Time delays between the soft and hard X-ray bands in GRS 1915+105

a r X i v :a s t r o -p h /0409671v 1 28 S e p 2004

Mon.Not.R.Astron.Soc.000,1–??(2004)Printed 2February 2008

(MN L A T E X style ?le v2.2)

Time delays between the soft and hard X-ray bands in

GRS 1915+105

A.Janiuk 1?,

B.Czerny 1

1

Nicolaus Copernicus Astronomical Centre,Bartycka 18,00-716Warsaw,Poland

2February 2008

ABSTRACT

The hard X-ray lightcurves exhibit delays of ～1s with respect to the soft X-ray

lightcurves when the microquasar GRS 1915+105is in the state of frequent,regular outbursts (states ρand κof Belloni et al.2000).Such outbursts are supposed to be driven by the radiation pressure instability of the inner disc parts.The hard X-ray delays are then caused by the time needed for the adjustment of the corona to changing conditions in the underlying disc.We support this claim by the computation of the time evolution of the disc,including a non-stationary evaporation of the disc and mass exchange with the corona.

Key words:accretion,accretion discs –black hole physics,instabilities,stars –binaries –close,X-rays

1INTRODUCTION

The observations of Galactic black hole binaries (GBH)im-ply the coexistence of a relatively cold,optically thick ac-cretion disc,responsible for a thermal disc-blackbody com-ponent in their soft X-ray spectra,with a hot,optically thin medium that is the source of power-law spectral tail in the hard X-ray band (see e.g.Done 2002for a review).At least in the High and Very High spectral states,the latter may have the form of a corona above the accretion disc,which means that the geometrical con?guration of these two me-dia is such that both are extending down to the last sta-ble orbit around the black hole,being vertically separated from each other.The hard X-rays are produced via Comp-ton upscattering of seed photons coming from the under-lying disc,which basically requires the energy dissipation within the corona.Apart from this radiative coupling,also a mass transfer,i.e.evaporation and condensation of matter between the disc and corona is possible (Meyer &Meyer-Hofmeister 1994;R′o ˙z a′n ska &Czerny 2000).

Radiation pressure instability (e.g.Taam &Lin 1984;Lasota &Pelat 1991)seems to be a plausible mechanism to account for the characteristic variability of the Galactic microquasar GRS 1915+105(Mirabel &Rodriguez 1994;Taam,Chen &Swank 1997;Belloni et al.2000).Exemplary lightcurves were analyzed recently by Naik et al.(2002).Some of these lightcurves exhibit a very regular shape of outbursts,that can be possibly related to the disc variabil-ity,while others are more chaotic and probably driven by other mechanism,e.g.jet emission.Time-dependent accre-?

E-mail:agnes@https://www.wendangku.net/doc/ef6492443.html,.pl

tion disc model with jet emission was studied e.g.by Nayak-shin et al.(2000)and Janiuk et al.(2002).Recently,Watarai &Mineshige (2003)analyzed the oscillations of this source allowing for the temporary evacuation of the inner disc.If the disc instability is the primary cause of regular outbursts,we may expect that the hard X-ray coronal emis-sion would be signi?cantly delayed with respect to the soft X-ray disc emission.In the present paper we check the hy-pothesis of the radiation pressure instability as the outburst driver by analyzing the observed time delays between the soft and hard X-ray emission.We study the behaviour of GRS 1915+105in various variability classes.We compare the observed time delays with the exemplary results of our theoretical model.In this model we compute numerically the time evolution of the disc-corona system with the mass exchange between the two media.

The structure of this article is as follows.In Section 2we present the results of analysis of the microquasar GRS 1915+106observations,obtained by the Rossi X-ray Timing Explorer (RXTE).The X-ray lightcurves were examined to determine the time lags between hard and soft X-ray bands in di?erent modes characteristic for the variability of this source.In Section 3we describe our model and assumptions about the disc and corona structure.The initial state of the disc plus corona system,as well as the prescription for the mass exchange,are described in Section 3.1,while the time-dependent equations,according to which this system subsequently evolves with time,are given in Section 3.2.The results of the evolution are given in Section 4.We discuss our model and compare its predictions with observations in Section 5.The conclusions are given in Section 6.

2 A.Janiuk &B.Czerny

0100200

300400500

2000

400060008000

time [s]

1.5-6 keV

0100200300400500

2000

40006000

8000 6.4-14.6 keV

Figure 1.The X-ray lightcurves of GRS 1915+105obtained from RXTE observation from June 202000(class ρ),in the energy bands 1.5-6keV and 6.4-14.6keV.

2TIME DELAYS

Fourier resolved time delays in the lightcurves of GRS 1915+105were analysed by Muno et al.(2001)for the periods of extended hard steady state.In the present pa-per we concentrate mostly on periods with signi?cant out-bursts.In this case Fourier resolved phase lag approach is not best suited since clear semi-periodic signal dominates each lightcurve.Therefore,we restore to the simplest direct delays,as measured by the cross-correlation function.

In order to determine the time lag between soft and hard X-ray emission we performed the Fourier analysis of these lightcurves by means of the Fast Fourier Transform (FFT)method.The cross-correlation function of two peri-odic functions F (t )and G (t )is de?ned as:Corr (?t )=

F (t )

G (t +?t )dt.(1)

For our analysis we select lightcurves representative for various variability classes,as studied in detail in Belloni et al.(2000).We choose the exemplary observations of GRS 1915+105made between 1996and 2000,available through the public RXTE archive.The data were binned to 0.256seconds and the lightcurves were generated for the two en-ergy bands separately:1.5-6keV (PHA channels 0-14)and 6.4-14.6keV (PHA channels 15-35).Each lightcurve consists of one or more intervals,and in the subsequent analysis we compare the single intervals between each other.

In Figure 1we show an exemplary lightcurve of the microquasar observed by RXTE on 20June,2000.This variability pattern belongs to the class ρof Belloni et al.(2000).The cross-correlation function is calculated for the two lightcurves obtained in this observation and shown in Figure 2.The maximum of this cross-correlation function de-?nes the time lag,?t ,between the two lightcurves.In

this

Figure 2.The normalized cross-correlation function of the two lightcurves,1.5-6keV and 6.4-14.6keV,shown in Fig.1.The main peak is shifted to -0.768s,which means that the hard X-ray lightcurve lags the soft X-ray one by 0.768s.

