The new ephemeris of MXB 1659-298

A possible solution of the puzzling variation of the orbital period of MXB 1659-298

Abstract

MXB 1659-298 is a transient neutron star Low-Mass X-ray binary system that shows eclipses with a periodicity of 7.1 hr. MXB 1659-298 went to outburst in August 2015 after 14 years of quiescence. We investigate the orbital properties of this source with a baseline of 40 years obtained combining the eight eclipse arrival times present in literature with 51 eclipse arrival times collected during the last two outbursts. A quadratic ephemeris does not fit the delays associated with the eclipse arrival times and the addition of a sinusoidal term with a period of yr is required. We infer a binary orbital period of hr and an orbital period derivative of s s. We show that the large orbital period derivative can be explained with a highly non conservative mass transfer scenario in which more than 98% of the mass provided by the companion star leaves the binary system. We predict an orbital period derivative value of s s and constrain the companion star mass between 0.3 and M. Assuming that the companion star is in thermal equilibrium the periodic modulation can be due to either a gravitational quadrupole coupling due to variations of the oblateness of the companion star or with the presence of a third body of mass M Jovian masses.

keywords:
X-rays: stars; X-rays: binaries; stars: neutron; binaries: eclipsing; ephemerides; stars: individual (MXB 1659-298)
12

1 Introduction

One of the most direct evidence for binary orbital motion is the presence of eclipse of the central source by a companion star. For Low Mass X-ray Binaries (LMXBs) with inclination angles between 75 and 80 the X-ray emission may be totally shielded by the companion star. As the companion transits between the X-ray central source and the observer the light curves show total eclipses. For inclination angles between 80 and 90 the LMXB is observed as an Accretion Disc Corona (ADC) source. In this case the observed X-ray emission comes from an extended corona that can reach the outer region of the accretion disc. The light curves of the ADC sources show an almost sinusoidal modulation and partial eclipses. The modulation of the light curve is generally explained with the presence of a geometrically thick disc whose height varies depending on the azimuthal angle and occults part of the X-ray emission. Since the companion star does not shield the whole extended corona the observed eclipses are partial; the prototype of the ADC sources is X1822-371 (see e.g. Iaria_11; Iaria_13; Iaria_15b, and references therein).

Total eclipses represent a good time reference, which is ideal to perform timing analysis of the binary orbital period, e.g. the O-C method is usually applied to refine the orbital period or trace orbital period changes (see Chou_14, for a recent review). To date, 12 LMXBs show total eclipses in their light curve. One of the best studied eclipsing X-ray source is EXO 0748-676, as it was active for more than 20 years (see Wolff_09, and references therein).

The eclipsing LMXB MXB 1659-298 was discovered by Lewin_76 in 1976. The light curve showed type-I X-ray bursts, thus revealing that the compact object was an accreting neutron star. The source was observed in outburst up to 1978 with SAS3 and HEAO (Com_83; Com_84; Com_89). Eclipses were firstly reported by by Com_84, which estimated a periodicity of 7.1 hr. Com_89 analysed two whole eclipses estimating two eclipse arrival times. From 1978 up to 1999 the region containing MXB 1659-298 was monitored by the X-ray observatories onboard Hakucho, EXOSAT and ROSAT, but the source was never detected (see Com_89; Verbunt_01). On April 1999 the Wide Field Cameras onboard BeppoSAX observed the source in outburst again (Zand_99). This new outburst continued up to September 2001. During the outburst MXB 1659-298 was observed with the Proportional Counter Array (PCA) onboard Rossi X-ray Timing Explorer (RXTE, see e.g. Wachter_00), with the Narrow Field Instruments (NFI) onboard BeppoSAX (Ost_01) and with XMM-Newton. From the analysis of the RXTE light curves of the source, Wachter_00 obtained four eclipse arrival times and found an orbital period derivative of s s suggesting that the orbit of the binary system is shrinking. Ost_01 obtained two eclipse arrival times from a BeppoSAX/NFI observation and combining their data with those present in literature found that the orbital period derivative, , is positive with a value of s/s. MXB 1659-298 turned again on outburst on 2015 August 21 (Negoro_15) and up to 2017 March is still X-ray bright. Using data of the X-ray Telescope (XRT) onboard Swift, Bahram_16 observed that the unabsorbed flux in the 0.5-10 keV energy range was , and erg cm s on 2016 January 28, February 2 and 11, respectively.

Figure 1: Light curve of MXB 1659-298 during the outburst occurred between 1999 and 2001 (left panel) and the latest started 2015 (right panel). The left panel shows the RXTE/ASM light curve in the 2-10 keV, the right panel shows the MAXI/GSC light curve in the 2-20 keV energy range; the bin time is five days for both the light curves. The eclipse arrival times are also indicated.

Com_89 measured an eclipse duration, , of s and an ingress/egress duration of s and s, respectively. They showed that, if the companion star is a main-sequence star of 0.9 M with a temperature close to 5000 K, the scale height of the stellar atmosphere should be around 200 km, corresponding to an ingress/egress duration close to 0.5 s. The authors concluded that the small value of the scale height cannot justify the large value of the measured ingress/egress durations. Furthermore, Com_89 suggested that the observed asymmetry between the ingress and egress duration could be caused by a one-sided extended corona of size km.

From the analysis of four eclipses obtained with RXTE/PCA, Wachter_00 estimated an average eclipse duration of s and average values of ingress/egress durations of s and s. The authors proposed that the large spread of values associated with the ingress/egress times could be caused either by flaring activity of the companion star or by the presence of an evaporating wind from the surface of the companion star created by irradiation from the X-ray source.

