# Accretion disk wind of Hercules X-1 during the Short High state

P. Kosec  MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139  E. Kara  MIT Kavli
Institute for Astrophysics and Space Research, Massachusetts Institute of
Technology, Cambridge, MA 02139  A. C. Fabian  Institute of Astronomy,
Madingley Road, CB3 0HA Cambridge, UK  C. Pinto  INAF - IASF Palermo, Via U.
La Malfa 153, I-90146 Palermo, Italy  I. Psaradaki  University of Michigan,
Dept. of Astronomy, 1085 S University Ave, Ann Arbor, MI 48109, USA  D.
Rogantini  MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139  R. Staubert
Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076
Tübingen, Germany  D. J. Walton  Centre for Astrophysics Research, University
of Hertfordshire, College Lane, Hatfield AL10 9AB, UK

###### Abstract

Hercules X-1 is a nearly edge-on X-ray binary with a warped, precessing
accretion disk, which manifests through a 35-day cycle of alternating High and
Low flux states. This disk precession introduces a changing line of sight
towards the X-ray source, through an ionized accretion disk wind. The
sightline variation allows us to uniquely determine how the wind properties
vary with height above the disk. All the previous wind measurements were made
in the brighter Main High state of Her X-1. Here, we analyze the only Chandra
observation during the fainter ‘Short’ High state, and significantly detect
blueshifted ionized absorption. We find a column density of
$2.0_{-0.6}^{+1.1}\times 10^{22}$ cm-2, an ionization parameter $\log(\xi$/erg
cm s${}^{-1})$=$3.41_{-0.12}^{+0.15}$ and an outflow velocity of $380\pm 40$
km/s. The properties of the outflow measured during the Short High state are
in good agreement with those measured at equivalent precession phases during
Main High. We conclude that we are sampling the same wind structure, seen
during both Main and Short High, which is precessing alongside with the warped
accretion disk every 35 days. Finally, the high spectral resolution of Chandra
gratings above 1 keV in this observation enabled us to measure the abundances
of certain elements in the outflow. We find Mg/O$=1.5_{-0.4}^{+0.5}$,
Si/O$=1.5\pm 0.4$ and S/O$=3.0_{-1.1}^{+1.2}$, whereas in our previous study
of Her X-1 with XMM-Newton, we found an over-abundance of N, Ne and Fe
compared with O. These peculiar abundance ratios were likely introduced by
pollution of the donor by the supernova which created Her X-1.

Accretion (14)

††facilities: Chandra††software: SPEX (Kaastra et al., 1996), Veusz

##  1 Introduction

Hercules X-1 (hereafter Her X-1, Giacconi et al., 1972) is one of the most
complex and fascinating X-ray binaries in the X-ray sky. Fortuitously, its
proximity (6.1 kpc, Leahy & Abdallah, 2014) and high intrinsic luminosity
($\sim 4\times 10^{37}$ erg/s) permit detailed X-ray studies and allowed us to
understand some of its mysteries. The compact object is a neutron star with a
rotation period of 1.24 s (determined from the presence of X-ray pulsations,
Tananbaum et al., 1972; Giacconi et al., 1973) and a surface magnetic field of
$4\times 10^{12}$ G, estimated from the energy of the cyclotron resonance
scattering feature (Truemper et al., 1978; Staubert et al., 2019). The neutron
star is fed from an intermediate mass donor with an orbital period of 1.7 days
(HZ Her, Davidsen et al., 1972; Bahcall & Bahcall, 1972; Middleditch & Nelson,
1976) through a tilted, twisted and precessing accretion disk (Gerend &
Boynton, 1976; Ogilvie & Dubus, 2001), oriented almost edge-on towards us,
such that eclipses by HZ Her are observed.

The disk shape and precession manifests in the long-term evolution of Her X-1
flux, by introducing a $\sim 35$ day cycle of High and Low flux states (Katz,
1973). The cycle begins by a turn-on into the Main High state, during which
the inner accretion flow (emitting the majority of X-rays) is directly
observable from our line of sight. About 10 days later, the precessing inner
edge of the accretion disk starts to obscure the X-ray source. This Low State
is followed by a second, fainter Short High state (Fabian, 1973), during which
the X-ray source is just barely uncovered by the accretion disk (Scott et al.,
2000; Leahy, 2002) and Her X-1 reaches about one third of the maximum Main
High state flux. Finally, a second Low State of the 35-day cycle follows, when
the X-ray source is again fully covered. For an illustration of these
variations, we refer to Fig. 2 of Scott et al. (2000), which shows an average
lighturve of the 35-day Her X-1 cycle, obtained by stacking many cycles
observed with the RXTE instrument (see also Fig. 4 in this paper for an
example of a single cycle).

The warped disk precession results in our line of sight passing through
different paths through the accretion flow, and specifically at different
heights above the warped disk. A unique opportunity is thus provided to us -
our line of sight samples and allows us to study any ionized material over a
range of heights above the disk. Namely, we are able to study the vertical
distribution of the properties of the accretion disk wind of Her X-1. A
schematic of this situation is shown in Fig. 1.

Figure 1: A schematic of the Her X-1 system, adapted from Fig. 1 in Kosec et
al. (2023). In this simplified scenario, the warped precessing disk is fixed,
and instead the observer line of sight is moving throughout the various states
of the 35-day precession cycle. The disk precession allows our line of sight
to pass through different parts of the accretion disk wind, at different
heights above the disk. While Kosec et al. (2023) focused on the Main High
state, this work focuses on the fainter, Short High state.

Accretion disk winds were detected via Doppler shifted emission or absorption
lines in X-ray spectra in both black hole (Kotani et al., 2000; Miller et al.,
2006b; Kubota et al., 2007) and neutron star X-ray binaries (Ueda et al.,
2001; Miller et al., 2011) and show typical velocities of 100s km/s (for a
recent review, see Neilsen & Degenaar, 2023). They appear to be common
especially in high inclination soft state black hole systems (Ponti et al.,
2012), suggesting an equatorial wind structure. More recently, outflows were
also detected in X-ray binaries outside the X-ray band - in the UV, optical
and IR bands, and during both hard and soft states (e.g. Muñoz-Darias et al.,
2019; Sánchez-Sierras & Muñoz-Darias, 2020; Castro Segura et al., 2022; Muñoz-
Darias & Ponti, 2022).

The origin and the launching mechanisms of these phenomena are widely debated
- they could be driven by Compton heating of the outer parts of the accretion
disk (Begelman et al., 1983; Tomaru et al., 2018), or via magnetic fields
(Miller et al., 2006a; Chakravorty et al., 2016; Fukumura et al., 2017), but
are unlikely to be driven by radiation pressure alone (Proga & Kallman, 2002).
These outflows have potential to drive away significant fractions of the
originally accreting mass (e.g. Lee et al., 2002; Kosec et al., 2020) and thus
affect the evolution of the X-ray binary (Verbunt, 1993; Ziółkowski &
Zdziarski, 2018). However, it is challenging to accurately constrain the wind
energetics as in most systems we only observe a single line of sight towards
the X-ray source. Therefore, X-ray absorption line spectroscopy gives us only
a very limited view of the 3D wind structure (although see Miller et al.,
2015a, for an approach using wind re-emission). As Her X-1 uniquely offers a
range of sightlines above the accretion disk, it is an ideal source to study
disk winds in order to determine their launching mechanisms and measure their
energetics.

An accretion disk wind was confirmed in Her X-1 using Main High state XMM-
Newton observations by Kosec et al. (2020), but previous UV observations
already showed the presence of an ionized outflow within the binary system
(Boroson et al., 2001). This outflow was predicted by Schandl & Meyer (1994)
and could be the driver of the disk warping (although see Ogilvie & Dubus,
2001, for an alternative radiation-driven warping mechanism). To follow-up on
the opportunity to study the vertical structure of a disk wind, we performed a
large XMM-Newton and Chandra campaign, densely sampling the wind properties
during a single Her X-1 Main High state (Kosec et al., 2023). We found that
the wind is weaker and clumpier at greater heights above the disk, and
combining the wind measurements with a Her X-1 warped disk model, we produced
a 2D map of the wind properties.

Both of the X-ray studies described above were based solely on Main High state
observations, where the high flux of Her X-1 results in the highest statistics
spectra. However, the fact that the wind is detected in all $\sim 30$ Main
High state observations suggests that the wind is most likely present in Her
X-1 at all times. As the outer accretion disk obscures the main X-ray source
during the Low States, the detection of an ionized absorber at these periods
is challenging. On the other hand, the central source is directly visible
again during the Short High state, meaning it should also be possible to see
the wind during this state too (Fig. 1).