Table 1.Time lag of the hard X-ray lightcurve (6.6-14.6keV)with respect to the soft X-ray lightcurve (1.5-6keV).?t 1is cal-culated for the non-smoothened lightcurves,binned by 0.256sec.,and ?t 2is calculated for the lightcurves smoothened over ?T =1s.

20402-01-28-0018/05/97α0.00.020402-01-41-0019/08/97δ0.00.020402-01-37-0017/07/97γ0.00.020402-01-44-0031/08/97β0.00.010408-01-15-0016/06/96θ0.00.020402-01-36-0010/07/97λ0.00.25620402-01-33-0018/06/97κ0.2560.51250703-01-15-0120/06/00ρ0.768 1.024

Time-delays in GRS 1915+105

3

sec.The resulting

time lags are given in the last column of Table 1.We see that the time lags between regular outbursts are even more pronounced,if these outbursts are cleared from the stochastic variability.On the other hand,the time lag ?t 2remains only marginally detectable,or equal to zero,in the lightcurves that exhibit mostly the stochastic varia-tions.

The measured lags calculated for the two latter obser-vations,κand ρ,seem to be exceptionally high for an X-ray binary (˙Zycki,private communication).However,this is

not surprising since only GRS 1915+105exhibits outbursts while other sources show only stochastic type of variability,which must be of di?erent nature.

3

TIME-DEPENDENT DISC/CORONA MODEL

Accretion discs are known to be locally unstable in cer-tain temperature (corresponding to the accretion rate)and density ranges,in which the cooling and heating balance is strongly in?uenced either by atomic opacities or by the ra-diation pressure.Such instabilities do not disrupt the disc completely,but lead to repetitive outbursts.

The ?rst type of instability,connected with the partial hydrogen ionization (Smak 1984;Meyer&Meyer-Hofmeister 1984),operates in accretion discs of binary systems in the range of radii of the order of 104?105R Schw and is responsi-ble for the luminosity changes on timescales of months,e.g.Dwarf Nova outbursts.

The other instability,caused by the radiation pressure domination over the gas pressure,operates in the innermost regions of the disc,where the temperature exceeds ～106K,and is responsible for the disc variability on the shortest timescales (of the order of tens -hundreds of seconds).This radiation pressure instability was ?rst noticed in Pringle,Rees &Pacholczyk (1974)and studied in Lightman &Eard-ley (1974)and Shakura &Sunyaev (1976).Abramowicz et al.(1988)found that radial advection has stabilizing e?ect on the disc at high accretion rates and the time-dependent calculations of the disc limit-cycle behaviour were performed by Honma et al.(1991)and by Szuszkiewicz &Miller (1998).

The innermost disc regions in which the radiation pres-sure instability is possible,are covered by the hot corona.Direct comparison of the extension of the radiation pres-sure domination zone and the corona covered zone,resulting from the model of the stationary,two-temperature corona,was performed by Janiuk &Czerny (2000).In the time-dependent model of radiation-pressure instability proposed in Janiuk,Czerny &Siemiginowska (2002)we used a simpli-?ed description of a stationary corona above the ?uctuating disc,parameterized by a constant value of the fraction of gravitational energy dissipated in the corona (f cor ).Here we extend our model to the case of a non-stationary corona,that forms due to the continuous evaporation of material form the disc surface.Due to the mass exchange with the disc the corona follows its time-dependent behaviour and therefore periodic changes are expected also in the hard X-ray luminosity.

3.1Assumptions and model parameters 3.1.1

Disc

First we describe the initial steady state disc model,from which we start our subsequent calculations.Throughout the calculations we use the vertically integrated equations of the disc structure,as the disc geometrical thickness H is always small (H/r ～0.01in the quiescent disc and H/r ～0.1in the outburst;see Section 4.1).

The angular velocity of the disc is assumed to be Kep-lerian,?= P/ρ=

?H .Here ρis the gas density in g/cm 3

,M is the mass of the accreting black hole and G is the gravitational constant.A non-rotating,Schwarzschild black hole is assumed and the inner radius of the disc is always at 3R Schw .The outer ra-dius is equal to 300R Schw ,and at this radius a constant mass

in?ow,parameterized by the external accretion rate ˙M

ext is assumed.Only the innermost zone up to ～100R Schw is the subject to radiation pressure instability,while the rest of the disc is stable (the exact value of the radial extension of the unstable zone depends on the model;see Section 4.1).The mass of the black hole is assumed to be 10M ⊙.

For the disc heating we assume that the viscous stress tensor is proportional to the total pressure P :τr?=?αP,

(2)

and the vertically integrated heating rate is Q +visc =

3

m p

ρT (5)

P rad =

1

(1+(ξ/ξ0)2

(7)

where ξ=P rad /P gas (for the discussion of this parameter-ization see Section 5.1).For the model parameters α0and ξ0we assumed the values 0.01and 8.0,respectively.This prescription implies that for small to moderate values of ξwe have e?ectively the disc heating proportional to the total pressure,while for large values of ξthe viscosity is propor-tional to the gas pressure.Therefore the radiation pressure

4 A.Janiuk&B.Czerny

instability may still operate,contrary to the so-calledβ-disc prescription(Lightman&Eardley1974).

The cooling in the disc is due to advection and radiation and the radiative cooling is equal to:

Q?rad=P rad c

κΣ

(8)

whereτis the optical depth,Σ=ρH is the gas column density in g cm?2,c andσare physical constants,and we adopt the electron scattering opacityκ=0.34cm2/g.

The advective cooling in a stationary disc is determined from the global ratio of the total advected?ux to the to-tal viscously generated?ux(e.g.Paczy′n ski&Bisnovatyi-Kogan1981;Muchotrzeb&Paczy′n ski1982;Abramowicz et al.1988)

Q?adv=F adv

3ρGM

(9)

and

q adv=(12?10.5β)?ln T

?ln r

(10)

Hereβis the ratio of the gas pressure to the total pressure β=P gas/P=1/(1+ξ).In the initial stationary disc we assume that q adv is approximately constant and of the or-der of unity(in the subsequent evolution the advection will be calculated more carefully,with appropriate radial deriva-tives).

