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It might be fossil fields (e.g.thespectralcharacteristicsof ?).. or fields produced through a dynamo mechanism. | It might be fossil fields \citep[e.g. the
spectral characteristics of Of?p stars are indicative of organized magnetic fields, most likely
of a fossil origin according to][]{Wade2010}, or fields produced through a dynamo mechanism. |
Recent simulations by ? of subsurface convective zone in massive stars show dynamo-generated magnetic fields of the order of one kG. According to these authors. these magnetic fields might reach the surface of OB stars. | Recent simulations by \citet{Cantiello2010}
of subsurface convective zone in massive stars show dynamo-generated magnetic fields of the order of one kG. According to these authors, these magnetic fields might reach the surface of OB stars. |
In the Sun. magnetic braking results from solar wind material following the magnetic field lines that extend well beyond the stellar surface. | In the Sun, magnetic braking results from solar wind material following the magnetic field lines that extend well beyond the stellar surface. |
This coupling exerts a torque on the surface layers of the Sun. and this slows down its rotation. | This coupling exerts a torque on the surface layers of the Sun, and this slows down its rotation. |
Could such a process also be active in massive stars showing a sufficiently strong surface magnetic field? | Could such a process also be active in massive stars showing a sufficiently strong surface magnetic field? |
has recently discovered that σ Ori E (B2Vpe. ~ 10 kG) is undergoing rotational braking. | \citet{Townsend2010} has recently discovered that $\sigma$ Ori E (B2Vpe, $\sim$ 10 kG) is undergoing rotational braking. |
The spin-down time of 1.34 Myr is in good agreement with theoretical predictions based on magnetohydrodynamical simulations of angular momentum loss from a magnetized line-driven. wind. | The spin-down time of 1.34 Myr is in good agreement with theoretical predictions based on magnetohydrodynamical simulations of angular momentum loss from a magnetized line-driven wind. |
This gives some support to the hypothesis that at least a few massive stars may indeed suffer magnetic braking. | This gives some support to the hypothesis that at least a few massive stars may indeed suffer magnetic braking. |
Such an effect has for the moment never been included in massive star models: however. by modifying the internal distribution of . the angular velocity. magnetic braking can significantly change the mixing of the elements inside the star. as Well as the evolution of its angular momentum content. | Such an effect has for the moment never been included in massive star models; however, by modifying the internal distribution of $\Omega$, the angular velocity, magnetic braking can significantly change the mixing of the elements inside the star, as well as the evolution of its angular momentum content. |
Although rotating models have improved the agreemer= between models and theory. some points remain to be clarified. | Although rotating models have improved the agreement between models and theory, some points remain to be clarified. |
A small subset of stars exhibit surface properties. such as low surface velocities and strong surface enrichments. which. when the star is not a giant or a supergiant. cannot be explained by current rotating stellar models for single stars (seediscussionsin ??).. | A small subset of stars exhibit surface properties, such as low surface velocities and strong surface enrichments, which, when the star is not a giant or a supergiant, cannot be explained by current rotating stellar models for single stars \citep[see discussions in][]{Brott2009, Hunter2009}. |
Although the fraction of these stars is small and some may be stars at the end of the MS phase (?).. 1t is worthwhile investigating what the physical cause of this behaviour could be. | Although the fraction of these stars is small and some may be stars at the end of the MS phase \citep{Maeder2009a}, it is worthwhile investigating what the physical cause of this behaviour could be. |
Theoretical models with rotation also predict. rotation velocities that are too high for the young pulsars (seethedis-cussionin. 22).. | Theoretical models with rotation also predict rotation velocities that are too high for the young pulsars \citep[see the discussion in][]{Heger2004, Heger2005}. |
Thus it does appear interesting to study any effects that can remove angular momentum from the star. | Thus it does appear interesting to study any effects that can remove angular momentum from the star. |
Can magnetic braking be an interesting explanation for the strongly mixed. slow rotators? | Can magnetic braking be an interesting explanation for the strongly mixed, slow rotators? |
