A precessing Be disk as a possible model for occultation events in GX 304-1
Matthias Kühnel, Richard E. Rothschild, Atsuo T. Okazaki, Sebastian Müller, Katja Pottschmidt, Ralf Ballhausen, Jieun Choi, Ingo Kreykenbohm, Felix Fürst, Diana M. Marcu-Cheatham, Paul Hemphill, Macarena Sagredo, Peter Kretschmar, Silvia Martínez-Núñez, José Miguel Torrejón, Rüdiger Staubert, Jörn Wilms
MMNRAS , 000–000 (2017) Preprint 20 March 2018 Compiled using MNRAS L A TEX style file v3.0
A precessing Be disk as a possible model for occultation events inGX 304 − M. Kühnel, (cid:63) R. E. Rothschild, A. T. Okazaki, S. Müller, K. Pottschmidt, , R. Ballhausen, J. Choi, I. Kreykenbohm, F. Fürst, D.M. Marcu-Cheatham, , P. Hemphill, M. Sagredo, P. Kretschmar, S. Martínez-Núñez, J. M. Torrejón, R. Staubert, and J. Wilms Dr. Karl Remeis-Observatory & ECAP, Universität Erlangen-Nürnberg, Sternwartstr. 7, 96049 Bamberg, Germany Center for Astrophysics and Space Sciences, University of California, San Diego, La Jolla, CA 92093, USA Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605, Japan CRESST/CSST/Department of Physics, UMBC, Baltimore, MD 21250, USA NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA Harvard Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA European Space Astronomy Centre (ESA/ESAC), Operations Departement, 28691 Villanueva de la Cañada, Madrid, Spain MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Ave, Cambridge, MA 02139, USA Instituto de Física de Cantabria, Avda. los Castros s/n, 39005 Santander, Spain Instituto Universitario de Física Aplicada a las Ciencias y las Tecnologías, University of Alicante, P.O. Box 99, 03690 Alicante, Spain Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We report on the
RXTE detection of a sudden increase in the absorption column density, N H ,during the 2011 May outburst of GX 304 −
1. The N H increased up to ∼ × atoms cm − ,which is a factor of 3–4 larger than what is usually measured during the outbursts of GX 304 − RXTE . Additionally, an increase in the variability of the hardness ratio ascalculated from the energy resolved
RXTE -PCA light curves is measured during this timerange. We interpret these facts as an occultation event of the neutron star by material in theline of sight. Using a simple 3D model of an inclined and precessing Be disk around the Betype companion, we are able to qualitatively explain the N H evolution over time. We are ableto constrain the Be-disk density to be on the order of 10 − g cm − . Our model strengthens theidea of inclined Be disks as origin of double-peaked outbursts as the derived geometry allowsaccretion twice per orbit under certain conditions. Key words:
X-rays: binaries – stars: Be – stars: neutron – occultations – pulsars: individual:GX 304 − The H α emission line observed in the optical spectra of many B-stars is believed to originate from a circumstellar disk (see, e.g.,Hanuschik 1996). The H α line profile in these stars often shows adouble-peaked structure due to intrinsic rotation of the disk. Theseemission features are denoted with an “e” in the stellar spectral type,which is why these stars are known as Be stars. The Be-star disksare supposed to form by the viscous diffusion of gas ejected fromthe central star (Lee et al. 1991). They are geometrically thin androtating at Keplerian velocities (see Rivinius et al. 2013, for a recentreview).Circumstellar disks around main-sequence stars have been pro- (cid:63) E-mail: [email protected] posed to explain the transient nature of some X-ray binaries (Rap-paport et al. 1978). In these so-called Be X-ray binaries (BeXRBs)a compact object is on a wide and eccentric orbit around a Be-typecompanion star. Around the periastron passage mass transfer fromthe Be-star disk becomes possible, which results in a sudden andluminous X-ray outburst. Furthermore, the Be disk in BeXRBs istidally truncated at a certain resonance radius due to the presence ofthe orbiting neutron star (Reig et al. 1997; Negueruela & Okazaki2001; Okazaki et al. 2002). This truncation leads to higher surfacedensities of BeXRB-disks compared to disks around isolated stars(Okazaki et al. 2002, in agreement with findings by Zamanov et al.2001). However, Zamanov et al. (2001) have drawn this conclu-sion based on measurements of H α equivalent widths and the peakseparation of the line profile.BeXRBs offer the opportunity to measure directly the Be- © 2017 The Authors a r X i v : . [ a s t r o - ph . H E ] J un M. Kühnel et al.
Table 1.
Orbital parameters of GX 304 − γ freemodel).Orbital period, P orb (d) = . ± . a sin i (lt-s) = ± e = . ± . τ (MJD) = . ± . ω ( ◦ ) = 130 . ± . disk density independently of other energy ranges such as opticalwavelengths. The particle density in the line of sight determines theabsorption of X-rays at energies below ∼
10 keV. As the X-ray lineof sight is fixed to the neutron star, which moves along its orbit, theparticle density around the Be star can be probed.The BeXRB GX 304 −
1, detected in 1967 by a balloon experi-ment (Lewin et al. 1968), consists of a shell Be-star of type B2Vne(Mason et al. 1978; Parkes et al. 1980) or B0.7Ve (Liu et al. 2006)and a neutron star which shows X-ray pulsations around ∼
272 s(McClintock et al. 1977). It is located at a distance of 2.4(5) kpc(Parkes et al. 1980) probably behind the Coalsack Nebula, a darkmolecular cloud on the Southern sky at a distance of ∼
150 pc (seeNyman 2008, and references therein).GX 304 − − P orb = . ( ) dbased on the outburst spacing in the MAXI-GSC light curves dur-ing this series. Together with pulse frequency measurements by Fermi -GBM (Finger et al. 2009) they derived the remaining orbitalparameters of GX 304 − − RXTE observations taken during four outbursts between March 2010and May 2011. Here, we particularly focus on the evolution of theabsorption column density, N H , over time. The paper is organized asfollows: in Section 2 we briefly describe the data reduction processperformed by R17. Section 3 gives a summary and discussion oftheir spectral results, which are then modelled by a simple 3D-modelof a precessing Be disk in Sect. 4. We present our conclusions inSect. 5. a b c d B A T - c t ss − c m − N H ( c m − ) H a r dn e ss Orbital Phase H a r dn e ss V a r i a b . Figure 1. a)
Count rate of GX 304 − Swift -BAT as a function of or-bital phase during the outbursts in 2010 March/April (green diamonds),2010 August (black circles), 2010 December (blue triangles), and 2011 May(red squares). b) Evolution of the absorption column density, N H , and c) evolution of the hardness ratio and d) its variability over the orbital phase. RXTE has observed GX 304 − − ∼
272 s (McClintock et al. 1977). These observations have beenexcluded from the spectral analysis, which focuses on pulse phaseaveraged spectra.
