{\lambda}s<_{\lambda}t\;\Rightarrow[rst].$ (1)

Having chosen a $\mu$ in $T$, let us label the elements of $V-T$ as $u,v,w$ in
such a way that $u<_{\mu}v<_{\mu}w$, and so $[uvw]$. Now for each of the
remaining two elements $\nu$ of $T$, property (1) implies $u<_{\nu}v<_{\nu}w$
or $w<_{\nu}v<_{\nu}u$; reversing the orders if necessary, we may assume that
$u<_{\nu}v<_{\nu}w$, and so

$u<_{\lambda}v<_{\lambda}w\;\text{ for all three $\lambda$ in $T$.}$ (2)

Next, let us label the elements of $T$ temporarily as $a,b,c$ in such a way
that $a<_{c}b$ and then permanently as $x,y,z$: If $b<_{a}c$, then
$x=a,y=b,z=c$; else either $x=a,y=c,z=b$ (in case $a<_{b}c$) or $x=c,y=a,z=b$
(in case $c<_{b}a$). Now we have

$x<_{z}y\;\text{ and }\;y<_{x}z.$ (3)

For future reference, let us note that

the restrictions of $\leq_{x}$ and $\leq_{z}$ on $\\{u,v,w,y\\}$ are identical
(4)

(Analogous statements apply to $x,y$ in place of $x,z$ and to $y,z$ in place
of $x,z$, but we will not need these variations). To see this, observe that,
by virtue of (2) and (1), our metric space $M$ determines the rank of $y$ in
the restriction of $\leq_{x}$ on $\\{u,v,w,y\\}$ in the same way as it
determines the rank of $y$ in the restriction of $\leq_{z}$ on
$\\{u,v,w,y\\}$.

The remainder of the proof relies on the fact that

$[\alpha\beta\gamma]\text{ and
}[\alpha\gamma\delta]\;\;\Rightarrow\;\;[\alpha\beta\delta]\text{ and
}[\beta\gamma\delta],$ (5)

which has been pointed out by Menger [5]. (By the way, extensions of (5) are
discussed in [3, Section 6].)

Finally, we are ready to prove that the triangle $\\{x,y,z\\}$, too, is
degenerate. Actually, we are going to prove $[xyz]$. For this purpose, let us
distinguish between three cases.

Case 1: $u<_{z}x$. By (3), we have $u<_{z}x<_{z}y$, and so $[uxy]$; by (4) and
(3), we have $u<_{x}y<_{x}z$, and so $[uyz]$. Now we have $[xyz]$ by (5).

Case 2: $z<_{x}w$. By (3), we have $y<_{x}z<_{x}w$, and so $[yzw]$; by (3) and
(4), we have $x<_{z}y<_{z}w$, and so $[xyw]$. Now we have $[xyz]$ by (5).

Case 3: $x<_{z}<u<_{z}v<_{z}w$ and $u<_{x}v<_{x}w<_{x}z$. In particular,

$[xuv],[xvw],[vwz],[uvz].$

Here we have neither $[vxz]$ (else $[uvz]$ and (5) would imply $[uvx]$,
contradicting $[xuv]$) nor $[xzv]$ (else $[xvw]$ and (5) would imply $[zvw]$,
contradicting $[vwz]$); since $\\{x,v,z\\}$ is degenerate, we must have

$[xvz].$

If $v<_{x}y$, then $[vyz]$ by (3); else $y<_{z}v$ by (4), and so $[xyv]$ by
(3); in either case, this implies $[xyz]$ by $[xvz]$ and (5). ∎

Proof of Theorem 4.  By definition, members of ${\rm cl}\,{\cal E}-{\cal E}$
can be enumerated as $T_{1},T_{2},\ldots T_{k}$ in such a way that, for each
$j=1,2,\ldots,k$, some six-point subset of $V$ contains precisely $19$ members
of ${\cal E}\cup\\{T_{i}:1\leq i<j\\}$. Induction on $j\;(=1,2,\ldots k)$
shows that all triangles belonging to ${\cal E}\cup\\{T_{i}:1\leq i<j\\}$ are
degenerate; the induction step relies on Lemma 1. $\square$

Proof of Theorem 3.  Given any three-point subset $T$ of $V$, we shall show
that $T\in{\rm cl}\,{\cal E}$. This is trivial when $T\in{\cal E}$; to prove
it when $T\not\in{\cal E}$, it suffices to find a six-point subset $S$ of $V$
such that $T$ is the unique three-point subset of $S$ not belonging to ${\cal
E}$. For this purpose, take one vertex from each $T^{\prime}-T$ such that
$T^{\prime}$ is a three-point subset of $V$ not belonging to ${\cal E}$ and
$T^{\prime}\neq T$. With $W$ standing for the set of all these vertices, $T$
is the unique three-point subset of $V-W$ not belonging to ${\cal E}$; since
$\lvert W\rvert\leq\binom{n}{3}-\lvert{\cal E}\rvert-1$, we have $\lvert
V-W\rvert\geq n-\binom{n}{3}+\lvert{\cal E}\rvert+1\geq 6$. $\square$

