Study on the magnetic field strength of NGC 300 ULX1

In an accreting binary system, the spin-up rate reflects the angular momentum being deposited on the NS (Shapiro & Teukolsky, 1983; Tong, 2015)

where $I$ is the moment of inertia of the NS, $\dot{\Omega}$ is the derivative of the angular velocity, $\dot{M}$ is the accretion rate, $G$ is the gravitational constant, $M$ is the mass of NS, and $R_{\rm A}$ is the Alfv$\acute{\rm e}$n radius :

where $R_{6}$ is the NS radius $R$ in units of $10^{6}\,\rm cm$, $m$ is the NS mass in units of solar mass, $L_{37}$ is the accreting luminosity ($L_{\rm acc}=GM\dot{M}/R$) in units of $10^{37}\,\rm erg\,s^{-1}$, $\mu_{30}=1/2\,B_{12}R_{6}^{3}$ is the magnetic moment in units of $10^{30}\,\rm G\,cm^{3}$, $B_{12}$ is the dipole magnetic field in units of $10^{12}\,\rm G$. With equations (1) and (2), the accretion rate can be written as

where $I_{45}$ is the moment of inertia $I$ in units of $10^{45}\,\rm g\,cm^{2}$. For a PSR with a certain spin period and spin-up rate, this equation shows an anti-correlation between the accretion rate and the magnetic field. We assume $P$ and $\dot{P}$ of a normal accreting PSR in high mass X-ray binary (HMXB) are $\sim 2\,\rm s$ and $\sim-1\times 10^{-10}\,\rm s\,s^{-1}$, respectively (Liu et al., 2006, 2007). Its $B-\dot{M}$ relation according to equation (3) is shown in the red dotted line in Fig. 1. We see that the magnetic field $\sim 10^{12}\,\rm G$ of such a PSR corresponds to the accretion rate $\sim 10^{18}\,\rm g\,s^{-1}$. If the magnetic field is as strong as $\sim 10^{14}\,\rm G$, the accretion rate reduces to $\sim 10^{17.5}\,\rm g\,s^{-1}$. And for NGC 300 ULX1, the $B-\dot{M}$ relation with the detected $P$ and $\dot{P}$ is plotted in the blue solid line in the same figure: if the magnetic field of NGC 300 ULX1 is $\sim 10^{12}\,\rm G$, the accretion rate is about $\sim 10^{19.7}\,\rm g\,s^{-1}$. When the magnetic field rises up to $\sim 10^{14}\,\rm G$, the accretion rate is slightly higher than $\sim 10^{19}\,\rm g\,s^{-1}$. The two accretion rates correspond to the luminosity of $\sim 9.4\times 10^{39}\,\rm erg\,s^{-1}$ and $>\sim 2.0\times 10^{39}\,\rm erg\,s^{-1}$, where the later one is closed to the observed value of $4.7\times 10^{39}\,\rm erg\,s^{-1}$. Thus we suggest that the magnetic field of NGC 300 ULX1 is about $\sim 10^{14}\,\rm G$, which will be investigated further in the following sections.

During the accretion, the total torque $N$ on the NS is $N=I\dot{\Omega}$ for neglecting the typically small effect of the change in the effective moment of inertia. It can also be expressed in terms of the accretion torque $N_{\rm acc}$ (Ghosh & Lamb, 1979; Ghosh, 1995):

where $R_{\rm 0}$ denotes the transition zone, that is composed of a broad outer zone where the angular velocity is Keplerian and a narrow boundary layer where it departs significantly from the Keplerian value. It is preferred to be $\sim 0.5R_{\rm A}$. The parameter $n(\omega_{\rm s})$ is the dimensionless torque related to the fastness parameter $\omega_{\rm s}$ (Ghosh & Lamb, 1979)

where $\dot{M}_{17}$ is $\dot{M}$ in units of $10^{17}\,\rm g\,s^{-1}$, $\omega_{\rm c}$ is the the critical fastness at which the torque changes direction and is taken to be 0.35 (Ghosh & Lamb, 1979; Ghosh, 1995). In case of $\omega_{\rm s}<\omega_{\rm c}$, the PSR is spun up. And for $\omega_{\rm s}>\omega_{\rm c}$, the PSR is spun down (Ghosh, 1995).

