On the high frequency spectrum of a classical accretion disc
OOn the high frequency spectrum of a classical accretiondisc
Steven A. Balbus ABSTRACT
We derive simple and explicit expressions for the high frequency spectrum ofa classical accretion disc. Both stress-free and finite stress inner boundaries areconsidered. A classical accretion disc spectrum with a stress-free inner boundarydeparts from a Wien spectrum at large ν , scaling as ν . (as opposed to ν )times the usual exponential cut-off. If there is finite stress at the inner discboundary, the maximum disc temperature generally occurs at this edge, even atrelatively modest values of the stress. In this case, the high frequency spectrumis proportional to ν times the exponential cut-off. If the temperature maximumis a local hot spot, instead of an axisymmetric ring, then an interior maximumproduces a ν prefactor while an edge maximum yields ν . . Because of beamingeffects, these latter findings should pertain to a classical relativistic disc. Theasymptotics are in general robust and independent of the detailed temperatureprofile, provided only that the liberated free energy of differential rotation isdissipated locally, and may prove useful beyond the strict domain of classical disctheory. As observations continue to improve with time, our findings suggest thepossibility of using the high energy spectral component of black hole candidatesas a signature prediction of classical theory, as well as an diagnostic of the stressat the inner regions of an accretion disc.
1. Introduction
Well into its fifth decade (e.g. Lynden-Bell 1969, Shakura & Sunyaev 1973), classical thindisc accretion theory has withstood the test of time reasonably intact. While it is clear thataccretion is considerably more complex than allowed for in classical disc theory (hereafterCDT), there are many examples in which its computed disc spectrum, a superposition ofblackbody rings over the surface of the disc, is realised with some fidelity in observations.Even when there are temporal changes in the spectra, there is generally a rather well-defined Dept. Physics, University of Oxford, Keble Rd., Oxford, UK OX1 3RH [email protected] a r X i v : . [ a s t r o - ph . H E ] J u l Letter , we show that the integral in questionmay, however, be performed analytically in the astrophysically relevant limit E γ (cid:29) kT max ,where E γ is the photon energy, k the Boltzmann constant, and T max the maximum disctemperature. The resulting spectra are similar to, but depart from, a Wien spectrum. Thecomputed frequency dependence is both robust and insensitive to the precise temperatureprofile. The form of the departure from a Wien spectrum changes (nearly discontinuously)beyond a moderate threshold value of the stress at the inner disc edge. It is this result thatmay prove useful to the analysis of the dynamical conditions at the inner disc edge. Thecalculational technique, however, should find application under any conditions in which thedisc has a well-defined temperature maximum. In principle, the effect of several isolatedmaxima may be superposed, but in general the spectrum will be dominated at high ν by thehighest temperature peak.The calculations are presented in the next section, which is followed by a brief discussionof our results. Both relativistic and nonrelativistic discs are considered.
2. Disc spectrum2.1. Preliminaries
In nonrelativistic CDT, the emitted flux per unit frequency ν from a thin disc is givenby the expression (e.g. Frank, King, & Raine 2002): F ν = 4 πhν cos ic r (cid:90) ∞ R ∗ R dR exp[ hν/kT ( R )] − h is Planck’s constant, c the speed of light, r the distance to the source, i the inclina-tion angle (from a face-on orientation), R ∗ the inner disc radius, and T ( R ) the temperatureprofile. The integral is over all radii R and formally extends to infinity.The time-steady surface temperature profile of a thin Keplerian disc is given by (Pringle1981, Balbus & Hawley 1998): T ( R ) = 3 GM ˙ M πR σ (cid:34) − (cid:18) R ∗ R (cid:19) / (cid:35) (2) 3 –where G is gravitational constant, σ the Stefan-Boltzmann constant, M the central mass, ˙ M the steady mass accretion rate. This result is calculated by assuming that the stress vanishesat the inner edge R = R ∗ . But R ∗ might equally well be regarded as an integration constant(arising from angular momentum flux conservation) whose value is determined by specifyingthe stress at some radius. To avoid confusion, we will write T ( R ) = 3 GM ˙ M πR σ (cid:34) − (cid:18) R R (cid:19) / (cid:35) (3)where R is an arbitrary constant with dimensions of a length whose value we leave unspec-ified. The choice R = R ∗ corresponds to what we shall view as the special case of the stressvanishing at the inner edge of the disc. Other notational conventions we will use areΩ = GMR , (4)for the Keplerian angular velocity (s − ), Σ for the height-integrated disc surface density, andthe characteristic temperatures T = 3 GM ˙ M πR σ , T ∗ = 3 GM ˙ M πR ∗ σ . (5)Care should be taken to distinguish T ∗ and T ( R ∗ ), which are quite distinct. The dominant Rφ component of the density-weigted velocity stress tensor (dimensions of velocity ) is denoted W Rφ .The temperature reaches a maximum, T max , when R max = (49 / R (Kubota et al.1998). This is near the inner edge if R also is, but there may be no temperature maximum if R (cid:28) R ∗ . This corresponds to the case of a significant stress at the inner edge. “Significant”means a value close to the local ˙ M Ω / (2 π Σ) (Balbus & Hawley 1998). We will show thatthe two cases of an interior temperature maximum and a boundary maxium lead to distinctobservational signatures.
