Theoretical investigation for the relation (supermassive black hole mass)–(spiral arm pitch angle): a correlation for galaxies with classical bulges

In this work, the determination of the masses of supermassive black holes (SMBHs) and the properties of their host spiral galaxies are focused for the purposes of constraining scaling relations and with the aim of understanding the role of SMBHs in the evolution of galaxies. The measurements of SMBHs mass for a sample of 40 spiral galaxies were studied by applying indirect techniques (the SMBHs mass versus stellar/gas velocity dispersion relation). In addition, spiral arm pitch angle of a sample of nearly face-on spiral galaxies were measured using IRAF (The Image Reduction and Analysis Facility) and two-dimensional Fast Fourier Transform (2DFFT) program. Finally, we present a new correlation between spiral arm pitch angle (a measure of the tightness of spiral structure) and the mass of supermassive black holes (BHs) in the nuclei of classical galaxies.


Introduction
Supermassive black holes (SMBHs) are common at the center of all or most of galaxies [1,2]. As observed at high sensitivities and resolution with the Hubble Space Telescope (HST). In addition, their masses are in the range of hundreds of thousands to billions of solar masses [3][4][5]. Over the last decade, studies of galaxies have led to the discovery that there are many strong or tight correlations locally between the masses of the SMBHs and the global properties of the spheroid components of their hosts. This suggests an intriguing link between galaxy formation and SMBH growth. As a result, astrophysicists believe that the energy released by growing SMBHs play important role in shaping the properties of the structure of their host galaxies [6] There is increasing evidence which indicates that relationships between the mass of the SMBH and almost or all the possible parameters of the host galaxy bulges. This suggests that SMBHs play an important role in galaxy formation.
Most galaxy bulges contain a central SBH whose mass strongly correlates with stellar velocity dispersion(σ * ) within the effective radius (r e ) ( M BH -σ * ); [2,7], with the bulge luminosity or spheroid luminosity of the galaxy (L bul ) (M-L Bulge ) [1,3 ,8,9,10 ], with the bulge mass (M bulge ) [1,9] and circular velocity [11], with the galaxy light concentration [12], the dark matter halo [11], with the effective radius [13], the Sersic index [14], with the gravitational binding energy and gravitational potential, combination of bulge velocity dispersion, effective radius and/ or intensity [15], with the radio core length [16] and the inner core radius [17]. As mentioned above, all scaling laws have led previous authors to the conclusion that SMBH, growth and bulge formation regulate each other [18]. That means that mass of the SMBH is somehow tied to the structural parameters of the rest of the galaxy.
The correlations between the mass of supermassive black holes (SMBHs) and properties of their host galaxies helped to understand the mechanism of nuclear energy by the formation and evolution of BHs [19].
In this work, first, we used a correlation to estimate the SMBH mass compared with other correlations (M BH -σ*), where σ* is the bulge velocity dispersion.
Second, we used the correlation with spiral arm pitch angle [1], to study a correlation between SMBHs mass and spiral arm pitch angle for bulges, pseudobulges, barred, nonbarred, non-AGN (Active galactic nuclei), and AGN galaxies.

Measuring spiral arm pitch angle
One of the more interesting methods to find SMBH masses in late-type galaxies use the relationship between SMBH mass in the nuclei of disk galaxies and spiral arm pitch angle (P) (a measure of the  Seigar et al. (2008) found that SMBHs are linked by a strong correlation with P. Additionally, a correlation between P and rotation curve shear (S) was discovered [1] Previous studies described logarithmic spiral in polar coordinates [20]. This is a special kind of spiral curve that describes the arm in disk galaxies: where r is radius, θ is central angle, r 0 is initial radius when θ = 0, and pitch angle is -90 ≤ Φ ≤ 90.
Because the spiral arm pitch angle has been shown to be independent of the wavelength at which it is measured, multi-band images were used to determine it for our sample of spiral galaxies [20]. The amplitude of each Fourier component is given by: Where r and θ are polar coordinates, I(ln r, θ) is the intensity at position (ln r, θ), m represents the number of arms or modes, and p is the variable associated with the pitch angle P defined by P =−(m/p max ).
IRAF (The Image Reduction and Analysis Facility) was used to determine the ellipticity values and major-axis position angle in order to deproject the 3.6 μm galaxy images to fully face-on by assuming circular outer isophotes. ELLIPSE in IRAF was used to derive inclination angle (α); [22], which is defined by: where (a) is the semi-major axis and (b) is the semi-minor axis. Where the value 0 о describes a faceon galaxy and 90 о describes an edge-on galaxy.

Measurement SMBHs using (M BHσ * ) relation:
The M BH -σ * relation supports the notion of regulated formation mechanisms and co-evolution for the galaxy's central black hole mass and the bulge velocity dispersion [23].
The M BH -σ * relation is one of the most common techniques used to estimate the mass of SMBH at the center of a spiral galaxy [15]. This is done by measuring the velocity dispersion of stars in the galactic bulge. This method was based on the observation that supermassive black hole masses correlate with the dispersion velocity of the surrounding stellar component (bulge) of spiral galaxies  [24]. Because BH masses found in late-type spirals and spheroidal galaxies have lower mass, we used the M BH -σ * relation since it has the least scatter [18].

Results and discussion
In this work, the correlation between SMBH mass and spiral arm pitch angle is re-examined. This Berrier et al (2013) that a significant correlation exists between SMBH masses and the spiral arm pitch angles [1,17,26 ]. On the other hand, our results are similar to some extent with Seigar's relation [17]. This part is dedicated to the results and discussion of the correlations between M BH and spiral arm pitch angle, where the study of the correlations is considered a benchmrk to research BH vs. host galaxy co-evolution, as well as the study of the location in the scaling relations for the classical bulges and pseudobulges of spiral galaxies. These results proved that pseudobulges do not follow the same M BH -P relation as classical bulge.
This means that M BH -P relation has the same properties of M BH -σ * relation as pseudobulge and classical bulge galaxies -due to the existence of a strong correlation between bulge central velocity dispersion and spiral arm pitch angle [1,17].
According to Pearson's linear correlation coefficient for a correlation between SMBH and P, the M BH − P relation for classical bulge galaxies is the tightest correlation that we measured; and the results is significantly consistent with the result of Seigar et al. (2008). On the contrary, theM BH -P for pseudobulges galaxies is significantly below the M BH −P relation.
The indirect MBH measurements for spiral galaxies would be reliable to study the important and analyse the behaviour of (M BH -P) for pseudobulges and classical bulge galaxies.  The best-fitting lines are shown for this diagram: respectively, for non-barred, Non-AGN, and AGN galaxies. use a new techniques of measuring the spiral arm pitch angle of the host galaxy such as the twodimensional fast Fourier transform decomposition program by several bands (K, 3.6um), and they found M BH -P relation [1,18,24].
The most prominent results of previous studies are as follows: 1-They found the differences of the best-fit parameters. of these, only a few were done on K-band and 3.6 µm which were available to us for comparison with our results. In Table ( [19,20], and our work.
In all cases, the previous studies found different values for the slope of the galaxies in the M BH -P correlation.

Conclusion
Despite the enormous interest in studying the scaling relations between SMBHs and the structural parameters of the host galaxies, based on direct and indirect methods of the different bands to measure In this work, the following conclusions are made: 1-Scaling relations were studied between SMBHs mass in the center of spiral galaxies, and spiral arm pitch angles.
2-Our findings are in agreement with Seigar et al. (2008) who found a strong correlation between M• and P using both direct and indirect methods.
3-Finally, we found the relation between SMBHs and spiral arm pitch angle (P), and the bestfitting lines of regression were: