An eccentric wave in the circumstellar disc of the Be/X-ray binary X Persei
R. K. Zamanov K. A. Stoyanov U. Wolter D. Marchev N. A. Tomov M. F. Bode Y. M. Nikolov V. Marchev L. Iliev I. K. Stateva


firstpage–lastpage
The exact form of the `Ockham factor'
in model selection
    Jonathan Rougier
Jonathan Rougier (corresponding author) is Honorary Professor, School
of Mathematics, University of Bristol (email:
); Carey Priebe is
Professor, Department of Applied Mathematics and Statistics, Johns
Hopkins University (email: ).

       School of Mathematics 

       University of Bristol, UK
       Carey E. Priebe 

       Department of Applied Mathematics and Statistics 

       Johns Hopkins University, USA
     
=====================================================================================================================================================================================================================================================================================================================================================================================================================================================




 


We present spectroscopic observations of the Be/X-ray binary X Per
obtained during the period December 2017 - January 2020 (MJD 58095 - MJD 58865).
In December 2017 the Hα, Hβ, and HeI 6678 emission lines 
were symmetric with violet-to-red peak ratio 
V/R ≈ 1.  
During the first part of the period (December 2017 - August 2018) the V/R-ratio decreased to 0.5
and the asymmetry developed simultaneously  in all three lines. 
In September 2018, 
a third component with velocity ≈ 250 km s^-1 appeared on the red side of the HeI line profile. 
Later this component emerged in Hβ, accompanied by the appearance of a red shoulder in Hα.



Assuming that it is due to an eccentric wave in the circumstellar disc, we find that 
the eccentric wave appeared first in the innermost part of the disc,
it spreads out  with outflowing velocity v_wave≈ 1.1 ± 0.2 km s^-1,
and the eccentricity of the eccentric wave is e_wave≈ 0.29 ± 0.07. 
A detailed understanding of the origin of such eccentricities would have applications 
to a wide range of systems from planetary rings to AGNs.







Stars: emission-line, Be – stars: winds, outflows – circumstellar matter – X-rays: binaries – stars: individual: X Per







§ INTRODUCTION

The relatively bright variable star 
X Persei (HD 24534)  is the optical counterpart of the X-ray source 4U 0352+309 
<cit.>   
and is classified in the Be/X-ray subclass of massive X-ray binary stars
<cit.>.  
It consists of an early type Be star and a slowly spinning neutron star. 
The X-ray  data revealed a neutron star with spin period ≈ 836 s, which  
exhibits quasi-periodic X-ray flares with a period of ∼ 7 years <cit.>.   
The pulse period shows episodes of spin-up and spin-down  <cit.>  
which indicates that  the neutron star  is close to a torque equilibrium 
and the long pulse period suggests a strong magnetic field 
<cit.>.     
<cit.>        
determined the orbital period ∼250 d,
orbital eccentricity e = 0.11, semi-major axis a = 2.2 a.u
and orbital inclination i = 26^0-33^0. 
     
During the last century, the visual brightness of X Per has varied in the range  V= 6.8 to  6.2 mag.

Spectrograms of X Per from 1913 to 2020 show X Per to have bright hydrogen
lines that are variable in structure, velocity, and intensity
<cit.>.   
The primary is a rapidly rotating main sequence Be star
which forms an outwardly diffusing gaseous disc.
It has projected rotational velocity  ≈ 215 km s^-1 
<cit.>, 
and is classified as O9.5 <cit.>,  
B0Ve <cit.>,
B1Ve <cit.>. 

  
In recent decades, X Per exhibited two disc-loss episodes (around 1977 and around 1989)
when the emission lines and the circumstellar disc  were missing 
<cit.>.   
The long-term variability of the emission lines and optical brightness 
indicate variable mass ejections from the donor star  into the circumstellar disc 
<cit.>.  


We present optical spectroscopic observations obtained in the last two years  
and discuss asymmetries in the circumstellar disc and the appearance of eccentric wave.