The condition of angular momentum conservation may quite generally be written (Bal-bus & Hawley 1998) − ˙ M R Ω2 π + Σ R W Rφ = constant = − ˙ M R ∗ Ω ∗ π + Σ ∗ R ∗ W (6) 4 –where W is the selected value of W Rφ at R = R ∗ . (Both Σ ∗ and Ω ∗ are evaluated at R = R ∗ .)This value of W Rφ is to be distinguished from a fiducial characteristic value W char = ˙ M Ω ∗ π Σ ∗ , (7)which will appear (as a normalization for W ) in the equations.If we now solve equation (6) for W Rφ , we obtain W Rφ = ˙ M Ω2 π Σ (cid:34) − (cid:18) R R (cid:19) / (cid:35) (8)where R = R ∗ (cid:18) − WW char (cid:19) (9)Equating the energy radiated by (each side of) the disc to the energy extracted from thedifferential rotation, we have (Balbus & Hawley 1998):2 σT = − Σ W Rφ d Ω d ln R = − ˙ M π d Ω d ln R (cid:34) − (cid:18) R R (cid:19) / (cid:35) , (10)leading immediately to (3). Equation (9) tells us precisely how the R constant is relatedto the imposed stress W and inner boundary R ∗ , and is for that reason very useful. Asnoted, the formal location of the temperature maximum from (3) is R max = (49 / R . Thequestion is, at what value of W does this radius move from within the disc proper to theinner edge? Setting R max = R ∗ and using (9) leads to W = W char W exceeds 0 . W char , the temperature maximum lies on the inner disc bound-ary, and when the stress drops below this, the temperature maximum moves off the boundaryto within the disc inerior. As we shall now see, the location of the temperature maximummakes a signficant difference to the emitted spectrum. ν limit of F ν . T max Consider the integral I = (cid:90) ∞ R ∗ R dR exp[ hν/kT ( R )] − T ( R ) given by (3). When hν (cid:29) kT max we may safely ignore the − I = (cid:90) ∞ R ∗ R exp[ − hν/kT ( R )] dR (13)The function β ≡ /kT ( R ) has a sharp minimum at R = R max , which renders the integral anideal candidate for an asymptotic expansion based on Laplace’s method (Bender & Orszag1978). Under these conditions, the entire contribution to the integral comes from a smallregion near R = R max . Expanding β and remembering the first derivative vanishes at R = R max : β = 1 kT max + β (cid:48)(cid:48) min ( R − R max ) ... (14)where β (cid:48)(cid:48) min = d dR (cid:20) kT ( R ) (cid:21) R = R max (15)The integral (13) transforms to I (cid:39) R max exp( − hν/kT max ) (cid:90) ∞−∞ exp( − hνβ (cid:48)(cid:48) min x / dx (16)where x = R − R max , and we set R = R max since only this neighborhood contributes.Extending the limits of integration introduces only exponentially small corrections. Hence, I (cid:39) R max (cid:18) πβ (cid:48)(cid:48) min hν (cid:19) / exp( − hν/kT max ) (17)For the distribution (3), β (cid:48)(cid:48) min works out to R β (cid:48)(cid:48) min = 3 / / (7 / × / × kT ) = 2 . / ( kT ) (18)This leads to an emission spectrum F ν = 2 .
