11institutetext: School of Physics and Astronomy, Queen Mary University of
London, London E1 4NS, UK
11email<EMAIL_ADDRESS>22institutetext: Department of Physics
and Astronomy, University of New Hampshire, Durham, NH 03824, USA
33institutetext: Department of Physics, Imperial College London, London SW7
2AZ, UK 44institutetext: NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA 55institutetext: Department of Aerospace, Physics and Space
Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
66institutetext: Johns Hopkins University Applied Physics Laboratory, Laurel,
MD 20723, USA 77institutetext: Space Sciences Laboratory, University of
California, Berkeley, CA 94720, USA 88institutetext: Lunar and Planetary
Laboratory, University of Arizona, Tucson, AZ 85719, USA 99institutetext:
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 92195 Meudon, France 1010institutetext: Climate and Space
Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA
1111institutetext: Smithsonian Astrophysical Observatory, Cambridge, MA 02138
USA 1212institutetext: Physics Department, University of California, Berkeley,
CA 94720, USA

# The Near-Sun Streamer Belt Solar Wind: Turbulence and Solar Wind
Acceleration

C. H. K. Chen 11 B. D. G. Chandran 22 L. D. Woodham 33 S. I. Jones-Mecholsky
44 J. C. Perez 55 S. Bourouaine 5566 T. A. Bowen 77 K. G. Klein 88 M.
Moncuquet 99 J. C. Kasper 10101111 S. D. Bale 1133771212

The fourth orbit of _Parker Solar Probe_ (_PSP_) reached heliocentric
distances down to 27.9 $R_{\odot}$, allowing solar wind turbulence and
acceleration mechanisms to be studied _in situ_ closer to the Sun than
previously possible. The turbulence properties were found to be significantly
different in the inbound and outbound portions of _PSP_ ’s fourth solar
encounter, likely due to the proximity to the heliospheric current sheet (HCS)
in the outbound period. Near the HCS, in the streamer belt wind, the
turbulence was found to have lower amplitudes, higher magnetic
compressibility, a steeper magnetic field spectrum (with spectral index close
to –5/3 rather than –3/2), a lower Alfvénicity, and a “$1/f$” break at much
lower frequencies. These are also features of slow wind at 1 au, suggesting
the near-Sun streamer belt wind to be the prototypical slow solar wind. The
transition in properties occurs at a predicted angular distance of $\approx
4^{\circ}$ from the HCS, suggesting $\approx 8^{\circ}$ as the full-width of
the streamer belt wind at these distances. While the majority of the Alfvénic
turbulence energy fluxes measured by _PSP_ are consistent with those required
for reflection-driven turbulence models of solar wind acceleration, the fluxes
in the streamer belt are significantly lower than the model predictions,
suggesting that additional mechanisms are necessary to explain the
acceleration of the streamer belt solar wind.

###### Key Words.:

solar wind – Sun: heliosphere – plasmas – turbulence – waves

##  1 Introduction

One of the major open questions in heliophysics is how the solar wind is
accelerated to the high speeds measured _in situ_ by spacecraft in the solar
system (Fox et al., 2016). Early models of solar wind generation, based on the
pioneering work of Parker (1958), were able to reproduce the qualitative
properties of the solar wind, although could not explain all of the measured
quantities seen at 1 au (see, e.g., the reviews of Parker, 1965; Leer et al.,
1982; Barnes, 1992; Hollweg, 2008; Hansteen & Velli, 2012; Cranmer et al.,
2015, 2017). Turbulence is now thought to be one of the key processes playing
a role in solar wind acceleration, providing both a source of energy to heat
the corona (Coleman, 1968) and a wave pressure to directly accelerate the wind
(Alazraki & Couturier, 1971; Belcher, 1971). Possible mechanisms for the
driving of the turbulence include reflection of the outward-propagating Alfvén
waves by the large-scale gradients (Heinemann & Olbert, 1980; Velli, 1993) and
velocity shears (Coleman, 1968; Roberts et al., 1992). Models that incorporate
these effects are now able to reproduce most solar wind conditions at 1 au
(e.g., Cranmer et al., 2007; Verdini et al., 2010; Chandran et al., 2011; van
der Holst et al., 2014; Usmanov et al., 2018; Shoda et al., 2019), but more
stringent tests come from comparing their predictions to measurements close to
the Sun.

The nature of the turbulence in the solar wind and plasma turbulence in
general is also a major open question (Bruno & Carbone, 2013; Alexandrova et
al., 2013; Kiyani et al., 2015; Chen, 2016). Initial results from _Parker
Solar Probe_ (_PSP_) have revealed many similarities, but also some key
differences in the near-Sun solar wind turbulence. Both the power levels and
cascade rates were found to be several orders or magnitude larger at $\sim
36\,R_{\odot}$ compared to 1 au (Bale et al., 2019; Chen et al., 2020;
Bandyopadhyay et al., 2020). The turbulence was also found to be less
magnetically compressible, more imbalanced, with a shallower magnetic field
spectral index of $\approx-3/2$. The low compressibility and polarisation is
consistent with a reduced slow mode component to the turbulence (Chen et al.,
2020; Chaston et al., 2020). Both the outer scale and ion scale spectral
breaks move to larger scales approximately linearly with heliocentric distance
(Chen et al., 2020; Duan et al., 2020), indicating that the width of the MHD
inertial range stays approximately constant over this distance range. The
steep ion-scale transition range, however, is more prominent closer to the
Sun, indicating stronger dissipation or increase of the cascade rate (Bowen et
al., 2020), or perhaps a build-up of energy at these scales (Meyrand et al.,
2020). The overall increase of the turbulence energy flux, compared to the
bulk solar wind kinetic energy flux, was found by Chen et al. (2020) to be
consistent with the reflection-driven turbulence solar wind model of Chandran
et al. (2011), showing that this remains a viable mechanism to explain the
acceleration of the open field wind. Other comparisons of the early _PSP_ data
to turbulence-driven models also report agreement (Bandyopadhyay et al., 2020;
Réville et al., 2020a; Adhikari et al., 2020).

Much of the solar wind measured in the early _PSP_ solar encounters has been
of open-field coronal hole origin (Bale et al., 2019; Panasenco et al., 2020;
Badman et al., 2020a, b), although short periods of streamer belt wind near
the heliospheric current sheet (HCS) were also identified (Szabo et al., 2020;
Rouillard et al., 2020; Lavraud et al., 2020). Encounter 4, however, was
different in that for the majority of the outbound portion, _PSP_ was
consistently in streamer belt plasma (Bale et al., 2020). In this Letter, the
properties of turbulence during this encounter are presented. These are
compared to the distance to the HCS to show the differences between the
streamer belt wind and the open field wind. The turbulence energy flux is also
compared to predictions of reflection-driven turbulence solar wind models to
investigate the acceleration mechanisms of the streamer belt wind.

##  2 Data

Figure 1: Time series of Encounter 4 solar wind parameters, with symbols
defined in the text. $\phi_{\mathrm{HCS}}$ is the predicted angular distance
of the spacecraft to the heliospheric current sheet. The vertical dashed lines
indicate the intervals used to calculate the spectra in Figure 6.

Data from _PSP_ (Fox et al., 2016), primarily from its 4th solar encounter,
but also from all of the first four orbits, were used for this study. Magnetic
field, $\mathbf{B}$, and electron density, $n_{\mathrm{e}}$, were obtained
from the MAG and RFS/LFR instruments of the FIELDS suite (Bale et al., 2016);
the electron density measurement is described in Moncuquet et al. (2020). Ion
(proton) velocity, $\mathbf{v}_{\mathrm{i}}$, and temperature,
$T_{\mathrm{i}}$ were obtained primarily from from the SPAN-I (Livi et al.,
2020), but also the SPC (Case et al., 2020), instruments of the SWEAP suite
(Kasper et al., 2016). The SPAN-I data consist of bi-Maxwellian fits to the
proton core population, described in Woodham et al. (2020), with the same
selection criteria used for excluding bad fits from the dataset, and this data
is used for $\mathbf{v}_{\mathrm{i}}$ and $T_{\mathrm{i}}$ unless stated
otherwise. Since the fluctuations investigated in this Letter are at MHD
scales, the solar wind velocity $\mathbf{v}$ is taken to be equal to
$\mathbf{v}_{\mathrm{i}}$.

Figure 2: Time series of Encounter 4 turbulence properties, with symbols
defined in the text. The red line is a 10-point running mean of $\alpha_{B}$.
The vertical dashed lines indicate the intervals used to calculate the spectra
in Figure 6.