Additionally, the binary system of Her X-1 and HZ Her is not oriented
perfectly edge-on, but has an inclination of $\sim 85^{\circ}$ (Leahy &
Abdallah, 2014), which is also the average inclination of the warped accretion
disk. Therefore, our line of sight during at least part of the Short High
state could sample lower heights above the warped disk compared with Main
High. This may happen because during the Main High, the outer disk quickly
moves out of our line of sight right after the Turn-on, whereas during the
Short High, different parts of the disk move away slower and remain relatively
close to our line of sight for prolonged periods of time. This situation and
the position of our line of sight over time with respect to different parts of
the warped disk is illustrated in Fig. 2 of Scott et al. (2000) and in Fig. 3
of Leahy (2002). Following the results of the Main High state wind studies,
the wind column density is expected to be higher at lower heights above the
disk.

Here we study an archival high-spectral resolution Chandra observation of Her
X-1 from Nov 2002, which reveals a number of narrow absorption lines, similar
to those detected in Her X-1 spectra taken during the Main High state. The
structure of this paper is as follows. Section 2 describes our data
preparation and Section 3 the spectral modelling methods. In Section 4 we
present the results of this study, and in Section 5 we discuss their
implications. Throughout the paper, we assume a distance of 6.1 kpc to Her X-1
(Leahy & Abdallah, 2014).

##  2 Data Reduction and preparation

Chandra (Weisskopf et al., 2002) observed Her X-1 once during its Short High
state, on November 3 2002 (observation ID 4375). This is the first time this
observation has been studied in detail and published. The observation exposure
was 20.4 ks, and the HETG gratings (Canizares et al., 2005) were used to
acquire X-ray data in high spectral resolution. The midpoint of the
observation was at MJD=52581.414. The data were taken in the continuous
clocking (CC) mode and the mean count rate (all instruments summed) was 9.4
ct/s.

The data were downloaded from the TGCAT archive (Huenemoerder et al., 2011) in
fully reduced form. We use both Medium Energy Grating (MEG) and High Energy
Grating (HEG) instruments, and stack the positive and negative first order
spectra for each instrument using the combine_grating_spectra routine in CIAO
(Fruscione et al., 2006). We analyzed the data in full spectral resolution
without any binning. We also tested our final spectral fit with the optimal
binning scheme (Kaastra & Bleeker, 2016), but found no differences in the
final results. The HEG and MEG spectra were fitted simultaneously using a
cross-calibration constant. The value of this parameter was always very close
to 1, indicating $<3$% calibration difference between the two instruments. We
used HEG data in the energy range between 1 and 9 keV (1.4 to 12.4 Å), and MEG
data in the range between 0.62 keV and 5 keV (2.5 to 20Å). These limits were
imposed by the instrument calibration and the signal-to-noise.

The spectra were fitted in the spex fitting package version 3.06.01 (Kaastra
et al., 1996). All reduced spectra were converted from ogip format into spex
format using the trafo routine. We used the Cash statistic (Cash, 1979), which
is appropriate for un-binned data, to analyze all spectra. The uncertainties
are provided at 1$\sigma$ significance.

##  3 Spectral modelling

We apply the spectral model from Kosec et al. (2022) to describe the Her X-1
continuum emission during Short High. Fortunately, we are able to make some
simplifications to the complex original model due to the different instrument
used here. For further details about the individual spectral components, we
refer the reader to section 4 of Kosec et al. (2022).

A comt model (Titarchuk, 1994) is used to describe the primary accretion
column emission from the X-ray pulsar, and a bb model describes the soft
(T$\sim 0.1$ keV) reprocessed emission. As our coverage of the soft X-ray band
($<1$ keV) is limited, we fix the blackbody temperature to 0.1 keV. We also
model Fe line emission in the Fe K band. We use two Gaussian emission lines, a
narrow line at 6.4 keV (low ionization Fe), and a medium-width line at 6.7 keV
(Fe XXV). A highly broadened (FWHM$\sim 2$ keV) Fe K line at $\sim 6.5$ keV,
seen during the Main High state of Her X-1 (Kosec et al., 2022), is not
required in our Short High state observation. This is likely a signal-to-noise
issue - our Chandra HETG spectrum has much lower statistics in the Fe K band
than the Main High state XMM-Newton EPIC-pn spectrum. On the other hand, we do
observe a strong excess at 1 keV (the ‘1 keV feature’ seen in the Main High
state, e.g. Fürst et al., 2013), which we model with a highly broadened
Gaussian emission line. This excess is likely due to a forest of Fe L lines,
possibly mixed with Ne IX-X emission. Finally, we also include a broadened
emission line at 18.96 Å (0.65 keV) to describe O VIII emission, seen also
during the Main High state. The O VIII position is very close to the lower
energy limit of our coverage, so we freeze the line wavelength to 18.96 Å. As
our band does not cover softer energies than 0.6 keV, we do not need to
include any further lines to describe O VII, N VII and N VI emission. Finally,
all of the source emission is obscured by Galactic absorption, which is
described by the hot model as an almost neutral absorber with a temperature of
$8\times 10^{-6}$ keV (de Plaa et al., 2004; Steenbrugge et al., 2005). We fit
for the column density of this absorber, but set a lower limit of $1\times
10^{20}$ cm-2 (HI4PI Collaboration et al., 2016).

The absorption lines from the disk wind are described with the pion spectral
model (Miller et al., 2015b; Mehdipour et al., 2016). pion takes the spectral
energy distribution (SED) from the currently loaded continuum spectral model
and self-consistently calculates the ionizing balance and the line depths
produced by an illuminated, photo-ionized slab of plasma. We use the model to
determine the disk wind properties such as its column density, ionization
parameter, systematic velocity, and its velocity width.

Additionally, pion can be used to fit for the elemental abundances in the
outflow. In Kosec et al. (2023) we fitted for the abundance of N, Ne and Fe,
while fixing the abundance of O to 1 (one of the elemental abundances must be
fixed as all the disk wind absorption lines are from metals). Here we use the
best-fitting abundances from that paper, determined from the higher statistics
Main High state XMM-Newton spectra: we fix N/O=3.4, Ne/O=2.3 and Fe/O=2.1.
However, the broader energy band in which HETG offers very good spectral
resolution (and does not suffer from instrumental issues as EPIC pn on XMM-
Newton) allows us to reliably fit for further elemental abundances: we can
determine the abundance of Mg, Si and S. We have previously used 3 Chandra
HETG spectra to study Her X-1 in the Main High state (Kosec et al., 2023), but
were not able to constrain the Mg, S and Si elemental abundances because the
wind was weaker in those 3 Chandra observations in comparison with the Short
High state exposure analyzed here. The abundances of these three elements are
left as free parameters in our spectral fit.

The final model, in symbolic form, is thus: hot x pion x (comt + bb + 4 ga).

##  4 Results

Figure 2: The 0.6-9 keV Short High state Chandra HETG spectrum of Her X-1
(heavily overbinned), showing the broadband continuum fit with the full
spectral model. MEG data are shown in blue, HEG data are in black, and the
best-fitting model is in red. Figure 3: Her X-1 spectrum from the Short High
state Chandra observation, focusing only on the narrow energy bands of
interest around the strongest elemental transitions. The absorption lines near
these transitions imply the presence of an ionized disk wind. MEG data are in
blue, HEG data are in black and the best-fitting spectral model (including the
ionized absorption component) is in red.

The high-resolution Chandra HETG spectrum reveals a number of absorption lines
located near strong Ly$\alpha$ elemental transitions of O, Ne, Si and S. The
HETG spectrum, focusing on only on the strongest elemental transitions, is
shown in Fig. 3. These absorption lines suggest the presence of a highly
ionized outflow with a low blueshift/velocity ($<1000$ km/s). We fit the
spectrum with the spectral model described in Section 3. This results in a fit
with C-stat=8599 for 7888 degrees of freedom (DoF). A continuum-only spectral
fit, where the pion component is omitted, results in fit statistic C-stat=8767
for 7895 DoF. The very large fit statistic improvement of $\Delta$C-stat=168
upon adding the ionized absorber to the baseline spectral fit, even compared
to observations during the brighter Main High state (Extended Data Table 1 of
Kosec et al., 2023), indicates a very high statistical detection ($>5\sigma$,
Kosec et al., 2018) of the disk wind during this Short High state observation
of Her X-1. The best-fitting spectral model is shown in Fig. 2 (which contains
the full $0.6-9$ keV broadband fit) and in Fig. 3 (focusing only on the narrow
energy bands around the strongest disk wind absorption lines).

The best-fitting disk wind properties are as follows. We determine a column
density of $2.0_{-0.6}^{+1.1}\times 10^{22}$ cm-2, an ionization parameter
$\log(\xi$/erg cm s${}^{-1})$ of $3.41_{-0.12}^{+0.15}$, a systematic velocity
of $570\pm 40$ km/s and a velocity width $240\pm 40$ km/s. The orbital motion-
corrected outflow velocity is $380\pm 40$ km/s, determined by following the
steps from Kosec et al. (2020). The measured elemental abundances are
Mg/O=$1.5_{-0.4}^{+0.5}$, Si/O=$1.5\pm 0.4$ and S/O=$3.0_{-1.1}^{+1.2}$. These
results are summarized alongside all the best-fitting continuum parameters in
Table 1.

