# Frequency Shift Algorithm: Application to a Frequency-Domain Multiplexing
Readout of X-ray Transition-Edge Sensor Microcalorimeters

D. Vaccaro<EMAIL_ADDRESS>NWO-I/SRON Netherlands Institute for Space
Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  H. Akamatsu
NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands  J. van der Kuur  NWO-I/SRON Netherlands Institute
for Space Research, Landleven 12, 9747 AD Groningen, The Netherlands  P. van
der Hulst  NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan
2, 3584 CA Utrecht, The Netherlands  A.C.T. Nieuwenhuizen  NWO-I/SRON
Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The
Netherlands  P. van Winden  NWO-I/SRON Netherlands Institute for Space
Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  L. Gottardi
NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands  R. den Hartog  NWO-I/SRON Netherlands Institute for
Space Research, Landleven 12, 9747 AD Groningen, The Netherlands  M.P. Bruijn
NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands  M. D’Andrea  NWO-I/SRON Netherlands Institute for
Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  J.R. Gao
NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands  Optics Group, Department of Imaging Physics, Delft
University of Technology, Delft, 2628 CJ, the Netherlands  J.W.A. den Herder
NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands  R.W.M. Hoogeveen  NWO-I/SRON Netherlands Institute
for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  B.
Jackson  NWO-I/SRON Netherlands Institute for Space Research, Landleven 12,
9747 AD Groningen, The Netherlands  A.J. van der Linden  NWO-I/SRON
Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The
Netherlands  K. Nagayoshi  NWO-I/SRON Netherlands Institute for Space
Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  K. Ravensberg
NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands  M.L. Ridder  NWO-I/SRON Netherlands Institute for
Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  E. Taralli
NWO-I/SRON Netherlands Institute for Space Research, Sorbonnelaan 2, 3584 CA
Utrecht, The Netherlands  M. de Wit  NWO-I/SRON Netherlands Institute for
Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands

###### Abstract

In the frequency-domain multiplexing (FDM) scheme, transition-edge sensors
(TES) are individually coupled to superconducting LC filters and AC biased at
MHz frequencies through a common readout line. To make efficient use of the
available readout bandwidth and to minimize the effect of non-linearities, the
LC resonators are usually designed to be on a regular grid. The lithographic
processes however pose a limit on the accuracy of the effective filter
resonance frequencies. Off-resonance bias carriers could be used to suppress
the impact of intermodulation distortions, which nonetheless would
significantly affect the effective bias circuit and the detector spectral
performance. In this paper we present a frequency shift algorithm (FSA) to
allow off-resonance readout of TES’s while preserving the on-resonance bias
circuit and spectral performance, demonstrating its application to the FDM
readout of a X-ray TES microcalorimeter array. We discuss the benefits in
terms of mitigation of the impact of intermodulation distortions at the cost
of increased bias voltage and the scalability of the algorithm to multi-pixel
FDM readout. We show that with FSA, in multi-pixel and frequencies shifted on-
grid, the line noises due to intermodulation distortion are placed away from
the sensitive region in the TES response and the X-ray performance is
consistent with the single-pixel, on-resonance level.

††preprint: AIP/123-QED

> The following paper has been accepted for publication in Review of
> Scientific Instruments.

##  I Introduction

Transition-edge sensors tes  (TES) are employed as very sensitive photon
detectors in a wide number of experimental applications, either ground-based,
balloon-borne or satellite telescopes, as bolometers or microcalorimeters for
mm-wave radiation up to gamma rays, such as LiteBIRDlitebird , SPICA-
SAFARIsafari , Athena X-IFUathena , LSPE-SWIPElspe , QUBICqubic .

To comply with the stringent thermal requirements to operate large arrays of
TES’s at sub-Kelvin temperatures and to reduce wiring complexity and cost,
their readout is performed using a multiplexing scheme, in which the signals
of the detectors are modulated with a set of orthogonal carriers in a limited
information bandwidth through a common transmission line. The sum of the
signals is fed to a cold superconducting quantum interference device (SQUID)
amplifier stage, then the output is retrieved back to warm, digital
electronics, where demodulation of the individual TES signals is performed.

To readout arrays of TES-based X-ray microcalorimeters, we have been
developing a frequency-domain multiplexing (FDM) readout technology with base-
band feedback bbfb  (BBFB). In the FDM scheme (see Figure 1), the detectors
are voltage biased with sinusoidal carriers at different frequencies in the
MHz range. A tuned, high-Q LC bandpass filter is put in series with each
detector, with the double function of limiting the information bandwidth and
allowing only one carrier to provide the AC bias. The TES’s signals are then
summed at the input of a SQUID and demodulated at room temperature by digital
electronics.

Figure 1: Diagram showing our FDM readout scheme with base-band feedback. AC
bias carriers are generated by the FPGA and converted to analog via the AC
bias DAC from our custom digital electronics (DEMUX) board at room
temperature. SQUIDs are controlled via a custom analog front-end electronics
(FEE). The SQUID output signal is first pre-amplified via a low noise
amplifier in the FEE, then fed via an ADC to the DEMUX board, where
demodulation is performed to retrieve the individual TES’s signals. These
signals are re-converted to analog via the Feedback DAC and used to null the
error signal at the SQUID summing point just around each bias carrier. Such
base-band feedback scheme allows an increase of the available dynamic range,
so that a large number of pixels can be accommodated in one readout chain.

The condition of independence between the carriers used to codify each pixel’s
information sets a minimum ratio between the carrier’s frequency spacing and
the individual pixel bandwidth, which in turn is set by the detector speed and
stability conditions under negative electro-thermal feedback irwin_tau . To
make use of the available readout bandwidth in an efficient way, the LC
filters are designed to fall on a regular grid, so that the bias carriers are
integer multiples of a frequency spacing that is constant between each
subsequent carrier. Due to unavoidable fabrication tolerances and stray
inductance from wire bonding, in practice resonators do not match with the
expected frequency, with a typical spread of $\approx$ 3 kHz around the design
value bruijn .

Non-linearities in the readout chain, due $e.g.$ to SQUID amplifiers or DACs,
can generate spurious signals appearing as line noises in the carrier
sidebands. With a high number of multiplexed detectors, such intermodulation
line noises (IMLN) are likely to fall within the sensitive region of the
detector response, degrading the detector performance. The frequencies of
IMLN’s are defined by linear combinations of the bias frequencies, hence
choosing the latter on a regular grid, so that also the IMLNs’ frequencies
would conveniently be on a regular grid outside of the TES sensitive band,
would avoid such performance degradation. Given the high Q of the resonators
however, off-resonance operation effectively changes the bias circuit, since
the reactive impedance of the LC filter becomes no longer negligible compared
to the TES resistance.

In a previous work hiroki2018  we showed that the spectral performance of one
X-ray TES microcalorimeter, spoiled by IMLN’s, could be recovered with off-
resonance operation by employing an early version of a frequency shift
algorithm (FSA). The original concept of FSAjan2018  contemplated the tuning
of the LC resonance frequency by adding an active impedance in TES bias
circuit via a feedback on the modulated bias signal. That version had the
drawback of requiring parameter fine tuning for each pixel and frequency
shift, thus not being scalable. In this paper we report on a new
implementation of FSA, designed to allow scalability to multi-pixel operation.
The technical aspects of such algorithm are discussed in a companion paper
paulpaper . The operation scheme is outlined at the beginning of Section III.

The paper is structured as follows. We first describe our experimental setup
in Section II, then in Section III the FSA working principle and requirements
for the application on an X-ray microcalorimeter readout are discussed. In
Section IV the reproducibility of the bias circuit in off-resonance readout is
tested and finally in Section V the results of X-ray experiments with FSA in
single-pixel and multi-pixel configurations are reported.

##  II Experimental setup

For our experiments we used a 5$\times$5 TES microcalorimeter array developed
at SRON Utrecht (for details about the fabrication, see K. Nagayoshi  _et
al._ken ). Each TES consists of a 100$\times$30 $\upmu$m2 Ti/Au bilayer
deposited on a SiN membrane and coupled to a 240$\times$240$~{}\upmu$m2 gold
absorber via two central pillars and four corner stems, as visible in Figure
2. These devices have $T_{C}~{}\simeq$ 72 mK and R${}_{N}\simeq$ 75 m$\Omega$.

Figure 2: Zoomed view through the Cu collimator of the SRON R1a 5$\times$5 TES
array, mounted on the Cu holder. Top right and bottom left pixels are not
connected, so that a total of 23 pixels is available.

To perform the FDM readout the TES’s are coupled to custom superconducting LC
filters and transformers, to achieve impedance matching between the TES and LC
filter. For this setup, LC filter and transformer chips result in a 2.5
$\upmu$H effective inductance. The detectors are voltage-biased through a 750
m$\Omega$ resistor and a capacitive divider with 1:26 ratio, for an effective
shunt resistance of 1.4 m$\Omega$. Remanent magnetic field is cancelled by
means of superconducting Helmholz coils. The cryogenic components are mounted
on a oxygen-free high-conductivity (OFHC) copper holder, as depicted in Figure
3, enclosed in a Nb shell and an outer Cryoperm shield. A 55Fe source is
hosted on the superconducting magnetic shield (Nb shell) to exploit the Mn-Kα
line complex for measuring the X-ray spectral performance of the detectors at
5.9 keV, with a count rate of $\approx$ 1.5 counts per second.