Com_84 discussed the nature of the optical counterpart of MXB 1659-298, named V2134 Oph, assuming an orbital period of 7.1 hr and an eclipse duration of 900 s. They constrained the mass of the companion star to be between 0.3 M and 0.9 M for an inclination angle of the binary system of 90 and 71.5, respectively. warner_95 inferred that the companion star mass is between 0.75 and 0.78 M if the companion fills its Roche lobe. This range of masses suggests that the companion is a K0 main-sequence star. During the quiescence of MXB 1659-298, Wachter_00 measured a magnitude in the -band of mag and Filippenko_99 measured a magnitude in the -band of mag. Wachter_00 found that the value of is compatible with an early K spectral type. Moreover, they suggested that, for a companion star belonging to the K0 class, the visual magnitude should be mag, value that is compatible with the measured lower limit of mag.

Gallo_08, analysing the type-I X-ray bursts observed with RXTE/PCA, inferred a distance to the source of and kpc for a hydrogen-rich and helium-rich companion star, respectively. Furthermore, Win_01 detected nearly coherent oscillations with a frequency around 567 Hz during type-I X-ray bursts suggesting that the neutron star could be an X-ray millisecond pulsar with a spin period of 1.8 ms.

The interstellar hydrogen column density, , was estimated by Cack_08 during the X-ray quiescence of MXB 1659-298. Combining Chandra and XMM-Newton observations collected between 2001 and 2008 they fitted the X-ray spectrum obtaining cm. Two more recents Chandra observations of the source, taken in 2012 Cack_13, seem to suggest an increase of the interstellar hydrogen column density at the value of cm. The authors proposed three different scenarios to explain the increase of : a) material is building up in the outer region of the accretion disc, b) the presence of a precessing accretion disc, and c) sporadic variability during quiescence due to low-level accretion.

Studying the XMM-Newton spectrum of MXB 1659-298, Sidoli_01 detected two absorption lines at 6.64 and 6.90 keV associated with the presence of highly ionised iron (\ionFexxv and \ionFexxvi ions) as well as absorption lines associated with highly ionised oxygen and neon (\ionOviii 1s-2p, \ionOviii 1s-3p, \ionOviii 1s-4p and \ionNeix 1s-2p transition) at 0.65, 0.77, 0.81 and 1.0 keV.

In this paper we report the updated ephemeris of MXB 1659-298 combining 45 eclipse arrival times obtained with XMM-Newton and RXTE during the outburst between 1999 and 2001 and six eclipse arrival times obtained with XMM-Newton, NuSTAR and Swift/XRT during the outburst started in 2015. The available temporal baseline allows to partially constrain the bizarre behaviour of the eclipse arrival times.

2 Observations

During the outburst occurred from 1999 to 2001, MXB 1659-298 was observed with XMM-Newton (Jansen_01) two times: on March 22 2000 and on Feb. 20 2001. The latter observation (obsid. 0008620701) was analysed by Sidoli_01 and Diaz_06, which studied the spectral properties of the source during the persistent emission, the dip and the eclipse, while the former observation (obsid. 0008620601) was never analysed. During the 2015 outburst, MXB 1659-298 was observed with XMM-Newton on September 26, 2015.

The European Photon Imaging Camera (Epic-pn, struder_01) onboard XMM-Newton collected data from the source in timing mode, with exposure times of 10, 32 and 34 ks, respectively. The Epic-pn light curve of the observation taken in 2001 shows two eclipses in the light curve (see Fig. 1 in Sidoli_01). To verify the presence of eclipses in the Epic-pn light curves of the observations taken in 2000 and 2015 we filtered the source events with the Science Analysis System (SAS) ver. 15.0.0. We reprocessed the Epic-pn events and applied the solar-system barycentre corrections adopting as coordinates RA and Dec (see Wij_03). During the observation taken in 2000, the light curve of MXB 1659-298 shows an eclipse with a duration of 900 s approximately 1400 s after the start time. The count rate is 32 c s and 1.4 c s outside and during the eclipse, respectively. During the observation taken in 2015, the light curve shows the presence of a type-I X-ray burst at 12 ks after the start of the observation. The count rate varies from 32 c s at the beginning of the burst up to 320 c s at the peak. An intense dipping activity is present at about 20 ks from the beginning of the observation, a complete eclipse is observed at 26 ks from the start time and an eclipse without the ingress is observed at beginning of the observation. The count rate out and during the eclipse is 32 and 1.4 c s, respectively.

The PCA instrument onboard RXTE (Jahoda_96) observed several times the source from 1999 to 2001. In our analysis we selected 43 RXTE/PCA observations showing the eclipse and for which it is possible to estimate the ingress and egress time accurately. To estimate the eclipse arrival times from the RXTE/PCA observations we analysed the standard product background-subtracted light curves with a bin time of 0.125 s and we applied the solar-system barycentric correction to the events using the ftool faxbary.

