11institutetext: 1European Space Agency (ESA), European Space Research and
Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ, Noordwijk, The Netherlands.
2University of Geneva, 7, route de Drize, 1227 Carouge, Switzerland.
3Astronomical Institute of the University of Bern, Sidlerstrasse 5, 3012 Bern,
Switzerland

# IRIS burst properties in active regions††thanks: Associated data available
at: 10.5281/zenodo.7016311

C. J. Nelson1 L. Kleint2,3

###### Abstract

Context. Interface Region Imaging Spectrograph (IRIS) bursts are localised
features thought to be driven by magnetic reconnection. Although these events
are well-studied, it remains unknown whether their properties vary as their
host active regions (ARs) evolve.

Aims. In this article, we aim to understand whether the measurable properties
(e.g. frequency, area, location, spectral characteristics) of IRIS bursts are
consistent during the evolution of their host ARs.

Methods. We study 42 dense 400-step rasters sampled by IRIS. These rasters
each covered one of seven ARs, with each AR being sampled at least four times
over a minimum of 48 hours. An automated IRIS burst detection algorithm is
used to identify IRIS burst profiles in this work. Data from the Solar
Dynamics Observatory’s Helioseismic and Magnetic Imager are also used to
provide context about the co-spatial line-of-sight magnetic field.

Results. Of the rasters studied, 36 ($86$ %) were found to contain IRIS burst
profiles. Five ARs (11850, 11909, 11916, 12104, and 12139) contained IRIS
burst profiles in each raster that sampled them whilst one AR (11871) was
found to contain no IRIS burst profiles at any time. A total of 4019 IRIS
burst profiles belonging to 752 connected objects, which we define as parent
IRIS bursts, were studied. IRIS burst profiles were only detected within
compact regions in each raster, with these regions appearing to increase in
size as the host ARs aged. No systematic changes in the frequency of IRIS
burst profiles or the spectral characteristic of IRIS burst profiles through
time were found for these ARs. Finally, $93$ % of parent IRIS bursts with
areas between $1$ arcsec2 and $4$ arcsec2 were observed to occur co-spatial to
bi-poles in the photosphere.

Conclusions. IRIS bursts are small-scale brightenings which have remarkably
consistent spectral and spatial properties throughout the evolution of ARs.
These events predominantly form within the cores of larger and more complex
ARs, with the regions containing these events appearing to increase in size as
the host region itself evolves.

###### Key Words.:

Sun: activity; Sun: atmosphere; Sun: transition region; Sun: UV radiation

††offprints<EMAIL_ADDRESS>
##  1 Introduction

Small-scale burst activity has been widely observed across the broad
temperature range of the solar atmosphere, from Ellerman bombs (EBs) in the
photosphere (see, for example, Ellerman 1917; Vissers et al. 2013; Nelson et
al. 2015), through UV bursts in the transition region (as discussed by Peter
et al. 2014; Judge 2015; Young et al. 2018), to the recently discovered
campfires in the corona (e.g. Berghmans et al. 2021; Mandal et al. 2021;
Panesar et al. 2021). Each of these dynamic events is thought to be driven by
localised magnetic reconnection between opposite polarity magnetic fields,
with the properties of the reconnection (e.g. height in the atmosphere, amount
of energy released) seemingly dictating which type of burst is formed. Recent
numerical simulations have supported this assertion (see: Danilovic 2017 for
EBs; Hansteen et al. 2017 for UV bursts; and Chen et al. 2021 for campfires)
providing us with a clearer understanding of the physical mechanisms
responsible for these events. Despite this general understanding, there is
still much we don’t know about these features, including whether UV bursts are
detected in all active regions (ARs) and whether their properties (potentially
diagnosing the physics of the magnetic reconnection driving these events)
change during the evolution of the host ARs.

A new sub-set of UV bursts, characterised by large increases in intensity in
spectral lines sampling the transition region (including the Si IV $1394$ Å
and $1403$ Å lines), were identified recently in data from the Interface
Region Imaging Spectrograph (IRIS; De Pontieu et al. 2014) by Peter et al.
(2014). Those authors studied a raster sampling AR $11850$ and found four
burst events, which were later named IRIS bursts in the literature, co-spatial
to bi-poles in the photosphere. It was suggested that these events could be
driven by magnetic reconnection heating the local plasma to temperatures close
to $80$ kK. The presence of several chromospheric absorption lines (e.g. Ni I
$1393$ Å) over-laid on the IRIS burst spectra, however, indicated that these
events occurred deep in the solar atmosphere, potentially at heights as low as
several hundred km. Subsequent research found that IRIS bursts could occur co-
spatial to EBs (Vissers et al. 2015; Tian et al. 2016), which are known to
have temperatures well below the typical formation temperatures of the Si IV
lines (see, for example, Fang et al. 2006; Hong et al. 2014), and cancelling
bi-poles (e.g. Nelson et al. 2016) in the photosphere. Dedicated 1D modelling
efforts have so far been unable to reproduce co-spatial EB and IRIS burst
spectra, in part due to their large temperature differences (e.g. Reid et al.
2017; Hong et al. 2017).

In order to better assess the overall impact of IRIS bursts on the global
solar atmosphere, Kleint & Panos (2022) conducted an extensive statistical
analysis of IRIS burst profiles using the more than $3500$ IRIS spectral
datasets sampled during 2013 and 2014 (utilising the methods detailed in Panos
et al. 2021 and Panos & Kleint 2021). Those authors considered a Si IV $1394$
Å spectra to be IRIS burst-like if absorption lines were over-laid on it,
agreeing with one of the key observational characteristics of these events as
noted by Young et al. (2018). More than $100$ k IRIS spectra satisfied this
criteria (within $287$ datasets), indicating that around $0.01$ % of all Si IV
$1394$ Å spectra recorded by IRIS are IRIS burst profiles. Of the sample
reported, no IRIS burst profiles were detected in the quiet-Sun or close to
the solar poles suggesting that these events occur exclusively in ARs. It is
likely, therefore, that the potential quiet-Sun IRIS burst presented by Nelson
et al. (2017) would not have been classed as being IRIS burst-like by this
algorithm (if it had been applied to that dataset) due to the lack of clear
absorption lines over-laid on the relatively narrow Si IV $1394$ Å line.
Finally, no systematic increase in emission co-spatial to IRIS burst profiles
was found either for the IRIS Fe XXI line or for the EUV lines sampled by the
Solar Dynamics Observatory’s Atmospheric Imaging Assembly (SDO/AIA; Lemen et
al. 2012) by Kleint & Panos (2022) suggesting that the magnetic reconnection
hypothesised to be driving these events does not heat the local plasma to
coronal (¿106 K) temperatures.

Although we focus specifically on IRIS bursts in this article, it should be
noted that a whole host of other burst events have also been detected in the
solar atmosphere in spectral windows sampling similar temperature ranges. This
wider family of events are collectively known as UV bursts, and include
features such as explosive events (Dere et al. 1989) and blinkers (Harrison
1997). Whilst one of the unique defining properties of IRIS bursts are the
characteristic overlying absorption lines on the Si IV $1394$ Å spectra, other
types of UV bursts have a plethora of other spectral shapes (some are
extremely narrow whilst some are extremely broad) and properties (some are
symmetric about the spectral line whilst some display broadenings in only the
red or blue wings). This large variance in spectral signatures means other UV
burst types will not be identified by the version of the algorithm developed
by Kleint & Panos (2022). The inclusion of more template spectra would be
required if these other types of UV bursts were to be studied in an automated
manner in the future. The spectral differences between distinct types of UV
bursts also means that the defining properties of this family of events must
be taken from other observable properties, with key signatures being that they
are: i) compact (areas typically below $2$″); ii) short-lived (lifetimes of up
to around one hour); iii) intense compared to the local background; iv)
relatively stationary; and v) not linked to larger-scale flares. Given these
properties it is, therefore, possible for an event to display an IRIS burst
profile but not be a UV burst. For a comprehensive review of both IRIS bursts
and the wider family of UV bursts see Young et al. (2018).

The algorithm developed by Kleint & Panos (2022) provides numerous
opportunities for advancing our understanding of the importance of IRIS bursts
in the solar atmosphere. In this article, we utilise a slightly modified
version of this algorithm to investigate how the measurable properties of IRIS
bursts (e.g. frequency, area, location, spectral characteristics) in large
IRIS rasters vary during the evolution of seven ARs. The motivation behind
this research is to identify whether there are preferential conditions for
when and where magnetic energy is released in ARs as they evolve. We structure
our work as follows: In Sect 2 we detail the data products analysed and
outline the changes made to the IRIS burst detection algorithm; In Sect. 3 we
present the results of our research and a brief discussion; before in Sect. 4
we draw our conclusions.

##  2 Methods

###  2.1 Observations

The primary data products analysed in this research were sampled by IRIS, a
highly versatile instrument which can be employed in a plethora of observing
modes allowing the user to vary the spatial, temporal, and spectral
resolutions of the returned data based on scientific need. This versatility
creates a comprehensive data catalogue but also means we must filter the
available datasets in order to isolate only those of interest for this
particular study. Here, we focus our analysis on ARs that were sampled by a
minimum of four separate $400$-step IRIS rasters in the time-period studied by
Kleint & Panos (2022). These observations must include the Si IV $1394$ Å
line, and have time differences between the first and final rasters of at
least 48 hours. Seven ARs (11850, 11856, 11871, 11909, 11916, 12104, and
12139) satisfied these criteria, forming the sample we analyse here. Of these,
two ARs (11871 and 11916) emerged onto the solar disk between one and three
days prior to their first IRIS raster scans, whilst the remaining five (11850,
11856, 11909, 12104, and 12139) rotated onto the solar disk as fully formed
ARs between four and six days prior to the first IRIS raster scans which
sampled them.

Figure 1: Locations on the Sun of each of the IRIS burst profiles studied in
this article, with different colours corresponding to different ARs. IRIS
burst profiles are detected in multiple rasters sampled over multiple days for
six of the seven ARs. Clearly, no IRIS burst profiles were detected at any
time within AR 11871. General information about the identified IRIS burst
profiles is presented in Table 1.

Figure 2: Example of IRIS burst profiles missed by the algorithm. (Left-hand
panel) Intensity of the FOV during the fourth IRIS raster sampling AR $12104$
at the core wavelength of the Si IV 1394 line (logarithmically scaled to
enhance contrast). Our algorithm did not identify any bursts in this raster.
The two times during which IRIS passed through the SAA, thereby causing noisy
spectra, during this observation are marked by red arrows. The orange circles
locate two pixels whose spectra are shown in the right-hand panels (black
lines), one containing a manually detected burst and the other containing no
burst spectra. (Right-hand panels) The two IRIS spectra selected to
demonstrate how a small number of IRIS burst profiles may be missed by the
algorithm. The top panel plots a manually detected IRIS burst with a clear,
albeit shallow, absorption blend over-laid on the Si IV $1394$ Å line, while
the bottom panel is a typical non-burst spectrum. Both spectra are normalised
to their peak intensity. We over-plot the mathematically closest reference
spectrum considered by the algorithm to each of these profiles (blue lines),
which in both cases is a non-burst reference spectrum. Therefore, no IRIS
burst profile is found for either pixel by the algorithm. The nearest IRIS
burst reference spectrum to the manually detected IRIS burst profile is also
over-plotted on the top panel for reference (orange line).

Overall, $42$ IRIS datasets were analysed in this study. Each of these
datasets had a raster step size of $0.35$″ and a pixel scale along the slit of
$0.17$″, allowing us to consistently measure the properties of IRIS bursts.
General information about each of these datasets is provided for reference in
Table 1. From this sample, $35$ datasets were found to contain a combined
total of $4019$ IRIS burst profiles using the modified IRIS burst detection
algorithm. This accounts for $0.02$ % of the total number of IRIS Si IV $1394$
Å profiles analysed here. To supplement this automated detection, we also
conducted a manual analysis of the remaining seven datasets using CRISPEX
(Vissers & Rouppe van der Voort 2012) in order to identify whether any clear
and obvious IRIS burst profiles were missed by the algorithm. IRIS burst
profiles, identified through the presence of absorption lines on the Si IV
$1394$ Å spectra for consistency, were detected in one further dataset through
these checks. The remaining six datasets did not appear to contain any clear
and obvious IRIS burst profiles (we cannot definitively state that a small
number of pixels did not evade our manual search which will have been biased
towards more intense pixels). The locations on the Sun of each of the IRIS
burst profiles detected using automated methods are plotted in Fig. 1. The
different coloured boxes correspond to different ARs, as detailed in the
legend.