Can it help in reducing the angular momentum of the core? | Can it help in reducing the angular momentum of the core? |
lr this first paper. we want to study these questions by presenting a first series of computations accounting for this effect. | In this first paper, we want to study these questions by presenting a first series of computations accounting for this effect. |
I section 2 we present the formalism we used to account for the magnetic braking effect. as well as the physical ingredients of the models. | In section 2 we present the formalism we used to account for the magnetic braking effect, as well as the physical ingredients of the models. |
The results are discussed in sections 3 and 4 and conclusions are given in section 3. | The results are discussed in sections 3 and 4 and conclusions are given in section 5. |
The formalism of the magnetie braking law used here follows theoretical developments first made for the Sun. starting with ? who used an idealized monopole field to model the angular momentum loss in the solar wind J. | The formalism of the magnetic braking law used here follows theoretical developments first made for the Sun, starting with \citet{Weber1967} who used an idealized monopole field to model the angular momentum loss in the solar wind $\dot J$. |
They found that where M is the mass loss rate and R4 the Alfvénn radius. 2/3MOR:.defined as the point where the ratio between the magnetic field energy density and the kinetic energy density of the wind is equal to 1. | They found that $
\dot J={2/3} \dot M \Omega R^2_{\rm A},
\label{WD}
$ where $\dot M$ is the mass loss rate and $R_{\rm A}$ the Alfvénn radius, defined as the point where the ratio between the magnetic field energy density and the kinetic energy density of the wind is equal to 1. |
? and ? have examined the angular momentum loss from magnetic hot stars with a line- stellar wind and a rotation-aligned dipole magnetic field using 2-D numerical MHD simulations. | \citet{UdDoula2002} and \citet{UdDoula2008} have examined the angular momentum loss from magnetic hot stars with a line-driven stellar wind and a rotation-aligned dipole magnetic field using 2-D numerical MHD simulations. |
They find that the total | They find that the total |
Tab. | Tab. |
1 ↽∙lists the complete log of. the observations. | \ref{obs} lists the complete log of the observations. |
. The weather conditions were good durug the might. which was photometric.phot.. andμή iedthe-? ousecing Uwas ~delay. 1. | The weather conditions were good during the night, which was stable and photometric, and the seeing was $\sim1\arcsec$. |
⋅⋅time⋅ ]processing UUwas(where carried out thewithinuud the withIRAE increasing: enuvironnieut. | The image processing was carried out within the environment. |
First. a anmap of the bad‘ features‘ of the chipP was‘ created‘ ‘and they were removed from the raw‘ naiages. | First, a map of the bad features of the chip was created and they were removed from the raw images. |
te Then. the bias stability was cliecked by comparing frames taken at different times during the ecutive run. and no significant. discrepaucies were found. | Then, the bias stability was checked by comparing frames taken at different times during the entire run, and no significant discrepancies were found. |
A 0.1% spatial eradicut was fouud aloug the. direction. thus a dnaster bias image was created by taking the median of all the bias inages. | A 0.4 spatial gradient was found along the $x$ direction, thus a master bias image was created by taking the median of all the bias images. |
This master bias image was subtracted from all the remaining frames. | This master bias image was subtracted from all the remaining frames. |
Sky flats were used to create master flat ficlds as medians of the single frames. | Sky flats were used to create master flat fields as medians of the single frames. |
Iu order to avoid the fall of quantum efficiency (QE) all around the border of the Loral CCD. we cut our images outside the limit where the QE was of the ceutral value. | In order to avoid the fall of quantum efficiency (QE) all around the border of the Loral CCD, we cut our images outside the limit where the QE was of the central value. |
Frou au inspection of the flats this limit iuposed au effective area of 1600« pixels νο, 1077&1077: Saviane eld 1998.. hereafter SITOS. give further details). | From an inspection of the flats this limit imposed an effective area of $1600\times 1600$ pixels (i.e. $10\farcm 7 \times 10\farcm 7$; Saviane Held \cite{s98}, hereafter SH98, give further details). |
The effective field is schematically represented iu Fig. 1.. | The effective field is schematically represented in Fig. \ref{field1}. |