##  3 A concluding remark

Following [2], let us refer to a $3$-uniform hypergraph $(V,{\cal E})$ as
metric if there is a metric space $M$ such that a triangle in $M$ is
degenerate if and only if it belongs to ${\cal E}$. Lemma 1 can be
reformulated as the statement that the $3$-uniform hypergraph with $6$
vertices and $19$ hyperedges is non-metric. Actually, this hypergraph is
minimal non-metric in the sense that the deletion of an arbitrary vertex from
it produces a metric hypergraph. To verify this, observe first that the
deletion produces a hypergraph with $5$ vertices and $10$ or $9$ hyperedges.
The former hypergraph is obviously metric. To see that the latter hypergraph
is metric, note that in the metric space on ground set $\\{u,v,x,y,z\\}$ with
metric dist defined by the chart

$\begin{array}[]{r|c|c|c|c|c|}&x&u&y&v&z\\\ \hline\cr x&0&1&2&3&2\\\ \hline\cr
u&1&0&1&2&3\\\ \hline\cr y&2&1&0&1&2\\\ \hline\cr v&3&2&1&0&1\\\ \hline\cr
z&2&3&2&1&0\\\ \hline\cr\end{array}$

$\\{x,y,z\\}$ is the only nondegenerate triangle. (By the way, infinitely many
minimal non-metric hypergraphs have been constructed in [4].)

## References

  * [1] P. Aboulker, X. Chen, G. Huzhang, R. Kapadia, and C. Supko, Lines, betweenness and metric spaces, Discrete & Computational Geometry 56 (2016), 427–448.
  * [2] L. Beaudou, V, Chvátal, A. Bondy, X. Chen, E. Chiniforooshan, M. Chudnovsky, N. Fraiman, and Y. Zwols, Lines in hypergraphs, Combinatorica 33 (2013), 633–654.
  * [3] V. Chvátal, Sylvester-Gallai theorem and metric betweenness, Discrete & Computational Geometry 31 (2004), 175–195.
  * [4] V. Chvátal and I. Kantor, Metric hypergraphs and metric lines,  arXiv:2207.11811 [math.CO]
  * [5] K. Menger, Untersuchungen über allgemeine Metrik, Mathematische Annalen 100 (1928), 75–-163.
  * [6] B. Richmond and T. Richmond, Metric spaces in which all triangles are degenerate, American Mathematical Monthly 104 (1997), 713–719.


#  NuSTAR hard X-ray observation of a sub-A class solar flare

Lindsay Glesener11affiliation: University of Minnesota, Minneapolis, USA , Säm
Krucker22affiliation: University of California at Berkeley, Berkeley, USA
33affiliation: University of Applied Sciences and Arts Northwestern
Switzerland, Windisch, Switzerland , Iain G. Hannah44affiliation: University
of Glasgow, Glasgow, UK , Hugh Hudson22affiliation: University of California
at Berkeley, Berkeley, USA 44affiliation: University of Glasgow, Glasgow, UK ,
Brian W. Grefenstette55affiliation: California Institute of Technology,
Pasadena, USA , Stephen M. White66affiliation: Air Force Research Laboratory,
Albuquerque, USA , David M. Smith77affiliation: University of California at
Santa Cruz, Santa Cruz, USA , Andrew J. Marsh77affiliation: University of
California at Santa Cruz, Santa Cruz, USA

###### Abstract

We report a NuSTAR observation of a solar microflare, SOL2015-09-01T04.
Although it was too faint to be observed by the GOES X-ray Sensor, we estimate
the event to be an A0.1 class flare in brightness. This microflare, with only
$\sim$5 counts s-1 detector-1 observed by RHESSI, is fainter than any hard
X-ray (HXR) flare in the existing literature. The microflare occurred during a
solar pointing by the highly sensitive NuSTAR astrophysical observatory, which
used its direct focusing optics to produce detailed HXR microflare spectra and
images. The microflare exhibits HXR properties commonly observed in larger
flares, including a fast rise and more gradual decay, earlier peak time with
higher energy, spatial dimensions similar to the RHESSI microflares, and a
high-energy excess beyond an isothermal spectral component during the
impulsive phase. The microflare is small in emission measure, temperature, and
energy, though not in physical size; observations are consistent with an
origin via the interaction of at least two magnetic loops. We estimate the
increase in thermal energy at the time of the microflare to be 2.4$\times
10^{27}$ ergs. The observation suggests that flares do indeed scale down to
extremely small energies and retain what we customarily think of as
“flarelike” properties.