Given longer exposure times may lead to an increase in the number of saturated
pixels (which are removed from the sample before the algorithm is applied), it
is also important to investigate the numbers of potential IRIS burst profiles
removed from each of these datasets before any additional analysis is
conducted. In order to test this, we identified all pixels which were
saturated (counts$>16180$) at one or more spectral positions in the wavelength
range $1393.75\pm 0.2$ Å. In total $239$ pixels matched this criterion across
all $42$ datasets with approximately $100$ of these appearing to be caused by
cosmic ray spikes and another $100$ appearing to be some other kind of burst,
with relatively narrow spectra and no absorption lines over-laid on the Si IV
$1394$ Å spectra. This, therefore, left around $40$ potential IRIS burst
profiles which were spread across four datasets. Around half of these
saturated potential IRIS burst profiles were found in the first dataset
sampling AR $11916$, which already contained $715$ IRIS burst profiles, whilst
the others were detected in the seventh raster to sample AR $11916$, the first
raster sampling AR $12104$, and the third raster sampling AR $12104$. Overall,
the small numbers of saturated pixels displaying potential IRIS burst profiles
(less than $1$ % of the sample studied here) and the centralisation of these
to only a small number of datasets indicates that the removal of saturated
pixels does not overly influence the results obtained in this manuscript.

We also made use of data from SDO’s Helioseismic and Magnetic Imager (SDO/HMI;
Scherrer et al. 2012) in order to study the line-of-sight (LOS) magnetic field
co-spatial to the IRIS observations. Full-disk LOS magnetic field maps were
downloaded with a cadence of one hour in the time-period spanning from six
hours prior to the start time of the first IRIS raster to six hours after the
end time of the last IRIS raster for each AR. A zoomed $180$″$\times 180$″
field-of-view (FOV) cut-out was then created co-temporal to the first IRIS
scan centred on the pointing co-ordinates of that raster. All other SDO/HMI
frames were then de-rotated to that time, before a co-spatial cut-out was
constructed in order to create a consistent FOV through the entire time-
period. Comparisons of these SDO/HMI maps with plage regions in the IRIS
rasters indicated that these alignments were reasonable (estimated errors
$<5$″). As we only study the overall behaviour of the magnetic field within
the entire ARs and the general structure of the magnetic field co-spatial to
the IRIS burst profiles through time, no further alignment between instruments
was conducted. These SDO/HMI data have a post-reduction pixel scale of $0.6$″
which is more than adequate for investigating the overall behaviour of these
ARs.

###  2.2 Modifications to the IRIS burst detection algorithm

To detect IRIS burst profiles, the algorithm of Kleint & Panos (2022) was used
with slight modifications applied in order to both be able to study rasters
that may have a small number of IRIS burst profiles and fix two minor bugs,
which became apparent during this analysis. To tackle the first issue, the
criterion of requiring at least 30 potential IRIS burst profiles applied by
Kleint & Panos (2022) was dropped and any observation that had at least one
candidate IRIS burst profile was analysed further. This increased the sample
of IRIS burst profiles detected within these $42$ rasters from $2462$ to
$3113$ (a $26.44$ % increase). Following this adaption, the first bug, related
to times when the IRIS orbit passes through the South Atlantic Anomaly (SAA)
and hence returns noisy spectra, became apparent in the results. Specifically,
the IRIS FITS SAA level2 keyword defaults to the value that is true for most
of the observation. Therefore, observations where more time was spent in the
SAA than outside of the SAA (which is very rare) will have this keyword set to
1 at all times and were, thus, excluded in the first version of the pipeline.
The first version of the pipeline correctly queried the level1 data, which
includes the actual flag for SAA/no SAA and updated the corresponding header
entry in the level2 data to 1 during times of SAA. However, it did not check
if the header entry may already be set to 1 due to the convention of the
level2 data for SAA dominant datasets, even though no SAA occurred at the
specific time when the spectra were sampled. The new version of the pipeline
does both checks.

The second bug is related to the fact that the Si iv observing window width
changed slightly with the evolution of the IRIS linelists, and between the
different types of linelists. The original pipeline had a bug with
extrapolations of spectra, leading to unintended values far away from the Si
iv lines. This did not affect strong bursts (since the signal is near the line
center and the continuum is close to zero due to the normalization), which
have always been classified properly, but it could mistakenly classify noisy
burst spectra into a non-burst category. Extrapolated values are now properly
set to zero and thus do not affect the classification. The classification in
Kleint & Panos (2022) focused on minimizing false-positives and if in doubt, a
spectrum was excluded from being of burst-type so this should not have
influenced their results excessively. Applying the algorithm following these
subsequent modifications increased the sample further from $3113$ IRIS burst
profiles to a total of $4019$ IRIS burst profiles (a further increase of
$29.10$ %). These new IRIS burst profiles mostly have very weak absorption
lines and/or are noisy, but do visually look like bursts so are included here
for completeness. Notably, these events were detected across a total of $35$
datasets, up from $15$ using the algorithm before removal of the $30$ spectra
criterion and the subsequent modifications.

Figure 3: Context plots for three of the ARs, namely ARs 11850, 11871, and
11916 from top to bottom, studied. (Left-hand column) Zoomed $100$″$\times
100$″ SDO/HMI magnetic context images. The plotted frames were selected as
they contained the peak total unsigned magnetic flux for these regions during
the time studied. (Middle column) The Si IV $1394$ Å line core intensity
sampled from the IRIS raster with start time closest to the SDO/HMI images
plotted in the left hand column. The pointing of the IRIS FOV was de-rotated
to the SDO/HMI observation time. (Right-hand column) The evolution of the
total unsigned (solid lines), positive (dashed lines), and negative (dotted
lines) magnetic flux within the wider $180$″$\times 180$″ FOV through the
time-period studied here for each AR. The red contours over-laid on the left-
hand column outline the IRIS burst profiles returned using automated methods
from the IRIS raster plotted in the respective middle column. The five ARs
with the highest peak magnitudes of the unsigned flux were found to contain
the majority of the IRIS burst profiles identified here. Figure 4: Structuring
of IRIS burst profiles in these datasets. (Left-hand panel) Binary map showing
the locations of IRIS burst profiles (black pixels) identified within a single
raster sampling AR 11850. Four of the parent IRIS bursts within this raster
correspond to the four events studied in Peter et al. (2014). The over-lying
cut-out displays how connected objects are combined to form parent IRIS
bursts, with nine events (all individually coloured) occurring in this small
region. (Middle panel) Histogram of parent IRIS burst area against frequency
constructed using all $42$ datasets, where the $y$-axis is plotted in
logarithmic scale. Each bar is binned over a range of $0.3$ arcsec2
(corresponding to five pixels). (Right-hand panel) Bar chart plotting the
number of parent IRIS bursts by AR. The different coloured sections of each
bar indicate the number of parent IRIS bursts identified in different rasters
sampling that AR, with the light yellow at the bottom corresponding to the
first raster within which IRIS bursts were detected using automated methods
for that specific AR. The progressively darker colours up the bars correspond
to the subsequent rasters within which IRIS bursts were detected. The physical
number of bursts plotted in each bar segment can be found in Table 1.

It should be mentioned, however, that the variety of burst spectra makes it
impossible to catch every single burst with a classification template of
“only” 423 different spectral types. We are confident, though, that the most
common types of burst spectra are identified reliably. An example of an IRIS
burst profile which was not detected by the pipeline is displayed in Fig. 2.
In the left-hand panel we plot the intensity of the Si IV $1394$ Å line core
for the fourth raster sampling AR 12104 (i.e. the raster where IRIS bursts
were detected manually but not automatically). The locations where IRIS passed
through the SAA during this observation are marked by the red arrows. The
orange circles outline two pixels, one containing a manually detected IRIS
burst profile and one containing a non-burst profile, chosen for reference. In
the right-hand panels, the black lines plot the observed spectra at each of
these pixels with clear, albeit shallow, absorption blends indicative of an
IRIS burst profile visible on the top panel. The blue lines plot the reference
spectra deemed to be the closest mathematically to the observed spectra which
in both cases is a non-burst reference spectra. The orange line included on
the top panel plots the closest IRIS burst reference spectrum to this
spectrum, which was mathematically ‘further’ from the observed spectrum than
the non-burst reference spectrum (distance of 0.83 compared to a distance of
0.52). The only way to improve detections of such bursts would be to increase
the number of template reference spectra from the 423 currently used. Given we
only found a very small number (around ten) of additional IRIS burst profiles
through our manual checks, and adding further reference spectra risks
increasing the rate of false detections, we did not deem this necessary for
this study.

##  3 Results and discussion

###  3.1 General AR properties

To begin our analysis, we investigated the general properties of each AR in
order to better contextualise any subsequent inferences about IRIS bursts.
Firstly, we identified the number of raster scans which displayed IRIS burst
profiles for each AR finding varying behaviour. Of the seven ARs studied here,
five (11850, 11909, 11916, 12104, and 12139) contained IRIS burst profiles in
each raster scan, including both IRIS burst profiles detected using automated
methods and those identified through the subsequent manual analysis. AR 11916
contained the most automatically detected IRIS burst profiles, with $2052$
($51.06$ % of the total sample studied here) being identified over the eight
rasters analysed, whilst the other four ARs each contained between $1144$ and
$168$ automatically detected IRIS burst profiles. On top of this, one AR
(11856) was found to contain IRIS burst profiles in some but not all rasters
and one AR (11871) was found to contain no IRIS burst profiles at any time.
This analysis indicates that the age of the AR is not a defining factor in
whether IRIS burst profiles are present with the two youngest ARs (11871 and
11916), which both emerged onto the solar disk in the three days prior to the
IRIS scans, displaying sharply contrasting behaviour.

Secondly, we calculated the total unsigned magnetic flux within a zoomed
$180$″$\times 180$″ FOV around each AR at each SDO/HMI time-step.
Interestingly, the five ARs which contained IRIS burst profiles in each raster
had the five highest peak fluxes ($1.44\times 10^{21}$ Mx, $1.66\times
10^{21}$ Mx, $1.91\times 10^{21}$ Mx, $2.23\times 10^{21}$ Mx, and $1.50\times
10^{21}$ Mx for ARs 11850, 11909, 11916, 12104, and 12139, respectively) with
each exceeding $10^{21}$ Mx. The remaining two ARs, which contained either a
small number or zero IRIS burst profiles, both had peak fluxes of less than
$10^{21}$ Mx ($8.60\times 10^{20}$ Mx and $5.38\times 10^{20}$ Mx for ARs
11856 and 11871, respectively). In the left-hand column of Fig. 3, we plot
$100$″$\times 100$″ zoomed SDO/HMI images sampled at the time at which the
peak unsigned magnetic flux was measured for three ARs (11850, 11871, and
11916 from top to bottom, respectively). The middle column of Fig. 3 plots the
IRIS Si IV $1394$ Å line core intensity across this FOV sampled at the closest
raster to the SDO/HMI images, with the pointing corrected for solar rotation.
The red contours over-laid on the SDO/HMI images outline the locations of the
IRIS burst profiles identified in these rasters using automated methods. The
right-hand column of Fig. 3 plots the evolution of the total unsigned magnetic
flux (solid line), positive flux (dashed line), and negative flux (dotted
line) calculated from the larger $180$″$\times 180$″ SDO/HMI FOV with time.
Clearly, ARs 11916 and 11850 are more complex in their magnetic structuring
(left-panels of Fig. 3) and have total unsigned magnetic fluxes which
initially increase before reaching a peak and then decreasing (right-panels of
Fig. 3). On the other hand, AR 11871 is less complex and has a total unsigned
magnetic flux which slowly decreases with time. Our results indicate that the
total unsigned magnetic flux and the complexity of the structuring of that
flux within an AR may be a dominant factor determining whether IRIS bursts are
widely detected or not, rather than the age of the AR.