The data reduction process and the spectral results analysed furtherin Section 3 are the same as in R17. In the following a brief summaryof the data reduction is given.The Proportional Counter Array (PCA; Jahoda et al. 2006) on-board
RXTE consisted of five identical Proportional Counter Units(PCU0–5). For spectral analysis only data from the top layer ofPCU2 have been extracted since this PCU is known to be the bestcalibrated one (Jahoda et al. 2006). We used the tools distributedby
HEASOFT v6.18 in order to extract spectra and lightcurves us-ing the standard2f data mode. We avoided PCU2-events up to 30
MNRAS000
MNRAS000 , 000–000 (2017) precessing Be disk as a possible model for occultation events in GX 304 − minutes from the start of the previous passage of the South AtlanticAnomaly and for elevation angles smaller than 10 ◦ above the Earth’slimb. Due to the high count rate of GX 304 − ≤
15 keV and of 1.5% at higher energies in order to achievea reduced χ , χ , near unity (see R17 for the affected ObsIDs).The PCU2 light curves were extracted with a 16 s time resolution inthe detector channels 4–15 and 15–60 (corresponding to the energybands 2.9–7.7 keV and 7.7–30.0 keV, respectively). These channelshave been chosen such that the ratio of the light curves, i.e., a hard-ness ratio, provides a handle on X-ray absorption (see Sect. 3.2 forfurther details).The High Energy X-ray Timing Experiment (HEXTE; Roth-schild et al. 1998) is sensitive for X-ray photons between 15 and250 keV and consisted of two identical clusters, HEXTE-A and -B.These clusters alternated between the on-source position and twobackground positions. This so-called “rocking” mechanism allowedone to simultaneously measure the X-ray source and background.Due to mechanical failures of this technique late in the mission, clus-ter A was fixed in the on-source position during all observations ofGX 304 −
1, while cluster B was fixed in one background position1.5 ◦ off-source. In order to investigate background corrected spec-tra for GX 304 −
1, we have extracted the source spectrum fromcluster A and estimated its corresponding background from thebackground spectrum of cluster B using the hextebackest tool asdescribed in Pottschmidt et al. (2006) and R17, Appendix A. Theresulting HEXTE-spectra of GX 304 − We have used the
Interactive Spectral Interpretation System (ISISHouck & Denicola 2000) v1.6.2-30 to perform a combined spectralanalysis of the PCA- and HEXTE-spectra for each
RXTE observa-tion. We use the results of R17, who describe the
RXTE spectra withtwo different continuum models. The cutoffpl model consists ofa power-law in combination with a multiplicative exponential andan additional blackbody (
CUTOFFPL + BBODY ). The second model highecut is a power-law continuum with an exponential roll-over,which sets in at higher photon energies (
POWERLAW × HIGHECUT ).Both models take low energy X-ray absorption into account usingthe model
TBnew which is and extended version of the model byWilms et al. (2000) with interstellar element abundances. Here,the corresponding cross-sections were set to those of Verner et al.(1996). The emission lines of iron at 6.40 keV and 7.06 keV weremodelled with Gaussians ( GAUSS ), and the cyclotron line known tobe present in GX 304 − GAUABS ).Furthermore, as discussed by R17 several systematic featuresare detectable in the
RXTE spectra. Although the background for http://pulsar.sternwarte.uni-erlangen.de/wilms/research/tbabs/ HEXTE-A can be estimated from HEXTE-B, the true backgroundstill differs from the estimated one. This results in four additionalbackground lines at 30.17, 39.04, 53.0, and 66.64 keV, which wehave modelled by additional Gaussian components. In some ob-servations slight residuals in absorption are seen in PCA below ∼ CORBACK in ISIS, which is similar to
RECOR inXSPEC). A detailed discussion of these systematic features in the
RXTE spectra, in particular the HEXTE background lines, is givenin R17, Appendix C.The resulting evolution of the spectral parameters, especiallythose of the power-law continuum and cyclotron line, for both con-tinuum models ( cutoffpl and highecut , see above) is presentedin R17. In this work, we focus on the time evolution of the absorptioncolumn density, N H , which tracks the number of particles along theline of sight to the neutron star. As already noticed by R17 a largeenhancement of the column density is detected during the 2011 Mayoutburst (see Fig. 1b), where N H is about × − RXTE . This enhancement eventlasts for around 3 days until N H suddenly drops to the usual value. Aweaker event ( × .
5) is also visible during the 2010 December out-burst. This result is independent of the choice of continuum model( cutoffpl and highecut ) and, thus, it is unlikely that the lowenergy continuum of the neutron star has been modelled improp-erly during these enhancement events. We will restrict the followingdiscussions to the results of the cutoffpl model.