Sun: corona — Sun: flares — Sun: X-rays, gamma rays

##  1 Introduction

Solar flares are impulsive transformations of magnetic energy into heating,
particle acceleration, and, occasionally, eruptions. They are of interest for
understanding the basic physics of the Sun because they represent a
restructuring of the coronal magnetic field, are often accompanied by coronal
mass ejections, accelerate a huge number of particles up to high energies
(e.g. Lin & Hudson, 1976), and may impulsively heat the corona (e.g. Klimchuk,
2006). While many flare investigations concentrate on the largest events due
to the rich multiwavelength observations and detailed phenomena that can be
studied, there is also extensive investigative opportunity on the smaller side
of the flare distribution, where flares are less dramatic but far more
frequent. In fact, if flares are responsible for coronal heating, it has been
shown that this heating must be in weak flare-like events, not in the form of
typical, larger flares (e.g. Hudson, 1991).

Hard X-rays (HXRs) are a useful tool in understanding energetics in flares of
any size, because they are produced via bremsstrahlung by high-energy
electrons that are either hot (millions to tens of millions of degrees) or
nonthermal (accelerated by the flare). For the smallest flares this can be
especially key – while the time necessary for ionization equilibrium means
that line emission in extreme ultraviolet (EUV) or soft X-rays (SXR) might be
suppressed for very short events (e.g. Bradshaw & Klimchuk, 2011), HXR
feedback is immediate.

Thorough statistical and case studies were undertaken on HXR flares and larger
microflares (also known as active region transient brightenings) using the
Hard X-ray Telescope on the Yohkoh spacecraft (Kosugi et al., 1991) in the
1990s and the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI)
(Lin et al., 2002) for solar cycles 23 and 24 (the current era). These
instruments employ indirect imaging and thus have been limited to flares of a
certain brightness – for RHESSI, temperatures $T\gtrsim$9 MK and emission
measure $EM\gtrsim 10^{45}$ cm-3, with particularly good coverage of flares
with $T\gtrsim$12 MK and $EM\gtrsim 10^{46}$ cm-3 (Hannah et al., 2008).

Nishio et al. (1997) examined 14 impulsive flares observed in microwaves (by
Nobeyama) and SXRs/HXRs (by Yohkoh) and found that 10 of the 14 displayed
evidence for at least two loops. The two loops were typically of very
different lengths, of order $\lesssim$20 arcseconds and 30–80 arcseconds. The
HXRs and SXRs came predominantly from the more compact loop and its
footpoints. Hanaoka (1997) also identified a two-loop configuration in many
radio/SXR flares, with many of the events showing a three-legged structure and
large angles between the loops. These observations have led to the modern
concept that flares generally consist of multiple, independently excited, loop
structures that may interact.

Battaglia et al. (2005) studied RHESSI flares from GOES class B to M classes,
finding a spectral softening of the nonthermal electron distribution with
smaller flare energy. Hannah et al. (2008) and Christe et al. (2008) studied
over 25,000 RHESSI microflares of GOES class A and B, finding that they all
arise in active regions and have properties similar to larger flares,
including impulsive rises and slow decays and the presence of thermal and
nonthermal spectral components.

At lower energies, active region microflares have been catalogued in soft
X-rays by Shimizu (1995) using Yohkoh data. In the quiet Sun, small transient
brightenings have been surveyed in the EUV using SOHO/EIT (Benz & Krucker,
2002) and TRACE (Parnell & Jupp, 2000; Aschwanden et al., 2000). While these
data sets are quite disparate in instrument, active region, and solar cycle
timing, a rough power law is evident, with small events occurring far more
frequently than large ones (see, e.g. Figure 2 in Hannah et al., 2011). It is
yet unclear if the smallest events release enough energy to play a major role
in heating the corona.

In recent years, new instruments have begun to demonstrate the dramatically
increased sensitivity available via direct HXR focusing as opposed to RHESSI’s
indirect imaging method (Hurford et al., 2002), with the first two flights of
the Focusing Optics X-ray Solar Imager FOXSI sounding rocket (Krucker et al.,
2014; Glesener et al., 2016) and occasional solar pointings by the Nuclear
Spectroscopic Array (NuSTAR) astrophysics spacecraft (Harrison et al., 2013;
Grefenstette et al., 2016). Focusing HXR instruments, with their larger
effective areas and drastically reduced detector backgrounds, can measure
flares of smaller temperatures, brightnesses, and total energies than those
available to indirect imagers.