Figure 3: Cryogenic components of our “XFDM" setup mounted on the Cu sample
holder: (a) SRON TES array, (b) transformer chip, (c) LC filter chip, (d) VTT
6-series array front-end SQUID, (e) RC snubber in parallel to SQUID input coil
and (f) SMD shunt resistor.

The setup is cooled to cryogenic temperature by means of a Leiden Cryogenics
dilution unit with 400 $\upmu$W of cooling power at 110 mK. For the
measurements reported in this paper, the base temperature is 60 mK. The
temperature of the setup is monitored via a Germanium thermistor anchored to
the Cu holder. Mechanical vibrations are damped by suspending the setup to the
mixing chamber with Kevlar wires. Thermal anchoring is performed by means of
OFHC Cu braids. A 2-stage VTT SQUID (Model J3 as Front-End SQUID and model F5
as Amplifier SQUID) is employed as trans-impedance amplifier to pre-amplify
the TES signals at cryogenic stage, which are demodulated and re-modulated for
base-band feedback by an FPGA board, at a sampling rate of 40 MSPS. An RC
snubber is placed in parallel to the SQUID input coil to lower the Q-factor of
the 10’s of MHz resonance created by the parallel between the SQUID input
inductance and parasitic capacitance.

Temperature drifts in the cooling system sensed by the detectors are corrected
in the data analysis pipeline exploiting the TES baseline current and the
pulse height information. The energy non-linearity is calibrated out by using
the zero energy information, the Mn-Kα and Mn-Kβ (6.5 keV) lines. The energy
of the X-ray event is evaluated by means of the optimal filtering technique.
The spectral performance is then assessed by fitting the Mn-Kα model holzer
to the collected events by using the Cash-statistics cstat  in the maximum-
likelihood method. The collected events are typically 3-4 thousand in order to
obtain sufficient statistics, with a typical statistical error on the energy
resolution of 0.2 eV.

Considering the typical energy resolution of the 23 connected pixels, measured
at R/RN typically in the range 0.15 to 0.25 and frequencies distributed to use
the full 1-5 MHz readout bandwidth, the average single-pixel X-ray spectral
performance is at a level of 2.7 eV at 5.9 keV.

##  III FSA for X-ray TES microcalorimeters

In this paper we focus on the application of FSA to the readout of X-ray TES
microcalorimeters. We point out however that the algorithm is designed to be
implementable also in different FDM readouts, such as for TES bolometer arrays
as the one of SPICA-SAFARI safari .

###  III.1 FSA working principle

The basic idea of FSA is that each carrier is shifted from the resonator
center frequency and the carrier amplitude is compensated for the signal
transfer loss due to the off-resonance operation of the LC filter. Such
compensation is achieved by means of a proportional-integral controller,
implemented in the FPGA. Similarly to the BBFB scheme, the FSA controller
works in base-band to avoid stability issues.

In off-resonance operation, the imaginary impedance of the LC filter causes a
phase difference between the TES current and bias voltage. The bias voltage is
known and the TES current is measured via the BBFB, whose outputs are the
demodulated in-phase ($I$) and quadrature ($Q$) orthogonal components of the
TES current. Since the phase difference is correlated with the quadrature
component, a convenient target for the controller is the $Q$ signal. The FSA
controls the $Q$ output of the BBFB via a feedback scheme, explained in Figure
4. In this "$Q$-nuller" scheme, the quadrature component of the TES current is
the input of the controller, which uses it as an error signal with respect to
a $Q=0$ reference.

For microcalorimeter application, a critical aspect is the speed of the
controller, which should be sufficiently fast to compensate the large decrease
in TES current when X-ray photons hit the detector. To make the controller
faster, the integrator gain is increased, an action which in principle can
make the controller unstable. To increase the stability margin of the
controller a proportional path is implemented. The regime changes from
integrating to proportional at frequencies higher than a crossover frequency
$f_{PI}$, calculated by the ratio of the integrating and proportional transfer
functions. In addition, a digital low-pass filter with cut-off frequency
$f_{LPF}$ is present in the loop to prevent interference between the
$Q$-feedback signals of neighbor pixels, fundamental for multi-pixel
operation.

Figure 4: Block diagram for the FSA controller paulpaper . The quadrature BBFB
output is the controller input, used as error signal with respect to the $Q=0$
reference. The controller loop is then split into two paths: an integrator
with gain $K_{i}$ and a proportional controller with gain $K_{p}$. The ratio
between the integrating and proportional transfer functions defines the
crossover frequency $f_{PI}$. The complex vector given by the bias signal
$U_{bias}$ (real) and the controller signal $U_{ctrl}$ (imaginary) is
modulated (_i.e._ rotated by an angle $\omega_{c}t$) and the real part of the
resulting output (the AC carrier) is fed to the LC filter. The SQUID output is
then demodulated, with only the $Q$-component of the BBFB output being used in
the controller loop. To suppress the out-of-band $Q$-signal, a second order
low-pass filter is employed.

Substantially, the FSA allows to handle the LC filter spread by moving the
issue from the cold components to the warm, digital readout electronics, which
are more feasible to control the generation of intermodulation products, whose
effect on performance is discussed as follows.

###  III.2 Effect of intermodulation distortions

The TES is sensitive to a IMLN when its frequency is close to the inverse of
the effective time constant under electro-thermal feedback (thermal
bandwidth). In this circumstance, the TES current is modulated and the energy
signal depends on the phase at which the X-ray photon hits the detector, with
the effect of severely degrading the measured energy resolution.

Ideally, it would be desirable to tune the carriers back to the original 100
kHz grid. Considering a 3$\upsigma$ scatter of the LC resonances around the
design values, this would require a shift of the bias frequencies of the order
of 10 kHz. A limitation of FSA is that the controller is expected to become
unstable if the applied shift from resonance is larger than the resonator
bandwidth. For the considered setup, the bandwidth of the RLC resonator with
the TES biased at typical working point is of the order of 1 kHz. Given this
boundary condition, shifting back the carriers to the 100 kHz grid is an
unrealistic goal.

In practice, such extreme frequency shifts are not necessary. FDM for
microcalorimeters requires TES’s with “slow" effective response time, so that
a larger inductance can be employed to shrink down the electrical bandwidth
$\approx$ R/L to prevent carrier leakage from neighbouring pixels hiroki2018 ,
while ensuring stable TES operation. For this reason, our detectors are
designed to have a thermal bandwidth of approximately 300 Hz, “slow" enough to
prevent carrier leakage at 100 kHz separation. This characteristic can be also
exploited to effectively use frequency shifts much smaller than 10 kHz.

The idea is then to place the bias carriers on a regular grid, so that the
IMLN’s would fall exactly at the same frequency (and higher harmonics) in the
TES response, sufficiently far from the thermal bandwidth. Therefore, we want
to assess what is the nearest frequency grid (taking into account DAC
resolution) for which no impact to the detector performance from
intermodulation distortions is expected.

To do so, we measure the energy resolution of one pixel with an external tone
applied to mimic the effect of an IMLN (Figure 5, left panel). The amplitude
of the tone is kept constant to a level of 3 orders of magnitude higher than
the TES response. The frequency $f_{tone}$ is changed from values within the
TES thermal bandwidth to the tail of the demodulated signal cut off by the
decimation filter. In the right panel of Figure 5 the measured performance
degradation with respect to the typical value for the investigated pixel is
shown, scaled in dB units.

Figure 5: _Left panel_ : X-ray pulse spectrum and TES noise spectrum with
applied external tone that mimics the effect of an IMLN. _Right panel_ :
measured degradation of energy resolution with respect to the typical value
for the considered pixel as a function of tone frequency $f_{tone}$, which is
defined from the difference to the LC resonator frequency. The X-ray pulse
spectrum is rescaled and overplotted in dashed red to show the thermal
bandwidth, evaluated at the 3 dB point.

We observe that the X-ray performance severely deteriorates when the tone
frequency falls into the thermal bandwidth and is recovered by shifting the
tone frequency outside the thermal bandwidth, with no significant degradation
for $f_{tone}=1.5$ kHz and a 0.6 dB degradation for $f_{tone}=1$ kHz, with
respect to the typical energy resolution for the active pixel. From this
experiment we conclude that, for such “slow" devices, a 1.5 kHz grid would be
necessary to completely tame out the impact of the applied external tone.

In Figure 6 we show, as an example of a practical multiplexing case with a
representative number of 20 pixels active, the noise spectra for 9 pixels in
groups of 3 neighbours in 100 kHz separation in different bias frequency
regimes. FSA is active on all the pixels and frequency shift is applied to
place the carriers on a 1 kHz grid. As can be seen, the amplitude for the
observed IMLN’s is at least one order of magnitude lower compared to the
applied tone in our experiment. Therefore, we expect the 0.6 dB degradation at
$f_{tone}=1$ kHz to become negligible in a practical multi-pixel
configuration. To minimize the frequency shifts necessary to arrange the
carriers on grid, thus reducing the risk of shifting the carriers outside of
the stability margin of the controller, we decide to perform our further tests
in the 1 kHz grid configuration. Such choice also minimizes the impact on the
bias voltage fed by the DAC, discussed in the next subsection.

Figure 6: TES noise spectra for 9 pixels (3 each for low-, mid- and high-
frequency regime in 100 kHz separation) acquired in a 20-pixel multiplexing
measurement. FSA is applied on all the pixels using a 1 kHz grid
configuration. As a result, the bias frequencies are shifted such that all the
IMLN’s appear at integer multiples of 1 kHz.