According to equation (4), the spin-up rate is (Ghosh & Lamb, 1979)

In the follow calculation, we take the mass and radius of NGC 300 ULX1 to be the typical values of a standard NS, that are $m=1.4$ and $R_{6}=1$.

The pulse profile of NGC 300 ULX1 is found to be almost sinusoidal, and dose not appear to be strong beamed (Carpano et al., 2018). Moreover, the accretion rate of the source was detected to be almost constant from 2016 to 2018 (Vasilopoulos et al., 2019). Thus we consider $L_{\rm acc}$ equals to $L_{\rm X}$ during the field strength calculation. Applying $P=31.6\,\rm s$ and $\dot{P}=5.56\times 10^{-7}\,\rm s\,s^{-1}$ into equation (7), the magnetic field of NGC 300 ULX1 is confound when $\omega_{\rm s}$ is considered in two cases:
$\bullet$ $\omega_{\rm s}$ is assumed to be zero for a slow rotator, and the dimensionless torque is $n(\omega_{\rm s})\simeq 1.4$, as defined by equation (5) (Ghosh, 1995; Parfrey et al., 2016). The estimated magnetic field is $B_{1}=1.19\times 10^{13}\,\rm G$, which is slightly stronger than the works by Carpano et al. (2018) and Vasilopoulos et al. (2018).
$\bullet$ $\omega_{\rm s}$ is determined by equation (6) which is associated with the calculated magnetic field, spin period and luminosity. The inferred magnetic field is $B_{2}=3.89\times 10^{14}\,\rm G$ and $\omega_{\rm s}\sim 0.25$, that indicates NGC 300 ULX1 is a highly magnetized NS.

We list the inferred magnetic field $B$ and the corresponding Alfv$\acute{\rm e}$n radius $R_{\rm A}$ in Table 1. The co-rotation radius $R_{\rm co}$ is $1.68\times 10^{9}\,\rm cm$ for $P=31.6\,\rm s$ according to $R_{\rm co}=(GM/4\pi^{2})^{1/3}\,P^{2/3}$. Both of the deduced magnetic fields by GL method imply the relations of $R_{\rm A}<R_{\rm co}$ and $\omega_{\rm s}<\omega_{\rm c}$. These results illustrate that the source is spinning up during the accretion, that is consistent with the observed case of NGC 300 ULX1.

More observations of NGC 300 ULX1 have been performed after it was confirmed as a PULX (Vasilopoulos et al., 2018). Similar to the above calculation, we calculate magnetic fields by 15 pairs of $P$ and $\dot{P}$ that are taken from Table B.1 in Vasilopoulos et al. (2018), as can be seen in Table 2. Vasilopoulos et al. (2018) suggested that the datasets, marked with the symbol “*” from the 2014 $\it Chandra$ and $\it Swift$/XRT observations, were with gaps and low statistics and therefore resulted in large uncertainties for determining the spin periods and spin-up rates. The magnetic fields inferred from those observed parameters are with the same symbol “*”. The mean value of the field solutions are $B_{1}\sim 10^{13}\,\rm G$ with $n(\omega_{\rm s})\simeq 1.4$ for the PSR as a slow rotator, and $B_{2}\sim 3.0\times 10^{14}\,\rm G$ with $n(\omega_{\rm s})$ according to equation (5). In Table 2, both $B_{1}$ and $B_{2}$ show a weak decay tendency in $3.4\,\rm yrs$ time span of the observation. In order to estimate a rational solution for NGC 300 ULX1, we perform a long term field decay analysis and compare the resulting field estimate with the magnetic field values in Table 2 inferred from observations.

Evolution of NS magnetic field during the accretion has been widely studied and discussed (Bisnovatyi-Kogan & Komberg, 1974; Geppert & Urpin, 1994; van den Heuvel & Bitzaraki, 1995; Melatos & Phinney, 2000; van den Heuvel, 2009; Ho, 2011; Igoshev & Popov, 2015; Igoshev et al., 2021). For estimating a reasonable magnetic field of NGC 300 ULX1 from the inferred solutions, we will simulate the magnetic field decay along with the accretion time $T_{\rm ac}$ by employing the model of the accretion induced the magnetic field decay of a NS by Zhang & Kojima (2006), hereafter ZK model. It had been used to test the magnetic field evolution of binary pulsars (BPSRs) (Pan et al., 2015).

The ZK model assumes that the PSR is evolved with a constant accretion rate in the accretion. Since the accreted material flows from the polar to the equator, the magnetic field lines in the polar caps are pushed aside and the polar-field strength are diluted. As the PSR magnetosphere is compressed onto the star surface with the accumulation of the accretion material, the accretion might become all over the star and the magnetic field of the PSR decays to the bottom field. The analytic solution of the evolved magnetic field is Zhang & Kojima (2006):

where $B_{\rm f}\simeq 1.32\times 10^{8}(\dot{M}/\dot{M}_{\rm Edd})^{1/2}m^{1/4}R_{6}^{-5/4}\phi^{-7/4}\,\rm G$, is the bottom magnetic field, $\dot{M}_{\rm Edd}\simeq 1.0\times 10^{18}\,\rm g\,s^{-1}$ is the typical Eddington accretion rate of a NS with a mass of $1.4M_{\odot}$ and radius of 10 km, $\phi$ is assumed to be 0.5, denoting the ratio of the magnetosphere radius to the Alfv$\acute{\rm e}$n radius. $C=1+[1-(B_{\rm f}/B_{0})^{4/7}]^{1/2}$, $B_{0}$ is the initial magnetic field when the PSR begins the evolution. The parameter $y=2\xi\Delta M/7M_{\rm cr}$, where $M_{\rm cr}\sim 0.2\,M_{\odot}$ is the crust mass, $\Delta M=\dot{M}T_{\rm ac}$ is the accreted material. The parameter $\xi$ $(0\leq\xi\leq 1)$ is the efficiency factor to express the frozen flow of the magnetic line due to the plasma instability. In this work, we assume $\xi=1$ for the completely frozen field lines that totally drift with the accreted material. The model illustrates that the field evolution of the NS is mainly affected by $\dot{M}$ and $T_{\rm ac}$.

During the evolution of $B$ with $T_{\rm ac}$, the initial magnetic field $B_{0}$ of NGC 300 ULX1 should be equaled to or stronger than the deducted magnetic field that are listed in Table 2. And $\dot{M}$ is assumed to be a constant corresponding to $L_{\rm X}=4.7\times 10^{39}\,\rm erg\,s^{-1}$. It should be noted that the observation time does not mean the accretion time. Hence, we plot the $B-T_{\rm ac}$ relation to find part of the evolved track, which has a similar distribution to the observed $B$ in a time span of $3.4\,\rm yrs$. After testing different values of $B_{0}$, two $B-T_{\rm ac}$ tracks with $B_{0}\sim 3.0\times 10^{15}\,\rm G$ for $B_{1}$ and $B_{2}$ are explored, that couple with the PSR positions well. The accretion time for the source field decaying from $B_{0}$ to the deduced value are $\sim 4000\,\rm yrs$ for $B_{1}$ and $\sim 400\,\rm yrs$ for $B_{2}$, as shown in the $B-T_{\rm ac}$ diagram in Fig. 2. The left panel displays the evolved track and the PSR positions with $B_{1}$, and the right panel reveals the ones with $B_{2}$. The sub-figure in each panel is the enlarged image for showing the detailed matching of the evolved track and positions. The red and black solid dots are the $B-T_{\rm ac}$ positions whose magnetic fields are with and without symbol "*" in Table 2, respectively. We see that the $B-T_{\rm ac}$ positions in the right panel are on or closed to the evolved track more tightly than that in the left panel. Moreover, the reduced chi-square of the simulated and calculated magnetic field are $\sim 45$ for $B_{1}$ and $\sim{\bf 2}$ for $B_{2}$, which also supports the higher magnetic field of NGC 300 ULX1. So we propose that the magnetic field of NGC 300 ULX1 would be $\sim 3\times 10^{14}\,\rm G$ rather than $\sim 10^{13}\,\rm G$.

With the preferred magnetic field strength $B\sim 3.0\times 10^{14}\,\rm G$ of NGC 300 ULX1, the B-P evolution is studied according to equations (7) and (8). We first consider a simple case that the source has been experiencing a steady accretion with the defined accretion time and rate. The accretion time is assumed to be the Hubble time $1.38\times 10^{10}\,\rm yr$ (Planck Collaboration et al., 2016). And the accretion rate is with two limitations: the maximum value corresponds to its ultra luminosity $L_{\rm X}=4.7\times 10^{39}\,\rm erg\,s^{-1}$. And the minimum value is $\dot{M}_{\rm min}\simeq 4.6\times 10^{15}\,\rm g\,s^{-1}$, which is also the minimum accretion rate required for a MSP forming in the binary system (Pan et al., 2018). The corresponding bottom magnetic fields are $B_{\rm f,max}\simeq 1.73\times 10^{9}\,\rm G$ and $B_{\rm f,min}\simeq 2.34\times 10^{7}\,\rm G$, respectively. Under these conditions, the B-P evolution began with $B\simeq 3.0\times 10^{14}\,\rm G$ and $P=31.6\,\rm s$ are shown in Fig. 3, where the blue dotted and dashed lines are with $\dot{M}_{\rm max}$ and $\dot{M}_{\rm min}$. We find that NGC 300 ULX1 will evolve to be a MSP if the source kept accreting throughout the Hubble time.

Since it is hard for a PSR to keep a constant accretion rate during the entire evolution, we now consider that the accretion rate of NGC 300 ULX1 is changeable and fluctuates between $\dot{M}_{\rm max}$ and $\dot{M}_{\rm min}$. And the source will evolve to different equilibrium status along with the accretion rate variation (Tong & Wang, 2019; Bhattacharya & van den Heuvel, 1991; Ho et al., 2014). The accretion time $T_{\rm ac}$ is restricted by the companion mass, which can be inferred from

where $m_{\rm c}$ is the companion mass in units of solar mass, $f$ is the accreted efficient factor that is usually taken to be 0.1 (Shapiro & Teukolsky, 1983). It has been detected NGC 300 ULX1 is with a $<20M_{\odot}$ companion (Carpano et al., 2018; Binder et al., 2020). And we consider a lower mass limit of the companion to be $\sim 10M_{\odot}$ as it is in HMXB. The corresponding accretion time is $\sim 10^{6}-10^{7}\,\rm yrs$ according to equation (9). with the variation of the accretion rate and the accretion time, two evolved B-P tracks are shown in Fig. 3, where the magenta solid line and green dot-dashed line are plotted with $T_{\rm ac}\sim 10^{6}-10^{7}\,\rm yrs$. Now we see that NGC 300 ULX1 evolves to a MSP whose spin period is $\sim 10\,\rm ms$.

Restricted by three B-P tracks that are the blue dotted line, blue dashed line and magenta solid line (for $M_{\rm c}=10\,M_{\odot}$) or green dot-dashed line (for $M_{\rm c}=20\,M_{\odot}$), an evolved range of NGC 300 ULX1 is got, that is covered in grey in Fig. 3. It shows all evolved possibilities of NGC 300 ULX1, e.g., a MSP with $B\sim 10^{9}\,\rm G$, or an evolved PSR with $P$ about tens of seconds and $B\sim 10^{10}\,\rm G$.

By the $B-\dot{M}$ relation and the GL model, we deduced the magnetic field of NGC 300 ULX1 with the X-ray luminosity, 15 pairs of the observed spin periods and spin-up rates. During the calculation with the GL method, we find that the magnetic field is strongly depends on the fastness parameter $\omega_{\rm s}$ in the GL model. For $\omega_{\rm s}$ defined by equation (6), $B\sim 3.0\times 10^{14}\,\rm G$. It is consistent with the work by Erkut et al. (2020), who estimated the maximum range of the magnetic dipole field to be $(45-240)\times 10^{13}\,\rm G$ from the spin-up rate. And in case of a slow rotator ($\omega_{\rm s}\sim 0$), $B\sim 10^{\bf 13}\,\rm G$, that is closed to the results of $\sim(0.7-10)\times 10^{12}\,\rm G$ by Carpano et al. (2018) and $\sim(3-20)\times 10^{12}\,\rm G$ by Vasilopoulos et al. (2018).

