# Influence of mass of black holes on radio/X-ray correlation diagram.

Avishek Dusoye
<EMAIL_ADDRESS>
###### Abstract

The radio/X-ray correlation diagram, for Black Hole X-ray Binaries (BHXBs) in
the hard state, depicts the connection that might exists between the radio
jets and X-ray emitting accretion discs. The current version of the
radio/X-ray correlation diagram shows two populations of BHXBs. These two
populations evolve along two different correlation tracks ($F_{rad}=\kappa
F_{x}^{b}$), namely the standard track and the outliers. Over the past years,
the key question has been to explain the existence of these two tracks. In
this paper, we investigate the impact of the black hole mass on the
radio/X-ray correlation for a sample of 17 BHXBs. We are led to conclude at
least one of the following with full consideration of large uncertainities in
mass estimates:(i) Most of the reported mass estimates of black holes are
incorrect or insufficiently accurate to infer their impact on radio/X-ray
correlation diagram. (ii) The estimated radio luminosities and X-ray
luminosites are still not reliable enough for two reasons. One reason is due
to the lack of associated errors in observational data. Another is that some
sources might still transit from one track to the other. (iii) The mass of BH
has a significant influence on which track, the source belongs to, on a
radio/X-ray correlation diagram.

###### keywords:

X-rays: binaries, Black hole physics, Accretion.

††pagerange: 1– 9††pubyear: 2015

##  1 Introduction

Black hole X-ray binaries (BHXBs) are stellar binary systems consisting of a
black hole (BH) and a companion star. They are known to produce X-ray emission
through the accretion of mass from the companion star onto the black hole via
an accretion disc (Remillard & McClintock, 2006), as well as radio emission
originating from their jets (Fender, 2006). Each component of a BHXB
(accretion disc and jets) dominates a specific frequency range, as observed in
its spectrum (Markoff et al., 2003). The jets and accretion disc appear
pronounced in radio and X-ray respectively. Therefore an approach to
investigate the connection between jets and accretion disc for these BHXBs is
to study the correlations between radio and X-ray emission, which should be
observed quasi-simultaneously.

In 2003, during the hard state of GX 339-4, Corbel et al. (2003) discovered a
very strong correlation between radio fluxes at 8.6 GHz using the Australian
Telescope Compact Array (ATCA), and X-ray fluxes in the band 3-9 keV using the
the Rossi X-ray Timing Explorer (RXTE) and BeppoSAX. This correlation takes
the form of a powerlaw,

$F_{rad}=\kappa\times F_{x}^{b}\;,$ (1)

with $b=0.706\pm 0.011$ and where $F_{rad}$ is the $8.6$ GHz radio flux
density, and $F_{x}$ is the $3-9$ keV X-ray flux density. The validity of this
correlation was extended to a larger sample of 10 BHXBs (Gallo et al., 2003).
The correlation between the radio luminosity and the X-ray luminosity for most
of the 10 BHXBs in the hard state seems to be consistent with a power law
slope of $b\simeq 0.7$. This suggested a universal correlation which could
imply that most BHXBs might behave similarly in terms of disc-jet coupling.

The work was extended then to larger mass scale using a compiled sample of 100
Active Galactic Nuclei (AGN) with supermassive BHs and 8 galactic BHXBs
(Merloni et al., 2003). The parameters considered for the AGN were similar to
BHXBs. However, in addition to the radio luminosities $L_{R}$
($\rm{erg\,s^{-1}}$) and X-ray luminosities $L_{X}$ ($\rm{erg\,s^{-1}}$), the
mass of the BH ($\rm{M_{\odot}}$) has been taken into account to study the
correlation, as it cannot be ignored anymore when comparing stellar mass BHs
and supermassive BHs. The results of Merloni et al. (2003) seem to indicate
the AGN behave very similarly to BHXBs when it comes to disc-jet coupling.
Equation (2) is a general form of Equation (1) when the mass scale is
considered,

$\log L_{R}=(0.6_{-0.11}^{+0.11})\log L_{X}-(0.78_{-0.11}^{+0.09})\log
M_{BH}+7.33_{-4.07}^{+4.05}\;.$ (2)