In order to calculate the initial steady-state con?gura-

tion,we solve the energy balance:F tot=Q+

visc =Q?

adv

+

Q?

rad .Here the total energy?ux F tot dissipated within the

disc at a radius r is calculated as:

F tot=

3GM˙M

r

)3/2 r?R Schw

m p

ρcor T cor(13)

The contribution from the electrons is neglected.The corona is hot and its ion temperature is assumed to be equal to the virial temperature:

T cor=T vir=GM

k

(14)

The initial con?guration of the corona is computed un-

der the assumption that its optical depth is equal to unity:

τcor=κΣcor=1.0(15)

and therefore the corona has a uniform surface density.

When the time evolution starts,the proper solution devel-

ops in the middle parts of the disc,but will be?xed at the

boundaries R in and R out by the above condition.

3.1.3Mass exchange(prescription I)

The mass exchange rate in the vertical direction,between

the disc and corona,is equal to the ratio between the locally

generated?ux used to evaporate the disc material and the

energy change per particle:

˙m z=

F

kT cor

(16)

(measured per surface unit,g/s/cm2).

In a stationary disc the generated?ux depends on the

accretion rate and the disc radius(see Eq.11).We as-

sume that during the time evolution the energy?ux lead-

ing to evaporation preserves this dependence.Since both

the energy dissipated within the corona and within the

disc can lead to disc evaporation,we assume that the en-

ergy?ux is proportional to the sum of the disc and corona

accretion rates,taken with di?erent numerical coe?cients:

F∝(0.5B1˙M cor+B2˙M disc).The coe?cients B1and B2are

in the range from0to1and express the fraction of the gen-

erated?ux that is used to drive the evaporation.The share

of the corona is always lower than the half of the total corona

?ux,since half of the?ux from the corona is directed toward

the observer,whereas the other half is directed towards the

disc and there reprocessed.

The total accretion rate is˙M=˙M cor+˙M disc and may

be locally constant in case of a stable disc.When the disc

is unstable,both local accretion rates strongly depend on

time and radius,and the relative contribution of the disc

and corona to the total?ow also vary.

Expressing the accretion rates through the local vari-

ables˙M cor=2πΣcor rv cor

r

and˙M disc=2πΣdisc rv disc

r

in the

corona and disc respectively,we obtain a useful formula for

the vertical mass transfer:

˙m z=

3

Time-delays in GRS1915+1055 3.2Time evolution

Having computed the initial disc and corona state we allow

the density and temperature of the disc and the density of

the corona to evolve with time.We solve the equation of

mass and angular momentum conservation:

?Σ

r ?

?r

(r1/2νΣ))?˙m z(18)

and the energy equation:

?T

?r =

T

12?10.5β

(

?Σ

?r

)(19) +

T

12?10.5β

(Q+?Q?).

Here

v r=3

?r

(νΣr1/2)(20)

is the radial velocity in the disc whileν=(2Pα)/(3ρ?)is the kinematic viscosity.The heating term is given by Equa-tion3and the cooling term Q?is given by Equation8, while the advection is included in the energy equation via the radial derivatives.

The evolution of the coronal density is given by mass and angular momentum conservation in the corona:

?Σcor

r ?

?r

(r1/2νcorΣcor))+˙m z.(21)

The radial velocity in the corona is calculated as:

v cor r =

3

?r

(νcorΣcor r1/2)(22)

withνcor=(2P corαcor)/(3ρcor?)and constant viscosity pa-rameter in the coronaαcor=0.01.There is no need to con-sider the thermal evolution of the corona,since its temper-ature is always equal to the virial temperature and does not vary with time.

We solve the above set of three time-dependent equa-tions using the convenient change of variables,y=2r1/2 andΞ=yΣ,at the?xed radial grid,equally spaced in y (see Janiuk et al.2002and references therein).The num-ber of radial zones is set to216.After determining the solutions for the?rst600time steps by the fourth-order Runge-Kutta method,we use the Adams-Moulton predictor-corrector method,allowing the time-step to vary,when needed.

We choose the no-torque inner boundary condition,Σin=T in=0for the disc.The outer boundary of the disc is parameterized by an external accretion rate˙M ext.If this accretion rate is high enough,the inner disc parts gradu-ally heat themselves and?nally end in the unstable regime, forcing the disc to oscillate.The boundary conditions in the corona are given by Equation15.

4RESULTS

4.1Surface density and temperature evolution The local solutions of the accretion disc model,in the sur-face density vs.temperature(Σ-T)plane,can be calculated for a stationary disc in the range of accretion rates.(Alter-natively,on the vertical axis we can have accretion rate˙M instead of the disc temperature.)These solutions lie along the S-shaped stability curve,whose position on the diagram depends on the model parameters:black hole mass,viscosity and radius(c.f.Janiuk et al.2002,Figs.1,3).

Both upper and lower branches of the S-curve are vis-cously and thermally stable.On the lower stable branch the gas pressure dominates;the middle branch is unstable(ra-diatively cooled and radiation pressure dominant),as shown in detail by Pringle,Rees&Pacholczyk(1974)and Light-man&Eardley(1974).The upper branch is stabilized in our model again by the dominant gas pressure,due to the mod-i?ed viscosity law.In case of the standard viscosity(with constantα)this branch would be stabilized mainly by ad-vection,as shown in Abramowicz et al.(1988).In our case the advection is also taken into account,but its role is never dominant.

The S-curve can also be plotted in the˙M?Σplane, for any chosen disc radius.This means that the temporary local solutions are determined by the mean(i.e.external) accretion rate in the disc.Whenever the external accretion rate is low,so that at all the radii in the disc the local solu-tion sits on the lower,stable branch,the accretion proceeds with this rate,which is constant throughout the disc and constant in time.But if the accretion rate is higher than some critical value,˙M ext>˙M crit,the solutions in the in-nermost annuli will?nd themselves on the unstable branch. The higher˙M ext,the more disc annuli will be unstable.This leads to the disc?uctuations,since the accretion cannot pro-ceed smoothly in the unstable mode.Therefore the local ac-cretion rate in the innermost strips changes periodically be-tween the lower and upper stable solutions,being no longer equal to˙M ext(the accretion rate starts to depend on radius and time).This is displayed in the local diagramsΣ?T that are resulting form the time-dependent model.