A time series of the data for Encounter 4 is shown in Figure 1. Additional
quantities plotted include the distance-normalised mass flux, $4\pi\rho
v_{\mathrm{r}}r^{2}$, where $\rho$ is the total mass density estimated as
$\rho=m_{\mathrm{p}}n_{\mathrm{e}}(1+3f_{\alpha})/(1+f_{\alpha})$, where
$f_{\alpha}=0.05$ is the assumed alpha fraction of the ion number density,
$v_{\mathrm{r}}$ is the radial solar wind speed and $r$ is the radial distance
of the spacecraft to the Sun, the distance-normalised kinetic energy flux,
$2\pi\rho v_{\mathrm{r}}^{3}r^{2}$, and the ion plasma beta,
$\beta_{\mathrm{i}}=2\mu_{0}n_{\mathrm{i}}k_{\mathrm{B}}T_{\mathrm{i}}/B^{2}$.
$\phi_{\mathrm{HCS}}$ is the angle (centred at the Sun) of the spacecraft to
the heliospheric current sheet (HCS) estimated using the Wang-Sheeley-Arge
(WSA) model, which consists of a PFSS model for the inner corona encased in a
Schatten current sheet shell (Arge & Pizzo, 2000; Arge et al., 2003, 2004;
Szabo et al., 2020). A zero-point corrected GONG synoptic magnetogram was
chosen for the model’s inner boundary condition, which was found to produce
solar wind predictions that correspond well with the IMF polarity inversion
observed by _PSP_ near the end of January. It can be seen that, unlike the
previous encounters, the inbound and outbound portions had different solar
wind properties: the outbound period had higher density, lower speed and
temperature, with a higher mass flux and plasma beta. These differences can be
accounted for by the fact that _PSP_ spent most of the outbound period close
to the HCS.

##  3 Results

###  3.1 Turbulence Properties

Figure 3: Average turbulence properties for times close to (blue) and far from
(red) the HCS as a function of the value $\phi_{\mathrm{cut}}$ used to define
close and far. Averages are arithmetic means except for quantities marked with
∗, which are geometric means. The error bars represent the standard error of
the mean. The largest overall difference in average values is at
$\phi_{\mathrm{cut}}\approx 4^{\circ}$.

The 11-day period of Encounter 4 (days 24-34 of 2020) was divided into
intervals of 300 s duration, roughly comparable to the outer scale (Chen et
al., 2020; Parashar et al., 2020; Bandyopadhyay et al., 2020; Bourouaine et
al., 2020), and in each a set of turbulence properties was calculated: the
total rms magnetic fluctuation amplitude

$\delta B_{\mathrm{rms}}=\sqrt{\left<|\delta\mathbf{B}|^{2}\right>},$ (1)

where $\delta\mathbf{B}=\mathbf{B}-\mathbf{B}_{0}$,
$\mathbf{B}_{0}=\left<\mathbf{B}\right>$, and the angular brackets denote a
time average, in this case over each 300 s interval, the normalised rms
fluctuation amplitude $\delta B/B_{0}$, the magnetic compressibility,

$C_{B}=\sqrt{\frac{\left<(\delta|\mathbf{B}|)^{2}\right>}{\left<|\delta\mathbf{B}|^{2}\right>}}$
(2)

the normalised cross helicity,

$\sigma_{\mathrm{c}}=\frac{2\left<\delta\mathbf{v}\cdot\delta\mathbf{b}\right>}{\left<|\delta\mathbf{v}|^{2}+|\delta\mathbf{b}|^{2}\right>},$
(3)

where $\mathbf{b}=\mathbf{B}/\sqrt{\mu_{0}\rho_{0}}$ and
$\delta\mathbf{v}=\mathbf{v}-\left<\mathbf{v}\right>$, the normalised residual
energy,

$\sigma_{\mathrm{r}}=\frac{2\left<\delta\mathbf{z}^{+}\cdot\delta\mathbf{z}^{-}\right>}{\left<|\delta\mathbf{z}^{+}|^{2}+|\delta\mathbf{z}^{-}|^{2}\right>},$
(4)

where the Elsasser fields are
$\delta\mathbf{z}^{\pm}=\delta\mathbf{v}\pm\delta\mathbf{b}$, and the angle
between the magnetic field and the radial direction,

$\theta_{\mathrm{BR}}=\cos^{-1}\left(\hat{\mathbf{B}}_{0}\cdot\hat{\mathbf{r}}\right).$
(5)

In the definition of $\mathbf{b}$, its sign is reversed if
$\theta_{\mathrm{BR}}<90^{\circ}$ so that positive $\sigma_{\mathrm{c}}$
corresponds to Alfvénic propagation away from the Sun. In addition, the MHD
inertial range magnetic field spectral index, $\alpha_{B}$, was calculated
from the FFT of 2-hour intervals and fitting a power-law function in the range
of spacecraft-frame frequencies
$10^{-2}\,\mathrm{Hz}<f_{\mathrm{sc}}<10^{-1}\,\mathrm{Hz}$.

Figure 4: Histograms of turbulence properties close to (blue) and far from
(red) the HCS. A clear difference can be seen in all properties.

A time series of these properties, along with the angular distance to the HCS,
$\phi_{\mathrm{HCS}}$, is shown in Figure 2. As expected, there is significant
variability of all quantities, but there are also consistent trends over the
encounter. The outbound portion appears to have lower fluctuations amplitudes,
higher magnetic compressibility, a steeper spectral index, and be less
dominated by pure outward Alfvénic fluctuations ($\sigma_{\mathrm{c}}$ is
closer to zero and $\sigma_{\mathrm{r}}$ further from zero). This does not
appear to be a consequence of radial distance (since the orbit is
geometrically symmetric) or the angle of the magnetic field: the distribution
of $\theta_{\mathrm{BR}}$ (reflected to lie in the range $0^{\circ}$ to
$90^{\circ}$) is the same to within uncertainties, with a mean value of
$\theta_{\mathrm{BR}}=26.8^{\circ}\pm 0.4^{\circ}$ in both cases. One key
difference, however, is the proximity to the HCS, the effect of which is
explored in the rest of this Letter.

One important consideration is whether the Taylor hypothesis remains valid as
_PSP_ gets closer to the Sun (Klein et al., 2015; Bourouaine & Perez, 2018,
2019, 2020). Figure 2 also contains the time series of the parameter
$\epsilon=\delta v_{\mathrm{rms}}/\sqrt{2}v_{\mathrm{sc}}$ calculated from 1
hour intervals, where $v_{\mathrm{sc}}$ is the magnitude of the solar wind
velocity in the spacecraft frame. This is the same parameter as in the model
of Bourouaine & Perez (2019), in which perpendicular sampling is assumed so
$v_{\mathrm{sc}}\sim v_{\mathrm{sc\perp}}$, and in which the Taylor hypothesis
is valid for $\epsilon\ll 1$. The model also assumes Gaussian random sweeping
and anisotropic turbulence $k_{\perp}\gg k_{\|}$, and is valid when
$\tan(\theta_{\mathrm{BV}})\gtrsim\delta v_{\mathrm{rms}}/v_{\mathrm{A}}$,
where $\theta_{\mathrm{BV}}$ is the angle between $\mathbf{B}_{0}$ and the
mean solar wind velocity in the spacecraft frame. Points for which the model
is valid are marked with blue dots and points for which it is not are marked
as red crosses. Bourouaine & Perez (2020) determined that within this model,
frequency broadening caused by the breakdown of the Taylor hypothesis does not
significantly modify the spectrum as long as $\epsilon\lesssim 0.5$, and as
shown in Figure 2, the data points satisfy this condition. Therefore, the
differences in turbulence characteristics investigated in this Letter are
likely not due to the differences in the validity of the Taylor hypothesis. A
more detailed analysis of the Taylor hypothesis for these first _PSP_ orbits
is given in Perez et al. (2020).

###  3.2 HCS Proximity Dependence

Figure 5: Distributions of normalised cross helicity, $\sigma_{\mathrm{c}}$,
normalised residual energy, $\sigma_{\mathrm{r}}$, Elsasser ratio,
$r_{\mathrm{E}}$, and Alfvén ratio, $r_{\mathrm{A}}$, for times close to
(blue) and far from (red) the HCS.

In the outbound portion of Encounter 4, _PSP_ spent significant time in the
streamer belt wind near the HCS (Bale et al., 2020). The width of the streamer
belt wind at these distances is not well known, so the dependence of the
turbulence properties on the distance to the HCS was investigated. Figure 3
shows average values close to and far from the HCS, as a function of the cut
value of the HCS angle used to define close and far, $\phi_{\mathrm{cut}}$.
For example, the first panel shows $\left<\delta
B_{\mathrm{rms}}\right>_{|\phi_{\mathrm{HCS}}|<\phi_{\mathrm{cut}}}$ as a
function of $\phi_{\mathrm{cut}}$ in blue and $\left<\delta
B_{\mathrm{rms}}\right>_{|\phi_{\mathrm{HCS}}|>\phi_{\mathrm{cut}}}$ as a
function of $\phi_{\mathrm{cut}}$ in red. Because the imbalance is so high,
plots for the Elsasser ratio,

$r_{\mathrm{E}}=\frac{1+\sigma_{\mathrm{c}}}{1-\sigma_{\mathrm{c}}},$ (6)

and Alfvén ratio,

$r_{\mathrm{A}}=\frac{1+\sigma_{\mathrm{r}}}{1-\sigma_{\mathrm{r}}},$ (7)

are also shown. All quantities show a difference at all cut angles, but the
largest overall difference is between $\approx 3^{\circ}$ and $\approx
5^{\circ}$, so a value of $\phi_{\mathrm{cut}}=4^{\circ}$ was used to define
the width of the region near the HCS in which the turbulence properties are
different.

Figure 4 shows the distributions of the turbulence properties over the
encounter both near to ($|\phi_{\mathrm{HCS}}|<4^{\circ}$) and far from
($|\phi_{\mathrm{HCS}}|>4^{\circ}$) the HCS. A clear difference can be seen in
each property: near the HCS there are lower amplitudes, higher magnetic
compressibility, a steeper spectrum, a lower level of imbalance and a broader
distribution of residual energy. The joint distributions of
$\sigma_{\mathrm{c}}$ with $\sigma_{\mathrm{r}}$ and $r_{\mathrm{E}}$ with
$r_{\mathrm{A}}$ are shown in Figure 5. The data are constrained
mathematically to lie within the regions

$\sigma_{\mathrm{c}}^{2}+\sigma_{\mathrm{r}}^{2}\leq 1$ (8)

and

$\left(\frac{r_{\mathrm{E}}-1}{r_{\mathrm{E}}+1}\right)^{2}+\left(\frac{r_{\mathrm{A}}-1}{r_{\mathrm{A}}+1}\right)^{2}\leq
1,$ (9)

respectively, marked as solid black lines. In general, it can be seen that in
both cases the fluctuations are highly Alfvénic ($\sigma_{\mathrm{c}}\approx
1$, $\sigma_{\mathrm{r}}\approx 0$, $r_{\mathrm{E}}\gg 1$,
$r_{\mathrm{A}}\approx 1$), more so than in previous encounters (Chen et al.,
2020; McManus et al., 2020; Parashar et al., 2020) but the near-HCS wind is
less so than the wind far from the HCS.

Table 1: Solar wind and turbulence properties close to
($|\phi_{\mathrm{HCS}}|<4^{\circ}$) and far from
($|\phi_{\mathrm{HCS}}|>4^{\circ}$) the heliospheric current sheet (HCS).
Quantities are arithmetic means, apart from those marked with ∗, which are
geometric means. $\begin{array}[]{ccc}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\mathrm{Property}&|\phi_{\mathrm{HCS}}|<4^{\circ}&|\phi_{\mathrm{HCS}}|>4^{\circ}\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr B\ \mathrm{(nT)}&56&88\\\ n_{\mathrm{e}}\
\mathrm{(cm^{-3})}&510&390\\\ v_{\mathrm{i}}\ \mathrm{(km\,s^{-1})}&240&300\\\
T_{\mathrm{i}}\ \mathrm{(eV)}&20&55\\\ 4\pi\rho v_{\mathrm{r}}r^{2}\
(10^{-14}\,M_{\odot}\,\mathrm{yr}^{-1})&3.1&2.1\\\ 2\pi\rho
v_{\mathrm{r}}^{3}r^{2}\ \mathrm{(10^{19}\,W)}&6.0&6.0\\\
\beta_{\mathrm{i}}^{*}&1.2&0.68\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\delta B^{*}\
\mathrm{(nT)}&7.0&21\\\ (\delta B/B_{0})^{*}&0.14&0.25\\\
C_{B}^{*}&0.036&0.0048\\\ \alpha&-1.63&-1.53\\\
\sigma_{\mathrm{c}}&0.55&0.88\\\ \sigma_{\mathrm{r}}&-0.12&-0.031\\\
r_{\mathrm{E}}^{*}&5.2&30\\\ r_{\mathrm{A}}^{*}&0.76&0.94\\\ f_{\mathrm{b}}\
\mathrm{(Hz)}&3\times 10^{-4}&4\times 10^{-3}\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

The averages for these different regions are given in Table 1. Aspects to note
are that near the HCS the wind is denser, slower, cooler, with a higher plasma
beta and mass flux, as expected for the streamer belt wind. The kinetic energy
flux, however, is very similar, as has been seen previously across different
wind types (Le Chat et al., 2012). The difference in turbulence properties is
consistent with the differences between “fast” and “slow” wind seen further
from the Sun (e.g., Tu & Marsch, 1995; Bruno & Carbone, 2013). Notably, the
magnetic field spectral index in the streamer belt wind is close to $-5/3$
whereas it is closer to $-3/2$ far from the HCS. This is consistent with
observed dependencies of the magnetic spectrum on the degree of Alfvénicity at
1 au (Podesta & Borovsky, 2010; Chen et al., 2013; Bowen et al., 2018) and in
the previous _PSP_ orbits (Chen et al., 2020). A similar difference of
spectral index and Alfvénicity was also seen when separating times of inverted
and non-inverted magnetic field (Bourouaine et al., 2020). The imbalance,
however, is larger than typically seen at 1 au, suggesting that the evolution
towards a more balanced state seen in the open field wind (Chen et al., 2020)
also occurs in the streamer belt wind.

###  3.3 Spectra

Spectra of representative intervals (marked by the vertical dashed lines in
Figures 1 and 2) are shown in Figure 6. The lower panel shows the local
spectral index, in which some important differences can be seen. Firstly, the
steeper magnetic field spectrum in the near-HCS interval can be seen
throughout the MHD inertial range. But more significantly, the break between
the “$1/f$” range and the MHD inertial range is at very different frequencies:
$f_{\mathrm{b}}\approx 3\times 10^{-4}$ Hz in the near-HCS wind and
$f_{\mathrm{b}}\approx 4\times 10^{-3}$ Hz far from the HCS. The origin of the
$1/f$ break is debated (Matthaeus & Goldstein, 1986; Velli et al., 1989;
Verdini et al., 2012; Perez & Chandran, 2013; Wicks et al., 2013a, b;
Chandran, 2018; Matteini et al., 2018), but this difference is also seen
between the “fast” and “slow” wind at 1 au (Bruno et al., 2019), where both
breaks are about a decade lower in frequency, consistent with the radial
evolution (Chen et al., 2020).

###  3.4 Energy Flux

Figure 6: Trace power spectra of magnetic, velocity, and total energy
fluctuations ($E_{b}$, $E_{v}$, and $E_{t}$) for intervals 1 and 2 (indicated
in Figures 1 and 2). The lower panel shows the local spectral index, $\alpha$,
calculated over a sliding window of a factor of 5.

Finally, the energy flux in the turbulent fluctuations was compared to the
Chandran et al. (2011) reflection-driven turbulence solar wind model,
following Chen et al. (2020). The Alfvénic turbulence enthalpy flux was
calculated as

$F_{\mathrm{A}}=\rho|\delta\mathrm{b}|^{2}\left(\frac{3}{2}v_{\mathrm{r}}+v_{\mathrm{A}}\right),$
(10)

where $v_{\mathrm{A}}=B_{0}/\sqrt{\mu_{0}\rho_{0}}$ is the Alfvén speed, and
the solar wind bulk kinetic energy flux as

$F_{\mathrm{k}}=\frac{1}{2}\rho v_{\mathrm{r}}^{3}.$ (11)

These quantities are similar to those plotted in Chandran et al. (2011) and
Chen et al. (2020) except $\delta\mathbf{b}$ is used instead of
$\delta\mathbf{z}^{+}$, which allows greater data coverage and is valid due to
the high degree of Alfvénicity (Figure 5). Figure 7 shows the ratio
$F_{\mathrm{A}}/F_{\mathrm{k}}$, calculated over 3 hour intervals, as a
function of solar distance over the first four orbits of _PSP_. The SPAN-I
data were used for $v_{\mathrm{r}}$ for times when reliable fits could be
made, and SPC data used otherwise. Also shown are two solutions to the
Chandran et al. (2011) model as described in Chen et al. (2020).

It can be seen that for Orbits 1-3, the data, on average, follow the model
solutions (although with some degree of spread), consistent with the findings
of Chen et al. (2020). For Orbit 4, however, there are a large number of
points that fall well below the model, originating from times during the
outbound portion of the encounter near the HCS when the turbulent amplitudes
were lower. Times when the spacecraft was close to the HCS during Encounter 4
($r<54\,R_{\odot}$) are marked with green crosses. These values are on average
3.0 times smaller than the slow wind model (whereas the average ratio to the
model for the other data points over this distance range is 1.0). This
suggests that the lower amplitude turbulence near the HCS does not contain
sufficient energy for reflection-driven Alfvénic turbulence models to provide
an explanation for the acceleration of the streamer belt wind.

Figure 7: Ratio of Alfvén wave energy flux, $F_{\mathrm{A}}$, to bulk solar
wind kinetic energy flux, $F_{\mathrm{k}}$, for Orbits 1-4. Times during Orbit
4 which are close to the HCS ($|\phi_{\mathrm{HCS}}|<4^{\circ}$) are marked
with green crosses. The fast and slow wind solutions to the Chandran et al.
(2011) model are shown with the red dashed and blue dash-dotted lines.

##  4 Discussion

In this Letter, it is shown that the turbulence properties in the near-Sun
streamer belt wind, from $\approx 28\,R_{\odot}$ to $\approx 54\,R_{\odot}$,
are significantly different to in the open field wind that has been measured
for most of the previous _PSP_ orbits. These differences include lower
amplitudes, higher magnetic compressibility, a steeper magnetic spectrum (–5/3
rather than –3/2), a lower degree of Alfvénicity, and a larger scale $1/f$
break. The differences are similar to the traditional fast/slow wind
differences reported at 1 au, suggesting the near-Sun streamer belt wind as
the prototypical slow solar wind.