We report here an observation of a small HXR microflare near the west limb on
2015 September 1 observed by NuSTAR. Since the microflare produced only 5
RHESSI counts s-1 detector-1 (not enough to reconstruct an image), we believe
it to be fainter than any HXR flare in the current literature.

Figure 1: AIA images of the 2015 Sep 01 microflare. (Top row) View of the
microflare in several different filters close to the peak time. (Bottom row)
estimates of the Fe XVIII contribution to AIA channels over time using a
linear combination of 94Å, 211Å, and 171Å emission, showing the evolution of
hot plasma ($\sim$4–10 MK).

##  2 Observations

###  2.1 Overview of the event

NuSTAR is a NASA Small Explorer that uses directly focusing HXR optics to
observe faint astrophysical sources (Harrison et al., 2013). While not
explicitly designed for solar observing, NuSTAR’s high-sensitivity telescopes
can measure faint phenomena on the Sun during low-activity times, when best
use is made of the instrument’s limited throughput (Grefenstette et al.,
2016). Solar pointings are coordinated as targets of opportunity and occur
several times per year for a few orbits at a time.111Summary plots can be
found at http://ianan.github.io/nsigh_all/. Prime conditions include (among
other scenarios) a productive active region (AR) at the limb with an otherwise
quiet disk. This condition was met in early September 2015, and NuSTAR
observed the Sun for 8 orbits spread out through the mornings of 2015
September 01 and 02. The majority of the AR of interest, 12403, had occulted
by that time, with only a small portion remaining on the visible disk.

Around 04:00 UT on 2015 September 01, a microflare occurred in the unocculted
part of AR 12403. The X-ray brightness of the microflare was below the
sensitivity limit of the GOES soft X-ray Sensor (XRS), which is typically used
for flare brightness classification. The microflare was observed by the
Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (Lemen et
al., 2012) and was independently identified in NuSTAR movies. Figure 1 shows
images from several of the AIA coronal bandpass filters (top row) and (bottom
row) a sequence of images over time computed using a linear combination of
three AIA filters (94Å, 171Å, and 211Å) in order to estimate the Fe XVIII
contribution (formation temperature $log(T)\approx 6.9$), as in Del Zanna
(2013).

###  2.2 RHESSI HXRs

The event was too faint to register in the RHESSI flare list, but manual
inspection of the data identified a small count rate rise in the 4–9 keV range
co-temporal with a NuSTAR peak in the same range. Detector 1 shows the
clearest detection of the nine RHESSI detectors since several other detectors
had thresholds set too high to register the relatively low-energy flare.
Detector 1 emission reached a peak of 8.8 counts second-1 over a background of
4.3 counts second-1, for a total of $\sim$120 photons, not enough to produce
an image.

Figure 2: NuSTAR images of the microflare in two energy bands. The top two
rows show images from FPMA at 2-4 keV and 4-8 keV, and the bottom two rows
show the same for FPMB. Images have been integrated for one minute, livetime-
corrected, and smoothed over 17 arcseconds using a Gaussian kernel to reduce
statistical noise. The same intensity scale is used for all images in each
row. A diagonal gap across the source in the FPMB images is due to the space
between detector quadrants. By 4:07 UT (last column), the microflare has
subsided and only quiescent emission is observed. The 2-4 keV images show
bright emission from the quiescent active region in addition to the
microflare.

###  2.3 NuSTAR HXR data and pointing corrections

NuSTAR, with its high effective area ($\sim$700 cm2 at 5 keV) and minuscule
detector background has higher sensitivity than ever before available at HXR
energies (Harrison et al., 2013). However, as X-ray flux from active regions
and flares is far larger than the instrument throughput (800 counts s-1
maximum) we can generally only record a fraction of the incoming X-rays. The
livetime (defined as the fraction of time for which the detector is ready to
acquire an event) when observing the Sun is typically limited to a small
percentage (Grefenstette et al., 2016). For the 2015 September 01 microflare,
the livetime for NuSTAR’s Focal Plane Module A (FPMA; one of two NuSTAR
detector arrays) was 1.57% at a nonflaring time, dropping to 1.24% at the
peak. The corresponding livetimes for Focal Plane Module B (FPMB) were 1.18%
and 1.01%, respectively. All data shown in this paper are corrected for
livetime, which is measured on a one-second cadence, though error bars are
derived from the raw counts. The effect of this low livetime is that a
nominally one-minute observation has an effective exposure time of less than
one second.