Since the value of $b\simeq 0.7$ in Equation (1) is within the range of values
of the coefficient for the first term in Equation (2), the validity of the
universal correlation at a even larger mass scale seems to be maintained. This
led to the definition of the fundamental plane of (hard state) black hole
activity. Plotkin et al. (2012) improved on the original work of Merloni et
al. (2003) to obtain the refined equation below describing the fundamental
plane of BH activity. The latter used the reciprocal Equation (3):

$\log L_{X}=(1.45_{+0.04}^{-0.04})\log L_{R}-(0.88_{+0.06}^{-0.06})\log
M_{BH}-6.07_{+1.10}^{-1.10}.$ (3)

Over the past ten years, many BHXBs have been detected with jets of lower
luminosity compared to what is predicted by the previous correlation (Equation
1) for the same X-ray luminosity (accretion-powered). Therefore this suggests
that some BHXB may have either a large range of jet power or a large range of
X-ray luminosity for similar accretion rates. More and more BHXBs in outburst
have been found to be located below the standard universal correlation. The
few examples include XTE 1650-500 (Corbel et al., 2004), IGR J17497-2821
(Rodriguez et al., 2007) and Swift J753.5-0127 (Soleri et al., 2010). Given
the fact that another population of BHXBs seems to situated off from the
standard correlation, these sources are sometimes referred as “outliers”.

Statistical evidence has been provided by Gallo et al. (2012) to discard the
hypothesis of a single universal correlation between the radio and X-ray
luminosity of hard state BHXBs. Corbel et al. (2013) updated the radio/X-ray
correlation plot with the new data set of GX 339-4 and some additional
sources. The correlation index of $b\sim 0.7$ still remains consistent for the
standard correlation on the updated plot. For the outliers, Coriat et al.
(2011b) found a steeper correlation with index of $b\sim 1.4$, using a large
sample of observations for H1743-322. Below a given luminosity, H1743-322 was
found to make a transition from the steep lower track to the upper (standard)
track of the correlation. Similarly since then, few other “outliers” sources
have been found to make this transition between the two correlation tracks.
These sources are MAXI J1659-152 and XTE J1752-223 (Jonker et al., 2012; Ratti
et al.,2012)

###  1.1 Definition of problem

The existence of dual correlation tracks completely turns down the original
paradigm of the scale-invariant and universal accretion disc jet coupling
model. Since this discovery, a key question is to explain the existence of the
two tracks. One school of thought proposes that black holes with the same
accretion inflow, can undergo outflows which can vary over a large range of
power. This suggests that some parameters of the BHXBs are actually adjusting
the mode of accretion disc-jet coupling. The BH is characterized by three
parameters which are its mass, spin and charge. The charge is considered
unimportant since the BH is electrically neutral on average on a macroscopic
scale. In addition, there are also parameters in the BHXBs as a binary system
such as orbital period, Roche lobe size and inclination which could play a
role. Therefore there is a need to constrain those possible parameters in
order to provide a plausible explanation for these two tracks.

The radio normalisation (parameter $\kappa$ in Equation 1) serves as an
indicator of the position of a source on the radio/X-ray correlation diagram.
By investigating the relations between the radio normalisation of a sample of
sources with any of the parameters mentioned above, one can study the
influence of this parameter on the dual track issue. Fender et al. (2010) show
evidence for no correlation between the radio normalisation and BH spin
assuming the measurements are correct. Soleri & Fender (2011) have
investigated the other fixed binary parameters such as orbital period,
accretion disc size, and inclination for 17 BHXB systems. In all cases, no
evidence for a correlation between these parameters and the radio
normalisation has been found. This suggests that they do not play a
significant role in the existence of the two correlation tracks.

In this paper, we investigate for plausible explanations for the existence of
the dual correlation tracks. This was done by taking the mass estimates for a
sample of 17 known BHXBs and attempt to correlate them with radio/X-ray
normalisation. Following the work of Soleri & Fender (2011) and Fender et al.
(2010), we have adopted the same methodology of using the radio/X-ray
normalisation to investigate the mass of the BH (analogous to spin, orbital
period, and other binary parameters) as a key parameter.