We tested for the multi-phase nature of the disk wind by adding a second pion
component to the previous spectral fit. A second ionization component could
potentially alter the best-fitting elemental abundances and shift them back
towards Solar ratios. However, by fitting this more complex spectral model for
a range of pion parameter values, we did not find any significant evidence
($\Delta$C-stat$>9$) for a second ionization zone. Similar results were found
in the Main High State (Kosec et al., 2023).

Table 1: Best-fitting properties of the continuum and the disk wind in the Short High spectrum of Her X-1. The continuum component luminosities are calculated from observed, absorption-corrected X-ray fluxes assuming a distance of 6.1 kpc.  Component | Parameter | Best-fitting value
---|---|---
Disk | column density |  $2.0_{-0.6}^{+1.1}\times 10^{22}$ cm-2
wind |  $\log(\xi$/erg cm s${}^{-1})$ | $3.41_{-0.12}^{+0.15}$
| outflow velocity |  $380\pm 40$ km/s
| velocity width |  $240\pm 40$ km/s
| Mg/O | $1.5_{-0.4}^{+0.5}$
| Si/O | $1.5\pm 0.4$
| S/O | $3.0_{-1.1}^{+1.2}$
|  $\Delta$C-stat | $168$
Primary | seed temp. |  $0.33\pm 0.02$ keV
Comptonization | electron temp. |  $3.8_{-0.3}^{+0.4}$ keV
| optical depth | $9.4\pm 0.3$
| luminosity |  $(6.5\pm 0.4)\times 10^{36}$ erg/s
Soft blackbody | temperature | 0.1 keV (fixed)
| luminosity |  $1.62_{-0.10}^{+0.09}\times 10^{36}$ erg/s
1 keV | energy |  $0.91\pm 0.02$ keV
feature | FWHM |  $0.46\pm 0.02$ keV
| luminosity |  $(2.1\pm 0.2)\times 10^{35}$ erg/s
Fe I | energy |  $6.397\pm 0.004$ keV
| FWHM |  $0.037_{-0.012}^{+0.015}$ keV
| luminosity |  $1.9_{-0.3}^{+0.4}\times 10^{34}$ erg/s
Fe XXV | energy |  $6.61_{-0.04}^{+0.05}$ keV
| FWHM |  $0.60_{-0.09}^{+0.13}$ keV
| luminosity |  $7.4_{-1.2}^{+1.4}\times 10^{34}$ erg/s
O VIII | energy | 0.654 keV (fixed)
| FWHM |  $0.034_{-0.010}^{+0.014}$ keV
| luminosity |  $8_{-3}^{+4}\times 10^{33}$ erg/s

##  5 Discussion and conclusions

We study an archival Short High state observation of Her X-1 taken with the
Chandra HETG instrument. The gratings allow us to resolve and detect the same
narrow absorption lines from an ionized disk wind that are present in the Main
High state observations. We model the absorber with the photo-ionization model
pion and determine the wind properties and the abundances of certain elements
within the outflow, leveraging the broader energy band of HETG in comparison
with XMM-Newton RGS.

To compare the properties of the disk wind during the Short High state
observation with our previous study during Main High (Kosec et al., 2023), we
determine the current precession phase as well as the time elapsed since the
Short High state Rise for the archival Chandra observation. In order to
estimate these quantities, we use archival RXTE/ASM observations. A lightcurve
of the relevant Her X-1 precession cycle is shown in Fig. 4.

Figure 4: RXTE/ASM lightcurve ($2-10$ keV) of Her X-1 super-orbital cycle
number 323, during which the archival Chandra observation of the Short High
state occurred. The cycle counting is according to Staubert et al. (1983),
following Davison & Fabian (1977). The midpoint of the Chandra exposure is at
MJD=52581.4 and is shown with the vertical blue dashed line.

The measurements of the Turn-on (into the Main High state) and the Rise (into
the Short High state) are based on formal best fits with a double exponential
function, which has been found to produce good fits to 35-day cycle
lightcurves of Her X-1 (Staubert et al., 2016). Following the method of
Staubert et al. (2016), we determined the Main High Turn-on times for the
current (MJD$=52559.3\pm 0.1$) and the following (MJD$=52594.6\pm 0.1$) 35-day
cycles. We then estimated the precession phase of the Chandra observation to
be $0.627\pm 0.004$. To find the phase elapsed since the Short High state
Rise, we also determined the Short High Rise time from the Her X-1 lightcurve
to be MJD$=52579.0\pm 0.2$. This means that the Short High Rise occurred at a
precession phase of $0.559\pm 0.009$. The mid-point of the Chandra observation
hence happened at phase of $0.068\pm 0.009$ since Her X-1 rose into the Short
High state.

We note that the Rise into the Short High state in Her X-1 does not occur
exactly one half of the 35-day cycle after the Main High Turn-on. This is
because the binary system is not aligned exactly edge-on towards us, but with
a more moderate inclination of about $85^{\circ}$ (Leahy & Abdallah, 2014).
For this reason, for any comparisons between the Main High and the Short High
state wind properties, it is necessary to correctly account for both Main High
Turn-ons and Short High Rise times in order to determine the correct
equivalent precession phases of both high states. Otherwise, the Short High
state properties will be shifted.

In Fig. 5, we compare the best-fitting wind column density and ionization
parameter during the Short High with previous measurements made during the
Main High state (presented in Kosec et al., 2023). We find that both the
column density and the ionization parameter are in good general agreement with
wind parameter measurements at equivalent precession phases during the Main
High state. This suggests that we are observing the same wind structure which
was detected during the Main High state, and that the structure does not vary
strongly between the two high states, as it precesses alongside with the
warped accretion disk.

Figure 5: Wind column density (top panel) and ionization parameter (bottom
panel) versus the equivalent disk precession phase. Main High state
observations (taken from Kosec et al., 2023) are shown in black (XMM-Newton)
and blue (Chandra), the only Chandra observation during the Short High is in
red. The X-axis value of the Short High state data point is the precession
phase elapsed since the Rise into the Short High state.

During the Main High, we found that the wind column density decreases with
increasing precession phase, i.e. with increasing height above the warped
accretion disk (Kosec et al., 2023). In other words, the wind is weakening at
greater heights above the disk. It is plausible that the same relationship is
followed during the Short High state. This will be probed by our upcoming XMM-
Newton and Chandra observation campaign (to occur in 2023).

We can also estimate the location of the outflow during the Short High state
observation and compare it with Main High measurements. Following the approach
of Kosec et al. (2023), we can determine the maximum distance of the outflow
from the ionizing source such as:

$R_{\rm max}=\frac{L_{\rm{ion}}}{N_{\rm{H}}\xi}$ (1)

where $L_{\rm{ion}}$ is the ionizing (13.6 eV - 13.6 keV) luminosity. The
estimated observed $L_{\rm{ion}}$ during the Chandra observation is $(6.2\pm
0.4)\times 10^{36}$ erg/s. Therefore, the maximum distance of the outflow from
the neutron star is $1.2_{-0.7}^{+0.5}\times 10^{11}$ cm. We compare this
value with the maximum distances of the wind from the ionizing source during
the Main High state (Fig. 3 of Kosec et al., 2023). We find that the maximum
distance of the outflow from the X-ray source is roughly comparable with the
maximum distances derived for wind measurements taken around phase
$\mathbf{0.04-0.05}$ during the Main High state.

In addition to the column density and the ionization parameter, we also
measure the wind outflow velocity and velocity width. These parameters are
again consistent with those measured during Main High. The outflow velocity
was observed to vary between 200 and 1000 km/s during the Main High (compared
with $380\pm 40$ km/s during Short High), while the velocity width was in most
observations measured to be between 70 and 500 km/s (versus $240\pm 40$ km/s
during the Short High observation).

Thanks to the Chandra HETG broader bandpass (observed in high-resolution), we
are additionally able to measure elemental abundances of certain elements
inaccessible to XMM-Newton. Those elements are Mg, Si and S. While technically
included in the RGS band, the Mg XII line is not as well detected and resolved
in the RGS data as with HETG data, preventing a direct abundance measurement.
We find that both Mg and Si are about 50% super-abundant compared to O, while
the S/O ratio is as high as 3.