NuSTAR’s solar pointing is uncertain to $\lesssim$1.5 arcmin due to the
forward-facing star camera being blinded by solar flux. This pointing offset
typically changes with orbital thermal changes. (More sudden pointing changes
due to changes in star camera combinations in use are not relevant for the
time period of this microflare.) In order to isolate the microflare region
over time (even at nonflaring times), NuSTAR images integrated over 12 seconds
were cross-correlated with AIA Fe XVIII images. Cross-correlations were
performed on a cadence of 12 seconds using an automated procedure. A smooth
curve was then interpolated through the points to determine the necessary
NuSTAR pointing adjustment as a function of time. This pointing correction
relies on the assumption that the majority of the NuSTAR and AIA Fe XVIII
emissions originate from the same location, a reasonable assumption given that
NuSTAR is highly sensitive to the Fe XVIII contribution to the AIA 94Å filter
($log(T)\approx 6.5-7.2$; see Lemen et al., 2012). The area used for cross-
correlation includes the entire active region, although the microflare
dominates at its peak time. The pointing adjustments (of 28–69 arcsec) derived
in this way have been applied to all images and regions selected for time
profiles and spectra shown in this paper.

###  2.4 NuSTAR HXR images, lightcurves, and spectra

NuSTAR images of the microflare over time are shown in Figure 2. FPMA and FPMB
data are shown separately, and two energy bands (2–4 keV and 4–8 keV) are
shown for each. Images have been integrated for one minute, livetime
corrected, and smoothed over 17 arcseconds using a Gaussian smoothing kernel.
A small space between detector quadrants produces a diagonal gap across the
FPMB image. By 4:07 UT (last column), the event has subsided and only
quiescent emission is observed. The 2-4 keV images show bright emission from
the quiescent active region in addition to the microflare, while the 4–8 keV
images show little emission except for the microflare itself.

Figure 3: (Lefthand panels) Evolution of NuSTAR spectral parameters throughout
the microflare, using the spectral fits shown to the right. Panels (a) and (b)
show the fit temperatures and emission measures. Panel (c) shows (red
triangles) the thermal energy of the flaring volume compared with (black
dashed line) the thermal energy of an equivalent quiescent volume based on
fits at non-flaring times. Panel (d) shows the GOES emission expected from the
microflare, with A0.1 and A0.01 levels shown for comparison (red dotted
lines). (Righthand panels) X-ray spectra fit simultaneously to FPMA and FPMB
during the microflare. Plots show FPMA and FPMB data points in blue and red,
respectively, as well as (dotted line) a fixed thermal component at the
quiescent level found in Section 2.4, held identically for all flaring
intervals, and (dashed line) a fit thermal component for the microflare. The
solid lines show the total fit models including all fit components. Vertical
dashed lines show the fit energy range. In the second two intervals there is
an excess in counts above the model above $\sim$5 keV; this excess emission
could be explained by a nonthermal component or by a small amount of much
hotter plasma.

We fit the NuSTAR spectra using the XSPEC spectral fitting software (Arnaud,
1996). Fits were performed simultaneously to FPMA and FPMB data, and the
pointing adjustments described in the previous section were applied in order
to select a consistent region for fitting. The region is a circle of radius 15
arcseconds centered at the microflare centroid location. First, we fit an
isothermal spectrum at 1-minute intervals throughout the time range shown in
the lefthand side of Figure 3. From these values, we identified seven
intervals that were obviously flaring. We excluded the flaring intervals and
computed the average temperature and emission measure for the nonflaring
times, which we call the quiescent parameters. Nonflaring fits are indicated
with black markers and error bars in panels A and B of Figure 3, and the
average quiescent values are indicated with a dashed line. Those quiescent,
nonflaring parameters were then held as a fixed thermal component during the
flaring times, while a second thermal component was fit to represent the
microflare (see righthand side of Figure 3). The resulting thermal fits at
flaring times are shown in red in panels A and B, while the fixed nonflaring
component is shown with black diamonds (and black dashed lines).

Next, we calculated the thermal energy $W_{T}$ of the microflare as
$W_{T}=3\sqrt{EM\,V}\,k_{B}T$, where $EM$ is the emission measure in cm-3, $V$
is the microflare volume, $k_{B}$ is Boltzmann’s constant, and $T$ is the
temperature. The volume was estimated from the AIA Fe XVIII images shown in
Figure 1 by considering the microflare loop to be a tube of roughly uniform
radius. We assumed the loop height to be perpendicular to the solar surface
and corrected the loop length for the projection due to its near-limb
position. The resulting volume is $\sim$3.2$\times 10^{26}$ cm3. We also
calculated the thermal emission of an equivalent volume of quiescent plasma,
i.e. using our quiescent background parameters and the same volume as the
microflare. In order to do this, we calculated a quiescent density from the
(nonflaring) emission measure using the area of the region included in the
NuSTAR spectroscopy (a circle of radius 15 arcseconds). We approximated the
line-of-sight extent from the longitudinal width of active region 12403 as
gleaned from the NOAA history; the resulting dimension of 102 Mm is a slight
overestimate. The calculated quiescent density is 5$\times 10^{8}$ cm-3. The
thermal energy of the plasma volume throughout the microflare (red triangles)
is shown in panel C of Figure 3, along with a dashed black line that indicates
the thermal energy of the equivalent volume of quiescent plasma at nonflaring
times. We find the microflare energy at its peak (2.4$\times 10^{27}$ ergs) to
be greater than, but within an order of magnitude of, the quiescent energy. A
unity filling factor was assumed in all density estimates. The quiescent
energy we have calculated is a lower limit since we observe only at the
highest temperatures; the likely presence of brighter but cooler plasma at
quiet times would increase the quiescent thermal energy.