NuSTAR (Harrison_13) observed MXB 1659-298 two times in 2015 and 2016 with both the independent solid state photon counting detector modules (FPMA and FPMB), with elapsed times of 96 ks and 50 ks, respectively. We processed the raw (Level 1) data with the ftool nupipeline (Heasoft ver. 6.19), obtaining cleaned and calibrated event data (Level 2). The solar-system barycentric corrected events of the FPMA and FPMB telescopes have been obtained applying the tool nuproducts on the Level 2 data. The corresponding light curves were created selecting a circular extraction region for the source events with a radius of and using the 1.6-20 keV energy range. The persistent emission has a count rate of 2 c s. A complete eclipse and an eclipse without the ingress are observed at 24 and 76 ks from the start time. The count rate during the eclipse is 0.02 c s. It is also evident the presence of the ingress to the eclipse at 49.7 ks from the start time. During the second observation MXB 1659-298 is brighter, with a persistent count rate of 20 c s, a whole eclipse is observed 30 ks after the start time of the observation. To increase the statistics of the NuSTAR light curve we summed the FPMA and FPMB light curve using the ftool lcmath.

During the 2015 outburst, MXB 1659-298 was observed several times with Swift/XRT (Gerels_04; Burrows_05), although only three observations show a complete eclipse. We obtained further Swift/XRT data as target of opportunity observations performed on February 8, 10 and 11, 2017 (obsid 0003400266, 0003400267 and 0003400268). All of the three observations cover the whole eclipse. The XRT data were processed with standard procedures (xrtpipeline v0.13.1), and with standard filtering and screening criteria with ftools. For our timing analysis, we also converted the event arrival times to the solar-system barycentre with the tool barycorr and subtracted the background using the ftool lcmath.

The All Sky monitor (ASM, Levine_96) onboard RXTE monitored the 1999-2001 outburst (Fig. 1, left panel). The two XMM-Newton observations were performed at a similar ASM count rate of 2.5 c s (about 30 mCrab in flux), corresponding to the source maximum flux. The outburst showed a sort of precursor lasting 100 d, afterwards the flux decreased up to a value compatible with zero for 86 d, and finally increased again rapidly reaching a constant flux of 30 mCrab for 700 d.

The Gas Slit Camera (GSC, Mihara_11) onboard the Monitor of All-sky X-ray Image (MAXI, Matsuoka_09) observed the recent outburst (see Fig. 1, right panel). The morphology of the outburst is similar to the previous one with a sort of precursor lasting 50 d, a new quiescent stage lasting 150 d and, after that, an increase of the flux at 30 mCrab lasting 150 d. The maximum GSC count rate is 0.12 c s. XMM-Newton and NuSTAR (obsid. 90101013002) observed the source when the GSC count rate was 0.05 c s; NuSTAR observed the source a second time when MXB 1659-298 was brighter with a corresponding GSC count rate of 0.1 c s.

3 Method and Analysis

Figure 2: Eclipse of MXB 1659-628 observed by the RXTE/PCA instrument (observation P40050-04-16-00). The superimposed red function is the step-and-ramp function adopted to estimate the eclipse arrival time.
Figure 3: From the top-left to the bottom-left the ingress, egress and eclipse duration, respectively, as function of time. The values are obtained from the RXTE/PCA eclipses analysed in this work. The red lines indicate the average values for each duration. From the top-right to the bottom-right we show the histograms of the occurrences of the ingress, egress and eclipse duration.
Point Eclipse Time Cycle Delay Ref. Point Eclipse Time Cycle Delay Ref.
(MJD;TDB) (s) (MJD;TDB) (s)
1 43058.7260(2) 0 0(13) [1],[2] 31 51769.43726(2) 29378 107.0(1.3) [4]
2 43574.6441(2) 1740 26(13) [1],[2] 32 51835.261275(9) 29600 106.9(7) [3]
3 51273.978079(2) 27707 96.46(13) [2] 33 51836.447292(6) 29604 106.8(5) [3]
4 51274.571102(8) 27709 97.7(7) [3] 34 51837.040274(5) 29606 104.4(4) [3]
5 51277.832626(4) 27720 95.4(3) [2] 35 51960.08961(2) 30021 101(2) [3]
6 51278.425648(10) 27722 96.5(9) [3] 36 51974.321836(6) 30069 101.4(5) [3]
7 51281.687174(4) 27733 94.5(3) [2] 37 51974.914836(8) 30071 100.6(7) [3]
8 51283.762726(3) 27740 96.2(3) [2] 38 51976.397381(8) 30076 102.5(6) [3]
9 51285.838220(11) 27747 93.0(9) [3] 39 51977.286855(12) 30079 99.1(1.0) [3]
10 51295.029855(5) 27778 92.5(4) [3] 40 52027.692627(9) 30249 99.2(7) [3]
11 51297.698476(8) 27787 99.4(7) [3] 41 52029.768118(8) 30256 95.8(7) [3]
12 51334.464970(12) 27911 93.6(1.1) [3] 42 52030.954185(12) 30260 100.0(1.1) [3]
13 51335.650973(6) 27915 92.2(5) [3] 43 52032.733167(8) 30266 96.1(7) [3]
14 51337.133479(6) 27920 90.8(5) [3] 44 52076.615786(4) 30414 91.7(3) [3]
15 51393.46935(4) 28110 92(4) [3] 45 52077.208801(7) 30416 92.2(6) [3]
16 51396.13784(4) 28119 87(3) [3] 46 52077.801847(6) 30418 95.3(5) [3]
17 51397.32378(3) 28123 81(3) [3] 47 52078.394837(7) 30420 93.7(6) [3]
18 51466.112958(9) 28355 91.5(8) [3] 48 52078.987831(8) 30422 92.4(7) [3]
19 51467.29898(12) 28359 92.2(1.0) [3] 49 52131.469068(10) 30599 86.8(9) [3]
20 51470.264016(9) 28369 91.1(8) [3] 50 52132.65509(2) 30603 87(2) [3]
21 51557.436333(6) 28663 89.8(5) [3] 51 52133.24811(8) 30605 88(7) [3]
22 51561.290937(6) 28676 93.7(5) [3] 52 52136.50958(8) 30616 81(7) [3]
23 51562.477008(6) 28680 98.3(5) [3] 53 52159.34046(2) 30693 83.9(1.4) [3]
24 51625.03951(2) 28891 104(2) [3] 54 57291.24010(2) 48001 17(2) [3]
25 51677.817305(4) 29069 101.3(4) [3] 55 57294.20513(2) 48011 16(2) [3]
26 51681.671901(6) 29082 104.5(5) [3] 56 57499.682737(14) 48704 13.7(1.2) [3]
27 51682.857903(7) 29086 103.2(6) [3] 57 57792.03631(3) 49690 23(3) [3]
28 51763.803676(5) 29359 106.3(4) [3] 58 57794.70484(5) 49699 22(4) [3]
29 51764.989711(8) 29363 107.7(7) [3] 59 57795.89087 (5) 49703 23(4) [3]
30 51768.84426(2) 29376 106.0(1.4) [4]

Note — Epoch of reference 43058.72595 MJD, orbital period 7.11610872 hr, the associated errors are at 68% confidence levels; [1] Com_89, [2] Wachter_00, [3] this work, [4] Ost_01 .

Table 1: Journal of the X-ray eclipse arrival times of MXB 1659-298

To estimate the eclipse arrival times, we folded the solar-system barycentric corrected light curves using a trial time of reference and orbital period, and , respectively. The value of the adopted corresponds to a time close to the start time of the corresponding observation. The adopted value of is 7.11610872 hr corresponds to the value of the orbital period at MJD obtained by Ost_01 adopting quadratic ephemeris.

We fitted the eclipse profiles with a simple model consisting of a step-and-ramp function, where the count rates before, during, and after the eclipse are constant and the intensity changes linearly during the eclipse transitions. This model involves seven parameters: the count rate before, during, and after the eclipse, called , , and , respectively; the phases of the start and stop times of the ingress ( and ), and, finally, the phases of the start and stop times of the egress ( and ). We show a typical eclipse of MXB 1659-298 in Fig. 2. The eclipse was observed during the RXTE/PCA observation P40050-04-16-00, the superimposed red function is the step-and-ramp best-fitting function. The phase corresponding to the eclipse arrival time is estimated as . The corresponding eclipse arrival time is given by . To be more conservative, we scaled the error associated with by the factor to take into account values of of the best-fit model larger than one. We show the obtained eclipse arrival times in Barycentric Dynamical Time (TDB), in units of MJD, in Tab. 1.

We used the 43 RXTE/PCA observations to estimate the average duration, , and of the eclipse, the ingress and the egress, respectively. The values of , and for each eclipse are shown as function of the eclipse arrival times in Fig. 3. We found that is scattered between 890 and 910 s. Fitting the values of eclipse duration with a constant we obtained a of 561(42) and a best-fit value of s at 68% confidence level (c.l.). The ingress duration is scattered between 10 and 30 s while the egress duration is scattered between 10 and 35 s. Fitting the ingress duration values with a constant we obtained a of 457 (38) and a best-fit value of s at 68% c. l., while, fitting the egress duration values we obtained a of 560 (39) and a best-fit value of s at 68% c. l.. The associated errors were scaled by the factor to take a value of of the best-fit model larger than one into account. We find that the average duration of the ingress and egress are similar. We also show in Fig. 3 the occurrences of the measured ingress, egress and duration using a bin of 3.1, 3.7 and 3.5 s, respectively.

We calculated the delays with respect to hr and to a reference epoch of MJD, corresponding to the first eclipse arrival time obtained by Com_89. The inferred delays, in units of seconds, of the eclipse arrival times with respect to a constant orbital period are reported in Tab. 1. For each point we computed the corresponding cycle and the eclipse arrival time in days with respect to the adopted . We show the delays vs. time in Fig. 4 (top panel).

Parameter LQ LQS LQCS
a (s)
b (s d)
c ( s d)
d ( s d) - -
A (s) -
P (d) -
t (d) -
(d.o.f.) 4083(56) 512(53) 455(52)
Table 2: Best-fit values
Figure 4: Top panel: delays with respect to the predicted eclipse arrival times, assuming as epoch of reference MJD and as orbital period hr, plotted versus time. The blue and red curves indicate the best-fit functions corresponding to eqs. 2 and 4, respectively. Middle panel: residuals in units of with respect to the blue curve. Bottom panel: residuals in units of with respect to the red curve.

Initially we fitted the delays with a quadratic function

(1)

where is the time in days (MJD-43058.72595), is the correction to in units of seconds, in units of s d with the correction to the orbital period, and finally, in units of s d, with representing the orbital period derivative. The corresponding best-fitting parameters are shown in the LQ column of Tab. 2. With a of 4083 for 56 d.o.f., we note that the quadratic function does not acceptably fit the data. Since the delays seem to show a periodic modulation we fitted them using the function

(2)

where is the amplitude in seconds of the sinusoidal function, is the period of the sine function in days, and, finally, represents the time in days at which the sinusoidal function is null. A clear improvement is obtained with a value of of 512 (53) that translates to a F-test probability chance improvement of . The best-fit function, indicated with a blue curve, and the corresponding residuals are shown in the top and middle panels of Fig. 4. The best-fit values are shown in the third column of Tab. 2. The corresponding ephemeris (hereafter LQS) is

(3)

where indicates the number of cycles, and . We obtained an orbital period derivative s s, a sinusoidal modulation characterised by a periodicity yr and a semiamplitude s.