Stellar photometry was performed usingDAOPLLOT. (Stetson. 1987)). audALLERAME. according to a standard procedure (see Paper I. | Stellar photometry was performed using, (Stetson, \cite{s87}) ), and, according to a standard procedure (see Paper I). |
Observatious of Laudolt's (1992)) standard stars were used to calibrate the photometry. | Observations of Landolt's \cite{l92}) ) standard stars were used to calibrate the photometry. |
In addition. the shutter was En1neasured withM a↴⋅ sequence↴ of imagesEM taken tablstable Iand exposure ⋅times. | In addition, the shutter delay time was measured with a sequence of images taken with increasing exposure times. |
. A value↽↽ of | (0.11↽ +£0.01slie‘Τι nuage ds the standard deviation)Ase was fouud. | A value of $\delay~\pm\errdelay$ s (where the error is the standard deviation) was found. |
owes . The: raw maenitudesau were first⋅ normalized∙ according∙↜ to the followiug:⋅↜ equatiou | The raw magnitudes were first normalized according to the following equation |
owes . The: raw maenitudesau were first⋅ normalized∙ according∙↜ to the followiug:⋅↜ equatiou∙ | The raw magnitudes were first normalized according to the following equation |
he relation presented. in Maiolino et al. ( | the relation presented in Maiolino et al. ( |
2007). who find a correlation between the fraction of obscured ACN as a 'unction of the optical Luminosity (i.e. at 5100 AY). | 2007), who find a correlation between the fraction of obscured AGN as a function of the optical luminosity (i.e. at 5100 ). |
Another indication of the custy torus was thought to be ratio between he accretion luminosity and the torus infrared. Luminosity. | Another indication of the dusty torus was thought to be ratio between the accretion luminosity and the torus infrared luminosity. |
In fact. the latter is just reprocessed radiation which scales incarly with the primary source: what makes the dillerence is the fraction of “heating radiation which is intercepted » the dust. which again depends on the covering factor. | In fact, the latter is just reprocessed radiation which scales linearly with the primary source: what makes the difference is the fraction of “heating radiation” which is intercepted by the dust, which again depends on the covering factor. |
We find a very similar trend. for these quantities. in both ype land tvpe 2 objects. also showing how differences in he amount of obscuration are very well explained byself-absorplion. Lo. thermal cust emission absorbed by dust iteself. | We find a very similar trend for these quantities, in both type 1 and type 2 objects, also showing how differences in the amount of obscuration are very well explained by, i.e. thermal dust emission absorbed by dust iteself. |
For the low-z sample. we found no evidence of emission [rom an AGN component for ~70% of the objects. | For the $z$ sample, we found no evidence of emission from an AGN component for $\sim 70$ of the objects. |
Although we cannot rule out the absence of an AGN from. these sources. we Can set an upper limit to its Luminosity since it is not observed at mid-infrared wavelengths where its emission is the strongest as compared to both that of the stellar and of the starburst-heated dust. | Although we cannot rule out the absence of an AGN from these sources, we can set an upper limit to its luminosity since it is not observed at mid-infrared wavelengths where its emission is the strongest as compared to both that of the stellar and of the starburst-heated dust. |
This sample sullers from high contamination from the stellar emission component. even at mid-inlrared) wavelengths where (on the contrary) its the AGN component that usually dominates when present. | This sample suffers from high contamination from the stellar emission component, even at mid-infrared wavelengths where (on the contrary) it's the AGN component that usually dominates when present. |
The question to ask. therefore. is what can be really constrained with three or four data points. | The question to ask, therefore, is what can be really constrained with three or four data points. |
One of the most. reliable «quantities should be the optical depth. since low values would make the torus emission stand over the stellar (SB or stars) in the MIB. | One of the most reliable quantities should be the optical depth, since low values would make the torus emission stand over the stellar (SB or stars) in the MIR. |
In this respect. the low-z sample may ος too. generous in selecting really active ACGN. | In this respect, the $z$ sample may be too generous in selecting really active AGN. |