The differences in turbulence properties occur at an angle to the HCS of
$|\phi_{\mathrm{HCS}}|\approx 4^{\circ}$, suggesting the total width of the
streamer belt wind at these solar distances to be $\approx 8^{\circ}$. This
interpretation is consistent with other studies of the streamer belt wind
during PSP Encounter 4, e.g., Badman et al. (2020b) found a reduced magnetic
flux during this period that did not fit the general radial scalings seen in
the rest of the PSP data. The inferred streamer belt wind width is also
consistent with coronal images that show the streamer rays to have a width of
around $10^{\circ}$ to $20^{\circ}$, which is likely a slight over-estimate
due to the line-of-sight integration (Rouillard et al., 2020; Poirier et al.,
2020).

The Alfvénic turbulence energy flux measured on _PSP_ ’s first three orbits is
generally in line with that required for the reflection-driven solar wind
model of Chandran et al. (2011), however, in the Encounter 4 streamer belt
wind it is several times lower than the model predictions. There are also
occasional periods in the previous orbits where this is the case, identified
by Chen et al. (2020) as the periods of quiet radial-field wind seen by Bale
et al. (2019). This raises the possibility that these may be small patches of
streamer belt wind; investigating this possibility would be an interesting
topic for further study. The difference to the model predictions implies that
purely reflection-driven Alfvénic turbulence solar wind models (e.g., Cranmer
et al., 2007; Chandran et al., 2011) may not be able to account for the
acceleration of the streamer belt wind, and additional processes are taking
place. One possible such processes is additional turbulence driving by
velocity shears (Usmanov et al., 2018; Ruffolo et al., 2020). One thing to
note about this possibility, however, is that it would be expected to produce
both inward and outward Alfvén waves, resulting in $\sigma_{\mathrm{c}}\approx
0$ and $r_{\mathrm{E}}\approx 1$ if the shears dominate the energy input. In
the streamer belt wind measured here, $\sigma_{\mathrm{c}}=0.55$ and
$r_{\mathrm{E}}=5.2$ on average, meaning that even though the imbalance is
less than in the open field wind, there is still 5 times more energy in the
outward waves compared to the inward ones, so such processes may play a role
but cannot be dominating the energy input overall. Another likely contribution
to the generation of the streamer belt wind is reconnection in the near-Sun
HCS (Lavraud et al., 2020) triggered by a tearing mode (Réville et al.,
2020b). The chain of processes involved in the acceleration of the streamer
belt wind, however, remains an open question. Future orbits of _PSP_ closer to
the Sun will allow more to be learnt about the nature of both plasma
turbulence near the Sun and solar wind acceleration.

###### Acknowledgements.

CHKC was supported by STFC Ernest Rutherford Fellowship ST/N003748/2 and STFC
Consolidated Grant ST/T00018X/1. LDW was supported by STFC Consolidated Grant
ST/S000364/1. JCP was partially supported by NASA grants NNX16AH92G,
80NSSC19K0275 and NSF grant AGS-1752827. SB was supported by NASA grants
NNX16AH92G, 80NSSC19K0275 and 80NSSC19K1390. KGK was supported by NASA ECIP
grant 80NSSC19K0912. SDB was supported in part by the Leverhulme Trust
Visiting Professorship programme. We thank the members of the FIELDS/SWEAP
teams and _PSP_ community for helpful discussions. _PSP_ data is available at
the SPDF (https://spdf.gsfc.nasa.gov).

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# Spectral signature of mass outflow in Two Component Advective Flow Paradigm

Santanu Mondal∗ Indian Institute of Astrophysics, II Block, Koramangala,
Bangalore 560034, India  Sandip K. Chakrabarti  Indian Centre for Space
Physics, 43 Chalantika,
Garia Stn. Road, Kolkata 700084, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Outflows are common in many astrophysical systems. In the Two Component
Advective Flow (TCAF) paradigm which is essentially a generalized Bondi flow
including rotation, viscosity and cooling effects, the outflow is originated
from the hot, puffed up, post-shock region at the inner edge of the accretion
disk. We consider this region to be the base of the jet carrying away matter
with high velocity. In this paper, we study the spectral properties of black
holes using TCAF which includes also a jet (JeTCAF) in the vertical direction
of the disk plane. Soft photons from the Keplerian disk are up-scattered by
the post-shock region as well as by the base of the jet and are emitted as
hard radiation. We also include the bulk motion Comptonization effect by the
diverging flow of jet. Our self-consistent accretion-ejection solution shows
how the spectrum from the base of the jet varies with accretion rates,
geometry of the flow and the collimation factor of the jet. We apply the
solution to a jetted candidate GS 1354-64 to estimate its mass outflow rate
and the geometric configuration of the flow during 2015 outburst using NuSTAR
observation. The estimated mass outflow to mass inflow rate is
$0.12^{+0.02}_{-0.03}$. From the model fitted accretion rates, shock
compression ratio and the energy spectral index, we identify the presence of
hard and intermediate spectral states of the outburst. Our model fitted jet
collimation factor ($f_{\rm col}$) is found to be $0.47^{+0.09}_{-0.09}$.

black hole physics — accretion disk — shock waves — hydrodynamics — ISM:jets
and outflows — star:individual GS 1354-64

##  1 Introduction

Observations show that jets are common in active galactic nuclei (AGN),
stellar mass black holes and neutron stars (e.g., Hjellming & Rupen, 1995;
Mirabel & Rodríguez, 1998; Miller-Jones et al., 2012; Blandford et al., 2019,
and references therein). A proper understanding of the origin and acceleration
of jets in compact objects is, however, lacking till date. Several models in
the literature attempt to explain the origin, powering, and collimation of the
jet. D’Silva & Chakrabarti (1994) suggested that only the dominant toroidal
field in the disk would be enough to accelerate and collimate the jets. This
has been recently verified using numerical simulation by Garain et al. (2020).
Fender et al. (2010) from their observational studies did not find any
evidence of jet powered by the spin energy of the black holes and the authors
speculated that this conclusion remained valid even for active galaxies and
quasar. On the contrary, Narayan & McClintock (2012) found a correlation
between jet power and black hole spin energy. Thus, the role of spin parameter
in powering jet is still unclear and more studies must be done to reach a firm
conclusion. The hydrodynamic jets can also be collimated due to the strength
of the jet-cocoon interaction and the collimation shock at the base of the jet
(Bromberg et al., 2011).

Therefore, accretion-ejection around black hole systems has three components:
the accretion disk, its static or dynamic corona, and the outflowing jet. It
is well established that the standard disk (Shakura & Sunyaev, 1973) emits
soft photons, up-scattered by the hot Compton cloud at the inner region of the
disk and emit as hard radiation. The emergent radiation may get down-scattered
by the diverging outflows for both sub- and super-Eddington accretion
(Kawashima et al., 2012; Titarchuk & Shrader, 2005, hereafter TS05). This
effect changes the intrinsic continuum spectra of accretion disks with an
additional bump (Sunyaev & Titarchuk, 1980) in the powerlaw spectrum. The
problem of photon propagation through a moving medium has been studied
extensively in several works (Blandford & Payne, 1981; Nobili et al., 1993;
Titarchuk et al., 2003). There are numerous studies on this context in the
literature, both in theory and radiation hydrodynamic simulation (Eggum et
al., 1988; King et al., 2001; Okuda, 2002; Ebisawa et al., 2003; Ohsuga et
al., 2005). In addition to down-scattering of hard radiation, jet can also
behave like an up-scattering Comptonizing medium of the soft photons from the
disk. The presence of soft-excess is a commonly observed spectral component
for both X-ray binaries and AGNs. This can be originated if there is some warm
Compton up-scattering medium is present is the system. The excess component
can also be associated as an intrinsic component with the disk, powered by the
mass accretion rate (Done et al., 2012), appeared as a reflection component
(Done et al., 2007, for a review). This distortion of the power-law continuum
above $\sim 10$ keV energy could also be due to down-scattering of hard
radiation by the outflowing plasma rather than from a static reflecting medium
(TS05). In addition to these spectral features, a converging flow falling onto
the black hole near the event horizon can show bulk motion effect (Chakrabarti
& Titarchuk, 1995; Titarchuk & Zannias, 1998, hereafter CT95). Under such a
complex radiation components coming from the accretion-ejection system, it is
important to ask, independent of how a jet is launched from the disk, what
would be the overall X-ray spectrum in the presence of a jet, and what is the
contribution from the base of the jet itself, specially the subsonic region
which is hot. Most importantly, whether there is any evidence of such a
contribution.