It is evident that the LQS ephemeris does not predict the first two eclipse arrival times. A possible explanation is that the orbital period derivative is changing from 1976 up to now. To take into account this fact, we added a cubic term to eq. 2, defining the new function

(4)

where includes the presence of a derivative of with . With the latter model we obtain a value of of 455 (52). By adding the cubic term we find a F-test probability chance improvement of 0.014 indicating that the improvement of the fit is between two and three of confidence level. The best-fit function, indicated with a red curve, and the corresponding residuals are shown in the top and bottom panel of Fig. 4. The best-fit parameters are shown in the fourth column of Tab. 2. The corresponding ephemeris (hereafter LQCS) is

(5)

from which we inferred the orbital period derivative at time MJD to be s s and the orbital period second derivative s s. The sinusoidal modulation has a period of yr and a semiamplitude of s.

4 Discussion

We analysed the eclipse arrival times of MXB 1659-298 with the main aim to estimate its ephemeris. Our baseline spans 40 years and covers the three outbursts of the source observed from 1976. We combined 51 eclipse arrival times, corresponding to the outbursts occurred in 1999-2001 and in 2015-2017, with the data already reported in literature. The campaign of observations made with Rossi-XTE/PCA during the 1999-2001 outburst seems to indicate a possible periodic modulation of 2.3 years; the delays associated with the six eclipse arrival times obtained during the most recent outburst agree with that periodic modulation. We find that the LQS ephemeris accounts for the eclipse arrival times except for the two eclipses observed in 1976-1978. The addiction of a cubic term (LQCS ephemeris) allows to account for all the available data, however the statistical improvement is less than three sigma, suggesting that a larger baseline is needed to confirm the more complex ephemerides. In both cases, a sinusoidal modulation with a period between 840 and 860 days is needed to obtain an acceptable fit of the eclipse arrival times. In the following we restrict our discussion to the LSQ ephemeris.

To estimate the eclipse arrival times we fitted the shape of the eclipse using a step-and-ramp function. We used the RXTE/PCA observations, covering 2.4 years during the second outburst of MXB 1659-298, to estimate the ingress/egress and eclipse durations. The obtained values are scattered, the mean values associated with the eclipse, ingress and egress are s, s and s, respectively. We find that the ingress and egress durations are similar contrarily to what reported by Com_89, that obtained an ingress and egress duration of s and s, respectively. Our different results can be explained by the larger sample and the higher quality of our dataset.

The ingress, egress, and eclipse durations show a jittered behaviour of the order of 15 s similarly to what observed in EXO 0748-676 (Wolff_02). Wolff_07 discussed the possibility that magnetic activity of the companion star generates extended coronal loops above the photosphere that could explain the amplitude of the observed jitter. This scenario may be plausible given the late K or early M type nature of the 0.3-0.4 M companion star in EXO 0748-676. Such stars can have magnetic activity if they rotate and if they have significant convective envelopes (see Wolff_07). The companion star in MXB 1659-298 is an early K type main-sequence star (see below), and hence it likely has similar magnetic activity. Ponti_17 showed that AX J1745.6-2901 has a different phenomenology. Although jitters are observed in the ingress and egress, the eclipse duration is nearly constant. The authors suggested that the matter ejected from the accretion disc could reach the companion star with a ram pressure comparable to the pressure in the upper layers of the companion star (that is a K type main-sequence star). This interaction could displace the atmosphere of the companion star delaying both the ingress and the egress times.

4.1 The masses of the binary system

We can estimate the companion star radius from the size of its Roche lobe, that can be expressed by using the formula of Pac_71

(6)

where is the orbital separation of the binary system and is the neutron star mass in units of solar masses. Combining the previous equation with the third Kepler’s law we find that

(7)

Assuming that the companion star fills its Roche lobe then the radius of the companion star coincides with . To estimate the mass of the companion star we adopted the mass-radius relation for a companion star in thermal equilibrium obtained by studying the cataclysmic variable systems (eq. 16 in knigge_11) although LMXBs lie in a somewhat different region of parameter space. We adopted the relation valid for large orbital periods that is

(8)

where M has a value of M and it is the mass of the convective region of the companion star. Combining the eqs. 7 and 8 and taking into account that the accuracy associated with the Roche lobe radius is 2% we find that the companion star has a mass of M and a radius of R. Hereafter we will assume a neutron star mass of M, this mass value was estimated by Ozel_12 analysing the mass distribution of neutron stars that have been recycled; the best value is the mean of the distribution and the associated error is the corresponding dispersion.

4.2 The mass accretion rate and the mass transfer rate

Using RXTE/PCA data taken during the outburst in 1999, Gallo_08 observed that the flux of MXB 1659-298 peaked at erg s cm in the 2-25 keV energy range during April 1999 , but it was between 4 and 6 erg s cm throughout the remainder of the outburst. To infer a good estimation of the flux in the 0.1-100 keV energy band, we adopted the broadband best-fit model of the persistent spectrum obtained by Ost_01, from which we extrapolate an unabsorbed flux of erg s cm.

From the analysis of the type-I X-ray bursts the distance to MXB 1659-298 was estimated to be and kpc for a hydrogen-rich and helium-rich companion star, respectively (see Gallo_08). We assume the average of the two values, kpc, considering that the source is accreting mixed H/He (Gallo_08).