Phere is. sometimes. no evidence for a ALL excess at all with respect. or example. not only to a “normal” starburst) emission. rut also to a passively evolving galaxy. | There is, sometimes, no evidence for a MIR excess at all with respect, for example, not only to a “normal” starburst emission, but also to a passively evolving galaxy. |
“Phe absence of any evidence for a hot dust emission in the MIT. which is the dace where the stellar component is less important and the AGN contribution is increasinglv. brighter. will remove the AGN component [rom the fits even though this could also » turned. into an upper limit. | The absence of any evidence for a hot dust emission in the MIR, which is the place where the stellar component is less important and the AGN contribution is increasingly brighter, will remove the AGN component from the fits even though this could also be turned into an upper limit. |
In the cases where there is a MIT. excess with respect to a pure stellar. emission. we explore two possibilities: torus and PALL emission. | In the cases where there is a MIR excess with respect to a pure stellar emission, we explore two possibilities: torus and PAH emission. |
In some cases à better Gt was obtained adding PALL -1e. | In some cases a better fit was obtained adding PAH -i.e. |
starburst component alone- to the model instead of AGN which should instead. cülute the PATI emission (Lutzetal.1998). | starburst component alone- to the model instead of AGN which should instead dilute the PAH emission \citep{lutz98}. |
. Comparison between clumpy and smooth tori models (e.g. Dullemond&vanDemmel 2005)) indicate that globally the SEDs produced by the two models are quite similar. but with some details characteristic for one or the other moclel: the silicate feature observed. in. absorption in objects seen edge-on is shallower for clumpy. models. the average near-IR Hus is weaker in smooth mocels and the clumpy models tend to produce slightly wider SEDs at certain inclinations. | Comparison between clumpy and smooth tori models (e.g. \citealt{dullemond05}) ) indicate that globally the SEDs produced by the two models are quite similar, but with some details characteristic for one or the other model: the silicate feature observed in absorption in objects seen edge-on is shallower for clumpy models, the average near-IR flux is weaker in smooth models and the clumpy models tend to produce slightly wider SEDs at certain inclinations. |
Furthermore. ¢lumpy models can produce very small tori sizes with Rowffi,~ο109 (Nenkovaetal.2008).. while still producing a broad. MIB. emission. | Furthermore, clumpy models can produce very small tori sizes with $R_{out}/R_{in} \sim 5-10$ \citep{nenkova08}, while still producing a broad MIR emission. |
Subsequently. the selection. of a smooth torus for the present study might have resulted in overestimated tori sizes without. however. jeopardising the estimates on the properties related to the IR emission or our conclusions on the Unification Scheme. | Subsequently, the selection of a smooth torus for the present study might have resulted in overestimated tori sizes without, however, jeopardising the estimates on the properties related to the IR emission or our conclusions on the Unification Scheme. |
(PAID) band. which is a useful though imperfect star formation rate (SER) indicator. | (PAH) band, which is a useful though imperfect star formation rate (SFR) indicator. |
It is therefore instructive to compare the ΜΟΙ luminosity with other star. formation indicators. such as Πα. radio. and total infrared humninosities ancl with these values lor other galaxies. | It is therefore instructive to compare the M31 luminosity with other star formation indicators, such as $\alpha$, radio, and total infrared luminosities and with these values for other galaxies. |
Wuetal.(2005). derived a calibration for SFR as a function of luminosity zL,[8jan(dust)] (hereafter Lx) using correlations between Ly and. L(Ia) and L(1.4GIIz). | \citet{wu05} derived a calibration for SFR as a function of non-stellar luminosity $\nu L_{\nu}[8\mu{\rm m(dust)}]$ (hereafter $L_8$ ) using correlations between $L_8$ and $L({\rm H}\alpha)$ and $L(1.4 {\rm GHz})$. |
The IRAC non-stellar flux. density measured for M31. corresponds to a luminosity of log(L4/L..)=8.8. | The IRAC non-stellar flux density measured for M31 corresponds to a luminosity of $\log(L_8/L_{\sun})={8.8}$. |
The Wuetal.(2005). calibration vields an SFR. οἱ 0M. |. | The \citet{wu05} calibration yields an SFR of $0.4 M_{\sun}$ $^{-1}$. |
I is possible to compare M3I's yan—derived SER with SFRs from other indicators. but since all such indicators have calibration uncertainties [e.g.. the Wuetal.(2005) /SER calibration is based on a small number of galaxies with a limited range of properties]. we instead. compare the observed properties directly. | It is possible to compare M31's -derived SFR with SFRs from other indicators, but since all such indicators have calibration uncertainties [e.g., the \citet{wu05} /SFR calibration is based on a small number of galaxies with a limited range of properties], we instead compare the observed properties directly. |