###  III.3 Impact on AC bias DAC

The TES current is maintained to its on-resonance value when changing the bias
frequency at the expense of an increase in the voltage fed by the bias DAC. A
requirement for practical application is that such increase in DAC output
should be acceptable from the point of view of (a) dynamic range and (b) noise
contribution to the energy resolution.

To assess the impact on the bias DAC output, we measure its raw RMS voltage
and rescale it to the full-range peak-to-peak value. We first characterize the
increment in voltage as a function of the frequency shift, as shown in Figure
7, left panel. The measurement is done for one pixel in three different bias
points in the typical operation range. For lower TES resistance values, the
$Q$-factor of the resonance is higher and so is the increase in the additional
impedances when moving away from resonance (0 frequency shift). For this
reason, the increment in DAC signal is larger for lower bias points.

Since for this setup the average shift to place the carriers on a 1 kHz grid
is approximately 250 Hz, from this measurement we expect a DAC voltage
increase of $\approx$ 5%. To verify this, we measure the DAC voltage as a
function of the number of multiplexed pixels, with bias frequencies shifted to
fall on the 1 kHz grid, up to the representative number of 20 pixels. The
result is reported in the right panel of Figure 7. We observe that the DAC
voltage increase is compatible with the expected value of 5% for
R/R${}_{N}\approx 0.2$ for an average shift of $\approx$250 Hz.

Figure 7: $Left$ $panel$: AC bias DAC voltage normalized to on-resonance value
as a function of frequency shift, for 3 different bias points in the typical
R/RN region. The resonance frequency used for reference is measured with the
TES biased at 20% of its transition (black points). The asymmetry of the
parabola for the red and blue points is due to the slight change in resonance
frequency at different bias points. $Right$ $panel$: voltage fed by the bias
DAC with FSA active normalized to the on-resonance value as a function of
active pixels. The normalised DAC voltage for one pixel active is close to 1
since for that pixel the shift towards its grid frequency is almost zero.

We conclude that the average frequency shift to place the carriers on grid
allows to make a fair prediction of the increase in DAC voltage, which can be
exploited to set a safety margin in the used dynamic range to avoid DAC
saturation.

By design, the controller provides a signal amplification that compensates for
the LC filter attenuation: however in this scheme also noise is passed through
the feedback loop and amplified, thus increasing the effective DAC noise
contribution to the spectral performance.

To make a quantitative evaluation of the impact on the DAC noise contribution
to the energy resolution, we refer to the estimated energy resolution budget
for the FDM readout of X-ray TES microcalorimeters in Athena X-IFU roland_dac
, where the total energy resolution of one pixel is considered as the root sum
square of several contributions, among which 2 eV come from the detector noise
in set point and 0.23 eV from AC bias DAC noise.

Considering only these two contributions, the pixel energy resolution is
affected as follows:

$\sqrt{\Delta E_{TES}^{2}+\Delta E_{DAC}^{2}}=\sqrt{(2\
\textrm{eV})^{2}+(0.23\ \textrm{eV})^{2}}\approx 2.01\ \textrm{eV}\ .$

When FSA is active and frequency shift is applied, the DAC noise is amplified
along with the signal. Calling $\varepsilon$ the increase in signal fed by the
bias DAC, the estimated spectral performance then becomes

$\Delta E_{FSA}\equiv\sqrt{\Delta E_{TES}^{2}+(\varepsilon\cdot\Delta
E_{DAC})^{2}}.$

To avoid underestimations, since we are not taking into account other effects
such as the increase of DAC non-linearity, we assume the worst possible case:
all the pixels need a 700 Hz shift (approximately the maximum observed shift
for a 1 kHz grid operation with this setup). Considering the blue curve from
Figure 7, this means a voltage increase of roughly 40%. With
$\varepsilon=1.40$ we then get $\Delta E_{FSA}\simeq 2.03$ eV.

Therefore, even making a substantial overestimation, we expect the increase in
DAC noise to not contribute significantly to the root sum square spectral
performance, in particular compared to the degradation observed due to (1)
off-resonance operation without FSA hiroki2018  and (2) the presence of IMLN’s
within the thermal bandwidth, as shown in previous subsection.

One more observation is that, to perform these measurements, the system could
handle stable FSA operation with more than 20 pixels active. This is a
fundamental step forward with respect to past FSA implementations and a
crucial point for practical FDM application on X-ray TES microcalorimeters.

##  IV Bias circuit reproducibility

The basic functionality of FSA is to preserve the electrical bias circuit when
shifting the carrier from the resonance frequency. In particular, the
controller must react if the measured TES current is not in-phase with the
bias voltage. Since even on resonance a phase angle can be present, due to
unavoidable delays in the signal path, the phase of the carrier is tuned to
compensate for such static offset. This is done with an automated routine that
scans for the phase angle that minimizes the demodulated $Q-$signal with the
TES in normal state. In this way, the controller correctly keeps the TES
current in phase with the bias voltage when frequency shift is applied.

To verify the conservation of the bias circuit, we measure the TES IV curve
with FSA under frequency shift and compare it with the original IV curve on-
resonance. Early tests showed that the IV curves diverged for increasing
frequency shifts. This was understood to be the effect of non-zero admittance
of the other resonators in parallel with the measured pixel. On resonance, the
TES impedance is much lower than the parallel impedance of other resonators
and the effect is negligible. However, when applying a frequency shift the
impedance of the LCR series gets higher and the contribution of the parallel
impedance becomes significant.

The compensation for such effect is made in the firmware, by multiplying the
bias signal for a scaling factor, essentially the parallel admittance
converted to adimensional units. This scaling factor is given by numerical
parameters depending on the setup (LC resonances, inductance value, SQUID coil
ratio, bias and feedback transfer function, conversion factors of warm
electronics) and is readily calculated for each pixel. In addition, we
developed an iterative procedure that empirically optimizes such scaling
factor, which we use to verify the goodness of the calculation. This approach
allows us to correctly compensate for the leakage of current though the
parallel resonators.

As previously stated, the electrical bandwidth at typical bias points for our
pixels is of the order of 1 kHz. Within this boundary ($\pm$1 kHz shift) the
controller should preserve the electrical circuit in an acceptable way. From
single-pixel X-ray measurements with this array we know that the best
performances are obtained by biasing the pixels approximately from 15% to 25%
of TES transition. From the IV curves we verified that in such transition
range, a change in current of the order of 1% is negligible compared to the
typical period of the $Q$-current oscillation due to weak-link effect. Since
no significant impact on spectral performance is expected from such small
effect for these devices, we set as a requirement that the controller should
maintain the electrical circuit within less than 1% deviation from reference.
To verify this, we measure how the TES current deviates from its on-resonance
value when frequency shift is applied. This is done by measuring the TES
current (1) as a function of bias voltage (IV curve) with fixed frequency
shift and (2) as a function of frequency shift (current scan) with fixed bias
voltage.

We verified that, by properly compensating the parallel admittance effect, we
are able to (1) reproduce the IV curve at shifted frequency and (2) keep the
TES current constant at the typical working point within our boundary
conditions ($\pm$1 kHz shift, $<$ 1% deviation). In Figure 8 the shifted IV
curves and in Figure 9 the TES current scan are representatively reported for
one pixel.

Figure 8: Comparison of the IV curves on-resonance and shifted with FSA
active. In the bottom panel, the residuals of the shifted curves with respect
to the on-resonance reference, zoomed in the superconducting transition region
are plotted scaled in percent. Some points, recorded in coincidence with an
X-ray hit, are removed from the plot for sake of clarity. Figure 9: TES
current change on typical bias point with FSA active across $\pm$1 kHz
frequency shift. Without FSA it is expected to have a similar parabolic shape
as the DAC compensation voltage (Figure 7, left panel), since both come from
the effect of the additional impedance due to off-resonance operation. Some
points, recorded when an X-ray hit the detector, are not visible in the shown
scale.

##  V X-ray spectral performance

As previously anticipated, the controller should be fast enough to follow the
rapid decrease of the TES current when an X-ray photon hits the detector. The
parameters to set are essentially three: the integrator gain, the crossover
frequency $f_{PI}$ (defined in Section III) and the low-pass filters cut-off
frequency $f_{LPF}$.

The procedure used to set these parameters is the following: to allow the
controller to correctly compensate the TES current during an X-ray event, we
increase the integrator gain until the TES response is stable and the X-ray
pulse shows no signs of oscillation. We find that the integrator gain should
be increased approximately in proportion with the TES bias frequency. In
practice, we use 3 sequential values for 3 corresponding regimes, which we
define as low- ($\approx$ 1 MHz to 2 MHz), mid- ($\approx$ 2 MHz to 3 MHz) and
high-frequency ($\approx$ 3 MHz to 5 MHz). To allow a sufficient integrator
gain while maintaining the feedback stable, the proportional gain is tuned so
that the crossover frequency $f_{PI}$ covers the electrical bandwidth of the
pixel, with a factor $\approx$ 2-3 of margin. The cut-off frequency $f_{LPF}$
of the low-pass filter is then set to $\approx$ 2-3$\times f_{PI}$, value for
which the out-of-band Q-feedback signal is sufficiently suppressed to prevent
interference with 100 kHz neighbour channels in multi-pixel operation, but
with enough margin to avoid attenuation of the in-band TES response. We find
that stable operation for all the pixels can be achieved by using the same
values of $f_{PI}$ and $f_{LPF}$.