###  3.2 Spatial properties of IRIS burst profiles

Figure 5: Spatial positions of IRIS burst profiles through time in two
datasets. (Left-hand panel) Map of the Si IV $1394$ Å line core intensity
(logarithmically scaled) within AR 11850 sampled during the raster starting at
05:59:43 UT on 26th September 2013. (Right-hand panel) Same as the left-hand
panel but for AR 11916 sampled during the raster starting at 03:09:49 UT on
7th December 2013. The contours and crosses over-laid on both panels denote
the IRIS burst profiles and their centres-of-mass, respectively, for each
raster containing more than 100 automatically detected IRIS burst profiles.
The colours indicate different rasters and are consistent with those used in
the right-hand panel of Fig. 4.

We continued our analysis by grouping the $4019$ IRIS burst profiles
identified here into separate, connected objects which we define as parent
IRIS bursts. To achieve this, we created binary maps of the locations of IRIS
burst profiles within each raster scan and then combined IRIS burst profiles
which were adjacent in the $x-y$ plane together into parent IRIS bursts. If
two parent IRIS bursts were separated by even one pixel, we considered that
they were different events. Additionally, if the observational routine had
more than one raster then only the middle raster was considered here in order
to avoid longer-lived parent IRIS bursts, potentially present in repeated
rasters, being counted multiple times. Importantly, our analysis of these
parent UV bursts does not include any temporal component. This stipulation
only removed $119$ IRIS burst profiles from our sample meaning it should not
have a major influence on our results. Overall, this analysis returned $752$
parent IRIS bursts spread across the $35$ IRIS rasters. In the left-hand panel
of Fig. 4, we plot an example of a binary map displaying the locations of
numerous IRIS burst profiles from one specific raster within AR 11850. This
plot is calculated from the dataset which was used by Peter et al. (2014) to
originally identify IRIS bursts, with the four largest parent IRIS bursts
corresponding to the four events studied by those authors. This clearly shows
the effectiveness of the algorithm developed by Kleint & Panos (2022) and
modified here. The zoomed cut-out over-laid on this panel more clearly
displays how adjacent IRIS burst profiles are combined to form separate parent
IRIS bursts. Nine parent IRIS bursts are present within this small FOV, with
each event being plotted in a different colour.

In the middle panel of Fig. 4, we plot a histogram of parent IRIS burst area
against frequency for all of the $752$ identified events. The histogram is
binned across $0.3$ arcsec2 (five pixels) and the $y$-axis is plotted using a
logarithmic scale. Clearly, the majority of the studied events are small, with
$383$ of these parent IRIS bursts having areas of only one pixel. The mean
area of this sample of parent IRIS bursts is $0.31$ arcsec2 (corresponding to
5.19 pixels); however, the distribution is non-Gaussian and the standard
deviation of this sample is $0.72$ arcsec2 (12.18 pixels), more than double
the mean, suggesting one should not interpret this as a precise estimate. Our
results do, though, clearly indicate that the majority of parent IRIS bursts
are extremely small-scale. Notably, several larger brightenings in the Si IV
$1394$ Å line core are found to contain numerous parent IRIS bursts through
our analysis. For example, one $6.30$″$\times 2.72$″ region in the second
raster sampling AR 12104 (around $x_{\mathrm{c}}\approx 200$″,
$y_{\mathrm{c}}\approx-296$″) was found to contain $20$ parent IRIS bursts,
with an average area of $0.12$ arcsec2 (corresponding to 1.57 pixels). From
the entire sample studied here, only 52 parent IRIS bursts (6.91 %) have an
area over $1$ arcsec2 (corresponding to 17 pixels) with 28 of these occurring
in AR 11916. The largest parent IRIS burst occurred in AR 11916 and had an
area of $8.98$ arcsec2 (corresponding to $151$ pixels), much larger than the
typical sizes of UV bursts in general (see Young et al. 2018).

Figure 6: Distributions of various spectral parameters for all IRIS burst
profiles identified within datasets sampling ARs 11850 (left-hand panels) and
11916 (right-hand panels). Only datasets which returned more than $100$ IRIS
burst profiles were considered. (Top panels) Logarithmic mean intensity from a
$\pm 0.075$ Å window surrounding the Si IV $1394$ Å line core calculated for
each IRIS burst profile. (Middle panels) Logarithmic maximum intensity for
each IRIS burst profile. (Bottom panels) Spectral location of the maximum
intensity of each IRIS burst profile. The diamonds indicate the mean for each
raster, while the upper and lower crosses indicate the $75$th and $25$th
percentiles, respectively. No systematic or monotonic changes in these
parameters through time are apparent.

In the right-hand panel of Fig. 4, we plot a bar chart displaying the number
of parent IRIS bursts identified per AR. The colours in each bar indicate
parent IRIS bursts identified in different rasters, with the bottom (light
yellow) portion corresponding to the first raster within which IRIS burst
profiles were identified for that specific AR. A high proportion of the parent
IRIS bursts were found within ARs 11850 and 11916 ($70.61$ % of the total
sample) whilst AR 11871 contains zero parent IRIS bursts. The highest number
of parent IRIS bursts in a single raster was 65, which were identified in the
first raster sampling AR 11916, with the mean number of parent IRIS bursts per
raster being 17.90. The average area of the parent IRIS bursts was found to
change in a seemingly random way from raster to raster (neither monotonically
increasing or decreasing) as the host ARs evolved. In total, ten rasters
within the sample analysed here contained more than $30$ parent IRIS bursts,
with five of these occurring in AR 11916, three occurring in AR $11850$, and
one occurring in both ARs $12104$ and $12139$. Overall, these results imply a
parent IRIS burst spatial frequency of $0.00069$ arcsec-2 in ARs, well below
the spatial frequency of, for example, spicules in the chromosphere. See Table
1 for a summary of these findings.

###  3.3 Locations of IRIS burst profiles within ARs

One of the main aims of this research is to identify whether IRIS burst
profiles are detected randomly across ARs or whether they are confined to
consistent localised regions through time. In order to analyse this, we
studied the locations of IRIS burst profiles within the two ARs (namely ARs
11850 and 11916) for which more than 100 IRIS burst profiles were detected
using the algorithm in more than one raster. We de-rotated each raster to a
consistent pointing which we converted as the origin (0″, 0″), constructed
binary maps of the locations of IRIS burst profiles, and then calculated the
centres-of-mass of IRIS burst profiles for each raster. In Fig. 5, we plot a
visualisation of this analysis on maps of the Si IV $1394$ Å line core
intensity (logarithmically scaled) from one representative raster for each AR
(11850 in the left-hand panel and 11916 in the right-hand panel). The over-
laid contours denote the locations of IRIS burst profiles and the crosses
indicate the centres-of-mass for each specific raster. The relationship
between colour and raster number is consistent with the colours used in the
right-hand panel of Fig. 4.

Figure 7: LOS magnetic field maps co-spatial to the $46$ parent IRIS bursts
with measured areas between $1$ arcsec2 and $4$ arcsec2. Each plot covers a
region of $5$″$\times 5$″ centred on the location of the parent IRIS burst
from the closest SDO/HMI image in time, de-rotated to the time of IRIS burst
profile detection. The red and blue contours outline regions of positive and
negative magnetic fields saturated at $20$ G and $-20$ G, respectively, whilst
the orange contours outline the approximate locations of the parent IRIS
bursts. Opposite polarity magnetic fields are found within the FOV for $43$
($93$ %) of these parent IRIS bursts.

For both ARs, IRIS burst profiles were found to be predominantly localised to
relatively small regions, with approximate areas of $100$″$\times 60$″. The
centres-of-mass of IRIS burst profiles also appeared to be consistent through
time for both ARs, only varying by a maximum of $31.11$″ in AR 11850 and
$37.96$″ in AR 11916. In general, the locations of IRIS burst profiles within
the ARs appeared to be limited to the most pronounced polarity inversion
lines, as would be expected for events hypothesised to be driven by magnetic
reconnection. Interestingly, IRIS burst profiles in both of these ARs appeared
to become more spatially separated through time, with the contours in Fig. 5
appearing further away from the centres-of-mass for subsequent rasters. In
order to investigate this, we calculated the average separation between the
IRIS burst profiles and the centres-of-mass through time for the rasters
containing more than $100$ IRIS burst profiles. For AR 11850, this separation
increased from $13.00$″ ($\pm 10.42$″) to $45.46$″ ($\pm 9.20$″) over the five
rasters studied here; whilst for AR 11916, the average separation increased
from $9.51$″ ($\pm 10.32$″) to $32.10$″ ($\pm 15.07$″) over the first five
rasters before decreasing again to $18.63$″ ($\pm 15.04$″) by the sixth
raster. Although this analysis indicates there may be an increase in the
separation of IRIS burst profiles within an AR as it evolves, we note that two
ARs were studied and large standard deviations were calculated, meaning this
result is still tentative and requires further analysis using a larger sample
of ARs in the future.

###  3.4 Spectral properties of IRIS burst profiles

To continue our analysis, we next investigated whether several spectral
properties of IRIS burst profiles varied systematically during the evolution
of ARs or whether they remained relatively consistent. Once again, only
rasters within ARs $11850$ (left-hand panels) and $11916$ (right-hand panels)
which contained more than $100$ IRIS burst profiles were studied in order to
provide appropriate statistics. In Fig. 6, we plot the distributions of three
parameters calculated from the Si IV $1394$ Å spectra sampled at all pixels
automatically identified as IRIS burst profiles, for the appropriate rasters.
The top panels plot the mean intensity (logarithmically scaled) from a $\pm
0.075$ Å window around the Si IV $1394$ Å line core, the middle panels plot
the logarithmically scaled peak intensity of the Si IV $1394$ Å line for the
identified IRIS burst profiles, whilst the bottom panels plot the spectral
positions where the maximum intensity was measured for each IRIS burst
profile. The diamonds on each distribution indicate the mean value calculated
for that raster, and the top and bottom crosses plot the $75$th and $25$th
percentile values, respectively. Overall, although some changes in the
distributions are observed between specific rasters, no systematic changes
(e.g. monotonic intensity increases, transitions from blue to red shifts) in
these parameters is detected as the ARs age.

Following on from this, we also studied which reference spectra were returned
the most frequently as the ‘closest’ fits to the IRIS burst profiles
identified in each of these rasters. This analysis allowed us to infer whether
any changes in the common spectral shapes of IRIS burst profiles are present
through time. In these datasets, no single reference spectrum was found to be
the most accurate mathematical match to more than $7.34$ % of all of the IRIS
burst profiles within a single raster. Additionally, unique reference spectra
were found to make up, on average across these datasets, $47.57$ % of all of
the observed spectra indicating that a huge variety of spectral shapes of IRIS
burst profiles exist for each dataset. These two combined facts suggest that
no preferential reference spectrum is present in any given raster, with the
identified IRIS burst profiles being randomly spread between different
reference spectra.

###  3.5 Relations to the local LOS magnetic field

Our final analysis focused on the relationship between the parent IRIS bursts
identified in this study and the co-spatial under-lying LOS magnetic field
structure as inferred by the SDO/HMI instrument. It is well known that IRIS
bursts often occur co-spatial to regions of opposite polarity magnetic field
(see, for example, Young et al. 2018) where magnetic reconnection is thought
to be possible in the lower solar atmosphere. As the SDO/HMI instrument has a
spatial resolution of approximately $1$″, we only studied parent IRIS bursts
with areas greater than $1$ arcsec2 (corresponding to approximately 17
pixels). Additionally, we removed the parent IRIS bursts with areas above $4$
arcsec2 (corresponding to 70 pixels) as the foot-points of those events would
be difficult to define in a consistent manner. These stipulations left $46$
parent IRIS bursts for analysis. In Fig. 7, we plot the LOS magnetic field
saturated at $\pm 100$ G sampled for a $5$″$\times 5$″ FOV surrounding the
centre of each parent IRIS burst. The red and blue contours outline positive
and negative polarity LOS magnetic fields at values of $20$ G and $-20$ G,
respectively, while the orange contours outline the locations of the parent
IRIS bursts. The magnetic field map for each event was constructed by de-
rotating the closest temporal frame in the respective one hour cadence SDO/HMI
dataset to the time at which the parent IRIS burst was detected, meaning the
locations of the bursts with respect to the magnetic field are only an
approximation. Although this method will have some inherent errors on small-
scales, the alignment appears to work well for the full FOV (as can be seen in
Fig. 3) giving us confidence that a closer alignment is not required for this
general analysis.

