Issue 
A&A
Volume 551, March 2013



Article Number  A4  
Number of page(s)  15  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201220378  
Published online  07 February 2013 
Accretion disks around black holes in modified strong gravity
^{1} Instituto Argentino de Radioastronomía, Camino Gral Belgrano Km 40 C.C.5, 1984 Villa Elisa, Buenos Aires, Argentina
email: danielaperez@iar.unlp.edu.ar
^{2} Facultad de Ciencias Astronómicas y Geofísicas, UNLP, Paseo del Bosque s/n CP, 1900 La Plata, Buenos Aires, Argentina
email: romero@iar.unlp.edu.ar
^{3} Departamento de Física Teórica, Instituto de Física, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, Maracanã, Rio de Janeiro, CEP: 20550900, Brasil
email: sepbergliaffa@gmail.com
Received: 13 September 2012
Accepted: 18 November 2012
Context. Stellarmass black holes offer what is perhaps the best scenario to test theories of gravity in the strongfield regime. In particular, f(R) theories, which have been widely discus in a cosmological context, can be constrained through realistic astrophysical models of phenomena around black holes.
Aims. We aim at building radiative models of thin accretion disks for both Schwarzschild and Kerr black holes in f(R) gravity.
Methods. We study particle motion in f(R)Schwarzschild and Kerr spacetimes.
Results. We present the spectral energy distribution of the accretion disk around constant Ricci scalar f(R) black holes, and constrain specific f(R) prescriptions using features of these systems.
Conclusions. A precise determination of both the spin and accretion rate onto black holes along with Xray observations of their thermal spectrum might allow to identify deviations of gravity from general relativity. We use recent data on the highmass Xray binary Cygnus X1 to restrict the values of the parameters of a class of f(R) models.
Key words: black hole physics / accretion, accretion disks / gravitation
© ESO, 2013
1. Introduction
General relativity (GR) is consistent, in some cases with good precision, with observational results (see for instance Will 2006). However, an energy momentum tensor representing exotic matter (loosely called “dark energy”, e.g. Li et al. 2011) must be introduced in the right hand side of Einstein’s equations to fit the currently available data when these are interpreted in the framework of the standard cosmological model (based on GR). Dark energy can be modelled by a cosmological constant, or by a scalar field with an equation of state given by p = ω_{DE}ρ, where ω_{DE} < −1/3 (Biswas et al. 2010a,b). None of these descriptions is free of problems, since the energy density associated with the cosmological constant that is inferred from astronomical observations is approximately 120 orders of magnitude lower than the value predicted by field theory (e.g. Weinberg 1989; Capozziello & Faraoni 2010), whereas the scalar field has features that are at odds with the scalar fields of particle physics (Sotiriou & Faraoni 2010).
A different approach to explain the cosmological data is to modify the field equations of the gravitational field, in such a way that the ensuing theory differs from Einstein’s in the lowcurvature regime. Since there is no a priori fundamental reason to restrict the gravitational Lagrangian to a linear function of the Ricci scalar R (see for instance Magnano et al. 1987), more general theories can be formulated using nonlinear functions of this scalar. The socalled f(R) theories (e.g. Capozziello & Faraoni 2010) were first used to mimic the inferred accelerated expansion of the universe by Capoziello (2002). Currently there is in the literature a handful of f(R)models in agreement with available data (De Felice & Tsujikawa 2010).
Although the present revival of f(R) theories is mainly due to their use in the description of phenomena that take place for low values of the Riemann curvature, these theories have also been applied to gravity in the opposite regime. As there is no direct evidence of the behaviour of the gravitational field for very large values of the curvature, the early universe and compact objects offer the possibility to find deviations from GR. Among the studies in modified gravity in the strong regime, we can mention the successful inflationary model based on the R + αR^{2} theory (Starobinsky 1980), and the related studies of reheating (Motohashi & Nishizawa 2012) and particle production (Arbuzova et al. 2012) in the early universe. Also of importance is the treatment of neutron stars (Cooney et al. 2010) and black hole solutions.
Different aspects of black hole physics in f(R) theories have been discussed in the literature by Psaltis et al. (2008a), Hendi & Momeni (2011), Myung (2011), Myung et al. (2011), Moon et al. (2011a,b), Hendi et al. (2012), and Habib Mazharimousavi et al. (2012). Static and spherically symmetric black hole solutions were obtained via perturbation theory by de la CruzDombriz et al. (2009), whereas black holes with these symmetries have been studied by means of a nearhorizon analysis by Perez Bergliaffa & De Oliveira (2011). Finally, f(R)KerrNewman black holes solutions with constant Ricci scalar have been recently studied by Cembranos and collaborators (Cembranos et al. 2011).
From an astrophysical point of view, thin accretion for the Schwarzschild spacetime in f(R)gravity has been discussed by Pun et al. (2008), without including the expected spectra of concrete astrophysical black holes or any comparison with observational data.
In the present work we investigate the existence of stable circular orbits in Schwarzschild and Kerr f(R) spacetimes with constant Ricci scalar and analyze the main features of accretion disks around these black holes. In particular, we present temperature and spectral energy distributions for PageThorne disks, and compare the results with those obtained using the standard ShakuraSunyaev model.
The paper is organized as follows. In Sect. 2 we provide a brief review of f(R) theories of gravity. Circular orbits in both Schwarzschild and Kerr f(R) spacetimes with constant Ricci scalar are studied in Sect. 3. Section 4 is devoted to the calculation of the properties of accretion disks in these spacetimes. Consequences for some specific prescriptions for the function f are discussed in Sect. 5. We close with some considerations on the potential of astronomical observations to test the strongfield regime of gravity.
2. f(R) gravity
In f(R) gravity, the Lagrangian of the HilbertEinstein action, given by: (1)is generalized to: (2)where g is the determinant of the metric tensor, and f(R) is an arbitrary function of the Ricci scalar. In the metric formalism the field equations are obtained varying Eq. (2) with respect to the metric: (3)where R_{μν} is the Ricci Tensor, □ ≡ ∇_{β}∇^{β}, f′(R) = df(R)/dR, and the energy momentum tensor is defined by: (4)Here, ℒ_{m} stands for the matter Lagrangian.
Equations (3) are a system of fourthorder nonlinear equations for the metric tensor g_{μν}. An important difference between them and the Einstein field equations is that in f(R) theories the Ricci scalar R and the trace T of the energy momentum tensor are differentially linked, as can be seen by taking the trace of Eq. (3), which yields: (5)Hence, depending on the form of the function f, there may be solutions with traceless energymomentum tensor and nonzero Ricci scalar. This is precisely the case of black hole spacetimes in the absence of a matter source.
Notice that in the case of constant Ricci scalar R_{0} without matter sources, Eqs. (3) can be rewritten as:
where: (6)and, by Eq.(5): (7)Hence, in this case, any f(R) theory is formally, but not physically, equivalent to GR with a cosmological constant given by Eq. (6)^{1}.
We shall investigate the existence of stable circular orbits in Schwarzschild and Kerr f(R) spacetimes with constant Ricci scalar in the next section.
3. Circular orbits around a black hole in f(R)
3.1. f(R)Schwarzschild spacetime
The Schwarzschild spacetime metric in f(R) theories with constant Ricci scalar R_{0} takes the form (Cembranos et al. 2011): (8)where R_{0}, given by Eq. (7), can take, in principle, any real value. Since we are looking for spacetime metrics that may represent astrophysical black holes, we shall select those values of R_{0} that lead to acceptable solutions.
The radius r_{0} of the horizon follows from the condition g_{00}(r_{0}) = 0. From Eq. (8), the values of r corresponding to the event horizon satisfy: (9)In terms of the following adimensional quantities: where r_{g} = GM/c^{2}, this equation takes the form: (12)We show in Fig. 1 the Ricci scalar as a function of the radial coordinate of the event horizon. We see that for R_{0} ∈ (0,4/9) there is an inner black hole event horizon and an outer cosmological horizon, whereas for R_{0} ≤ 0 there is only one black hole event horizon. The event and cosmological horizons collapse for R_{0} = 4/9 and for larger values of the Ricci scalar naked singularities occur and hence, no black holes are possible^{2}. We shall, then, restrict ourselves to the study of trajectories for R_{0} ∈ ( − ∞,4/9] .
Fig. 1 Ricci scalar as a function of the radial coordinate of the event horizon in f(R)Schwarzschild spacetime. 