Having fixed such parameters, we first perform X-ray measurements in single-
pixel with 200 Hz frequency shift for several channels, confirming that the
energy resolution does not degrade under FSA operation. In this way we confirm
that the controller is able to compensate for fast changes in TES current,
while maintaining stable operation. In Table 1 the comparison between the
typical single pixel performance and the measured performance with FSA is
representatively reported for one pixel per each frequency regime.

Ch nr. | Freq (MHz) |  $\Delta$E${}_{\textrm{SP}}$ (eV) |  $\Delta$E${}_{\textrm{SP}}$ FSA (eV)
---|---|---|---
1 | 1.1 | 2.57 $\pm$ 0.12 | 2.57 $\pm$ 0.17
17 | 2.9 | 2.76 $\pm$ 0.11 | 2.82 $\pm$ 0.17
25 | 4.2 | 2.75 $\pm$ 0.12 | 2.83 $\pm$ 0.20
Table 1: Comparison with typical single-pixel performance (averaged over two
different bias points) and measured single-pixel performance with FSA under a
200 Hz frequency shift.

We then perform multiplexing experiments to verify that the energy resolution
is not degraded in multi-pixel configuration. We first use three neighbouring
channels in the 100 kHz separation, namely Ch1, Ch2 and Ch3, whose single-
pixel average performance is at level of $\approx$ 2.5 eV. The bias
frequencies are placed on a regular 1 kHz grid, resulting in shifts of
approximately 150 Hz for Ch1 and Ch2 and 600 Hz for Ch3. Measuring the X-ray
performance in multiplexing we find a summed energy resolution of 2.54 $\pm$
0.10 eV, consistent with the expected value from single-pixel measurements.
The energy spectra for these channels in single-pixel and 3-pixel multiplexing
are compared in Figure 10.

Figure 10: Measured spectral resolutions for Ch1-2-3 in single pixel, on
resonance and without FSA (left panels), and in multi-pixel at 100 kHz
separation, with FSA active and bias frequencies shifted on a 1 kHz grid
(right panels).

Using the same FSA parameters, we repeat the measurement considering other
pixels in the 100 kHz separation (namely Ch5, Ch6 and Ch7). In this case, the
frequency shifts necessary for on-grid operation are $\approx$ 130 Hz, 210 Hz
and 420 Hz, respectively. We measure a summed average energy resolution of
2.55 $\pm$ 0.09 eV, which again is consistent with the expected value from
single-pixel measurements. The results of such measurements are summarized in
Table 2.

Ch nr. | Freq (MHz) |  $\Delta$E${}_{\textrm{SP}}$ (eV) |  $\Delta$E${}_{\textrm{3MUX}}$ FSA (eV)
---|---|---|---
1 | 1.1 | 2.57 $\pm$ 0.12 | 2.63 $\pm$ 0.17
2 | 1.2 | 2.47 $\pm$ 0.11 | 2.47 $\pm$ 0.16
3 | 1.3 | 2.64 $\pm$ 0.12 | 2.51 $\pm$ 0.18
5 | 1.5 | 2.65 $\pm$ 0.10 | 2.53 $\pm$ 0.15
6 | 1.6 | 2.73 $\pm$ 0.11 | 2.71 $\pm$ 0.14
7 | 1.7 | 2.27 $\pm$ 0.10 | 2.42 $\pm$ 0.15
Table 2: Energy resolutions at 5.9 keV measured in single pixel (no FSA) and
comparison with the individual performances obtained in 3-pixel multiplexing
experiments with FSA, with 100 kHz neighbour channels and bias frequencies
shifted on grid.

These experiments allow us to verify that (1) FSA is stable in multi-pixel
operation with 100 kHz separated pixels and (2) once the optimal values for
$f_{PI}$ and $f_{LPF}$ are fixed, no further fine tuning is required for the
different pixels and/or frequency shifts. With respect to the previous version
of FSA, these are major points of improvement that consent to scale the
applicability of FSA up to an entire readout chain.

Figure 11: Summed energy spectrum from 20 pixels measured in a 22-pixel
multiplexing experiment (2 outliers excluded in the analysis) with FSA. Figure
12: Individual energy histograms of 20 pixels measured in a 22-pixel
multiplexing experiment (2 outliers excluded in the analysis) with FSA. Bias
frequency, energy resolution and shift from resonance to on-grid frequency are
reported for each pixel in the bottom panel.

To show FDM results is outside the focus for this paper, nonetheless we
mention that we verified that in $\geq$ 20-pixel multiplexing we can obtain
stable operation of the system and summed spectral resolution at a level
better than 3 eV, where the typical average energy resolution measured in
single pixel with this array is at a level of 2.7-2.8 eV, with one outlier at
a level of 4.4 eV and two at a level of 3.5 eV. In Figure 11 and Figure 12 we
report, respectively, the summed histogram and individual energy spectra of 20
pixels obtained in a 22-pixel multiplexing experiment (the 4.4 eV outlier not
being active), with a measured summed energy resolution at 5.9 keV of
2.91$\pm$0.03 eV. The summed energy resolution degrades to 2.97$\pm$0.03 eV if
the two 3.5 eV outliers are included in the analysis.

A further, more detailed discussion on FDM results, obtained with new SRON
arraysmartin2020 , will be reported separately in a paper currently under
preparation.

##  VI Summary

We have developed a frequency shift algorithm to perform FDM readout of TES
detectors in off-resonance operation. In this scheme, by shifting the carriers
on a regular grid, line noises due to intermodulation distortion can be placed
sufficiently far away from the sensitive region on the TES response, avoiding
degradation in spectral performance. By applying the FSA up to 20 pixels in
multiplexing, we evaluated that the increase in bias DAC output is acceptable
in terms of DAC dynamic range and noise contribution to the energy resolution.

After finding and fixing the optimal FSA parameters for our experimental
setup, we have verified that the on-resonance bias circuit can be reproduced
with acceptable margin when the bias frequency is shifted.

Finally, we have measured the X-ray spectral performance in multi-pixel
configuration in 100 kHz neighboring pixels with off-resonance, on-grid bias
frequencies, observing no significant degradation from single-pixel energy
resolutions. This is achieved without the necessity of fine-tuning the
controller parameters for the different pixels and frequency shifts.

Having shown scalability towards and stability in multi-pixel operation, we
consider the FSA mature enough to be implemented on the FDM readout of X-ray
TES microcalorimeters, to finalize the demonstration of our FDM technology.

## Acknowledgements

SRON is financially supported by the Nederlandse Organisatie voor
Wetenschappelijk Onderzoek.

The SRON R1a TES array used for the measurements reported in this paper was
developed in the framework of the ESA/CTP grant ITT AO/1-7947/14/NL/BW.

## Data availability

The corresponding author makes available the data presented in this paper upon
reasonable request.

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Viscous Dark Energy Accretion Activities : Sonic Speed, Angular Momentum and
Mach Number Studies

Sandip Dutta<EMAIL_ADDRESS>Promila Biswas
<EMAIL_ADDRESS>and Ritabrata<EMAIL_ADDRESS>
Department of Mathematics, The University of Burdwan, Golapbag Academic
Complex, Burdwan -713104, Dist.:- Purba Barddhaman, State:- West Bengal,
India.

###### Abstract

In this present article, we study different accretion properties regarding
viscous accretion of dark energy. Modified Chaplygin gas is chosen as the dark
energy candidate. Viscosity is encountered with the help of Shakura-Sunyaev
viscosity parameter. We study sonic speed vs radial distance curves. We
compare between adiabatic and dark energy dominated cases and follow that
sonic speed falls as we go nearer to the central gravitating object. As
viscosity is imposed, a threshold drop in accretion sonic speed is followed.
Average rate of fall in accretion sonic speed is increased with black hole's
spin. This is signifying that this kind of accretion is weakening the overall
matter/energy infall. Specific angular momentum to Keplerian angular momentum
ratio is found to fall as we go far from the black hole. Accretion Mach number
turns high as we go towards the inner region and high wind Mach number is not
allowed as we are going out. Combining, we conclude that the system weakens
the feeding process of accretion.

Keywords : Black Hole Accretion Disc, Dark Energy.

PACS Numbers : 95.30.Sf, 95.36.+x, 95.35.+d, 98.80.-k.

##  1 Introduction

Just one Gyr after the Big Bang, super massive black holes (SMBHs) with masses
of order $M_{BH}\sim 10^{9}M_{\odot}$ started to exist. This is clear from
studies of luminous quasars which are as early as $z\sim 6-7$ [1]. To become
such massive, they required continuous and exponential growth at the order of
Eddington limit throughout the time universe has already experienced by then.
For early SMBHs, these conditions might not be necessarily ubiquitous [2].

Mass accretion history of $\sim 14000$ different haloes at $z=0$ using the ART
code [3] in WMAP1 cosmology is identified by Wechsler et al [4]. The authors
have found that the accretion histories of their present day haloes, on
average, were well fitted by

$M_{H}(z)=M_{0}e^{-\alpha(z_{f})z}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (1)

where $M_{0}$ is the present day mass of a halo, $\alpha(z_{f})$ is a
parameter which describes its formation epoch. For Einstein deSitter universe,
the average mass accretion $\left<\dot{M}_{H}\right>$ onto a halo of mass
$M_{H}$ has the same $M_{H}\dot{z}$ dependence like the equations

$\left<\dot{M}_{H}\right>\simeq
M_{H}\bigg{\rvert}\frac{d\delta_{c}}{dt}\bigg{\rvert}f(M_{H})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d\delta_{c}}{dt}=\frac{d\delta_{c}}{dD}\frac{dD}{dz}\frac{dz}{dt}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (2)

(where $\delta_{c}(t)$ = critical density above which an object will collapse
to form a bound structure, $D(z)$ is linear growth factor and $f(M_{H})$ = a
weak function of halo mass) differing only in normalization
$(\frac{d\delta_{c}}{dz}=1.686)$. With this idea, it is found to treat
equation (1) to be chosen as a sensible fit if we ignore slight mass
dependencies of $\alpha(z_{f}(M_{H}))$ and $f(M_{H})$ term.

Somerville & Kolatt's Nbranch merger tree algorithm [5] was used by Vanden
Bosch (2002)[6] to find a two parameter fit which better described the mass
accretion history of concerned haloes. The relation provided was

$\alpha=\left(\frac{z_{f}}{1.43}\right)^{-1.05}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ .$

Though it is more common to present the definition of $z_{f}$ as an epoch
where present day halo of interest had half of its present day mass

$z_{f}=\frac{ln(2)}{\alpha}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ .$

Mass accretion history of $\sim\leavevmode\nobreak\ 5,00,000$ haloes from the
Millennium Simulation with $M_{H}>10^{12}M_{\odot}$ and $0\leq z\leq 6$ was
investigated by Mc Bride et. al.[7] who found that approximately $25\%$ can be
well described by equation (1). A second parameter $\beta$ has been introduced
by them which requires another relation

$M_{H}(z)\propto(1+z)^{\beta}exp\\{-\nu_{z}\\}$ (3)

and this has provided a better fit to halo mass accretion history.

More recently, a joint dataset from the Millennium I and II simulations are
used by Fakhori et al. [8, 9, 10] to support equation (3) to hold accross five
decades in mass upto $z=15$. These works state very strongly that accretion
rates on different haloes depend on the redshift zone in time it is embedded
with. So different time in past we look back, different accretion rates might
be possible to observe. Cosmic history of our universe is led by recent rapid
progressions of precision cosmology. Simplest justifying model among many is
$\Lambda$CDM which agrees with many cosmological observations [11]. Though
this model suffers from different theoretical issues like fine tuning problem
and cosmic coincidence problem [12, 13].

Alternative dynamical dark energy models have been proposed [14, 15]. As the
dark sector comprises of dark matter and dark energy, interaction between dark
matter particles and dark energy is studied [16, 17, 18, 19, 20, 21, 22]. It
is assumed that these two components stay coupled and hence we are motivated
to study dark energy dominated accretion. As coupling with dark energy halo is
considered, disc accretion model is preferred to be built.

Dark energy requires highly relativistic structure to fall gravitationally in.
So we consider our disc around a black hole. First ever relativistic accretion
was studied by F. Michel [23] where he has shown, by solving relativistic
momentum and energy conservation equations, that the radiation speed gradient
can be expressed as a ratio of two expressions. Denominator of the said
expression turns zero when radial inward speed equals the sound speed through
the fluid and to obtain a physical flow, we need to take help of L'Hospital's
rule and a second degree first order differential equation will be
constructed. This will give birth of two flows from the sonic point namely the
accretion and the wind flows. Shakura and Sunyaev [24] added another milestone
to the literature where shear viscosity, $\omega_{r\phi}$ is replaced by
$\alpha_{ss}\rho c_{s}^{2}$, with $\alpha_{ss}$ is a parameter to regulate as
per the model's viscosity requirement.

Accretion of dark energy was first studied by Babichev et al [25]. They
assumed some fluid permeated all over in the universe and responsible for
late-time cosmic acceleration and the field equation followed by such fluid is
enforcing it to weaken the corresponding accretion. Point to be noted, this
article has considered spherical accretion. In between, we followed
hierarchical clustering of dark matter is responsible for different structure
creation [26]. Also, different works predicted interactions between dark
matter and dark energy [27, 28]. These support the presence of dark energy
coupled with dark matter in different regions of galaxies. Does this ensure
the presence of dark sectors in the vicinity of galactic cores? Very recently,
the possibility of dark matter in the form of bosons are proposed to form
self-gravitating bound structures in different galaxies [29]. Besides, authors
of the reference [30] compared the motion of test particles in the
gravitational fields of supermassive black holes and dark matter core to find
a noticeable discrepancy as the radial distance from the center is less than
$100\leavevmode\nobreak\ AU$. In future, different finer observations like
VLBI or Black Hole Cam Project might be able to more clear picture how much
different the actual shadow of a black hole and the effect of a black hole
mimicker is. As of now, we can not just exclude the theories that predict the
existence of SMBH candidates like gravastars, boson stars etc. Dark energy and
bulk viscosity can be formed out of delayed decay of dark matter [31]. This,
particularly, motivates us to examine the dark energy’s effect around a
supermassive black hole. It is evident again accretion disc takes cylindrical
structures [32].

First study of disc accretion of dark energy is found in 2011 [33] where it
was shown that if modified Chaplygin gas type agent is accreting upon a black
hole, at a finite distance from the gravitating center, it will increase the
wind branch’s radial velocity so much that the speed will be equal to that of
light. Physically, this interprets that matter is being thrown out with
extreme speed and nothing is allowed to fall in. Viscosity is added to this
model and studied in [34], [35], [36] and [37] to establish that viscosity
will catalyze the whole process of accretion and as a result, the density of
the accreting fluid drops drastically as we proceed towards the central black
hole. Other physical quantities like Mach number, spin speed etc were not
studied till date. However, the observations of the event horizon telescope
enrich us a lot regarding these physical quantities. This motivates us towards
the construction of this present article.

In this article, we are motivated to study viscous accretion onto a SMBH and
to find its different consequences. In the next section, we will construct the
mathematical problem. After that the solutions will be obtained and presented.
Finally, we will conclude.

##  2 Construction of the Model

In the time interval 1980 to 2002, we observe a series of literatures trying
to replace the general relativistic effect by pseudo Newtonian potentials. In
this genre, Mukhopadhyay proposed one pseudo Newtonian force as [38]

$\mathscr{F_{G}}\mathpzc{(x,J)}=\mathpzc{\frac{(x^{2}-2J\sqrt{x}+J^{2})^{2}}{x^{3}\\{\sqrt{x}(x-2)+J\\}^{2}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (4)

where $\mathpzc{x}=\left(\frac{\mathpzc{R}}{\frac{GM}{c^{2}}}\right)$,
$\mathpzc{J}=\left(\frac{\mathpzc{A}}{c}\right)$, $\mathpzc{R}$, $G$, $M$, $c$
and $\mathpzc{A}$ are radial distance from the central object, Newton's
gravitational constant, mass of the central gravitating object, speed of light
and Kerr spinning parameter respectively.

Axisymmetric accretion study was constructed by Novikov and Thorne [39] in
1973. Quasi spherical model [40, 41, 42], however, is treated to be the
process which is involved in a large number of high energy astrophysical
situations. These models support both galactic and extragalactic compact X-ray
sources, novae, dwarf novae, active galactic nuclei, globular cluster X-ray
sources, X-ray and $\gamma$-ray bursts and quasars [42, 43]. Chakrabarti
studied transonic properties of conical shaped accretion flows, known as
Wedge-Shaped flow [44].

Accretion model keeping hydrostatic equilibrium in vertical direction is also
found in literature. To keep simplicity, necessity to construct models having
all the salient features of the original problem should be given priority.
Cylindrical model, conical model and flow in vertical equilibrium model have
identical physical properties, in spite of the fact that they are constructed
based on fundamentally different assumptions. Results of one model can be
obtained by changing the physical parameters of the other [45]. Alongside, the
reference [46] and the references therein are also the relevant literature.

In our article, we assume axisymmetric flow and radiative losses of internal
energy. Hence the disc turns geometrically thin. In this case, accretion
occurs owing to the overflow of the effective potential barrier near the black
hole. This is equivalent to the case of the Roche lobe overflowing star
present in a binary. Viscosity considered is due to the small scale turbulence
and described by $\alpha_{SS}$-prescription. In Kerr space-time, vertical
hydrodynamical equilibrium of stationary thin and slim accretion discs is also
studied in [47]. Cylindrical polar coordinates are best to explain
axisymmetric models. Hence we consider cylindrical polar coordinates and
construct the model. So at a distance $\mathpzc{x}$, quantity of matter inflow
should be $\dot{\mathscr{M}}=-4\pi\mathpzc{xU}\rho\mathpzc{h(x)}$ and if this
rate is taken as constant, i.e., equation of continuity leads us to

$\frac{d}{d\mathpzc{x}}\bigg{\\{}\mathpzc{xU}\rho\mathpzc{h(x)}\bigg{\\}}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (5)

here $\mathpzc{U}=\frac{\mathpzc{V_{r}}}{c}$, $\mathpzc{V_{r}}$ is radial
inward speed and $\rho$ (accretion disc height) presents the vertically
averaged density as we are not considering changes in density or any other
parameters with height.

Next we will consider Navier Stokes equation for steady state given as

$\rho(\overrightarrow{\mathpzc{U}}.\overrightarrow{\nabla})\overrightarrow{\mathpzc{U}}=-\overrightarrow{\nabla}p+\rho\mathpzc{\nu}{\nabla}^{2}\overrightarrow{\mathpzc{U}}-\overrightarrow{\mathscr{F_{G}}\mathpzc{(x,J)}}$
(6)

and the radial momentum balance equation turns out as

$\mathpzc{U}\frac{d\mathpzc{U}}{d\mathpzc{x}}+\frac{1}{\rho}\frac{dp}{d\mathpzc{x}}-\frac{\lambda^{2}}{\mathpzc{x}^{3}}+\mathscr{F_{G}}\mathpzc{(x,J)}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (7)

where $p$ is the accretion fluid pressure, $\mathpzc{\nu}$ is the viscous
coefficient and $\lambda$ is specific angular momentum.