Overall, $43$ ($93$ %) of the panels plotted in Fig. 7 contain bi-poles in the
LOS magnetic field within the $5$″$\times 5$″ surrounding the parent IRIS
bursts. For the three remaining events, one occurs co-spatial to an apparently
uni-polar region of positive polarity flux and the other two occur co-spatial
to apparently uni-polar regions of negative polarity flux. As the magnetic
field maps could have been measured up to $\pm 30$ minutes around the
detection of the parent IRIS burst, we note that these magnetic field maps may
not be entirely representative of the exact magnetic field structure at the
time of potential reconnection as, for example, cancellation could have
occurred removing evidence of any bi-poles (see for example, Nelson et al.
2016) for the three uni-polar events. Additionally, the spatial resolution of
the SDO/HMI instrument may be insufficient for resolving all of the magnetic
field elements at these locations (as discussed by Reid et al. 2016) which
could cause us to detect a uni-polar region when a bi-polar region was
actually present. It is clear, however, that the majority of parent IRIS
bursts with areas between $1$ arcsec2 and $4$ arcsec2 occur co-spatial to
regions of opposite polarity magnetic field in the lower solar atmosphere,
supporting the assertion that these events could be driven by magnetic
reconnection.

##  4 Conclusions

In this article, we used a slightly modified (see Sect. 2.2) version of the
algorithm originally applied in Kleint & Panos (2022) in order to investigate
whether changes in the measurable properties of IRIS burst profiles occurred
as their host ARs evolved. We studied $42$ dense IRIS rasters which each
sampled one of seven host ARs, with each AR being sampled at least four times.
We found:

  * •

The maximum total unsigned magnetic flux within an AR, and the complexity of
the structuring of that flux, may be important as an indicator of whether IRIS
burst profiles are present. The two ARs with the lowest peak total unsigned
magnetic fluxes contained far fewer IRIS burst profiles than the five ARs with
the highest unsigned magnetic fluxes. Specifically, AR $11871$ (with a peak
unsigned magnetic flux of $5.38\times 10^{20}$) was found to contain no IRIS
burst profiles in the four rasters which sampled it, whilst AR $11856$ (with a
peak unsigned magnetic flux of $8.60\times 10^{20}$ was found to contain only
$25$ IRIS burst profiles ($0.62$ % of the total number of IRIS burst profiles
studied here) across the five rasters which sampled it. The age of the AR was
not found to influence the number of IRIS burst profiles returned by the
algorithm, with the two youngest ARs ($11916$ and $11871$) being found to
contain the most and least IRIS burst profiles, respectively.

  * •

In total, $4019$ IRIS burst profiles were returned by the algorithm from these
$42$ rasters, with these being found to belong to $752$ parent IRIS bursts.
These parent IRIS bursts had a mean area of $0.31$ arcsec2, with $93.09$ % of
the identified events having areas below $1$ arcsec2 (centre panel of Fig. 4).
We note that the mean area should not be over-interpreted due to the large
standard deviation from this dataset ($0.72$ arcsec2) but it can be used to
show that these events are typically small-scale. The frequencies and areas of
these features did not monotonically increase or decrease as the host ARs
evolved but instead varied in a seemingly randomly manner from raster to
raster. The largest parent IRIS burst found through this analysis had an
apparent area of $8.98$ arcsec2 (corresponding to $151$ pixels). The overall
spatial frequency of IRIS burst profiles across the studied ARs was calculated
to be $0.00069$ arcsec-2, well below the coverage of spicules, for example.

  * •

IRIS burst profiles are typically confined to localised regions (around
$100$″$\times 60$″ for the two ARs studied here) close to the cores of ARs (as
is displayed in Fig. 5). The average spatial separation between the centre-of-
mass of all IRIS burst profiles within a raster and the individual IRIS burst
profiles themselves appeared to increase with time in ARs $11850$ and $11916$
(the only ARs which contained more than $100$ IRIS burst profiles in multiple
rasters), suggesting IRIS bursts occurred across a larger area as their host
ARs evolved. This increase in area potentially corresponds to the growth of
the ARs themselves. These results suggest that one must be very careful when
selecting appropriate targets for co-observations of IRIS burst-Ellerman bomb
pairs with ground-based telescopes with limited FOVs, e.g. with the Daniel K.
Inouye Solar Telescope (DKIST; Rimmele et al. 2020; Rast et al. 2021).

  * •

The spectral properties (e.g. peak intensities, spectral locations of the peak
intensities) of IRIS burst profiles varied in a non-monotonic way in datasets
sampling ARs $11850$ and $11916$ which contained more than $100$ IRIS burst
profiles, and then only slightly, through time. On top of this, the
automatically detected IRIS burst profiles were found to be spread across a
large number of the reference spectra used by the detection algorithm with, on
average, unique reference spectra equalling $47.57$ % of the total number of
IRIS burst profiles in any given raster. Additionally, no individual reference
spectrum was found to be the best fit for more than $7.34$ % of the IRIS burst
profiles returned for any given raster. These combined result indicate that no
‘typical’ IRIS burst spectra exists for the rasters studied here.

  * •

$93$ % of all parent IRIS bursts with areas between $1$ arcsec2 and $4$
arcsec2 occurred co-spatial to bi-poles in the solar photosphere. This result
further supports the assertion that these events are driven by magnetic
reconnection in the lower solar atmosphere (Peter et al. 2014; Vissers et al.
2015; Hansteen et al. 2017).

Expanding this research to include a larger number of ARs (i.e. those sampled
after 2014) will allow us to make further inferences about IRIS bursts in the
future.

###### Acknowledgements.

CJN is thankful to ESA for support as an ESA Research Fellow. LK gratefully
acknowledges funding via a SNSF PRIMA grant. IRIS is a NASA small explorer
mission developed and operated by LMSAL with mission operations executed at
NASA Ames Research Center and major contributions to downlink communications
funded by ESA and the Norwegian Space Centre. SDO/HMI data provided courtesy
of NASA/SDO and the HMI science team. This research has made use of NASA’s
Astrophysics Data System Bibliographic Services.

## References

  * Berghmans et al. (2021) Berghmans, D., Auchère, F., Long, D. M., et al. 2021, A&A, 656, L4
  * Chen et al. (2021) Chen, Y., Przybylski, D., Peter, H., et al. 2021, A&A, 656, L7
  * Danilovic (2017) Danilovic, S. 2017, A&A, 601, A122
  * De Pontieu et al. (2014) De Pontieu, B., Title, A. M., Lemen, J. R., et al. 2014, Sol. Phys., 289, 2733
  * Dere et al. (1989) Dere, K. P., Bartoe, J. D. F., & Brueckner, G. E. 1989, Sol. Phys., 123, 41
  * Ellerman (1917) Ellerman, F. 1917, ApJ, 46, 298
  * Fang et al. (2006) Fang, C., Tang, Y. H., Xu, Z., Ding, M. D., & Chen, P. F. 2006, ApJ, 643, 1325
  * Hansteen et al. (2017) Hansteen, V. H., Archontis, V., Pereira, T. M. D., et al. 2017, ApJ, 839, 22
  * Harrison (1997) Harrison, R. A. 1997, Sol. Phys., 175, 467
  * Hong et al. (2017) Hong, J., Carlsson, M., & Ding, M. D. 2017, ApJ, 845, 144
  * Hong et al. (2014) Hong, J., Ding, M. D., Li, Y., Fang, C., & Cao, W. 2014, ApJ, 792, 13
  * Judge (2015) Judge, P. G. 2015, ApJ, 808, 116
  * Kleint & Panos (2022) Kleint, L. & Panos, B. 2022, A&A, 657, A132
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17
  * Mandal et al. (2021) Mandal, S., Peter, H., Chitta, L. P., et al. 2021, A&A, 656, L16
  * Nelson et al. (2016) Nelson, C. J., Doyle, J. G., & Erdélyi, R. 2016, MNRAS, 463, 2190
  * Nelson et al. (2017) Nelson, C. J., Freij, N., Reid, A., et al. 2017, ApJ, 845, 16
  * Nelson et al. (2015) Nelson, C. J., Scullion, E. M., Doyle, J. G., Freij, N., & Erdélyi, R. 2015, ApJ, 798, 19
  * Panesar et al. (2021) Panesar, N. K., Tiwari, S. K., Berghmans, D., et al. 2021, ApJ, 921, L20
  * Panos & Kleint (2021) Panos, B. & Kleint, L. 2021, ApJ, 915, 77
  * Panos et al. (2021) Panos, B., Kleint, L., & Voloshynovskiy, S. 2021, ApJ, 912, 121
  * Peter et al. (2014) Peter, H., Tian, H., Curdt, W., et al. 2014, Science, 346, 1255726
  * Rast et al. (2021) Rast, M. P., Bello González, N., Bellot Rubio, L., et al. 2021, Sol. Phys., 296, 70
  * Reid et al. (2016) Reid, A., Mathioudakis, M., Doyle, J. G., et al. 2016, ApJ, 823, 110
  * Reid et al. (2017) Reid, A., Mathioudakis, M., Kowalski, A., Doyle, J. G., & Allred, J. C. 2017, ApJ, 835, L37
  * Rimmele et al. (2020) Rimmele, T. R., Warner, M., Keil, S. L., et al. 2020, Sol. Phys., 295, 172
  * Scherrer et al. (2012) Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207
  * Tian et al. (2016) Tian, H., Xu, Z., He, J., & Madsen, C. 2016, ApJ, 824, 96
  * Vissers & Rouppe van der Voort (2012) Vissers, G. & Rouppe van der Voort, L. 2012, ApJ, 750, 22
  * Vissers et al. (2013) Vissers, G. J. M., Rouppe van der Voort, L. H. M., & Rutten, R. J. 2013, ApJ, 774, 32
  * Vissers et al. (2015) Vissers, G. J. M., Rouppe van der Voort, L. H. M., Rutten, R. J., Carlsson, M., & De Pontieu, B. 2015, ApJ, 812, 11
  * Young et al. (2018) Young, P. R., Tian, H., Peter, H., et al. 2018, Space Sci. Rev., 214, 120

##  Appendix A Summary of observations and results

Table 1: Summary of the $42$ datasets studied in this article.