In order to confirm the enhancement in N H in an independent wayof any phenomenological modelling of the X-ray spectra, we haveinvestigated the energy resolved RXTE light curves of GX 304 − (cid:46)
10 keV, the ratio between two light curvesin different energy bands, a so-called hardness ratio, then tracksabsorption variability without analyzing the full X-ray spectrum.We have computed the ratio of two PCA light curves (1 s timeresolution) in the energy bands 2.9–7.7 keV and 7.7–30.0 keV. Dueto the energy dependence of GX 304 − ∼ .
5, i.e., by a factor of ∼ .
4. Outside of the enhancementevent the hardness is again consistent with the data from the otheroutbursts. This confirms the results of our spectral analysis.We also found that the standard deviation of the hardness ratioover time during each observation (Figure 1d) shows a very similarincrease by a factor of ∼ . MNRAS , 000–000 (2017)
M. Kühnel et al. r T a -20 r T b -20 x disk (R ⋆ ) y d i s k ( R ⋆ ) x disk (R ⋆ ) z d i s k ( R ⋆ ) -13-15 log( ρ ) (g cm − ) Figure 2.
Assumed density profile of the Be disk as calculated after Eqs. 1–4for a base density of ρ = − g cm − , a truncation radius r T / a = . a =
601 lt-s in Cartesian coordinates ( x disk , y disk , z disk ). The white circle in the disk’s centre is the Be star. The arrow inthe density scale marks the density at the truncation radius. a) Densityprofile within the disk plane ( z disk =
0) showing the radial dependence. b) Height dependence of the density profile, i.e., perpendicular to the diskplane ( y disk = In order to investigate the origin of these enhancement events, wehave used the orbital parameters by Sugizaki et al. (2015, see Ta-ble 1) to convert the date of each
RXTE observation into an orbitalphase. Figure 1b shows the measured absorption column densityduring all four outbursts over the resulting orbital phases. The twoenhancement events nicely align between the orbital phases − . − .
01, which is right before the periastron passage of the neu-tron star. In these events no significant flare was detected on top ofthe
Swift -BAT light curve of GX 304 − RXTE light curves (see Fig. 1 of R17). Thus,it is unlikely that the matter, which is responsible for the enhance-ment in N H , was accreted by the neutron star as this would result inan additional increase in X-ray luminosity.From GX 304 − π ( a sin i ) ∆ t / P orb ∼
86 lt-s as the di-mension of the cloud with the duration ∆ t = − ω = ◦ (seeTable 1), which means that this point, as seen from Earth, is be-hind the companion’s tangent plane of the sky. That is, the neutronstar is farther away from Earth during the periastron passage thanthe Be companion. Thus, the X-ray line of sight, which is alwaysfixed on the neutron star, might has passed through or behind thecircumstellar material of the companion, such as its equatorial disk.In this Section, we investigate the possibility of an occultationevent of the neutron star by the Be disk. Therefore we introducea simple 3D-model of a rigid Be disk with a physically motivateddensity profile and apply this model to the observed N H evolution. According to the viscous decretion disk scenario (Lee et al. 1991),Be disks are formed by viscous diffusion of gas ejected from thecentral star. They are Keplerian disks, radially supported by the rotation, and in hydrostatic equilibrium in the vertical direction,supported by the gas pressure. As the model of the Be disk inGX 304 −
1, we assume for simplicity that the disk is isothermal at T d = . T eff (Carciofi & Bjorkman 2006). We also assume that thedisk does not change over the X-ray activity period and its densityprofile, ρ ( r , h ) , is cylindrically symmetric, i.e., depending on thedistance, r , to the symmetry axis of the disk and the height, h ,above the disk plane: ρ ( r , h ) = ρ (cid:18) rR ∗ (cid:19) − n exp (cid:20) − h H ( r ) (cid:21) , (1)where ρ is the base density, i.e., the density at ( r , h ) = ( R ∗ , ) , n is a constant that characterizes the radial density distribution (see,e.g., Okazaki et al. 2013), R ∗ is the radius of the Be star, and H ( r ) is the vertical scale-height given by H ( r ) = c s Ω K ( r ) = (cid:32) kT d R ∗ µ m H GM ∗ (cid:33) / (cid:18) rR ∗ (cid:19) / , (2)where c s = ( kT d / µ m H ) / is the isothermal sound speed, with µ and m H being the mean molecular weight and the mass of thehydrogen atom, respectively, and Ω K = ( GM ∗ / R ∗ ) / is the Kep-lerian rotation velocity with the mass, M ∗ , of the Be star. We adopt R ∗ = (cid:12) , M ∗ =
10 M (cid:12) , and T eff = µ = .
62 for fully ionizedplasma with cosmic abundances. We fixed the density profile in-dex n = . −
1, the Be disk is thought to be trun-cated at a radius smaller than the binary separation at periastron.If the Be disk is coplanar with the binary orbital plane, the trunca-tion occurs at a resonance radius, r T , which mainly depends on theorbital eccentricity and the disk viscosity (Negueruela & Okazaki2001; Okazaki & Negueruela 2001). For instance, for the viscos-ity parameter α ∼ . −
1, the disk is truncated at the 6:1 resonance radius( r T / a = . −
1, however, it has a significantly larger radius thanthe coplanar disks, because of the weaker resonant torques, and islikely to fill the Roche lobe of the Be star (Lubow et al. 2015; Mi-randa & Lai 2015). Given that there are large uncertainties in theviscosity parameter (according to Clark et al. 2001 and Wisniewskiet al. 2010, α ∼ .
1, while Carciofi et al. 2012 assume α ∼
1) andthe misalignment angle, we examine the disk obscuration effect fortwo extreme disk cases, one truncated at the 6:1 resonance radius( r T / a = .
29) and the other that fills the Roche lobe radius averagedover the binary orbit ( r T / a = . d = a ( − e ) / , as r T = . q / d . q / + ln ( + q / ) , (3)where q = M ∗ / M X is the binary mass ratio assuming the canonicalneutron star mass of M X = . (cid:12) . For radii larger than the trun-cation radius, r T , we assume that the density distribution of the Bedisk is ρ ( r > r T , h ) = ρ T ( h ) (cid:18) rr T (cid:19) − m (4)with the density at the truncation radius ρ T ( h ) = ρ ( r T , h ) as calcu-lated after Eq. 1. The constant m > n leads to a faster decrease of thedensity profile than for radii r < r T and we assume m =
10, which isconsistent with numerical results obtained by Okazaki et al. (2002).
MNRAS000
MNRAS000 , 000–000 (2017) precessing Be disk as a possible model for occultation events in GX 304 − Figure 2 shows the radial (a) and height (b) dependence of thedensity profile as defined above in the case of r T / a = .
49 with a =
601 lt-s and ρ = − g cm − . In order to calculate the absorption column density, N H , we needto transform the line of sight to the neutron star, given in the bi-nary’s reference frame, into the Be-disk’s reference frame, i.e., intocylindrical coordinates. Figure 3 shows a sketch of the followingdefinitions. We define the origin of the binary’s reference frame tobe fixed at the position of the Be-type companion star. The xy-planeis equivalent to the tangent plane of the sky, i.e., the z -axis is parallelto the line of sight to the companion and pointing away from Earth.The position of the neutron star on its orbit is found by solvingKepler’s equation, E − e sin E = M , (5)for the eccentric anomaly, E . Here, e is the eccentricity of the orbitand M = π (cid:18) t − τ P orb (cid:19) (6)is the mean anomaly with the time of the observation, t , the timeof periastron passage, τ , and the orbital period, P orb (see Table 1for the orbital parameters of GX 304 − r ns = ( x ns , y ns , z ns ) T , is then given by (cid:169)(cid:173)(cid:171) x ns y ns z ns (cid:170)(cid:174)(cid:172) = R ( e y , ◦ − i ) (cid:169)(cid:173)(cid:171) a cos ω ( cos E − e ) − b sin E sin ω a sin ω ( cos E − e ) + b sin E cos ω (cid:170)(cid:174)(cid:172) (7)with the rotation matrix, R , around the unit vector along the y -axis, e y , and the angle 90 ◦ − i with the inclination, i , of the orbital planewith respect to the tangent plane of the sky. The semi-major axis, a ,is found using the measured value of a sin i , b = (cid:112) a ( − e ) is thesemi-minor axis, and ω is the argument of periastron.In order to transform any vector in the binary’s reference frameinto the reference frame of the Be disk with cylindrical coordinates( r , h ), we first calculate the normal vector, n disk , of the disk by takingthe inclination of the orbit, i , the misalignment angle of the disk, δ ,and the position angle of the disk, θ , into account, n disk = R ( n orb , θ ) R ( e y , ◦ − i + δ ) e x . (8)Here, e x , is the unit vector along the x -axis (of the binary’s referenceframe), and n orb is the normal vector of the orbital plane, n orb = R ( e y , ◦ − i ) e x . (9)Due to the fact that no enhancement event was detected in the earlieroutbursts of 2010, the Be disk was not in the X-ray line of sight,while a strong occultation event occurred in 2011 May. A likelyexplanation is that the Be disk in GX 304 − θ = ω disk − (cid:219) ω disk ( t − t ) , (10)with the initial position angle ω disk at the time t and the precessionfrequency, (cid:219) ω disk . Note that in the context of the binary’s referenceframe defined above, the position angle, θ , is measured between theline of sight, e z , and the highest point of the disk, both projected ontothe orbital plane (see Fig. 3). Finally, the transformation, T , of any Table 2.
Parameters of the Be-disk occultation model as defined in Eqs. 5–12. See the text for a detailed description and for the assumptions of fixingcertain parameters.Be-star radius R ∗ = (cid:12) (fixed)Be-star mass M ∗ =
10 M (cid:12) (fixed)Be-disk temperature T eff = r T (fixed)Be-disk base density ρ Density profile index n = . m =
10 (fixed)Orbital parameters (see Table 1) P orb , a sin i , e , τ , ω (fixed)Orbit inclination i Be-disk inclination δ Be-disk position angle at t ω disk Be-disk precession frequency (cid:219) ω disk Precession reference time t = MJD 55690 (fixed)Foreground absorption N H , frgrd vector, r , given in the binary’s reference frame into the cylindricalcoordinates of the reference frame of the Be disk is T : r → ( r , h ) = (| r − ( n disk · r ) n disk | , n disk · r ) . (11)The absorption column density is finally found by computingthe integral of the Be-disk density, ρ (Eqs. 1–4), along the line ofsight, i.e., along the z -axis up to the neutron star’s position: N H = N H , frgrd + m H ∫ z ns −∞ ρ ( T (( x ns , y ns , z (cid:48) ) T )) d z (cid:48) (12)Here, N H , frgrd accounts for constant interstellar foreground absorp-tion. Table 2 summarizes the fixed and free parameters of the fullmodel. Initial attempts to apply the model defined in Eq. 12 to the observed N H evolution over time shown in Fig. 1b revealed several issues.First, finding the best-fit using a commonly used χ -minimizationis complicated by several local minima within the χ -landscape,where the actual observed enhancement event in 2011 May is notmodelled at all. Furthermore, χ is much larger than unity evenduring outbursts where no occultation event is detected due to asignificant scattering of the observed N H values (see discussionbelow). Finally, from the calculation of the disk’s normal vector, n disk (Eq. 8), we expect a parameter degeneracy between the orbitinclination, i , and the Be-disk inclination, δ .In order to solve these issues, we applied a Bayesian analy-sis of the model to the N H evolution in form of a Markov chainMonte Carlo (MCMC) sampling approach after Goodman & Weare(2010). We used the emcee algorithm as implemented by Foreman-Mackey et al. (2013) and ported into ISIS by M.A. Nowak, whichis distributed via the ISISscripts . The advantages of this approachare that parameter degeneracies are automatically error propagatedand the algorithm always provides the most probable answer even incases of a bad goodness of the fit. The result of an emcee run is theprobability distribution for each free parameter of the model used.The most probable parameters, i.e., the best-fit parameters, corre-spond to the maxima found in the probability distributions. Theseare sampled by so-called walkers , which move within the parameterspace and are distributed uniformly at the beginning. MNRAS , 000–000 (2017)
M. Kühnel et al. t o E a r t h li n e o f s i g h t r e r h n d i s k r ns ω A P i o f t h e s ky t a n g e n t p l a n e o r b i t a l p l a n e z y x a) r e r h n d i s k r ns δ θ H B e d i s k P o r b i t a l p l a n e z b) Figure 3.