In order to compare with a common measure of flare brightness, we estimate the
GOES XRS flux in the long and short wavelength bands that would be expected
given our measured temperatures and emission measures; see panel D of Figure
3. For reference, the A0.1 and A0.01 levels (long wavelength channel) are
indicated with red dotted lines. The microflare peaks at an estimated GOES
1.0–8.0 Å level of 1.4$\times$ 10-9 W m-2. In actuality, the microflare is not
observable by GOES in either channel due to the background contribution of the
rest of the solar disk and due to sensitivity and/or sampling limitations. We
have also estimated the emission observable by the six AIA coronal filters,
and find that the NuSTAR microflare peak brightness is consistent with the
measured emission in the 94Å filter, while the images in the other filters are
dominated by their responses to plasma cooler than the 6 MK microflare.

Figure 4: (a) NuSTAR time profiles in 30-second bins for the entire active
region in several energy bands. Data from FPMA and FPMB have been livetime-
corrected, background-subtracted, and added together. (b) The same profiles,
when normalized, show that the emission appears, on average, earlier with
higher energy. (c) RHESSI time profile for Detector 1 (D1) integrated over 4–9
keV. (d) Evolution of AIA Fe XVIII line emission for the four regions shown in
Panel (f). (e) NuSTAR contours, with preflare emission subtracted, coaligned
to and overlaid onto a reference AIA Fe XVIII image (at 04:00 UT). The NuSTAR
images are integrated across the flare peak (0358–0402 UT) and have been
smoothed over 17 arcseconds using a Gaussian kernel. Contour levels are 50,
70, and 90%. (f) Selected AIA regions in a flare-integrated Fe XVIII image.

###  2.5 High-energy excess in the impulsive phase

NuSTAR fits of binned count spectra for several intervals throughout the
microflare are shown in the righthand side of Figure 3. Fits were performed
simultaneously to FPMA and FPMB (blue and red data points, respectively)
including a fixed thermal component (dotted lines) at the background level and
a fitted thermal component for the microflare (dashed lines). Isothermal flare
components (plus the isothermal background component) fit the data well except
during the impulsive phase, $\sim$03:59–04:01 UT, where at high energies there
is an evident excess in counts above the model. This excess emission could be
explained either by a nonthermal component due to flare-accelerated electrons
with a rather flat power-law index of 3 or by a small amount of hotter plasma
(temperature 13 MK, emission measure 3$\times$1043 cm-3). However, both fits
are poorly constrained given the low statistics above 5 keV.

Since the pileup of photons arriving in quick succession could, in principle,
produce a high-energy excess, we checked the pileup probability as indicated
by the “non-physical” event grades; see Appendix C of Grefenstette et al.
(2016) for an explanation. Since no events associated with the microflare were
found to have non-physical grades, we conclude that pulse pileup does not
affect our spectra.

Spectral fitting was also performed to RHESSI data using the
OSPEX222https://hesperia.gsfc.nasa.gov/ssw/packages/spex/doc/ospex_explanation.htm
SSWIDL package. Detectors 1, 3, and 5 detected enough photons to produce a
spectrum. A thermal fit to the spatially-integrated, detector-summed data
result in a temperature of 13.1$\pm$4 MK and an emission measure of
(1.7$\pm$3.2$)\times$1044 cm-3 during the time interval of 03:59–04:00 UT in
the energy range 4–9 keV. This is a somewhat hotter temperature than NuSTAR
finds, but this is probably due to the high-energy excess present in this time
interval.