Chakrabarti (1999, hereafter C99) postulated that the jets in the black hole
systems are formed out of the post-shock region, which is also called the
CENtrifugal pressure dominated BOundary Layer or CENBOL. This is the subsonic,
hot region between the shock and the inner sonic point (Chakrabarti, 1989)
located just inside the marginally stable orbit. C99 also proposed that the
entire outflow is produced by the CENBOL and the mass outflow rate directly
depends on the compression ratio $R=\frac{\rho_{+}}{\rho_{-}}$ of the shock
itself, where $\rho_{-}$ and $\rho_{+}$ are the densities of the flow in the
pre-shock and post-shock regions. It was shown that the ratio of the mass
outflow rate to inflow rate is indeed a function of $R$ varies in a non-linear
way apart from a geometric factor coming from the solid angles of the outflow
and the inflow. For a very strong shock, which forms farther out as in a hard
state, the thermal driving force is low and the jet is not powerful.
Similarly, in a soft state, when the shock is the weakest, the size of the
base of the jet is negligibly small, thermal driving force is also low and
hence the ratio is also low. Only in between the two states, the ratio is
significant. Several hydrodynamic simulations showed that a few percent of
accreting matter is indeed present in the jet (Molteni et al., 1994;
Chattopadhyay et al., 2004; Garain et al., 2012). Singh & Chakrabarti (2011)
studied jet properties of the flow when the energy dissipation is present at
the shock and found lower efficiency of the jet formation when viscosity is
high.

In Chakrabarti (1997) two component flow scenario, it is established that the
emission from the CENBOL decreases as the accretion rate of the Keplerian disk
is increased, both because its size and temperature are reduced. This
reduction of thermal pressure of CENBOL reduces the mass outflow rate. Thus,
accretion rates, spectral states, and jet are interlinked. Observations
evidenced prominent jets in hard and hard-intermediate spectral states (HS and
HIMS, Fender et al. (2004)). Transonic solution including real cooling (Mondal
& Chakrabarti, 2013) and outflows (Mondal et al., 2014a) showed that spectral
states are related to mass outflow rate and a significant change in energy
spectral index are observed in HS and HIMS. These solutions are self-
consistent in the sense that cooling and mass loss are incorporated in the
flow equations before obtaining the spectra. Recent work by Nagarkoti &
Chakrabarti (2016) using a viscous solution confirms that a significant amount
of mass loss is centrifugally driven as suggested by Blandford & Payne (1982)
though the disk may be sub-Keplerian in nature.

The interdependency between the accretion rate, CENBOL size and QPOs is also
well known in the literature During an outburst, when a source moves from HIMS
to SIMS, the QPO type switches from type-C to type-B (e.g., Mondal et al.,
2014b; Debnath et al., 2015; Chatterjee et al., 2016, and references therein).
However, jet properties also changed drastically, which is observed in large
amplitude radio flares (Fender et al., 2004, 2009). Radhika & Nandi (2014)
reported the accretion-ejection mechanism for the object XTE J1859+226 and
pointed out that the quasi-periodic-oscillations (QPOs) disappear during the
high ejection of jets. All these are expected from the TCAF paradigm, since
QPOs are known to be formed due to the resonance of various time scales inside
CENBOL which also produces jets. Interestingly, the highest outflow to inflow
rate from the theoretical solution of C99 happens for intermediate compression
ratio of the shock, i.e., in hard and soft-intermediate states. Retaining
CENBOL is possible only in a low viscosity state (Mondal et al., 2017).

The original TCAF solution of CT95 and C97 does not include the contribution
from outflows and therefore, presence of significant outflows require a
modification of the model. This is the primary goal of this paper. Here, we
concentrate on the effects on the emergent spectra when the base of the
outflow also participates in Comptonization of injected photons along with the
CENBOL. We study the jet spectrum when the accretion rates, the geometry of
the flow and jet parameters vary. We also see the variation of the spectra
from the jet with the variation of CENBOL size and shock compression ratio.
The study of optical and radio observations (Brocksopp et al., 2001; Pahari et
al., 2017) inferred the presence of jet in GS 1354-64 system, however, there
is very little study on this object as far as the effects of jets are
concerned. We apply the solution to study this object, in order to throw some
light on accretion-ejection behavior during its 2015 outburst. In fact, from
the fits of the modified TCAF model, we extract the jet parameters, such as
the mass outflow rate, as well. In future, we aim to apply the solution to
other well known candidates which exhibit prominent jets.

The paper is organized as follows: in the next Section, we present the
configuration of the jet and the equations we used to calculate jet
temperature and internal number density, etc. In §3, we present the results of
the effects of mass loss on normal in TCAF output, mainly the spectral
variation with location of the shock, shock compression ratio and accretion
rates. In §4, we calculate the mass outflow rate of the BHC GS 1354-64 during
its 2015 outburst using NuSTAR observation. Finally, in §5, we briefly present
our concluding remarks.

##  2 Equations and jet configuration

To extract the jet properties from a spectral fit, we consider a conical jet
in the vertical direction of the flow, which is originated from the post-shock
region. The base of the jet is a part of the Compton cloud along with the
CENBOL. In Figure 1, we present a cartoon diagram of the two-component flow
with a jet where a Keplerian disk resides at the equatorial plane, and it is
surrounded by the low angular momentum flow (dynamic halo). At the center, a
black hole of mass $M_{\rm BH}$ is located. We show also the solid angles
$\Theta_{\rm o}$ and $\Theta_{\rm in}$ subtended by the two outflow components
and the axisymmetric torus created by the CENBOL respectively.

Figure 1: A cartoon diagram of the geometry of an accretion flow with a jet in
the vertical direction. Here, $\Theta_{\rm o}$ and $\Theta_{\rm in}$ are solid
angles subtended by the outflow and the CENBOL respectively. The base of the
jet is treated as a Comptonizing medium, which also contributes in computing
model spectra. Zigzag trajectories are the typical paths followed by soft
photons from the Keplerian disk. These photons are Comptonized by the CENBOL
and the jet base (marked) and are radiated as hard X-rays.

The spectrum emitted from the system contains (a) modified blackbody
component, which is coming from the Keplerian disk, (b) the soft photons from
the disk are up-scattered by the hot electrons at the inner region of the disk
and come out as hard radiation, (c) disk soft photons also get up-scattered by
the jet base, which modifies the spectrum on the shoulder of the blackbody
bump, and (d) hard radiation from the CENBOL are down-scattered by the
outflowing jet due to Doppler effect. The components (a) and (b) are already
computed in the original Chakrabarti & Titarchuk (1995) paper. For the
component (c) we use the same radiative transfer calculation incorporating the
temperature and optical depth of the jet. Once we have the physical quantities
of the jet medium and the spectral component (b), we use it to pass through
the jet medium following TS05. Below, we discuss the equations used to
estimate different physical quantities of the jet medium.

The radiation from the CENBOL may constantly heat up the jet and the jet may
not get enough time to cool at least up to the sonic point. Thus, the ratio of
the mass outflow rate to the mass inflow rate is derived for an isothermal
base of the jet (C99), which is given by,

$R_{\dot{m}}=f_{\rm
col}f_{0}^{3/2}\frac{R}{4}exp\left[\frac{3}{2}-f_{0}\right],$ (1)

where, $f_{\rm 0}=\frac{R^{2}}{R-1}$ and $f_{\rm col}$(=$\Theta_{\rm
o}/\Theta_{\rm in}$) is the collimation factor. Here, $\Theta_{\rm o}$ and
$\Theta_{\rm in}$ are the solid angles subtended by the two outflow components
and the inflow. We should mention that $f_{0}$ factor diverges when R$\sim$1\.
However, that does not create any complications in producing the spectrum,
since when R$\sim$1, the flow is not a TCAF and produces the blackbody
spectrum only. It is clear that the mass outflow rate is a function of only
$R$ and $f_{\rm col}$. Here, $f_{\rm col}$ is a parameter and its value is
varied in the range (see Table 1) to fit the observed data. One of the values
of $f_{\rm col}$ is $0.1$, which corresponds to $\Theta_{\rm o}\sim
10^{\circ}$. Molteni et al. (1994) discussed that $\Theta_{\rm o}$, should
depend on the strength of the centrifugal barrier. If the barrier is stronger,
$\Theta_{\rm o}$ and consequently the mass loss rate is higher. The total
accretion rate of the in-flowing matter at the CENBOL after mixing of the
Keplerian and sub-Keplerian components is given by,

$\dot{M}_{\rm in}=\dot{M}_{\rm d}+R\dot{M}_{\rm h},$ (2)

here, $\dot{M}_{\rm d}$ and $\dot{M}_{\rm h}$ are the disk and halo rates
respectively of the flow in gm sec-1 unit. Since in TCAF, the boundary layer,
i.e., CENBOL, is emitting the jet, the initial launching velocity of the
ejecta comes from the temperature of the post-shock region, i.e., ${\rm
v}_{\rm j}\sim\sqrt{T}_{\rm shk}$ $\sim 0.1c$. However, it changes depending
on the CENBOL location. The density at the base of the jet is calculated
using,

$\rho_{\rm j0}=\frac{\dot{M}_{\rm j}}{\pi X_{\rm s}^{2}{\rm v}_{\rm j}},$ (3)

where, $\dot{M}_{\rm j}(=R_{\dot{m}}\dot{M}_{\rm in})$ is the rate of the mass
outflow. As the height increases, density of the outflowing matter falls as,

$\rho_{\rm j}=\rho_{\rm j0}\left(\frac{r_{\rm j}}{r_{\rm jb0}}\right)^{-2},$
(4)

where, $r_{\rm j}$ and $r_{\rm jb0}$(=hshk) are the (running) height of the
jet and the fixed height of the base of the jet respectively. The latter is
basically the height of the shock (hshk). We assume that the jet matter moves
away subsonically till the sonic surface, the radius of which is given by
(C99),

$r_{\rm c}=\frac{f_{\rm 0}X_{\rm s}}{2}.$ (5)

We assume that the temperature of the jet ($T_{\rm j}$) varies as $\sim$
$1/r_{\rm j}$. In this work, we also consider the effects of down-scattering
of the Comptonized photons by the bulk motion of the outflowing jet as
discussed in TS05. There are a number of sources which show evidence of strong
reflection above $\sim 10$ keV. This distortion of the power-law continuum
above this energy can also be originated from the down-scattering of the hard
radiation by the outflowing plasma rather than from a static reflecting
medium. This is discussed in CT95 and TS05. Here, we implement this to explain
the excess radiation component present in the spectrum. We use the following
equation from TS05 for the final down-scattered jet spectrum:

$F_{\rm BMC}\propto\Phi_{\rm
Comp}\\{1+N_{av}z[(1-\alpha)-\frac{\epsilon}{3z_{\ast}}-\frac{z}{z_{\ast}}]\\},$
(6)

where the second term in the bracket describes the pile-up and softening of
the input Comptonization spectrum ($\Phi_{\rm Comp}$), resulting from the down
scattering effect by the jet electrons. Here, $N_{\rm av}(=3\tau_{0}^{2}/8)$
is the number of average scattering, $\epsilon$ is the efficiency of the
energy loss in the divergent flow $(\sim 2{\rm v}_{\rm j}/c\tau_{0})$, $z$ is
the dimensionless photons energy, $z_{\ast}$ is the dimensionless temperature
and $\tau_{0}$ is the optical depth of the scattering medium, namely, the base
of the jet. The $\Phi_{\rm Comp}$, we used here, is the spectrum emitted from
the CENBOL due to up-scattering of disk soft photons and $\alpha$ is the
energy spectral index estimated from the original TCAF code (CT95) to make the
computation to be consistent. However, TS05 assumed this spectral component to
be a cut-off power-law for a given $\alpha$. The subscript BMC implies the
bulk motion Comptonization.

Table 1:  JeTCAF model parameters range Pars. | Units | Default | Min. | Min. | Max. | Max. | Increment
---|---|---|---|---|---|---|---
MBH |  M⊙ | 6.0 | 4.0 | 4.0 | 15.0 | 15.0 | 2.0
$\dot{m}_{\rm d}$ | $\dot{M}_{\rm Edd}$ | 0.01 | 0.001 | 0.001 | 2.0 | 2.0 | 0.05
$\dot{m}_{\rm h}$ | $\dot{M}_{\rm Edd}$ | 0.1 | 0.001 | 0.001 | 5.0 | 5.0 | 0.2
$X_{\rm s}$ | $r_{\rm g}$ | 50.0 | 6.0 | 6.0 | 400.0 | 400.0 | 8.0
R | “ ” | 2.1 | 1.1 | 1.1 | 7.0 | 7.1 | 0.2
fcol | “ ” | 0.1 | 0.05 | 0.05 | 0.5 | 0.5 | 0.01

The units of mass, length, speed, and accretion rates are $M_{\odot}$,
$r_{g}=2GM/c^{2}$, $c$, and $\dot{M}_{\rm Edd}$ respectively, where
$M_{\odot}$, $G$, $M_{\rm BH}$, $c$ and $\dot{M}_{\rm Edd}$ are the mass of
the Sun, gravitational constant, mass of the central object, speed of light
and Eddington accretion rate. Throughout the paper we follow these units for
model parameters and physical quantities.

##  3 Results and Discussions

We show how the spectral properties change when the base of the jet is also a
part of the Compton cloud. In Figure 2a, we show the spectral variation of the
jet component from the total emitted spectrum for different accretion rates
($\dot{m}_{\rm d}$), when other flow parameters ($\dot{m}_{\rm h}$, $M_{\rm
BH}$, $X_{\rm s}$ and $R$) are kept constant. We increase the accretion rates,
i.e., the number of soft photons and observe that the jet spectrum softens.
Here, we used $f_{\rm col}$=0.1 and other parameters are marked with figure.
Increase in soft photon number also cools down the Compton cloud rapidly.
Thus, the emitted jet does not have sufficient thermal energy to up-scatter
the soft photons significantly. In Figure 2b, we present the spectra of
different spectral states by varying the accretion rates only, keeping all
other parameters fixed. Here the spectra vary from HS to SS and the same
variation is reflected in the jet contribution in the spectra as well.
Spectral states may change due to different sizes of the cloud and compression
ratio of the shock, since, these two physical parameters are also related to
density and optical depth of the system.

Figure 2: (a) Scattering of soft photons from the disk by the jet when disk
accretion rate varies from 0.001 to 0.1 when halo rate is 1.0, keeping all
other flow parameters fixed. The spectrum softens with increasing accretion
rate. (b) Similar spectral variation as in (a) for different sets of disk and
halo accretion rates, when other parameters are the same as in (a). The
spectrum softens when the accretion rate changes to change the spectral state
from hard to intermediate. Other parameters are $M_{\rm BH}$=10.0, $X_{\rm
s}$=20.0, R=2.5 and $f_{\rm col}$=0.1. Here, frequency is in Hz unit.

Figure 3a shows the effects of the size of the Compton cloud ($X_{\rm s}=10,\
20,\ 40$ and $60$) on the jet component of the total emitted spectrum. We see
that as the size of the cloud increases, the jet spectral component becomes
softer. As the base of the jet is the CENBOL, the jet component varies with
shock location. Larger CENBOL being farther from the black hole, its
temperature is already less (roughly $\propto 1/X_{s}$). Also, it intercepts
more soft photons from the Keplerian disk which reduces the overall CENBOL
temperature (Chakrabarti, 1997). Thus, the resulting temperature of the base
of the jet becomes lower. Thus, the soft primary photons are not up-scattered
many times. This causes the jet spectrum to be softer. In Figure 3b, we show
the jet component variation for different values of $R$ (=$2.0$, $2.5$, $3.5$,
$4.5$ and $6.0$). We observe significant changes in the jet component, which
becomes softer with increasing R. From C99 solution, we see that the mass
outflow rate decreases when $R$ is higher than $\sim 4.0$. This affects the
spectral shape, for instance, when $R=6.0$, mass outflow rate is very low and
consequently the spectrum is also softer. In Figure 3c, we see the spectral
hardening when jet collimation factor ($f_{\rm col}$) has changed from $0.05$
to $0.3$.

Figure 3: Same as Figure 2, (a) for varying size of the Compton cloud
($X_{s}$) when R=2.5 and $f_{\rm col}$=0.1, (b) for different values of
compression ratio R when $X_{s}$=20.0 and $f_{\rm col}$=0.1, and (c) for
different values of collimation factor ($f_{\rm col}$) when $X_{s}$=20.0 and
R=2.5. In (a), as the location/size increases, the spectrum becomes softer. In
(b) as the value of R increases, mass outflow rate also increases, which
increases the optical depth and the average number of the scattering. Thus,
the spectrum becomes harder. However, mass outflow rate starts to decrease
when R becomes higher than 4.5 and the spectrum again gets softer. This can be
seen for $R=6.0$. In (c), spectrum becomes harder with increasing collimation
factor. The other parameters for both the plots are $M_{\rm BH}$=10.0,
$\dot{m}_{\rm d}$=0.001, and $\dot{m}_{\rm h}$=1.0.

Figure 4a shows the variation of the net spectrum with (solid) and without jet
(dashed line) contribution as a function of the shock compression ratio ($R$).
We note that due to the presence of the jet, the total spectrum becomes harder
as compared to that without the jet. The variation of the total spectrum with
$R$ is significant. In Figure 4b, we show the total emitted spectra for
different values of collimation factor. The dashed line shows the total
spectrum when jet contribution is not included, and the solid colored lines
are the total spectrum with the jet contribution. One can see that as the
collimation factor increases, the spectrum becomes harder when the other
parameters are fixed.

Figure 4: (a) Total emitted spectra with (solid lines) and without the jet
contribution (dashed lines, same as TCAF spectra), for different values of
$R$, and (b) $f_{\rm col}$. In (a), we see that the spectrum becomes harder
with increasing $R$. The new parameter ($f_{\rm col}$) does not affect the
CENBOL spectrum and thus the total spectrum without the jet contribution is
the same for all (dashed line). The other parameters are $M_{\rm BH}$=10.0
$M_{\odot}$, $\dot{m}_{\rm d}=0.001$, $\dot{m}_{\rm h}=1.0$ and $X_{\rm
s}=20.0$. Here, frequency is in Hz unit.