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3.1.1. Equations of motion and effective potential in f(R)Schwarzschild spacetime
The geodesic equations in the metric given in Eq. (8) can be obtained by means of the EulerLagrange equations using the Lagrangian: (13)where ẋ^{μ} ≡ dx^{μ}/dσ, and σ is an affine parameter along the geodesic x^{μ}(σ). The resulting geodesic equations for t and φ are (setting θ = π/2): where k and h are constants. An equation for r that is simpler than the one obtained from the Lagrangian follows from the modulus of the 4momentum p, given by g_{μν}x^{μ}x^{ν} = ϵ^{2}, where ϵ^{2} = c^{2} for massive particles, and ϵ^{2} = 0 for photons. It takes the form: (16)with ṫ and given by Eqs. (14) and (15), respectively. The set of Eqs. (14), (15), and (16) completely determine the motion of a particle in the f(R)Schwarzschild spacetime.
3.1.2. Trajectories of massive particles
Equations (14), (15), and (16) can be used to obtain the socalled energy equation (e.g. Hobson et al. 2006): (17)The constant k is defined as k = E/m_{0}c^{2}, where E represents the total energy of the particle in its orbit, and m_{0}c^{2} its rest mass energy. The constant h stands for the angular momentum of the particle per unit mass. From Eq. (17) we can identify the effective potential per unit mass as: (18)The extrema of the effective potential are obtained by looking for the roots of the derivative of the latter equation with respect to the radial coordinate. In terms of x, R_{0}, and the adimensional angular momentum per unit mass of the particle h = h(cr_{g})^{1}, this reads: (19)The derivative of Eq. (19) with respect to x gives: (20)where we have replaced h by (Harko et al. 2009): (21)where x_{c} corresponds to the radius of a circular orbit. We have performed the numerical calculation of the extrema of the effective potential for different values of R_{0}.
Location of the event and cosmological horizon, and of the innermost and outermost stable circular orbits for R_{0} > 0 in f(R)Schwarzschild spacetime.
Fig. 2 Function given by Eq. (23). The absolute maximum corresponds to x = 15/2 and R_{0} = 2.85 × 10^{3}. 

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Fig. 3 Effective potential for different values of R_{0} > 0 and h in f(R)Schwarzschild spacetime. The dots indicate the location of the innermost stable circular orbit. 

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For R_{0} > 0, as shown by Stuchlík et al. (1999) and Rezzolla et al. (2003) in Schwarzschildde Sitter spacetime, stable circular orbits exist for values of the specific angular momentum that satisfy: (22)where h_{isco} and h_{osco} stand for the local minimum and local maximum of the specific angular momentum at the inner and outer marginally stable radii. From Eq. (20) we see that the existence and location of the circular orbits depend on the Ricci scalar R_{0}. By equating Eq. (19) to zero and isolating R_{0}, we obtain the Ricci scalar as a function of the radial coordinate of the circular orbits: (23)In Fig. 2 we show the plot of the latter equation. We see that there is an upper limit for the Ricci scalar, R_{0} = 2.85 × 10^{3}, for which circular orbits are possible. In Fig. 3 we plot the effective potential as a function of the radial coordinate. The dots indicate the location of the innermost stable circular orbits. The corresponding values of the event and cosmological horizons, radii of the innermost and outermost stable circular orbits, for six different values of R_{0} ∈ (0,2.85 × 10^{3}) are shown in Table 1. We see that for increasing values of the Ricci scalar the event horizon becomes larger than in Schwarzschild spacetime in GR as well as the location of the innermost stable circular orbit.
Fig. 4 Ricci scalar as a function of the radial coordinate of the event horizon (line) and of the Ricci scalar as a function of the radial coordinate of the innermost stable circular orbits (dashed line) for R_{0} ∈ [ − 1.5,0.45] in f(R)Schwarzschild spacetime. 