Combining these measurements with previous XMM-Newton results (Kosec et al.,
2023) on other elemental abundances (N/O$=3.4_{-0.8}^{+0.6}$,
Ne/O$=2.3_{-0.5}^{+0.4}$, Fe/O$=2.1\pm 0.3$), we find that all the elements
are super-abundant compared with O. This is seen at low statistical
significance for Mg and Si, but is significant for the remaining 4 elements.
These peculiar abundances are presumed to have originated by pollution of the
donor by the supernova which created Her X-1 (Jimenez-Garate et al., 2002,
2005). An alternative explanation may be that O is depleted rather than other
elements being over-abundant. However, it is not likely that O would be
strongly under-abundant since the CNO cycle (presumably operating in the Her
X-1 progenitor) will create a strong over-abundance of N and a strong under-
abundance of C, without heavily depleting the O abundance (Przybilla et al.,
2010). The UV spectra of Her X-1 indeed show over-abundance of N and depletion
of C (Boroson et al., 1997). On the other hand, how the heavier elements such
as S and Fe become super-abundant by a factor of 2-3, is unclear. We note that
Allen et al. (2018) found evidence for non-Solar elemental ratios of certain
heavier elements including Mg and Ca in the neutron star X-ray binary GX 13+1.
Similarly, Kallman et al. (2009) found an over-abundance of Fe-group elements
in the black hole X-ray binary GRO J1655-40 and suggested that these findings
are consistent with enrichment by a core-collapse supernova. In any case,
further spectroscopic observations (of Her X-1 and other X-ray binaries) are
necessary to decrease the currently significant uncertainties on the measured
abundances to confirm these findings.

Finally, we also measure the properties of the various prominent emission
lines of Her X-1 during Short High. The position of the narrow Fe line is
fully consistent with 6.4 keV, i.e. with low ionization Fe (Fe I up to Fe
XIII), while the position of the medium-width Fe line is consistent at
$1\sigma$ with Fe XXV. These line energies as well as the line widths are
consistent with the lines detected in Her X-1 during Main High (Kosec et al.,
2022). The same is found for the width of the O VIII line, and for both the
energy and the width of the 1 keV broad residual.

The upcoming XRISM observatory (XRISM Science Team, 2020), expected to be
launched in 2023 will revolutionize the X-ray spectroscopic studies of X-ray
binaries thanks to its Resolve calorimeter with a resolution of just 5-7 eV
(e.g. Tomaru et al., 2020). To investigate how XRISM will be able to improve
our measurements of the disk wind in Her X-1, we simulated a brief 10 ks
observation of the Short High state using the best-fitting spectral parameters
from the Chandra observation used in this work. We used publicly available
XRISM simulation files111https://heasarc.gsfc.nasa.gov/docs/xrism/proposals/
assuming the goal 5 eV spectral resolution. The simulated data are shown in
Fig. 6, focusing just on the Fe K energy band where XRISM will achieve the
best results.

The XRISM statistics are much higher than those of Chandra HETG across the
full energy range, in agreement with its superior effective area over Chandra.
The simulated spectrum has a count rate of about 45 ct/s. At soft energies
(e.g. the Ne X transition at $\sim 1$ keV), the spectral resolution of XRISM
is lower than that of Chandra HETG. The resolutions of the instruments are
comparable at medium energies (e.g. around the S XVI transition at 2.6 keV).
However, XRISM shines at high energies, particularly in the Fe K band, where
it easily resolves individual wind absorption lines, their widths and their
shapes. We particularly note that the broadening of the Fe XXV and XXVI
absorption lines shown in Fig. 6 is not instrumental but due to the velocity
width of the disk wind measured from the Chandra spectral fit.

Figure 6: Simulated 10 ks XRISM spectrum of Her X-1 in the Short High state,
focusing only on the Fe K energy band. XRISM data are in blue and the
simulated spectral model (including the ionized absorption component) is in
red. The data are binned to oversample the instrumental resolution by a factor
of 2.

We performed a spectral fit of this 10 ks XRISM mock simulation. The disk wind
is detected over the baseline continuum with a fit improvement of
$\Delta$C-stat=507, indicating a much stronger detection than in the longer 20
ks archival Chandra dataset (where $\Delta$C-stat=168 was measured). This
indicates that we will be able to detect the wind during Short High and
measure its properties in even shorter XRISM snapshots, with exposures
possibly as low as 2-3 ks. Such low exposure requirement for measurements will
allow us to search for any short-term wind variability.

The wind parameters are also significantly better constrained in the 10 ks
XRISM exposure in comparison with Chandra data: the column density will be
determined with a precision of $2\times 10^{21}$ cm-2 (10% precision), the
ionization parameter $\log(\xi$/erg cm s${}^{-1})$ will be measured with a
precision of 0.04, and the outflow velocity and velocity width with a
precision of 20 km/s. The uncertainties on the abundance of Mg and Si will
shrink by roughly a third, while the uncertainties on the S abundance will
decrease by a factor of 2. Clearly, even very brief XRISM snapshots will allow
detailed measurements of the accretion disk wind in Her X-1 and in comparably
bright X-ray binaries.

In conclusion, the Chandra observation of Her X-1 during the Short High state
reveals the same outflow detected during the Main High. The best-fitting wind
properties during Short High are broadly consistent with those determined at
an equivalent precession phase during the Main High, after taking into account
the delayed Short High state Rise time. By combining the results from both the
Main High and the Short High, we will be able to probe the vertical structure
of the disk wind of Her X-1 in great detail over a broad range of heights.
This will allow us to understand, for the first time, the 3D structure and
energetics of such an outflow, inaccessible in other systems with fixed
sightlines without full 3D modelling of the wind re-emission. Future
observations with XRISM (to be launched in 2023) will allow precision
measurements of the Her X-1 wind properties, particularly thanks to its
superior resolution in the Fe K band. Her X-1 may thus be the Rosetta stone of
X-ray binary accretion disk winds, which will allow us to fully determine
their physics, energetics and 3D geometry.

This work is based on data, obtained by the Chandra X-ray Observatory,
available at https://doi.org/10.25574/cdc.155 (catalog DOI: cdc.155). Support
for this work was provided by the National Aeronautics and Space
Administration through the Smithsonian Astrophysical Observatory (SAO)
contract SV3-73016 to MIT for Support of the Chandra X-Ray Center and Science
Instruments. PK and EK acknowledge support from NASA grants 80NSSC21K0872 and
DD0-21125X.

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# Charged Strange Star Model with Stringy Quark Matter in Rainbow Gravity

Wasib Ali<EMAIL_ADDRESS>Department of Mathematics, Division of Science
and Technology, University of Education, Lahore, Pakistan.  Umber Sheikh
<EMAIL_ADDRESS>Department of Mathematics, Division of Science and
Technology, University of Education, Lahore, Pakistan.

###### Abstract

This study deals with the formation and evolution of a strange star in the
Krori Barua Rainbow spacetime from collapsing charged stringy quark matter.
The dynamical variables are explored from the field equations, taking into
account the effects of particle’s energy on the mass density, pressure, and
string tension. The electric field is also computed using the MIT Bag model.
The real time data of $SAX~{}J1808.4-3658$ is used to analyzed the physical
properties including gradients, energy conditions, anisotropy, stability,
Tolman Oppenheimer Volkoff equation, mass function, compactness, and red-
shift. The graphical analysis has been made according to both the theories of
rainbow gravity and general relativity. The energy conditions and anisotropy
are found to be satisfied, indicating the physical existence of suggested
model. Tolman Oppenheimer Volkoff equation is satisfied indicating equilibrium
of forces and stability of the compact object. Overall, our model is
consistent with the observational information of $SAX~{}J1808.4-3658.$

###### pacs:

04.20.Jb, 04.50.Kd, 04.40.Dg

Keywords: Strange Star; Rainbow Gravity; Krori Barua Spacetime; Charged
Stringy Quark Matter; Dynamical Variables.

##  I Introduction

Compact stars are astrophysical objects that captivate researchers due to
their intriguing properties and their potential to shed light on the
fundamental physics of matter and gravity. These celestial objects,
characterized by their high mass and compactness, possess immense
gravitational fields that challenge our current understanding of the laws
governing the universe. They provide a unique laboratory for studying matter
under extreme conditions and exploring the nature of gravity itself.

Among the various types of compact stars, strange stars have attracted
significant attention. Strange stars are objects composed primarily of up,
down, and strange quark matter. These exotic objects are believed to exist in
the universe and have been proposed as an alternative to neutron stars or
black holes. The existence of uncharged and charged strange stars is analyzed
in the background of torsion based $f(T)$ modified gravity with the discussion
on maximum mass limit 1 ; 2 .

The physical properties and existence of charged anisotropic strange stars are
investigated within the Rastall framework 3 . The presence of strange stars
within the $f(T,T)$ gravity reveals the properties under the influence of
torsion 4 . A unique anisotropic solution for the strange star model in the
context of 5D Einstein-Gauss-Bonnet gravity is examined considering the
implications of higher-dimensional gravitational theories 5 . Solving the
field equations within the context of $f(R,T)$ gravity and the discoveries
contributed significantly to the theoretical understanding of strange stars 6
. A comprehensive analysis is conducted of various candidates for strange
stars, including $EXO1785-248,LMCX-4,SMCX-1,SAXJ1808.4-3658,4U1538-52,$ and
$HerX-1$. The physical features of these candidates, considering them as
uncharged and charged, static, spherically symmetric, and isotropic fluid
spheres, thus valuable insights are provided into their unique characteristics
7 ; c1 ; c2 ; c3 ; c4 .