AR | Date | Start (UT) | End (UT) | xc″ | yc″ | Exp (s) | Rasters | BPs | Bs | OBSID
---|---|---|---|---|---|---|---|---|---|---
AR 11850 | 2013-09-24 | 05:09:45 | 05:29:12 | -327 | 92 | 2 | 1 | 109 | 39 | 4000254145
AR 11850 | 2013-09-24 | 11:44:43 | 12:04:10 | -265 | 88 | 2 | 1 | 181 | 21 | 4000254145
AR 11850 | 2013-09-24 | 15:39:43 | 15:59:10 | -264 | 70 | 2 | 1 | 5 | 3 | 4000254145
AR 11850 | 2013-09-25 | 06:39:43 | 06:59:10 | -88 | 85 | 2 | 1 | 41 | 11 | 4000254145
AR 11850 | 2013-09-25 | 11:09:43 | 11:29:10 | -53 | 85 | 2 | 1 | 312 | 46 | 4000254145
AR 11850 | 2013-09-26 | 05:59:43 | 06:58:04 | 130 | 90 | 2 | 3 | 258 | 28 | 4000254145
AR 11850 | 2013-09-26 | 11:09:43 | 11:29:10 | 178 | 98 | 2 | 1 | 26 | 9 | 4000254145
AR 11850 | 2013-09-27 | 05:24:32 | 05:48:21 | 327 | 82 | 2 | 1 | 156 | 45 | 3800254046
AR 11850 | 2013-09-27 | 06:24:43 | 06:44:10 | 335 | 78 | 2 | 1 | 56 | 15 | 4000254145
AR 11856 | 2013-10-09 | 05:31:44 | 06:31:59 | 177 | 35 | 8 | 1 | 0 | 0 | 3820259446
AR 11856 | 2013-10-09 | 09:35:44 | 10:35:44 | 258 | 36 | 8 | 1 | 13 | 5 | 3820259446
AR 11856 | 2013-10-10 | 05:00:58 | 06:01:13 | 376 | 29 | 8 | 1 | 7 | 4 | 3820259446
AR 11856 | 2013-10-11 | 05:20:57 | 06:22:17 | 622 | 74 | 8 | 1 | 5 | 2 | 3820259646
AR 11856 | 2013-10-11 | 10:21:27 | 11:22:47 | 627 | 51 | 8 | 1 | 0 | 0 | 3820259646
AR 11871 | 2013-10-18 | 13:42:56 | 17:14:43 | -244 | 257 | 30 | 1 | 0 | 0 | 3820013446
AR 11871 | 2013-10-18 | 19:50:30 | 23:22:17 | -194 | 263 | 30 | 1 | 0 | 0 | 3820013446
AR 11871 | 2013-10-19 | 17:55:46 | 18:56:16 | -48 | 225 | 15 | 1 | 0 | 0 | 3820009446
AR 11871 | 2013-10-20 | 12:15:00 | 17:48:53 | 91 | 273 | 15 | 3 | 0 | 0 | 3820011446
AR 11909 | 2013-12-03 | 06:24:38 | 06:57:55 | -38 | -317 | 4 | 1 | 30 | 17 | 3800256046
AR 11909 | 2013-12-03 | 07:54:14 | 08:14:07 | -76 | -273 | 2 | 1 | 126 | 20 | 3800254046
AR 11909 | 2013-12-03 | 11:14:38 | 11:47:55 | 18 | -305 | 4 | 1 | 43 | 12 | 3800256046
AR 11909 | 2013-12-03 | 17:54:44 | 18:14:37 | 27 | -272 | 2 | 1 | 64 | 18 | 3800254046
AR 11909 | 2013-12-03 | 21:14:38 | 21:47:55 | 109 | -314 | 4 | 1 | 54 | 8 | 3800256046
AR 11909 | 2013-12-04 | 03:25:09 | 03:45:02 | 162 | -306 | 2 | 1 | 57 | 15 | 3800254046
AR 11909 | 2013-12-05 | 18:44:51 | 19:18:11 | 495 | -327 | 4 | 1 | 54 | 11 | 3800256196
AR 11909 | 2013-12-05 | 20:22:31 | 20:55:51 | 453 | -297 | 4 | 1 | 6 | 2 | 3800256196
AR 11916 | 2013-12-06 | 07:34:51 | 08:08:11 | 36 | -223 | 4 | 1 | 715 | 65 | 3800256196
AR 11916 | 2013-12-06 | 15:49:31 | 16:22:51 | 97 | -222 | 4 | 1 | 120 | 25 | 3800256196
AR 11916 | 2013-12-06 | 22:20:41 | 22:54:01 | 158 | -226 | 4 | 1 | 288 | 58 | 3800256196
AR 11916 | 2013-12-07 | 03:09:49 | 03:43:09 | 202 | -225 | 4 | 1 | 236 | 39 | 3800256196
AR 11916 | 2013-12-07 | 11:14:51 | 11:48:11 | 273 | -220 | 4 | 1 | 283 | 45 | 3800256196
AR 11916 | 2013-12-07 | 22:46:41 | 23:20:01 | 390 | -216 | 4 | 1 | 273 | 59 | 3800256196
AR 11916 | 2013-12-09 | 07:10:51 | 07:44:11 | 633 | -216 | 4 | 1 | 73 | 2 | 3800256196
AR 11916 | 2013-12-09 | 21:56:54 | 22:30:14 | 721 | -214 | 4 | 1 | 64 | 21 | 3800256196
AR 12104 | 2014-07-04 | 11:40:30 | 15:10:55 | -114 | -237 | 30 | 1 | 42 | 14 | 3824263396
AR 12104 | 2014-07-05 | 23:00:30 | 02:30:55∗ | 203 | -238 | 30 | 1 | 72 | 32 | 3824263396
AR 12104 | 2014-07-07 | 23:35:30 | 03:05:55∗ | 562 | -226 | 30 | 1 | 54 | 7 | 3824263396
AR 12104 | 2014-07-08 | 19:26:13 | 22:56:38 | 702 | -220 | 30 | 1 | 0† | 0 | 3824263396
AR 12139 | 2014-08-15 | 07:08:03 | 08:10:51 | -439 | 120 | 8 | 1 | 94 | 34 | 3800258196
AR 12139 | 2014-08-15 | 22:36:09 | 02:09:34∗ | -305 | 112 | 30 | 1 | 10 | 7 | 3880012196
AR 12139 | 2014-08-16 | 21:17:49 | 22:20:37 | -106 | 103 | 8 | 1 | 45 | 12 | 3800258196
AR 12139 | 2014-08-17 | 21:44:03 | 22:46:51 | 105 | 104 | 8 | 1 | 47 | 1 | 3800258196

111General information about the $42$ datasets studied here, including: The AR
sampled; the date the observation started on; the start time of the
observation; the end time of the observation (∗ indicates the following day);
the initial x-coordinate of the centre of the FOV; the initial y-coordinate of
the centre of the FOV; the exposure time per raster step; the number of
rasters sampled during the observation; the number of IRIS burst profiles
(BPs) detected using automated methods († indicates that IRIS burst profiles
were subsequently apparent in our manual analysis); the number of parent IRIS
bursts (Bs) identified; and the OBSID of the experiment.


# Temperature effects on core g-modes of neutron stars

Nicholas Lozano<EMAIL_ADDRESS>Vinh Tran
<EMAIL_ADDRESS>Prashanth Jaikumar<EMAIL_ADDRESS>Department of Physics and Astronomy, California State University Long Beach,
Long Beach, California 90840, USA

###### Abstract

Neutron stars provide a unique physical laboratory to study the properties of
matter at high density. We study a diagnostic of the composition of high-
density matter, namely, g-mode oscillations, which are driven by buoyancy
forces. These oscillations can be excited by tidal forces and couple to
gravitational waves. We extend prior results for the g-mode spectrum of cold
neutron star matter to temperatures that are expected to be achieved in
neutron star mergers using a parameterization for finite-temperature effects
recently proposed by Raithel, Özel and Psaltis. We find that the g-modes of
canonical mass neutron stars ($\approx$1.4$M_{\odot}$) are suppressed at high
temperature, and core $g$-modes are supported only in the most massive
($\geq$2$M_{\odot}$) of hot neutron stars.

The study of pulsations of stars has a rich history, beginning with the
observations of variable stars (eg., Cepheids and RR Lyrae) and continuing
through the last century with treatises by Ledoux & Walraven Ledoux and
Walraven (1958), Cox Cox (1980), Unno Unno  _et al._ (1989) and others
Gautschy and Saio (1995). Helioseismology and asteroseismology are considered
mature fields that have provided important insight into the inner structures
of the star as well as the mechanisms that power their variability. Compact
stars such as neutron stars and black holes can support pulsation modes that
couple to gravitational waves, providing a fingerprint of the equation of
state (for neutron stars) Kokkotas  _et al._ (2001), or global properties such
as mass and spin rate (for black holes) Kokkotas and Schmidt (1999). Within
the context of neutron stars, which is the focus of this work, of particular
interest is the $g$-mode: a type of non-radial fluid oscillation that is
driven by buoyancy forces arising from temperature and composition-related
gradients in the star Reisenegger and Goldreich (1992). Strong tidal effects
during the inspiral phase of neutron star mergers could excite $g$-modes that
are potentially observable with the next generation of gravitational wave
detectors.

While many studies of the $g$-mode abound in the literature (e.g., Jaikumar
_et al._ (2021) and references therein), most hew to the discussion of
discontinuity $g$-modes at zero temperature, for e.g. Tonetto and Lugones
(2020); Sotani  _et al._ (2001). In contrast, fewer works have studied the
composition $g$-modes pioneered in the work of Reisenegger and Goldreich
Reisenegger and Goldreich (1992) or the effect of high temperatures (${\cal
O}$(MeV)) on them. This work is a focused study of how temperature affects the
composition $g$-mode through lifting of degeneracy of neutron star matter. In
general, the detection of $g$-modes through gravitational wave astronomy could
play an important role in uncovering the composition of neutron stars in the
near future, including various forms of exotic matter in them.

For the benefit of the busy reader or non-expert, we summarize the main
conclusion of this work up front. We find that $g$-modes of hot neutron stars
are very different in frequency than those of cold neutron stars, and that the
former are suppressed for canonical mass neutron stars due to thermal effects
that lead to an increase in the equilibrium sound speed at typical core
densities. A detailed discussion is provided in Sec 4.4 and Sec 5. The paper
is organized as follows: we begin with a description of the core $g$-mode in
neutron stars and its relation to the equilibrium and adiabatic sound speeds
in the star through the principal equations used to compute the $g$-mode
frequencies within the relativistic Cowling approximation Ranea-Sandoval  _et
al._ (2018), which neglects the associated perturbations of the background
metric. In order to extend our analysis of $g$-mode frequencies to high
temperatures, we find it convenient to utilize an analytic parameterization
for temperature effects on the equation of state for hot and dense matter
developed by Raithel, Özel and Psaltis Raithel  _et al._ (2019, 2021a) and
applied to simulations of neutron star mergers. The microscopic physics
underlying this parameterization is general enough to apply to most nuclear
equation of state, and so we choose two EOS, namely, the Akmal-Pandharipande-
Ravenhall (APR) equation of state Akmal  _et al._ (1998) and the Zhao-Lattimer
(ZL) equation of state Zhao and Lattimer (2020) for our study. Finally, we
interpret our results for the $g$-modes of hot neutron stars and motivate our
conclusions.

##  I $g$-modes and sound speeds in neutron stars

In general, the oscillatory displacement of a fluid element in a spherically
symmetric star is represented by a vector field
${\vec{\xi}}^{nlm}(\vec{r}){\rm e}^{-i\omega t}$ with $n,l$ and $m$ denoting
the radial, azimuthal and magnetic mode indices Jaikumar  _et al._ (2021). To
be precise, the frequency $\omega$ also carry subscripts $nlm$ implicitly
understood, with degeneracies that are broken in more realistic cases such as
with rotation or magnetic fields (not considered here).