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The extrema of the effective potential for R_{0} < 0 are all located outside the event horizon, as shown in Fig. 4. The value of the radial coordinate for the event horizon is less than 2 (i.e. smaller than for Schwarzschild black holes in GR). The location of the innermost stable circular orbit is closer to the horizon than that of the Schwarzschild solution in Einstein’s gravity. The limit R_{0} → − ∞ in Eq. (20) yields: (24)The plot of the effective potential corresponding to the values of the parameters of Table 2 is shown in Fig. 5.
Location of the event horizon, and of the innermost stable circular orbit for R_{0} < 0 in f(R)Schwarzschild spacetime.
Fig. 5 Effective potential for different values of R_{0} < 0 and h in f(R)Schwarzschild spacetime. The dots indicate the location of the innermost stable circular orbit. 

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3.2. Kerr spacetime in f(R) theories
The axisymmetric, stationary and constant Ricci scalar geometry that describes a black hole with mass, electric charge and angular momentum was found by Carter (1973), and was used to study f(R) black holes by Cembranos et al. (2011). The form of the metric is the following: (25)where: Here M and a denote the mass and angular momentum per unit mass of the black hole, respectively, and R_{0} is given by Eq.(7).
Fig. 6 Ricci scalar as a function of the radial coordinate of the event horizon for R_{0} ∈ [ − 0.3,1] and a = 0.99 in f(R)Kerr spacetime. 

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Because of the constancy of p^{t} and p^{φ} along the trajectories, and of the reflectionsymmetry of the metric through the equatorial plane, the orbit of any particle with initial condition p^{θ} = 0 will remain in the plane π/2, where the metric has the form: (30)Here, If R_{0} → 0, Eq. (30) represents the Kerr spacetime metric in GR as expected.
The equation that yields the position of the event horizon is obtained by setting 1/g_{rr} = 0: (33)In terms of x, R_{0}, and a ≡ a(r_{g})^{1}, this equation takes the form: (34)In Fig. 6 we plotted the Ricci scalar as a function of the radial coordinate of the event horizon for R_{0} ∈ [−0.3,1] and a = 0.99 (i.e. a nearly maximally rotating black hole, such as Cygnus X1, Gou et al. 2011). If R_{0} ∈ (0,0.6] , there are 3 event horizons: the inner and outer horizons of the black hole and a cosmological horizon; for R_{0} > 0.6 there is a cosmological horizon that becomes smaller for larger values of R_{0}. If R_{0} ∈ (−0.13,0) there are 2 event horizons. For R_{0} ≤ −0.13 naked singularities occur. In the following we shall analyse the existence of stable circular orbits for R_{0} ∈ (−0.13,0.6] .
3.2.1. Equations of motion and effective potential in f(R)Kerr spacetime
In order to obtain an expression for the effective potential, we make use of the invariant length of the 4momentum p: (35)where ϵ^{2} = c^{2} for massive particles and ϵ^{2} = 0 for photons. Since we are only interested in trajectories on the equatorial plane, we set p_{θ} = 0, and Eq. (35) gives: (36)where: Substituing Eqs. (37), (38), and (39) into (36), the equation for ṙ^{2} takes the form: (40)The contravariant components of the spacetime metric given by Eq. (30) are: Hence, the energy equation for a massive particle is given by: (45)where the effective potential is: (46)and: (47)
3.2.2. Equatorial circular orbits of massive particles
If a massive particle is moving in a circular orbit of radius r_{isco}, the value of the effective potential at any point of the orbit satisfies the equation: (48)so, (49)For a stable orbit, the equation: (50)must also be satisfied. As shown in Sect. 3.2, there are black holes if R_{0} ∈ ( − 0.13,0.6] .
Radii of event horizons and circular orbits for a f(R)Kerr black hole of angular momentum a = 0.99, for some values of R_{0} < 0.
Numerical calculations of the radius of the innermost stable circular orbit for several values of R_{0} < 0, with a = 0.99, show that the stable circular orbits lay outside the event horizon (see Table 3). Notice that as the value of the scalar decreases, the radius of the innermost stable circular orbit becomes smaller. The radius of the innermost stable circular orbit in Kerr spacetime in GR (r_{isco} = 1.4545r_{g}, in the prograde case) is always larger than in f(R)Kerr. In Fig. 7, the effective potential that correspond to the values of Table 3 are shown.
Location of the event and cosmological horizon, and of the innermost and outermost stable circular orbits for R_{0} > 0 and a = 0.99 in f(R)Kerr spacetime.
We follow Stuchlík & Slaný (2004) to study the existence of stable circular orbits for R_{0} ∈ (0,0.6] . The specific angular momentum of a massive particle in a corotating circular orbit yields (Stuchlík & Slaný 2004): (51)We see from the latter equation that circular orbits must satisfy the following two conditions: (52)which is the same for f(R)Schwarzschild spacetime with positive Ricci scalar, and: (53)The minimum and maximum of Eq. (51) give the values of the specific angular momentum that correspond to the innermost and outermost stable circular orbit respectively, once the angular momentum a of the black hole and R_{0} are fixed. We show in Table 4 the values of such radii for different values of R_{0}, and a = 0.99, and in Fig. 8 the corresponding plot of the effective potential. As expected, the radius of inner circular orbit is larger than in Kerr spacetime in GR. We also found, by equating to zero the derivative of Eq. (51), that stable circular orbits only exist if R_{0} ∈ (0,1.45 × 10^{1}).
Fig. 7 Effective potential as a function of the radial coordinate (R_{0} < 0, a = 0.99), in f(R)Kerr spacetime. The dots indicate the location of the innermost stable circular orbit. 

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Fig. 8 Effective potential as a function of the radial coordinate (R_{0} > 0, a = 0.99), in f(R)Kerr spacetime. The dots indicate the location of the innermost stable circular orbit. 