There will be two interpretations which can follow, once the results are made.
Firstly, for a given accretion rate, jets are fainter due to reduce power or
reduce radiative efficiency. Soleri & Fender (2011) mentioned that many BHXBs
have been found to produce fainter jets at a given accretion-powered
luminosities than expected from the earlier correlation, and suggested that
BHXBs with similar accretion flows can produce a wide range of jet power due
to some parameter or factor (possibly the BH mass). Secondly, for a given
accretion rate, the accretion flow is radiatively more efficient. The
radiative efficiency of an accretion flow is given by
$\epsilon=\frac{L}{\dot{M}c^{2}}$, where $L$ is the total luminosity and
$\dot{M}$ is the accretion rate (Shakura & Sunyaev, 1973). Coriat et al.
(2011b) proposed that those hard state BHXBs which are radiating efficiently
occupies the outliers’ track whereas the BHXBs on the standard track are
actually radiating inefficiently. Consequently, it would be interesting to see
if the radiative efficiency is influenced by the BH mass.

###  1.2 Mass measurement techniques

Accurate measurements of BH masses are critical to verify whether the position
of sources on radio/X-ray correlation diagram has dependence on their BH mass.
The measurement of mass for a BH is not trivial and still have large
uncertainities associated to it. From the literature, we have compiled mass
estimates of a sample of BHXBs, which were obtained from two main measurement
techniques:

(i) the Dynamical Method; The method is based on Kepler’s third law of motion.
Since BHXBs are similar to single-lined spectroscopic binaries, only the
radial velocity curve of the companion star (extracted from a timing series of
spectra) is available. From the radial velocity curve, the orbital period
($P_{orb}$) and the radial velocity semi-amplitude ($K_{c}$) may be extracted
and used in the mass function, which is defined by Equation (4). This is used
to obtain the mass of the BH, $M_{X}$. The mass function is given by,

$f(M)=\frac{K_{c}^{3}P_{orb}}{2\pi
G}=\frac{M_{X}^{3}\sin^{3}i}{(M_{X}+M_{c})^{2}}=\frac{M_{X}\sin^{3}i}{(1+q)^{2}}\>.$
(4)

In Equation (4), $M_{c}$ is the mass of the companion star, $q$ is the mass
ratio $q=M_{c}/M_{X}$ and G is the Newtonian gravitational constant. This
expression assumes a circular orbit at an inclination $i$. If the inclination
of the system is not constrained, the mass function provides a solid lower
limit to the mass of the black hole even for a mass of the companion $M_{c}=0$
and an edge-on geometry (Casares & Jonker, 2013).

(ii) the Spectral-timing correlation scaling technique; The inherent features
of the X-ray spectrum and the rapid variability of X-ray luminosity can be
correlated to determine the mass of the black hole and its distance from us
(Shaposhnikov et al., 2010). This technique of analysis requires (1) a
reference source, whose BH mass and distance is known, and (2) a large
scalable number of observations which records how the target and the reference
source evolves in its spectra over time as they make spectral transitions.
There exists a spectral index associated to each frequency at which quasi
periodic oscillations (QPO) occur. By plotting an empirical function to the
X-ray variablity data, one can infer both a gradient (index) and a
normalisation of the power law function used to fit the data. The index and
normalisations are correlated to find any coherence arising from the QPO and
these are known as index$-$normalisation correlation patterns. The spectral
index$-$QPO frequency is the frequency of the quasi periodic oscillation at
which the spectral index is being taken or considered. The spectral index–QPO
frequency and index–normalisation correlation patterns are used to calculate
the scaling factors with respect to the reference source. The BH mass can then
be extracted since the QPO frequency is assumed to depend on the BH mass only.
But this is a highly debated assumption since we still do not understand the
exact origin of QPOs.