Definition of the reference frames as described in the text. a) The tangent plane of the sky (yellow) defines the cartesian reference frame ( x , y , z ;red arrows) with the z -axis and the line of sight (blue arrow) perpendicular to this plane. Dashed lines are within the orbital plane (orange). P and A denotethe position of the periastron and apastron on the neutron star orbit, respectively. b) The Be disk (blue) is inclined with respect to the orbital plane (orange).The position vector of the neutron star ( r ns ) is decomposed using the unit vectors of the Be-disk plane ( e r and n disk ). Dash-dotted lines are within the Be-diskplane. H marks the highest point of the disk above the orbital plane. Note that for clarity we show different orbital phases in plots a) and b). For each possible truncation radius ( r T / a = .
49 or 0.29, seeSect. 4.1), we have performed an emcee run with 1 000 walkers perfree parameter and 10 000 iteration steps. The resulting parameterchains were investigated on convergence, i.e, the acceptance rate isstable around 0.25 after ∼
500 iteration steps, which is in the range of0.2–0.5 as expected in convergence (Foreman-Mackey et al. 2013).Thus, we ignored the first 500 iteration steps in the following analysisof the parameter distributions.Almost all probability distributions, which are found by sortingthe corresponding parameter chain into a histogram, show two orthree distinct maxima, i.e., multiple possible solutions. These solu-tions are best seen in the 2D-probability distribution (i.e., similar toa χ contour map) between the orbit inclination, i , and the Be-diskinclination, δ . Figure 4 shows this distribution for a Be-disk trun-cation radius of r T / a = .
49. The peak with the highest probability(solution δ , anda moderate orbit inclination, i , while its the other way around forthe second highest peak (solution r T / a = .
29 even a thirdsolution appears (solution r T / a = .
49. In this way, the number of peaks in the probabilitydistribution of each parameter reduces to one, i.e., the solution isunique and we are able to provide the final parameter values forall possible solutions and the two extreme cases for the truncationradius. The most probable parameter value is found by determiningthe peak’s position using a polynomial fit around the maximum. Thelower and upper confidence levels are determined such that 68% ofthe peak’s total area is within this confidence interval. Thereby, a W a l k e r s B e - d i s k I n c li n a t i o n , δ ( d e g r ee s ) Orbit Inclination, i (degrees) solution solution Figure 4. i ,and the Be-disk inclination, δ , with respect to the orbital plane for the case r T / a = .
49. The line style indicates the 68% (solid), 90% (dashed), and99% (dotted) confidence level. In order to check the emcee result, the un-derlying colour map shows the corresponding χ at each point, calculatedindependently of the emcee run. The vertical and horizontal histogramscorrespond to the 1D-probability distributions of both parameters, i.e., sum-ming up the columns or rows of the map, respectively. The white, dashedboxes mark the regions which have been used to split the parameter chainsinto the two distinct solutions (000
49. The line style indicates the 68% (solid), 90% (dashed), and99% (dotted) confidence level. In order to check the emcee result, the un-derlying colour map shows the corresponding χ at each point, calculatedindependently of the emcee run. The vertical and horizontal histogramscorrespond to the 1D-probability distributions of both parameters, i.e., sum-ming up the columns or rows of the map, respectively. The white, dashedboxes mark the regions which have been used to split the parameter chainsinto the two distinct solutions (000 , 000–000 (2017) precessing Be disk as a possible model for occultation events in GX 304 − Table 3.
Most probable parameters for both extreme cases of Be-disk truncation and for all solutions discovered in the parameter space. See Table 2 for a briefdescription of the model parameters. r T / a [solution no.] 0 .
49 [ .
49 [ .
29 [ .
29 [ .
29 [ i (degree) 54 . + . − . . + . − . . + . − . . + . − . . + . − . δ (degree) 76 . + . − . + − . + . − . . + . − . . + . − . ω disk (degree) 121 . + . − . . + . − . . + . − . . + . − . . + . − . (cid:219) ω disk (degree yr − ) 190 + − + − + − + − + − log ( ρ / g cm − ) − . + . − . − . + . − . − . + . − . − . + . − . − . + . − . N H , frgrd (10 cm − ) 3 . + . − . . + . − . . + . − . . + . − . . + . − . χ / d.o.f. 252.03 / 63 303.11 / 63 313.08 / 63 290.51 / 63 307.33 / 63 linear continuum has been subtracted from the probability distribu-tion in order to investigate the area of the peak only. Table 3 liststhe resulting most probable parameters and uncertainties for eachinvestigated truncation radius and solution found in the parameterspace. For the case of r T / a = .
49, we compare in Fig. 5 the mod-elled evolution of N H for both solutions with the data. We also showthe resulting geometries of the binary and Be disk projected ontothe tangent plane of the sky, i.e., as seen from Earth. In this paper we have presented a study of the N H behaviour ofGX 304 − r T / a = .
49 and three for r T / a = .
29. However,these really reflect only three basic physical scenarios, which werefer to as solutions r T / a = .
49 and r T / a = .
29, while Solution r T / a = .