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The analysis of the circular orbits presented in this section will be applied next to the construction of accretion disks around black holes.
4. Accretion disks in strong gravity
4.1. Standard disk model in general relativity
The first realistic model of accretion disks around black holes was formulated by Shakura & Sunyaev (1973). They considered that the matter rotating in circular Keplerian orbits around the compact object loses angular momentum because of the friction between adjacent layers and spirals inwards. In the process gravitational energy is released, the kinetic energy of the plasma increases and the disk heats up, emitting thermal energy.
Novikov, Thorne, and Page (Novikov & Thorne 1973; Page & Thorne 1974) made a relativistic analysis of the structure of an accretion disk around a black hole. They assumed the background spacetime geometry to be stationary, axiallysymmetric, asymptotically flat, and reflectionsymmetric with respect to the equatorial plane. They also postulated that the central plane of the disk coincides with the equatorial plane of the black hole. This assumption entails that the metric coefficients g_{tt}, g_{tφ}, g_{rr}, g_{θθ}, and g_{φφ} depend only on the radial coordinate r.
The disk is supposed to be in a quasisteady state (Novikov & Thorne 1973), so any relevant quantity (for example the density or the temperature of the gas) is averaged over 2π, a proper radial distance of order 2H^{3}, and the time interval Δt that the gas takes to move inward a distance 2H. During Δt, the changes in the spacetime geometry are negligible.
Particles move in the equatorial plane in nearly geodesic orbits; consequently, the gravitational forces exerted by the black hole completely dominate over the radial accelerations due to pressure gradients.
The expression of the energy flux for a relativistic accretion disk takes the form (Novikov & Thorne 1973; Page & Thorne 1974): (54)where stands for the mass accretion rate, Ω for the angular velocity and and represent the specific energy and angular momentum, respectively. The lower limit of the integral r_{isco} corresponds to the location of the innermost stable circular orbit.
The angular velocity Ω, the specific energy , and the specific angular momentum of the particles moving in circular orbits are given by (Harko et al. 2009): Equations (56), and (57) can be derived by writing the effective potential V_{eff}(r) in terms of the metric coefficients and solving for and the equations V_{eff}(r) = 0 and V_{eff}_{,r}(r) = 0. The formula for the angular velocity Ω = dφ/dt is obtained by substituing , and into the geodesic equations dt/dτ and dφ/dτ (Harko et al. 2009).
In the next subsections we calculate the energy flux, temperature and luminosity of an accretion disk around a Schwarzschild and a Kerr black hole in GR and f(R) gravity with constant Ricci scalar, adopting the following values for the relevant parameters: M = 14.8 M_{⊙}, Ṁ = 0.472 × 10^{19} g s^{1}, and a = 0.99, which are the best estimates available for the wellknown galactic black hole CygnusX1 (Orosz et al. 2011; Gou et al. 2011).
4.1.1. Relativistic accretion disk around Schwarzschild and Kerr black holes
In order to obtain an expression of the energy flux and temperature of the disk, for the Schwarzschild black hole, we calculate the angular velocity Ω, the specific energy and angular momentum of the particles in the disk, using the metric (Schwarzschild 1916): (58)From Eqs. (55)–(57) we get: By replacing these equations in Eq. (54), we obtain: (62)where x_{isco} = 6r_{g} is the location of the innermost stable circular orbit in Schwarzschild spacetime.
By means of StefanBolzmann’s law, (63)(where z stands for the correction due to the gravitational redshift), the temperature of the disk can be obtained as a function of the radial coordinate r.
In Figs. 9 and 10 we plot the energy flux and the temperature as a function of the radial coordinate for a Schwarzschild black hole in both Keplerian and relativistic accretion disk models. We see that GR effects introduce a decrease of the peak of the energy flux by a factor ≈ 2, and that the temperature distribution is also diminished.
Fig. 9 Energy flux as a function of the radial coordinate of an accretion disk around a Schwarzschild black hole in ShakuraSunyaev (SS) and PageThorne (PT) models, respectely. 

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Fig. 10 Temperature as a function of the radial coordinate of an accretion disk around a Schwarzschild black hole in SS and PT models, respectively. 

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The Kerr spacetime metric (Kerr 1963) in BoyerLindquist coordinates for θ = π/2 is: (64)where: (65)The expression of the energy flux now becomes: (66)where: and the coefficients p_{i} are given by: Here x = r/r_{g} is an adimensional radial coordinate, and a = a/r_{g} is the angular momentum of the black hole in adimensional units.
Fig. 11 Energy flux as function of the radial coordinate for an accretion disk around a Kerr black hole of angular momentum a = 0.99 in the PT model. 

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In Figs. 11 and 12 we show the plots of the energy flux and temperature of an accretion disk around a Kerr black hole of angular momentum a = 0.99, whose innermost stable circular orbit is at r_{isco} = 1.4545r_{g}. In Fig. 13 we show the luminosity of relativistic accretion disks around both Schwarzschild and Kerr black holes. For comparison, we also present the Schwarzschild/ShakuraSunyaev luminosity.
The values of the maximum temperature, luminosity, and the energy of the peak of the emission for all these models are shown in Table 5. As expected, the highest luminosity corresponds to a prograding accretion disk around a Kerr black hole. Since the last stable circular orbit is located closer to the black hole than in Schwarzschild spacetime, thermal radiation is emitted at higher energies.
Fig. 12 Temperature as function of the radial coordinate of an accretion disk around a Kerr black hole of angular momentum a = 0.99 in the PT model. 

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Fig. 13 Luminosity as function of the energy for a relativistic accretion disk around a Schwarzschild and a Kerr black hole (a = 0.99). We also plot the luminosity as a function of the energy of an accretion disk around a Schwarzschild black hole that corresponds to the SS model. 

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Values of the energy of the peak of the emission, the maximum temperature, and luminosity of an accretion disk around Schwarzschild and Kerr black holes (a = 0.99) in the SS and PT models.
4.2. f(R)gravity
4.2.1. f(R)Schwarzschild black holes
The energy flux of an accretion disk around a f(R)Schwarzschild black hole with metric given by Eq. (8) takes the form: (74)where (75)We adopt for the radius of the outer edge of the disk (Dove et al. 1997): (76)According to the latter equation, if we take for the innermost stable circular orbit r_{isco} = 6.3r_{g}, the outer egde of the disk yields approximately 70r_{g}. If larger values of r_{out} are considered, there are no major differences in the temperature and luminosity distributions.
We first compute the temperature and luminosity spectra distributions for R_{0} < 0, and adopt the values given in Table 2. In Figs. 14 and 15 we show the plots of the temperature as a function of the radial coordinate, and of the luminosity as a function of the energy, respectively. Notice that the corrections due to the gravitational redshift have been taken into account.
The maximum temperature as well as the luminosity increase for smaller values of R_{0} (Figs. 14 and 15). In the four cases presented, the accretion disk is hotter than in GR, e.g. in a f(R)Schwarzschild disk for R_{0} = −1.5, the maximum temperature and luminosity are a factor 1.9 and 3.7, respectively, higher. The energy corresponding to the peak of the emission shifts to higher values, reaching 1359.20 eV for the set of adopted parameters.
Fig. 14 Temperature as a function of the radial coordinate for some values of R_{0} < 0, for a f(R)Schwarzschild black hole. 