##  2 Results and analysis

In this section, we present our results on whether the BH mass in a BHXB has
an influence on the track it follows on the radio/X-ray correlation diagram.
This was done by comparing the radio normalisation of a sample of sources
against their corresponding estimated BH mass. Futhermore, we reviewed the
dependence of binary parameters (orbital period and inclination) as in Soleri
& Fender (2011), for our sample of BHXBs which includes additional and
recently monitored sources.

The quasi-simultaneous radio and X-ray observations of the sources used in
Corbel et al. (2013) were revisited and updated. The X-ray luminosity is in
the 1-10 keV band. Sources having only an upper limit recorded ($\leq
10^{33}\;\rm{erg\,s^{-1}}$ in luminosity) are excluded. Table 1 shows 17
BHXBs, for which the radio/X-ray normalisation values were determined. This
was done by minimising our fitting function
($\log\,L_{rad}=\log\,\kappa+\log\,L_{X}^{0.6}+\log\,10^{34}$), which will
best fit the evolution of both fluxes for all the observations of each source.
The value of index $b=0.6$ has been fixed. The last term $\log\,10^{34}$ is a
normalisation factor for the X-ray luminosity, similar to Fender et al.
(2010). The values $\log\kappa$ are recorded in Table 1. The best-fit
parameters are determined using Chi-square value, which is defined as follows:

$\chi^{2}=\sum_{i=1}^{N-1}D_{i}^{2}=\sum_{i=1}^{N-1}\dfrac{({y_{i}-f(x_{i})})^{2}}{\sigma_{yi}^{2}+\sigma_{xi}^{2}f^{\prime^{2}}(x_{i})}$
(5)

The values of $y_{i}$, $x_{i}$, $\sigma_{yi}$ and $\sigma_{xi}$ correspond to
the radio luminosity, X-ray luminosity, and their corresponding errors
respectively for the $i^{th}$ observation of one source. The uncertainties on
the best-fit parameters are derived from the diagonal elements of the
covariance matrix. All the sources whose data have been used are referenced in
Table 1.

Figure 1: Radio/X-ray normalisation values against inclination (top panel) and
orbtial period (bottom panel). The Spearman correlation coefficients for these
two data sets do not indicate any significant correlation between the
parameters (see text). Figure 2: Radio/X-ray normalisation as a function of BH
mass for the full sample of sources (see Table 1). The Spearman correlation
coefficient is $\Omega\sim 0.28\pm 0.25$.

###  2.1 Inclination and orbital period

Using these values of radio normalisation, we can probe which parameter
influences the BHXB to follow a particular track. Figure 1 shows that there is
no clear correlation between the radio/X-ray normalisation with neither the
binary inclination nor the orbital period (from literature; see Table 1). The
Spearman correlation coefficient was calculated for the inclination against
radio/X-ray normalisation data set (top panel of Figure 1) and found to be
$-0.20\pm 0.55$. Similarly the Spearman correlation coefficient was calculated
to be $-0.27\pm 0.40$ for the orbital period against radio/X-ray normalisation
data set (bottom panel of Figure 1). Our results are consistent with the work
of Soleri & Fender (2011) since the values of the coefficient do not imply a
correlation between the studied parameters.

###  2.2 Black hole mass estimates - Full sample

In order to search for any link between BH mass and radio/X-ray normalisation,
we have compiled the most up-to-date mass estimates from the literature for
the 17 BHXBs. The values and corresponding references are listed in Table 1.
Figure 2 shows the radio/X-ray normalisation as a function of the BH mass for
all the source of our sample without applying any selection criteria. The
Spearman correlation coefficient for the Figure 2 is $\Omega\sim 0.28\pm 0.25$
(without selection criteria). It can noticed that if we include the sources
with non-dynamical mass estimates (see Figure 3), the HMXBs, and sources with
quiescent measurement only, the correlation between the normalisation and the
mass is much weaker or non-existent. Therefore we require to narrow down this
full sample to a selected sample, which is more reliable

###  2.3 Black hole mass estimates - Selected sample

When our sample is restricted using well defined selection criteria, the
correlation becomes clearer. The selection criteria which have been adopted
are discussed in Section 3.2. Figure 4 shows our correlation diagram for the
radio/X-ray normalisation $\log(\kappa)$ against the BH mass but chosen within
the set of full selection criteria mentioned above. The Spearman correlation
coefficient was found to be $\Omega\sim 0.78\pm 0.01$ which means that the set
of selection criteria has improved this correlation.

