Study of temperature of black holes in terms of Chandrasekhar limit

Stephen Hawking gave the formula for the temperature for the black holes as  . In the present research article, we have converted this formula in terms of Chandrasekhar limit [M ch ] and also calculated their values for different test black holes existing in XRBs and


INTRODUCTION
A black hole is a solution of Einstein's gravitational field equations in the absence of matter that describes the space time around a gravitationally collapsed star. Its gravitational pull is so strong that even light cannot escape from it [1,2].
In classical theory black holes can only absorb and not emit particles. However, it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature 6 ( / ) 10 MM , where  is surface gravity of black holes and k is the Boltzmann constant. This thermal emission leads to show a decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 10 15 gram would have evaporated by now [3]. Hawking (1974Hawking ( , 1975 introduced what is now called Hawking radiation as the effective black body radiation from a black hole in terms of the 4th power of the black hole temperature and the Stefan-Boltzmann constant [3,4]. Stephan Hawking provided a theoretical argument for its existence in 1974, and sometimes also after the physicist Jacob Bekenstein who -44-predicted that black holes should have a finite, non-zero temperature and entropy and also stated that it is proportional to the black hole area A [5].
Later on, Bardeen, Carter and Hawking performed calculations using a semi-classical approximation, putting Bekenstein conjecture on a firm basis. They established that the black hole temperature is proportional to its surface gravity [6]. Silva proposed intuitive derivations of the Hawking temperature and the Bekenstein-Hawking entropy of a Schwarzschild black hole [7].
Ved Prakash et al. has discussed the statistical analysis of lifetime and temperature of the black holes have been studied existing in X-ray binaries and active galactic nuclei [8]. Mehta et al. derived an expression for the variation of temperature of the black holes with respect to mass and also calculated their values of different test black holes existing only in X-ray binaries [9].
The present research article discusses the conversion of Stephen Hawking temperature of black holes as given by formula and also calculates their values of different black holes existing in XRBs and AGN.

DISCUSSION
Silva proposed intuitive derivations of the Hawking temperature of a black hole of mass in terms of fundamental parameters is given by the following equation [7]: The solar mass M  in terms of S.I. units is given by 1.99x10 30 kg [2]. The temperature of a black hole of mass M in terms of solar mass can be written by using the values of , c, and k G in the above equation and solving, we have

DATA IN SUPPORT OF MASS OF BLACK HOLES
There are two categories of black holes classified on the basis of their masses clearly very distinct from each other, with very different masses M ~ 5 20 M ʘ for stellarmass black holes in X-ray binaries and M ~ 10 6 -10 9.5 M ʘ for super massive black holes in galactic nuclei [11][12][13].
The mass of spinning black holes in terms of Chandrasekhar limit corresponding to the masses M ~ 5 20 M ʘ for stellarmass black holes in X-ray binaries are 3 M ch to 13 M ch and for the masses M ~ 10 6 -10 9.5 M ʘ in super massive black holes in the active galactic nuclei are 1 x 10 6 M ch to 3 x 10 9 M ch .

RESULTS AND DISCUSSION
In the present work, we have converted the formula for temperature of black holes in terms of Chandrasekhar limit and calculated their values for the different test black holes in X-ray binaries (XRBs) and Active galactic nuclei (AGN). To discuss the temperature of black holes with the mass (M ch ), graphs have been plotted between: (i) the mass of different test black holes(M ch ) and their corresponding values of temperature in XRBs ( Fig. 1 ) (ii) mass of different test black holes(M ch ) and their corresponding values of temperature using logarithmic scale in AGN (Fig. 2 ) -49- The Figure 1 shows that the temperature of black holes decreases with the increase of the mass of the different test black holes in XRBs, while in the Figure 2, the variation of temperature of black holes with the mass in the case of AGN is not be the same to that of XRBs, but differing in the some manner.
In this case, the temperature of black holes decreases with increase in the mass of black holes. From the graph, it is clear that the nature of variation of temperature of black holes with their corresponding mass in AGN is repeated in the same manner after certain different mass. These certain different masses are 10 6 M ch , 10 7 M ch , 10 8 M ch and 10 9 M ch .
These data of masses of black holes are in the multiple of ten (10) and hence these masses have special characters.

CONCLUSIONS
During the study of present research work, we can draw the following conclusions such as: (i) The temperature of black holes decreases with the increase in the mass of the different test black holes in XRBs showing that heavier mass of black hole has smaller temperature and vice-versa.
(ii) The nature of variation of the temperature of black holes with their corresponding mass in AGN is repeated in the same manner after certain different mass.
(iii) The mass of black holes in the multiple of ten (10) has special character.