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Fig. 15 Luminosity as a function of the energy for some values of R_{0} < 0, for a f(R)Schwarzschild black hole. 

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Values of the location of the last stable circular orbit, location in the radial coordinate of the maximum temperature, maximum temperature and luminosity, and the energy of the peak of the emission for an accretion disk around a f(R)Schwarzschild black hole with R_{0} < 0.
We showed in Sect. 3.1.2 that for R_{0} > 0, stable circular orbits are possible within a minimum and maximum radius. We see in Table 1 that only for R_{0} = 10^{12} and R_{0} = 10^{6} accretion disks are possible, if we take for the radius of the outer edge of the disk r_{out} = 70r_{g}. The values of the location of the innermost stable circular orbit, location in the radial coordinate of the maximum temperature, maximum temperature and luminosity, and the energy of the peak of the emission are displayed in Table 7. We conclude that for R_{0} ∈ (0,10^{6}] the temperature and energy distributions have no significant differences with Schwarzschild’s distributions in GR.
Values of the location of the innermost stable circular orbit, location in the radial coordinate of the maximum temperature, maximum temperature and luminosity, and the energy of the peak of the emission for an accretion disk around a f(R)Schwarzschild black hole with R_{0} > 0.
4.2.2. f(R)Kerr black holes
We calculate next the energy flux of an accretion disk around a f(R)Kerr black hole: (77)where: and If we adopt for the radius of the inner edge of the disk 1.4545r_{g}, according to Eq. (76), r_{out} ≈ 16r_{g}. From Eq. (77) we numerically calculate the temperature and luminosity for the values shown in Table 3, taking into account the corrections coming from the gravitational redshift. The results are displayed in Figs. 16, 17, and Table 8. We see that the temperature of the disk increases for smaller values of R_{0}. The ratio of the maximum temperature between the GR and f(R) cases, with R_{0} = −1.25 × 10^{1}, is 1.20. The peak of the emission rises a factor of 2, and the corresponding energy is shifted towards higher energies.
Fig. 16 Temperature as a function of the radial coordinate for some values of R_{0} < 0 of a f(R)Kerr black hole of angular momentum a = 0.99, corrected by gravitational redshift. 

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Fig. 17 Luminosity as a function of the energy for some values of R_{0} < 0, for a f(R)Kerr black hole of angular momentum a = 0.99. 

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Location of the last stable circular orbit and maximum temperature, maximum temperature, luminosity, and energy of the peak of the emission for an accretion disk around a f(R)Kerr black hole with R_{0} < 0 and a = 0.99.
Values of the location of the last stable circular orbit, location in the radial coordinate of the maximum temperature, maximum temperature and luminosity, and the energy of the peak of the emission for an accretion disk around a f(R)Kerr black hole with R_{0} > 0 and a = 0.99.
Since the radius of the outer edge of the disk is 16r_{g}, we see from Table 4 that accretion disks are only possible for R_{0} = 10^{6} and R_{0} = 10^{4}, until up R_{0} = 6.67 × 10^{4}. For such values of R_{0}, we show in Table 9 the values of the location of the last stable circular orbit, maximum temperature, luminosity, and the energy of the peak of the emission, and in Figs. 18 and 19, the temperature and luminosity distributions respectively. As in f(R)Schwarzschild black holes with positive Ricci scalar, these differences are minor.
Fig. 18 Temperature as a function of the radial coordinate for some values of R_{0} > 0, for a f(R)Kerr black hole of angular momentum a = 0.99. 

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Fig. 19 Luminosity as a function of the energy for some values of R_{0} > 0, for a f(R)Kerr black hole of angular momentum a = 0.99. 

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We proceed now to examine some specific forms of the function f, and the constraints imposed on them by the previous analysis.
5. Limits on specific prescriptions for f(R)
As discussed in Sects. 3.1.2 and 3.2.2, the existence of PageThorne disks around f(R) black holes imposes the following limits on R_{0}:
As we shall see in Sect. 6, contemporary observations of Cygnus X1 rule out accretion disks around f(R)Schwarzschild black holes, since the maximum temperature obtained in such models is lower than the inferred through observations (Gou et al. 2011). Hence, we will only consider the values of R_{0} given by expression (83). We shall show in this section how these values lead to limits on the parameters of two examples of f(R) theories via Eq. (7). We shall also impose the following viability conditions, to be satisfied by any f(R) (Cembranos et al. 2011): (84)(85)5.1. f(R) = α R^{β}
The parameters α, β and the Ricci scalar are related by Eq. (7) as follows: (86)Introducing the adimensional parameter with , this equation can be written as (87)Notice the condition β > 0 to ensure the GR limit for small values of the Ricci scalar R. Let us consider first the case of a positive Ricci scalar, which leads to:
Case II (90)or (91)and 1/(β − 1) an even number, that is: (92)with n ∈ Z. By isolating α′ from Eq. (87), we obtain the function: (93)We show in Fig. 20 the plot of α′ as a function of β (with β > 0) for fixed values of the Ricci scalar. We see that for β ∈ (0,2), α′ ∈ (−∞,0). For β = 2, Eq. (93) is not defined, while for β > 2, α′ takes large positive values. Since α needs to be small in order to recover GR for small values of the Ricci scalar, the case α′ > 0, β > 2 is discarded. Hence, we obtain the following restrictions on the parameters:
For negative values of the Ricci scalar, from Eq. (93) we require that: (94)or: (95)where , so that β > 0. If m = 0, β = 1 and α′ = α = −1. These values of the parameters lead to f(R) = −R, which does not reduce to GR. For β = 2, Eq. (93) is not defined. If β ≥ 3 and is an odd number, α′ takes positive large values, while if β ≥ 4 and is an even number α′ is large and negative. Since α′ has to be small to recover GR for small values of the Ricci scalar, we conclude that negative values of R_{0} are not allowed in this theory.
Fig. 20 α′ as a function of β for different values of R_{0}. 