Again we get the azimuthal momentum balance equation as

$\mathpzc{U}\frac{d\lambda}{d\mathpzc{x}}=\frac{1}{\mathpzc{x}\rho\mathpzc{h(x)}}\frac{d}{d\mathpzc{x}}\left[{\mathpzc{x}^{2}}\alpha_{ss}(p+\rho{\mathpzc{U}^{2}})\mathpzc{h(x)}\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ .$ (8)

We will assume that the vertical equilibrium and be able to write the vertical
momentum balance equation as
$\mathpzc{h(x)}=c_{s}\sqrt{\frac{\mathpzc{x}}{\mathscr{F_{G}}\mathpzc{(x,J)}}}.$

As we have stated earlier, we will choose dark energy as accreting agent. We
take modified Chaplygin gas as the candidate of dark energy. This model of
dark energy was proposed within the framework of FLRW cosmology [48]. The
equation of state is given by $p=\alpha\rho-\frac{\beta}{\rho^{n}}$, where
$\alpha$, $\beta$ and $\mathpzc{U}$ are free parameters.

This equation of state is able to interpolate from matter dominated era to the
cosmological constant dominated era. Xu and Lu have constrained the parameter
$n$ and related cosmological parameters like $\Omega_{b}{h^{2}}$, $H_{0}$ as

$n=0.033^{+0.066}_{-0.071}(1\sigma)^{+0.096}_{-0.087}(2\sigma)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$
$\Omega_{b}h^{2}=0.0233^{+0.0023}_{-0.0016}(1\sigma)^{+0.0029}_{-0.0020}(2\sigma)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ and$
$H_{0}=69.97^{+2.87}_{-2.78}(1\sigma)^{+3.48}_{-3.08}(2\sigma)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$

with a minimum chi squared $=519.342$.

From the equation of state of modified Chaplygin gas, we obtain the sound
speed squared as

${c_{s}}^{2}=\frac{\partial p}{\partial\rho}=\alpha+\frac{\beta
n}{\rho^{n+1}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ .$ (9)

With the help of the continuity equation, we obtain

$\frac{1}{\rho}\frac{dp}{d\mathpzc{x}}=-\frac{2{c_{s}}^{3}}{(n+1)({c_{s}}^{2}-\alpha)}\frac{dc_{s}}{d\mathpzc{x}}=-\frac{1}{n+1}\frac{d{c_{s}}^{2}}{d\mathpzc{x}}-\frac{\alpha}{n+1}\frac{d}{d\mathpzc{x}}\\{ln({c_{s}}^{2}-\alpha)\\}$
(10)

and the gradient of sound speed is obtained as

$\frac{dc_{s}}{d\mathpzc{x}}=\bigg{\\{}\frac{3}{2\mathpzc{x}}-\frac{1}{2\mathscr{F_{G}}\mathpzc{(x,J)}}\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}+\frac{1}{\mathpzc{U}}\frac{d\mathpzc{U}}{d\mathpzc{x}}\bigg{\\}}\left\\{\frac{(n+1)c_{s}({c_{s}}^{2}-\alpha)}{(1-n){c_{s}}^{2}+\alpha(n+1)}\right\\}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ .$ (11)

Azimuthal momentum balance equation turns to be

$\frac{d\lambda}{d\mathpzc{x}}=\frac{\mathpzc{x}\alpha_{ss}}{\mathpzc{U}}\left[\frac{1}{2}\bigg{\\{}\frac{5}{\mathpzc{x}}-\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\bigg{\\}}\bigg{\\{}\frac{(n+1)\alpha-{c_{s}}^{2}}{n}+\mathpzc{U^{2}}\bigg{\\}}\right.$
$\left.+2\mathpzc{U}\frac{d\mathpzc{U}}{d\mathpzc{x}}+\left\\{\left(\frac{(n+1)\alpha-{c_{s}}^{2}}{n}+\mathpzc{U^{2}}\right)\frac{1}{c_{s}}-({c_{s}}^{2}+\mathpzc{U^{2}})\left(\frac{1}{n+1}-\frac{2c_{s}}{{c_{s}}^{2}-\alpha}\right)\right\\}\frac{dc_{s}}{d\mathpzc{x}}\right]$
(12)

and replacing $\frac{1}{\rho}\frac{dp}{d\mathpzc{x}}$ in (10), we obtain,

$\frac{d\mathpzc{U}}{d\mathpzc{x}}=\frac{\frac{\lambda^{2}}{\mathpzc{x^{3}}}-\mathscr{F_{G}}\mathpzc{(x,J)}+\left\\{\frac{3}{\mathpzc{x}}-\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right\\}\frac{{c_{s}}^{4}}{\\{(1-n){c_{s}}^{2}+\alpha(n+1)\\}}}{\mathpzc{U}-\frac{2{c_{s}}^{4}}{\mathpzc{U}\\{(1-n){c_{s}}^{2}+\alpha(n+1)\\}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ .$ (13)

After we have completed the constructions of the required differential
equations (11),(12) and (13), we will require to set the initial conditions.
To do so, we will take help of a basic crucial nature of accretion around
black holes. Unlike neutron stars and other accretors, black holes do not
possess solid surfaces. Near the event horizon, the accretion speed becomes
almost equal to speed of light. Besides it is obvious that very far from the
accretion disc, where a particle just enters into the disc due to the angular
momentum transportation process, the radial inward speed is very low, almost
equal to zero. Now throughout the accretion disc, the radial speed starts from
zero and reaches $1$ where $c$ is taken to be unit. In between at some
$\mathpzc{x}={\mathpzc{x}}_{c}$ the speed should be equal to
$\frac{\sqrt{2}{c_{s}}^{2}}{\\{(1-n){c_{s}}^{2}+\alpha(n+1)\\}^{\frac{1}{2}}}$
and the denominator will vanish, i.e., $D({\mathpzc{U}}_{c},c_{sc})=0$. Now to
have a continuous physical flow, the numerator should vanish as well,
$N({\mathpzc{x}}_{c},\leavevmode\nobreak\ \mathpzc{J},\leavevmode\nobreak\
c_{sc})=0$ which provides the value of sound speed at ${\mathpzc{x}}_{c}$.
Using this, we obtain ${\mathpzc{U}}_{c}$ at ${\mathpzc{x}}_{c}$ from
$D({\mathpzc{U}}_{c},c_{sc})=0$. $\lambda_{c}($ at
$\mathpzc{x}={\mathpzc{x}}_{c})$ is assumed such that a physical flow is
obtained.

Now applying the L'Hospital's rule in equation (13) at the critical point (say
$\mathpzc{x}=\mathpzc{x}_{c}$) we get a quadratic equation of
$\frac{d\mathpzc{U}}{d\mathpzc{x}}$ in the form,

$A\left({\frac{d\mathpzc{U}}{d\mathpzc{x}}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}^{2}+B\left(\frac{d\mathpzc{U}}{d\mathpzc{x}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}+C=0.$
(14)

Where
$A=1+\frac{1}{c_{sc}^{2}}-\frac{4\left(n+1\right)\left(c_{sc}^{2}-\alpha\right)}{{\left\\{\left(1-n\right)c_{sc}^{2}+\alpha\left(n+1\right)\right\\}}^{2}}$,
$B=\frac{4\lambda\mathpzc{U}}{\mathpzc{x}_{c}^{3}}+\frac{2\mathpzc{U}_{c}c_{sc}\left(1-n\right)}{\left\\{\left(1-n\right)c_{sc}^{2}+\alpha\left(n+1\right)\right\\}}-\frac{2\mathpzc{U}_{c}\left(n+1\right)\left(c_{sc}^{2}-\alpha\right)}{{\left\\{\left(1-n\right)c_{sc}^{2}+\alpha\left(n+1\right)\right\\}}^{2}}\left\\{\frac{3}{\mathpzc{x}_{c}}-\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}\right\\}+\\\
\left[\left(c_{sc}^{2}+\mathpzc{U}_{c}^{2}\right)\left(\frac{1}{n+1}\frac{2c_{sc}}{c_{sc}^{2}-\alpha}\right)-\left\\{\frac{3}{\mathpzc{x}_{c}}-\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}\right\\}\frac{4\mathpzc{U}_{c}}{c_{sc}{\left\\{\left(1-n\right)c_{sc}^{2}+\alpha\left(n+1\right)\right\\}}}-\left(\frac{\left(n+1\right)\alpha-
c_{sc}^{2}}{n}+{\mathpzc{U}}^{2}\right)\frac{1}{c_{sc}}\right]\left\\{\frac{\mathpzc{U}_{c}\left(n+1\right)\left(c_{sc}^{2}-\alpha\right)}{2c_{sc}^{3}}\right\\}$
and $C=D+E+F$.
The values of $D$, $E$ and $F$ are,
$D=\left[\left(c_{sc}^{2}+\mathpzc{U}_{c}^{2}\right)\left(\frac{1}{n+1}\frac{2c_{sc}}{c_{sc}^{2}-\alpha}\right)-\left\\{\frac{3}{\mathpzc{x}_{c}}-\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}\right\\}\frac{4\mathpzc{U}_{c}}{c_{sc}{\left\\{\left(1-n\right)c_{sc}^{2}+\alpha\left(n+1\right)\right\\}}}-\left(\frac{\left(n+1\right)\alpha-
c_{sc}^{2}}{n}+{\mathpzc{U}}^{2}\right)\frac{1}{c_{sc}}\right]\\\
\left\\{\frac{3}{2\mathpzc{x}_{c}}-\frac{1}{2\mathscr{F_{G}}\mathpzc{(x,J)}}\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}\right\\}\left\\{\frac{\left(n+1\right)c_{sc}\left(c_{sc}^{2}-\alpha\right)}{\left(1-n\right)c_{sc}^{2}+\alpha\left(n+1\right)}\right\\}$,
$E=\left\\{\frac{3}{\mathpzc{x}_{c}}-\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{dx}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}\right\\}\frac{\mathpzc{U}_{c}c_{sc}\left(1-n\right)}{{\left\\{\left(1-n\right)c_{sc}^{2}+\alpha\left(n+1\right)\right\\}}}+\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}$