 	     






§ OBSERVATIONS


We have 101 optical spectra of X Per on 57 nights secured with the 
ESpeRo Echelle spectrograph <cit.>    
on the 2.0 m RCC telescope of the Rozhen  National Astronomical Observatory, Bulgaria


and  
with the HEROS spectrograph  <cit.>   
on the 1.2 m  TIGRE telescope in the astronomical observatory La Luz in Mexico.
The variability of  Hα,  Hβ and HeI 6678 emission lines
of X Per is presented in Fig. <ref>. The spectra are normalized 
to the local continuum and a constant is added to each spectrum.
In this figure are plotted only 15 out of 101 spectra obtained. 





On the spectra we measure the following parameters: equivalent width of the line, 
radial velocities of the peaks, intensity of the peaks. 
To measure the radial velocity we applied Gaussian fitting at the top of the peak. 
On a few spectra we see three peaks – one violet and two red peaks.
In such cases we use the stronger red peak to calculate the distance between the peaks 
and V/R ratio. The measurements are given in a few tables in the Appendix. 






§ RESULTS

During the period December 2017  - January 2020, 
the equivalent width of  the Hα emission line 
(Wα) varies in the range  -26 Å ≤ Wα≤ -10 Å,
the equivalent width of the Hβ emission line  
(Wβ) varies in the range  -4.4 Å ≤ Wβ≤  -1.4 Å,
and the equivalent width of the HeI 6678 emission line 
varies in the range  
-2.0 Å  ≤ W(HeI6678) ≤ -0.4 Å. 
The variability of the equivalent widths is presented in Fig. <ref>,
together with V band magnitude and X-ray flux in  2 keV  – 10 keV.
The V-band data are from  the American Association of Variable Star Observers (AAVSO). 
The X-ray data are from  MAXI  <cit.>.
The vertical dashed lines indicate the three periods discussed in Sect. <ref>.  
There is a correlation between equivalent widths of these three lines.
The correlation analysis between 
Wα and  Wβ, Wα - W(HeI6678), Wβ - W(HeI6678) gives 
correlation coefficient ≥ 0.80, significance p-value <  10^-15,  
in other words a very strong correlation between the equivalent widths of the three lines.

It can be seen from Fig. <ref> that when the W_α decreases the V band brightness 
of X Per increases. The connection between  W_α and V band magnitude 
might be caused by the mass ejection from the Be star as discussed in  Sect. 4.1 by <cit.>.
There are no large changes in the X-ray flux. 
This indicates that the variability of the emission lines during the period
December 2017 - January 2020 does not affect the mass accretion rate onto the neutron star. 
  






















 §.§ Disc size



The variability of the distance between the peaks is presented in Fig. <ref>. 
The emission lines form  in the disc surrounding the Be star.  

The discs of the Be stars are Keplerian supported by the rotation 
[e.g. <cit.>, <cit.> and references therein].     
For a Keplerian circumstellar disc the peak separation can be regarded as a measure of 
the outer radius (R_disc) of the emitting disc <cit.>:   

 
    R_disc = R_1  (2 v sini)^2/Δ V ^2 ,


where  R_1 is the radius of the primary and v sini is its projected rotational velocity. 

Radii estimation through this method, is a good approximation for symmetric profiles. 
The projected rotational velocity of the primary 
is estimated  to be  v  sin  i =200 km s^-1  <cit.>,   
v  sin  i =215 ± 10 km s^-1  using the HeI λ 4026 Å  absorption line 
<cit.>  
and  v  sin  i = 191 ± 12 km s^-1
from the width of the Hα emission  <cit.>.     
For the primary we adopt  R_1=10.5 R_⊙  and mass  M_1=13.5   M_⊙
<cit.>.   
This gives the Keplerian velocity on the surface of the star V_Kepl = 495 km s^-1. 
Adopting inclination i=30^0 <cit.>,  
and that the star rotates with 0.8 of the critical velocity
<cit.>,  
we estimate v  sin  i  ≈ 198 km s^-1, which is in agreement with the above values. 
This agreement also is a clue that there is no considerable deviation between the orbital plane of the binary
and the equatorial plane of the Be star.
If it exists at all it should be less then 5^∘.  