002 4 π cos ic ( hkT ) / R r ν / exp( − hν/kT max ) (19)where T max = 0 . T . The solution for a classical zero-stress inner boundary is obtainedby setting by using the inner edge of the disc R ∗ for R and T ∗ for T . The key point is thatthe frequency dependence differs from a Wien spectrum by a factor of ν − / . 6 – -6 -4 -2 F ν e h ν / ( k T m a x ) Frequency/(kT max )W/W char =0 ν Fig. 1.—
Plot comparing a F ν e hν/kT max , renormalised for display, with the large ν asymptoticresult for the case of vanishing stress. Solid line is from numerical evaluation; dotted line is large ν asymptotic form (eq. [17]). T max If the stress W exceeds W char /
7, the temperature maximum moves to the boundary. Inthat case, equations (3) and (9) may be combined to yield the temperature at the inner discedge, T ( R ∗ ): T ( R ∗ ) = T ∗ w / , with w ≡ W/W char . (20)Another quantity of interest we shall require is the temperature gradient at R = R ∗ . This ismost conveniently expressed in the form (cid:18) d ln Rd ln β (cid:19) R = R ∗ = − (cid:18) d ln Rd ln T (cid:19) R = R ∗ = w w − . (21)The high frequency behaviour of (13) is obtained by a simple integration by parts. Nowthe first derivative β (cid:48) term dominates and one finds I (cid:39) (cid:18) R ∗ hν (cid:19) (cid:18) exp[ − hν/kT ( R ∗ )] β (cid:48) ( R ∗ ) (cid:19) = R ∗ (cid:18) kT ∗ hν (cid:19) w / w −
1) exp( − hν/ [ kT ∗ w / ]) , (22)where β (cid:48)∗ = − kT ∗ (cid:18) dTdR (cid:19) R = R ∗ . (23) 7 –This gives a spectral flux of F ν = (cid:18) R ∗ r (cid:19) π cos ic ( kT ∗ ) w / w − ν exp( − hν/ [ kT ∗ w / ]) (24)This is a less steep frequency dependence than is present in (19). The case of small stressand an interior maximum corresponds to a flatter region of high temperature, and a cor-respondingly larger disc area is able to contribute. A boundary maximum is more steeplycut-off as one moves outward, and less of the disc is able to contribute, reducing the high ν emission relative to the interior maximum case.Figures (1) and (2) show two representative examples illustrating the region of validityof our approximation. Deviations from the leading asymptotic behavior are expected to be O ( kT max /hν ) in both cases (Bender & Orszag 1978). Quantitatively, the agreement betweenasymptotic and exact integrals is excellent when hν/kT max (cid:38) kT max . When the ratio is 10,the results are indistinguishable, at which point the flux is about 5 × − of its peak value. -6 -4 -2 F ν e h ν / ( k T m a x ) Frequency/(kT max )W/W char =2/3 ν Fig. 2.—
As in figure (1) for the finite stress case W = (2 / W char . The axisymmetric form of the emission integral (12) of classical disc theory is not pre-served when relativistic physics is included. The most important deviation is due to beaming 8 –from the portion of the disk approaching the observer. To extract the asymptotic behaviourof the spectrum, however, we need not explicitly invoke the full machinery of general rela-tivity. It will suffice to note that the emission integral will be of the form I = ν (cid:90) S F ( s , s ) dS exp[ hν/kT ( s , s )] − S is the effective “working surface,” s and s surface coordinates, F an unspecifiedfunction of coordinates (but, importantly, not ν ) and dS an area element. If T now has amaximum within the disc interior localised at some particular location ( s , s ), the high ν contribution is dominated by a small two-dimensional neighbourhood of this point. Even ifthe disc shape is globally complicated, locally it can always be represented as a flat planewith Cartesian ( x, y ) coordinates.Near the temperature maximum T max we proceed as in subsection 2.3.1, expanding the β function around the coordinates x max , y max : β = 1 kT max + β xx ( x − x max ) β yy ( y − y max ) ... (26)where the subscript x or y denotes partial differentiation with the other variable held fixed.The first order partial derivatives vanish at the maximum, as does the mixed derivative β xy for the proper choice of coordinates. The second order (nonmixed) derivatives are understoodto be evaluated at x max , y max . Exactly the same reasoning as before leads to the emission(double) integral I = ν F max exp[ − hν/kT max ] (cid:90) ∞−∞ (cid:90) ∞−∞ exp( − hνβ xx x /