For even-parity or spheroidal modes, separation into radial and tangential
components yields $\xi_{r}^{nlm}(\vec{r})$ =
$\eta_{r}^{nl}(r)Y_{lm}(\theta,\phi)$ and $\vec{\xi}_{\perp}^{nlm}(\vec{r})$ =
$r\eta_{\perp}^{nl}(r)\nabla_{\perp}Y_{lm}(\theta,\phi)$, respectively, where
$Y_{lm}(\theta,\phi)$ are the spherical harmonics Thorne and Campolattaro
(1967). From the perturbed continuity equation for the fluid, the tangential
function $\eta_{\perp}$ can be traded for fluid variables as $\delta
p/\epsilon$ = $\omega^{2}r\eta_{\perp}(r)Y_{lm}(\theta,\phi){\rm e}^{-i\omega
t}$, where $\delta p$ is the corresponding local (Eulerian) pressure
perturbation and $\epsilon$ the local energy density. Within the relativistic
Cowling approximation111The Cowling approximation neglects the back reaction
of the gravitational potential and reduces the number of equations we have to
solve. While this approximation is not strictly consistent with our fully
general relativistic (GR) treatment of the equilibrium structure of the star,
it does not change our conclusions qualitatively or even quantitatively that
much, since this approximation is accurate for $g$-mode frequencies at the few
% level Gregorian (2014); Flores and Lugones (2017); Zhao  _et al._ (2022).,
the equations of motion to be solved to determine the frequency of a
particular mode are (in $c$=1 units) McDermott  _et al._ (1983); Reisenegger
and Goldreich (1992); Kantor and Gusakov (2014)

$\displaystyle-\frac{1}{\mathrm{e}^{\lambda/2}r^{2}}\frac{\partial}{\partial
r}\left[\mathrm{e}^{\lambda/2}r^{2}\xi_{r}\right]+\frac{l(l+1)\mathrm{e}^{\nu}}{r^{2}\omega^{2}}\frac{\delta
p}{p+\epsilon}-\frac{\Delta p}{\gamma p}=0$ $\displaystyle\frac{\partial\delta
p}{\partial r}+g\left(1+\frac{1}{c_{\mathrm{s}}^{2}}\right)\delta
p+\mathrm{e}^{\lambda-\nu}h\left(N^{2}-\omega^{2}\right)\xi_{r}=0\,,$ (1)

where enthalpy $h=p+\epsilon$, $\gamma$ the adiabatic index of the fluid, and
we have suppressed the indices on $\omega$ and $\xi$. The Lagrangian variation
of the pressure enters as $\Delta p$, and is related to the Eulerian variation
$\delta p$ through the operator relation $\Delta\equiv\delta+\xi\cdot\nabla$.
The symbol $c_{s}$ denotes the adiabatic sound speed, the square of which is
given as $c_{s}^{2}=\gamma_{1}p/(\mu_{n}n_{B})$ where $\mu_{n}$ is the neutron
chemical potential222In beta-equilibrated charge neutral neutron star matter,
the neutron chemical potential is sufficient to determine all other chemical
potentials., $n_{B}$ the local baryon density and
$\gamma_{1}{=}(n_{B}/p)\partial p(n_{B},Y_{p})/\partial n_{B}$ is the
adiabatic index. The advantage of employing the analytic parameterization of
the finite temperature EOS which separates out the dependence on $n_{B}$ and
$Y_{p}$ is that the partial derivative $\partial p(n_{B},Y_{p})/\partial
n_{B}$, and hence the adiabatic sound speed at any density can be computed
analytically. On the other hand, the equilibrium sound speed squared
$c_{e}^{2}=(dp/d\epsilon)$ is a total derivative that can be evaluated
numerically from the tabulated EOS. Both sound speeds enter through the Brunt-
Väisälä frequency ($N$) which is given by

${\it N}^{2}\equiv
g^{2}\Big{(}\frac{1}{c_{e}^{2}}-\frac{1}{c_{s}^{2}}\Big{)}{\rm
e}^{\nu-\lambda}\,,$ (2)

where the local gravitational field $g$=-$\nabla\phi$=-$\nabla p/h$, $\nu(r)$
and $\lambda(r)$ are metric functions of the unperturbed star which feature in
the Schwarzschild interior metric. Eq.(1) can be analyzed in the short-
wavelength limit ($kr\gg 1$) where the local dispersion relation has two
distinct branches, with the lower frequency branch corresponding to the
$g$-modes. The local $g$-mode frequency is then $\omega^{2}\propto{\rm
e}^{\lambda}N^{2}$ McDermott  _et al._ (1983), highlighting the importance of
the two sound speeds (in particular, the difference of their inverse squares,
as in Eq.(2)). The global g-mode frequency is constant for a given stellar
configuration (see Fig.4) and can be thought of as an average of the local
g-modes (although it is still sensitive to phase transitions).

In this work, we study the fundamental $g$-mode with $n$ = 1 and fix the
mode’s multipolarity at $l$ = 2. This is because the $l=2$ mode is quadrupolar
in nature, and can couple to gravitational waves. Higher $l$ values (octupole
and higher) are generally weaker than the quadrupole. The reason to study the
$n=1$ (fundamental) $g$-mode is that the local dispersion relation for
$g$-modes $\omega^{2}\propto 1/k^{2}$ implies that the $n=1$ excitation has
the highest frequency, whereas higher values of $n$ are known to have a
smaller amplitude of excitation and a weaker tidal coupling coefficient
Constantinou  _et al._ (2021). For the non-rotating stars we consider here,
solutions are degenerate in $m$. Note that our definition of the “fundamental”
mode refers to the lowest radial order of the $g$-mode which also has the
highest frequency. This should not be confused with the qualitatively
different $f$-mode which is also referred to sometimes as the fundamental
mode. Furthermore, overtones with lower frequency exist, but we do not perform
any computations with them here, since the fundamental $g$-mode is likely the
most strongly excited during a tidal perturbation and is within the
sensitivity range of current generation of gravitational wave (GW) detectors
Lai (1994); Zhao and Lattimer (2022).

The system of equations in Eq. (1) cannot be solved analytically even with a
simple model of a neutron star. Our aim will be to solve this numerically as
an eigenvalue system for the $g$-mode frequency $\omega$. Physically, the
solution to this system of equations, under the boundary conditions $\Delta
p=0$ at the surface and $\xi_{r},\,\delta p/\epsilon$ regular at the center,
only exists for discrete values of the mode frequency $\omega$. These values
represent the $g$-mode spectrum for a chosen stellar model. Because we have
employed the Cowling approximation and ignored the perturbations of the metric
that must accompany fluid perturbations, we cannot compute the imaginary part
of the eigenfrequency (damping time) of the $g$-mode333The damping time of
$g$-modes due to viscosity and gravitational wave emission, estimated in some
works Lai (1999); Wei  _et al._ (2020), suggests that the $g$-mode can become
secularly unstable for temperatures $10^{8}~{}{\rm K}<T<10^{9}~{}{\rm K}$ for
rotational speeds exceeding twice the $g$-mode frequency of a static star..

The effect of temperature on the $g$-mode in degenerate matter inside neutron
stars is relatively unexplored. Fu et al. Fu  _et al._ (2017) calculated the
time-evolution of the $g$-mode spectrum of a newly born neutron star/strange
quark star as it cools after formation in a supernova explosion, within a
specific model (the relativistic mean-field/MIT Bag model respectively). Our
purpose in this work is to explore the behavior of $g$-modes with increasing
temperature (decreasing degeneracy) in an evolved neutron star. For this
exercise, we employ two parameterized models for a purely nucleonic finite-
temperature equation of state that satisfy current observational constraints
and are described in more detail below.

##  II Finite temperature model for the neutron star equation of state

References Raithel  _et al._ (2019, 2021a) introduced a finite-temperature
model (henceforth, ROP) in parameterized form that captures thermal effects in
$npe$ matter at arbitrary density, temperature and proton fraction, and can
describe a wide range of microscopic realistic EOS. In particular, the model
can describe thermal effects on degeneracy and is therefore more widely
applicable to the temperature and density regimes relevant to neutron star
mergers than the ideal fluid approximation. The authors also tabulate fit
parameters for different classes of thermal EOS, such as the single nucleus
approach of Lattimer & Swesty Lattimer and Swesty (1991), Shen et al. Shen
_et al._ (1998) as well as nuclear statistical equilibrium, e.g., Hempel &
Schaffner-Bielich Hempel and Schaffner-Bielich (2010) and Steiner et al
Steiner  _et al._ (2013)444Not all fit parameters in the ROP model are
consistent with the combined set of experimental constraints Lattimer and Lim
(2013) or with error bands from chiral effective field theory Krüger  _et al._
(2013); Drischler  _et al._ (2021) but a subset of parameters always are, with
a reasonable compromise to allow for a wide range of EOS in their initial
study..

The relevant aspects of the ROP model for this work are the various thermal
contributions to the energy and pressure of neutron star matter, which are all
expressed in analytic (exact or parameterized) form, with explicit dependence
on temperature, density and compositional dependence (in this case, proton
fraction only), facilitating a computation of the equilibrium and adiabatic
sound speed required for the finite temperature $g$-mode. Thermal
contributions come from the relativistic species (photons and leptons only;
neutrinos are ignored), the ideal gas component (non-degenerate nucleons) and
a thermal-degenerate component calculated to leading order ($\propto T^{2}$).
Each of these three components dominates in different density regimes, from
low-density to high-density respectively, which can be approximated by a
smoothed expression that recovers the dominant contribution to the energy and
pressure of the strongly interacting matter. In the ROP prescription, the
Dirac effective mass of the nucleon (neutron or proton) is also parameterized
by a power-law form, which, at the expense of introducing additional fit
parameters, reproduces the behavior of the effective mass in the microscopic
theory at densities and temperatures relevant to the core of neutron stars
Constantinou  _et al._ (2015a, b). The nucleon’s effective mass has been shown
to be important for other oscillation modes, such as $f$-modes Pradhan and
Chatterjee (2021). Thus, the ROP model is well suited to a computation of the
sound speeds and the finite-temperature $g$-mode. For the convenience of the
reader, we reproduce the generic expressions for the pressure and energy
within the ROP model in the Appendix.

##  III Results

We present our results for the $g$-mode in this section, beginning with the
inputs required for their computation, namely, the EOS, the mass-radius
relation, and the sound speeds.

Figure 1: (a) The APR EOS with thermal contributions included as per the
parameterizations in Raithel  _et al._ (2019). (b) Thermal effects are most
pronounced in the low-density regime (0.05-3.0) $n_{\rm sat}$ and are thus
expected to inflate the radius of the hot neutron star relative to a cold one.
This is proportionate to the impact of thermal effects, shown here for a
temperature $T$=50 MeV, on lifting the degeneracy of dense matter.
Corresponding results for the ZL EOS are qualitatively similar.

###  III.1 Pressure and Energy

We choose two representative nuclear models for the EOS of dense matter in the
core of the neutron star. One is the APR EOS Akmal  _et al._ (1998). To
construct this cold EOS for use in Eq.(5) & (6), we fit the parabolic
approximation

$\displaystyle E(n,Y_{p},T{=}0)$ $\displaystyle=E_{B}(n,Y_{p}{=}1/2,T{=}0)$
(3) $\displaystyle\qquad+(1{-}2Y_{p})^{2}E_{\rm sym}(n,T{=}0)$

to the results of Akmal  _et al._ (1998) with

$\displaystyle
E_{B}(n,Y_{p}{=}1/2,T{=}0)=E_{0}(u{-}2{-}\delta)/(1{+}u\,\delta)$ (4)

with $u=n/n_{\rm sat}$ ($n_{\rm sat}$ = 0.154 fm-3 is the saturation density),
$E_{0}=-16$ MeV the binding energy of uniform nuclear matter and $\delta$ a
parameter that is fixed by choosing the value of the compressibility $\kappa$
of nuclear matter at saturation555 Compressiblity $\kappa$=
18$E_{0}/(1+\delta)$ for the fitted APR EOS. It is easy to check that the
pressure of symmetric matter vanishes at saturation. The fit for $E_{\rm
sym}(n,T{=}0)$ is given by Eq.(14) of Raithel  _et al._ (2019) and the
parameters required for this are the symmetry energy at saturation
$E_{sym}(n_{sat},T{=}0)$, the density dependence of the potential contribution
to the symmetry energy, typically parameterized by a power-law index $\gamma$
and the slope parameter $L$ at saturation. Our parameter values are within the
typical range allowed by terrestrial and astrophysical data, as detailed in Li
_et al._ (2015). A common nucleon mass is chosen for convenience. With these
choices fixed, one can obtain $Y_{p,\beta}(n)$, the proton fraction in charge-
neutral $npe$ matter in $\beta$-equilibrium. The compressibility at saturation
density $\kappa$ is tuned to satisfy the maximum mass constraint of 2.08
${}_{-0.07}^{+0.07}M_{\odot}$ derived from observations of PSR J0740+6620 in
Fonseca  _et al._ (2021). All parameter values for the APR EOS are listed in
Table 1.