The exemplary stability curves of the accretion disc, calculated at several radii from the stationary disc model, are shown in Figure3(thin solid lines).The thick points represent the subsequent solutions of the time-dependent model.

The evolution of the disc on the surface density-temperature plane proceeds at?rst along the lower stable branch,up to the instability region.This is forced by the value of the external accretion rate parameter,which has to be large enough to drive the disc to the unstable con?gu-ration.Here we assumed˙M ext=1.5×1019g s?1,which is equal to0.45of the Eddington rate(for black hole mass of 10M⊙and e?ciency of1/16).The critical accretion rate in our model depends on whether the corona covers the disc or not;˙m crit=0.05for the plain disc,while in the case of a disc/corona system the corona has a stabilizing role and the critical accretion rate is about˙m crit=0.2.The accretion rates within the range7.5×1018?2.6×1019g s?1,were obtained for soft states of GRS1915+105by Sobolewska&˙Zycki(2003).

Firstly,in the starting,steady con?guration we assumed the accretion rate of˙m～1.5×10?2of the Eddington rate throughout the disc.Therefore at the beginning of the subse-quent time-dependent calculations the model has to saturate at the temperatures and densities imposed by the value of ˙M

ext,imposed at the outer disc radius.

Next,the evolution proceeds in a form of loops between the lower and upper branches.Each loop refers to a single cycle of the instability,and the size of this loop depends on

6 A.Janiuk &B.

Czerny

Figure 3.Local evolution of the disc on the surface density -temperature plane,plotted for 4values of radius:3.53,10.14,21.47and 80.71R Schw .The thin solid line marks the stability curve resulting from the initial steady disc model,and the solid points are the time-dependent solutions.

the location in the disc.For larger radii the loops become smaller and ?nally,in the outer disc regions (above ～80?100R Schw ),there are no instabilities and the disc remains stable all the time.The exact value of the maximal radius of the instability zone depends again on the model.In case of a plain disc,for the external accretion rate ˙m ext =0.45it is R max =100R Schw ,while for ˙m ext =0.56it is R max =110R Schw .In case of the disc with corona the extension of the unstable zone is respectively R max =80R Schw and R max =90R Schw .

Note,that in the initial steady disc model we use a sim-pli?ed

parameterization of advection,with q adv =1.0(see Equation 10).Therefore the upper stable branch does not represent exactly the advective branch that results from the time-dependent calculations,which are based on the equa-tions with radial derivatives.In fact,q adv is not constant throughout the disc and should depend on radius.However,this initial simpli?cation does not in?uence our results,since our starting model is located on the lower,gas pressure dom-inated branch.Here the advection is negligible,and the sub-sequent time-dependent solutions match the stability curve perfectly.

In Figure 4we show the radial pro?les of the surface density in the accretion disc,in several snapshots during such a loop (one instability cycle).In the minimum of a cycle the surface density in the inner parts of the accre-tion disc has a ?at radial distribution.When the outburst starts,there appears a sharp density peak in the outer part of the unstable zone,which then moves outward.In the max-imum of the cycle this peak is accompanied by the largest ?uctuation in the density distribution.The reason for these ?uctuations are the viscosity and angular momentum trans-

Figure 4.Radial pro?les of the disc surface density during the cycle of the evolution.The snapshots were made every 80seconds and the whole cycle lasts for 625seconds.he solid curve (1)refers to the minimum of the cycle (lowest disc luminosity,see also Figs.6and 8),the dashed curve (2)refers to the rise phase and the solid curve (3)refers to the maximum of the cycle.The dotted line (4)and other dotted lines (unnumbered)refer to the decay phase.

fer changes inside this propagating “density wave”.When the innermost radii of the disc switch to the hot state,the geometrical thickness of this region also rises,thus giving the rise to the kinematic viscosity.The increased transport rate results in the temporary density decay in the unstable zone,as the material starts to fall faster into the black hole.Simultaneously,at the inner edge of the disc there forms a temporary “bump”of material,forced by the no-torque inner boundary condition.

The density ?uctuation subsequently vanishes in the end of the cycle and during the decay phase the surface density in the unstable zone gradually rises,to reach the starting con?guration.

The corona evolves on a timescale much shorter than the disc.First,we investigate the corona formation in case of no mass exchange with the disc (B 1=B 2=0).The initial distribution of the surface density in the corona was ?at,as determined by Equation 15.When the evolution starts,the corona very quickly achieves its ?nal shape –the surface density distribution saturates after ～125sec.At the same time the disc evolves very slowly,being ready to start its oscillations after ～104sec.of a gradual rise in density and temperature.Since there is no coupling with the corona,these oscillations do not in?uence the corona structure.

Secondly,we proceed with the evolution in case of a substantial mass exchange between the disc and corona (B 1=B 2=0.5).In this case,the evaporation of the disc accelerates the corona formation,and after 100seconds the coronal surface density in the maximum of the radial

Time-delays in GRS 1915+105

7

Figure 5.Radial pro?les of the corona surface density in case of no mass exchange with the disc (B 1=B 2=0).The pro?les were calculated every 10seconds and the density saturated at its ?nal pro?le after ～125seconds of the evolution.distribution (around 10R Schw )exceeds 10g/cm 2.Then the corona is further fed with material by the disc as its evolu-tion proceeds along the stability curve and the coronal sur-face density further gradually increases.The maximum sur-face density saturates at ～80g/cm 2after the disc reaches the critical point on the stability curve,which determines the maximum density in the disc.

When the disc oscillations start,

the corona also follows its time-dependent behavior.In Figure 6we plot the surface density distribution in the corona during one cycle of the disc instability.The curves are plotted every 80seconds in case of a full cycle lasting ～625seconds.The solid line (1)is the coronal density in the minimum of the disc limit cycle,the dashed line (2)corresponds to the rise phase of the disc outburst and the solid line (3)corresponds to the maximum of the cycle (outburst).Immediately after the outburst the coronal density distribution comes back to the initial con-?guration (dotted lines)and remains there for the most of the cycle (4).Therefore during the decay phase of the disc,when the disc surface density gradually changes to reach a ?at distribution,there is no change of the density in the corona.