To convert the X-ray luminosity in mass accretion rate we used the relation , where is the efficiency of the accretion and is the speed of the light. We take into account that the neutron star is rapidly spinning with a frequency of 567 Hz (Win_01) adopting the relation proposed by Sibga_00

(9)

where is the spin frequency of the neutron star in units of kHz. The latter relation is valid assuming a gravitational mass of the neutron star of 1.4 M and the commonly adopted FPS equation of state for a neutron star. Using a spin frequency of 567 Hz we find that . Our assumption implies that all of the released gravitational energy is converted to X-ray emission and that negligible amount of energy is carried away by bulk outflows. This is confirmed by the spectral studies of the source; in fact, the absorption lines associated with the presence of \ionFexxv and \ionFexxvi ions are narrow suggesting that it is not possible to associate to the source a superluminal jet (see Sidoli_01). Furthermore Diaz_16 suggested that MXB 1659-298 could have a mild thermal wind but only static atmospheres have been reported.

Using we find M yr. Considering a quiescence duration of almost 14.5 yr and a mean outburst duration of 1.5 yr we find that the average value of the observed mass accretion rate is M yr.

On the other hand, from theoretical considerations, we can estimate the rate of mass that has to be transferred from the companion star in order to explain the quadratic term of the LQS ephemeris interpreted as the orbital period derivative of the system. From the long-term orbital evolution we can estimate the mass transfer rate using the eq. 4 in Burderi_10

(10)

where is the mass transfer rate in units of M yr, is the mass-radius index of the companion star, is the companion star mass in units of solar masses, is the orbital period derivative in units of s s and is the orbital period in units of 5 hr. This is derived combining the third Kepler law with the contact condition, that is (where is the Roche Lobe radius of the secondary and is the radius of the secondary). Adopting , , s s and hr, we find that the mass transfer rate implied by the observed orbital period derivative is M yr, that is almost two orders of magnitude higher that the observed averaged mass accretion rate. This means that in order to explain the observed orbital period change rate we have to invoke a highly not conservative mass transfer for this system.

The above described scenario assumes a mass transfer rate of M yr and a companion star mass of M in thermal equilibrium. The time scale associated with the mass transfer rate, , is yr. The companion star is in thermal equilibrium if is longer than the thermal time scale of the companion star (Pac_71). To estimate the thermal timescale we need to infer the luminosity of the companion star. For a star close to the lower main sequence it holds the relation (see Salaris_05). For a companion star mass of M we obtain that yr which is comparable with , for this reason we cannot exclude the the companion star is less massive of M.

4.3 The prediction of the orbital period derivative for a highly non conservative mass transfer

We can define a parameter in the following way, , where is the mass accretion rate. Hence in a conservative mass transfer scenario and in a non conservative mass transfer scenario. Comparing the observed averaged mass accretion rate with the mass transfer rate implied by the observed orbital period derivative, we obtain , suggesting that only of the mass transferred from the companion star is indeed accreted onto the neutron star.

According to the orbital evolution theory, orbital period changes are expected to be driven by mass transfer from the companion to the compact object, by emission of gravitational waves (GR) and/or by magnetic braking (MB). For orbital periods larger than two hours the effects of MB dominate the orbital evolution of the binary system. Following Verbunt_81, Verbunt_93 and Tauris_01 the torque associated with MB can be parametrised as

(11)

where is a dimensionless parameter for which a value of either 0.79 (Skumanich_72) or 1.78 (Smith_79) has been assumed, is the gyration radius for a star with mass of 0.8 M (Claret_90), is the orbital period in units of two hours, is the mass ratio and, finally, is the mass of the compact object in units of solar masses. Because depends on the effects of the MB on the derivative of the angular momentum of the binary system will be larger for than for .

We can calculate the secular orbital period derivative expected from the non-conservative secular evolution of the system using the relation

(12)

where

(13)

(see Disalvo_08; Burderi_09; Burderi_10), where is the orbital period derivative in units of s s and is a dimensionless parameter that quantifies the specific angular momentum of the ejected matter in the case of a non-conservative mass transfer scenario. The specific angular momentum, , with which the transferred mass is lost from the system can be written in units of the specific angular momentum of the secondary, that is , where is the distance of the secondary star from the centre of mass of the system, is the orbital separation and is the orbital period of the binary system. For a neutron star mass of M we obtain an orbital period derivative of s s, which is compatible within one with the value s s inferred from the eclipse arrival times.

A highly non-conservative mass transfer in this source may be justified by the fact that MXB 1659-298 is a fast spinning neutron star (Win_01). During the quiescent periods, if the region around the neutron star is free from matter up to the light cylinder radius, the radiation pressure of the rotating magnetic dipole, given by the Larmor formula, may be able to eject from the system the matter transferred by the companion star at the inner Lagrangian point, according to the mechanism termed radio ejection and described in detail in Burderi_01. Once significant temporary reduction of the mass accretion rate occurs, the neutron star can emit as a magnetic-dipole rotator and the radiation pressure from the pulsar may be able to eject the matter transferred from the companion out of the system. We note that the disc instability model (see the review of Lasota_01) may have a role in triggering the radio ejection and starting a non conservative mass transfer. The radio ejection has been invoked to explain the high orbital period derivative observed in the accreting millisecond pulsar (AMSP) SAX J1808.4-3658 (see Disalvo_08; Burderi_09), and, more recently, for the AMSP SAX J1748.9-2021 for which a high orbital period derivative is also observed (Sanna_16). We therefore suggest that a similar mechanism could be also at work for MXB 1659-298.