Krori Barua (KB) metric is exemplary to present a compact star as this metric
does not contain any singularity. It is found to be feasible to present the
famous neutron, strange, and quark stars using this metric in different
theories of gravity. The physical properties and stability of anisotropic
compact stellar objects in energy-momentum squared gravity using KB metric are
verified for $HerX-1,~{}4U1538-52,$ and $SAXJ1808.4-3658$ compact stars c10 .
The values of the Rastall parameter for $SAXJ1808.4-3658,~{}VelaX-12,$ and
$HerX-1$ are investigated for their physical existence using KB metric c11 .
The real-time data for $PSRJ1614-2230,~{}4U1608-52,~{}CenX-3,~{}EXO1785-248,$
and $SMCX-1$ has been verified using KB metric in $f(T,T)$ theory c12 . The
physical properties of
$Her-1,~{}SAXJ1808.4-3658,~{}4U1820-30,~{}PSRJ1614-2230,~{}VelaX-1,$ and
$CenX-3$ are verified using KB metric with Hu-Sawicki and logarithmic model in
$f(G,R)$ gravity c13 . The same has been done for generic data in higher-order
torsion theory c14 .

Rainbow gravity (RG) is a framework that proposes modifications to general
relativity by amalgamating energy-dependent spacetime curvature. The notion of
rainbow gravity stems from the concept that the effects of quantum gravity may
manifest at energy scales close to the Planck scale. A ubiquitous conformal
coupling results from the effective switch-off of RG at higher or at the very
least energies. This suggests a significant departure from the standard
behavior of gravity in the high-energy regime 9 . The hydrostatic equilibrium
and structural properties of white dwarfs in Rastall-Rainbow gravity are
investigated and aimed to understand the effects of this modified
gravitational theory 10 . The thermodynamics and phase transition phenomena
associated with rainbow Schwarzschild black holes were examined utilizing the
rainbow functions, shedding light on the modified black hole properties 11 .
The thermodynamic properties and behavior of black holes are discussed 13 .
The dynamics and properties of isotropic quantum cosmological perfect fluid
provide insights into the modified gravitational effects at the quantum
cosmological level 12 .

RG theory is an effort to merge general relativity and quantum mechanics. When
we deal with strange matter c21 , the radiation emerging from the black hole
c22 ; c24 ; c25 , and information paradox like problems F1 -F4 . The
equilibrium configurations of neutron stars in RG demonstrated that the
energy-dependent curvature can significantly affect the mass-radius
relationship A1 . The stability of strange quark stars in RG, concluding that
the modified curvature leads to a more stable configuration A2 . These studies
highlight the relevance and potential of RG in understanding the behavior of
compact stars.

In this manuscript, we will examine the effect of probing particle’s energy,
which, when approaching zero, yields results consistent with general
relativity. The paper is structured as follows: The next section provides a
detailed explanation of the KB Rainbow geometry, focusing on its application
to the star and string fluid matter distribution. Section 3 will present the
formulation of the field equations, along with the solutions for the dynamical
variables and the analysis of energy conditions. In Section 5, we will model
the physical characteristics of strange stars, utilizing the specific physical
data of $SAXJ1808.4-3658$. Lastly, the concluding section will summarize our
findings and provide insights into the overall study.

##  II Geometry of Spacetime and Fluid Description

We have assumed a star filled with charged stringy quark matter. This matter
obeys the energy-momentum tensor F3 :

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}-lx_{\mu}x_{\nu}+\mathbb{E}^{2}g_{\mu\nu},$
(1)

where $\rho,~{}p,~{}l,~{}E^{2}$ represents the matter density, fluid pressure,
string tension, and squared electric field intensity respectively. The
electric field intensity can be described in terms of charge $Q$ as
$\mathbb{E}^{2}=\frac{Q^{2}}{4\pi r}.$ Position and velocity unit vectors are
presented by $x_{\mu}$ and $u_{\mu}$ respectively. It is noteworthy that for
Eq.(1):

  * •

$p=0$ gives a string cloud,

  * •

$l=0$ leads to quark matter,

  * •

$\rho-p=0$ provides stringy stiff fluid,

  * •

$\rho+p=0$ exhibits stringy anti-stiff fluid,

  * •

$\rho+l=0$ presents a Reddy strings oriented fluid,

  * •

$\rho-l=0$ shows a Nambu strings oriented fluid respectively.

Stringy quark fluid is expected to be present in early Universe PKS . The
stringy quarks are bound under pressure and form the stringy quark fluid. We
have assumed radially stretched one-dimensional strings attached with each
quark in the fluid.

This matter is assumed to collapse in the interior of a star with the Karori
Barua rainbow (KBR) metric:

$ds_{-}^{2}=-\frac{1}{s^{2}\left(\frac{E}{E_{P}}\right)}e^{AR^{2}+C}dt^{2}+\frac{1}{m^{2}\left(\frac{E}{E_{P}}\right)}\left(e^{YR^{2}}dr^{2}+r^{2}d\Omega^{2}\right).$
(2)

with $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}.$ As the interior
spacetime is filled with charged fluid, this leads to a charge exterior
spacetime for matching at the surface of the star. Thus, we have assumed the
exterior as Reissner Nordstrom line element expressed as:

$ds_{+}^{2}=-\frac{1}{s^{2}\left(\frac{E}{E_{P}}\right)}(1-2\frac{M}{r}+\frac{Q^{2}}{r^{2}})dt^{2}+\frac{1}{m^{2}\left(\frac{E}{E_{P}}\right)}\left((1-2\frac{M}{r}+\frac{Q^{2}}{r^{2}})^{-1}dr^{2}+r^{2}d\Omega^{2}\right),$
(3)

where $M$ and $Q$ demonstrate the configuration’s mass and charge.

It is worth noting that in both the line elements, the functions
$s\left(\frac{E}{E_{P}}\right)=1$ and
$m\left(\frac{E}{E_{P}}\right)=\sqrt{1-\eta\frac{E}{E_{P}}}$ are the rainbow
functions. Here the rainbow parameter $\eta>0,$ $E,~{}E_{P}$ are the probing
particle’s and Planck’s energies. The pair of rainbow functions can be used to
illustrate the dynamics of black holes and the information paradox.
Furthermore, the quantum radiations produced by gamma-ray bursts are
anticipated to reveal detailed information on the strange star’s behavior.

Comparing Eqs. (2) and (3), we reached at the following relations at the
boundary of the strange star:

$1-2\frac{M}{R}+\frac{Q^{2}}{R^{2}}=e^{AR^{2}+C},~{}(1-2\frac{M}{R}+\frac{Q^{2}}{R^{2}})^{-1}=e^{YR^{2}},~{}\frac{M}{R^{2}}-\frac{Q^{2}}{R^{3}}=ARe^{AR^{2}+C},$
(4)

leading to

$Y=-\frac{1}{R^{2}}\ln\left(1-2\frac{M}{R}+\frac{Q^{2}}{R^{2}}\right),~{}A=\frac{\left(\frac{M}{R}-\frac{Q^{2}}{R^{2}}\right)}{R^{2}\left(1-2\frac{M}{R}+\frac{Q^{2}}{R^{2}}\right)},~{}C=\ln\left(1-2\frac{M}{R}+\frac{Q^{2}}{R^{2}}\right)-\frac{\left(\frac{M}{R}-\frac{Q^{2}}{R^{2}}\right)}{\left(1-2\frac{M}{R}+\frac{Q^{2}}{R^{2}}\right)}.$
(5)

##  III Formulation of Field Equations and Dynamical Variables

The famous Einstein field equations of general relativity relate astronomical
objects’ geometry to their physics. Because RG incorporates all of the
essential ideas of general relativity, with minor departures from quantum
physics, Einstein’s field equations are valid in this theory to describe
collapse.

The Einstein field equations for collapsing stringy quark fluid (Eq.(1)) in
KBR spacetime (Eq.(2)) are computed in geometric units as:

$\displaystyle
G_{tt}:\frac{-m^{2}e^{Ar^{2}+C}(2Yr^{2}+e^{Yr^{2}}-1)}{s^{2}r^{2}e^{Yr^{2}}}=-\frac{8\pi
e^{Ar^{2}+C}}{s^{2}}(\rho+\mathbb{E}^{2}),$ (6) $\displaystyle
G_{rr}:\frac{-2Ar^{2}+e^{Yr^{2}}-1}{r^{2}}=\frac{-8\pi
e^{Yr^{2}}}{m^{2}}(l+p-\mathbb{E}^{2}),$ (7) $\displaystyle
G_{\theta\theta}:\frac{r^{2}(YAr^{2}-A^{2}r^{2}+Y-2A)}{e^{Yr^{2}}}=\frac{8\pi
r^{2}}{m^{2}}(\mathbb{E}^{2}-p),$ (8) $\displaystyle
G_{\phi\phi}:r^{2}\sin^{2}\theta\frac{YQr^{2}-Q^{2}r^{2}+Y-2Q}{e^{Yr^{2}}}=\frac{8\pi
r^{2}\sin^{2}\theta}{m^{2}}(\mathbb{E}^{2}-p).$ (9)

It is noteworthy that Eqs.(8) and (9) are found to be the same.