The field strength $\sim 10^{13}\,\rm G$ of NGC 300 ULX1 is excluded according to the evolution of the magnetic field decaying with the accretion time and the chi-square test for the simulated and the calculated magnetic fields. Based on these study, we conclude that the magnetic field of NGC 300 ULX1 is $\sim 3.0\times 10^{14}\,\rm G$, and suggest the fastness parameter $\omega_{\rm s}$ to be a variate instead of a constant during the magnetic field deduction.

CRSF is a direct way for constraining the magnetic field of a PSR. In the confirmed PULXs, M51 ULX8 is the only source that was detected with a certain absorption feature at $\sim 4.5\,\rm keV$. It was preferred to be with a proton CRSF which implied a magnetic field of $\sim 10^{15}\,\rm G$ (Brightman et al., 2018). NGC 300 ULX1 is the second PULX that was probed a potential CRSF at $\sim 13\,\rm keV$ based on the phase-resolved broadband spectroscopy using the data of $\it XMM-Newton$ and $\it NuSTAR$, which implied the magnetic field to be $\sim 10^{12}\,\rm G$ with the assumption of scattering by electrons (Walton et al., 2018). The recent work favored a proton CRSF that could lead to a magnetar field strength of NGC 300 ULX1. It was consistent with the high spin up rate of the source to estimate the plausible ranges for the beaming fraction (Erkut, 2021).

Meanwhile, such a CRSF needed a further confirmation, since Koliopanos et al. (2019) found a model without the absorption feature could successfully describe the spectra of NGC 300 ULX1. Therefore, the magnetic field of NGC 300 ULX1 has to be derived through the theory models, such as the accretion torque or the model of the magnetic field decay by accretion. And We look forward to an undoubtable detection of CRSF to determine the magnetic field of NGC 300 ULX1 directly.

Limited by the accretion rate and accretion time, NGC 300 ULX1 will evolve to an recycled PSR, whose magnetic field is $\lesssim 10^{12}\,\rm G$ and the spin period is $\lesssim 10\,\rm s$, as shown with the grey area in the B-P diagram in Fig. 3. We plot the B-P positions of magnetars and binary PSRs in the same figure, whose data are from McGill Online Magnetar Catalog and ATNF pulsar catalogue (Manchester et al., 2005; Olausen & Kaspi, 2014). It is found that: (1) the present position of NGC 300 ULX1 is closed to the magnetars, (2) its possible B-P evolved area covers not only the magnetars with low magnetic field ($\sim 10^{12}\,\rm G$), but also the gap between magnetars and BPSRs.

It is known that the magnetars are always found as the isolated PSRs whose magnetic field are $\sim 10^{14}-10^{15}\,\rm G$ (Olausen & Kaspi, 2014; Popov, 2016; Kaspi & Beloborodov, 2017). And there were also a few magnetars whose magnetic field is slightly weaker $\sim 10^{12}\,\rm G$, that were suspected to undergo the accretion induced their fields decay (Rea et al., 2010; Zhou et al., 2014; Mereghetti et al., 2015; Esposito et al., 2021). Meanwhile, there is no binary magnetar found until now. Although an accreted magnetar was doubtful existed in the $\gamma$-ray binary system LS 5039 with a period of $\sim 9\,\rm s$ (Yoneda et al., 2020). It was denied later since the statistical significant bursts or quasiperiodic variability was not found with the same data (Volkov et al., 2021). If NGC 300 ULX1 is the accreting magnetar, it might provide us a chance for probing the magnetar evolution and explaining the formation of the magentars with the low magnetic fields.

Our work suggests that NGC 300 ULX1 is an accreting magnetar whose magnetic field is about $3.0\times 10^{14}\,\rm G$, which would be with even more strong multi-pole magnetic field. The strong magnetic field can reduce the electron scattering cross-section of the PSR, and in turn supports the super-critical accretion and ultra X-ray luminosity to occur.

The ultra X-ray luminosity of PULXs can be accounted for by the beaming effect (King & Lasota, 2020; Song et al., 2020). However, some confirmed PULXs showed nearly sinusoidal pulse profile which means the beaming effect is negligible. Mushtukov et al. (2021) also suggested that the large pulsed fraction of PULXs excluded the strong beaming (Eksi et al., 2015; Fürst et al., 2017; Carpano et al., 2018). Since NGC 300 ULX1 was observed with a similar property (Carpano et al., 2018), we propose that the accretion luminosity is indeed supper-Eddington.