On the spectrum  20190325  (see Fig. <ref>),  
the HeI emission extends upto ≈ 370 km s^-1  at zero intensity.  
We note that velocities in the profiles of the emission lines 
> 495 sin i km s^-1, which is ≈ 250 km s^-1,   
are probably super-Keplerian velocities, and indicate  eccentric motion in the disc, 
where these parts of the emission line are formed (see Sect. <ref>).

In December 2017 the emission lines have double-peaked symmetric profiles
(see Fig. <ref>). 
For 5 spectra obtained in December 2017 we measure
Δ Vα =120 ± 1 km s^-1,
Δ Vβ  =150 ± 1 km s^-1, and
Δ V_HeI =180 ± 2 km s^-1.
Using Eq. <ref>, these values correspond  to disc size  
R_disc (Hα) = 134 R_⊙, 
R_disc (Hβ) =  86 R_⊙, and 
R_disc (HeI6678) = 60 R_⊙. 

During the period of our observations, 
the average distance between the peaks 
of the lines is 
Δ Vα = 111 ± 14 km s^-1,
Δ Vβ  = 142 ± 13 km s^-1,
Δ V_HeI = 213 ± 52 km s^-1.






These values correspond to the
following average disc size for different emission lines (calculated using Eq. <ref>): 
R_disc (Hα) = 156 R_⊙, 
R_disc (Hβ) =  96 R_⊙, 
R_disc (HeI6678) = 43 R_⊙. 
The typical errors are about ± 5 %.
For the calculations of the outflowing velocity (Sect. <ref>)
we will use R_disc (Hβ) = 86 - 96 R_⊙. 

The average ratios between disc sizes (which is equivalent to the ratio of the peak separations)  are: 
R_disc (Hα) / R_disc (Hβ)    = 1.62,
R_disc (Hα) / R_disc (HeI6678)   = 3.63.









 	    



   





 §.§ V/R ratio and  asymmetries in the disc



In Fig.<ref> it is seen that the  emission lines are 
symmetric  in December 2017, with  V/R ratio ≈ 1. 
After that  all three lines become asymmetric.
Fig. <ref> presents the V/R ratio, calculated as V/R=(I_V-1)/(I_R-1), where I_V and
I_R are the intensity of the blue (violet) and red peak, respectively. The spectra are normalized before 
the measurements and the continuum level is ≡ 1.0.


(1)   During the first part of the period (MJD 58095 - MJD 58365) 
in all three lines the V/R-ratio varies in practically the same manner –
it decreased from V/R  ≈ 1 to  V/R  ≈ 0.5, 
with  speed  Δ (V/R)/Δ t≈ 2 × 10^-3 d^-1.
The fact that the changes in V/R-ratio are practically identical in 
all three lines (Hα, Hβ and HeI,
see Fig.<ref>)
means that the asymmetry develops in the entire disc simultaneously.

(2) During the second  part  of the period 
(MJD58365 - MJD58600) the V/R-ratio of Hα and 
Hβ goes up from 0.5 to ∼ 1.0. 
However the behaviour of the V/R-ratio of HeI6678  deviates from the behaviour of the Hα
and Hβ lines (see Fig. <ref>).  V/R (HeI) changes from  0.5 to 1.5, having a faster rate  
of change Δ (V/R)/Δ t≈ 4 × 10^-3 d^-1. 
For comparison, the  Hα emission peaks 
have rate of change Δ (V/R)/Δ t≈ 1.7 × 10^-3 d^-1
and the Hβ peaks – Δ (V/R)/Δ t≈ 2.3 × 10^-3 d^-1 .
In addition to that, a third component appeared in the red side of the HeI profile
(Fig. <ref>). This third component emerged in HeI in November 2018. 




This  is an indication 
that  the structural changes begin in the innermost region of the disc. 

(3) The behaviour of the V/R-ratio of Hβ is similar to that of Hα
till MJD 58600. After  it it begins to deviate (see Fig. <ref>). 
This is an indication that the asymmetry spreads out in the disc. 


The development of asymmetries visible in the Hα peaks corresponds to processes in the outer parts,
while the variability of HeI is connected with changes in the inner parts of the circumstellar disc. 