To estimate the //o. Iuminositv of 101. we convert the value given bv Devereuxοἱal.(1994) to the 780 kpe distance. multiply by 0.8 to account for [NII] contamination. and multiply by 3.4 (asdonebyWalterbos&Braun1994). to correct [or extinction. | To estimate the $H\alpha$ luminosity of M31, we convert the value given by \citet{dev94} to the 780 kpc distance, multiply by 0.8 to account for [NII] contamination, and multiply by 3.4 \citep[as done by][]{wb94}
to correct for extinction. |
The resulting L(lla)=2.6x10*L.. which predicts log(Ls/L..)=9.1. | The resulting $L({\rm H}\alpha)=2.6\times 10^7 L_{\sun}$, which predicts $\log(L_8/L_{\sun})=9.1$. |
The 1.4 GIIz radio flix density measured by Becketal.(1998).. W +. vields a predicted log(L4/L..)=8.4. | The 1.4 GHz radio flux density measured by \citet{bbh98}, $3.34\times 10^{20}$ W $^{-1}$, yields a predicted $\log(L_8/L_{\sun})=8.4$. |
Figure 12 of Daleetal.(2005). gives the ratio of pμπας//(3—EE00fan) as a function of f,Cr0j)/f(16050) for SINGS ealaxies. | Figure 12 of \citet{dale05} gives the ratio of $\nu f_{\nu}[8\mu{\rm m(dust)}]/f(3-1100\mu{\rm m})$ as a function of $f_{\nu}(70 \mu{\rm m})/f_{\nu}(160 \mu{\rm m})$ for SINGS galaxies. |
From Gordonetal.(2006).. M31 has (το)f(160)=0.12. with the corresponding [(3—1100jm)=2.57x10.M Wm 7. vielding a predicted log(La4/L..)=9.0. | From \citet{gordon06}, M31 has $f(70)/f(160)=0.12$, with the corresponding $f(3-1100\mu {\rm m})=2.57\times10^{-10}$ W $^{-2}$, yielding a predicted $\log(L_8/L_{\sun})=9.0$. |
The Ho and predictions are in reasonably good agreement. aad are consistent with the observed huminosity given (he scatter in the relationships. | The $\alpha$ and far-infrared predictions are in reasonably good agreement, and are consistent with the observed luminosity given the scatter in the relationships. |
The luminosity predicted from the radio [lux is lower (han observed. perhaps indicating that some extended emission was resolved out of the radio maps. | The luminosity predicted from the radio flux is lower than observed, perhaps indicating that some extended emission was resolved out of the radio maps. |
All of the ‘star-lormation’ huminosities are at the low end of the galaxy distribution. | All of the `star-formation' luminosities are at the low end of the galaxy distribution. |
Despite its prominent star forming ring. M31 is a predominantly quiescent. galaxy. | Despite its prominent star forming ring, M31 is a predominantly quiescent galaxy. |
We thank the releree for helpful suggestions. | We thank the referee for helpful suggestions. |
This work is based on observations made wilh the Spitzer Space Telescope. which is operated bv the Jet Propulsion Laboratory. California Institute of Technology under à contract with NASA. | This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. |
Support for Chis work was provided by NASA through an award issued by JPL/Caltech. | Support for this work was provided by NASA through an award issued by JPL/Caltech. |
Facilities: | Facilities: |
iis a peculiar low-mass powered by a double (wind plus disk) accretion onto a recently-born compact object (in this case the 24-ks signal would result from the orbital motion of the system) or hosting a magnetar (DeLucaetal.2006;Pizzolato2008;Bhadkamkar&Ghosh 2009). | is a peculiar low-mass powered by a double (wind plus disk) accretion onto a recently-born compact object (in this case the 24-ks signal would result from the orbital motion of the system) or hosting a magnetar \citep{deluca06,pizzolato08,bhadkamkar09}. |
. Both scenarios require nonstandard assumptions about the formation and evolution of compact objects in supernova explosions. | Both scenarios require nonstandard assumptions about the formation and evolution of compact objects in supernova explosions. |
Here we report on the results from a 5-year monitoring of wwith the ssatellite (Gehrelsetal.2004). | Here we report on the results from a 5-year monitoring of with the satellite \citep{gehrels04short}. |
. This unique data set allowed us to obtain a phase-coherent timing solution encompassing also aand aarchival observations. | This unique data set allowed us to obtain a phase-coherent timing solution encompassing also and archival observations. |
Thanks to this, we are able to derive an accurate period for aand to set the first upper limits on the period derivative for this puzzling source. | Thanks to this, we are able to derive an accurate period for and to set the first upper limits on the period derivative for this puzzling source. |