###  2.6 HXR and EUV evolution over time

Figure 4 examines the time behavior of the NuSTAR emission in various energy
bands. Panel (a) shows the evolution in 1-keV wide energy bins from 2 to 6
keV. Data are summed from FPMA and FPMB and have been livetime-corrected and
background subtracted. These same lightcurves are shown, normalized to their
respective maxima, in Panel (b). The emissions show a slightly earlier peak
time and faster decay with higher energy. While data from FPMA and FPMB have
been summed for better statistics, the FPMs individually show this trend.
Panel (c) shows 4-9 keV HXRs detected by RHESSI’s Detector 1, though the
emission is primarily $>5$ keV. The small numbers of counts are statistically
significant (5$\sigma$), and the time of the peak is roughly consistent with
that of the higher NuSTAR energy bands. Panel (d) shows time profiles from
four different regions of the AIA Fe XVIII images that are indicated in panel
(f). The most impulsive emission emanates from a compact source. The northern
section of the primary loop brightens next, and the lower and upper sections
of the long loop show more gradual behavior. Visual comparison of the NuSTAR
and AIA lightcurves suggest that the HXR emission emanates from the compact
sources in the northern region of the microflare, which are not resolved by
NuSTAR. See Panel (e) of Figure 4, though note that NuSTAR images have been
coaligned to AIA.

##  3 Discussion

To summarize the observations, NuSTAR successfully observed an extremely weak,
$\sim$A0.1 class microflare that reached a peak thermal energy of
2.4$\times$1027 ergs. NuSTAR data show a clear trend of earlier peak time with
higher HXR energies. The temperature rises quickly and falls slowly, while the
emission measure has a gradual rise and fall, suggesting impulsive energy
release early in the event, followed by gradual filling and draining of the
flare loop(s). Spectral HXR fits show a high-energy excess in the impulsive
phase. All of these features are consistent with those observed in larger
flares (e.g. those observed by RHESSI).

Figure 5 shows the NuSTAR temperature and emission measure at seven times
throughout the microflare compared with GOES (blue) and RHESSI (red)
microflares studied by Hannah et al. (2008) and Christe et al. (2008). The
NuSTAR microflare is cooler or fainter than all of the RHESSI microflares, and
the temperature at the peak time (6.4 MK at 4:00–4:01 UT) is approximately
half that of the typical RHESSI microflare (12.6 MK). In further comparison,
the SphinX instrument observed flares during its spatially-integrated
observations in 2009, including two small flares with emission measures of
$\sim$1046cm$-3$ and temperatures of 4–5 MK (Engell et al., 2011). The NuSTAR
microflare has an estimated loop length and volume of 70 Mm and 3.2$\times
10^{26}$ cm3. When compared with the loop lengths and volumes for the RHESSI
microflares (Figure 4 in Hannah et al. (2008)), we find that the small energy
release takes place in a loop that is not unusually short or low-volume.

Based on the Shimizu (1995) study of Yohkoh SXR transients, the expected
occurrence rate for microflares on the order of $10^{27}$ ergs is $10^{-52}$
s-1 cm-2 erg-1, which works out to $\sim$5.5 flares hour-1 for the entire
disk. Dividing this rate between the four named active regions on the disk on
2015 September 01 yields an expected rate of $\sim$1 flare hour-1 active
region-1. (Each NuSTAR orbit yields approximately one hour of solar observing
time.) No other AIA brightenings were visible by eye during the given orbit,
although the microflare region does exhibit a precursor brightening just
before the NuSTAR observation begins. Therefore, our observation of a single
microflare in this orbit is not unusual.

Figure 5: NuSTAR spectral fit temperature and emission measure compared with
GOES (blue) and RHESSI (red) microflares studied by Hannah et al. (2008). Data
points with error bars show the NuSTAR fit parameters for seven consecutive
1-minute intervals throughout the 2015 September 01 microflare. Black contours
give expected RHESSI counts s-1 detector-1.

While the microflare is fainter than any previously reported HXR flare, it is
clear that the event is not a single energy release. The AIA Fe XVIII emission
displays complexity, with a small, compact source brightening first, followed
by another, nearby, compact region. Either or both of these compact sources
could be small loops. In the longer loop, brightening progresses from low to
high altitudes. This is commonly observed in larger flares and, in those
cases, is usually interpreted as reconnecting of field at progressively higher
altitudes within a flare arcade (e.g. Gallagher et al., 2002).

Where does high-energy emission lie within this dynamic picture? The NuSTAR
time profiles in Figure 4 and the spectra in Figure 3 show an impulsive phase
of the microflare from 03:59–04:00 UT. It is here that the most significant
high-energy NuSTAR emission is observed, along with a high-energy excess that
cannot be accounted for by an isothermal model of the flaring emission. This
high-energy excess ($\gtrsim$5 keV in Figure 3) could be provided by
nonthermal electrons, as are commonly observed in the impulsive phase of
larger flares; if present the nonthermal power law must be quite flat, with an
index of $\sim$3\. Alternatively, the emission could be provided by a hot,
faint, thermal component (temperature 13 MK, emission measure 3$\times$1043
cm-3). In either case, the NuSTAR emission is likely associated with energy
release early in the microflare, and the compact AIA source discussed next
responds immediately to this energy release.