 §.§ Third component in He I



In November 2018 a third component appears in the red side of the HeI line.
In March 2019, this component peaks at  radial velocity  ∼ 250  km s^-1, 
and at zero intensity  it is in the range 130-390 km s^-1. 
The three component structure is demonstrated on Fig. <ref>,  left panel.
The Keplerian velocity around the primary is
V_Kepl = √( G M_1 /r ),
 where G is the gravitational constant and r is the distance. 

We estimate that the observed velocity of the third component  corresponds to
a distance from the centrum of the primary of 40 - 20 R_⊙, respectively. 
Bearing in mind that R_1 ≈ 10.5 R_⊙, 
it means that this component 
originates somewhere  about 1-3 stellar radii above the stellar surface. 
The velocity of this component is similar to the velocities
observed during the double disc formation in 1994 
<cit.>,     
when the blue and red peaks of the inner disc were
at -308 km s^-1 and +258 km s^-1, respectively.  







§ ECCENTRIC WAVE


The discs around stars, or other central massive bodies,
can support disturbances in which the fluid motion is nearly Keplerian with non-zero eccentricity.
The eccentric waves are expected not only in Be discs but also 
in accretion discs, protoplanetary discs, and in
discs with an embedded planet (e.g. <cit.> and references therein).   
In X Per, the appearance of a significant density enhancement in the disc near the stellar surface 
and its evolution was also observed and discussed  by <cit.>
and 
<cit.>. 


Assuming  that the observed behaviour of the emission lines of X Per is due to 
an eccentric wave, we calculate its velocity and eccentricity.  



 §.§ Velocity of the eccentric wave


We first see the appearance of the third component in HeI lines on our spectrum in November 2018  (20181124).
It is not visible on our spectrum obtained in mid-October 2018 (20181010). 
In March 2019 this component starts to be visible in the red side of H β. 
The profile of H β on 5 December 2019 (20191205) with three peaks 
is  similar to that of HeI obtained 8 months earlier (20190325), see Fig.<ref>. 
We suppose that the eccentric wave began somewhere about MJD 58360 (about end of August 2018)
when the behaviour of the HeI peaks started to deviate from Hβ and Hα 
(see the arrow on Fig. <ref>). 
At this time it is likely to be at a distance r ≥ R_1.  
It reached  near 
to the outer edge  of the Hβ disc in January 2020
(when the two red peaks of Hβ  blended).
This indicates that the eccentric wave spreads from inside out.
Bearing in mind the disc size in the lines
(Sect. <ref>)
this corresponds to an  outflowing velocity of the wave 
v_wave≈ 1.2 ± 0.1 km s^-1.  




In the above calculation we supposed that  the motion of the density wave is linear. 
We also supposed that the peak merging corresponds 
to the moment when the wave reached the outer edge.
We performed numerical experiments with Gaussian fitting of 
the peaks and their blending (merging). 
The experiments showed that to observe two peaks the difference between 
their velocities must be ≳  30 km s^-1. If it is less than 30 km s^-1 then we will see one peak. 
This value depends  on the signal-to-noise ratio of the spectra, width of the peaks, their intensity, etc.
This limit of  30 km s^-1 was found with numerical experiment with two Gaussian peaks, 
having width and intensity similar to those observed. 
This also indicates that the above estimation should be considered as an upper limit 
of  v_wave.
Assuming that the merging of the peaks corresponds to the moment when 
 the wave is at 0.85 ± 0.15 of the disc size, this will 
give v_wave≈ 1.1 ± 0.2 km s^-1.


In Fig. <ref> are plotted the heliocentric radial velocities of the peaks
for Hα  (V_r α),  for Hβ (V_r β), and for HeI 6678 (V_r HeI). 
The radial velocities of the HeI peaks indicate that the wave begins somewhere about MJD 58350 (August 2018). 
The radial velocities of the Hβ peaks  indicate that the two red peaks  
merged somewhere about MJD 58850 (January 2020). 




















In Fig. <ref> we plot distance between the peaks of Hα normalized with the stellar 
rotation  versus Wα. These two parameters correlate in the Be stars, representing
the fact that the disc grows as W_α increases. 
The black open circles are data for Be stars taken from 
<cit.>,    
<cit.>,    
<cit.>,    
<cit.>,     
<cit.>,      
and <cit.>.   
The red plus signs are our measurements of X Per. 
We see that X Per is close
to the average behaviour of the Be stars. This indicates that the eccentric wave does not change the 
overall structure of the circumstellar disc.   