The famous MIT bag model suggests the quark matter density and fluid’s
pressure admits the relation: $p_{r}=\frac{1}{3}(\rho-4B_{g}),p_{r}=p+l.$
Using this model, Eqs.(6),(7) and (8) respectively lead to the density $\rho$,
pressures $p,$ string tension $l,$ and squared electric field intensity
$E^{2}$ of the stringy quark matter is computed as:

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{3m^{2}}{16\pi
e^{Yr^{2}}}(A+Y)+B_{g},$ (10) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{m^{2}}{8\pi
e^{Yr^{2}}}\left(\frac{e^{Yr^{2}}-1}{r^{2}}+\frac{1}{2}(A-Y)+(A-Y)Ar^{2}\right)-B_{g},$
(11) $\displaystyle l$ $\displaystyle=$ $\displaystyle\frac{m^{2}}{8\pi
e^{Yr^{2}}}(YAr^{2}-A^{2}r^{2}+Y+\frac{1-e^{Yr^{2}}}{r^{2}}),$ (12)
$\displaystyle\mathbb{E}^{2}$ $\displaystyle=$ $\displaystyle\frac{m^{2}}{8\pi
e^{Yr^{2}}}\left(\frac{e^{Yr^{2}}-1}{r^{2}}+\frac{1}{2}(Y-3A)\right)-B_{g}.$
(13)

It is visible that all these dynamical variables are radius dependent
quantities. The rainbow function $m=m\left(\frac{E}{E_{P}}\right)$ depends on
the probing particle’s energy $E.$ The MIT bag constant is $B_{g}=2.543\times
10^{-4}$. The literature reveals that $SAXJ1808.4-3658$ admits mass
$M=1.435\bigodot$ where $\bigodot$ presents sun’s mass, and radius $R=7.07km$
22b. This leads to the calculation of the KB constants as $A=1.282\times
10^{-2}$ and $Y=1.702\times 10^{-1}.$

The density $\rho$, pressures $p,$ string tension $l$ and squared electric
field intensity $E^{2}$ of the stringy quark matter are exhibited graphically
for the region $0\leq R\leq 5,~{}0\leq E\leq 5.$

Figure 1: Progression of mass density in RG for $SAX.J1808.4-3658$.

FIG. 1 depicts the mass density of a collapsing fluid. The mass density is
highest and finite in the star’s center. However, as one moves radially out
from the center, mass density drops. Furthermore, particles with more energy
have a higher mass density than particles with less energy

Figure 2: Progression of pressure in RG for $SAX.J1808.4-3658$.

The pressure behavior of a collapsing fluid is seen in FIG. 2. The graph shows
that the pressure is limited and is greatest close to the star’s core. When it
gets close to the star’s outer edge, it descends. As the energy of the probing
particle increases, pressure increases.

Figure 3: Progression of string tension in RG for $SAX.J1808.4-3658$.

A collapsing fluid’s string tension is seen in FIG. 3. String tension is
negative and increases with decreasing radius of the star. As the energy of
the probing particle rises, string tension rises as well.

Figure 4: Progression of string tension in RG for $SAX.J1808.4-3658$.

FIG. 4 exhibits the squared electric field intensity behavior of a collapsing
fluid. The graph indicates that electric field intensity is finite and highest
near the star’s center. It decreases when it approaches the star’s outer
boundary.

In general relativity, we may also examine the behavior of these variables by
calculating the energy of a vanishing probing particle. The following is
general relativistic interpretation of these quantities:

Figure 5: Progression of matter density, pressure and string tension in GR for
$SAX.J1808.4-3658$.

FIG. 5 exhibits matter density, pressure, and string tension evolution. It is
obvious that all of the quantities except electric field intensity reach their
highest levels in the distribution center. However, when the radius rises,
these variables diminish, where as the electric field intensity increases with
the increase of radius. This results in the formation of a compact star as a
result of collapsing string fluid.

The distribution of mass at the star’s periphery and the massive concentration
at its center are seen in FIGS. 1-4. The creation of a compact star is
indicated. The fact that the center of resulting astronomical object has a
finite maximum output of mass density, pressure, and string tension indicates
that it is a compact star rather than a black hole.

###  III.1 Gradients

For the sake of a viable strange star, the dynamical variables are required to
possess the following gradients:

  * •

$\frac{d\rho}{dr}\leq 0,$

  * •

$\frac{dp}{dr}\leq 0,$

  * •

$\frac{dl}{dr}\leq 0.$

By calculating the derivative of the three dynamical quantities
$\rho,~{}p,~{}l$ (Eqs.(10)-(12)) corresponding to radius, we can derive the
following gradients $\frac{d\rho}{dr},~{}\frac{dp}{dr},~{}\frac{dl}{dr}$ as:

$\displaystyle\frac{d\rho}{dr}$ $\displaystyle=$
$\displaystyle-\frac{3Yrm^{2}\left(Y+A\right)}{8{\pi}\mathrm{e}^{Yr^{2}}},$
$\displaystyle\frac{dp}{dr}$ $\displaystyle=$
$\displaystyle-\frac{m^{2}\left(2\mathrm{e}^{Yr^{2}}+\left(-Y^{2}+3AY-2A^{2}\right)r^{4}+\left(2A^{2}Y-2AY^{2}\right)r^{6}-2Yr^{2}-2\right)}{8{\pi}r^{3}\mathrm{e}^{Yr^{2}}},$
$\displaystyle\frac{dl}{dr}$ $\displaystyle=$ $\displaystyle-\frac{m^{2}}{4\pi
r^{3}e^{Yr^{2}}}(1+Yr^{2}-e^{Yr^{2}}+(A^{2}-YA+Y^{2})r^{4}-YA(A-Y)r^{6}).$

The general relativistic analogue of these equations are:

$\displaystyle\frac{d\rho}{dr}|_{E=0}$ $\displaystyle=$
$\displaystyle-\frac{3Yr\left(Y+A\right)}{8{\pi}\mathrm{e}^{Yr^{2}}},$
$\displaystyle\frac{dp}{dr}|_{E=0}$ $\displaystyle=$
$\displaystyle-\frac{\left(2\mathrm{e}^{Yr^{2}}+\left(-Y^{2}+3AY-2A^{2}\right)r^{4}+\left(2A^{2}Y-2AY^{2}\right)r^{6}-2Yr^{2}-2\right)}{8{\pi}r^{3}\mathrm{e}^{Yr^{2}}},$
$\displaystyle\frac{dl}{dr}|_{E=0}$ $\displaystyle=$
$\displaystyle-\frac{\left(1+Yr^{2}-e^{Yr^{2}}+(A^{2}-YA+Y^{2})r^{4}-YA(A-Y)r^{6}\right)}{4\pi
r^{3}e^{Yr^{2}}}.$

The first order derivatives of the dynamical quantities $\rho,~{}p,~{}l$ with
respect to the radius are shown as follows:

Figure 6: Progression of $\frac{d\rho}{dr},~{}\frac{dp}{dr},~{}\frac{dl}{dr}$
in RG for $SAX.J1808.4-3658$.

FIG. 6 shows the progression of
$\frac{d\rho}{dr},~{}\frac{dp}{dr},~{}\frac{dl}{dr}$ corresponding to the
radius and probing particle’s energy. The graph clearly shows that this change
is greatest near the center. As $r$ increases, this change decreases
exponentially. Furthermore, when $E$ grows, the gradients of dynamical
quantities reduce. As a result, a particle with higher energy sees a bigger
change in dynamical quantities associated with $r$ than a particle with less
energy.

Figure 7: Progression of $\frac{d\rho}{dr},~{}\frac{dp}{dr},~{}\frac{dl}{dr}$
in GR for $SAX.J1808.4-3658$.

FIG. 7 depicts the progression of
$\frac{d\rho}{dr},~{}\frac{dp}{dr},~{}\frac{dl}{dr}$ in GR. The graph
indicates that the change is greatest in the center and lessens as we approach
the star’s outer border.

###  III.2 Energy Conditions

The imposition of energy conditions identifies the physical significance of
matter presented by the stress-energy tensor. Null, weak and string energy
conditions for the string fluid are $\rho+p-l>0,~{}\rho>0,$ and $\rho+3p-l>0$
respectively U . These can be computed for $SAX~{}J1808.4-3658$ taking account
of the three computed dynamical quantities, mass density $\rho,$ fluid’s
pressure $p,$ and string tension $l$ from Eqs.(10)-(12) as:

Figure 8: Progression of null, weak and strong energy expressions in RG for
$SAX.J1808.4-3658$.