$+\left\\{\frac{1}{{\mathscr{F_{G}}\mathpzc{(x,J)}}^{2}}\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right)^{2}_{\mathpzc{x}=\mathpzc{x}_{c}}-\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\left(\frac{d^{2}\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}^{2}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}-\frac{3}{\mathpzc{x}_{c}^{2}}\right\\}\frac{\mathpzc{U}_{c}^{2}}{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$

$F=\frac{\lambda\alpha_{ss}}{\mathpzc{x}_{c}^{2}\mathpzc{U}_{c}}\left\\{\frac{1}{\mathscr{F_{G}}\mathpzc{(x,J)}}\left(\frac{d\mathscr{F_{G}}\mathpzc{(x,J)}}{d\mathpzc{x}}\right)_{\mathpzc{x}=\mathpzc{x}_{c}}-\frac{5}{\mathpzc{x}_{c}}\right\\}\left\\{\frac{\left(n+1\right)\alpha-
c_{sc}^{2}}{n}+\mathpzc{U}_{c}^{2}\right\\}$.

While we are working with a particular type of EoS we supply $n$ and $\alpha$,
i.e., vanishing denominator, $D\left(\mathpzc{U}_{c},c_{sc},n,\alpha\right)=0$
will be reduced to an equation $D\left(\mathpzc{U}_{c},c_{sc}\right)=0$, which
tells us the relation between $\mathpzc{U}_{c}$ and $c_{sc}$, on the other
hand vanishing numerator, i.e.,
$N\left(\lambda_{c},\mathpzc{x}_{c},\mathpzc{J},c_{sc},n,\alpha\right)=0$ will
be reduced to $N\left(\lambda_{c},\mathpzc{x}_{c},c_{sc}\right)=0$ once we
provide the information about the accreting fluid and the rotation of the
black hole. We will choose where the sonic point will be formed and adjust the
value of angular momentum at the critical point which ultimately form an
algebraic equation of $c_{sc}$, $N\left(c_{sc}=0\right)$ From there we can
easily solve $c_{sc}$ and hence calculate the value of $\mathpzc{U}_{c}$.
These will be used as the initial values to solve (14).

##  3 Solutions and Graphical Interpretations

Fig 1.1(a)

Fig 1.1(b)

Figure 1.1 : Plots of accretion and wind branches for $log(c_{s})$ (log of
sound speed) vs $log(\mathpzc{x})$ for non-rotating black hole with different
viscosity parameter (showed in different colours). (a) For adiabatic fluid
accretion (b) For MCG accretion.

Fig 1.2(a)

Fig 1.2(b)

Figure 1.2 : Plots of accretion and wind branches for $log(c_{s})$ (log of
sound speed) vs $log(\mathpzc{x})$ for rotating black hole
($spin\leavevmode\nobreak\ parameter\leavevmode\nobreak\ \mathpzc{J}=0.5$)
with different viscosity parameter (showed in different colours). (a) For
adiabatic fluid accretion (b) For MCG accretion.

Fig 1.3(a)

Fig 1.3(b)

Figure 1.3 : Plots of accretion and wind branches for $log(c_{s})$ (log of
sound speed) vs $log(\mathpzc{x})$ for rotating black hole
($spin\leavevmode\nobreak\ parameter\leavevmode\nobreak\ \mathpzc{J}=0.9$)
with different viscosity parameter (showed in different colours). (a) For
adiabatic fluid accretion (b) For MCG accretion.

Fig 2.1(a)

Fig 2.1(b)

Figure 2.1 : Plots of accretion and wind branches for
$log(\frac{\lambda}{\lambda_{k}})$ vs $log(\mathpzc{x})$ for non-rotating
black hole with different viscosity parameter (showed in different colours).
(a) For adiabatic fluid accretion (b) For MCG accretion.

Fig 2.2(a)

Fig 2.2(b)

Figure 2.2 : Plots of accretion and wind branches for
$log(\frac{\lambda}{\lambda_{k}})$ vs $log(\mathpzc{x})$ for rotating black
hole ($spin\leavevmode\nobreak\ parameter\leavevmode\nobreak\
\mathpzc{J}=0.5$) with different viscosity parameter (showed in different
colours). (a) For adiabatic fluid accretion (b) For MCG accretion.

Fig 2.3(a)

Fig 2.3(b)

Figure 2.3 : Plots of accretion and wind branches for
$log(\frac{\lambda}{\lambda_{k}})$ vs $log(\mathpzc{x})$ for rotating black
hole ($spin\leavevmode\nobreak\ parameter\leavevmode\nobreak\
\mathpzc{J}=0.9$) with different viscosity parameter (showed in different
colours). (a) For adiabatic fluid accretion (b) For MCG accretion.

Fig 3.1(a)

Fig 3.1(b)

Figure 3.1 : Plots of accretion and wind branches for
$log(\frac{\mathpzc{U}}{c_{s}})$ (Mach number) vs $log(\mathpzc{x})$ for non-
rotating black hole with different viscosity parameter (showed in different
colours). (a) For adiabatic fluid accretion (b) For MCG accretion.

Fig 3.2(a)

Fig 3.2(b)

Figure 3.2 : Plots of accretion and wind branches for
$log(\frac{\mathpzc{U}}{c_{s}})$ (Mach number) vs $log(\mathpzc{x})$ for
rotating black hole ($spin\leavevmode\nobreak\ parameter\leavevmode\nobreak\
\mathpzc{J}=0.5$) with different viscosity parameter (showed in different
colours). (a) For adiabatic fluid accretion (b) For MCG accretion.

Fig 3.3(a)

Fig 3.3(b)

Figure 3.3 : Plots of accretion and wind branches for
$log(\frac{\mathpzc{U}}{c_{s}})$ (Mach number) vs $log(\mathpzc{x})$ for
rotating black hole ($spin\leavevmode\nobreak\ parameter\leavevmode\nobreak\
\mathpzc{J}=0.9$) with different viscosity parameter(showed in different
colours). (a) For adiabatic fluid accretion (b) For MCG accretion.

Solving this problem numerically, we plot $log(c_{s})$ vs $log(\mathpzc{x})$
curves in figure 1.1(a)-(b), 1.2(a)-(b) & 1.3(a)-(b).

In fig 1.1(a)-(b), we have plotted $log(c_{s})$ vs $log(\mathpzc{x})$ for
nonrotating black holes. Among these, 1.1a is for adiabatic accretion and
1.1(b) is for Chaplygin gas as accreting agent. Purple solid, Blue dashed and
Red double dot dashed lines are for the accretion. Yellow solid, green dashed
and blue double dot dashed lines are for the wind branch.

First, we will discuss the graphs of adiabatic flow for different viscosity.
Very far from the black hole, sonic speed in wind flow is less than that of
acrretion flow. As we move towards the black hole, these speeds increase. But
as we cross the critical point, sonic speed for wind becomes higher than that
of accretion. In inner zone of the disc, sonic speed increases very slowly.
Pseudo Newtonian formalism is observed to empower sonic speed to diverge to
the speed of light in the near vicinity of Schwarzschild radius. However, in
contrast to 3-velocities, in extreme relativistic regimes, sonic speed is
supposed to converge to the upper limit $c/\sqrt{3}$ [49]. J. Fukue has
proposed a modified sonic speed for Paczinsky Wiita PNF as $c_{s}^{PN}$ and
forced the new sonic speed to converge to the said upper limit as approaching
to the Schwarzschild radius [50]. Redefining a sonic speed for Mukhopadhyay
PNF is kept for a future scope. For now, we consider the form proposed by
Fukue and redraw the curves. These are found to stay below the upper limit of
$c/{\sqrt{3}}$.

Speed of light in vacuum $c=299792458\leavevmode\nobreak\ m/s$. This points
that the upper limit of sonic speed should be
$c/{\sqrt{3}}=173085256.3\leavevmode\nobreak\ m/s$. Now taking log, we get
$log(c/{\sqrt{3}})=8.238260076$. In the figures of $log(c_{s})$ vs
$log(\mathpzc{U})$, the sound speed for our accretion system lies below the
limit of $c/\sqrt{3}$.

Conclusions regarding the temperature of the flow can be drawn from this
information. As a flow preceedes towards the central gravitating object,
temperature $\sim{c_{s}}^{2}$ increases, but slowly. On the other hand, wind,
i.e., the flow which is going out of the disc is with very high temperature
near the black hole and looses the temperature quickly as it moves outwards.
As we increase the viscosity, the slope of sonic speed increment of both the
branches increases towards black hole. This means angular momentum transport
is a quick process generated from viscosity and temperature increases rapidly
due to this.

