Importance of electron-positron pairs on the maximum possible luminosity of the accretion columns in ULXs

One of the models explaining the high luminosity of pulsing ultra-luminous X-ray sources (pULXs) was suggested by Mushtukov et al. (2015). They showed that the accretion columns on the surfaces of highly magnetized neutron stars can be very luminous due to opacity reduction in the high magnetic field. However, a strong magnetic field leads also to amplification of the electron-positron pairs creation. Therefore, increasing of the electron and positron number densities compensates the cross-section reduction, and the electron scattering opacity does not decrease with the magnetic field magnification. As a result, the maximum possible luminosity of the accretion column does not increase with the magnetic field. It ranges between 10$^{40} - 10^{41}$ erg s$^{-1}$ depending only slightly on the magnetic field strength.

Introduction At present a few pULXs are known (see e.g.Fabrika et al. 2021).One of the models describing their observed high luminosities is a strongly magnetized neutron star (NS) with high mass-accretion rate (Mushtukov et al. 2015) in a binary system.This model predicts increasing of the pULX luminosity with the strength of magnetic field, and, therefore, NSs with a magnetar-like magnetic fields are necessary for explaining the observed pULX luminosities (see also Eksi et al. 2015, Brice et al. 2021).Here we improve the model by changing of some geometry assumptions and considering the contribution of electron-positron pairs to the opacity.
Model We made following improvements in the cross-section model of the accretion column.The magnetospheric radius R m in radiation-dominated discs as well as the propeller conditions were computed according to Chashkina et al. (2019).We fixed the relative thickness of the transition layer between the disc and the magnetosphere z d = ∆R/R m .We took into account the radiation friction between the radiation and the plasma outside the column.We changed the velocity law along the column height, V (h) = V 0 (h/H x ) ξ , where ξ > 0 is a parameter, and H x is the current column height, H x=0 = H 0 .The dependence of the horizontal radiation flux was taken as F (x) = F 0 (1 − τ x /τ 0 ) 1/β with a parameter β 1. e + -e − pairs were considered assuming thermodynamic equilibrium according to Kaminker & Yakovlev (1993) and Mushtukov et al. (2019).It was shown in these papers that a number density of the pairs strongly depends on the magnetic field strength and increases significantly if B > B cr = 4.414 × 10 13 G.We assume the solar 1 arXiv:2208.14237v1[astro-ph.HE] 25 Aug 2022 2 V. F. Suleimanov, A. Mushtukov, I. Ognev, V. A. Doroshenko & K. Werner H/He mix as a chemical composition of the accreting plasma, and take into account all the opacity sources namely free-free transitions, electron scattering, cyclotron absorption, opacity due to two-photon annihilation and one-photon annihilation in a strong magnetic field.
Electron number density vs. temperature for three magnetic field strengths and fixed plasma density.The thermodynamical equilibrium positron number density for the non-magnetic case is shown with the black dotted curve.
Dependences of the Rosseland mean opacities on the plasma temperature for various magnetic field strengths and two plasma densities, ρ = 0.1 (dashed curves) and 10 g cm −3 (solid curves).The magnetic field strengths are marked at the curves.
Continuum and cyclotron opacities in two modes for the given plasma parameters and fixed angle between direction of photon propagation and magnetic field.The two-photon and one-photon annihilation opacities are also shown.The latter opacity is given for three magnetic field strengths.
The accretion column shapes at different values of the parameter β=1, 0.1, and 0.01.The column shapes were computed with (solid curves) and without (dashed curves) taking radiation drag force into account.Other model parameters are fixed and shown in the plot.
Pairs prevent huge luminosity of the accretion columns in magnetars.
Strong fields are not necessary for pulsing ULXs even at low beaming (see also Chashkina et al. 2019).
Positions of the pULXs on the luminosity -NS magnetic field strength plane.The upper limits on the magnetic field strength are found from the propeller conditions.The corresponding conditions for five spin periods are shown with the dotted lines.The approximate boundaries for the magnetospheres located in the radiation-and the advection-dominated parts of the discs are also plotted with the dashed and the dot-dashed lines respectively.The new recomputed maximum possible luminosities of the accretion columns (with H0=RNS) are plotted with the thick solid (β=0.1) and dashed (β=0.01)red curves.The corresponding old curve (Mushtukov et al. 2015) is drawn with the thin dashed red curve.
• Differences in the observed properties can be explained by different accretion disk states (Fig. 5).
• Observed for a long burst dependense K 1/4 F can be fitted by theoretical f c L/L Edd dependences (Fig. 6).  Figure 6: Comparison of the X-ray burst data for 4U 1724 307 to the theoretical models of NS atmosphere.The crosses present the observed dependence of K 1/4 vs. F for the long burst, while diamonds represent two short bursts.
Figure 7: Constraints on the mass and radius of the NS in 4U 1724 307 from the long burst spectra.These correspond to the three chemical compositions: green for pure hydrogen, blue for the solar ratio of H/He and subsolar metal abundance Z = 0.3Z appropriate for Terzan 2, and red for pure helium.