It should be noted that the already known dependence of the radio and X-ray
luminosities on the BH mass (i.e. from the fundamental plane of BH
activity;Merloni et al.,2003) could slightly bias our study. Therefore, we
remove this bias by subtracting this mass dependence that originates from the
fundamental plane. The Corrected normalisation $\eta$ is simply defined as:

$\eta=log(\kappa)+Klog(M/M_{\odot})\;,$ (6)

where $K=\frac{\xi_{M}}{\xi_{R}}$ is the ratio of the fundamental plane
coefficients as in Plotkin et al. (2012); $\xi_{M}=-0.88\pm 0.06$ and
$\xi_{R}=1.45\pm 0.04$. Having subtracted the mass dependence from the
fundamental plane, we can now test for the existence of a correlation between
the BH mass and the radio/X-ray normalisation.

Figure 5 shows our correlation diagram for the corrected normalisation $\eta$
against the mass of BH for our sample of 11 sources as in Table 1 (within the
same selection criteria). The Spearman correlation coefficient was calculated
which gives $\Omega\sim 0.73\pm 0.01$. Although by comparing the corrected
values and the actual values of radio/X-ray normalisation in Table 1, one can
notice that those corrections, as indicated in Equation 6, are actually small.
The correlation still remains when the correction from the fundamental plane
of BH activity is taken into account. It should also be noted that the
uncertainties of the fundamental plane coefficient (Equation 3) have been
propagated into the error of the corrected normalisations. The later are
therefore larger than the uncorrected normalisations which reduces the
Spearman correlation coefficient. However, the correlation coefficient
$\Omega\simeq 0.73\pm 0.01$ still indicates a significant correlation between
the two tested parameters. This result should be further confirmed once
additional and more accurate mass measurements will be available.

##  3 Discussion

###  3.1 Methodology

Following Soleri & Fender (2011) and Fender et al. (2010), we have adopted the
same methodology of using the radio/X-ray normalisation to investigate the
influence of the mass of the BH. This methodology seems valid as long as we
have (i) accurate BH masses, and (ii) reliable radio/X-ray normalisation
values.

The BH mass estimates have a cubic dependence on the inclination (Equation 4).
Therefore BH mass estimates may not be reliable, given the presence of
systematic errors in the measurement of inclination. The two main sources of
systematics on light curves are superhump modulation and contamination from
rapid aperiodic variability. If there is a large spread of inclination values,
there will be more larger range of BH masses Casares & Jonker (2013). The BH
mass estimates has also two limitations for High Mass X-ray Binaries (HMXBs).
Firstly, the value of mass ratio ‘q’ is very small (given the spectral type of
their companion star; Podsiadlowski et al.2003), and therefore the BH mass
would not be very accurate. Secondly, the mass transfer is powered by stellar
winds Ninkov et al. (1987).

The estimated radio/X-ray normalisations could be debated for its accuracy due
lack of errors in observational data especially where only one observation was
recorded for one source e.g. A0620-00 and IGRJ17091-3624, IGRJ17177-3656 and
XTEJ1720-318. An error corresponding to 10$\%$ of observed fluxes was assigned
to these 4 sources which corresponds to the typical (conservative) accuracy
obtained with the RXTE satellite. It is also worth discussing the significance
for the numerical value of the estimated radio/X-ray normalisation. The exact
value of the normalisation is highly dependent on the number of data points
(observations) used to calculate it. This is particularly true since we are
fitting a power-law with a fixed index $b=0.6$ which does not match the
observed slope of the lower track. Given the observations of each source is
different, the number of data points associated to an observed source is also
different. Hence the estimated radio/X-ray normalisation is not uniformly
significant, even over the selected sample. Therefore, the correlation we
observe should rather be considered qualitatively rather than quantitatively
in the sense that sources evolving on the lower track tend to have less
massive BH. Those facts should be taken into consideration before taking any
stance regarding this correlation and any inference about it.