Figure 5a shows the total spectral variation of different states with (solid
lines) and without (dotted lines) the jet contribution. Here the spectral
state changes with accretion rates when other parameters of the model are kept
fixed. As the accretion rate increases, more soft photons from the Keplerian
disk are intercepted by the Compton cloud and the spectrum becomes softer. On
the other hand, an increase in disk accretion rate cools down the CENBOL which
reduces the thermal pressure and thus reduces the mass outflow rate.
Consequently, the system moves to the soft state through intermediate states.
In Chakrabarti (1996), it was proposed that viscosity is mainly responsible
for changing the relative rates, which is indirectly taken into account by two
accretion rates in TCAF. In Mondal et al. (2017), it was shown that the
observed viscosity parameter indeed rises in the rising phase and decreases in
the declining phase, thereby completing the full cycle of the observed states.
In Figure 5b, we choose typical model parameters for soft states. One can see
that the jet contribution is negligible. There are many observations in the
literature which show the correlation between observed spectral states with
the jet (Fender et al., 2004; Belloni et al., 2005; Sriram et al., 2012). They
also discussed correlations between the QPOs and the jet emissions in the HID
diagram. All these correlations fall in place once the computation of the
outflow rate to inflow rate ratio as a function of the properties of the
centrifugal barrier was made (C99). The theoretical result directly shows that
if indeed the outflows are produced from CENBOL, then, the soft and extremely
hard states are going to produce very little outflows. Only intermediate
states have higher outflow rates for a given inflow rate. When the optical
depth is higher at the base of the jet, it produces blobby jets Chakrabarti et
al. (2002); Nandi et al. (2001). In the soft state, due to cooling effects,
this region is quenched and the mass outflow rate is reduced (see also, Garain
et al., 2012). Recently, Jana et al. (2017) estimated the X-ray flux from the
base of the jet for Swift J1753.5-0127 BHC using TCAF model.

Figure 5: (a) Total emitted spectrum with (solid lines) and without (dashed
lines) jet contribution for different spectral states depending on the
accretion rates when other parameters are fixed ($X_{\rm s}=20.0$ and
$R=3.0$). There is a significant change in the spectrum as compared with the
soft state in presence of jet. (b) Total spectrum in the soft state in
presence of jet (solid line) component when $X_{\rm s}=10.0$ and $R=1.2$. In
the soft state, the jet contribution is negligible and both the spectra are
the same. For both the plots $M_{\rm BH}$=10.0 and $f_{\rm col}=0.2$. Here,
frequency is in Hz unit.

Till now, we did not show the contribution of BMC component in the spectral
plots. In Figure 6 we show the emitted spectrum when hard photons from the
CENBOL pass through the outflowing diverging jet. In Figure 6, we show the
spectral components including jet when the down-scattering effects of the
CENBOL photons due to the bulk motion of the jet is considered. The black line
shows the net spectrum when the BMC effect is also included, the blue and
indigo lines show the jet component with BMC effect respectively. Here, green
and red lines are disk blackbody and thermal Comptonization spectra obtained
from the original C97 solution. The net spectrum clearly shows a bump, which
is slightly higher in JeTCAF spectra as compared to the TCAF. In, TS05
explained that the bump is due to the BMC effects of the CENBOL photons by the
diverging outflows discussed in CT95. In the next Section, we use both TCAF
and JeTCAF model to fit the observed data, where we find the signatures of the
BMC effect.

Figure 6: Different components including the jet which contribute to the net
spectrum. Black line shows the total spectrum with a jet, the blue line is the
jet component and the indigo line indicates the BMC effect of the outgoing
jet. Red and green lines show thermal Comptonization and blackbody spectra as
in the original TCAF solution. The disk and the jet parameters for this plot
are $M_{\rm BH}$=10.0, $\dot{m}_{\rm d}=0.001$, $\dot{m}_{\rm h}=1.0$, $X_{\rm
s}=20.0$, $R=3.0$ and $f_{\rm col}=0.1$. Here, frequency is in Hz unit.

##  4 GS 1354-64

We now use our computed spectra to fit the data of the black hole candidate GS
1354-64. GS 1354-64 is a dynamically confirmed low mass X-ray binary with a
black hole mass $>$ 7 $M_{\odot}$, $\sim 2.5$ day orbital period, and a
distance of $\sim 25-61$ kpc (Casares et al., 2004, 2009). Reynolds & Miller
(2011) reported the possible distance to be $\sim 15$ kpc and this large
difference is due to the extinction effects. Very recently, Gandhi et al.
(2019) estimated the distance of this source to be less than or around 1 kpc
from Gaia DR2 observation. The DR2 distance would make the source under-
luminous in both optical and X-ray wavebands. Apart from that, a large
discrepancy also makes a large uncertainty in estimating the donor star
classification and its intrinsic brightness. This candidate showed two
confirmed outbursts: one in 1987 (Makino, 1987), when the source entered into
the soft state (Kitamoto et al., 1990) and in 1997 (Brocksopp et al., 2001),
when the source was in a pure hard state (Revnivtsev et al., 2000). In the
same outburst for the first time, Fender et al. (1997) observed a radio
counterpart. In late May 2015, the optical brightness of this source was twice
higher than the quiescent values reported by Russell & Lewis (2015). Recently,
Koljonen et al. (2016) showed that the source has again entered into the hard
state, studying multi-wavelength observation during its 2015 outburst. They
also confirmed that optical/UV are tightly correlated with X-ray, which is
consistent with the emission from the jet.

We consider the NuSTAR (Harrison et al., 2013) observation of 2015 June 13
(90101006002, hereafter X02) and July 11 (X04) outbursts. We consider FPMA
observation with 4.0-65.0 keV energy range data. The exposure times for these
observations are $\sim 24$ks and $\sim 30$ks respectively. For pipeline
reduction and spectral file generation we use nupipelne and nuproducts tasks.
After successfully processing the data we do spectral fitting using (1)
Tbabs*(Gauss+TCAF), and (2) Tabs*(Gauss+JeTCAF). We follow the standard data
extraction procedures using NuSTARDAS and HEASoft packages and the spectral
fitting using XSPEC. The detail of the analysis procedure and the related
fitting issues are discussed in Mondal & Chakrabarti (2019). For the spectral
fitting of X02, we use TCAF model generated fits file (Debnath et al., 2014)
as a local additive table model, following atable command in XSPEC. For the
generation of model fit file, large no of spectra are generated for the range
between $10^{14}-10^{21}$ Hz. During the fitting, we keep the black hole mass
as a free parameter and for both the observations we keep $N_{H}$ fixed at
$0.5\times 10^{22}{\rm cm}^{-2}$ following HEASARC column density tool (Dickey
& Lockman, 1990). Data fitting with TCAF gives a very good fit with reduced
$\chi^{2}=1.046$. After successful fitting, we get the value of $M_{\rm
BH}=7.22^{+0.25}_{-0.89}$. In Figure 7, we show the TCAF model fitted spectra
of X02.

Figure 7: NuSTAR FPMA data (X02) of GS 1354-64, fitted with TCAF model as a
local additive table model. Gray points with error bar shows the observed data
and red line is the model spectrum. Data/model (D/M) ratio is shown in the
lower panel. Data are rebinned for clarity.

For the fitting of X04, first, we attempt to fit the data using TCAF model
generated fits file. We tried with different combinations of model parameters,
however, the quality of the fit is poor, with a reduced $\chi^{2}$ value
$>2.5$, as that version of the fits file does not include the jet
contribution. After this, we fit the data with JeTCAF model keeping all
parameters free provided the $N_{\rm H}$. We see that JeTCAF model fits the
observed data with a much better statistics. In this case, we run the source
code directly in XSPEC as a table model using initpackage and lmod tools
(Arnaud, 1996). The model parameters range used in lmod to run JeTCAF in XSPEC
is shown in Table 1. The range of energy considered for the generation of
model spectrum to fit the data is between $10^{17}-10^{20}$ Hz. For both the
models we assumed accretion efficiency 0.1.

In Figure 8, we show the JeTCAF model fitted spectra. In the upper panel, the
red curve shows the JeTCAF model spectrum and the gray dots with error bars
show the observed spectra. In the lower panel, we show the ratio of data and
model, which shows a good agreement of fit with a reduced $\chi^{2}=1.112$.
For comparison with TCAF model fit to the data, we overplotted the model
spectrum (green line) and corresponding D/M ratio (in green) in the bottom
panel. The model fitted parameters are shown in Table 2 for both X02 and X04
observations. The model fitted mass of the black hole for X04 is
$6.7^{+0.54}_{-0.68}$. Comparing mass obtained from both the observations, we
conclude that the mass of the central black hole is $\sim 7\,M_{\odot}$. After
reaching a good fit, we use the model fitted parameters to recalculate the
energy spectral index ($\alpha$) for X02 and X04. The calculated value of
$\alpha$ for X02 and X04 are $0.63$ and $0.87$ respectively. The values of
$\alpha$ and accretion rates confirm that on X02 day the source was in the
“hard spectral state” and on X04 day it was in the “intermediate spectral
state” when the jet was observed. The ratio of outflowing and inflowing matter
rate for this observation is $0.12^{+0.02}_{-0.03}$, recalculated from the
model fitted parameters using Eq. 1. The estimation of error in $R_{\dot{m}}$
measurement is discussed in Appendix. The above value of mass loss rate and
the values of $R$ during the outburst, which is common in centrifugally driven
flows (Molteni et al., 1994, C99), show the strong signature of an outflowing
jet. Our model fitted mass is well within the range reported earlier for this
source.