Conclusions
• The neutron star radius in 4U 1724 307is larger than 14 km for masses below 2.3 solar mass (Fig. 7).
• Most possible the inner core of the neutron star has a stiff equation of state.
Figure 1.Left panel: Dependences of the Rosseland mean opacities on the plasma temperature for various magnetic field strengths and two plasma densities, ρ= 0.1 (dashed curves) and 10 g cm −3 (solid curves).The magnetic field strengths are marked at the curves.Right panel: Positions of the pULXs on the luminosity -NS magnetic field strength plane.The upper limits on the magnetic field strength are found from the propeller conditions.The corresponding conditions for five spin periods are shown with the dotted lines.The approximate boundaries for the magnetospheres located in the radiation-and the advection-dominated parts of the discs are also plotted with the dashed and the dot-dashed lines respectively.The new recomputed maximum possible luminosities of the accretion columns (with H0 = RNS) are plotted with the thick solid (β=0.1) and dashed (β=0.01)red curves.The corresponding old curve (Mushtukov et al. 2015) is drawn with the thin dashed red curve.

Results
Electron scattering on e + -e − pairs increases the Rosseland opacity at kT > 30 keV significantly (right panel in Fig. 1).We note that high luminosity accretion columns are optically thick at the considered parameters.Therefore, there are conditions for pair thermodynamical equilibrium and the pair numerical densities are significant along all the column height at high magnetic fields B > B cr .The opacity increase due to electron scattering on the pairs overcomes the opacity reduction due to magnetic field increase, and pairs prevent a huge luminosity of the accretion columns in magnetars at B > 10 14 G. (left panel in Fig. 1).The considered model provides the high enough accretion column luminosities at some parameters even at low magnetic fields, and strong NS fields are not necessary for pULXs even at low beaming due to the new propeller conditions derived by Chashkina et al. (2019).Appendix The poster itself presented as Fig. Electron number density vs. temperature for three magnetic field strengths and fixed plasma density.
The thermodynamical equilibrium positron number density for the non-magnetic case is shown with the black dotted curve.
Dependences of the Rosseland mean opacities on the plasma temperature for various magnetic field strengths and two plasma densities, ρ = 0.1 (dashed curves) and 10 g cm −3 (solid curves).The magnetic field strengths are marked at the curves.
Continuum and cyclotron opacities in two modes for the given plasma parameters and fixed angle between direction of photon propagation and magnetic field.The two-photon and one-photon annihilation opacities are also shown.The latter opacity is given for three magnetic field strengths.
It leads to significant increasing of Rosseland opacity at kT > 30 keV and B > 10 13 G.Pairs prevent huge luminosity of the accretion columns in magnetars.
Strong fields are not necessary for pulsing ULXs even at low beaming (see also Chashkina et al. 2019).
Positions of the pULXs on the luminosity -NS magnetic field strength plane.The upper limits on the magnetic field strength are found from the propeller conditions.The corresponding conditions for five spin periods are shown with the dotted lines.The approximate boundaries for the magnetospheres located in the radiation-and the advection-dominated parts of the discs are also plotted with the dashed and the dot-dashed lines respectively.The new recomputed maximum possible luminosities of the accretion columns (with H0=RNS) are plotted with the thick solid (β=0.1) and dashed (β=0.01)red curves.The corresponding old curve (Mushtukov et al. 2015) is drawn with the thin dashed red curve.
Importance of electron-positron pairs on the maximum possible luminosity of the accretion columns in ULXs.
V. Suleimanov 1,2,3 , A. Mushtukov 4,3 , I. Ognev 5 , V. Doroshenko 1,3 , and K. Werner 1 1 Method • Observed spectra of X-ray bursting NSs are fitting by blackbody spectra with two parameters: color temperature T BB and normalization K = R 2 BB [km]/D 2 10 .• Real spectra of X-ray bursting NSs are close to diluted blackbody F E = f 4 c B E (f c T e↵ ) with color correction factrors f c in range 1.4 -1.8 (Fig. 1), therefore K connected to real NS radius via f c and gravitational redshift z: K = (R(1 + z)) 2 /(f 2 c d 10 ) 2 .• In late phases of photospheric radius expantion bursts (when a photosphere radius equal to the neutron star radius) K depends on f c only.
• We suggest to fit observed dependences K 1/4 F by theoretical dependences f c L/L Edd (Fig. 2), where F is an observed bolometric flux.
• Three parameters can be obtained from fitting: , where X is a hydrogen mass fraction, and Edd, 7 K, where F Edd, 7 = F Edd /10 7 erg cm 2 s 1 .
• Some curve on a M R plane corresponds to each obtained parameter (Fig. 3).Crossing points give necessary solutions for M and R.