###  3.2 Selection Criteria of sources:

  1. 1.

No sources having just quiescent measurement or single datapoint at low
luminosity
Single observation at low luminosity or even quiescent measurements do not
show and cannot predict the evolution of the source on a radio/X-ray
correlation. It can be biased to associate the sources which are at low
luminosity to lie on either the upper or the lower track. The reason is that
we have also observed cases where a source has changed from one track to
another e.g. H1743-322. We excluded such sources in our analysis because they
cannot be evaluated on which track they lie.

  2. 2.

Low-mass X-ray binaries only
We have excluded Cyg X-1 as it is a HMXB with a very peculiar behaviour partly
due to the fact that it transfers mass through winds. Since Cyg X-1 is also
the only HMXB of our sample and it might be biased to include it.

  3. 3.

Dynamically estimated masses
We additionally excluded sources for which the mass measurement has been
obtained through the spectral-timing correlation technique (see Section 1.2).
The latter relies on assumptions about a direct link between the BH mass and
the spectral and timing properties which are not demonstrated yet. The dynamic
method which simply relies on direct observation of lightcurves, is a more
reliable and accurate method to obtain the BH masses. Figure 3 shows the
sources whose mass were estimated dynamically, and the sources whose mass were
estimated through spectral-timing correlation.

Figure 3: Radio/X-ray normalisation as a function of BH mass for the full
sample of sources where the measurement technique of the BH mass is specified.
These are by (i) Dyanmical method (indicated with black symbols) and (ii)
Spectral-timing correlation scaling method (indicated with red pointers).

###  3.3 Inference of results

We have found a good correlation for the mass of BH with the radio/X-ray
normalisation (see Figure 4), where the Spearman rank correlation coefficient
is found to be $\Omega=0.78\pm 0.01$ ($\Omega=0.0$ means no correlation and
$\Omega=\pm 1.0$ implies an ideal correlation). When we reviewed the
correlation of radio/X-ray normalisation with inclination and orbital period,
we find the values $\Omega\simeq-0.20\pm 0.55$ and $\simeq-0.27\pm 0.40$
respectively. The results are consistent with the work of Soleri & Fender
(2011) since the values of the coefficient do not indicate a significant
correlation. The Spearman correlation coefficient is considered as a reliable
tool to verify the strength of a correlation. Therefore, this suggests that
the mass of BH has a significant influence on which track (standard or
outliers) the source belongs to on a radio/X-ray correlation diagram.

Our sample consists of only 17 sources (and 10 sources once the selection
criteria have been applied) and may not be fully representative of the general
behaviour of all BHXBs that exist. However, we can deduce the following. From
the correlation diagram in Figure 4, one can infer that sources with more
(less) massive BH tend to have higher (lower) radio/X-ray normalisation
$\log(\kappa)$. This may imply that BHXBs with a massive BH will tend to
populate the standard track whereas BHXBs with a less massive BH will tend to
populate the outliers’ track.

We find that, at a given X-ray luminosity, BHXBs with a more massive BH (on
the standard track) are observed with higher emission of radio luminosities.
Similarly, BHXBs with less massive BH (on the outliers’ track) are observed
with lower emission of radio luminosities. This may imply that BHXBs with more
(less) massive BH will have more (less) powerful jets.

An alternative possibility to the less/more powerful jets (as mentioned above)
is that hot inner accretion flow seems to be radiatively more (less) efficient
if the BHXB has a less (more) massive accreting BH. This should be valid, at
least, within a given range of luminosity (typically above $10^{-3}L_{edd}$),
since at low luminosity, sources have been observed to change track (e.g.
H1743-322). At the same time, although there exists transitions between the
two correlation tracks, further investigations are needed to understand why
such a small difference of mass could lead to such noticeable differences in
correlation index on the radio/X-ray correlation diagram. A theoretical
approach would be necessary to investigate this possibility but that it is out
of the scope of this paper.