Open with DEXTER 
We now restrict the values of α and β according to Eqs. (84) and (85). The first and second derivative for the given f function are: The restrictions over α and β that satisfy Eq. (85) are: (98)or (99)The condition given by Eq. (98) is discarded because it does not satisfy Eq. (93). The viability condition expressed by Eq. (84) takes the form: (100)We can constrain the values of α using the latter inequality as follows: By multiplying by R_{0}^{β − 1}α the latter restrictions yields: (101)In order to satisfy Eq. (100): If β = 0, α > − R_{0}, and for β = 1, α > − 1. The set of values for α is α ∈ ( − R_{0},0).
We conclude that the values of α and β that are permitted by our model as well as by the two viability conditions are:
5.2. f(R) = R ϵ
In this case, the parameters ϵ and α, and the Ricci scalar are related by Eq. (7) in the simple form: (102)Dividing by R_{g}, we obtain . For R_{0} > 0, the function α′(ϵ) is always positive, while it is negative for all ϵ if R_{0} < 0. The constrains over ϵ and α that follow from Eq. (7) are:
The first and second derivative of the function f take the form: The condition f′′(R_{0}) > 0 is satisfied if ϵ > 0 ∧ R_{0} > 0, or ϵ < 0 ∧ R_{0} < 0. Equation (84) in adimensional form is: (105)This equation, together with (104), yields:
For ϵ > 0 and R_{0} > 0, .