The above described scenario assumes a mass transfer rate of M yr and a companion star mass of M. The time scale associated with the mass transfer rate, , is yr. The companion star is in thermal equilibrium if is longer than the thermal time scale of the companion star (Pac_71). To estimate the thermal timescale we need to infer the luminosity of the companion star. For a star close to the lower main sequence it holds the relation (see Salaris_05). Since the companion star mass is M we obtain that yr which is comparable with . Since the two timescales are comparable we cannot exclude that the companion star is out of the thermal-equilibrium; hence, the value of M has to be considered an upper limit to the companion star mass.

We note that for a mass of the companion star lower than the mass transfer rate would be also lower, because of the linear dependence of on in eq. 10. Therefore, the minimum mass transfer rate is reached for a M. This has to be considered as a lower limit to the mass of the companion since below this mass the companion star is expected to become fully convective and the magnetic braking switches off (Rappa_83). For this limiting mass, the mass transfer rate is M yr. However, a detailed study of the evolution of this system is beyond the aims of this paper and will be reported elsewhere. Here we note that the results presented in this paper do not change significantly adopting a lower mass for the companion star. Therefore, we will continue our discussion assuming a companion star mass of M, keeping in mind that lower masses for the companion star are also possible.

The changes of the equivalent hydrogen column density during the X-ray quiescence

The mass ejected from the system can explain the variable equivalent hydrogen column density measured during the X-ray quiescence of the source. Cack_08; Cack_13 measured two different values of cm and cm, respectively, at different times, while the estimation of obtained by Dickey_90 is cm. Here we suggest that the matter ejected from the system can account for the additional absorption. Most of the matter provided by the companion is ejected from the inner Lagrangian point forming a circumbinary ring of matter around MXB 1659-298. Because of the large inclination angle of the system it is possible that the ejected matter interposes between the source and the observer. Local density inhomogeneities and/or changes in the mass transfer rate could produce changes in the equivalent hydrogen column as observed by Cack_08; Cack_13 during quiescence.

We use the eq. 4 of Iaria_13 to estimate the density of the ejected matter

(14)

where is the density in units of cm, is the distance from the inner Lagrangian point, is a parameter that takes into account a non-spherical distribution of matter, a parameter larger than 1, is the mass transfer rate in units of Eddington mass accretion rate and is the orbital separation of the binary system. Adopting a mass transfer rate of M yr, an orbital period of 7.116 hr, a companion star mass and a neutron star mass of M and M, respectively, we obtain cm. Supposing a constant particle density along the line of sight, we can determine the equivalent hydrogen column density associated with the neutral matter using , where cm. We find cm. Since the quantity is close to unity (see Iaria_13) we find that the equivalent hydrogen column of the cold matter is cm, that is of the same order of magnitude of the changes observed during quiescence of the source and, furthermore, it explains the discrepancy by a factor of two between the values measured by Cack_13 and Dickey_90.

The inclination angle of the source

From our estimate of the duration of the eclipse ingress, that is s, we can estimate the size of the corona, , surrounding the central source using the relation

(15)

we find cm. Assuming a neutron star mass of M and a companion star mass of M we infer that the Roche lobe radius, of the compact object is cm. The radius of the accretion disc, , corresponds to the tidal radius (see Frank_02, eq. 5.122), hence the accretion disc radius is cm. This result suggests that the corona is much smaller than the accretion disk, and therefore it is a relatively compact corona around the neutron star.

Figure 5: Schematic geometry of MXB 1659-298 not in scale.

Using our estimate of the eclipse duration we can also estimate the inclination angle, , of the system finding the angle represented in Fig. 5. Knowing that the eclipse duration is s we can estimate the size of the occulted region as before using

(16)

We obtain cm, where is the green segment shown in Fig. 5. The angle , representing the angle between the line of sight and the equatorial plane of MXB 1659-298, is obtained from

We infer degrees. Our result is compatible with the presence in the light curve of the source of dips and total eclipses that can be observed for inclination angles in the approximate range –80 (see Fig. 5.10 in Frank_02). We note that for a companion star mass of 0.35 M the inclination angle of the system is degree, that is marginally compatible with the value obtained for a companion star mass of M.

Sidoli_01 detected absorption lines associated with the presence of \ionOviii, \ionNeix, \ionFexxv and \ionFexxvi ions in the XMM spectrum of MXB 1659-298. The authors, assuming an inclination angle of 80 inferred the distance of the absorbing plasma from the central source, finding cm, cm and cm, respectively. Revisiting the results obtained by Sidoli_01 for an inclination angle of we find cm, cm and . Since we have estimated a size of the corona of cm, we suggest that the absorbing plasma is located in the outer regions of the corona.

4.4 The 2.31-yr periodic modulation: possible explanations

Our ephemeris of MXB 1659-298 also includes a sinusoidal modulation with a period of yr. One possibility is that this periodic modulation observed in the delays may be produced by the gravitational coupling of the orbit with changes in the shape of the magnetically active companion star. These changes are thought to be the consequence of the torque applied by the magnetic activity of a sub-surface magnetic field in the companion star with the convective envelope. The convective envelope induces a cyclic exchange of angular momentum between the inner and outer regions of the companion star causing a change in the gravitational quadrupole moment (see Apple_92; Apple_94). A similar mechanism has been proposed for the eclipsing LMXBs EXO 0748-676 (Wolff_09) and AX J1745.6-2901 (Ponti_17).