Melrose (1997) considered a model of reconnection between current-carrying
loops and found an energetically favorable configuration to be the
reconnection of two current-carrying loops at large angles to one other, with
closely-spaced footpoints of opposite polarity, so that the by-products are a
small, compact loop and a longer, overarching one. Our observations of this
microflare are consistent with that geometry. The compact source located at
approximately [928,-185] arcseconds could be the (unresolved) short post-
reconnection thermal loop. This loop is hot but small, and could quickly fill
with chromospheric plasma. The longer loop may take longer to fill due to its
length, accounting for the differing timescales in the lightcurves of the two
loops. Observationally, this microflare is consistent with the survey
performed by Nishio et al. (1997), which found two-loop interactions in most
of the 14 events they examined, as well as large asymmetries in the two loop
lengths.

Studies of such small events address the relationship between nanoflares and
flares. Since individual nanoflares are not currently observable, their
contribution to coronal heating is often addressed by assuming that they are
the low-energy end of a single power-law distribution of larger, already-
observed flares (see, e.g., Hannah et al., 2011). This extrapolation
explicitly requires that the physics of nanoflares be the same as the physics
of much larger flares. At a crude level, the mechanism for flares in the solar
atmosphere has a source of free energy, believed to be in the form of magnetic
fields, that accumulates slowly until it exceeds some threshold, at which time
the energy is released impulsively. This is consistent with the original idea
of Parker (1988) for nanoflares, in which current sheets arise due to
footpoint motion and increase in strength until the coronal magnetic field
change across the current sheet exceeds a threshold and triggers rapid
magnetic reconnection. The question of a trigger is crucial: for many years
magnetic reconnection theory focused on how to enable fast reconnection in
order to match the energy release rates observed in large flares, but
successfully doing so then raises the question of how to suppress reconnection
while energy is being stored between flares (e.g., Cassak et al 2006). An
obvious difference between nanoflares and larger events is in the volume of
magnetic field whose energy is released. Nanoflares are assumed to occur on
very small scales between almost parallel field lines, whereas the amount of
energy released in a large flare requires a large volume for storage and could
involve large field angles. It is not clear that the same trigger mechanism
should operate over such different regimes, and so it is not obvious that
flares and nanoflares should form opposite ends of a single distribution. It
is therefore critical to push flare measurements to smaller and smaller scales
in order to probe the transition between flare triggering regimes.

##  4 Summary

We have presented a NuSTAR microflare fainter than any in the previous HXR
literature. We estimate the microflare to be class $\sim$A0.1, with a peak
thermal energy of 2.4$\times 10^{27}$ ergs, similar to the quiescent energy of
an equivalent plasma volume. We observe several qualities common to larger HXR
flares, such as early, impulsive energy release followed by a gradual thermal
response, and even more gradual at higher altitudes. HXRs peak earlier with
higher energy and show a high-energy excess during the impulsive phase that is
due to either nonthermal electrons or faint, hotter plasma. We conclude that
flares do indeed scale down to extremely small energies and retain what we
customarily think of as “flarelike” properties.

We wish to emphasize that, while new to the literature, this microflare is not
unique. Other NuSTAR microflares will be the subject of upcoming papers,
including one by Wright et al. (2017) that shows a detailed differential
emission measure obtained via Hinode XRT coordination. We expect future NuSTAR
events with higher livetime and correspondingly better statistics. In
addition, the FOXSI sounding rocket has observed three microflares in its
first two flights (Glesener et al., 2016), one of which is estimated to be an
A0.5 class flare (Vievering et al., 2016). This set of observations suggests
that as more sensitive instruments are developed, even smaller flares will be
discovered, allowing more thorough understanding of flare energetics and
triggering, and the relationship between flares, microflares, and nanoflares.
Thorough discovery of this new flare regime must wait until a solar-dedicated
spaceborne mission with focusing HXR optics is realized.

Support for this work was provided by a NASA HSR grant NNX14AG07G, as well as
an NSF Faculty Development Grant (AGS-1429512) to the University of Minnesota.
S.K. acknowledges funding from the Swiss National Science Foundation
(200021-140308 and 200020-169046). I.G.H. is supported by a Royal Society
University Fellowship. The authors are grateful to the NuSTAR Science and
Operation teams for promoting and supporting solar observations, to Matej
Kuhar and Kathy Reeves for insightful comments on paper drafts, and to the
anonymous reviewer for excellent comments and questions that undoubtedly
improved the paper.

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