In Figure 7we show the rate of the mass exchange be-tween the disc and corona,˙m z ,during the disc instability cy-cle.In the minimum of the cycle the evaporation rate is very low and the maximal mass supply to the corona is achieved at r ～6R Schw .When the disc outburst starts,the evapora-tion rate grows dramatically in the middle of the unstable zone,while dropping to ˙m z <0at the outer edge of this zone.This is why the corona collapses locally at the outer parts,while expanding slightly in the inner parts during the disc outburst.In the decay phase the evaporation rate again

Figure 6.Radial pro?les of the corona surface density during the cycle of the evolution.The snapshots were made every 80seconds and the whole cycle lasts for 625seconds.The labels are the same as in Fig.4.

becomes low,with a decaying ?uctuation at the outer edge of the instability zone.4.2

Lightcurves

The lightcurves represent the luminosity of the disc and corona separately.For the optically thick disc the luminosity is given by:L disc =

R max

R min

Q ?rad 2πrdr =

4σ

Σ

2πrdr (23)

The luminosity of the corona is calculated under the

assumption that the corona is in the thermal equilibrium.The ions are heated by the viscous dissipation and either cool by advection or transfer their energy to the electrons,which in turn radiate e.g.in the Inverse Compton process.Therefore we have:

Q ?cor =Q +cor ?Q ?

adv =

32

?ln T cor ?r

)

and L cor =

R max

R min

Q ?cor 2πrdr

(25)

In Figure 8we show an exemplary lightcurve calculated for several cycles of the disc outburst.The disc limit cycle

is very strong since ˙M

ext =1.5×1019g s ?1substantially exceeds the critical value.Therefore the loops marked by the time-dependent solutions on the Σ?T plane encompass substantial range of temperatures and densities,and the un-stable region of the disc has the radial extension up to about

8 A.Janiuk &B.

Czerny

Figure 7.Radial pro?les of the mass exchange rate during the cycle of the evolution.The snapshots were made every 80seconds and the

whole cycle lasts for 625seconds.The labels are the same as in Fig.4.80R Schw .This results in regular,large amplitude outbursts

of ?log L =0.9and ?t =625s.The luminosity variations in the corona,however of the same frequency,are not that strong.Also,they are sometimes anti-correlated with the lu-minosity changes in the disc,since at the disc rise phase the coronal luminosity at ?rst drops and then rises slightly,to drop and rise again during the disc decay phase.Character-istically,between the disc outbursts the coronal luminosity decreases very slowly,with ?log L cor /?t ≈10?5,whereas the disc luminosity rises gradually,forming a ’wing’preced-ing the main outburst.4.3

Accretion rate

The mass exchange rate in our prescription depends on the accretion rates in the disc and corona (Equation 17).Since they strongly vary with time during the cycle of the disc evo-lution,the mass exchange can have locally negative value.This has been shown in Figure 7.

In Figures 9and 10we show the radial pro?les of the ac-cretion rates in the disc and corona,taken in di?erent phases of the cycle:between the disc outbursts (corresponding to phase “1”in Fig.7)and in the outburst peak (phase “3”).

In the minimum of the cycle,i.e.between the outbursts,the accretion rate in the inner parts of the disc,where most of the dissipation takes place,is relatively low.It rises sub-stantially in the maximum of the cycle,therefore causing the disc luminosity outburst.The coronal accretion rate does not change that much,and the corresponding luminosity changes are not very pronounced.This is because in certain radii,at the outer edge of the instability strip,the radial ve-locity becomes negative.In consequence,the both accretion rates and mass exchange rate,˙m z ,also become negative,

Figure 8.The time evolution of the disc and corona luminosity.Lower curve shows the luminosity outbursts of the disc,while the upper curve shows the simultaneous coronal ?uctuations.The

external accretion rate is ˙M

ext =1.5×1019g/s.which means that some amount of material that has been

evaporated to the corona,now goes back and sinks in the disc.Negative values of ˙m z are not unphysical:stationary models based on the disc-corona mass exchange of R′o ˙z a′n ska &Czerny (2000)predict coronal condensation at some radii for high accretion rates.The coronal surface density,and in turn the dissipation rate,is therefore reduced,even in the in-ner regions,as the mass is spread out over the whole corona.The temporary loss of the accretion rate can be estimated

as ?˙M =4πr ˙m z ?r and it can be as high as 1.9×1018g/s.Therefore almost the whole rise of the coronal accretion rate (and luminosity),triggered by the disc outburst,is immedi-ately compensated by this loss due to the sinking of material in the strip around ～50R Schw .

In addition,the accretion rate in the corona is negative in the outermost radii.The extension of this zone of nega-tive ˙M

cor depends on the outer boundary condition in the corona,which in our case is ?xed by Σcor (r out )=1/κ≈2.94.

It implies that ˙M

cor <0for r >100R Schw ,so the material slowly ?ows out from the corona at its outer radius.4.4

Alternative prescription for the mass exchange (prescription II)

In the above used prescription for the rate of mass exchange between the disc and corona we expressed the total locally generated ?ux as proportional to the sum of the accretion rates in the disc and corona.In case of a steady-state disc this is equivalent to the sum of the locally generated ?uxes by the viscous energy dissipation.However,when we con-sider the time evolution of an unstable disc,the radial ve-locity,and in turn the accretion rate,can have locally neg-ative values in some parts of the disc.This forces the mass

Time-delays in GRS 1915+105

9

Figure 9.Radial pro?les of the accretion rate in the disc (trian-gles)and in the corona (squares)between the disc outbursts.The time refers to the lightcurve shown in Fig.

8.

Figure 10.Radial pro?les of the the accretion rate in the disc

(triangles)and in the corona (squares)in the outburst.The time refers to the lightcurve shown in Fig.8.

exchange rate to locally decrease in the state of the disc out-burst,which is not the case for the viscously generated ?ux.In other words,the prescription for the mass exchange rate in the outburst of an unstable disc is no longer equivalent to the sum of the locally dissipated ?uxes.