Table 2: Best fitted Gauss+TCAF and Gauss+JeTCAF model fitted parameters. The
Gaussian model fitted line energy is $E_{\rm g}$ and width is $\sigma_{\rm
g}$. Here, $N_{\rm g}$ is the Gaussian model normalization in Photons cm-2 s-1
unit.

| TCAF | JeTCAF
---|---|---
Paramaters | X02 | X04
$E_{\rm g}$ [keV] | $6.40^{+0.06}_{-0.06}$ | $6.45^{+0.05}_{-0.06}$
$\sigma_{\rm g}$ [keV] | $0.24^{+0.06}_{-0.07}$ | $0.60^{+0.04}_{-0.04}$
$N_{\rm g}[10^{-4}]$ | $1.41^{+0.22}_{-0.25}$ | $17.28^{+1.17}_{-1.05}$
$M_{\rm BH}$ [$M_{\odot}$] | $7.22^{+0.25}_{-0.89}$ | $6.70^{+0.54}_{-0.68}$
$\dot{m}_{\rm d}$ [$\dot{M}_{\rm Edd}$] | $0.038^{+0.005}_{-0.003}$ | $0.084^{+0.003}_{-0.003}$
$\dot{m}_{\rm h}$ [$\dot{M}_{\rm Edd}$] | $1.346^{+0.092}_{-0.076}$ | $0.265^{+0.009}_{-0.013}$
$X_{\rm s}$ $[r_{\rm g}$] | $140.24^{+18.07}_{-5.89}$ | $37.74^{+3.49}_{-2.74}$
$R$ | $1.94^{+0.13}_{-0.18}$ | $4.22^{+0.19}_{-0.28}$
$f_{\rm col}$ | $-$ | $0.47^{+0.09}_{-0.09}$
| $\chi_{\rm r}^{2}=1.046$ | $\chi_{\rm r}^{2}=1.112$

Figure 8: NuSTAR FPMA data (X04) of GS 1354-64, fitted with JeTCAF model as a
local model directly in XSPEC. Gray points with error bars show the observed
data and the red line is the model spectrum. The green line shows the TCAF
model spectrum for comparison with JeTCAF model. The difference can be noted
between $\sim 15-25$ keV and near the tail of the spectrum due to BMC of the
photons scattered off the base of the jet. Data/model (D/M) ratio is shown in
lower panels. Data are rebinned for clarity.

Earlier reports confirmed this outburst as a “failed outburst” due to the
presence of only hard state (El-Batal et al., 2016; Stiele & Kong, 2016). From
our model fitted parameters, $\alpha$ value and high mass outflow rate, we
identify the spectral state to be a hard (intermediate) state, like earlier
findings. It is to be noticed that disk accretion rate increased by $\sim 2.2$
times, and entered into the region from where catastrophic cooling becomes
significant according to TCAF. As we discussed earlier, from the model fitted
parameters, mass loss rate, and energy spectral index, we confirm the presence
of both hard and intermediate states.

This candidate showed a low-frequency QPO ($\sim 0.2$ Hz) with increasing
centroid frequency (Koljonen et al., 2016), which might be the indication of
some instability occurring periodically. From the high flux behavior several
authors argued this source to be similar to GRS 1915+105. Another possibility
could be a periodic feedback from jet (Chakrabarti et al., 2002; Vadawale et
al., 2001). There is no detailed dynamical lightcurve analysis on this topic
till date. If we consider the burst time ($t_{\rm b-b}$) theory of C99 and
estimate the periodicity of the burst from the model fitted parameters, it
gives $\sim 20$ sec (see Eq. 24 of C99). The $t_{\rm b-b}$ value can also be
estimated from the observed QPO frequency for this source (0.2 Hz), using
(C99) $t_{\rm b-b}\propto\nu_{\rm QPO}^{-2}$ $\sim 25$ s. Both estimates are
consistent with each other. This estimate also matches with 20 s periodicity
observed in the lightcurve of this source during RXTE era (see Fig. 2 of
Janiuk & Czerny, 2011). There are several works in the literature (Belloni et
al., 2000; Rao et al., 2000; Naik et al., 2002; Chakrabarti et al., 2004; Pal
& Chakrabarti, 2015) which discussed changes in different classes of
lightcurves in black hole candidates in a matter of few seconds. Such quick
transitions are possible only if the changes in accretion/ejection processes
are due to local effects at the base of the jet or due to the return flow onto
the disk (e.g., see Chakrabarti et al., 2002), whereas burst-on and burst-off
states of GRS 1915+105 were discussed as due to interacting outflow with the
inner disk. Here we suggested cycles inside a lightcurve due to the feedback
effects from the outflows in a timescale of 20 sec. It is possible and is seen
in GRS 1915+105 and IGR 17091-3624 (Belloni et al., 2000; Rothstein et al.,
2005; Pal & Chakrabarti, 2015) very often. There has to be changes in the
local disk rather than the global disk.

##  5 Summary and Conclusions

In this paper, we study the spectral properties of the accretion flows around
black holes when the jet is also included in the TCAF solution. We apply this
JeTCAF solution to study the transient source GS 1354-64 during its 2015
outburst using the data from NuSTAR satellite. We treat the base of the jet to
be an additional Compton cloud component which also scatters soft photons from
the disk. We also include the effects of bulk motion of the diverging jet. Our
jet configuration is self-consistent with the accretion dynamics in the sense
that it estimates mass loss solving flow equations, rather than considering it
separately. In this model, the jet is originated from the post-shock region
(CENBOL) of the flow, and the velocity of the outflowing matter is calculated
from the average temperature of the CENBOL We estimate the mass outflow rate
from the inflow rate. We use a realistic outflow to obtain the optical depth
and the temperature of the Compton cloud. The mass outflow rate and the
velocity are high in hard and intermediate states, thus the jet contribution
to the spectrum is also modified significantly (due to bulk motion) during
that time. In the soft state, the temperature is not high enough to accelerate
the mass outflow. Thus, the spectrum remains nearly blackbody. We also see
that the contribution of the jet to the overall spectrum varies with the size
of the Compton cloud, the shock compression ratio and the jet collimation
factor. The overall fits using our simple consideration is acceptable, and we
do not require presence of magnetic field or synchrotron emission to fit the
data.

In this manuscript, we applied the model on the BHC GS 1354-64 and calculated
the mass outflow rate during its 2015 outburst. In future, we aim to consider
other jetted candidates using the same model. Our study shows that during the
outburst, this candidate had a ratio of mass outflow to mass inflow rate of
about $0.12^{+0.02}_{-0.03}$. We suspect the reason behind the presence of
very low QPO frequency with varying centroid frequency is also an indication
of micro outbursts or feedback from jet to the disk occurring in very small-
time scales (20 sec). This is similar to what happened in GRS 1915+105 (also
in IGR 17091-3624). It is true that the spin of the black hole is important
for both the disk and jet study, however its effects cannot spread beyond a
few gravitational radii, while the physics we discuss is of farther out
regions. As we discussed earlier, the spin may or may not contribute in
powering the jet. However, the present TCAF or JeTCAF model does not consider
the spin effect. In future, we aim to implement that effect self-consistently
in the model. A further study of this object is required to understand the jet
feedback mechanism, class transition in lightcurve to understand the dynamics
of the disk.

##  6 Acknowledgment

We thank the referee for making insightful comments and suggestions. SM is
thankful to Keith A. Arnaud for helping in model inclusion during his visit to
NASA/GSFC as a student of COSPAR Capacity-building Workshop Fellowship program
jointly with ISRO, India. SM acknowledges Ramanujan Fellowship (#
RJF/2020/000113) by SERB-DST, Govt. of India for this research. This research
has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly
developed by the ASI Science Data Center (ASDC, Italy) and the California
Institute of Technology (Caltech, USA).

## APPENDIX

For the estimation of error in $R_{\dot{m}}$, we derived the relation:

$\sigma_{R_{\dot{m}}}=\sqrt{\left(\frac{\partial R_{\dot{m}}}{\partial
R}\right)^{2}\sigma_{R}^{2}+\left(\frac{\partial R_{\dot{m}}}{\partial f_{\rm
col}}\right)^{2}\sigma_{fcol}^{2}},$

where, $\sigma_{R}$ and $\sigma_{f_{\rm col}}$ are the error in measurement of
shock compression ratio (R) and collimation factor ($f_{\rm col}$)
respectively. After doing a few steps of algebra we get the value of
$\frac{\partial R_{\dot{m}}}{\partial R}$=-0.05 and $\frac{\partial
R_{\dot{m}}}{\partial f_{\rm col}}$=0.24, which give the value of
$\sigma_{R_{\dot{m}}}$ = {+0.02, -0.03}.

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