In addition, Xie & Yuan (2012) states that the radiative efficiency of
accretion flow increases with increasing accretion rate, which gets
intensified by viscous interactions. However, since the viscosity parameter of
the disc is not related to mass of BH, it could suggest that viscosity plays
no critical role in explaining the dual radio/X-ray correlations, if the mass
dependence reported here holds.

##  4 Conclusion

We investigated three parameters (inclination, orbital period, mass of compact
core) with our sample of 17 BHXBs in an attempt to explain the dual tracks on
radio/X-ray correlation diagram. Figure 1 shows that there is no clear
correlation of these radio/X-ray normalisation values with inclination and
orbital period respectively. Our results are consistent with the work of
Soleri & Fender (2011). Figure 4 shows all our sources for which we have a
radio/X-ray normalisation $\log(\kappa)$ and mass of the compact object. The
Spearman correlation coeficient for the Figure 4 is $\Omega\sim 0.78\pm 0.01$
(selected sample). We are led to conclude at least one of the following with
full consideration of large uncertainities in mass estimates:(i) Most of the
reported mass estimates of black holes are incorrect or insufficiently
accurate to infer their impact on radio/X-ray correlation diagram. (ii) The
estimated radio luminosities and X-ray luminosites are still not reliable
enough for two reasons. One reason is due to the lack of associated errors in
observational data. Another is that some sources might still transit from one
track to the other. (iii) The mass of BH has a significant influence on which
track, the source belongs to, on a radio/X-ray correlation diagram.

## Acknowledgments

This research has been funded by NRF/SKA Masters Scholarship Grant No. 82750,
Reference: SKH1208229663. The author also acknowledges the financial support
from NRF Grants No.120390, Reference: BSFP190416431035, and No.120396,
Reference: CSRP190405427545, and No 101775, Reference: SFH150727131568.

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Figure 4: Radio/X-ray normalisation against the mass of the accretor for
sources listed in Table 1 chosen within selection criteria (see text). The
Spearman rank correlation coefficient is calculated as $\Omega\sim 0.78\pm
0.01$. Figure 5: The correlation diagram for the corrected normalisation
$\eta$ against the mass of the accretor for sources listed in Table 1 chosen
within selection criteria. These radio/X-ray normalisations are corrected from
any mass dependence from fundamental plane of BH activity (see Equation 6).
The Spearman rank correlation coefficient for the corrected version is found
to be $\Omega\sim 0.73\pm 0.01$.

Table 1: Parameters for the sample of black hole X-ray binaries used in this
study; The sources are identified by their names. $\eta$ and $\log\kappa$,
represent the corrected and uncorrected radio/X-ray normalisation.
$Err_{\eta}$ and $Err_{\log\kappa}$ are the associated errors. $M_{X}\pm
dM_{X}$ denotes the mass of the source X. $i\pm di$ denotes inclination and
$D\pm dD$ represents the distance to the source. The references for mass
estimates as well as radio and X-ray luminosity are under RefA and RefB
columns respectively.