					  






 §.§ Eccentricity  of the eccentric wave



The vis-viva equation, connects 
the instantaneous orbital speed of a body at any given point in its trajectory
with distance. For periastron and apastron we have:

    v_per  = √( GM_1 ( 2/a(1-e) - 1/a))



    v_ap  = √( GM_1 ( 2/a(1+e) - 1/a) )

where a is the length of the semi-major axis of the elliptical orbit, 
and a(1-e) and a(1+e) are the distances at which the speed is to be calculated
(in our case periastron and apastron, respectively).

Using three spectra on which Hα is symmetric 
(20161211, 20170317, 20171207) we measure the velocity 
at the half maximum of the Hα  emission  -11 ± 3 km s^-1. 
The measurement is done using the position of the bisector at the half maximum intensity 
of the emission, as shown in Fig. 1 of <cit.>. 
 The old spectroscopic data of Hutchings (1977) give an  average 
 value for the radial velocity of 
 the Balmer absorption ≈ -20 km s^-1 and for  HeI lines -19 ±  10 km s^-1.
<cit.>  
give systemic velocity 
γ = 1.0 ±  0.9 km s^-1 on the basis of
absorption lines in the IUE spectra.



On the spectra where 3 peaks are visible we measure the heliocentric radial 
velocities of the peaks. They are given in Table <ref>. 
For HeI and Hβ these are  measurements of the peaks. 
For Hα it is the velocity of the shoulder.  
Correcting these values for the systemic velocity: 
  
V_r1 - γ  = v_apsin i   

  and 

V_r3 - γ = v_persin i,    



and using Eq. <ref> and  Eq. <ref>, we estimate 
the eccentricity of the wave, which is given in the last column of Table <ref>. 
The average eccentricity of the wave is e_wave = 0.29 ± 0.07. 

In the above calculations we adopted  γ = -11 km s^-1.
If we assume   γ = -1 km s^-1, the estimated e_wave will increase 
up to  e_wave =  0.36 ± 0.07. 
There should be two blue peaks corresponding to the two red peaks. In our spectra we see
only one, see Fig. <ref> (left panel). It means that the two blue peaks are blended
(and have approximately equal velocities). 
To estimate the possible influence of the blending 
on our value of e_wave, we fitted the 
profile with four Gaussian (Fig. <ref>, right panel). During the fitting 
we explicitly assumed that the two inner peaks are identical. 
The fitting suggests that the peak (corresponding to the wave) could be blue 
shifted with 10 – 20 km s^-1. A correction  of -10 km s^-1 
to the measured radial velocity  produces a lower value e_wave = 0.24 ± 0.06.  


These two sources of  uncertainty  give  the range 0.16 < e_wave < 0.42
for the eccentricity of the wave.





§ DISCUSSION


Eccentric discs, in which planet, or star, or fluid elements
follow elliptical orbits of variable eccentricity around a central mass,
have  applications in various astrophysical objects: Be stars <cit.>,
galactic nuclei <cit.>, 
planetary rings and protoplanetary discs <cit.>.
A more deep understanding of the appearance and  the evolution of asymmetries and  eccentricities 
in the circumstellar disc of X Per and other Be/X-ray binaries 
would therefore be of general interest.
There are different theories about the origin of the long-term  variability
observed in Be stars
[e.g. <cit.>, Section 4.2].   
The most accepted is the global oscillation scenario, 
which proposes that a Keplerian disc around a Be stars 
is subject to global distortion – a one-armed density wave
<cit.>,  
which is an updated  and  sophisticated
version of the old elliptical disc model    <cit.>.  
Eccentric mode in Be/X-ray binaries can be excited in the disc through direct driving 
as a result of a one-armed bar potential of the binary <cit.>.


In our opinion, during the first part or our observations (MJD 58095 – MJD 58365) 
we observe
a global distortion (Hα, Hβ, HeI vary together)
similar to that observed in  H 1145-619 (see Fig. 7 by
<cit.>).   