2) exp( − hνβ yy y / dx dy (27)where F max is F evaluated at x max , y max . This yields I = 2 πF max h ( β xx β yy ) / ν exp[ − hν/kT max ] (28)The frequency dependence is exactly that of a classical axisymmetric disc with a temperaturemaximum at the inner boundary. On the other hand, for a disc with a localised hot spot atthe inner boundary, exactly the same techniques we have been using show that the large ν asymptotic form is I ∼ ν / exp( − hν/kT max ) (Hot spot on inner boundary . ) (29)
3. Discussion
We have shown that at large photon energies, CDT yields a mathematically simple—and possibly observationally interesting—difference in the frequency dependence evinced by 9 –a thin disc and a true Wien spectrum. For a Novikov-Thorse relativistic disk, the differenceis yet more pronounced. Of greater astrophysical significance, perhaps, is our finding thata spectral changes of comparable magnitude and simplicity occur in going from zero tomoderate stress at the disc’s inner boundary. This arises because, unless the stress is verysmall (cf eq.[11]), the maximum disc temperature is reached on this inner boundary . Byconstrast, a zero stress constraint always results in a temperature peak within the discinterior. These different locations of the maxima cause measurably different high energyspectra: a ν / ( ν ) power law multiplying an exponential for the case of an interior (edge)maximum. (Recall that a Wien spectrum has a ν power law prefactor.) For a relativisticdisc, or any other disc in which the maximum is a localised hot spot, the scalings are ν ( ν / ). These findings stand on their own, but since the question of the presence or absenceof an inner stress, which is likely to be magnetic in origin (e.g. Agol & Krolik 2000), is alively and contested issue (Beckwith, Hawley, & Krolik 2008), the current results may notbe devoid of practical significance.At the very least, our findings are useful benchmarks for numerical calculations of discspectra. Real discs, onthe other hand, live in messy accretion environments, often with manyspectral components. Principal sources of confusion at high frequencies include emissionfrom a hot corona and Comptonisation of soft photons. Even here, there is some utility inknowing the precise frequency dependence of a thermal disc, if only as a baseline from whichto mark differences. Moreover, there are discs with minimal coronal components, and themodels studied by Shimura & Takahara (1995; see also the discussion of Davis et al. 2005)indicate that the effects of Comptonisation can be well-modelled by replacing the Planckfunction in equation (1) by a “dilute blackbody” form. This modification would not changethe frequency dependence of our formulae. Asymptotic expansions in the high frequencylimit of spectral integrals, together with data that promises to be ever more accurate, willboth strengthen and deepen our understanding of compact X-ray sources. Development andobservational applications of these findings, as well as detailed comparisons with relativisticdisc models (Novikov & Thorne 1973), are currently being pursued. This is true provided the stress follows the precepts of CDT and causes local dissipational heating asdescribed by equation (10). Whether magnetic stresses behave in this manner is currently being investigated.(I thank J. Krolik for drawing my attention to this important point.)
10 –
Acknowledgements
It is a pleasure to thank Shane Davis, Julian Krolik, and Chris Done for extended cor-respondence and important advice. I would also like to acknowledge detailed conversationswith Omer Blaes, Mari Kolehmainen, Will Potter (who also kindly prepared figures 1 and2), and helpful comments from an anonymous referee. Support from the Royal Society inthe form of a Wolfson Research Merit Award is gratefully acknowledged.
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