The inferred periodicity of 843 d and the amplitude of 9.6 s correspond in this case to an orbital period variation of . We estimate that the transfer of angular momentum needed to produce an orbital period change is g cm s (see Apple_92, eq. 27). The asynchronism of the companion, quantified through the quantity , is , where is the orbital angular velocity of the binary system and is the variation of the orbital angular velocity needed to produce (see Apple_94, eq. 3). The variable part of the luminosity of the companion star required to power the gravitational quadrupole changes is erg s. Considering that we obtain , in agreement with the prediction of obtained for magnetic active stars (see Apple_92, and references therein). Our results suggest that a change in the magnetic quadrupole of the companion star can produce the observed sinusoidal modulation. The energy required to transfer the angular momentum from the interior of the companion star to a thin shell, with a mass of 10% of M, at the surface (and viceversa) is furnished by ten percentage of the thermonuclear energy produced by the companion star. Furthermore, we obtain that the mean sub-surface magnetic field of the companion star is close to G (see Apple_92, eq. 23).

The origin of the sinusoidal modulation could also be explained by the presence of a third body orbiting around the binary system, similarly to what is found for the LMXB XB 1916-053 (Iaria_15). Adopting the inclination angle of we find that the orbital separation between the centre of mass of MXB 1659-298 and the centre of mass of the triple system is , where is the speed of light. Using the values in the third column of Tab. 2 we obtain that cm. Assuming a non-eccentric and coplanar orbit of the third body and that the companion star is in thermal equilibrium, the mass M of the third body is obtained from

(17)

where M is the mass of the binary system and is the revolution period of the third body around the binary system (see e.g. Bozzo_07). We obtain that the mass of the third body is M, where M indicates the Jovian mass; the distance of the third body from the centre of mass of the triple system is AU. Releasing the constrain of a co-planar orbit the mass of the third body is larger than 21 M. We have checked that the derived orbit of the third body is stable in the formalism by Kiseleva_94. If this result will be confirmed, this will be the first circumbinary Jovian planet spotted around a LMXB. In the case of a no-coplanar orbit we find that the mass of the third body should be larger than 21 M.

5 Conclusions

We have estimated 51 eclipse arrival times for MXB 1659-298 when the source was in outburst in 2000, 2001 and 2015 using Rossi-XTE, XMM-Newton, NuSTAR and Swift/XRT data. Combining these times to the previous ones reported in literature we obtain a baseline of 40 years, from 1976 to 2017, to constrain the ephemeris of the source. The data are clustered in three temporal intervals covering six years corresponding to the periods when the source was in outburst. In the hypothesis that the companion star is in thermal equilibrium and fills its Roche Lobe, we estimate that the companion star mass is M, in agreement with the possibility that the companion is an early K-type main-sequence star as reported in literature.

Using RXTE/PCA data we have studied the profile of the total eclipse observing jitters in the ingress/egress duration and eclipse duration of about 10-15 s. The average values of the ingress, egress and eclipse durations are s, s and s, respectively. Using the average ingress and eclipse duration values we find that the size of the corona surrounding the neutron star is cm and the inclination angle of the binary system is degree assuming a companion star in thermal equilibrium.

We find that the eclipse arrival times are well described by ephemeris composed of a linear, a quadratic and a sinusoidal term. We find an orbital period derivative of s s. We are able to explain the value of assuming a highly non conservative mass transfer scenario. We find that the mass transfer rate is M yr, and only of this mass is observed to accrete onto the neutron star. We also suggest that the ejected matter produces a local absorber with an equivalent hydrogen column density of cm.

The sinusoidal modulation has a period of yr and an amplitude of s. The 2.3-yr periodic modulation of the orbital period can be explained either with the presence of a gravitational quadrupole coupling of the orbit to a variable deformation of the magnetically active companion star or with the presence of a third body orbiting around the binary system. In the second scenario we find that the mass of the third body is larger than M.

Finally, we note that the first two eclipse arrival times, measured during the outburst occurred in 1976-1978, are marginally accounted for the quadratic ephemeris. To fit them we adopted a more complex ephemeris taking into account the second derivative of the orbital period. However, the statistical improvement is smaller than three . A larger baseline is needed to confirm or discard more complex ephemerides.

Acknowledgements

This research has made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. This research has made use of MAXI data provided by RIKEN, JAXA and the MAXI team.We are grateful to the Swift team, and especially Kim Page, for their assistance and flexibility in the scheduling of our ToO observations. This work was partially supported by the Regione Autonoma della Sardegna through POR-FSE Sardegna 2007-2013, L.R. 7/2007, Progetti di Ricerca di Base e Orientata, Project N. CRP-60529. We also acknowledge a financial contribution from the agreement ASI-INAF I/037/12/0. AR and AS gratefully acknowledge the Sardinia Regional Government for its financial support (P.O.R. Sardegna F.S.E. Operational Programme of the Autonomous Region of Sardinia, European Social Fund 2007-2013 - Axis IV Human Resources, Objective l.3, Line of Activity l.3.1.). We also acknowledge fruitful discussions with the international team on “The disk magnetosphere interaction around transitional ms pulsars” supported by ISSI (International Space Science Institute), Bern”.

References

Footnotes

  1. pubyear: 2016
  2. pagerange: A possible solution of the puzzling variation of the orbital period of MXB 1659-298A possible solution of the puzzling variation of the orbital period of MXB 1659-298
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