!
Figure 1: Color-correction factor as a function of the NS luminosity for different chemical composition (see Suleimanov et al. 2010a).The surface gravity is taken to be g = 10 14.0 cm s 2 .The dashed curve shows the results for a hydrogen atmosphere at larger gravity of log g = 14.3.  2 Application to a long burst of 4U 1724 307, LMXB in a globular claster Terzan 2 • RXTE has observed three photospheric radius expanshion bursts (Fig. 4), one long, and two short.
• Differences in the observed properties can be explained by different accretion disk states (Fig. 5).
• Observed for a long burst dependense K 1/4 F can be fitted by theoretical f c L/L Edd dependences (Fig. 6).  Figure 6: Comparison of the X-ray burst data for 4U 1724 307 to the theoretical models of NS atmosphere.The crosses present the observed dependence of K 1/4 vs. F for the long burst, while diamonds represent two short bursts.
Figure 7: Constraints on the mass and radius of the NS in 4U 1724 307 from the long burst spectra.These correspond to the three chemical compositions: green for pure hydrogen, blue for the solar ratio of H/He and subsolar metal abundance Z = 0.3Z appropriate for Terzan 2, and red for pure helium.

Conclusions
• The neutron star radius in 4U 1724 307is larger than 14 km for masses below 2.3 solar mass (Fig. 7).

Figure 4 :
Figure 4: Evolution of the observed blackbody fluxes, color temperatures and normalizations for a long (black circles) and short bursts from 4U 1724 307.
Figure 5: Different spectral states of 4U 1724 307accretion disk before a short and a long bursts.

Figure 2 :
Figure 2: Illustration of the suggested new cooling tail method.Figure 3: Constraints on M and R from various observed values.If the assumed distance is too large, there are no solutions (the corresponding curves for F Edd =const and R1=const shown by thin lines do not cross).

Figure 3 :
Figure 2: Illustration of the suggested new cooling tail method.Figure 3: Constraints on M and R from various observed values.If the assumed distance is too large, there are no solutions (the corresponding curves for F Edd =const and R1=const shown by thin lines do not cross).

Figure 4 :
Figure 4: Evolution of the observed blackbody fluxes, color temperatures and normalizations for a long (black circles) and short bursts from 4U 1724 307.
Figure 5: Different spectral states of 4U 1724 307accretion disk before a short and a long bursts.