Now we check whether the other,more

’conservative’

Figure 11.Radial pro?les of the corona surface density during the cycle of the evolution for the case of the mass exchange rate given by Eq.26.The snapshots were made every 80seconds and the whole cycle lasts for 625seconds.The labels are the same as in Fig.4.

prescription for the mass exchange rate can lead to di?erent results of the time evolution of the disc plus corona system.Instead the formula for the vertical mass transfer given by Equation 17,we use the following:˙m z =

3GM

f (r )

0.5B 1αcor Σcor

GM

10 A.Janiuk &B.

Czerny

Figure 12.The time evolution of the disc and

corona luminosi-ties,for the case of the mass exchange rate given by Eq.26.Lower curve shows the luminosity outbursts of the disc,while the up-per curve shows the simultaneous coronal outbursts.The external

accretion rate is ˙M

ext =1.5×1019g/s.GRS 1915+105,we adopt this prescription in further anal-ysis.

We calculated the cross-correlation function,as de?ned by Equation (1),for the theoretical lightcurves plotted in Figure 12.Our points were separated at least by 0.039s,and we found that the corona lightcurve lags the disc by 1.16seconds.This is shown in Figure 13.The time lag in this case is equal to about 0.5%of the duration of an out-burst.We note that the outburst duration depends on the

external accretion rate ˙M

ext and the lag depends on the coronal viscosity parameter α,so we could obtain a range of values for di?erent model parameters.

In Figure 14we show the luminosity-color diagram for one of the observations of GRS 1915+105,that shows the pronounced outbursts (class ρ,cf.Table 1).On the horizon-tal axis we plot the soft X-ray ?ux in the range 1.5-6keV,and on the vertical axis we plot the X-ray color,i.e.the ra-tio of the hard to soft ?uxes,F 6.4?14.6keV /F 1.5?6keV .During its evolution,the source follows a characteristic track,in the form of a loop on this diagram.In the lower region (the “ba-nana”shape)the soft luminosity is low,and the X-ray color is soft.The upper region (“island”)shape is characterized by substantial luminosity both in hard and soft X-ray bands.

In Figure 15we show the result of our modeling.On the horizontal axis of the theoretical luminosity-color diagram we plot the disc luminosity,while in the vertical axis we plot the ratio of the corona to disc luminosities.The “banana”shape is reproduced in our model quite well.However,we do not obtain the other distinct region on this diagram (the “island”).In our simulations the source moves only along the track shown in Fig.15,and after reaching the end of the

Figure 13.The normalized cross-correlation function of the disc

and corona lightcurves.The main peak is shifted to -1.16s,which means that the corona lightcurve lags the disc one by 1.16s.

“banana”region immediately jumps to the top-left corner of the diagram.

5DISCUSSION

The main simpli?cation of our model was the assumption that the temperature in the corona is constant and equal to the virial one.Therefore the corona is heated only via increase of its surface density and is not radiatively coupled to the disc.We do not consider here any speci?c radiative processes that are important in cooling of the coronal gas,https://www.wendangku.net/doc/ef6492443.html,ptonization.

On the other hand,this approach lets us solve the set of three time-dependent equations for the disc plus corona evo-lution,accompanied by the formula for the mass exchange rate,and avoid numerical problems with the fourth equa-tion for the corona https://www.wendangku.net/doc/ef6492443.html,ing a ?nite time-step we are able to follow the time evolution of the system as long as it saturates in a quasi-stationary state,with oscillations of constant amplitude and duration.We qualitatively and quantitatively check,how the behaviour of the underlying,unstable disc a?ects the corona and vice versa.

The full,two-or three dimensional treatment to the global long-term evolution of an accretion disc (or disc/corona system)is very complex.Up to now,the so-phisticated 2-D and 3-D accretion disc simulations (e.g.Agol et al.2001;Turner et al.2003)either treated the problem locally,or did not include radiative cooling.Turner (2004)made the ?rst attempt to compute the 3-D simulation in the ?ux-limited approximation;in this case however the plasma temperatures are still much too low to form the hot corona.Since the simulations are not able to reach the vis-cous timescale of the disc,the long-term evolution of an

Time-delays in GRS 1915+10511

2000

4000

6000

8000

0.3

0.4

0.5

0.6

0.7

0.8

soft L

Figure 14.The

luminosity-color diagram of the two observed lightcurves,shown in Fig.1.On the horizontal axis is the soft X-ray ?ux in the range 1.5-6keV,and in the vertical axis is the ratio of the hard to soft ?uxes,F 6.4?14.6keV /F 1.5?6keV .

Figure 15.The luminosity-color diagram of the two theoretical lightcurves,shown in Fig.12.On the horizontal axis is the soft X-ray (disc)luminosity,and in the vertical axis is the ratio of the hard to soft (corona to disc)luminosities.

X-ray source cannot be followed and any observational con-sequences of such a model are di?cult to be checked.There-fore the simple approach presented in this paper can still be valuable.5.1

Viscosity parameterization

In our choice of the viscosity law we follow the paper of Nayakshin,Rappaport &Melia (2000).The motivation of these authors was mostly observational:they needed a vis-cosity law which allows for a disc instability at the inter-mediate accretion rates but provides the stable solution at low and high accretion rates.Moreover,the stable upper branch should appear at accretion rates smaller or compa-rable to the Eddington rate.The standard αP tot viscosity law of Shakura &Sunyaev (1973)satis?es in a natural way all requirements but the last one.Upper stable branch in such solution forms due to advection (i.e.the “slim disc”;Abramowicz et al.1988),and the e?cient advection develops only for accretion rates much larger than the Eddington val-ues,de?nitely too high too account for the time behaviour e.g.of the microquasar GRS 1915+105.What is more,in most cases the unstable branch should cease to exist at all since most of the X-ray binaries accreting at high (but sub-Eddington)rates are quite stable and well described by the classical disc (Gierli′n ski &Done 2004).

Therefore,observations tell us that some modi?cations are absolutely needed.Either we must postulate very strong out?ow (which e?ectively cools the disc),or a modi?cation of the dissipation law itself.