For ϵ < 0 and R_{0} < 0,
6. Discussion
The results presented in Sect. 4 can be compared with current observational data to derive some constraints on a given f(R) theory. In order to illustrate this assertion we shall consider Cygnus X1, which is the most intensively studied black hole binary system in the Galaxy. A series of recent highquality papers (Reid et al. 2011; Orosz et al. 2011; Gou et al. 2011) have provided an unprecedented set of accurate measurements of the distance, the black hole mass, spin parameter a, and the orbital inclination of this source. This opens the possibility to constrain modified theories of gravity with rather local precision observations of astrophysical objects in the Galaxy.
Cygnus X1 was discovered at Xrays by Bowyer et al. (1965). Early dynamical studies of the compact object suggested the presence of an accreting black hole (e.g. Bolton 1972). The distance to Cygnus X1 is currently estimated to be kpc (Reid et al. 2011). This value was determined via trigonometric parallax using the Very Long Baseline Array (VLBA). At this distance, the mass of the black hole is (Orosz et al. 2011) M = 14.8 M_{⊙}. This is the value adopted in all calculations presented in the previous sections.
The source has been observed in both a lowhard state, dominated by the emission of a hot corona (e.g., Dove et al. 1997; Gierlinski et al. 1997; Poutanen 1998), and a highsoft state, dominated by the accretion disk, which in this state goes all the way down to the last stable orbit. In the lowhard state, where the source spends most of the time, a steady, nonthermal jet is observed (Stirling et al. 2001). The jet is absent in the thermal state. Therefore, in this latter state a clearer Xray spectrum can be obtained.
The accretion rate and the spin parameter of the hole are ~ 0.472 × 10^{19} g s^{1} and 0.99, respectively, according to estimates from a Kerr plus blackbody disk model (Gou et al. 2011). These GR models yield a spectral energy distribution with a maximum at E_{max} ~ 1.6 keV. On the contrary, f(R)models with negative curvature correspond to a low maximum temperature, lower even than what is expected for the (unrealistic) case of a Schwarzschild black hole. Therefore, we can presume that a fit of f(R)Kerr models to the data would also prefer high values of maximum temperature, i.e., ones with nonnegative curvature. Models with accretion rates and spin close to those obtained by Gou et al. (2011) and small positive curvature seem viable, something that is consistent with an asymptotic behaviour corresponding to a de Sitter spacetime endowed with a small and positive value of the cosmological constant.
Deep Xray studies with Chandra satellite might impose more restrictive limits, especially if independent constraints onto the accretion rate become available.
7. Conclusions
We have studied stable circular orbits and relativistic accretion disks around Schwarzschild and Kerr black holes in f(R) gravity with constant Ricci curvature in the strong regime. We have found that stable disks can be formed only for curvatures in the ranges of ( − ∞,10^{6}] and [ − 1.2 × 10^{3},6.67 × 10^{4}] in the cases of Schwarzschild and Kerr black holes, respectively. Current observations of Cygnus X1 in the soft state rule out curvature values below − 1.2 × 10^{3}. Additional constrains can be imposed on specific prescriptions of f(R) gravity. In particular, logaritmicgravity prescriptions are strongly constrained by observational data. Future highprecision determination of the parameters of other black hole candidates can be used to impose more restrictive limits to extended theories of gravity.
Acknowledgments
We are grateful to an anonymous referee for useful suggestions. BH astrophysics with G.E. Romero is supported by grant PIP 2010/0078 (CONICET). Additional funds comes from Ministerio de Educación y Ciencia (Spain) trhough grant AYA 201021782C0301. S.E.P.B. acknowledges support from UERJ, FAPERJ, and ICRANetPescara. We thank Gabriela Vila and Florencia Vieyro for comments on accretion disks.
References
 Abdelwahab, M., Goswami, R., & Dunsby, P. 2012, Phys. Rev. D, 85, 083511 [NASA ADS] [CrossRef] [Google Scholar]
 Arbuzova, E. V., Dolgov, A. D., & Reverberi, L. 2012, JCAP, 02, 04 [Google Scholar]
 Biswas, T., Cembranos, J. A. R., & Kapusta, J. I. 2010, Phys. Rev. Lett., 104, 021601 [NASA ADS] [CrossRef] [Google Scholar]
 Biswas, T., Cembranos, J. A. R., & Kapusta, J. I. 2010, JHEP, 1010, 048 [NASA ADS] [Google Scholar]
 Bolton, C. T. 1972, Nat. Phys. Sci., 240, 124 [NASA ADS] [CrossRef] [Google Scholar]
 Bowyer, S., Byram, E. T., Chubb, T. A., & Friedman, H. 1965, Science, 147, 394 [NASA ADS] [CrossRef] [Google Scholar]
 Capozziello, S. 2002, Int. J. Mod. Phys. D, 11, 483 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Capozziello S., & Faraoni V. 2010, Beyond Einstein Gravity. A survey of Gravitational Theories for Cosmology and Astrophysics (DordrechtHeidelbergLondonNew York: Springer) [Google Scholar]
 Carter, B. 1973, in Les Astres Occlus eds. C. DeWitt, & B. DeWitt (New York, Gordon & Breach) [Google Scholar]
 Cembranos, J. A. R., de la CruzDombriz, A., & Jimeno Romero, P. 2011 [arXiv:1109.4519v1] [Google Scholar]
 Cooney, A., DeDeo, S., & Psaltis, D. 2010, Phys. Rev. D, 82, 064033 [NASA ADS] [CrossRef] [Google Scholar]
 De Felice, A., & Tsujikawa, S. 2010, Living Rev. Relat., 13 3 [Google Scholar]
 de la CruzDombriz, A., Dobado, A., & Maroto, A. L. 2009, Phys. Rev. D, 80, 124011; Erratumibid, 2011, 83, 029903 [NASA ADS] [CrossRef] [Google Scholar]
 Dove, J. B., Wilms, J., Maisack, M., & Begelman, M. 1997, ApJ, 487, 759 [NASA ADS] [CrossRef] [Google Scholar]
 Gierlinski, M., Zdziarski, A. A., Done, C., et al. 1997, MNRAS, 228, 958 [NASA ADS] [CrossRef] [Google Scholar]
 Gou, L., McClintock, J. E., Reid, Mark, J., et al. 2011, ApJ, 742, 85 [NASA ADS] [CrossRef] [Google Scholar]
 Harko, T., Kovács, Z., & Lobo, F. N. N. 2009, Phys. Rev. D, 79, 064001 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Hendi, S. H., & Momeni, D. 2011, Eur. Phys. J. C, 71, 1823 [NASA ADS] [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
 Hendi, S. H., Panah, B. E., & Mousavi, S. M. 2012, Gen. Rel. Grav., 44, 835 [NASA ADS] [CrossRef] [Google Scholar]
 Hobson, M. P., Estathoiou, G. P., & Lasenby, A. N. 2006, General Relativity. An Introduction for Physicists (Cambridge: Cambridge, University Press) [Google Scholar]
 Kerr, R. P. 1963, Phys. Rev. Lett., 11, 237 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Li, M., Li, X., & Wang, S. 2011, Commun. Theor. Phys., 56, 525 [NASA ADS] [CrossRef] [Google Scholar]
 Magnano, G., Ferraris, M., & Francaviglia, M. 1987, Gen. Rel. Gravit., 19, 465 [NASA ADS] [CrossRef] [Google Scholar]
 Mazharimousavi, H. S., Halilsay, M., & Tahamtan, T. 2012, Eur. Phys. J. C, 72, 1851 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Moon, T., Myung, Y. S., & Son, E. J. 2011a, Gen. Rel. Gravit., 43, 3079 [NASA ADS] [CrossRef] [Google Scholar]
 Moon, T., Myung, Y. S., & Son, E. J. 2011b, Eur. Phys. J., C71, 1777 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Motohashi, H., & Nishizawa, A. 2012, Phys. Rev. D, 86, 083514 [NASA ADS] [CrossRef] [Google Scholar]
 Myung, Y. S. 2011, Phys. Rev. D, 84, 024048 [NASA ADS] [CrossRef] [Google Scholar]
 Myung, Y. S., Moon, T., & Son, E. J. 2011, Phys. Rev. D, 83, 124009 [NASA ADS] [CrossRef] [Google Scholar]
 Novikov, I. D., & Thorne, K. S. 1973, in Black Holes, eds. C. DeWitt, & B. DeWitt (New York: Gordon & Breach) [Google Scholar]
 Orosz, J. A., McClintock, J. E., Aufdenberg, J. P., et al. 2011, ApJ, 742, 84 [NASA ADS] [CrossRef] [Google Scholar]
 Page, D. N., & Thorne, K. S. 1974, ApJ, 191, 499 [NASA ADS] [CrossRef] [Google Scholar]
 Perez Bergliaffa, S. E., & Yves De Oliveira, E. 2011, Phys. Rev. D, 84, 084006 [NASA ADS] [CrossRef] [Google Scholar]
 Poutanen, J. 1998, Theory of Black Hole Accretion Disks, eds. M. A. Abramowicz, Gunnlaugur Bjornsson, and James E. Pringle. (Cambridge: Cambridge University Press) [Google Scholar]
 Psaltis, D., Perrodin, D., Dienes, K. R., & Mocioiu, I. 2008a, Phys. Rev. Lett., 100, 091101 [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
 Psaltis, D., Perrodin, D., Dienes, K. R., & Mocioiu, I. 2008b, Phys. Rev. Lett., 100, 119902 [NASA ADS] [CrossRef] [Google Scholar]
 Pun, C. S., Zovács, Z., & Harko, T. 2008, Phys. Rev. D, 78, 024043 [NASA ADS] [CrossRef] [Google Scholar]
 Reid, M. J., McClintock, J. E., Narayan, R., et al. 2011, ApJ, 742, 83 [NASA ADS] [CrossRef] [Google Scholar]
 Rezzolla, L., Zanotti, O., & Font, J. A. 2003, A&A, 412, 603 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Schwarzschild, K. 1916, Sitzungsberichte der KöniglichPreussischen Akademie der Wissenschaften, 3, 189 [Google Scholar]
 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 377 [NASA ADS] [Google Scholar]
 Sotiriou, T. P., & Faraoni, V. 2010, Rev. Mod. Phys., 82, 451 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Starobinsky, A. A. 1980, Phys. Lett. B, 91, 99 [NASA ADS] [CrossRef] [Google Scholar]
 Stirling, A. M., Spencer, R. E., de la Force, C. J., et al. 2001, MNRAS, 327, 1273 [NASA ADS] [CrossRef] [Google Scholar]
 Stuchlík, Z., & Hledík, S. 1999, Phys. Rev. D, 60, 044006 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Stuchlík, Z., & Slaný, P. 2004, Phys. Rev. D, 69, 064001 [NASA ADS] [CrossRef] [Google Scholar]
 Slaný, P., & Stuchlík, Z. 2005, Class. Quant. Grav., 22, 3623 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Weinberg, S. 1989, Rev. Mod. Phys., 61, 1 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Will, C. M. 2006, Living Rev. Relat., 9, 3 [NASA ADS] [CrossRef] [Google Scholar]
All Tables
Location of the event and cosmological horizon, and of the innermost and outermost stable circular orbits for R_{0} > 0 in f(R)Schwarzschild spacetime.
Location of the event horizon, and of the innermost stable circular orbit for R_{0} < 0 in f(R)Schwarzschild spacetime.
Radii of event horizons and circular orbits for a f(R)Kerr black hole of angular momentum a = 0.99, for some values of R_{0} < 0.
Location of the event and cosmological horizon, and of the innermost and outermost stable circular orbits for R_{0} > 0 and a = 0.99 in f(R)Kerr spacetime.
Values of the energy of the peak of the emission, the maximum temperature, and luminosity of an accretion disk around Schwarzschild and Kerr black holes (a = 0.99) in the SS and PT models.
Values of the location of the last stable circular orbit, location in the radial coordinate of the maximum temperature, maximum temperature and luminosity, and the energy of the peak of the emission for an accretion disk around a f(R)Schwarzschild black hole with R_{0} < 0.
Values of the location of the innermost stable circular orbit, location in the radial coordinate of the maximum temperature, maximum temperature and luminosity, and the energy of the peak of the emission for an accretion disk around a f(R)Schwarzschild black hole with R_{0} > 0.
Location of the last stable circular orbit and maximum temperature, maximum temperature, luminosity, and energy of the peak of the emission for an accretion disk around a f(R)Kerr black hole with R_{0} < 0 and a = 0.99.
Values of the location of the last stable circular orbit, location in the radial coordinate of the maximum temperature, maximum temperature and luminosity, and the energy of the peak of the emission for an accretion disk around a f(R)Kerr black hole with R_{0} > 0 and a = 0.99.
All Figures
Fig. 1 Ricci scalar as a function of the radial coordinate of the event horizon in f(R)Schwarzschild spacetime. 