FIG. 8 shows that mass density, pressure and the string tension we derived for
$SAX~{}J1808.4-3658$ satisfy the null, weak, and strong energy conditions.
This means that matter is non-negative when it encounters a time-like or
light-like vector. Furthermore, matters is drawn towards the gravity. These
characteristics validate the physical reality of our stringy quark matter
model.

Figure 9: Progression of null, weak and strong energy expressions in GR for
$SAX.J1808.4-3658$.

All of the energy conditions have been satisfied in GR, as shown in FIG. 9,
confirming the presence of string fluid in GR.

##  IV Physical Properties

Anisotropy, EOS parameters, the TOV equation, Harrera’s cracking condition,
mass function, compactness, and redshift are just a few of the physical
characteristics of $SAX.J1808.4-3658$ that will be covered.

###  IV.1 Anisotropy

Anisotropy is the property of compact star models which assume various
characteristics in different directions. It is the essential component of the
accurate compact star model. The existence of distinct fluid characteristics,
such as a magnetic, external field, rotation, viscosity, and phase
transitions, can cause anisotropy in a fluid’s pressure. In a collapsing
process, anisotropy is the force necessary to keep the compact star from
falling farther. It should be noted that raising anisotropy parameters
improves zone stability. Using Eqs. (11) and (12) anisotropy of the spherical
distribution of matter in KBR metric is determined to be:

$\Delta=p_{t}-p_{r}=p-(p+l)=-\frac{m^{2}}{\kappa
e^{Yr^{2}}}\left(YAr^{2}-A^{2}r^{2}+Y-\frac{e^{Yr^{2}}+1}{r^{2}}\right).$ (15)

Anisotropy has the same general relativistic equivalent as described above.

Following is a representation of $SAX.J1808.4-3658$’s anisotropy:

Figure 10: Progression of anisotropy in RG and GR for $SAX.J1808.4-3658$.

FIG. 10 shows that at the center of the star, the anisotropy is decreasing. It
steadily grows as we approach the star’s outskirts. An analogue of anisotropy
in GR is obtained by replacing $E=0$, which yields the same result as in RG.

###  IV.2 Equation of State Parameters

The EOS describes the state of fluids under certain conditions. An efficient
knowledge of the EOS parameters helps to understand the processes in extreme
conditions. There are two EOS parameters for string fluid:

$\displaystyle\textmd{Radial~{}Parameter:}w_{r}$ $\displaystyle=$
$\displaystyle\frac{m^{2}(A+Y)-16\pi e^{Yr^{2}}B_{g}}{3m^{2}(A+Y)+16\pi
e^{Yr^{2}}B_{g}},$ $\displaystyle\textmd{Tangential~{}Parameter:}w_{t}$
$\displaystyle=$
$\displaystyle\frac{2m^{2}\left(\frac{e^{Yr^{2}}-1}{r^{2}}+(A-Y)Ar^{2}+\frac{1}{2}(A-Y)\right)-16\pi
e^{Yr^{2}}B_{g}}{3m^{2}(A+Y)+16\pi e^{Yr^{2}}B_{g}}.$

The plot of these parameters are shown as follows:

Figure 11: Progression of radial and tangential EOS parameters in RG for
$SAX.J1808.4-3658$.

The EOS radial and transverse parameters, $w_{r}$ and $w_{t}$ respectively are
shown in FIG. 11. The radial parameter is largest in the center and
monotonically decreases as we get closer to the star’s outer edge. The
transverse component grows monotonically from the star’s center to its outer
edge, with the transverse component being lowest at the center. These two
choices both take values in the range of $[0,0.15].$

Figure 12: Progression of radial and tangential EOS parameters in GR for
$SAX.J1808.4-3658$.

FIG. 12 exhibits the two parameters in GR behave similarly.

###  IV.3 Tolman Openheimer Volkoff Equation

A compact star’s stability is determined using the TOV equation. It presents a
thorough examination of the equilibrium condition between gravitational,
hydrostatic, and anisotropic forces ($F_{g},F_{h},F_{a}$). The vanishing
resultant of all system-related forces is required for stable equilibrium. A
non-vanishing outcome indicates a system in disequilibrium. The following is
the general form of the TOV equation:

$\frac{dl}{dr}+\frac{2Qr(\rho+p-l)}{r}-2\frac{l}{r}=0.$ (16)

Here the gravitational force is $F_{g}=\frac{dl}{dr},$ the hydrostatic force
is $F_{h}=\frac{2Qr(\rho+p-l)}{r}$ and anisotropic force is
$F_{a}=2\frac{l}{r}$ respectively. For the strange star model, the
formulations for these forces are as follows:

$\displaystyle\textmd{Gravitational~{}Force:}F_{g}$ $\displaystyle=$
$\displaystyle\frac{dl}{dr}=-\frac{m^{2}}{4\pi
r^{3}e^{Yr^{2}}}(1+Yr^{2}-e^{Yr^{2}}-YA(A-Y)r^{6}$ (17) $\displaystyle+$
$\displaystyle(A^{2}-YA+Y^{2})r^{4}),$
$\displaystyle\textmd{Hydrostatic~{}Force:}F_{h}$ $\displaystyle=$
$\displaystyle\frac{Am^{2}}{2\pi
e^{Yr^{2}}}\left(\frac{e^{Yr^{2}}-1}{r^{2}}+A(1+Ar^{2}-Yr^{2})\right),$ (18)
$\displaystyle\textmd{Anisotropic~{}Force:}F_{a}$ $\displaystyle=$
$\displaystyle\frac{m^{2}}{4\pi
re^{Yr^{2}}}\left(YAr^{2}-A^{2}r^{2}+Y+\frac{1-e^{Yr^{2}}}{r^{2}}\right).$
(19)

These forces have the following general relativistic analogue:

$\displaystyle\textmd{Gravitational~{}Force:}F_{g}|_{E=0}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi
r^{3}e^{Yr^{2}}}(1+Yr^{2}-e^{Yr^{2}}-YA(A-Y)r^{6}$ (20) $\displaystyle+$
$\displaystyle(A^{2}-YA+Y^{2})r^{4}),$
$\displaystyle\textmd{Hydrostatic~{}Force:}F_{h}|_{E=0}$ $\displaystyle=$
$\displaystyle\frac{A}{2\pi
e^{Yr^{2}}}\left(\frac{e^{Yr^{2}}-1}{r^{2}}+A(1+Ar^{2}-Yr^{2})\right)$ (21)
$\displaystyle\textmd{Anisotropic~{}Force:}F_{a}|_{E=0}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi
re^{Yr^{2}}}\left(LAr^{2}-A^{2}r^{2}+L+\frac{1-e^{Yr^{2}}}{r^{2}}\right)$ (22)

The gravitational, hydrostatic, and anisotropic force graphs for
$SAX~{}J1808.4-3658$ are shown below:

Figure 13: Progression of gravitational, hydrostatic and anisotropic forces in
RG for $SAX.J1808.4-3658$.

The gravitational, hydrostatic, and anisotropic forces are shown in FIG. 13.
The graphic unequivocally demonstrates that hydrostatic force is positive
whereas gravitational and anisotropic forces are both negative. All three
forces are at their greatest in the star’s center. They get less intense as we
go radially in the direction of the star’s periphery. As the energy of the
probing particle rises, these forces also do.

Figure 14: Progression of gravitational, hydrostatic and anisotropic forces in
GR for $SAX.J1808.4-3658$.

FIG. 14 exhibits a similar behavior of gravitational, hydrostatic and
anisotropic forces in GR as RG.

###  IV.4 Harrera’s Cracking Condition

Harrera’s cracking condition explores the stability of strange star
configurations. When a stable equilibrium star structure is perturbed, either
from incoherent radiation emission or a system anisotropy, the cracking
occurs. As a result, it requires a consistent nature for matter distribution.
Thus, $v^{2}_{l}\leq 1$ is also known as the no-cracking notion. Moreover,
literature reveals that $v^{2}_{l}$ is also labeled as the squared speed of
sound. Here $v^{2}_{l}=\frac{dl}{d\rho}$ can be evaluated for the strange star
as follows:

$v^{2}_{l}=\frac{dl}{d\rho}=\frac{2\left(1+Yr^{2}-e^{Yr^{2}}+(A^{2}-YA+Y^{2})r^{4}-YA(A-Y)r^{6}\right)}{3Yr^{4}(Y+A)}$
(23)

The general relativity case, i.e., for vanishing energy of probing particles
$E,~{}v^{2}_{l}$ would be the same as that calculated in RG.

The quantity $v^{2}_{l}$ in RG and GR is shown as follows:

Figure 15: Progression of $v^{2}_{l}$ in RG and GR for $SAX.J1808.4-3658$.