We have much better understanding and numerous ob-servational constraints for the viscosity law when the gas pressure dominates.3-D MHD simulations of the magneto-rotational instability (MRI)well explain the nature of the angular momentum transfer and the rough value of the vis-cosity parameter.Still,some ad hoc modi?cations,like ad-ditional dependence on the disc thickness,are used to model the development of the ionization instability responsible for outbursts in cataclysmic variables and numerous X-ray tran-sients (e.g.Cannizzo et al.1995).

The theoretical background of the viscosity law is still quite poor in case when radiation pressure is important.Sev-eral authors in the past suggested that actually the whole idea of αP tot scaling is inappropriate,and instead the αP gas should perhaps be used even if the radiation pressure dom-inates (Lightman &Eardley 1974;Stella &Rosner 1984).Sakimoto &Coroniti (1989)argued that the magnetic ?eld will be expelled from the radiation pressure dominated disc by the buoyancy so the e?ectiveness of the angular mo-mentum transfer must decrease with an increase of radia-tion pressure.Other authors argued that whenever radiation pressure dominates,new kinds of instabilities develop which may modify the disc structure considerably.Gammie (1998)discussed the photon bubble instability,and Ruszkowski &Begelman (2003)argued this instability leads to the disc clumpiness which in turn in?uences the radiative transfer.

We expect some improvement of the viscosity model-ing with the future development of MHD simulations of ra-diation pressure dominated discs.So far,two such simula-tions were performed.2-D computations of Agol et al.(2001)lasted only for a fraction of the disc thermal timescale and as a result the disc collapsed to gas pressure dominated state.

12 A.Janiuk&B.Czerny

A3-D simulation of the disc which reached the thermal balance and lasted for about8thermal timescales was re-cently completed by Turner(2004).The disc did not achieve the full stability and showed long-lasting variations by a factor of a few but the instability was not as violent as pre-dicted by the standardαP tot mechanism(Szuszkiewicz& Miller1998;Janiuk et al.2002).The magnetic?eld lines were indeed partially expelled from the disc interior.In this simulation,the initial Shakura&Sunyaev(1973)state with α=0.01evolved to a complex state with mean dissipation level equivalent toα=0.0013.

Therefore,the modi?ed viscosity law,given by Eq.7, seems to be a plausible option in case of the accretion disc. Since in the corona the radiation pressure does not con-tribute to the total pressure,the classical viscosity parame-terization of the corona with constantαis also justi?ed. 5.2Comparison with observations

Our time dependent model of the non-stationary accretion onto a black hole gives at least a partial explanation of the complex variability of the microquasar GRS1915+105.Di-rect evidence for variable accretion rate in this microquasar comes from the spectral analysis(Migliari&Belloni2003), and a plausible mechanism that leads to the local accre-tion rate variations in the inner parts of a disc,giving the outbursts of appropriate amplitudes and durations,is the radiation pressure instability.

The microquasar is unique because of its large ampli-tude,regular outbursts.Such a behaviour is not observed in other Galactic X-ray sources,probably because the Ed-dington ratio in this microquasar is higher than in other black hole systems(Done,Wardzi′n ski&Gierli′n ski2004). Our model,at least qualitatively,explains this phenomenon. The radiation pressure instability leads to the strong out-bursts only if the accretion rate is higher than some critical value.Also,the corona has a stabilizing role and can sup-press the disc oscillations completely or make them less pro-nounced.Quantitatively,however,the reliable determina-tion of the critical accretion rate above which the outbursts occur,would require a more detailed modeling of viscosity within the disc and essentially the2-dimensional calcula-tions.This is beyond the scope of the present work.

The physical coupling between the disc and corona via the mass exchange leads in a natural way to the regular time delays between the disc(soft X-ray)and corona(hard X-ray)emission.The lag of the order of1s is comparable with the viscous timescale in the corona and is required by this hot?ow to adjust to the variable conditions in the underly-ing disc.Such lags are present in some observed lightcurves of the microquasar,i.e.these exhibiting outbursts that are possibly connected with the radiation pressure instability.

The strong variability of GRS1915+105manifests it-self also by characteristic tracks on the color-color and luminosity-color diagram.The bottom-left corner of the luminosity-color plot is occupied by the source during its gradual rise(the“wing”),preceding the outburst state. This,so called here“banana”shape is well modeled in our calculations.While the disc luminosity(soft X-ray?ux)is gradually rising before the outburst,the rise in the coronal luminosity is much?atter,leading therefore to the decreas-ing hard-to-soft?ux ratio(color).On the other hand,in the outburst peak,when the soft luminosity is the highest,the color starts rising.Just after the outburst there appears a somewhat?at maximum in both hard and soft X-rays,lead-ing to the other distinct region occupied by the source on the luminosity color-diagram:so called here“island”shape. This is not present in our model calculations,since our out-burst maximum is a sharp peak.Therefore we do not obtain a state of both high disc and corona?uxes,which in conse-quence would give the high color in the disc emission peak. Instead,the corona lags the disc emission and its maximum corresponds to the already decaying state of the disc.

6CONCLUSIONS

We presented the?rst results of the time-dependent calcu-lations of thermal-viscous evolution of accretion disc that is coupled to the corona by the mass exchange.Contrary to the accretion disc,the corona is stable,since there is only gas pressure included in its equation of state.However,it is also to a certain extent subject to similar changes as the accretion disc during its instability cycle,since it actively responds to the behaviour of the underlying disc.

The main conclusions of the present work are as follows:

?The luminosity outbursts in the disc are correlated with similar outbursts in the corona if the mass exchange rate is proportional to the sum of the locally dissipated?uxes( prescription II).The assumption that it is proportional to the sum of the local accretion rates(prescription I)is equiv-alent,but only during the disc quiescence.In the outburst this prescription leads to the luminosity dips in the corona rather than outbursts.Speci?cally,the lightcurves of the disc and corona are anti-correlated,with two dips in the corona corresponding to the peak of the disc luminosity.

?The luminosity pro?le between the outbursts in the corona is gradually decreasing,while in the disc it is grad-ually rising(prescription I).Alternatively,the prescription II leads to the slow rise of the coronal?ux,correlated with the steep rise of the disc.

?The coronal outbursts are of much smaller amplitude than the disc ones,but of similar duration.

?The outbursts in the corona lag the disc evolution by ～1second,which is in good agreement with observations of the microquasar GRS1915+105.