Now, looking towards fig.1.1(b) graphs for Chaplygin gas, we do not exactly
see the previous case explained. Both the sonic speed decreases as we go
towards the black hole. Sonic speed for accretion decrease slowly whereas wind
decreases rapidly. Wind sonic speed is higher than accretion sonic speed if
$\mathpzc{x}>{\mathpzc{x}}_{c}$ and opposite in inner region.

Physically, this interpretes that the temperature falls as we move towards the
central black hole via an accretion branch. Energy to be fallen is radiated
away before it falls if the accretion fluid is of Chaplygin gas type. Dark
energy wind shows abrupt increment in temperature as it goes off far from the
black hole. When dark energy is winded, it carries much energy out of the
disc. If viscosity is increased, a threshold drop in accreting sonic speed is
observed. It moves nearer with the increment in viscosity.

Fig. 1.2(a) is for sonic speed for a accretion disc where the central black
hole is rotating with spin parameter $\mathpzc{J}=0.5$. For same gravitational
scenario, dark energy accretion cases are enlisted in fig. 1.2(b).

With nonzero spin of black hole, sonic speed for adiabatic accretion and wind
is rapidly increasing than a non rotating black hole as we go towards with
black hole. On the other hand, these values decreases rapidly if dark energy
accretes in. So fall of energy is higher for a rotating case than a non
rotating counterpart. Same tendencies are carried for fig. 1.3(a)-(b) where
$\mathpzc{J=0.9}$. It is seen rotation increases the flow of energy towards
black hole for adiabatic case and decreases the flow and vacuums it outward as
dark energy is accreting. Viscosity helps in this activities. As we move far
from the central black hole, pressure increases as density is low for dark
energy while enough gravitational pull is not working. As a result, the sonic
speed increases suddenly at a larger distances.

To study and understand the disc properties deeply, we will study the ratio of
specific angular momentum to Keplerian angular momentum in fig. 2.1(a)-(b)(for
nonrotating black holes), fig. 2.2(a)-(b) ($\mathpzc{J}=0.5$), fig. 2.3(a)-(b)
($\mathpzc{J}=0.9$). Non rotating adiabatic black hole accretions shows both
accretion and wind's ratio of $\frac{\lambda}{\lambda_{k}}$ decrease as we
move far from the black hole. This supports normal angular momentum transport
method. If $\mathpzc{J}=0.5$, accretion $\frac{\lambda}{\lambda_{k}}$ reduces
as $\mathpzc{x}$ increases and after reaching a local minima, it increases and
becomes equal to 1 at $\mathpzc{x}=\mathpzc{x_{1}}$ (say). Here the disc is
terminated. In figure 2.1(a), $\mathpzc{J}=0.9$, same pattern is followed and
$\frac{\lambda}{\lambda_{k}}$ turns equal to 1 at
$\mathpzc{x}=\mathpzc{x_{2}}$ (say) where $\mathpzc{x_{2}}>\mathpzc{x_{1}}$.
We find the reason if spcific angular momentum becomes equal to Keplerian
angular momentum, the disc is truncated hereafter. No physical disc is found
beyond that. Viscosity increases the accretion rotation as compared to the
Keplerian angular momentum in outer disc regions and this causes a centrifugal
outward disturbances and thus the disc terminates. Nonrotating nonviscous
Chaplygin gas case shows almost the same nature of 2.1a. Difference is noted
once viscosity is introduced, the ratio starts to fall in the outward region.
More viscosity forces to drop the accretion angular momentum to fall in nearer
distance to the black hole. So dark energy opposes the rotation of the disc as
we go outwards and hence the disc is truncated. Rotating black holes
($\mathpzc{J}=0.5$ and $0.9$) show almost similar tendencies in fig.
2.2(a)-(b) and fig. 3.2(a)-(b). But rotation drops down the dark energy
$\frac{\lambda}{\lambda_{k}}$ in nearer regions than nonrotating cases.

Now we will study the variations in Mach number with distance from the black
hole in the fig. 3.1(a)-(b), 3.2(a)-(b) and 3.3(a)-(b).

Compressibility of a fluid flow can be best understood by the measure of its
Mach number. Mach number's graph are not making after each other like
$\frac{\lambda}{\lambda_{k}}$ curves for nonrotating, nonviscous cases for
adiabatic and Chaplygin gas flow. Accretion Mach number increases steeply as
we go towards the central black hole. On the other hand wind Mach number
increases as we go far from the black hole. Accretion is transonic in inner
region whereas wind shows the opposite nature. As the rotation is increased,
adiabatic wind turns hypersonic in the outer regions. Besides Chaplygin gas
accretion and wind both turn supersonic or hypersonic in the inner disc part.
It is to be noted that density changes faster than the change of velocity by a
factor of Mach number squared.

In general, when a system starts to move with hypersonic speed, above shock
created by this is followed by an increment of density. This is again followed
by a decrease in volume and gives birth of small shock stand off distance.
Changes in entropy across the shock also increases and a highly vertical flow
should be observed. Relation between radial inward speed, Mach number and
density of the flow can be related as $-(Mach\leavevmode\nobreak\
number)^{2}\frac{dv}{v}=\frac{d\rho}{\rho}$. Hence when Mach number is small
(in farthest accretion and nearest wind) compressibility can be ignored. On
the other hand compressibility withold a good effect if Mach number is high
(inner accretion and outer wind).

Mathematically, the slim disk equations form an eigen value problem, with the
eigen value being the angular momentum of matter crossing the black hole
horizon. The physical reason for the eigen value nature of the problem is that
the black hole accretion must necessarily be transonic, with the sonic point
being the critical (saddle) point of the slim disk differential equations. The
regularity conditions at the sonic point assure that its location is very
close to ISCO (Innermost Stable Circular Orbit).

To locate the density jumps across the shock, we take the help of the article
[36]. At each radius, the value of $\partial_{\phi}\rho({\mathpzc{x}},\phi)$
can be calculated to identify the region where the value exceeds one. Number
of shocks, jump across the shock, azimuthal width of the wake etc can be
watched. In this present article we try not to illustrate the issue to avoid
unnecessary volumetric increase of the article.

##  4 Brief Discussions and Conclusions

In this article, we study different properties of viscous dark energy
accretion. Variations of sonic speed, specific angular momentum to Keplerian
angular momentum ratio and Mach number, i.e., fluid flow to sonic speed ratio
are plotted and analysed. To set the relativistic attractive force's effects
towards the central super massive black hole, we have chosen the pseudo
Newtonian force method and precisely a particular pseudo Newtonian potential
which at a distance $\mathpzc{x}$ from the center of a black hole spinning
with parameter $\mathpzc{J}$, measures the force applicable on a particle.
Navier Stokes equations, equation of continuity and equation of state are used
to form a system of three simultaneous differential equations to study a thin
viscous accretion disk around a rotating super massive black hole. We collect
the initial conditions from the continuity and physical properties of the flow
in the disc. We study the sonic speed vs radial distance curves in three
different sets of two rows of three graphs. Sets are for different spin
parameters $\mathpzc{J}=0$, $0.5$ and $0.9$. Rows are for adiabatic and
Chaplygin gas cases. Columns are for non viscous, slightly viscous and highly
viscous cases. For sonic speed, we observe accretion and wind both sound
speeds increase as we go towards the black hole. Wind sonic speed is less than
accretion sound speed before the critical point
($\mathpzc{x}>\mathpzc{x_{c}}$) and opposite if we are inside it
($\mathpzc{x}<\mathpzc{x_{c}}$). The whole scenario is completely opposite if
Chaplygin gas acrretion is taken. As we go towards a black hole the speed
decreases in both the branches. Fall in wind branch is steeper as we go
towards the black hole. Keeping spin parameter same, if we increase viscosity,
we find a fall in accretion branch and this threshold like fall moves towards
the black hole as we increase viscosity. Except the fall, there is a general
verage tendency of fall in accretion branch sonic speed. This tendency
increases as we increase spin.

We have seen in the works [34, 35, 36] that the accretion disc is truncated as
Chaplygin gas is accreted. The cause is better understood if we study the
$\frac{\lambda}{\lambda_{k}}$ curves. For adiabatic case this ratio turns
equal to $1$ as we go far from the black hole in the wind branch. For
Chaplygin gas this ratio falls abruptly. It shows Chaplygin gas resists even
the disc to rotate. Resistance increases as we increase spin as well as
viscosity.

Next we move towards the study of Mach number. Mach number is high in inner
region of accretion and outer region of wind branch. For adiabatic fluid, wind
Mach number is sufficiently high as we go far from the black hole. This
signifies a low accretion speed. For dark energy wind, this is not
sufficiently high and is truncated at a sufficiently lower value near to $1$.
This signifies ocurrance of high speed at outer wind branch for dark energy
accretion. Dark energy accretion, as we go nearer to the central gravitating
object, shows high Mach number. This indicates a weak inflow indeed.

Achkowledgements :  This research is supported by the project grant of
Government of West Bengal, Department of Higher Education, Science and
Technology and Biotechnology (File no:- $ST/P/S\&T/16G-19/2017$). SD and PB
thank to Department of Higher Education and Technology, Government of West
Bengal for awarding Swami Vivekananda Merit Cum Means Junior Research
Fellowship. RB thanks IUCAA for providing Visiting Associateship.

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