29 case. Solution χ is a large Be diskwith a high misalignment angle ( r T / a = .
49, χ = . χ valuesmay have several reasons: first, our simple model assumes a rigidand cylindrically symmetric Be disk. In particular, we ignore anystructures within the Be disk, such as warping and spiral densitywaves, which are known to be present due to the tidal interactionwith the neutron star (see, e.g., Okazaki et al. 2002 and Martin et al.2011). Such a behaviour might explain the observed scattering of the N H values, especially during the 2010 December outburst. Here, ourmost probable model suggests that the line of sight crosses the out-ermost parts of the Be disk near the truncation radius (Fig. 5), wherethe tidal effects are expected to be most prominent as discussed forthe BeXRB 4U 0115 +
634 by Negueruela & Okazaki (2001) andReig et al. (2007) and for A0535+262 by Moritani et al. (2013).Secondly, the N H measurements itself might be influenced by sys-tematic effects, such as the Xe L-edge, which R17 had to introducein order to model calibration uncertainties in RXTE -PCA. Furtherevidence for these systematics is that the emcee runs prefer a largeinterstellar foreground absorption of N H , frgrd ∼ . × cm − .Although GX 304 − Suzaku observations during its 2010 August and 2012 January out-bursts revealed N H ∼ × cm − (Jaisawal et al. 2016), whichis consistent with the foreground absorption found in 21 cm sur-veys (Kalberla et al. 2005; Dickey & Lockman 1990). Assuming asystematic uncertainty of 2 × cm − to be consistent to thesealternative foreground absorption measurements indeed results in a χ around unity. This does not, however, affect the significance ofthe occultation event in 2011 May.We have found that the base density of the Be disk, ρ , which ismainly determined by the measured N H values during the occulta-tion event in 2011 May, is on the order of 10 − –10 − g cm − . Thisresult is in very good agreement with the commonly accepted valueof 10 − g cm − (e.g., Okazaki et al. 2013) for disks of isolated Bestars. Although the Be disks in binaries are expected to be approxi-mately twice as dense as compared to isolated stars (Zamanov et al.2001), the solutions with densities significantly above 10 − g cm − would be unlikely, such as solution r T / a = . (cid:219) ω disk , of solution P disk , due to tidal forces on a misaligned disk is P disk = (cid:18) + qq (cid:19) / (cid:18) ar T (cid:19) / P orb cos δ (13)with q = M X / M ∗ . Using the resulting disk misalignment an-gle, δ , of solution r T / a = .
49 wefind P disk ∼
80 yr, which is ×
42 longer than the emcee result of360 ◦ / (cid:219) ω disk ∼ . emcee solutions does not weakenthis disagreement significantly. Since the material forming the Bedisk originates from the central Be star, the disk’s precession mightbe induced by an intrinsically precessing Be star. This precession isthe result of tidal forces on an oblate star such as a rapidly rotatingBe star. Using the theoretical investigations by Kopal (1959) andAlexander (1976), we have estimated the precession period of the MNRAS , 000–000 (2017)
M. Kühnel et al. S o l u t i o n N H ( c m − ) y (lt-s) x ( l t - s ) y (lt-s) y (lt-s) y (lt-s) N H (atoms cm − ) S o l u t i o n N H ( c m − ) y (lt-s) x ( l t - s ) y (lt-s) y (lt-s) y (lt-s) Figure 5.
Absorption column and projected geometry of the Be disk for the case of r T / a = .
49. The upper row shows the results for solution r T / a = .
29 are not shown as they are very similar. The top panels in each subfigure show measured(green/purple) and modelled (black) absorption column densities, N H , for all four outbursts of GX 304 − N H , frgrnd (see Table 3). The bottom panels in each subfigure illustrate the modelled Be-disk geometry and neutron star position as seen from Earth,i.e., the components of the binary are projected onto the tangent plane of the sky. The white circle in the centre of the reference frame is the Be star, the crossesmark the positions of the neutron star along its orbit (black line) during all observations. The blueish region is proportional to the N H along the line sightthrough the Be disk, and the blue circle marks the truncation radius, r T / a = .
49, of the Be disk within the disk plane. The highest point on the disk’s rimabove the orbital plane is connected with the central Be star by one dashed line and to the orbital plane by another line perpendicular to the latter. The dashedline from the Be star to the neutron star’s orbit marks the orbital phase of the highest point of the disk, i.e., the position angle of the disk, ω disk . Be star in GX 304 − r T / a = .
49 and r T / a = .
29 and similar mis-alignment angles, δ , as listed in Table 3. As a result from the KLmechanism, the Be-disk particles precess much faster around thebinary’s orbital momentum vector than compared to the circular ap-proximation of Eq. 13. For a Be-disk radius of r T / a = .
29 the pre-cessing time-scale is ∼
10 yr and decreases to 4–5 yr for r T / a = . α line profile, they have derived a period of ∼
674 d, which theyinterpret as the precession period of a warped Be-disk.The misalignment angles of the Be disk found by our analysisare in the range of 15 ◦ < δ < ◦ . The most probable solution( r T / a = .
49, ◦ . As discussed by Brandt & Podsiadlowski (1995) ahigh-velocity supernova kick of the neutron star’s progenitor mightresult in a large misalignment angle. In fact, they suggested that theBe disk in BeXRBs are typically not aligned with the orbital plane.An inclined Be disk in GX 304 − MNRAS000
49, ◦ . As discussed by Brandt & Podsiadlowski (1995) ahigh-velocity supernova kick of the neutron star’s progenitor mightresult in a large misalignment angle. In fact, they suggested that theBe disk in BeXRBs are typically not aligned with the orbital plane.An inclined Be disk in GX 304 − MNRAS000 , 000–000 (2017) precessing Be disk as a possible model for occultation events in GX 304 − B A T - c t ss − c m − a b shifts by R17 A v a il a b l e M a ss ( g ) Time (MJD) O r b i t a l P h a s e Figure 6. a)
The available Be-disk mass within the Roche radius of theneutron star as a function of time for r T / a = .