During the period MJD 58365 –  MJD 58865, 
we observe an eccentric wave, which
starts in the innermost parts of the disc and spreads out with velocity 
v_wave≈ 1.1 km s^-1. 
Once the wave is in the disc, it should begin to rotate. Its rotation 
is expected to  be much slower than the disc rotation <cit.>.   
The radial velocities on Fig. <ref> indicate that the period of rotation of the density wave 
is  probably ∼ 800 d.   


The interesting behavior of the HeI emission of X Per was studied 
by <cit.>
and 
by <cit.>. 
They observed four peaks in HeI emission in 1995 and 
suggested the formation of a double circumstellar disc. 
<cit.>
were first to argue about the existence of a two component structure of 
the circumstellar envelope around a Be stars following the example of the well known star Pleione (BU Tau). 
They proposed a model consisting of two layers 
and also considered a separate fast rotating layer closer to the stellar equator. 
The formation of a new envelope, coexisting with the previous one
is also detected by <cit.> on the basis of four peak structure of 
Hα emission. 
<cit.> presented an extensive data set of X Per, 
covering  the period 1987 to 2001, 
and interpreted the HeI variability as a formation of a density wave
which moves towards the outer parts faster than the disc formation. 
It is likely that the four peaks profiles of X Per
in 1995 represented a wave with eccentricity almost zero. 

 



Following
<cit.>   
the periastron passages of the neutron star is expected at 

MJD 51353 (± 7)  + 250.3 (± 0.6)  × E. 

The excitation of the eccentric wave could be connected with the periastron passage 
at about MJD 58361.  The periastron passage of the neutron star 
initiates (generates) a higher tidal wave on the stellar surface. Superposition of 
a high tide with another mechanism  (e.g. non-radial pulsations)    
could cause a mass ejection event from the star surface into the inner disc. 
A similar mass ejection was already discussed 
by  <cit.> in connection between the long term variability
of the Hα line and the X-ray emission from the neutron star.     


The recent radiative transfer calculations on the structure of Be discs in coplanar circular binary 
systems suggest a V/R cycle every half orbital period 
<cit.>.  
We do not see signs of such modulation in X Per in our data, 
which covers 770 days, i.e. more than three orbital periods.



§ CONCLUSIONS


We report 100 spectral observations of the Be/X-ray binary X Per 
during the period December 2017 - January 2020. 
We study 
the evolution of the profiles of the emission lines 
Hα, Hβ, and HeI6678, 
which are formed in the Be circumstellar disc.
Their evolution suggests an eccentric wave in the circumstellar disc, for which we
find that:  (1) it spreads from inside out; (2) its velocity is v_wave = 1.1 ± 0.2 
km s^-1;
(3) the eccentricity of the eccentric wave is in the range 0.17 ≤ e_wave≤ 0.41. 


The development of 
asymmetries in the inner and outer parts of the circumstellar disc
in the Be/X-ray binary X Per provides a laboratory to test 
the theoretical models of eccentric waves in the Be discs.    
                          
We encourage high-resolution spectroscopic observations 
of this relatively  bright object 
(e.g. Echelle spectrographs on 2.0m class telescopes)
to monitor the evolution of the emission lines.








§ ACKNOWLEDGMENTS


This work was supported by the  Bulgarian National Science Fund project  number KΠ-06-H28/2 08.12.2018
"Binary stars with compact object".
The TIGRE telescope is a collaboration of the Hamburger
Sternwarte, the Universities of Hamburg, Guanajuato and Liége.
UW acknowledges funding by DLR, project 50OR1701.
DM acknowledges partial support by grant RD-08-122/2020 from Shumen University.
This research has made use of  the MAXI data provided by RIKEN, JAXA, and the MAXI team 
and  observations from the AAVSO International Database 
contributed by observers worldwide.
We are very grateful to the anonymous referee for the very helpful and  constructive comments
on the original manuscript. 


Data availability: The spectra are available upon request from the authors: rkz@astro.bas.bg, 
kstoyanov@astro.bas.bg. 

mnras














§ APPENDIX


		    			    

		    			    

			    

		    			    

		    			    



		    			    

		    			    



		    			    

 				      							 

	    											 
	    											 

										 
	    											 