The squared speed of sound is greatest near the center of a compact star, as
seen in FIG. 15. As we move along the radius towards the star’s periphery, it
gets smaller. The condition does not alter when $E$ changes since the
expression is independent of the probing particle’s energy. However, across
the domain, the squared sound speed lies inside $[0,1]$, supporting Harrera’s
intriguing theory.

###  IV.5 Mass Function, Compactness and Redshift

The strange star model, distinguishing it from a black hole, requires a
maximum mass limit for dynamical stability. The following formula can be used
to calculate the mass function of a collapsing spherical distribution inside
the radius $r$:

$\displaystyle M(r)$ $\displaystyle=$ $\displaystyle\int_{0}^{r}4\pi r^{2}\rho
dr,$ (24) $\displaystyle=$ $\displaystyle\int_{0}^{r}\left(4\pi
r^{2}\frac{3m^{2}}{16\pi e^{Yr^{2}}}(A+Y)+B_{g}\right)dr$ $\displaystyle=$
$\displaystyle\dfrac{3m^{2}(A+Y)\left(\sqrt{{\pi}}\,\operatorname{erf}\left(\sqrt{Y}\,r\right)-2\mathrm{e}^{-Yr^{2}}\sqrt{Y}\,r\right)}{16Y^{\frac{3}{2}}}+\dfrac{4\pi
B_{g}r^{3}}{3}$

The mass function in general relativity can be derived for vanishing probing
particle energy as:

$M(r)|_{E=0}\dfrac{3(A+Y)\left(\sqrt{{\pi}}\,\operatorname{erf}\left(\sqrt{Y}\,r\right)-2\mathrm{e}^{-Yr^{2}}\sqrt{Y}\,r\right)}{16L^{\frac{3}{2}}}+\dfrac{4\pi
B_{g}r^{3}}{3}$ (25)

In spherical rainbow geometry, the mass function of collapsing string fluid is
expressed graphically as:

Figure 16: Progression of mass function in RG and GR for $SAX.J1808.4-3658$.

FIG. 16 shows that at the strange star’s center, the mass function is
approximately equal to zero. It grows higher as we approach the star’s
boundary. The mass function is affected by probing a particle’s energy. The
mass function grows as the probing particle’s energy grows. A low mass in the
center shows that the resultant is a strange star rather than a black hole.
Furthermore, the function behaves similarly in RG and GR.

The maximal mass of the strange star defines the maximum values of
compactness. The compactness factor is defined as the mass function to radius
ratio. Compactness of an anisotropic string fluid model is given as:

$\displaystyle u(r)$ $\displaystyle=$ $\displaystyle\frac{M}{r}$ (26)
$\displaystyle=$
$\displaystyle\dfrac{3m^{2}(A+Y)\left(\sqrt{{\pi}}\,\operatorname{erf}\left(\sqrt{Y}\,r\right)-2\mathrm{e}^{-Yr^{2}}\sqrt{Y}\,r\right)}{16rY^{\frac{3}{2}}}+\dfrac{4\pi
B_{g}r^{2}}{3}$

The general relativistic analogue of the compactness factor can be calculated
by reducing the energy of the probing particle to zero, i.e.,

$u(r)|_{E=0}=\dfrac{3(A+Y)\left(\sqrt{{\pi}}\,\operatorname{erf}\left(\sqrt{Y}\,r\right)-2\mathrm{e}^{-Yr^{2}}\sqrt{Y}\,r\right)}{16rY^{\frac{3}{2}}}+\dfrac{4\pi
B_{g}r^{2}}{3}.$ (27)

The compactness factor is represented as:

Figure 17: Progression of compactness factor in RG and GR for
$SAX~{}J1808.4-3658$.

FIG. 17 exhibits the strange star’s compactness factor $SAX~{}J1808.4-3658.$
The center of the spherical configuration has the lowest compactness factor.
This factor increases as $r$ and $E$ increase. The Buchdahl limit, which says
that the compactness factor should be less than $\frac{4}{9}\approx 0.444.$,
is followed by the compactness factor in RG and GR.

Surface redshift is the term used to describe the shift in electromagnetic
radiation caused by a source in the gravitational field, notably near the
surface of a peculiar star. The surface redshift can be computed as follows:

$\displaystyle Z_{s}$ $\displaystyle=$
$\displaystyle(1-2u(r))^{-\frac{1}{2}}-1$ $\displaystyle=$
$\displaystyle\left[1-\dfrac{3m^{2}(A+Y)\left(\sqrt{{\pi}}\,\operatorname{erf}\left(\sqrt{Y}\,r\right)-2\mathrm{e}^{-Yr^{2}}\sqrt{Y}\,r\right)}{8rY^{\frac{3}{2}}}-\dfrac{8\pi
B_{g}r^{2}}{3}\right]^{-\frac{1}{2}}-1$

The general relativistic analogue of surface redshift can be computed as:

$Z_{s}|_{E=0}=\left[1-\dfrac{3(A+Y)\left(\sqrt{{\pi}}\,\operatorname{erf}\left(\sqrt{Y}\,r\right)-2\mathrm{e}^{-Yr^{2}}\sqrt{Y}\,r\right)}{8rL^{\frac{3}{2}}}-\dfrac{8\pi
B_{g}r^{2}}{3}\right]^{-\frac{1}{2}}-1.$ (29)

The surface redshift in RG its general relativistic analogue is exhibit as
follows:

Figure 18: Progression of surface redshift in RG and GR for
$SAX.J1808.4-3658$.

The center of the star has the smallest redshift, as seen in FIG. 18. It gets
stronger as we go closer to the star’s boundary. Furthermore, the redshift
grows as $E$ increases. The positive anisotropy factor has no upper limit on
the redshift. As a result, the particle with more energy will have a more
significant redshift. The redshift behavior is observed to be the same in both
GR and RG.

##  V Conclusions

This research investigates the dynamics of a collapsing charged stringy quark
matter under the influence of rainbow gravity. The geometry of the KBR metric
is used to understand gravitational collapse and the MIT Bag model is
utilized. The real-time data of charged star $SAX~{}J1808.4-3658$ is used to
determine the physical properties like energy conditions, first-order
gradients of mass density, pressure and string tension, anisotropy, equation
of state parameters, Tolman Oppenheimer Volkoff equation, Harrera’s cracking
condition, mass compactness, and redshift. The following key findings were
obtained:

  * •

As one proceeds outward, the mass density and pressure of $SAXJ1808.4-3658$
decrease from their maximum points in the center.

  * •

The string tension is negative, reaching a maximum near the center of
$SAX~{}J1808.4-3658$ and lowering as the radius increases.

  * •

The squared electric field intensity $\mathbb{E}^{2}$ is highest at the center
of $SAX~{}J1808.4-3658$ and decreasing as radius increases.

  * •

Observers with higher energy experience the dynamical quantities (mass
density, pressure, string tension) more intensely compared to those with lower
energy.

  * •

First-order gradients of dynamical quantities are negative, showing all these
functions are decreasing, compatible with a strange star model.

  * •

The energy conditions are satisfied for charged stringy quark fluid,
supporting its physical presence.

  * •

The anisotropy of the fluid increases towards the boundary of the strange
star, particularly for observers with higher energy.

  * •

The equation of state parameters is positive and larger for particles with
higher energy.

  * •

The hydrostatic force is positive, but the gravitational and anisotropic
forces are both negative, leading to a balanced and stable model of
$SAXJ1808.4-3658$.

  * •

The squared sound speed is greatest close to the star’s centre and falls down
towards the edge, with higher energy viewers reporting a quicker speed of
sound.

  * •

The mass function, compactness, and redshift increase with the increasing
radius, with higher values for more energetic observers.

General relativistic analogs of all the properties of the RG model for
$SAXJ1808.4-3658$ were computed to validate the data, and all properties were
determined to be satisfactory. Thus, this study presents a generalized model
of charged $SAX~{}J1808.4-3658$ in RG, incorporating quantum effects through
the inclusion of rainbow functions in the expressions of the dynamical
quantities of the string fluid. This model supports the evolution and presence
of charged strange stars such as $SAX~{}J1808.4-3658$ in stringy quark matter,
probably in the early phases of the Universe.

## Declarations

### Funding

Not Applicable

### Conflicts of Interest/Competing Interests (Include Appropriate
Disclosures)

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

### Availability of Data and Material

Not Applicable

### Code Availability

Not Applicable

### Authors’ Contributions

Umber Sheikh: Conceptualization, Design of Analysis, Writing - Reviewing and
Editing, Supervision, Software, Validation.
Wasib Ali: Data Collection, Methodology, Formal Analysis and Investigations,
Writing - Original draft preparation, Investigation, Software, Visualization.

### Acknowledgements

The authors acknowledge University of Education Lahore and HEC Pakistan for
the needful support.

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