49 and solution
Swift -BAT light curve of GX 304 − b) The orbital phases (black crosses)for the maxima in a) around the observed outbursts. R17 introduced shiftsin orbital phase (purple circles) in order to match the falling edges of theoutburst light curves. tion for the occurrence of the two observed double-peaked outburstsof the source in 2010 June and 2010 October. We have investigatedthis idea further and calculated the available Be-disk mass withinthe Roche radius of the neutron star as calculated after Eq. 3 for themass ratio q = M X / M ∗ and d = a ( − e )/( + e cos ( f )) with thetrue anomaly, f , of the neutron star’s position. As shown in Fig. 6athe resulting mass within the Roche radius around the observeddouble-peaked outburst of GX 304 − − Swift -BAT light curve (shown on top of Fig. 6a). Our model failsto explain the observed double-peaked outburst around MJD 56080and the available mass during each outburst does not scale withthe observed count rate of GX 304 − Swift -BAT.The available mass can be modified, however, due to a warped disk,density fluctuations, or tidal streams. The latter results from over-flowing gas of the Be disk close to periastron, which gets unboundas soon as the Roche radius of the companion star decreases belowthe disk’s truncation radius, which is the case for r T / a (cid:38) .
30 inGX 304 −
1. Finally, deriving the mass accretion rate and, thus, theluminosity from the available mass requires further geometrical and hydrodynamical investigations, which is beyond the scope of thispaper.Figure 6b shows the orbital phases at which the available masswithin the neutron star’s Roche lobe was at a maximum (includingboth maxima for the double-peaked outburst around MJD 56220).The orbital phases scatter around the periastron passage between − .
08 and + .
12. For the 2010 August, 2010 December, and 2011May outbursts of GX 304 −
1, R17 noticed orbital phase shifts be-tween the falling edges of these outbursts. Interestingly, our derivedorbital phases match those of R17 to some extent (Fig. 6b, purplecircles).In summary, a precessing and inclined Be disk explains theobserved N H -evolution of GX 304 − − α line profiles, would helpsolving these issues in the future. ACKNOWLEDGEMENTS
MK acknowledges support by the Bundesministerium für Wirtschaftund Technologie under Deutsches Zentrum für Luft- und Raum-fahrt grants 50OR1113 and 50OR1207 and by the European SpaceAgency under contract number C4000115860/15/NL/IB. SMN ac-knowledges support by research project ESP2016-76683-C3-1-R. JMT acknowledges research grant ESP2014-53672-C3-3P. Wethank the Deutscher Akademischer Austauschdienst (DAAD) forfunding JC through the Research Internships in Science and Engi-neering (RISE). Parts of this work can be found in the Ph.D. thesisof Müller (2013). All figures within this paper were produced usingthe
SLXfig module, which was developed by John E. Davis. Wethank the
RXTE -team for accepting and performing our observa-tions of GX 304 −
1. Finally, we acknowledge the comments by thereferee, which helped improving the content of our paper.
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APPENDIX A: STELLAR PRECESSION IN BINARIES
In BeXRBs, if the rotation axis of the Be star is inclined with respect to theorbital plane, the compact object will exert a tidal torque on the Be star anddisk. The torque on the Be star will be non-zero if the Be star is oblate (dueto, e.g., fast rotation). In the following we estimate the resulting precessionperiod of the Be star using two formulations by Kopal (1959, hereafter K59)and by Alexander (1976, hereafter A76). For all calculations we assume theorbital parameters as listed in Table 1, a stellar mass and radius for the Bestar of M ∗ =
10 M (cid:12) and R ∗ = (cid:12) , respectively (taken from Table 2), anda neutron star mass of M X = . (cid:12) . The radial density profile of the Bestar as shown in Fig. A1 was calculated using the Evolve ZAMS (EZ) code (Paxton 2004) for an initial stellar mass and metallicity of 10 M (cid:12) and 0.02,respectively. We simulated stellar evolution up to an age of 15.6 Myr in orderto let the star expand up to the assumed radius of 6 R (cid:12) . When discussingother stellar radii or masses as shown in Fig. A2, we have linearly scaled thedensity profile for simplicity. In case of a different stellar mass, we scaledthe density such that the total enclosed mass, i.e., the integral of the densityprofile over the full radius results in the assumed mass. A1 Formulation by K59
Assuming that the inclination of the rotation axis of the primary star to thebinary orbit axis is small, K59 derived the period of stellar precession as(K59, Eq. II.5-62) P orb U = Γ + Π , (A1)where P orb is the orbital period, U is the period of precession of the primarystar, and Γ and Π are quantities defined by (K59, Eq. II.5-25) Γ = C (cid:48)(cid:48) C (cid:48) γ (A2) We have used the online tool EZ-Web developed by RichardTownsend, . MNRAS000
Assuming that the inclination of the rotation axis of the primary star to thebinary orbit axis is small, K59 derived the period of stellar precession as(K59, Eq. II.5-62) P orb U = Γ + Π , (A1)where P orb is the orbital period, U is the period of precession of the primarystar, and Γ and Π are quantities defined by (K59, Eq. II.5-25) Γ = C (cid:48)(cid:48) C (cid:48) γ (A2) We have used the online tool EZ-Web developed by RichardTownsend, . MNRAS000 , 000–000 (2017) precessing Be disk as a possible model for occultation events in GX 304 − P/ d 20050 e e =0 → a/ lt-s 800200 ω /ω crit r / R ⊙
85 155 m / M ⊙ a → b Relative Parameter Value P r e c e ss i o n P e r i o d ( y r ) Relative Parameter Value P r e c e ss i o n P e r i o d ( y r ) Figure A2.
