Predicting spin of compact objects from their QPOs: A global QPO model
aa r X i v : . [ a s t r o - ph ] S e p Predicting spin of compact objects from theirQPOs: A global QPO model
Banibrata Mukhopadhyay
Astronomy and Astrophysics Programme, Department of Physics, Indian Institute of Science,[email protected]
Abstract.
We establish a unified model to explain Quasi-Periodic-Oscillation (QPO) observed fromblack hole and neutron star systems globally. This is based on the accreting systems thought to bedamped harmonic oscillators with higher order nonlinearity. The model explains multiple propertiesparallelly independent of the nature of the compact object. It describes QPOs successfully for severalcompact sources. Based on it, we predict the spin frequency of the neutron star Sco X-1 and thespecific angular momentum of black holes GRO J1655-40, GRS 1915+105.
Keywords: black hole physics; gravitation; relativity; stars: neutron
PACS:
INTRODUCTION
The origin of Quasi-Periodic-Oscillation (QPO) and its properties are still ill-understood. The observed QPO frequencies of compact objects are expected to berelated to the spin parameter of the compact object itself. In certain neutron star systemsa pair of QPO forms and QPO frequencies appear to be separated either by the order ofthe spin frequency of the neutron star, apparently for slow rotators e.g. 4U 1702-429,or by half of the spin frequency, for fast rotators e.g. 4U 1636-53. Their frequencyseparation decreases with the increase of one of the QPO frequencies. The black holesystems, on the other hand, e.g. GRO J1655-40, XTE J1550-564, GRS 1915+105,exhibit the kHz QPO pairs which seem to appear at a 3 : 2 ratio [1].Interaction between the surface current density in the disk and the stellar magneticfield generating warps and subsequent precession instability in the inner accretion diskproduces motions at low frequencies. This is similar to the Lense-Thirring precession[2]. This can explain the mHz QPOs for strongly magnetized neutron stars [3]. Theeffect of nonlinear coupling between g-mode oscillations in a thin relativistic disk andwarp has also been examined [4] for a static compact object. Recent observations [5]indicate a strong correlation between low and high frequency QPOs holding over mHz tokHz range which strongly supports the idea that QPOs are universal physical processesindependent of the nature of the compact object. The correlation is also explained interms of centrifugal barrier model [6]. Indeed earlier the correlation was shown in termsof the effective boundary wall created by the strong centrifugal force in the disk [7].Accretion dynamics is a nonlinear hydrodynamic/magnetohydrodynamic phe-nomenon. It was already shown that QPOs may arise from nonlinear resonancephenomena in an accretion disk (e.g. [8, 9]) occurred due to resonance betweenpicyclic motions of accreting matter. However, the separation between vertical andradial epicyclic frequencies increases mostly as a function of either of the epicyclicfrequencies contradicting the observed QPO feature.We aim at establishing a global model based on higher order nonlinear resonancetheory to describe black hole and neutron star QPOs together and then try to predict thespin parameter/frequency of compact objects. The model predicts, alongwith importantQPO features, the spin parameter of black holes as well as spin frequency of a neutronstar.
MODEL
A system of N degrees of freedom has N linear natural frequencies n i ; i = , , ... N [10].These frequencies have commensurable relations which may cause the correspondingmodes to be strongly coupled and yield an internal resonance. If the system is excited byan external frequency n ∗ , then the commensurable relation exhibiting resonance mightbe a n ∗ = N (cid:229) i = b i n i , with a + N (cid:229) i = | b i | = j , (1)apart from all the primary and secondary resonance conditions c n ∗ = d n m , where a , b i , c , d are integers and j = k + k is the order of nonlinearity.Now we consider an accretion disk with a possible higher order nonlinear resonance.The resonance is driven by the combination of the strong disturbance created by therotation of the compact object with spin n s and the existent (weaker) disturbances in thedisk at the frequency of the radial ( n r ) and vertical ( n z ) epicyclic oscillations given by(e.g. [11]) n r = n o r q D − ( √ r − a ) , n z = n o r q r − a √ r + a , (2)where D = r − Mr + a , a is the specific angular momentum (spin parameter) of thecompact object and n o is the orbital frequency of the disk particle given by 2 pn o = W = / ( r / + a ) .We describe the system schematically in Fig. 1. It is composed of two oscillators dueto radial and vertical epicyclic motion with different spring constants. The basic idea isthat the mode corresponding to n s will couple to the ones corresponding to n r and n z igniting new modes with frequencies n r , z ± p n s [10], where p is a number e.g. L / L ,when L is an integer, if the effect is nonlinear or linear [10, 8] respectively. While L = L are very weak to exhibitany observable effect. Now at a certain radius in the nonlinear regime where n s / n s in the linear regime) is close to n z − n r [10, 8], and n s / n s ) is also coincidentallyclose to the frequency difference of any two newly excited modes, a resonance mayoccur which locks the frequency difference of two excited modes at n s / n s ).As the neutron star has a magnetosphere coupled with the accretion disk, the modewith n s can easily disturb the disk matter. A black hole, on the other hand, with magnetic IGURE 1.
Cartoon diagram describing coupling of various modes in an accretion disk and corre-sponding nonlinear oscillator. The oscillators describing by spring constant k and k indicate respectivelythe coupling of spin frequency n ∗ of the compact object with mass M to radial ( x ) epicyclic frequency andvertical ( z ) epicyclic frequency of a disk blob with mass m , where b and b represent correspondingdamping factors respectively. field connecting to the surrounding disk, may transfer energy and angular momentumto the disk by the variant Blandford-Znajek process [12] what was already verified byXMM-Newton observation (e.g. [13]). This confirms possible ignition of new modes ina disk around a spinning a black hole.Therefore, rewriting (1) for an accretion disk we obtain n r + n n s = n z − n s + m n s (3)with − b = b = m , n are integers. Now we propose the higher and lower QPOfrequency of a pair respectively to be n h = n r + n n s , n l = n z − n s . (4)Hence, we understand from (1) and (3) the order of nonlinearity in accretion disksexhibiting QPOs k = n − m + n z − n r ∼ n s / D n = n h − n l ∼ n s / n = m =
1, and
D n ∼ n s for n = m = n = D n locking in the nonlinear regime with m =
1. On theother hand, n = D n locking in the linear regime with m =
2. For a black hole, however, in absence of its magnetosphere, disturbance and thencorresponding coupling may not be strongly nonlinear and occurs with the condition n z − n r < ∼ n s resulting the resonance locking at the linear regime with n = m =
2, whichproduces
D n < ∼ n s (sometimes ∼ n s / n < ∼ n s / ∼ n s /
5) and n z − n r < ∼ n s for n = m =
1. In principle, there maybe possible resonances with other combination of n and m (e.g. n = , m =
1) which areexpected to be weak to observe.Once we know the spin frequency n s of a neutron star from observed data we candetermine specific angular momentum a (spin parameter) with the information of equa-torial radius R , spin frequency n s , mass M , radius of gyration R G . If we consider theneutron star to be spherical in shape with equatorial radius R , then the moment of inertiaand spin parameter are computed as I = MR G , a = I W s c / GM , where W s = pn s , G isthe Newton’s gravitation constant and c is speed of light. We know that for a solid sphere R G = R / R G = R /
3. However, in practice for a neutronstar R G should be in between. Moreover, the shape of a very fast rotating neutron staris expected to be deviated from spherical to ellipsoidal. Hence, in our calculation werestrict 0 . ≤ ( R G / R ) ≤ . a is the most natural quantity what wesupply as an input. The corresponding angular frequency of a test particle at the radiusof marginally stable circular orbit r ms in spacetime around it is then given by [11] W BH = pn s = − g f t ( r = r ms ) g ff ( r = r ms ) = ar ms + r ms a + a , (5)where and light inside r ms is practically not expected to reach us. RESULTSNeutron stars
The choice of mass M = . M ⊙ and ( R G / R ) = . M < . M ⊙ , then our theory has an excellent agreement withobserved data with realistic R [14] shown in Fig. 2b. Similarly for other neutron starsshowing twin kHz QPOs, results from our model have good agreement with observeddata (not discussed in the present paper but would be described elsewhere in future),which are the beyond scope to discuss in the present paper. Estimating spin of Sco X-1
The spin frequency of Sco X-1 is not known yet. We compare in Fig. 2c the observedvariation of frequency separation as a function of lower QPO frequency with thatobtained from our model. We find that mass of Sco X-1 must be less than 1 . M ⊙ andresults with sets of inputs with smaller n s and M fit the observed data better and arguethat Sco X-1 is a slow rotator with n s ∼ − IGURE 2.
Variation of QPO frequency difference in a pair as a function of lower QPO frequency for(a) 4U 1636-53, (b) 4U 1702-429, (c) Sco X-1. Results for parameter sets given in TABLE 1 from top tobottom row for a particular neutron star correspond to solid, dotted, dashed (4U 1702-429 and Sco X-1),long-dashed (Sco X-1) lines. The triangles are observed data points.
TABLE 1. n s is given in unit of Hz, M in M ⊙ , R in km, radial coordinate whereQPO occurs, r QPO , in unit of Schwarzschild radius.neutron star n s M ( R G / R ) n , m R range of r QPO
4U 1636-53 581 .
75 1 . . . . − .
74U 1636-53 581 .
75 1 . . . . − .
34U 1636-53 581 .
75 1 . .
35 1 16 . . − .
74U 1702-429 330 .
55 1 . . . . − .
55 0 .
83 0 .
35 2 18 . . − . n s Sco X-1 422 . . . . . − . . . . . − . . . . . . − . . .
81 0 .
35 2 18 . . − Black holes
The possible mass or range of mass of several black holes is already predicted fromobserved data. However, the spin of them is still not well established. By supplyingpredicted mass and arbitrary values of a our theory reproduces observed QPOs forGRO J1655-40 and GRS 1915+105 with their 3 : 2 ratio for n = m = r ms given in TABLE 2. However, if we enforce QPOs to produce at r ms strictly,then they produce at a higher a for n = m =
1. Similarly, results from our model for otherblack holes showing twin kHz QPOs have good agreement with observed data (wouldbe discussed elsewhere in future), which are the beyond scope to discuss in the presentpaper.
ABLE 2. n l , h are given in unit of Hz, M in M ⊙ , r QPO and its distance from marginally stable orbit, D r , are expressed in unit of Schwarzschild radius. black hole M a n h n l r QPO D r n , m GRO J1655-40 estimated theory/observation theory/observation6 − . − .
778 450 /
450 300 /
300 4 . − .
25 1 . − .
23 27 .
05 0 .
95 451 . /
450 299 . /
300 1 .
94 0 1
GRS 1915+105 estimated theory/observation theory/observation10 −
20 0 . − .
797 168 /
168 113 /
113 7 . − . . − .
98 218 . .
95 167 . /
168 114 . /
113 1 .
94 0 1
SUMMARY
We have prescribed a global QPO model based on nonlinear resonance mechanismin accretion disks. Based on this we have predicted the spin parameter/frequency ofcompact objects. The model has addressed, for the first time to best of our knowledge,the variation of QPO frequency separation in a pair as a function of the QPO frequencyitself for neutron stars successfully. We argue that Sco X-1 is a slow rotator.We have addressed QPOs of black holes as well and predict their spin parameter ( a )which is not well established yet. According to the present model, none of them is anextremally rotating black hole. As our model explains QPOs observed from several blackholes and neutron stars including their specific properties, it favors the idea of QPOs tooriginate from a unique mechanism, independent of the nature of compact objects. ACKNOWLEDGMENTS
The work was supported partly by a project, Grant No. SR/S2/HEP12/2007, funded byDST, India.
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