The X-Ray Telescope (XRT; Burrowsetal. 2005)) on-board uuses a front-illuminated CCD detector sensitive to photons between 0.2 and 10 keV with an effective area of about 110 cm? (at 1.5 keV) and a field view of 23-arcmin in diameter. | The X-Ray Telescope (XRT; \citealt{burrows05short}) ) on-board uses a front-illuminated CCD detector sensitive to photons between 0.2 and 10 keV with an effective area of about 110 $^2$ (at 1.5 keV) and a field view of 23-arcmin in diameter. |
Two main readout modes are available: photon counting (PC) and windowed timing (WT). | Two main readout modes are available: photon counting (PC) and windowed timing (WT). |
PC mode provides two dimensional imaging information and a 2.5073-s time resolution; in WT mode only one-dimensional imaging is preserved, achieving a time resolution of 1.766 ms. | PC mode provides two dimensional imaging information and a 2.5073-s time resolution; in WT mode only one-dimensional imaging is preserved, achieving a time resolution of 1.766 ms. |
Between April 2006 and April 2011, wwas observed by XRT 49 times, for a total net exposure time of 102.8 ks in PC The distribution of the oobservations can be seen in the long-term light curve in Fig. 1,, | Between April 2006 and April 2011, was observed by XRT 49 times, for a total net exposure time of 102.8 ks in PC The distribution of the observations can be seen in the long-term light curve in Fig. \ref{swiftlc}, |
while in Fig. | while in Fig. |
2 we show the image of aand rresulting from all the XRT data gathered so far. | \ref{ds9} we show the image of and resulting from all the XRT data gathered so far. |
Except for periods in which the source was not visible by the XRT because of pointing constraints of the sspacecraft, approximately one 2-ks observation in imaging mode was collected per month. | Except for periods in which the source was not visible by the XRT because of pointing constraints of the spacecraft, approximately one 2-ks observation in imaging mode was collected per month. |
On a few occasions, when sshowed hints of consistent flux variations, we requested target-of-opportunity observations. | On a few occasions, when showed hints of consistent flux variations, we requested target-of-opportunity observations. |
For example this happened at the end of October 2010 (see Fig. | For example this happened at the end of October 2010 (see Fig. |
1 around MJD 555500), when the source count rate remained at a relatively high level of ~0.09 counts s! for a few consecutive pointings spanning ~5 days. | \ref{swiftlc}
around MJD 500), when the source count rate remained at a relatively high level of $\approx$ 0.09 counts $^{-1}$ for a few consecutive pointings spanning $\sim$ 5 days. |
The Observations were not time-constrained, so the monitoring can be considered a sampling of the phase of1613. | The observations were not time-constrained, so the monitoring can be considered a sampling of the phase of. |
. The XRT data were uniformly processed with (version 12, in the software package version 6.9), filtered and screened with standard criteria. | The XRT data were uniformly processed with (version 12, in the software package version 6.9), filtered and screened with standard criteria. |
In order to reduce the contamination from the SNR, the source counts were energy-selected in the 1-10 keV band and extracted within a 10-pixel radius (one XRT pixel corresponds to about 2: 96). | In order to reduce the contamination from the SNR, the source counts were energy-selected in the 1–10 keV band and extracted within a 10-pixel radius (one XRT pixel corresponds to about $2\farcs36$ ). |
To convert the photon arrival times to the Solar system barycentre for the timing analysis, we used the task and the pposition of DeLucaetal.(2008). | To convert the photon arrival times to the Solar system barycentre for the timing analysis, we used the task and the position of \citet{dmz08}. |
. For the spectroscopy, we used the latest spectral redistribution matrices in (20091130), while the ancillary response files were generated withXRTMKARF,, which accounts for different extraction regions, vignetting and point-spread function corrections. | For the spectroscopy, we used the latest spectral redistribution matrices in (20091130), while the ancillary response files were generated with, which accounts for different extraction regions, vignetting and point-spread function corrections. |
We complemented the Swift//XRT data-set with the few aand oobservations long-enough to contain a minimum of two modulation cycles of ((Table 1)). | We complemented the /XRT data-set with the few and observations long-enough to contain a minimum of two modulation cycles of (Table \ref{obs-log}) ). |
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