Table 1: Parameters of the cold EOS employed in this work. EOS | $\kappa$ |  $E_{\rm sym}(n_{\text{sat}})$ (MeV) |  $L$ (MeV) | $\gamma$
---|---|---|---|---
APR | 254 | 32 | 57.6 | 0.6
ZL | 250 | 31.6 | 60.1 | 0.8

The second EOS model is that of Zhao and Lattimer Zhao and Lattimer (2020). In
this case, the data for $E_{B}(n,Y_{p}{=}1/2,T{=}0)$ is fitted to the
compressibility of nuclear matter, along with a binding energy of -16 MeV and
vanishing pressure, all at saturation666Data for the ZL EOS was obtained in
private communication with author of Constantinou  _et al._ (2021), while data
for $E_{\rm sym}(n,T{=}0)$ is again fit by Eq.(14) of Raithel  _et al._
(2019). The parameters values $\kappa,E_{sym}(n_{sat}),L,\gamma$ for the ZL
EOS are listed in Table 1.

For the cold ($T$=0) APR EOS, the dimensionless tidal polarizability
$\Lambda_{1.4}^{\rm APR}$=492 while for the more compact ZL EOS,
$\Lambda_{1.4}^{\rm ZL}$=357. Both are within the constraint 70
$\leq\Lambda_{1.4}\leq$ 580 obtained using uncorrelated priors on the binary
components of the merger event GW170817 obtained by Abbott et al. Abbott  _et
al._ (2018), and also within the constraints of De et al. De  _et al._ (2018)
which used correlated priors. For both EOS, we will treat the crust as a
homogeneous fluid so that $c_{s}$=$c_{e}$ and no crustal $g$-modes are
supported. The assumption of homogeneity is reasonable given that the crust
melting temperature is estimated to be well below temperatures of 10 MeV or
more that we are considering in our study of finite temperature $g$-modes.

In either case, adding the thermal contributions gives us the complete EOS at
any temperature and density relevant to this study; Eqs.(8) and (9). Figure
1(a) shows the pressure-energy relation for the APR EOS in $\beta$-equilibrium
for varying temperatures including the density regime of relevance to the core
of neutron stars, while Figure 1(b) shows the relative importance of thermal
contributions as a function of density for a fixed temperature. The smoothed
pressure is the sum of the relativistic term and the harmonic mean of the
ideal and the degenerate terms, and is employed to avoid artificial
discontinuities in the calculation of sound speeds. It also has the right
limits in that it approaches the ideal gas pressure at low densities and the
degenerate pressure at high density. The corresponding results for the ZL EOS
are qualitatively similar, hence not shown here.

###  III.2 Mass-radius relation

Figure 2: The effects of temperature on a neutron star’s structure. As the
temperature increases, degeneracy is lifted. This means that the stars will
essentially “puff” up without gaining more mass.

The mass-radius relations of cold and hot neutron stars described by the
parameterized APR EOS and the ZL EOS are shown in Figure 2. The maximum mass
of the $T$=0 neutron star is 2.12 $M_{\odot}$ for the APR EOS and 2.15
$M_{\odot}$ for the ZL EOS, representative of the heavier neutron stars
observed to date, giving us a sufficiently wide range to study the $g$-mode
dependence on the mass. With increasing temperature ($T$ from 0 to 30 MeV),
the maximum mass for the APR EOS changes only slightly to 2.15 $M_{\odot}$,
while the size of the star increases, with dramatically larger radii near the
upper limit of this range. Such high temperatures can be achieved a few tens
of milliseconds after the merger in the core of the remnant, as suggested by
results for the thermal profile emerging from merger simulations Raithel  _et
al._ (2021b). It should be noted that at this stage, the remnant is not in a
state of mechanical or chemical equilibrium, so our mass-radius profiles are
not directly applicable to such a scenario. However, it does demonstrate that
the radial extent of the star is quite sensitive to the treatment of thermal
pressure. Including the thermal terms leads to radius inflation and as we note
below, affects the sound speeds and the behavior of the $g$-mode frequency.

###  III.3 The equilibrium ($c_{e}$) and adiabatic ($c_{s}$) sound speeds

Figure 3: The difference between the inverse of the squares of the equilibrium
and adiabatic sound speeds as a function of density in the neutron star core
for varying temperatures, for the APR EOS. A positive difference indicates
stable $g$-modes. (a) For $Y_{p}$ the same as that given by the cold EOS
(i.e., in $\beta$-equilibrium), at temperatures of 50 MeV or more, no stable
$g$-modes are supported . (b) For a fixed, arbitrary $Y_{p}=0.1$, this
difference, though positive at low density eventually turns negative at large
density irrespective of the temperature. Interpolation effects at the core-
crust boundary ($n_{B}$=0.08 fm-3) cause the peaks in the left panel figure
and do not have any physical significance. In practice, the sound speed
difference in the core rises smoothly towards the core-crust boundary and
drops abruptly to 0 in the crust in accordance with our assumption of a
homogeneous fluid crust (see text for additional discussion on this point).
The corresponding results for the ZL EOS are qualitatively similar.

The difference between the (inverse squares of) equilibrium and adiabatic
sound speeds determines the driving frequency of the $g$-modes and depends on
the composition in $\beta$-equilibrium. As pressure and energy density vary
with temperature, we find that the sound speed difference also changes
strongly with temperature, which affects the $g$-mode frequency. If we assume
the proton fraction remains unchanged from the cold EOS value
$Y_{p}$=$Y_{p,\beta}$, we observe from Figure 3(a) that with increasing
temperature, this sound speed difference gradually decreases and turns
negative, switching off the $g$-modes. At low temperatures, the equilibrium
sound speed squared ($c_{e}^{2}$) is smaller than the adiabatic sound speed
squared ($c_{s}^{2}$) at all densities of relevance to the neutron star’s
core. With increasing temperature, $c_{e}^{2}$ exceeds $c_{s}^{2}$ at a
threshold density which is within the realm of a neutron star’s core
densities. This increase in $c_{e}^{2}$ is principally due to the thermal
pressure from the ideal gas term at lower densities, and the degeneracy term
at higher densities. On average, the sound speed difference
$c_{e}^{2}-c_{s}^{2}$ is most affected by the lifting of the degeneracy. We
conclude that composition $g$-modes in neutron stars should be suppressed at
high temperatures. Both figures terminate at a lower density of $n_{b}$=0.08
fm-3 (core-crust interface). In principle, the crust can also support
$g$-modes and these can be calculated given a particular model of the crust
with shells of varying composition and shear moduli Finn (1990), but crustal
$g$-modes couple very weakly or not at all to core $g$-modes Reisenegger and
Goldreich (1992), hence for our purposes, it suffices to treat the crust as a
homogeneous fluid with $c_{s}$=$c_{e}$ without any $g$-modes there.

As the model in ROP is a good fit to realistic EOS across a range of
asymmetries 0.01 $\leq Y_{p}\leq 0.5$ in $npe$ matter, we also study the
behavior of the sound speed difference with baryon density at fixed $Y_{p}$.
Although the fixed $Y_{p}$ scenario is not realized in neutron stars and does
not apply to the $g$-mode perturbations which assume small deviations from
$\beta$-equilibrium, we display the results here to demonstrate the marked
effects of composition on sound speeds. For example, Figure 3(b) shows that
$c_{e}^{2}<c_{s}^{2}$ at low densities, even up to high temperatures, as
matter in the core of a neutron star is, on average more degenerate than a
configuration with fixed $Y_{p}$=0.1 implies. The corresponding results for
the ZL EOS are qualitatively similar, hence not shown here.

###  III.4 $g$-modes at finite temperature

For composition-driven $g$-modes of zero-temperature neutron stars, the scale
of the characteristic frequency is set by the Brunt-Väisälä frequency, and is
of the order of 100-200 Hz. The $g$-mode frequency increases slowly with
stellar mass, until a sharp increase occurs near the maximum mass. The
$g$-mode frequency has been found to scale with the central lepton fraction
Zhao  _et al._ (2022), and is sensitive (in $npe$ matter) to the symmetry
energy and its density derivative Wei  _et al._ (2020). As Figure 4 shows,
these modes are strongly dependent on the star’s temperature as well, with
g-modes for canonical neutron stars (1.4 $M_{\odot}$) being suppressed at or
above temperatures of 30 MeV or so. This is a direct result of an increase in
the equilibrium sound speed with increasing temperature. Neutron stars close
to the maximum mass of 2$M_{\odot}$ in the APR or ZL EOS model can support
stable $g$-modes even at higher temperature, but eventually at temperatures of
50 MeV or more, no stable $g$-modes can be found.

Figure 4: Plots of the g-mode frequencies as a function of the star’s mass (in
solar masses). As temperature increases, g-mode frequencies are suppressed. At
temperatures of 30 MeV, g-modes vanish except near the maximum mass, and
eventually these disappear as well with increasing temperature.

##  IV Discussion

We studied the effects of temperature on the composition-driven $g$-modes of a
neutron star, which could be excited during or in the aftermath of a neutron
star merger, and possibly be detected in future gravitational wave detectors.
Toward this end, we employed a convenient parameterization of finite
temperature effects that accurately reproduces exact results from a variety of
microscopic calculations of finite-temperature equations. In particular, the
separation of the density and composition dependence of the parameterized
form, for both the cold and thermal contributions to the pressure, facilitates
the computation of the $g$-mode frequencies.

Several effects can be seen with temperatures of order 10 MeV or more which
characterize hot neutron stars. In general, higher temperatures alter the
equation of state, leading to higher energy density and pressure. As a result,
the mass-radius relation of the stars changed as a function of temperature
(see Figure 2). For example, with the APR EOS, a star with a mass of 1.4 solar
mass at low temperature has a radius of about 12.45 km. At 30 MeV, a star with
similar mass has a radius of 13.68 km. The increased temperature lifts the
degeneracy and puffs up the star creating an overall slightly less dense
neutron star on average when compared to lower temperatures.

Additionally, the calculated frequencies of the $g$-mode decreased as
temperature increased (Figure 4). One observes that at high temperatures
$g$-mode oscillation frequencies are suppressed and disappear almost
completely beyond 30 MeV, except in the most massive of neutron stars near the
maximum mass configuration, due to the stronger effect of degeneracy in the
cores of such stars. This suppression is due to specific thermal contributions
in the equation of state which makes the equilibrium sound speed larger than
the adiabatic sound speed.

Since the merger of two neutron stars results in a hotter and more massive
neutron star than the merging components, one may speculate that $g$-modes can
still be marginally supported in the remnant due to its highly degenerate
core. Such remnant stars vibrate strongly for a few seconds, so gravitational
wave detectors may detect $g$-modes from them, assuming viscosity does not
damp the $g$-mode too quickly. The detection of zero-temperature $g$-modes
from neutron stars in the pre-merger phase has been studied extensively, and
may also be a promising avenue to constrain the composition of cold neutron
stars.

Some caveats to our study are to be noted. The composition $g$-modes
introduced by Reisenegger and Goldreich Reisenegger and Goldreich (1992) are,
strictly speaking, excited in non-convective (cold) neutron stars. If
temperatures become large enough for convective forces, such $g$-modes can
become unstable. Convective forces are not taken into account in this work,
nor are temperature gradients. For a varying temperature profile (e.g.,
assuming a constant entropy/baryon), a more general expression for the
adiabatic sound speed such as eq.(56) of Albright and Kapusta (2016) is
required. A more detailed investigation of the $g$-mode with a realistic
temperature profile and a locally temperature dependent sound speed is much
more involved and left to future work. Neutrino trapping is also ignored, but
if we nominally include them as a thermal component by setting $f_{s}=43/8$
above typical trapping temperatures of 10 MeV, we find a systematic reduction
of approximately 15-20% in the $g$-mode frequency. This is significant and
implies that neutrinos can affect the $g$-mode frequency in hot neutron stars,
a finding worthy of additional study. The inclusion and impact of additional
species (such as hyperons or quarks) can all cause a substantial enhancement
of the $g$-mode frequency by about 500 Hz, though in each case the reason for
the enhancement is different (see, e.g., discussion in Constantinou  _et al._
(2021)). As the $g$-mode is expected to be swamped by the $f$-mode, it has not
been the focus of attention in the early LIGO/VIRGO studies (e.g., Pratten
_et al._ (2020)) but the important compositional information carried by the
$g$-mode may be teased out in future detectors such as the Cosmic Explorer or
the Einstein Telescope.