Dependencies of the precession period on stellar and binary parameters after a) K59 and b) A76. The parameter values are normalized to the onesassumed for GX 304 − e = Π = a C (cid:48) − A (cid:48) m . (A3)Here, m is the mass of the primary, a is the semi-major axis of the binary,and γ is the ratio of the rotation frequency of the primary, ω , to the binaryorbital frequency (K59, Eq. II.5-11), γ = ω n = ω (cid:115) a G ( m + m ) , (A4)with the mass, m , of the secondary and the gravitational constant, G . Themoment of inertia of the primary perpendicular, A (cid:48) , and parallel, C (cid:48) , to itsrotational axis are given by (K59, Eq. II.3-34) A (cid:48) = π ∫ r ρ r (cid:48) d r (cid:48) − ( ∆ − ) ω r G (A5)and by (K59, Eq. II.3-39) C (cid:48) = π ∫ r ρ r (cid:48) d r (cid:48) + ( ∆ − ) ω r G , (A6)respectively. Here, ρ is the density of the primary star as a function of thedistance, r (cid:48) , to its centre up to the full radius, r . In each equation, the firstterm on the right hand side is the moment of inertia of the primary at rest,which is calculated for the stellar structure as shown in Fig. A1. The secondterm describes the contribution of the rotational deformation to the momentof inertia, where ∆ is related to the apsidal constant, k , as ∆ = + k . (A7)We assume k = .
01 given that the apsidal constant of 10 M (cid:12) main-sequence stars are typically in the range of (5–8) × − (Torres et al. 2010).The tidal bulge forming inside the primary star due to the gravitationalpull of the secondary modifies the moment of inertia. In the rotating referenceframe of the orbital plane, this moment of inertia, C (cid:48)(cid:48) , is calculated by (K59,Eq. II.3-46) C (cid:48)(cid:48) (cid:39) ( ∆ − ) m r d , (A8)where d is the distance of the secondary from the primary, which is givenby d = a ( − e ) + e cos f , (A9)with e and f being the eccentricity and the true anomaly. When orbitaveraged, Eq. A8 is reduced to C (cid:48)(cid:48) (cid:39) ( ∆ − ) m r a ( − e ) / . (A10)Using Eqs. A4–A6, the orbit average of C (cid:48)(cid:48) can also be written as C (cid:48)(cid:48) (cid:39) ( C (cid:48) − A (cid:48) ) Gm ω a ( − e ) / = m m + m ( C (cid:48) − A (cid:48) )( − e ) / γ . (A11) Note that the factor ( − e ) / is set to 1 in K59, where the orbit is assumedto be almost circular. We retain this factor in Eq. A11 since the orbits ofBeXRBs are generally eccentric.With Eq. A11, we can write Eq. A2 as Γ = C (cid:48)(cid:48) C (cid:48) γ = m m + m C (cid:48) − A (cid:48) C (cid:48) ( − e ) − / γ − . (A12)Note that the power of γ in Eq. A12 of − +
1, which has been noticed by, e.g., Walter (1975)already.Assuming that the Be star is rotating at the critical speed, ω crit , as anupper limit, ω = ω crit = (cid:113) Gm / r , (A13)and setting r = R ∗ , m = M ∗ , and m = M x we finally find U ∼ − A2 Formulation by A76
Alexander (1976) generalized the formulation of Kopal (1959) to an arbitraryangle, Θ , between the angular momentum vectors of the primary and thebinary orbit. If the angular momentum vectors of the Be star and its diskalign, Θ is then equal to the disk misalignment angle, δ , introduced inSect. 4.2. In comparison to Eq. A1, Alexander (1976) derived (A76, Eq. 2.21) − µ h cos Θ HH = π U , (A14)for the precession period, U , of the primary, which is negative for retrogradeprecession. Here, µ is a constant given by (A76, Eq. 2.18) µ = C + C R1 − C T1 C T1 C + C T1 , (A15)with the moment of inertia, C , of the primary star at rest, which is calculatedby (A76, Eq. 2.4) C = π ∫ r ρ r (cid:48) d r (cid:48) (A16)and equal to the first term on the right hand side in Eq. A6. The rotationaland tidal distortions to the total moment of inertia are are given by (A76,Eq. 2.5) C R1 = k ω r G (A17)and (A76, Eq. 2.14) C T1 = k m r a ( − e ) / , (A18)MNRAS , 000–000 (2017) M. Kühnel et al. respectively.The total angular momentum, H , of the binary is defined as (A76,Eq. 2.19) (cid:174) h + (cid:174) h + H (cid:174) s = (cid:174) H . (A19)The angular momentum of the primary, (cid:174) h , is expressed as (A76, Eq. 2.12) (cid:174) h = ( C + C R1 − C T1 ) (cid:174) ω + C T1 ( (cid:174) ω · (cid:174) s )(cid:174) s , (A20)with its angular velocity vector, (cid:174) ω , and the normal vector, (cid:174) s , of the orbitalplane. Since the secondary is a compact object, which is a point mass ingood approximation, its angular momentum is (cid:174) h = (cid:174)
0. The orbital angularmomentum of the binary, H , is given by (A76, Eq. 2.11) (cid:174) H = H (cid:174) s = m m m + m (cid:113) G ( m + m ) a ( − e )(cid:174) s , (A21)Assuming a misalignment angle Θ = − U ∼ − A3 Parameter dependencies
The precession period of the stellar companion star in GX 304 −
1, which wederived following K59 and A76, is around 5000 yr and both formulationsagree within a few hundred years. This period does not explain the shortperiod of ∼ m (red curve in Fig. A2), and the semi-major axis, a (green curve). This isdue to the fact that A76 uses Kepler’s third law, P a = π G ( m + m ) , (A22)to calculate the binary orbital period, P orb , while the equations in K59 havea direct dependency. Indeed, when substituting P orb in Eq. A1 with Eq. A22,both formulations agree very well. MNRAS000