Sources | $\eta$ | $Err_{\eta}$ | $\log\kappa$ | $Err_{\log\kappa}$ | Porb/hrs | $M_{X}/M_{\odot}$ | $dM_{X}$ | i$/\circ$ | di | D/kpc | dD | RefA | RefB
---|---|---|---|---|---|---|---|---|---|---|---|---|---
GX339-4 | 28.39 | 0.07 | 28.848 | 0.002 | 42.14 |  $>$5.73 | 0.29 | 40 | 20 | 8 | 0 | [1] | (a)(b)
V404Cyg | 28.15 | 0.274 | 28.76 | 0.22 | 155.31 | 10.00 | 2.0 | 55 | 4 | 2.39 | 0.14 | [2][3] | (d)
GROJ1655-40 | 27.59 | 0.39 | 28.09 | 0.38 | 62.92 | 6.60 | 0.5 | 68.7 | 1.5 |  $<$1.7 | - | [4] | (f)(g)
IGRJ17091-3624 | 28.27 | 0.33 | 28.56 | 0.04 | - | 3.00 | 1.5 | - | - | 20.0 | - | [5][6] | (h)
A0620-00 | 28.20 | 0.10 | 28.70 | 0.07 | 7.75 | 6.60 | 0.3 | 51 | 0.9 | 1.2 | 0.4 | [7] | (i)
4U1543-47 | 28.58 | 0.12 | 29.17 | 0.05 | 26.95 | 9.42 | 0.97 | 21 | 2 | 7.5 | 0.5 | [8][20] | (j)
GROJ0422+32 | 27.26 | 0.18 | 27.62 | 0.04 | 5.09 | 3.97 | 0.95 | 45 | 2 | 2.49 | 0.3 | [9] | (j)
H1743-322 | 27.61 | 0.26 | 28.29 | 0.17 | - | 13.30 | 3.2 | 70 | - | 9.1 | 1.5 | [10] | (c)
GRS1915+105 | 28.81 | 0.13 | 29.42 | 0.11 | 812.40 | 10.10 | 0.6 | 70 | 2 | 11.2 | 0.8 | [11] | (e)
SwiftJ1753.5-0127 | 27.42 | 0.19 | 27.77 | 0.06 | 3.20 | 4.0 | 1.0 | 85 | 0 | 8 | - | [18] | (k)(l)(m)
XTEJ1118+480 | 27.93 | 0.32 | 28.46 | 0.30 | 4.08 | 7.55 | 0.65 | 74 | 5.5 | 1.71 | 0.05 | [12] | (j)(n)
XTEJ1550-564 | 27.92 | 0.13 | 28.50 | 0.10 | 36.96 | 9.10 | 0.6 | 72 | 5 | 4.1 | 0.8 | [13] | (j)
XTEJ1650-500 | 27.02 | 0.21 | 27.479 | 0.008 | 7.69 | 5.65 | 1.65 | 50 | 3 | 2.6 | 0.7 | [19][20] | (o)
XTEJ1752-223 | 27.79 | 0.14 | 28.385 | 0.003 | - | 9.50 | 1.5 | - | - | 8.0 | - | [14] | (p)(q)
GRS1758-258 | 26.86 | 0.38 | 27.33 | 0.05 | 442.80 | 6.00 | +3.4-1.45 | - | - | 8.5 | - | [15] | (r)
MAXIJ1659-152 | 27.52 | 0.15 | 28.309 | 0.004 | - | 20.00 | 3 | - | - | 8.0 | 0.0 | [16] | (s)
CygX-1 | 27.51 | 0.15 | 28.22 | 0.12 | 134.40 | 14.80 | 1 | 35 | 5 | 2.1 | 0.4 | [17] | (b)

RefA: (1):Muñoz-Darias et al. (2008), (2): Casares (1996), (3):Wagner et al.
(1992), (4): Shahbaz (2003), (5): Altamirano et al. (2011), (6): Rebusco et
al. (2012), (7): Cantrell et al. (2010), (8): Orosz2003, (9): Gelino &
Harrison (2003), (10): Shaposhnikov & Titarchuk (2009) , (11): Steeghs et al.
(2013), (12): Khargharia et al. (2013), (13): Orosz et al. (2011b),
(14):Shaposhnikov et al. (2010), (15): Bezayiff (2006), (16): Shaposhnikov et
al. (2011), (17): Orosz et al. (2011a), (18) Neustroev2014, (19): Orosz2004,
(20): Ozel2010. The references of the radio and X-ray luminosities, RefB:
(a)Corbel2003 (b)Corbel et al. (2013) (c)Coriat et al. (2011b) (d)Corbel et
al. (2008) (e)Rushton et al. (2010) (f)Coriat et al. (2011a) (g)Calvelo et al.
(2010) (h)Rodriguez et al. (2011) (i) Gallo et al. (2006) (j) Gallo2003
(k)Cadolle Bel et al. (2007) (l)Soleri et al. (2010) (m) Coriat et al. (2011b)
(n) Fender et al. (2010) (o)Corbel et al. (2004) (p)Brocksopp et al. (2013)
(q)Ratti et al. (2012) (Gallo2003) (s) Jonker et al. (2012).