Open with DEXTER  
In the text 
Fig. 2 Function given by Eq. (23). The absolute maximum corresponds to x = 15/2 and R_{0} = 2.85 × 10^{3}. 

Open with DEXTER  
In the text 
Fig. 3 Effective potential for different values of R_{0} > 0 and h in f(R)Schwarzschild spacetime. The dots indicate the location of the innermost stable circular orbit. 

Open with DEXTER  
In the text 
Fig. 4 Ricci scalar as a function of the radial coordinate of the event horizon (line) and of the Ricci scalar as a function of the radial coordinate of the innermost stable circular orbits (dashed line) for R_{0} ∈ [ − 1.5,0.45] in f(R)Schwarzschild spacetime. 

Open with DEXTER  
In the text 
Fig. 5 Effective potential for different values of R_{0} < 0 and h in f(R)Schwarzschild spacetime. The dots indicate the location of the innermost stable circular orbit. 

Open with DEXTER  
In the text 
Fig. 6 Ricci scalar as a function of the radial coordinate of the event horizon for R_{0} ∈ [ − 0.3,1] and a = 0.99 in f(R)Kerr spacetime. 

Open with DEXTER  
In the text 
Fig. 7 Effective potential as a function of the radial coordinate (R_{0} < 0, a = 0.99), in f(R)Kerr spacetime. The dots indicate the location of the innermost stable circular orbit. 

Open with DEXTER  
In the text 
Fig. 8 Effective potential as a function of the radial coordinate (R_{0} > 0, a = 0.99), in f(R)Kerr spacetime. The dots indicate the location of the innermost stable circular orbit. 

Open with DEXTER  
In the text 
Fig. 9 Energy flux as a function of the radial coordinate of an accretion disk around a Schwarzschild black hole in ShakuraSunyaev (SS) and PageThorne (PT) models, respectely. 

Open with DEXTER  
In the text 
Fig. 10 Temperature as a function of the radial coordinate of an accretion disk around a Schwarzschild black hole in SS and PT models, respectively. 

Open with DEXTER  
In the text 
Fig. 11 Energy flux as function of the radial coordinate for an accretion disk around a Kerr black hole of angular momentum a = 0.99 in the PT model. 

Open with DEXTER  
In the text 
Fig. 12 Temperature as function of the radial coordinate of an accretion disk around a Kerr black hole of angular momentum a = 0.99 in the PT model. 

Open with DEXTER  
In the text 
Fig. 13 Luminosity as function of the energy for a relativistic accretion disk around a Schwarzschild and a Kerr black hole (a = 0.99). We also plot the luminosity as a function of the energy of an accretion disk around a Schwarzschild black hole that corresponds to the SS model. 

Open with DEXTER  
In the text 
Fig. 14 Temperature as a function of the radial coordinate for some values of R_{0} < 0, for a f(R)Schwarzschild black hole. 

Open with DEXTER  
In the text 
Fig. 15 Luminosity as a function of the energy for some values of R_{0} < 0, for a f(R)Schwarzschild black hole. 

Open with DEXTER  
In the text 
Fig. 16 Temperature as a function of the radial coordinate for some values of R_{0} < 0 of a f(R)Kerr black hole of angular momentum a = 0.99, corrected by gravitational redshift. 

Open with DEXTER  
In the text 
Fig. 17 Luminosity as a function of the energy for some values of R_{0} < 0, for a f(R)Kerr black hole of angular momentum a = 0.99. 

Open with DEXTER  
In the text 
Fig. 18 Temperature as a function of the radial coordinate for some values of R_{0} > 0, for a f(R)Kerr black hole of angular momentum a = 0.99. 

Open with DEXTER  
In the text 
Fig. 19 Luminosity as a function of the energy for some values of R_{0} > 0, for a f(R)Kerr black hole of angular momentum a = 0.99. 

Open with DEXTER  
In the text 
Fig. 20 α′ as a function of β for different values of R_{0}. 

Open with DEXTER  
In the text 
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