###### Acknowledgements.

Authors acknowledge helpful comments from an anonymous referee. P.J. is
supported by the U.S. National Science Foundation Grant No. PHY-1913693.

## Appendix A

###  A.1 ROP parameterization for the temperature dependent EOS

A note about units: Following convention, in the governing $g$-mode equations
of Sec. 2, we have set $c$=1 and chosen MeV/fm3 for units of energy density
and pressure. For consistency, in this section, $E$ is in MeV and $p$ in
MeV/fm3, so one should use as appropriate the conversions MeV.fm = 1/197.33,
while measuring $T$ in MeV (1 MeV = 1.16$\times 10^{10}$K) and number
densities in fm-3. The energy and pressure are decomposed as

$\displaystyle
E_{\text{total}}(n,Y_{p},T)=E_{\text{cold}}(n,Y_{p})+E_{th}(n,Y_{p},T)\,.$ (5)
$\displaystyle
p_{\text{{total}}}(n,Y_{p},T)=p_{\text{{cold}}}(n,Y_{p})+p_{th}(n,Y_{p},T)\,.$
(6)

where $E_{th}(E_{cold})$ and $p_{th}(p_{cold})$ represent the total respective
thermal (cold) components. In Eq.(5) & (6)

$E_{\text{cold}}(n,Y_{p})=E\left(n,Y_{p,\beta},T=0\right)+E_{\text{sym}}(n,T=0)\left[\left(1-2Y_{p}\right)^{2}-\left(1-2Y_{p,\beta}\right)^{2}\right]+3K\left(Y_{p}^{4/3}-Y_{p,\beta}^{4/3}\right)n^{1/3}\\\
p_{\text{cold}}(n,Y_{p})=p\left(n,Y_{p,\beta},T=0\right)+p_{\text{sym}}(n,T=0)\left[\left(1-2Y_{p}\right)^{2}-\left(1-2Y_{p,\beta}\right)^{2}\right]+3K\left(Y_{p}^{4/3}-Y_{p,\beta}^{4/3}\right)n^{4/3}$
(7)

where the cold symmetry energy or symmetry pressure is denoted by $E_{\rm
sym}$ or $p_{\rm sym}$ and $K=(3\pi^{2})^{1/3}$. The thermal terms in the
smoothed approximation are provided in Eq. (8) and (9) below.

$\displaystyle E_{th}(n,Y_{p},T)=\frac{4\sigma
f_{s}T^{4}}{n}+\biggl{(}\left(\frac{3T}{2}\right)^{-1}+[a(0.5n,0.5M_{SM}^{*})+a(Y_{p}n,m_{e})Y_{p}]^{-1}T^{-2}\biggr{)}^{-1}\,,$
(8)

where $\sigma$=$\pi^{2}$/60 is the Stefan-Boltzmann constant, $m_{e}$ is the
electron mass, $f_{s}$ is the number of ultra-relativistic species that
contribute to the thermal energy (taking the value of either $1$ for $T\leq 1$
MeV or $(11/4)$ for $T>1$ MeV provided neutrinos are neglected above $T\sim$
10 MeV), and $a(n,M_{SM}^{*})$ is a level density parameter that relies on the
Dirac effective mass of the nucleon in symmetric matter, $M_{SM}^{*}$, and the
baryon density. This parameter is defined in Eq. (10). Similarly,

$p_{\text{th}}(n,Y_{p},T)=\frac{4\sigma f_{s}T^{4}}{3}\\\
\qquad+\biggr{(}\left({nT}\right)^{-1}+\biggr{[}\frac{\partial
a(0.5n,0.5M_{SM}^{*})}{\partial n}\\\ \qquad+\frac{\partial
a(Y_{p}n,m_{e})Y_{p}}{\partial n}\biggr{]}^{-1}n^{-2}T^{-2}\biggr{)}^{-1}\,,$
(9)

$a\left(n_{q},M^{*}_{q}\right)\equiv\frac{\pi^{2}}{2}\frac{\sqrt{\left(3\pi^{2}n_{q}\right)^{2/3}+M^{*}_{q}\left(n_{q}\right)^{2}}}{\left(3\pi^{2}n_{q}\right)^{2/3}}\,,$
(10)

$M_{q}^{*}\left(n_{q}\right)=\left\\{\left(mc^{2}\right)^{-b}+\left[mc^{2}\left(\frac{n_{q}}{n_{0}}\right)^{-\alpha}\right]^{-b}\right\\}^{-1/b}\,,$
(11)

where $n_{q}$ refers to the density of a single species $q$ (nucleon or
lepton), $m$ is its vacuum mass and $n_{0}$ is a fiducial density. We chose
$b=2$,$\alpha=0.87$ and $n_{0}=0.13$ as in Raithel  _et al._ (2019) since
these provide best fits to the density dependence of the common nucleonic
effective mass in symmetric matter $M_{SM}^{*}$ for a wide variety of EOS.
Eqs.(8), (9), (10) and (11) were used for all our numerical computations
involving thermal terms in the energy and pressure.

## References

  * Ledoux and Walraven (1958) P. Ledoux and T. Walraven, Handbuch der Physik 51, 353 (1958).
  * Cox (1980) J. P. Cox,  _Theory of stellar pulsation_ (Princeton University Press, 1980).
  * Unno  _et al._ (1989) W. Unno, Y. Osaki, H. Ando, H. Saio, and H. Shibahashi,  _Nonradial oscillations of stars_ (1989).
  * Gautschy and Saio (1995) A. Gautschy and H. Saio, Annual Review of Astronomy and Astrophysics 33, 75 (1995).
  * Kokkotas  _et al._ (2001) K. D. Kokkotas, T. A. Apostolatos, and N. Andersson, Monthly Notices of the Royal Astronomical Society 320, 307 (2001).
  * Kokkotas and Schmidt (1999) K. D. Kokkotas and B. G. Schmidt, Living Reviews in Relativity 2 (1999).
  * Reisenegger and Goldreich (1992) A. Reisenegger and P. Goldreich, Astrophys. J. 395, 240 (1992).
  * Jaikumar  _et al._ (2021) P. Jaikumar, A. Semposki, M. Prakash, and C. Constantinou, Phys. Rev. D 103, 123009 (2021).
  * Tonetto and Lugones (2020) L. Tonetto and G. Lugones, Phys. Rev. D 101, 123029 (2020).
  * Sotani  _et al._ (2001) H. Sotani, K. Tominaga, and K.-i. Maeda, Phys. Rev. D 65, 024010 (2001).
  * Ranea-Sandoval  _et al._ (2018) I. F. Ranea-Sandoval, O. M. Guilera, M. Mariani, and M. G. Orsaria, Journal of Cosmology and Astroparticle Physics 2018, 031 (2018).
  * Raithel  _et al._ (2019) C. A. Raithel, F. Özel, and D. Psaltis, The Astrophysical Journal 875, 12 (2019).
  * Raithel  _et al._ (2021a) C. A. Raithel, F. Özel, and D. Psaltis, The Astrophysical Journal 915, 73 (2021a).
  * Akmal  _et al._ (1998) A. Akmal, V. R. Pandharipande, and D. G. Ravenhall, Phys. Rev. C 58, 1804 (1998).
  * Zhao and Lattimer (2020) T. Zhao and J. M. Lattimer, Phys. Rev. D 102, 023021 (2020).
  * Thorne and Campolattaro (1967) K. S. Thorne and A. Campolattaro, Astrophys. J. 149, 591 (1967).
  * Gregorian (2014) P. Gregorian,  _Nonradial neutron star oscillations_ , Master’s thesis, Universiteit Utrecht in collaboration with Theoretical Astrophysics Eberhard-Karls-University Tuebingen (2014).
  * Flores and Lugones (2017) C. V. Flores and G. Lugones, Phys. Rev. C 95, 025808 (2017).
  * Zhao  _et al._ (2022) T. Zhao, C. Constantinou, P. Jaikumar, and M. Prakash, Phys. Rev. D 105, 103025 (2022), arXiv:2202.01403 [gr-qc] .
  * McDermott  _et al._ (1983) P. N. McDermott, H. M. van Horn, and J. F. Scholl, Astrophys. J. 268, 837 (1983).
  * Kantor and Gusakov (2014) E. M. Kantor and M. E. Gusakov, Monthly Notices of the Royal Astronomical Society: Letters 442, L90 (2014).
  * Constantinou  _et al._ (2021) C. Constantinou, S. Han, P. Jaikumar, and M. Prakash, Phys. Rev. D 104, 123032 (2021).
  * Lai (1994) D. Lai, Monthly Notices of the Royal Astronomical Society 270, 611 (1994).
  * Zhao and Lattimer (2022) T. Zhao and J. M. Lattimer, “Universal relations for neutron star f-mode and g-mode oscillations,”  (2022).
  * Lai (1999) D. Lai, Monthly Notices of the Royal Astronomical Society 307, 1001 (1999).
  * Wei  _et al._ (2020) W. Wei, M. Salinas, T. Klähn, P. Jaikumar, and M. Barry, The Astrophysical Journal 904, 187 (2020).
  * Fu  _et al._ (2017) W.-j. Fu, Z. Bai, and Y.-x. Liu, “Identifying the qcd phase transitions via the gravitational wave frequency,”  (2017).
  * Lattimer and Swesty (1991) J. M. Lattimer and F. D. Swesty, Nuclear Physics A 535, 331 (1991).
  * Shen  _et al._ (1998) H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Nuclear Physics A 637, 435 (1998).
  * Hempel and Schaffner-Bielich (2010) M. Hempel and J. Schaffner-Bielich, Nuclear Physics A 837, 210 (2010).
  * Steiner  _et al._ (2013) A. W. Steiner, M. Hempel, and T. Fischer, The Astrophysical Journal 774, 17 (2013).
  * Lattimer and Lim (2013) J. M. Lattimer and Y. Lim, The Astrophysical Journal 771, 51 (2013).
  * Krüger  _et al._ (2013) T. Krüger, I. Tews, K. Hebeler, and A. Schwenk, Phys. Rev. C 88, 025802 (2013).
  * Drischler  _et al._ (2021) C. Drischler, S. Han, J. M. Lattimer, M. Prakash, S. Reddy, and T. Zhao, Phys. Rev. C 103, 045808 (2021).
  * Constantinou  _et al._ (2015a) C. Constantinou, B. Muccioli, M. Prakash, and J. M. Lattimer, Phys. Rev. C 92, 025801 (2015a).
  * Constantinou  _et al._ (2015b) C. Constantinou, B. Muccioli, M. Prakash, and J. M. Lattimer, Annals of Physics 363, 533 (2015b).
  * Pradhan and Chatterjee (2021) B. K. Pradhan and D. Chatterjee, Phys. Rev. C 103, 035810 (2021).
  * Li  _et al._ (2015) B.-A. Li, W.-J. Guo, and Z. Shi, Phys. Rev. C 91, 044601 (2015).
  * Fonseca  _et al._ (2021) Fonseca _et al._ , The Astrophysical Journal Letters 915, L12 (2021).
  * Abbott  _et al._ (2018) Abbott _et al._ , Phys. Rev. Lett. 121, 161101 (2018), arXiv:1805.11581 [gr-qc] .
  * De  _et al._ (2018) S. De, D. Finstad, J. M. Lattimer, D. A. Brown, E. Berger, and C. M. Biwer, Phys. Rev. Lett. 121, 091102 (2018).
  * Raithel  _et al._ (2021b) C. A. Raithel, V. Paschalidis, and F. Özel, Phys. Rev. D 104, 063016 (2021b).
  * Finn (1990) L. S. Finn, Monthly Notices of the Royal Astronomical Society 245, 82 (1990).
  * Albright and Kapusta (2016) M. Albright and J. I. Kapusta, Phys. Rev. C 93, 014903 (2016).
  * Pratten  _et al._ (2020) G. Pratten, P. Schmidt, and T. Hinderer, Nature Communications 11 (2020), 10.1038/s41467-020-15984-5.