?The outer boundary condition in the corona determines if the material?ows out at the outer edge.For small surface densityΣout～3g/cm2the accretion rate in the corona is negative above～100R Schw.

?If the mass exchange between the disc and corona is proportional to the sum of the local accretion rates(pre-scription I),the material may also locally sink into the disc in the region around50R Schw.

Our model is only in part able to reproduce the tracks of the system on the luminosity-color diagram,having a“ba-nana”shape without the“island”one.This should be mod-eled in more detail,including also the spectral analysis.In addition,the tracks on the color-color diagram,not analyzed here,in some observed sources may have a form of hysteresis (Maccarone&Coppi3002;Zdziarski et al.2004).Since the phenomenon is connected with much longer timescales,than those considered here,the underlying instability mechanism

Time-delays in GRS1915+10513

should be di?erent.A plausible one seems to be the partial hydrogen ionization instability,which operates essentially in the same way,causing the disc to oscillate between the cold and hot phases.This is the subject of our future investiga-tions.

ACKNOWLEDGMENTS

We thank Ma l gosia Sobolewska for help in data reduc-tion and Agata R′o˙z a′n ska,Piotr˙Zycki,Friedrich Meyer and Emmi Meyer-Hofmeister for helpful discussions.This research has made use of data obtained through the HEASARC Online Service,provided by NASA/Goddard Space Flight Centre.This work was supported in part by grant2P03D00124of the Polish State Committee for Scien-ti?c Research.

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感官动词和使役动词

感官动词和使役动词 默认分类2010-05-28 23:14:26 阅读46 评论0 字号：大中小订阅 使役动词，比如let make have就是3个比较重要的 have sb to do 没有这个用法的 只有have sb doing.听凭某人做某事 have sb do 让某人做某事 have sth done 让某事被完成（就是让别人做） 另外： 使役动词 1.使役动词是表示使、令、让、帮、叫等意义的不完全及物动词,主要有make(使,令), let(让), help(帮助), have(叫)等。 2.使役动词后接受词,再接原形不定词作受词补语。 He made me laugh. 他使我发笑。 I let him go. 我让他走开。 I helped him repair the car. 我帮他修理汽车。 Please have him come here. 请叫他到这里来。 3.使役动词还可以接过去分词作受词补语。 I have my hair cut every month. 我每个月理发。 4.使役动词的被动语态的受词补语用不定词,不用原形不定词。 (主)He made me laugh. 他使我笑了。 (被)I was made to laugh by him. 我被他逗笑了。 使役动词有以下用法： a. have somebody do sth让某人去做某事 ??i had him arrange for a car. b. have somebody doing sth.让某人持续做某事。 ??he had us laughing all through lunch. 注意：用于否定名时，表示“允许” i won't have you running around in the house. 我不允许你在家里到处乱跑。 ******** 小议“使役动词”的用法 1. have sb do 让某人干某事 e.g:What would you have me do? have sb/sth doing 让某人或某事处于某种状态，听任 e.g: I won't have women working in our company. The two cheats had the light burning all night long. have sth done 让别人干某事，遭受到 e.g:you 'd better have your teeth pulled out. He had his pocket picked. notes: "done"这个动作不是主语发出来的。 2.make sb do sth 让某人干某事 e.g:They made me repeat the story. What makes the grass grow?

惠斯通电桥实验报告南昌大学

南昌大学物理实验报告 课程名称：_____________ 大学物理实验 实验名称：_______________ 惠斯通电桥 学院：___________ 专业班级： 学生姓名：_________ 学号： 实验地点：___________ 座位号： 实验时间：第11周星期4上午10点开始

、实验目的: 1. 掌握电桥测电阻的原理和方法 2. 了解减小测电阻误差的一般方法 、实验原理: (1) 惠斯通电桥原理 惠斯通电桥就是一种直流单臂电桥，适用于测中值电阻，其原理电路如图 7-4所示。若调节电阻到合适阻值时, 可使检流计 G 中无电流流过，即 B 、D 两点的电位相等，这时称为“电桥平衡”。电桥平衡，检流计中无电流通过， 相当于无BD 这一支路，故电源 E 与电阻R ,、R x 可看成一分压电路；电源和电阻 R 1 上面两式可得 R 2 桥达到平衡。故常将 R 、R 2所在桥臂叫做比例 臂,与R x 、R S 相应的桥臂分别叫做测量臂和比 较臂。 V B C 点为参考，贝y D 点的电位V D 与B 点的电位V B 分别为 R 2 R S R S V D R X 因电桥平V B V D 故解 R 2、R S 可看成另一分压电路。若以 R x 为 E 待测电阻，则有 R>< R X R S 上式叫做电桥的平衡条件，它说明电桥平衡时，四个臂的阻值间成比例关系。如果 1 10,10 1等)并固定不变，然后调节 金使电

(2)电桥的灵敏度

n R S R S 灵敏度S 越大，对电桥平衡的判断就越容易，测量结果也越准确。 此时R s 变为R s ，则有：R x R2 R s ，由上两式得R x . R s R s 三、 实验仪器： 线式电桥板、电阻箱、滑线变阻器、检流计、箱式惠斯通电桥、待测电阻、低压直流电源 四、 实验内容和步骤： 1. 将箱式电桥打开平放，调节检流计指零 2. 根据待测电阻(线式电桥测量值或标称值)的大小和 R 3值取满四位有效数字原则，确定比例臂的取值，例如 R 为数千欧的电阻，为保证 4位有效数字，K r 取 3. 调节F 3的值与R ＜的估计 S _____ S 的表达式 R S R S S-i S 2 _____________________ ES R i R 2 R s R x 1 R E % R i R 2R X Rg 2 R x R s R 2 R - R E 2 R R s R x (3) 电桥的测量误差 电桥的测量误差其来源主要有两方面，一是标准量具引入的误差, 二是电桥灵敏度引入的误差。为减少误差传递, 可采用交换法。 交换法：在测定R x 之后，保持比例臂 R -、R 2不变，将比较臂 R s 与测量臂R x 的位置对换，再调节 R s 使电桥平衡，设 电桥的灵敏程度定义: R i

第四部分 习题与思考题

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